Research Article Multilevel Tunnelling Systems and Fractal ... · systems (ATS) proposed by one of...

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 263742 20 pageshttpdxdoiorg1011552013263742

Research ArticleMultilevel Tunnelling Systems and Fractal Clustering in theLow-Temperature Mixed Alkali-Silicate Glasses

Giancarlo Jug12 and Maksym Paliienko1

1 Dipartimento di Fisica e Matematica Universita dellrsquoInsubria Via Valleggio 11 22100 Como Italy2 INFNmdashSezione di Pavia and IPCFmdashSezione di Roma Italy

Correspondence should be addressed to Giancarlo Jug giancarlojuguninsubriait

Received 7 March 2013 Accepted 31 March 2013

Academic Editors A Ovchinnikov and K Prokes

Copyright copy 2013 G Jug and M Paliienko This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The thermal and dielectric anomalies of window-type glasses at low temperatures (119879 lt 1K) are rather successfully explained by thetwo-level systems (2LS) standard tunneling model (STM) However the magnetic effects discovered in the multisilicate glasses inrecent times magnetic effects in the organic glasses and also some older data from mixed (SiO

2)1minus119909

(K2O)

119909and (SiO

2)1minus119909

(Na2O)

119909

glasses indicate the need for a suitable extension of the 2LS-STMWe show thatmdashnot only for the magnetic effects but also for themixed glasses in the absence of a fieldmdashthe right extension of the 2LS-STM is provided by the (anomalous) multilevel tunnellingsystems (ATS) proposed by one of us for multicomponent amorphous solids Though a secondary type of TS different from thestandard 2LS was invoked long ago already we clarify their physical origin and mathematical description and show that theircontribution considerably improves the agreement with the experimental data In spite of dealing with low-temperature propertiesour work impinges on the structure and statistical physics of glasses at all temperatures

1 Introduction

Glasses are ubiquitous materials of considerable importancefor many practical applications however for physicists thenature of the glass transition and the ultimate microscopicstructure of glasses determining their physical propertiesremain to this day issues of considerable intellectual challenge[1] Glasses are normally regarded as fully homogeneouslydisordered amorphous systems much alike liquids exceptfor the glassy arrested dynamics close and below the glasstransition temperature 119879

119892 which leads to an increase of

several orders of magnitude in the viscosity for 119879 rarr 119879+

119892

Nevertheless this homogeneity is most probably only auseful idealization for real glasses must always contain somesmall (in ceramic glasses not so small) concentration of tinyordered or nearly ordered regions of variable size with theirown frozen dynamics Indeed the thermodynamically stablephase of an undercooled liquid would be the perfect crystalthus every substance in approaching the crystallizationtemperature 119879

119888(119879

119888gt 119879

119892) from above would spontaneously

generate local regions of enhanced regularity (RER) muchlike a system (a vapour or a paramagnet) approaching itscritical temperature is known to develop regions (droplets)resembling the ordered low-temperature phase These RERare of course to be distinguished from the concept of short-ranged atomic order which is typical of ideal glasses and isrestricted to the first few atomic spacingsWe are consideringin this paper realistic glasses in which a degree of devitri-fication has occurred The size and concentration of theseRER will depend for example on the rapidity of the quenchleading to the formation of the glass but also on the chemicalcomposition of the substance the presence of impurities andso onHowever on general grounds even the purest of glassesshould contain RER in non-zero concentration and size

This case has been demonstrated recently for the structureof the metallic glass Zr

50Cu

45Al

5[2] where a combination

of fluctuation electron spectroscopy (FEM) andMonte Carlosimulation (MC) has revealed the presence of crystallineregions of subnanometer size embedded in an otherwisehomogeneously amorphous mass of the same composition

2 The Scientific World Journal

It is believed that other metallic glasses should presentsimilar structural features and thusmdashon general groundsmdashone would expect that nonmetallic window glasses too likepure SiO

2and all the more so the commercial multisilicates

of complex chemical composition should present a mul-tiphased structure with the size and concentration of thenear-crystalline regions or RER depending for exampleon composition quench rate and the presence of impuritiesacting as nucleation centres for the RER Indeed materials ofthe general composition (MgO)

119909(Al

2O

3)119910(SiO

2)1minus119909minus119910

(MASin short) are termed ceramic glasses (one of the best knowncommercial examples being Schottrsquos Ceran where Li

2O

replaces MgO and of course CaO or BaO can also replaceor be added to MgO and still yield a ceramic glass) Thesematerials are known to contain microcrystals embedded inan otherwise homogeneously amorphous matrix [3] This isnot surprising for materials made up of a good glass-former(eg SiO

2 Al

2O

3 etc) and good crystal-formers (eg BaO

K2O ) are known to be multiphased [4] with the good

crystal-formers generating their own pockets and channelscarved out within the otherwise homogeneously amorphousnetwork of the good glass-formerrsquos ions [5] Within thesepockets and channels incipient nano- or even microcrystalsmay form but the point of view will be taken in this workthat on general grounds even the purest single-component(eg As SiO

2) glass-former will be rich in RER unless the

quench-rate from themelt is so large as to avoid the formationof crystalline regions or RER

These refined structural details of glasses are evidentlyhard to reveal in all and especially the near-ideal cases(no good crystal-formers no impurities added and rapidquenches) with the available spectroscopic techniques Forexample X-ray spectroscopy does not reveal nano-crystalsbelow the nanometer size However at low and very lowtemperaturesmdashwhere all said structural features remain basi-cally unalteredmdashsome recent experimental findings mightnow improve perspectives with what would appear setto become a new spectroscopy tool Indeed a series ofremarkable magnetic effects have recently been discovered innonmagnetic glasses (multisilicates and organic glasses) [6ndash13] with in the opinion of the present authors a most likelyexplanation for the newphenomena stemming precisely fromthe multiphase nature of real glasses and the presence ofthe RER or microcrystalline regions in their microscopicstructure In turn when the multiphase theory shall befully developed the magnetic effects could represent a validnew spectroscopic tool capable of characterizing micro- ornanocrystals or even incipient crystals and RER in the realglasses The key to this possible development is some newexciting physics of the cold glasses in the presence (and evenin the absence as shown in the present paper) of a magneticfield The magnetic effects in the cold glasses could becomeeventually the amorphous counterpart of the de Haas-vanAlphen and Shubnikov-de Haas effects in crystalline solidsin determining the real structure of amorphous solids

Systematic research on the low-temperature propertiesof glasses has been ongoing for more than 40 years andsome significant theoretical and experimental progress has

been made in the understanding of the unusual behaviourof glasses and of their low-temperature anomalies [14ndash16]This temperature range (119879 lt 1K) is deemed importantfor the appearance of universal behaviour (independent ofcomposition) as well as for the effects of quantummechanicsin the physics of glasses However to make progress in theunderstanding of the low-temperature physics of glassesthere remains a wide range of important questions that arestill open or only partially answered particularly in thelight of some still poorly understood recent and even olderexperiments in cold composite glasses

It is well known that cold glasses show somewhat univer-sal thermal acoustic and dielectric properties which are verydifferent from those of crystalline solids at low temperatures(below 1K) [17 18] The heat capacity 119862

119901of dielectric glasses

is much larger and the thermal conductivity 120581 is orders ofmagnitude lower than the corresponding values found intheir crystalline counterparts119862

119901depends approximately lin-

early and 120581 almost quadratically on temperature119879 (in crystalsone can observe a cubic dependence for both properties)The dielectric constant (real part) 1205981015840 and sound velocity atlow frequencies display in glasses a universal logarithmicdependence in 119879 These ldquoanomalousrdquo and yet universalthermal dielectric and acoustic properties of glasses arewell explained (at least for 119879 lt 1K) since 1972 whenPhillips [19] and also Anderson et al [20] independentlyintroduced the tunnelling model (TM) the fundamentalpostulate of which was the general existence of atoms orsmall groups of atoms in cold amorphous solids which cantunnel like a single quantum-mechanical particle betweentwo configurations of very similar energy (two-level systems(2LS)) The 2LS-TM is widely used in the investigation ofthe low-temperature properties of glasses mostly becauseof its technical simplicity In fact it will be argued in thispaper that tunneling takes place in more complicated localpotential scenarios (multiwelled potentials) and a situationwill be discussed where the use of a number of ldquostatesrdquogreater than two is essential Moreover new insight will begiven on the role of percolation and fractal theory in theTM of multicomponent glasses We present in this paper thejustification and details of the construction of an extendedTM that has been successfully employed to explain theunusual properties of the cold glasses in a magnetic field [21]as well as in zero field when systematic changes in the glassrsquocomposition are involved [22]

The linear dependence in ln(119879) of the real part ofthe dielectric constant 1205981015840(119879) makes the cold glasses usefulin low-temperature thermometry and normally structuralwindow-type glasses are expected to be isotropic insulatorsthat do not present any remarkable magnetic-field responsephenomena (other than a weak response in 119862

119901to the

trace paramagnetic impurities) For some multicomponentsilicate glass it has become possible to measure observablemuch larger than expected changes in 1205981015840(119879 119861) (12057512059810158401205981015840 sim

10minus4) already in a magnetic field as weak as a few Oe

[6 7] A typical glass giving such strong response has thecomposition Al

2O

3-BaO-SiO

2thus a MAS ceramic-glass

herewith termed AlBaSiO The measurements were made

The Scientific World Journal 3

on thick sol-gel fabricated films a fabrication procedurefavoring microcrystal formation [4] cooled in a 3He- 4Hedilution refrigerator reaching temperatures as low as 6mKMagnetic effects have been reported for both the real andimaginary part of 120598 at low frequency (120596 sim 1 kHz) for theheat capacity 119862

119901(see eg [21]) and for the polarization

echo (where changes in the presence of a magnetic fieldhave been the strongest [10ndash12]) as well This behavior wasconfirmed in other multicomponent glasses like borosilicateoptical glass BK7 and commercial Duran [8] and moreoversimilar effects on 1205981015840(119879) have been confirmed in studies of thestructural glass 119886-SiO

2+119909C119910H

119911in the range 50 lt 119879 lt 400mK

and 119861 le 3119879 [9] Although the dielectric magnetocapacitanceenhancement is not dramatic (1205751205981015840(119861)1205981015840 is typically in the10

minus6ndash10minus4 range) the available measurements show that anunusual effect of the magnetic field is indeed present in theabove glasses yet not measurable in ultrapure SiO

2(Suprasil

W) and cannot be ascribed to spurious agents (The presenceof incipient- or microcrystals in real glasses (sometimescalled devitrification) should not be considered a spuriouseffect On the contrary the magnetic effects should be a wayof characterising the two-phase structure of real glasses) orto trace paramagnetic impurities (always present in silicateglasses although in lt6 ppm concentration in the case ofBK7) Polarization-echo experiments in the AlBaSiO Duranand BK7 glasses have also shown considerable sensitivity inthe response of the echo amplitude to very weak magneticfields and the magnetic effects clearly do not scale withthe concentration of paramagnetic impurities [8 10ndash12]Striking magnetic effects the presence of a novel isotopeeffect and remarkable oscillations in the dephasing timehave also been reported in studies of the polarization echoesin organic glasses (amorphous glycerol) [13] However interms of a detailed theoretical justification for all of theobserved magnetic effects (and the lack of an observablemagnetic effect in the acoustic response [23] so far) anexplanation relying on a single theoretical model for all ofthe available experimental data is still missingWe believe thetwo-phase model reproposed in this paper to be the correctgeneralization of the standard 2LS-TM that is being soughtand here we work out its predictions in zero magnetic fieldbut for different controlled concentrations of glass-formingand crystal-forming components In this way we put ourapproach to a new test

The essential behavior of the dielectric response of glassesat low temperatures is well known [17 18] According to thestandard 2LS-TM (STM fromnowon) the dielectric constantis predicted to vary like minus ln119879 due to the constant density ofstates of the TS Above a certain temperature 119879

0(120596) relax-

ational absorption of the TS becomes important resultingin an increase of the dielectric constant with temperatureproportional to +ln119879 according to the STM This has beenchecked experimentally for several glasses The temperature1198790of the resulting minimum depends on the frequency 120596

and occurs around 50 to 100mK in measurements at around1 kHz

Somemore interesting behavior has been shown by someas yet unexplained data from experiments on the mixed

(SiO2)1minus119909(K

2O)

119909and (SiO

2)1minus119909(Na

2O)

119909glasses studied as a

function of the concentration 119909 of the good crystal-formerat low temperatures [24] The heat capacity 119862

119901(119879) for these

glasses is larger than that for pure vitreous silica and thebehavior as a function of 119879 is very peculiar for differentmolar concentrations 119909 of potassium or sodium oxide andis not explained by the STM The heat capacity decreasesand then increases againwith increasingmolar concentration119909 of K

2O The minimum in the dielectric constant 1205981015840(119879) is

observed for 1198790near 100mK as is typical for these glassy

solids The temperature dependence of 1205981015840 both above andbelow 119879

0 shows however a slope in plusmn ln119879 qualitatively

increasing with increasing concentration 119909 of K2O One can

notice moreover that above the minimum 1198790the relaxation

part of 1205981015840 is increasing faster in slope than the resonantpart below 119879

0for the same 119909 [24] a feature completely

unexplained thus far This work is an indication that notonly the magnetic and electric fields influence the propertiesof glasses but the concentration of chemical species inthe composite materials too (a fact not accounted for bythe STM) In this paper we show in detail how the verysame approach that explains the magnetic properties in themultisilicates [21] also provides a quantitative explanationfor the above-mentioned composition-dependent physicalproperties The picture that emerges regarding the natureof the TS in the multicomponent glasses provides a noveland detailed description of the micro- and nanostructureof the glassy state In turn the linear dependence of theconcentration119909ATS of anomalous TS (ATS)mdashthat responsiblefor themagnetic and composition effects in our theorymdashon 119909fully corroborates the founding assumptions of our approach

The paper is organised as follows In Section 2 we presenta detailed justification for the two-phase approach and theconstruction of the two-species TS model for the amorphoussolids at low temperatures In Section 3 we present thedetailed predictions of this model for the dielectric constant1205981015840(119879 119909) as a function of temperature 119879 and composition 119909 of

alkali oxide (good-crystal former) for the mixed glasses andwe compare the predictions with the experimental data [24]In Section 4 we present the detailed predictions of ourmodelfor the heat capacity 119862

119901(119879 119909) for the mixed glasses and we

compare the predictions with the available experimental data[24] Section 5 contains our conclusions about the nature ofthe TS namely we show how the tunneling ldquoparticlerdquo must infact represent a whole cluster of 119873 correlated real tunnelingions in thematerial Finally in the appendix wework out howthe effective tunneling parameters of our model are relatedvia 119873 to more standard microscopic tunneling parametersA short preliminary account of this work was published in[22]

2 Building Up a Suitable Tunneling Model

The traditional picture [17 18] viewed the TS present inlow concentration (sim1016 g minus1) in the material associatedwith the nonequivalence of two (or more) bonding-angleconfigurations per atomic unit in the amorphous solidrsquosatomic structure Each TS is represented in the standard


case by a particle in an asymmetric (one-dimensional (1D))double-well potential where at low-119879 only the ground statesof the two constituent single wells are assumed to be relevantConsequently only the two lowest-lying double-well statesare taken to determine the physics of each single TS A 2LSsimplified picture then applies and one can describe thelow energy Hamiltonian of each independent TS in termsof an equivalent notation with spin-12 pseudospin matrices120590119909and 120590

119911(Pauli matrices) leading to the compact notation

119867(2)

0= minus(12)(Δ120590

119911+ Δ

0120590119909) for the Hamiltonian of a single

2LS TS In matrix form (the so-called well- or position-spacerepresentation ⟨119894|119867(2)

0|119895⟩ |119894⟩ being the two unequivalent

wells 119894 = 1 2 or 119894 = 119871 119877) this then reads

119867(2)

0= minus

1

2(Δ Δ

0

Δ0minusΔ) (1)

Here the phenomenological parameters Δ and Δ0(known

as the energy asymmetry and (twice) the tunnelling matrixelement resp) represent away of describing the essential low-119879 relevant features of the full and yet unknown in its detailsTS single-particle Hamiltonian in the effective single-wellmatrix representation One obtains E

12= plusmn(12)radicΔ2 + Δ

2

0

for the two lowest-lying energy levels and the physics ofthe glass is then extracted by assuming (initially) the 2LS tobe independent entities in the glass are averaging physicalquantities over a probability distribution for the parametersΔ Δ

0of the standard form (119875 being a material-dependent

constant)

119875 (Δ Δ0) =

119875

Δ0

(2)

This distribution reflects the generally accepted opinion thatΔ and minus ln(Δ

0ℏΩ) (the latter proportional to the double-

well potential barrier 1198810divided by the single-well attempt

frequencyΩ 1198810ℏΩ) should be rather broadly distributed in

a homogeneously disordered solid This leads to an almostconstant density of states (DOS) and the above STM has beenemployed with considerable success in order to explain awide range of physical properties (thermal dielectric (ac andpulsed) acoustic etc [17 18]) of nonmetallic glasses below1K

There are however several drawbacks with the STMas thoughtfully pointed out by Leggett and coworkers [25ndash27] For a start the nature of the TS (and of the two wellsof a single 2LS) and that of the motion inside a singleTS remain to date completely unknown (We remark thatrecently thanks also to the efforts towards the explanation ofthe magnetic effects [21 28] and from the study of quantumdomain-wall excitations in the cold glasses [29] a picture isemerging of a correlated (or coherent) tunneling cluster ofsome 119873 (charged) particles (atoms or molecules) which isbeing represented in the TM by a single fictitious tunnelingparticle In this paper we argue that 119873 sim 200 in agreementwith [29]) Much easier is the diagnostic for the nature of2LS in the case of disordered crystals such as Li-KCl orKBr-KCN solutions [30] (we shall come back to disorderedcrystals later) On general grounds other types of (multilevel)

excitations are always possible in glasses and it is not clearwhy their distribution of parameters should be so similar (andgiven by (2)) in all of the amorphous solids Next the STMhas gathered great consensus for the explanation of manyexperiments at low temperatures but in its simplest form(1)-(2) it fails to explain sound velocity shift and adsorptiondata at low-119879 and the origin of the ldquobumprdquo in 119862

119901(and

ldquoplateaurdquo in 120581) well above 1198790that goes under the name of

boson peak (see eg the references in [25ndash27]) Moreoverthe STM fails to explain the remarkable universality of theultrasonic attenuation coefficient 119876minus1 (roughly independentof every external parameter and glass chemical composition)below 1K [31] To resolve these (and other) difficulties withthe STM Leggett and collaborators have proposed a genericmodel in the context of anharmonic elasticity theory whichcan account for all of the significant features of glasses below1K including the super universality of 119876minus1 [25ndash27]

However it is hard to see how this generic elasticmodel can be extended to account for the magnetic andcomposition-dependent effects in glasses also consideringthat in the multicomponent (ie real non model) glassesmost of the said universality features (eg in 119862

119901(119879 119861) and

1205981015840(119879 119861) [6 7 21] or in 119862

119901(119879 119909) and 1205981015840(119879 119909) [22 24]) are

lost Therefore here we adopt the strategy of resumingthe TS approach by means of a completely different (andmore modern) justification for the TM and then extend theSTM to take the presence of a magnetic field into accountand to explain composition-dependent features (this work)In a rather general fashion the TS can be thought of asarising from the shape of the theoretical potential-energylandscape 119864(r

119894) of a glass as 119879 is lowered well below

the glass freezing transition 119879119892 The concept of free-energy

landscape was introduced for example by Stillinger [32 33]and successfully employed in the study of glasses (eg [1])and spin-glasses (eg [34 35]) A large number of local andglobal minima develop in 119864(r

119894) as 119879 rarr 0 the lowest-

energy minima of interest being made up of 119899119908= 2 3

local wells separated by shallow energy barriers At low-119879these configuration-space localmultiwelled potentials are ourTS and it seems reasonable to expect that the 119899

119908= 2-

welled potentials (2LS) should be ubiquitous in this pictureThese should be thought of as an effective representationof local ldquotremblementsrdquo of the equilibrium positions r(0)

119894

of some of the glass atomsionsrsquo positions spanning overa large number of near-neighborsrsquo distances (unlike in thecase of disordered crystals where the TS are known tobe rather well-localized dynamical entities) Hence just asthe 119899

119908= 2-welled case is possible so ought to be the

119899119908= 3 4 -welled situations which would also be local

rearrangements involving several atomsionsmolecules Theconcentration of these local potentials should not necessarilydecrease exponentially with increasing 119899

119908 in glasses as it

is known to happen for the disordered crystals (2LS presentwith probability 1198882 3LS with 1198883 4LS with 1198884 etc 119888 beingthe defectsrsquo percent concentration)

We can reason this out over the quantitative description ofthe glassy energy landscape of a model situation as was stud-ied by Heuer [36] who considered the molecular-dynamics


minus084

minus086

minus088

minus09

minus119864119864

crys

t

Figure 1 (Color online) The energy landscape (for 120588 = 1

Lennard-Jones density adapted from [36]) of a toy glass model withhighlightedmultiwelled potentials (black the 2LS light blue the 3LS4LS )

(MD) simulation data of a toy glass made up of several (13 or32) particles interacting through a Lennard-Jones potentialand with periodic boundary conditions applied Adopting asuitable 1D projection procedure where a ldquodistancerdquo betweentwo local total energy minima is (not completely unam-biguously) defined the 1D position of a local minimum issomehow attained and the energy landscape of the modelsystem can be charted out Figure 1 reports this chart for thetotal energy landscape for a given density (from [36]) Besidethe deep minimum of the crystalline configuration a largenumber of local minima are visualized and then a suitabledefinition of local double-welled potentials (2LS) is adoptedto classify couples of adjacent minima constituting a singletunneling 2LS (highlighted in black Figure 1)This definitionguarantees that at low temperatures a ldquoparticlerdquo subjected toany such local potentials will switch between both minimawithout escaping to a third minimum Interestingly the dis-tribution of the tunneling parametersΔΔ

0(suitably defined)

for these 2LS could also be evaluated fromMD simulations ofthe above toymodel and this119875(Δ Δ

0) turned out to be not so

perfectly flat as a function of Δ as implied by (2) Rather anincrease (though no divergence) of probability for 2LS withΔ rarr 0 was measured in previous MD simulations [37] StillFigure 1 also allows for tunneling multiwelled local potentialsto be identified and we have highlighted (in light blue)some of them (three- and four-welled local potentials) Therequirement that a ldquoparticlerdquo subjected to such multiwelledlocal potentials should not escape (at low-119879) to foreignminimahas been equally respected and one can see that thesemultiwelled situations are not at all rare We therefore believethat 3LS 4LS and so on should also be considered in the TMThe reduced Hamiltonians (well- or position-representation)for these local multiwelled potentials can be easily writtendown as generalizations of (1) For 119899

119908= 3 (3LS)

119867(3)

0= (

1198641119863

0119863

0

11986301198642119863

0

1198630119863

01198643

) (3)

where 1198641 119864

2 119864

3are random energy asymmetries between

the wells chosen to satisfy sum3

119894=1119864119894= 0 and taken from

an appropriate probability distribution (see below) together

with the tunneling parameter1198630gt 0 (see below) For 119899

119908= 4

(4LS)

119867(4)

0= (

1198641119863

1119863

2119863

1

11986311198642119863

1119863

2

1198632119863

11198643119863

1

1198631119863

2119863

11198644

) (4)

where 1198641 119864

2 and 119864

3 119864

4are random energy asymmetries

taken from an appropriate probability distribution togetherwith the tunneling parameters 119863

1(nn well hopping) and

1198632(nnn hopping |119863

2| ≪ |119863

1|) These are simple possible

choices clearly other special-purpose generalizations of the2LS matrix Hamiltonian are possible and we believe thatthe 3LS of (3) is the minimal generic multiwelled potentialwhich can take the magnetic field into account [21] (the 2LSHamiltonian of (1) could also be adjusted for this purposehowever the energy spectrum would be totally insensitiveto 119861) One can easily convince oneself at this point thatas long as the energy parameters of the above multiwelledeffective Hamiltonians obey the usual uniform distribution(see (2) suitably reformulated) as is advocated by the STMthe DOS 119892(119864) will remain (roughly) a constant It is then tobe expected that all thesemultiwelled local potentials will giverise to the very same physics as in the 119899

119908= 2 case and that

thus in practice the 2LS choice represents the appropriateminimal model for all of the extra low-energy excitationscharacterising amorphous solids at low-119879 It is clear from theabove discussion however that the 2LS tunneling ldquoparticlerdquois not atomic particle at all but on general grounds it ratherrepresents the local rearrangements of a good number of realparticles (ions or molecules)

All changes if the glass is made up of a mixture ofnetwork-forming (NF) ions (like those of the good glass-forming SiO

4or (AlO

4)minus tetrahedral groups) as well as of

network-modifying (NM) ions (like those of the good crystal-forming K+ or Na+ or Ba2+ from the relative oxides)which these last ones could act as nucleating centres for apartial devitrification of the glass as is known to occur inthe multicomponent materials [38ndash41] Indeed the NM-ionsof the good crystal-formers are termed ldquoglass modifiersrdquo inthe glass chemistry literature [42] since they do not becomepart of the interconnected random network but carve outtheir own pockets and channels within the glassy network[5 43] Figure 2 (courtesy from Meyer et al [5]) shows asnapshot of a MD simulation of the glass having compositionNa

2Osdot3(SiO

2) (or (Na

2O)

025(SiO

2)075

) at 2100K (above 119879119892

in fact) in which the nonnetworking NM Na-atoms are putin evidence (big blue spheres) Simulations and experimentsin the multisilicates definitely show that the NM-speciesin part destroy the networking capacity of the NF-ionsand form their own clusters inside the NF-network [5]The chance for these NM-clusters to be the nest of RERincipient- or actual microcrystals is obviously very goodconsidering that these clusters are made of good crystal-forming atoms However on general grounds and as dis-cussed in Introduction we shall take the attitude that even thepurest single-component glasses will contain RER in somemeasure Figure 3 (from [2]) shows one such RER within a


Figure 2 (Color online) Molecular dynamics snapshot of thestructure of sodium trisilicate at 2100K at the density 120588 = 22 g cmminus3The big blue spheres that are connected to each other represent theNa atoms The SindashO network is drawn by yellow (Si) and red (O)spheres that are connected to each other by covalent bonds shownas sticks between Si and O spheres (from [5] by permission)

Figure 3 (Color online) A region including the crystal-like super-cluster from a snapshot of the model simulationmdashincorporatingfluctuation electron microscopy datamdashof the Zr

50Cu

45Al

5metallic

glass at 300∘C (from [2]) The atomic separation distances of themiddle zone are about 025 nm This is a first realistic image of acrystal embryo in a glass this object should not be confused withthe concept of short-range order in ideal glasses

snapshot from a jointMC-simulationFEM-measurement onthe metallic glass Zr

50Cu

45Al

5 The picture clearly shows an

embryo crystal which could not grow to macroscopic sizedue to the arrested dynamics below 119879

119892 such structures are

expected to ubiquitous in all glassesmetallic and nonmetallic[44] except that they are difficult to observe with theavailable spectroscopic tools when subnanometric in sizeThe concentration and size of these RER will dictate whethermagnetic- or composition-effects become measurable in thelow-119879 experiments 119886-SiO

2in its purest form (Suprasil W)

revealed no measurable magnetic effects [6 7 9ndash12]It goes without saying that TS forming in the proximity

and within these RER ormicrocrystalline regions will requirea completely differentmathematical description in particularthe possibility of having more than two wells affords a more

realistic description of the energy landscape Hence 119899119908gt 2

multiwelled systems inside the glass-modifying NM-pocketsand -channels should follow some new energy-parametersrsquodistribution formwhen some degree of devitrification occursleading to entirely new physics One of the present authorshas proposed that precisely this situation occurs inside themagnetic-sensitive multicomponent glasses [21] and in thispaper we show how this theory explains the 119861 = 0

composition-dependent dielectric and heat capacity data of[24] as well Instead of the standard 1D double-welled (W-shaped) potential leading to (1) which continues to describethe ordinary tunneling 2LS inherent to the homogeneouslydisordered 119886-SiO

2network we take for the TS nested in or

near the RER crystal embryos or micro-crystals the modelof a ldquoparticlerdquo having charge 119902 and moving in a 119899

119908-welled 3D

potential of the shape displayed for 119899119908= 3 in Figure 4 for

the 2D (119909 119910)-space The hopping Hamiltonian of a singlenon interacting tunneling 3LS has therefore the form (for afictitious second-quantization particle in the well-coordinaterepresentation)

119867(3)

0=

3

sum

119894=1

119864119894119888dagger

119894119888119894+ sum

119894 = 119895

1198630119888dagger

119894119888119895+ hc (5)

and is described inmatrix formby (3) (where in fact ⟨119894|119867(3)

0|119895⟩

is displayed |119894⟩ (119894 = 1 2 3) denoting the single-well groundstates) This is our minimal generic model for a multiwelledTS The parameter119863

0is chosen positive (contrary to custom

in the STM indeed minus(12)Δ0lt 0 in (1)) for a good number

of reasons First due to the possible softness of the local NM-potential since indeed in general [17 18] 119863

0≃ 119886ℏΩ119890

minus1198871198810ℏΩ119886 and 119887 being numbers such that for 119881

0≳ ℏΩ119886 gt 0 and 119887 =

119874(1) can arise [17 18 21] This choice is still compatible withthe concept of tunneling and at the same time yields ratherlarge values of119863

0asymp ℏΩ On more general grounds however

one should take into account that the tunneling ldquoparticlerdquois not moving in a vacuum but is embedded in a solidthat is for the most part deprived of microscopic dynamicsat low-119879 Thus the surrounding frozen atoms are taking apart in the determination of the tunneling particlersquos loweststationary states In the case of a perfectlyD

3-symmetric local

119899119908= 3 welled potential of the type depicted in Figure 4

Hamiltonian (3) leads to a doubly degenerate ground stateand a first excited nondegenerate state (as is easily verifiedfrom (3) if 119864

1= 119864

2= 119864

3) This may seem unphysical and

yet Sussmann has demonstrated in a remarkable paper [45]that for electrons trapped in a crystal (or equivalently in aglass) the situation above described is realised whenever thetrapping potential is multiwelled with a triangular (119899

119908= 3)

or tetrahedral (119899119908= 4) well-centers geometryThe binding of

the seemingly antibonding ground state is then guaranteed bythe TS interaction with the rest of the solid This reasoning isirrelevant for the STM-2LS parameter Δ

0 since both positive

and negative signs for this parameter yield the same physicsIf 119899

119908gt 2 the sign will matter and Sussmannrsquos work shows

that the choice1198630gt 0 is physically justified for an embedded

particle (or vacuum) in the glass Finally it will be shownin Conclusions that in fact the tunneling ldquoparticlerdquo cannot


(a) (b)

OSi

Ba Al

119881(119909 119910)

(c)

Figure 4 Two-dimensional representation of the plausible source of magnetic-field sensitive (anomalous) tunneling systems in for examplethe AlBaSiO glass The tight vitreous-SiO

2structure (a) is broken up by the Al- and large Ba-atoms (b) thus leaving many metal ions free to

move in a 119899119908-minima (soft) tunneling potential with 119899 ge 3 (c) The unbroken SindashOndashSi bond dynamics if any is of the usual 2LS-type

be considered a single atom ion or molecule but ratherit represents a cluster of 119873-correlated tunneling atomic-scale particles with 119873 asymp 200 Then it is reasonable toexpect that the ground state of such a cluster might be near-degenerate so our choice 119863

0gt 0 for the effective single

tunneling ldquoparticlerdquo is sound and not in conflict with anygeneral quantum-mechanical principle This 119863

0gt 0 is the

major assumption for the multiwelled TS theory It shouldbe mentioned however that multiwelled potentials appearalso in the Jahn-Teller quantum phenomena [46] and that inthat context degenerate ground states are also commonplaceIn the present situation however the disorder inherent inglasses does not allow for a detailed symmetry analysis

At this point we make a choice for the probability distri-bution of the parameters 119864

1 119864

2 119864

3 and 119863

0of a tunneling

3LS nesting in the proximity of a RER crystal embryo ormicro-crystal (one could also work with a 119899

119908= 4 model

potential in the appendix we show that essentially the sameresults can be attained) This is dictated by the fact that near-degeneracy (119864

1= 119864

2= 119864

3) must be favored yet not

fully attained for the wellsrsquo energy asymmetries of one such3LS We thus choose assuming again the tunneling potentialbarriers to be broadly distributed

119875ATS (1198641 1198642 1198643 1198630) =

119875lowast

(1198642

1+ 119864

2

2+ 119864

2

3)119863

0

(6)

which has the advantage of making use of a dimension-less material-dependent parameter 119875lowast 119875ATS(1198641 1198642 1198643 1198630

)multiplied by the concentration 119909ATS of these anomalous(multiwelled and now near-degenerate) tunneling systems(ATS) is the probability of finding one such ATS per unitvolume In the following 119909ATS will be absorbed in theparameter 119875lowast This choice for 119875ATS has provided a gooddescription of the experimental data for the multisilicatesin a magnetic field [21] when in the Hamiltonian (3) (orequivalently (5)) 119863

0at position (119894 119895) is replaced with 119863

0119890119894120601119894119895

(120601119894119895being the appropriate Peierls phase) As was shown in

[21] the spectrum of this 119861 gt 0 modified 3LS Hamiltonian

(3) is formally given by (using Vietersquos formula for the cubicequationrsquos solutions)

E119896

1198630

= 2radic1 minus

sum119894 = 119895119864119894119864119895

61198632

0

cos(13120579 + 120579

119896)

cos 120579 = (cos120601 +119864111986421198643

21198633

0

)(1 minus

sum119894 = 119895119864119894119864119895

31198632

0

)

minus32

(7)

(with 119896 = 0 1 2 and 120579119896= 0 +(23)120587 minus(23)120587 distinguishing

the three lowest eigenstates) and for a choice of 1198641 119864

2 119864

3

and 1198630≫ radic119864

2

1+ 119864

2

2+ 119864

2

3(near-degenerate limit) this is

shown in Figure 5 One can see that for very small 120601 (theAharonov-Bohm phase proportional to the magnetic field 119861120601 = 2120587Φ(B)Φ

0 Φ(B) = S

Δsdot B being the flux through the

single ATS (see also the appendix)) the spectrum consists ofan isolated near-degenerate doublet which is well separatedfrom the higher excited statesWe shall exploit the 120601 = 0 limitof this description for an explanation of the composition-dependent experiments

It should be stressed at this point that in the absenceof a magnetic field like in this work one could make useof a 2LS minimal model for the description of the ATS119867

(2)

0(119864

1 119864

2 119863

0) and with the distribution 119875(119864

1 119864

2 119863

0) =

119875lowast119863

0radic119864

2

1+ 119864

2

2ensuing from the proximity of RER or

incipient microcrystallites It was shown in [21] that at leastfor the heat capacity this leads to the same physics as obtainedfrom the 3LS multiwelled model There is no harm in usingfor theATSnesting in the incipient crystalline regions amorerealistic minimal generic multiwelled model like the above3LSHamiltonian119867(3)

0which better approximates the physical

reality of the energy landscape Moreover the model for thecomposition-dependent effects remains the very same usedfor the magnetic effects and many results already obtainedfor that theory can be exploited by setting simply 119861 = 0We remark also that a distribution of the type (6) for theenergy asymmetry was already proposed for the explanation


0 05 1 15 2 25 3

0

05

1

15

2

Dim

ensio

nles

s ene

rgy

leve

ls

minus05

minus1

minus15

minus2

Aharonov-Bohm phase

Energy levels 1198641 + 1198642 + 1198643 = 0

Figure 5 Variation within the magnetic Aharonov-Bohm phase 120601of the energy spectrum (units119863

0= 1) for a choice of 119864

1 119864

2 and 119864

3

with 1198631198630= 001 In this work we are interested in the 120601 = 0

limit of this spectrum which can be treated at low-119879 as that of aneffective 2LS

Figure 6 A 2D cartoon of the chocolate-like ceramic-glass struc-ture of a real glass in which partial devitrification has occurred withthe location of its low-119879 two-species TS In the randomly networkedbulk of the material sit the STM-2LS with their own concentration1199092LS whilst within and in the proximity of the incipient crystallites

nest the ATS with their own bulk concentration 119909ATS each beingdescribed by (3) and (6) We expect 119909ATS lt 1199092LS and that 119909ATS rarr 0

in the best glasses

of low-119879 experiments with mesoscopic Au and Ag wires [47]where TS (of standard 2LS type) were advocated and wherethe polycrystallinity of metals must be accounted for

In summary we have fully justified the extended TMwhich we have used in [21] and which we exploit alsoin this paper The realistic glass is recognized to have astructure resembling that of chocolate [48] (or of opals) andas is pictured in the cartoon in Figure 6 a homogeneously-disordered networked solid in which (at low-119879 in the glass)only standard 2LS are present with their own concentration1199092LS and in which incipient crystallites are embedded (for

chocolate these would be sugar crystals) In the proximityor within these crystallites are nested the ATS with theirown concentration 119909ATS in the solid and with their ownquantum mechanics and statistics defined by the minimalgeneric model represented by (3) and (6)This is by nomeansan ad hoc model since the very same model would describeTS in all types of real metallic and nonmetallic glassesand quantitatively explain all of the low-119879 experiments innonmetallic glasses tackled so far

3 Predictions for the Dielectric Constant

The 2LS-STM has been successful in the semiquantitativeexplanation of a variety of interesting thermal dielectric andacoustic anomalies of structural glasses at temperatures 119879 lt1 K [14ndash18] the physics of cold glasses being important notonly for their universalities but also because of their linkwith the physics of the glass transition (see eg [49 50])Beside the linearity in 119879 behavior of the heat capacity 119862

119901

it is believed that the linearity in plusmn ln119879 behavior of the realpart of the frequency-dependent dielectric constant 1205981015840(119879 120596)represents a cogent characterization of the glassy state at lowtemperaturesWe begin by deriving this behavior and puttingit to test on data for 1205981015840 for pure amorphous silica (ie nomeasurable ATS effects)

In the presence of an applied electric field F we mustadd the dipole energy minusF sdot p

0to the parameter (12)Δ in the

expression (1) for the low-energy Hamiltonian 119867(2)

0 We can

express the permittivity (strictly speaking the polarization)as 120598 = minus120597

2119891(119865)120597119865

2|119865=0

where 119891(119865) = minus(1119896119861119879) ln119885(119865)

represents the free energy per unit volume The statisticalaverage implies also an integration over the two parametersof the 2LS Δ and Δ

0 according to the distribution given by

(2)We canwrite the partition function in terms of the energylevels 119864

12 119885 = 119890minusE1119896119861119879 + 119890minusE2119896119861119879

Figure 7 (inset) shows the behavior of the 119879-dependentpart of 1205981015840(119879 120596) Δ12059810158401205981015840 = [120598

1015840(119879) minus 120598

1015840(119879

0)]120598

1015840(119879

0) (where

1198790(120596) is a characteristic minimum) for pure vitreous SiO

2

(Spectrosil) It can be seen that linear regimes in minus ln119879 for119879 lt 119879

0and + ln119879 for 119879 gt 119879

0are observed and roughly

with slopes 119878minus= minus2119878 and 119878

+= +119878 gt 0 or in a minus2 1 ratio

According to the 2LS-STM in fact we have the expressions[14ndash18 51]

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162LS=Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES+Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES=2119875119901

2

0

31205980120598119903

int

119911max

119911min

119889119911

119911

radic1 minus (Δ

0min2119896

119861119879119911)

2

tanh 119911

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL=119875119901

2

0

31205980120598119903

times int

119911max

119911min

119889119911int

120591max

120591min

119889120591

120591radic1 minus

120591min120591

times coshminus2 (119911) 1

1 + 12059621205912

(8)


160

140

120

100

80

60

40

20

0

60

40

20

0

0001

0001

001

001

01

01

1

1

Temperature (K)

Temperature (K)

105middotΔ

120598998400120598

998400

105middotΔ

120598998400120598

998400

Al2O3-BaO-SiO2

Glass (AlBaSiO)

AlBaSiO

ATS

2LSKFS2LS + ATS

SiO2 (Spectrosil)

Figure 7 Dielectric signature of pure 119886-SiO2(inset) and AlBaSiO

(main) glasses SiO2data [53] fitted with (8) display a minus2 1 2LS-TM

behavior AlBaSiO data [54] display rather aminus1 1 behavior yet couldbe fitted with (8) (dashed line) [54] with a large Δ

0min = 122mK2LS tunneling parameterWe have fitted all data with amore realisticΔ

0min = 39mK and best fit parameters from Table 1 using (8) and(14) (driving frequency 120596 = 1 kHz)

where we neglect (for low 120596) the frequency dependence inthe RES part where 119911minmax = Δ 0minmax2119896119861119879 and where 120591is the phenomenological 2LS relaxation time given by (with119864 = 2119896

119861119879119911) [17 18]

120591minus1=119864Δ

2

0

120574tanh( 119864

2119896119861119879) (9)

In these expressions Δ0min and Δ

0max are Δ0rsquos phe-

nomenological bounds 120574 is an elastic material parameterof the solid and 120591

minus1

min = 1198643120574 tanh(1198642119896

119861119879) 120591minus1max =

119864Δ2

0min120574 tanh(1198642119896119861119879) 119875 (containing the 2LS volume con-centration 119909

2LS) is the probability per unit volume andenergy that a 2LS occurs in the solid (it appears in (2)) and 1199012

0

is the average square 2LS electric dipole moment Moreoverthe strategy of dielectric relaxation theory has been adoptedwhereby the full complex dielectric constant 120598(119879 120596) has beenwritten as for 120596120591 ≪ 1 [51 52]

120598 (119879 120596) = 1205981015840

RES (119879) + 1205981015840

REL (119879)1

1 + 119894120596120591 (10)

the subscripts RES and REL refer to the zero relaxation-timeresonant and relaxational contributions to the linear response1205981015840 at zero frequency respectively

Presently from expressions (8) we deduce that (1) the so-called resonant (RES) contribution has the leading behavior

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES≃

minus2

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0max) if 119879 lt

Δ0max2119896

119861

0 if 119879 gtΔ

0max2119896

119861

(11)

(2) the relaxational (REL) contribution has instead theleading behavior

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL≃

0 if 120596120591min ≫ 1

1

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0min) if 120596120591min ≪ 1

(12)

Thus the sum of the two contributions has a V-shaped formin a semilogarithmic plot with the minimum occurring ata 119879

0roughly given by the condition 120596120591min(119896119861119879) ≃ 1 or

1198961198611198790(120596) ≃ ((12)120574120596)

13 1205980120598119903is here the bulk of the solidrsquos

dielectric constant and we see that a minus2 1 characteristicbehavior is justified by the STM with the 119879 gt 119879

0slope given

by 119878 = 119875119901203120598

0120598119903

This behavior is observed in pure 119886-SiO2[53] (Figure 7

(inset) with the fitting parameters of Table 1 119909 = 0 from ourownbest fit to (8))However inmostmulticomponent glassesone more often observes a V-shaped curve with a (roughly)minus1 1 slope ratio Figure 7 (main) shows this phenomenon forthe multisilicate AlBaSiO glass (in fact a MAS-type ceramic-glass) which has been extensively investigated in recent timesdue to its unexpected magnetic field response [6 7 9ndash1221] Also Figure 8 shows the remarkable behavior of thedielectric constant versus 119879 for the glasses of composition(SiO

2)1minus119909(K

2O)

119909containing a molar concentration 119909 of

alkali oxide [24] It is seen that a 119878minus119878

+slope ratio of roughly

minus1 1 is observed with the slope definitely changing with 119909(and faster for 119879 gt 119879

0) These data from the Anderson

group [24] thus far unexplained by the 2LS-STM call for anextension of the accepted STM and we show below that asimple explanation can be given in terms of the very sameATS that have been justified in Section 2 and advocatedby one of us in order to explain the magnetic response ofAlBaSiO and other multicomponent glasses [21] In view ofthe interest for these materials in low-119879 metrology and onfundamental grounds such explanation appears overdue tousMoreover ldquoadditionalrdquo TS (beside the standard 2LS) of thetype here advocated were already called for in [24] and othertheoretical papers [55ndash57]

For themultiwelled (3LS in practice)Hamiltonian (3) wehave 119899

119908= 3 low-lying energy levels with E

0lt E

1≪ E

2

In the 119864119894rarr 0 and 119863 equiv radic119864

2

1+ 119864

2

2+ 119864

2

3≪ 119863

0limits

(due to the chosen near-degenerate distribution (6)) wecan approximate the 119899

119908= 3-eigenstate system through an

effective 2LS (though sensitive to all three well asymmetriesand their distribution) having gap ΔE = E

1minusE

0

limΔE ≃ radic1198642

1+ 119864

2

2+ 119864

2

3equiv 119863 (13)

We have also exploited the condition 1198641+ 119864

2+ 119864

3= 0

Using the theory of [21] to work out the 3LS contributionsto 1205981015840RES and 1205981015840REL we arrive at the following expressions for


140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400

(SiO2)1minus119909 (K2O)119909 glass

119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS

Figure 8 Dielectric signature of mixed (SiO2)1minus119909(K

2O)

119909glasses as

function of 119879 and 119909 [24] Fitting parameters from Table 1 using (8)and (14) from our theory (driving frequency 120596 = 10 kHz)

the contribution to the dielectric anomaly from the advocatedATS

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910

119910coshminus2 (119863min

2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)

Here we have again neglected for low-120596 the frequencydependence in the RES part we have put 119910 = 119863119863min and120591119860max is the largest phenomenological ATS relaxation timegiven by [59]

120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)

Moreover 119863min is the lowest energy gap of the multilevelATS Γ is another appropriate elastic constant and lowast isthe (slightly renormalised) probability per unit volume (afterinclusion of 119909ATS) that an ATS occurs within theNM-pocketsand channels with1199012

1the average squareATSdipolemoment

lowast and 119875lowast are so related

lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630

rsquos lower and upper bounds respec-tively This description is intimately linked to the chosendistribution function (6) for these ATS which is favoringnear-degenerate energy gaps119863 bound frombelowby119863min Inturn this produces an overall density of states given by ([21]for 119861 = 0)

119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)

and that is now roughly of the form advocated by Yu andLegget [25ndash27] and by some other preceeding authors (eg[60]) to explain anomalies not accounted for by the standard2LS-TM 120579(119909) is the step function

Manipulation of the expressions in (14) shows that (1)the RES contribution from the ATS has the leading behavior(note that for 119879 lt 119863min2119896119861 120598

1015840|ARES is roughly a constant)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)

(2) the REL contribution is instead characterised by theleading form

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)

Thus the V-shaped semilogarithmic curve is somewhat lostHowever adding the 2LS (8) and ATS (14) contributionstogether one does recover a rounded V-shaped semilog witha slope 119878

minus≃ minus2119878 basically unchanged for 119879 lt 119879

0and an

augmented slope 119878+= 119878 + 119878ATS for 119879 gt 119879

0with 119878ATS =

7120587lowast1199012

16120598

0120598119903119896119861119879 that for 119879 lt 119863min119896119861 may approach 2119878

and thus (qualitatively) explain a minus1 1 slope ratioWe have fitted the full expressions (8) and (14) to the

data for AlBaSiO in Figure 7 (main) and to the 119909-dependentdata for (SiO

2)1minus119909(K

2O)

119909in Figures 8 and 9 obtaining in all

cases very good agreement between theory and experiments[24] Figure 9 shows the fit of our theory to the frequency-dependent data for 119909 = 02 In all of these best fits we havekept the value of Δ

0min = 39mK fixed as obtained fromour pure SiO

2fit and the value of 119863min also independent

of 119909 and 120596 The idea is that these parameters are ratherlocal ones and should not be influenced by NFNM dilutionTable 1 gathers the values of all the (2LS and ATS) parametersused for our best fits and Figure 10 shows the dependenceof the prefactors (containing 119909

2LS in 119875 and 119909ATS in lowast)

with 119909 It can be seen that as expected the ATS prefactor119860ATS = 120587

lowast1199012

1120598

0120598119903119863min scales linearly with 119909 an excellent

confirmation that the ldquoadditionalrdquo TS of [24 55ndash57] are thoseATS proposed by us and modelled as 3LS forming nearand inside the microcrystallites that may nucleate within theNM-pockets and channels It can be seen instead that the2LS prefactor 119860

2LS = 1198751199012

0120598

0120598119903of our fits also increases

though less rapidly with increasing 119909 (a decrease like 1 minus 119909


would be expected)Wepropose (adopting aNF-NM-clusterpercolation picture) that new ldquodilution-inducedrdquo 2LS formwith alkali mixing near the NFNM interfaces of the NFpercolating cluster(s) as 119909 is increased from 0This reasoningleads to the expression 119860

2LS = 119860bulk(1 minus 119909) + 119860 surf119909119891 for

the 2LS prefactor with119860bulk 119860 surf and119891 fitting parametersOur best fit leads to the value119891 = 081 in fair agreement withthe euristic expression

119891 = 1 minus (119863 minus 119863119904) ] (20)

(where 119863 is the fractal dimension of the percolating cluster119863

119904with 119863

119904le 119863 is that of its ldquobridgingrdquo surface (not

necessarily the hull) and ] is the connectedness lengthrsquosexponent) that one would deduce from elementary fractalor percolation theory (see eg [61 62]) 119863

119904is the fractal

dimension of that part of the NM random-clusterrsquos surfacewhere formation of TS takes place and we expect 2 le 119863

119904le 119863

It is indeed reasonable to expect newTS to be forming at theseNMNF random interfaces for these are surfaces of chemicaldiscontinuity in the materialThe above expression is derivedas follows Imagine (as is shown in the cartoons in Figure 11)the NM-clusters percolating through the NF-bulk with a siteconcentration 119909 so that their volume scales like V sim ℓ

119863where ℓ sim 119909] is their typical linear size The number of 2LSon the surface of these clusters will scale like 119873(119904)

2LS sim 119909ℓ119863119904

and so their density like 119873(119904)

2LSV sim 119909119909(119863119904minus119863119891)] = 119909

119891 withthe given expression (20) for 119891 If we consider clusters of2D percolation and assume 119863

119904= 119863

ℎ= 74 (the fractal

dimension of the hull of the spanning cluster) then with119863 =9148 and ] = 43 [61 62] we would get 119891 = 2936 = 08055More realistically on the assumption of percolating 3D NM-clusters in the mixed glasses we can make use of the values[61ndash63] 119863 ≃ 252 119863

119904= 119863

ℎ= 214 and ] ≃ 088 to arrive

at the value 119891 = 067 using (20) (We are well aware thatthe quoted fractal dimensions apply to percolation clustersstrictly only at the percolation threshold 119909 = 119909

119888 Our attitude

is that fractal-type clusters can be used to model the NM-lumps from the good crystal-formers even for 119909 lt 119909

119888and

with non-integer fractal dimensionsThis assumption failingwe have no explanation for the extracted non-zero value ofthe 119891-exponent and 119860 surf-prefactor) It is however not at allclear where at the NMNF fractal interfaces the new 2LS willform (ie what the exact definition of 119863

119904ought to be hull

surface sites screening sites dead-end sites etc) If all of thehull sites are involved then for 3D 119909 = 119909

119888percolation119863

119904= 119863

and one then expects 119891 = 1 Thus this new phenomenologyopens a tantalizing new investigation avenue for research onthe applications of fractal theory to low-119879 physics At thesame time the knowledge of which type of NMNF fractalinterface sites are involved in the TS-formation would greatlyimprove our understanding about the microscopic nature ofthe TS (see also [28])

4 Predictions for the Heat Capacity

We now come to the explanation of the also rather anoma-lous heat-capacity data for themixed glasses (SiO

2)1minus119909

(K2O)

reported in [24] as a function of 119879 and for different 119909 The

120

80

40

0

002 01 1Temperature (K)

105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS

(SiO2)08 (K2O)02 glass5kHz

10kHz

30kHz


2O)

119909glasses as

function of119879 and120596 for 119909 = 02 [24] Fitting parameters from Table 1using (8) and (14) from our theory

300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS

Figure 10 The 2LS and ATS dimensionless prefactor parameters(times105) for all glasses (from Table 1) as a function of 119909 Our data fitwell with our theoretical expectations (full lines)

heat capacityrsquos low-temperature dependence in zeromagneticfield is for pure glasses usually given by the followingexpression

119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)

The first term accounts for the Debye-type contribution fromthe acoustic phonons and dominates above 1 K the secondterm is usually attributed to the low-energy excitationsspecific of all vitreous solidsmdashthe tunneling 2LS119861ph and1198612LSare material-dependent constants This expression describeswell the experimental data for pure silica glass at zero field(Figure 12 black circles 119909 = 0 with fit parameters fromTable 2) but it fails for the multicomponent glasses like


Table 1 Extracted parameters for the glasses K-Si stands for the (SiO2)1minus119909(K2O)119909 glasses In all of the best fits we have employed the values

Δ0min = 39mK and Δ

0max = 10K extracted from fitting the pure SiO2data of Figure 7 (inset)

Glass type 119909 1198602LS 120574 119860ATS 119863min Γ

mol 10minus5

10minus8 sJ3 10

minus5 K 10minus6 sK5

SiO2

0 472 530 mdash mdash mdashAlBaSiO mdash 1162 1340 2647 065 6973K-Si 005 1041 133 755 087 355K-Si 008 1465 123 1300 087 397K-Si 010 1585 115 1600 087 508K-Si 020 2395 082 2819 087 644

SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1

Figure 11 (Color online) A cartoon of the fractal (presumably percolating) geometry of the NM-pockets and channels (green) these NM-clusters grow with increasing 119909

AlBaSiO BK7 Duran (see eg [21] and references therein)and for the mixed glasses (SiO

2)1minus119909(K

2O)

119909for 119909 gt 0 [24]

Typically the heat capacityrsquos experimental data for themulticomponent glasses in zero field denote a kind ofldquoshoulderrdquo at intermediate-low temperatures This suggests adensity of states for at least some of the independent TS inthe glass of the form 119892(119864) prop 1119864 in contrast to the standard2LS-TM prediction 119892(119864) ≃ const which ensues from thestandard TM distribution of parameters Indeed this 1119864contribution to the DOS was the very first observation thathas led to the hypothesis of the ATS formulated in [21]

To find out the precise expression for the heat capacitydue to the ATS we make use of the 3LS formulation for theATS described in [21] and in more detail in Section 2 Theheat capacity is determined from the second derivative of thefree energy with respect to temperature

119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)

where 119865ATS(119879) is the free energy of the ATS given by if weneglect the third highest energy level in the spectrum ofHamiltonian (3) (effective 2LS approximation)

119865ATS (119879) = minus119896119861119879 ln (119890minusE0119896119861119879 + 119890minusE1119896119861119879)

= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)

with 119864 = E1minus E

0 The heat capacity is then obtained by

averaging over the parameter distribution or equivalently bya convolution with the DOS

119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)

where density of states 119892ATS(119864) has the following form [21]

119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)

and119863min is the lower cutoffThe final expression for the ATSheat capacity results in [21]

119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)

where the prefactor for the ATS is 119861ATS = 2120587lowast119896119861119899ATS120588(119909)

lowast as in Section 3 119899ATS being the ATS mass concentration

and 120588(119909) the glassrsquo mass density Of course 119909ATS = 119899ATS120588(119909)For 119896

119861119879 ≳ 119863min this is indeed roughly a constant and

gives the observed ldquoshoulderrdquo in119862119901(119879)when the contribution

119861ph1198793 (from virtual phonons) as well as the STM linear term

1198612LS119879 are taken into accountBoth prefactors for the 2LS and ATS contributions are

dependent on the molar concentration 119909 of alkali-oxide justas we found in Section 3 for the prefactors of the dielectricconstant 119861

2LS ≃ 119861bulk(1 minus 119909) + 119861surf119909119891 119861ATS ≃ 119861119909 Also

119861ph requires to be reevaluated With increasing K2O molar

concentration 119909 for the (SiO2)1minus119909(K

2O)

119909glass the number

of phonons from the NM-component (K2O in this case)

increases linearly with the concentration 119909 and for the NF-component (SiO

2) it should also decrease linearly like (1 minus

119909) Just as we assumed in the previous section there are


fractalpercolation effects between the NM- and NF-clusterswhich makes room for some percolation clustersrsquo interfaceswhere the phonons also might contribute somehow with aterm proportional to 119862ph119909

119891 (119862ph being an 119909-independentconstant)

For these glasses moreover a nonnegligible concentra-tion of Fe3+ (or according to coloring Fe2+) impurities isreported a side effect of the industrial production processEstimates give 102 ppm for AlBaSiO and 126 ppm for Duran6 ppm for BK7 100 ppm for Pyrex 7740 and 12 ppm for Pyrex9700 (see eg the discussion in [21]) All glassesmay indeedcontain some [FeO

4]0 impurity-substitution F-centers (in the

glass similar to a liquid in concentrations however muchmuch lower than the nominal Fe bulk concentrations [43])The Fe3+ cation and the O2minus anion on which the hole islocalized (forming the Ominus species ie the O2minus + hole subsys-tem) form a bound small polaron In this configuration theFe3+ cation is subject to a crystal field with an approximate1198623symmetry axis along the Fe3+-O minus direction This axis

plays a quantization role for the Fe3+ electronic spin Thehole is assumed to be tunneling between two neighboringoxygen ions switching the quantization axis between twodirections and therefore entangling its spin states This islikely to give some tiny contribution to the heat capacity andwe should therefore also take it here into account [64] Thespin Hamiltonian of the [FeO

4]0 F-center is 119867

119904minus119878= 119881

119911119904119911119878119911

where 119881119911is the principal value of the dipole interaction

matrix and 119904119911and 119878

119911are the spin operators of the hole and of

the Fe3+ ion respectively In the absence of a magnetic fieldthere are only two low-lying energy levels 119864

12= plusmn(54)|119881

119911|

The unknown distribution function G(119881119911) must approach

zero when its argument approaches either zero or infinityand have a maximum at a definite argument value 119881

0 The

simplest one-parameter function displaying such propertiesis a Poisson distribution

119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810

) 119881119911isin (minusinfin 0] 1198810 lt 0

(27)

The contribution from the [FeO4]0 ensemble to the heat

capacity is as usual

119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)

where119865Fe3+(119879) is the free energy of the [FeO4]0 ensemble that

one evaluates as

119865Fe3+ = minus119896119861119879 ln (119890minus1198641119896119861119879 + 119890minus1198642119896119861119879)

= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)

here 119864 = (54)|119881119911| Using said distribution function for

119866(119881119911) (27) as well as the expression for 119862Fe3+(119879) from

119865Fe3+(119879) one can obtain an expression for the heat capacity

from the trace [FeO4]0 centres in the glass and which should

be added to the total heat capacity 119862119901

119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)

times coshminus2 (5119881

119911

8119896119861119879)

(30)

where 119899119895= 119909

119895120588(119909) is the mass concentration of the tiny

amount of Fe3+ ions (a very small fraction of the total bulkFe-concentration) substituting the Si4+ in the network

Hence the total heat capacity will be the sum of allthese contributions (21) (26) and (30) (Regarding the roleof the Fe-impurities at zero magnetic field this was totallyoverlooked in [21] where to provide a good fit of the datathe existence of a weak stray magnetic field was wronglyadvocated)

119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)

Making use of expression (31) we have fitted the experimentaldata for the heat capacity of the (SiO

2)1minus119909(K

2O)

119909glasses from

[24] In order to fit the pure 119886-SiO2data we use only formula

(21) that fits the pure silicarsquos data well within the 2LS-STMThe heat capacity 119862

119901(119879 119909) data [24] for the


2O)

119909glasses were obtained using a signal-

averaging technique and for these samples the data arepresented in Figure 12 As one can see the heat capacityfor the (SiO

2)1minus119909(K

2O)

119909glasses at low temperatures is

larger than that for pure silica glass as is typical for themulticomponent glasses already with the smallest 5concentration of K

2O The heat capacity decreases and

then again increases with increasing molar concentration119909 of K

2O The additional heat capacity arises from the

addition of ATS in the K2O NM-clusters and also from

the presence of Fe3+ impurities contained in small (andunknown) concentrations but contributing to the low- andmiddle-range of the temperature dependence

Both prefactors for 2LS and ATS are indeed dependenton the molar concentration 119909 from our data analysis andin the same way as we did in Section 3 we have fittedthe extracted prefactors with the forms 119861

2LS ≃ 119861bulk(1 minus

119909) + 119861surf119909119891 119861ATS ≃ 119861119909 (119861 being some constant) These

dependencies are shown in Figure 13 Also 119861ph is found tochange by increasing the concentration 119909 of the good crystal-former K

2O and in the way we anticipated

With increasing concentration119909 for the (SiO2)1minus119909(K

2O)

119909

glass the number of phonons from the NM-component(K

2O) increases linearly with the concentration and for the

NF-component (SiO2) it should be decreasing linearly like

(1 minus 119909) As we reasoned for the dielectric constant there


(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)

Figure 12 The temperature dependence of the heat capacity for 119886-SiO

2(black circles) and for the (SiO

2)1minus119909(K

2O)

119909glasses [24]The full

lines are our theoretical curves as generated by (31)

0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891

Figure 13 The 2LS and ATS prefactor parameters (times108) for allglasses (from Table 2) as a function of 119909 The experimental data fitwell with our theoretical expectations with 119891 = 081 (full lines)

are percolation mixing effects between the NM- and the NF-systems which create percolation clusters and their NFNMinterfaces where phonons also might be populated in a wayproportional to 119862ph119909

119891 As it turns out the very same value119891 = 081 can be extracted from all our fits just as was donein Section 3 for the dielectric constant data

5 Summary and Conclusions

We have demonstrated that there is direct evidence in zeromagnetic field already for the existence of multiwelled ATS(modelled as tunneling 3LS) and with the new distributionfunction advocated to explain the magnetic field effectsin the multicomponent glasses (see [21]) The relevance ofnear-degenerate multiwelled TS in glasses is a new andunexpected finding in this field of research Our workpredicts in particular that the magnetic response of themixed alkali-silicate glasses should be important and scalelike the molar alkali concentration 119909 At the same time theminus1 1 slope-ratio problem of the standard TM in comparisonwith experimental data for 1205981015840(119879) has been given a simpleexplanation in terms of our two-species tunneling modelThe main result of this work is that the concentration 119909ATS(absorbed in lowast and thus in the 119860ATS- and 119861ATS-prefactors)of ATS indeed scales linearly with 119909 for both 1205981015840(119879 119909) and119862119901(119879 119909) This is supported by our analysis of the existing

experimental data [24] very well indeed Our analysis isin our view strong evidence that the ATS are nesting inthe NM-clusters of the good crystal-formers Our fractalmodeling of the phase-separation NFNM cluster interfacesin the multicomponent glasses gives a strong indicationthat the TS are forming in correspondence to the chemicaldiscontinuities in the structure of amorphous materialsThe justification of our mathematical modeling implies theexistence of incipient crystallites in all amorphous solidswhere the relevant degrees of freedom appear to be correlatedover decades or even hundreds of atomic spacings Thiscooperativity now seems to be a commonplace occurrence inthe glassy state at all temperatures below 119879

119892

Using the results of this analysis (and for AlBaSiO theresults of the experimental data analysis in a magnetic field[21]) we can estimate the value of the dipole momentassociated with the ATS 119901eff = radic11990121 For AlBaSiO using thevalue of lowast extracted from 119862

119901[21] and that of 119860ATS given in

Table 1 we extract 119901eff = 041D For (SiO2)1minus119909(K

2O)

119909 we

notice from the definitions in Section 3 that the ratio of thedielectric and heat capacity prefactors

119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)

is almost independent of the K2O concentration 119909 From

our extracted values in Tables 1 and 2 and the measuredvalues of 120588(119909) [24] we estimate 119901eff = 0045D for the mixedglasses independently of 119909 Considering the elementaryatomic electric-dipolersquos value is 119890119886

0= 254D these small

values of 119901eff for the ATS confirm that their physics mustcome from the coherent (or correlated) tunneling of smallionic clusters (the very same origin for the large values of119863min and for1198630minmax see the appendix) Indeed a cluster of119873 coherently tunneling particles has a dipole moment 119901eff =|sum

119873

119894=1p119894| that can becomemuch smaller than 119890119886

0(the order of

magnitude of each |p119894| in the sum) as119873 grows largeThe fact

that we extract values of 119901eff much smaller than 1198901198860 confirms

the picture of a correlated tunneling cluster in the 119861 = 0 casealready


Table 2 Extracted parameters for fits to the heat capacity data for SiO2and (SiO

2)1minus119909(K

2O)

119909glasses with119863min = 087K and 119881

0= minus042K as

fixed

Glass type 119909 119861ph times 108

1198612LS times 108

119861ATS times 108

119909119895

mol Jmminus3 Kminus4 Jmminus3 Kminus2 Jmminus3 Kminus4 ppmSiO

20 24555 7065 mdash mdash

K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300

It is noteworthy that several papers from the Andersongroup have proved that the addition of any NM-species in anetworking pure glass causes significant (and thus far unex-plained) deviations from the predictions of the 2LS-STM [2465 66] We have explained the origin of these deviations for119862119901(119879 119909) as well as for 120598(119879 119909) However experiments do show

that the thermal conductivity 120581(119879 119909) prop 1198792 remains (below

1K) remarkably universal and composition independent [24]This is connected with the superuniversality of the internalfriction coefficient 119876minus1 in the cold glasses these and otherremarkable findings will be addressed elsewhere within thecontext of our approach

In summary we have shown that there is direct evi-dence in zero magnetic field already for the multiwelledATS advocated to explain the magnetic field effects in themulticomponent glasses Similar 119909-dependent phenomenaare to be expected for the low-119879 anomalies of the MAS-type ceramic-glass of composition (SiO

2)1minus119909(MgO)

119909 which

should also respond to the magnetic field (Experimentson these glasses using different isotopes Mg24-Mg26 andMg25 could also serve to confirm the nuclear quadrupoleexplanation for the magnetic effects) (just like the mixedalkali-silicates of this work should) One may remark atthis point that any extension of the 2LS-STM enlarging theadjustable-parameter space is bound to improve agreementwith the experimental data In this paper we have shownthat it was not just a matter of quantitative agreement butqualitative as well Whilst agreeing that the TM remainsunsatisfactory we stress that it is the only approach we knowof which is versatile enough to allow for an interestingexplanation of rather puzzling phenomena at low-119879 in thereal glasses Furthermore our two-species multilevel TSmodel has been able to consistently explain a good numberof different experimental data [21 59] It cannot be a merecoincidence that the same phenomenological model withrather similar material parameters in different experimentsis capable of explaining so much new physics Far frombeing an ad hoc model our approach reveals the intimatemicroscopic structure of the real glasses which cannot beconsidered as being homogeneously disordered anymoreand this must have some important consequences also for abetter understanding of the mechanisms underlying the glasstransition

As for the possibility of estimating the size and density ofthe incipient crystals in glasses from our theory we remarkthat the simplified geometric-averaging procedure adopted

for the physics of the ATS so far [21] does not allow anythingmore than an estimate of the119875lowast parameter (sim197times1017 cmminus3

for AlBaSiO [21] this being in fact the value of 119909ATS119875lowast

119875lowast being the unknown dimensionless parameter of the ATS

distribution in (6))However the geometric-averaging proce-dure should be performed in two stages (within the incipientmicrocrystals first and then within the glassy matrix in whichthe crystallites are embedded) at the price of making thetheory considerably more complicated When this is donewith a more efficient and complete theoretical formulationthen information on the size distribution of the incipientcrystallites could be gained from further low-119879 experimentsin magnetic fields and at different controlled compositions

Appendix

We first show how the spectrum of the 4LS Hamiltonian (4)is similar to that of the 3LS Hamiltonian (3) in the near-degenerate limit We rewrite the119867(4)

04LS Hamiltonian in the

presence of a magnetic field coupled orbitally to the chargedtunneling particle

119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and

nnn hopping energies respectively and where

120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)

is the Aharonov-Bohm phase resulting from the magneticflux Φ(B) threading the square-loop (having area 119878

) closed

trajectory of the particle The above Hamiltonian should infact be symmetrised over its permutations since the signof the nnn Peierls phase is ambiguous (in practice onereplaces 119863

2119890plusmn1198941206012 with 119863

2cos(1206012) in the appropriate matrix


entries) The eigenvalues equation giving the energy levels isthen as follows

E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)

More instructive than numerically extracting the four exactroots E

0123(with E

0lt E

1lt E

2lt E

3) is for us the

physically interesting limit case in which |119864119894119863

1| ≪ 1 and

|1198632119863

1| ≪ 1 (near-degeneracy of the four-welled potential)

The above eigenvalue equation then becomes much easier tostudy

E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634

1(1 minus cos120601) asymp 0

(A4)

this being the eigenvalue equation of the reduced 4LS Hamil-tonian

119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863

1| ≪ 1 and which has the following

solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)

The perhaps surprising result is an energy spectrum whereonly the middle doublet (E

1 E

2in our notation) becomes

near-degenerate at weak (or zero) magnetic fields (120601 rarr 0)This is shown in the inset of Figure 14 and is reminiscentof the situation with dimerized 2LS considered in [67] inorder to account for the oscillations of the dielectric constantwith 119861 Beside there being no evidence for a dimerizationof TS in glasses (unlike perhaps in mixed and disorderedcrystals) one would have to explain why the ground stateE

0is prohibited for the tunneling particle (the real energy

gap being in fact ΔE = E1minus E

0) The way out can be

found again in Sussmannrsquos paper [45] since the 119899119908= 4

welled trapping potential giving rise to the same physicsas our 3LS must in fact have tetrahedral and not squaregeometry The tetrahedral 4LS in a magnetic field will beconsidered elsewhere [59] Here we only want to remark

that the tetrahedral situation can be mimicked by a squaremultiwelled potential in which |119863

1119863

2| ≪ 1 and always

in the limit case |119864119894119863

2| ≪ 1 This corresponds to the

counterintuitive situation in which it is easier for the particleto tunnel across the square to the nnn site rather than to ann site as if the middle potential barrier had collapsed Inthis limit case (A4) becomes instead

E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)

with once more easily found solutions

E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)

as exemplified in Figure 14 (main) We therefore obtainthat the lowest-lying gap remains near-degenerate for weak

fields and we can conclude therefore that the lowest-lyingeigenvalues display at low-119879 almost the same physics as in


0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls

Figure 14 (Main) Variation with the magnetic Aharonov-Bohmphase 120601 of the energy spectrum (units 119863

2= 1) for the case 119863

1= 0

and a choice of 1198641 119864

2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001

This is to be compared with the 3LS energy spectrum Figure 5(Inset) The energy spectrum in the opposite case 119863

2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863

1about 100 times stronger

the case of a 3LS (Figure 5) This shows the important role ofthe frozen solid surrounding the tunneling ldquoparticlerdquo (whichcould be perhaps a vacuum in fact) and that when nestedwithin an incipient crystallite a magnetic-field-sensitive TSis well described by a tunneling 3LS as the minimal genericmodel potential

Next we seek a description of a cluster of 119873-correlatedtunneling particles (atoms ions or molecules) and derive thetransformation rules for the tunneling parameters (includingalso those involved in the theory for themagnetic effects [21])when the cluster is replaced by a single ldquotunneling particlerdquo asthe result of the coherent tunneling (CT) of the particles in thecluster

The ldquotunneling particlerdquo in question is only a fictitiousone as was inferred in Section 2 by examining local minimain the energy landscape representing the CT of a cluster of119873 true tunneling particles (which in the real glasses mightbe the lighter species involved in the material Li+ in thedisordered crystal Li KCl O2minus in the multisilicates and H+

andor D+ in 119886-glycerol) and for which we have to make upappropriate renormalized tunneling parametersThe conceptof CT in separate local potentials is distinct from that of thejoint tunneling of119873 particles in the same local potential forin the latter case the tunneling probabilitywould be depressedexponentially 119863

0ℏ asymp Ω(Δ

0ℏΩ)

radic119873 (Δ0being the real

particlesrsquo common tunneling transparency) As we shall showbelow at least for moderate values of 119873 for CT in separatepotentials we expect instead

1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)

and for the fictitious particlersquos charge and flux-threaded area

(see [21] for the magnetic effects)

119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)

where 1199020= 119874(119890) is the charge of the real tunneling particles

and 1198860Bohrrsquos radius) In the latter relations less obvious is the

renormalization of the flux-threaded area 119878Δof a 3LS ATS It

is however the direct consequence of our multiphase modelof a real glass thought of as made up of regions of enhancedatomic ordering (RER) or microcrystals (Figures 3 and 6)embedded in a homogeneously disordered host matrix Themagnetic flux appears quadratically in our theory [21] eachelementary flux adding up within each microcrystallite orRER and then appearing squared multiplied by cos2120573 in theglassy matrix in a magnetic field (120573 being the random angleformed by S

Δwith the magnetic field B) a factor averaging

out to 12 in the bulk From these considerations andfrom (A9) and (A10) the renormalization of the compositephenomenological parameter119863

0|119902119890| 119878

Δwould be as follows

(if 119902 = 2119890 appropriate for the multisilicates)

1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)

Setting Δ0= 1mK one gets a value of 119873 ranging from

about 25 coherent-tunneling particles in a cluster at the lowesttemperatures [59] to about 600 at the higher temperaturesThese estimates are somewhat speculative since the realvalues of the elementary flux-threaded area and of theelementary tunneling barrier transparency Δ

0are unknown

we are however inclined to support the value 119873 asymp 200 thatwas proposed by Lubchenko and Wolynes [29] This wouldyield a value of Δmin ranging from 80 120583K to 4mK also forthe mixed alkali-silicate glasses (for which 119863min asymp 800mK)The above considerations show all in all the tendency forthe coherent-tunneling cluster size119873 to be also temperaturedependent

We now come to the justification of (A9) At low tem-peratures the interactions between true tunneling particlesbecome important and coherent-tunneling motion can takeplace Coherentmotion in the context of the tunnelingmodelis a state in which all of the particles in each local potentialcontribute to the overall tunneling process in a correlatedway We exemplify our ideas in the context of the simplest2LS situation first Let us consider two interacting 2LS Letthe positions of the particles in the two wells be left (119871) andright (119877) The tunneling particles in the cluster interact via aweak potential119880whichmayhave its origin for example fromeither a strain-strain interaction having the form 119880 sim 119860119903

3

(dipole-dipole interaction) [29 68] where 119903 is the distancebetween a pair of tunneling particles either in the 119871 or 119877 welland 119860 is a constant or it could be due to electrostatic dipole-dipole interaction The tunneling of the particle in one 2LSfrom 119871 to 119877 (or vice versa) influences via the interactionthe particle in the other 2LS forcing it to jump into the freewell The hopping Hamiltonian of two interacting 2LS can


be written as follows (with Δ119894119871= minusΔ

119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc

minus 119880 (119888dagger

11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)

which favors coherent 119871119871 rarr 119877119877 and 119877119877 rarr 119871119871 jointtunneling and acts on the joint states |119886119886

1015840⟩ =

|119871119871⟩ |119871119877⟩ |119877119871⟩ |119877119877⟩ The coherent motion of the tworeal particles can now be replaced by the tunneling of a newfictitious particle in its own double well In order to writethe renormalized Hamiltonian of two coherent-tunnelingparticles we are interested only in the matrix elements⟨119871119871|119867

2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)

instead of the latter two the pair ⟨119877119871|1198672|119871119877⟩ and

⟨119871119877|1198672|119877119871⟩ having the very same value Δ

01+ Δ

02

could have served the purpose These matrix elementsrepresent the Hamiltonian of the fictitious particle whichcorresponds to both real particles tunneling coherentlytogether

1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)

The conditionΔ1015840

1+Δ

1015840

2= 0 is to be fixed through the addition

of an overall constant Next we consider the case of threeinteracting 2LS and repeat the previous considerations TheHamiltonian of three interacting 2LS has the form

1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)

The matrix elements of a single replacing fictitious particlethat correspond to CT are obtained as follows

⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)

(the choice of the latter two not excluding the remain-ing coherent matrix elements pairs ⟨119877119877119871|119867

3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and

Figure 15 A cluster of 119873 = 4 weakly interacting (real) tunnelingparticles that is being replaced with a (fictitious) single 3LS (Fig-ure 4(c)) having renormalised parameters according to (A9) and(A10)

⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867

3|119871119877119877⟩ which are all equivalent)

One can notice that the renormalized tunneling parameteris the sum of the Δ

0119894of each 2LS The energy asymmetry is

also the arithmetic sum of the Δ119894of each 2LS but one must

add the interaction energy minus119880 multiplied by 119873(119873 minus 1)2Thus for a coherently tunneling cluster of 119873 2LS we findthat the diagonal matrix element becomes generalizing toarbitrary 119873 Δ = sum

119873

119894=1Δ

119894minus (119873(119873 minus 1)2)119880 and the off-

diagonal element that corresponds to the CT-splitting for all119873 particles becomes simply Δ

0= sum

119894Δ

0119894

Applying the previous considerations to our model for anumber 119873 of ATS with three wells (see Figure 15) we canwrite the interacting Hamiltonian in the form

119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)

If we represent the group of 119873 coherently tunneling parti-cles as a single fictitious particle moving in a three-welledpotential which is characterized by its own ground-stateenergies119864

119860and tunneling parameter119863

0 we can describe this

renormalized 3LS by the following Hamiltonian

1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)

The ground-state energies 119864119860

in the wells and tunnelingparameter 119863

0for the fictitious particle in line with the

calculations above can be obtained through

119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)


(and the remaining variants of the second definition lineall equivalent) In analogy with the 2LS considerations onecan see that the renormalized tunneling parameters 119863 =

radic1198642

1+ 119864

2

2+ 119864

2

3and especially 119863

0can be replaced by the

arithmetic sums of those of the bare coherently tunnelingparticles 119863 asymp 119873119863

119894(neglecting the correction for a suffi-

ciently weak 119880 and moderate values of 119873) and 1198630asymp 119873119863

0119894

respectively Indeed the tunneling probabilities of weaklycorrelated events should add up for values of119873 not too largeTherefore since 119873 can attain values as large as 200 [29](independently of the solidrsquos composition) in some modelsand as corroborated by our reasoning in this appendix thisleads to values of 119863

119894and 119863

0119894(as extracted from our theoryrsquos

fitting parameters) comparable to those characteristic of the2LS-TM The large values of 119863min and especially of 119863

0minand 119863

0max as extracted from our fits of our theory to theavailable experimental data find therefore an interesting andphysically plausible explanation

Acknowledgments

Special thanks are due to M F Thorpe for advice to ABakai M I Dyakonov P Fulde and S Shenoy for theirinterest in this work and for illuminating discussions andto A Borisenko for a fruitful collaboration M Paliienko isgrateful to the Italian Ministry of Education University andResearch (MIUR) for support through a PhD Grant of theProgetto Giovani (ambito di indagine n 7 materiali avanzati(in particolare ceramici) per applicazioni strutturali)

References

[1] E J Donth The Glass Transition Springer Berlin Germany2001

[2] J Hwang Z H Melgarejo Y E Kalay et al ldquoNanoscalestructure and structural relaxation in Zr

50Cu

45Al

5bulkmetallic

glassrdquo Physical Review Letters vol 108 no 19 Article ID 1955055 pages 2012

[3] H Bach and D Krause Analysis of the Composition andStructure of Glass and Glass Ceramics Springer New York NYUSA 1999

[4] G Schuster G Hechtfischer D Buck and W HoffmannldquoThermometry below 1Krdquo Reports on Progress in Physics vol57 no 2 p 187 1994

[5] A Meyer J Horbach W Kob F Kargl and H SchoberldquoChannel formation and intermediate range order in sodiumsilicate melts and glassesrdquo Physical Review Letters vol 93 no2 Article ID 027801 1 page 2004

[6] P Strehlow C Enss and S Hunklinger ldquoEvidence for a phasetransition in glasses at very low temperature a macroscopicquantum state of tunneling systemsrdquo Physical Review Lettersvol 80 no 24 pp 5361ndash5364 1998

[7] P StrehlowMWohlfahrt A GM Jansen et al ldquoMagnetic fielddependent tunneling in glassesrdquo Physical Review Letters vol 84no 9 pp 1938ndash1941 2000

[8] M Wohlfahrt P Strehlow C Enss and S HunklingerldquoMagnetic-field effects in non-magnetic glassesrdquo EurophysicsLetters vol 56 no 5 pp 690ndash694 2001

[9] J Le Cochec F Ladieu and P Pari ldquoMagnetic field effect onthe dielectric constant of glasses evidence of disorder withintunneling barriersrdquo Physical Review B vol 66 no 6 Article ID064203 5 pages 2002

[10] S Ludwig C Enss P Strehlow and S Hunklinger ldquoDirectcoupling of magnetic fields to tunneling systems in glassesrdquoPhysical Review Letters vol 88 no 7 Article ID 075501 4 pages2002

[11] S Ludwig P Nagel S Hunklinger and C Enss ldquoMagnetic fielddependent coherent polarization echoes in glassesrdquo Journal ofLow Temperature Physics vol 131 no 1-2 pp 89ndash111 2003

[12] A Borisenko and G Jug ldquoParamagnetic tunneling systems andtheir contribution to the polarization echo in glassesrdquo PhysicalReview Letters vol 107 no 7 Article ID 075501 4 pages 2011

[13] P Nagel A Fleischmann S Hunklinger and C Enss ldquoNovelisotope effects observed in polarization echo experiments inglassesrdquo Physical Review Letters vol 92 no 24 Article ID2455111 4 pages 2004

[14] P Esquinazi Ed Tunneling Systems in Amorphous and Crys-talline Solids Springer Berlin Germany 1998

[15] C Enss ldquoBeyond the tunneling model quantum phenomena inultracold glassesrdquo Physica B vol 316-317 pp 12ndash20 2002

[16] C Enss ldquoAnomalous behavior of insulating glasses at ultra-lowtemperaturesrdquo inAdvances in Solid State Physics B Kramer Edvol 42 p 335 Springer Berlin Germany 2002

[17] W A Phillips EdAmorphous Solids Low-Temperature Proper-ties Springer Berlin Germany 1981

[18] W A Phillips ldquoTwo-level states in glassesrdquo Reports on Progressin Physics vol 50 no 12 p 1657 1987

[19] W A Phillips ldquoTunneling states in amorphous solidsrdquo Journalof Low Temperature Physics vol 7 no 3-4 pp 351ndash360 1972

[20] P W Anderson B I Halperin and C M Varma ldquoAnomalouslow-temperature thermal properties of glasses and spin glassesrdquoPhilosophical Magazine vol 25 no 1 pp 1ndash9 1972

[21] G Jug ldquoTheory of the thermal magnetocapacitance of multi-component silicate glasses at low temperaturerdquo PhilosophicalMagazine vol 84 no 33 pp 3599ndash3615 2004

[22] G Jug and M Paliienko ldquoEvidence for a two-componenttunnelling mechanism in the multicomponent glasses at lowtemperaturesrdquo Europhysics Letters vol 90 no 3 Article ID36002 2010

[23] M LayerMHeitz J Classen C Enss and SHunklinger ldquoLow-temperature elastic properties of glasses in magnetic fieldsrdquoJournal of Low Temperature Physics vol 124 no 3-4 pp 419ndash429 2001

[24] W M MacDonald A C Anderson and J Schroeder ldquoLow-temperature behavior of potassium and sodium silicate glassesrdquoPhysical Review B vol 31 no 2 pp 1090ndash1101 1985

[25] C C Yu and A J Leggett ldquoLow temperature properties ofamorphous materials through a glass darklyrdquo Comments onCondensed Matter Physics vol 14 p 231 1988

[26] D C Vural and A J Leggett ldquoUniversal sound absorptionin amorphous solids a theory of elastically coupled genericblocksrdquo Journal of Non-Crystalline Solids vol 357 no 19-20 pp3528ndash3537 2011

[27] D C Vural Universal sound attenuation in amorphous solidsat low-temperatures [PhD thesis] University of Illinois 2011httparxivorgabs12031281

[28] A Fleischmann andC Enss ldquoGeheimnis der Tunnelsysteme imGlas geluftetrdquo Physik Journal vol 6 no 10 pp 41ndash46 2007


[29] V Lubchenko andPGWolynes ldquoIntrinsic quantumexcitationsof low temperature glassesrdquo Physical Review Letters vol 87 no19 Article ID 195901 4 pages 2001

[30] J J de Yoreo W Knaak M Meissner and R O Pohl ldquoLow-temperature properties of crystalline (KBr)

1minus119909(KCN)

119909 a model

glassrdquo Physical Review B vol 34 no 12 pp 8828ndash8842 1986[31] RO Pohl X Liu andEThompson ldquoLow-temperature thermal

conductivity and acoustic attenuation in amorphous solidsrdquoReviews of Modern Physics vol 74 no 4 pp 991ndash1013 2002

[32] T A Weber and F H Stillinger ldquoInteractions local orderand atomic-rearrangement kinetics in amorphous nickel-phosphorous alloysrdquo Physical Review B vol 32 no 8 pp 5402ndash5411 1985

[33] F H Stillinger ldquoA topographic view of supercooled liquids andglass formationrdquo Science vol 267 no 5206 pp 1935ndash1939 1995

[34] K H Fischer and J A Herz Spin Glasses CambridgeUniversityPress Cambridge UK 1993

[35] J A Mydosh Spin Glasses An Experimental Introductionchapter 8 Taylor amp Francis London UK 1993

[36] A Heuer ldquoProperties of a glass-forming system as derived fromits potential energy landscaperdquo Physical Review Letters vol 78no 21 pp 4051ndash4054 1997

[37] A Heuer and R J Silbey ldquoMicroscopic description of tunnelingsystems in a structuralmodel glassrdquo Physical Review Letters vol70 no 25 pp 3911ndash3914 1993

[38] C -Y Fang H Yinnon and D R Uhlmann ldquoA kinetictreatment of glass formation VIII critical cooling rates forNa

2O-SiO

2and K

2O-SiO

2glassesrdquo Journal of Non-Crystalline

Solids vol 57 no 3 pp 465ndash471 1983[39] N Pellegri E J C Dawnay and E M Yeatman ldquoMultilayer

SiO2-B

2O

3-Na

2O films on Si for optical applicationsrdquo Journal

of Sol-Gel Science and Technology vol 13 no 1ndash3 pp 783ndash7871998

[40] GDeA Licciulli CMassaro et al ldquoSilver nanocrystals in silicaby sol-gel processingrdquo Journal of Non-Crystalline Solids vol 194no 3 pp 225ndash234 1996

[41] X L Duan D Yuan Z Sun et al ldquoSol-gel preparation of Co2+MgAl

2O

4nanocrystals embedded in SiO

2-based glassrdquo Journal

of Crystal Growth vol 252 no 1ndash3 pp 311ndash316 2003[42] J E Shelby Introduction to Glass Science and Technology

chapter 5The Royal Society of Chemistry Cambridge UK 2ndedition 2005

[43] B Henderson and G F Imbush Optical Spectroscopy of In-Organic Solids Oxford University Press New York NY USA1989

[44] L Lichtenstein M Heyde and H J Freund ldquoCrystalline-vitreous interface in two dimensional silicardquo Physical ReviewLetters vol 109 no 10 Article ID 106101 5 pages 2012

[45] J A Sussmann ldquoElectric dipoles due to trapped electronsrdquoProceedings of the Physical Society vol 79 no 4 p 758 1962

[46] S Sugano Y Tanabe and H Kamimura Multiplets ofTransition-Metal Ions in Crystals chapter 9 Academic PressLondon UK 1970

[47] F Pierre H Pothier D Esteve and M H Devoret ldquoEnergyredistribution between quasiparticles in mesoscopic silverwiresrdquo Journal of Low Temperature Physics vol 118 no 5-6 pp437ndash445 2000

[48] H Schenk and R Peschar ldquoUnderstanding the structure ofchocolaterdquo Radiation Physics and Chemistry vol 71 no 3-4 pp829ndash835 2004

[49] M H Cohen and G S Grest ldquoOrigin of low-temperaturetunneling states in glassesrdquo Physical Review Letters vol 45 no15 pp 1271ndash1274 1980

[50] M H Cohen and G S Grest ldquoA new free-volume theory of thegvolass transitionrdquoAnnals of the New York Academy of Sciencesvol 371 pp 199ndash209 1981

[51] H M Carruzzo E R Grannan and C C Yu ldquoNonequilibriumdielectric behavior in glasses at low temperatures evidence forinteracting defectsrdquo Physical Review B vol 50 no 10 pp 6685ndash6695 1994

[52] S Hunklinger and W Arnold ldquoUltrasonic properties of glassesat low temperaturesrdquo in Physical Acoustics R N Thurston andW P Mason Eds vol 12 p 155 1976

[53] S A J Wiegers R Jochemsen C C Kranenburg and GFrossati ldquoComparison of some glass thermometers at lowtemperatures in a high magnetic fieldrdquo Review of ScientificInstruments vol 58 no 12 article 2274 5 pages 1987

[54] S Kettemann P Fulde and P Strehlow ldquoCorrelated persistenttunneling currents in glassesrdquo Physical Review Letters vol 83no 21 pp 4325ndash4328 1999

[55] J L Black and B I Halperin ldquoSpectral diffusion phononechoes and saturation recovery in glasses at low temperaturesrdquoPhysical Review B vol 16 no 6 pp 2879ndash2895 1977

[56] V G Karpov M I Klinger and F N Ignatrsquoev ldquoTheory ofthe low-temperature anomalies in the thermal properties ofamorphous structuresrdquo Soviet PhysicsmdashJETP vol 57 no 2pp 439ndash448 1983 translated from Zhurnal Eksperimentalrsquonoii Teoreticheskoi Fiziki vol 84 p 760 1983

[57] R Jankowiak J M Hayes and G J Small ldquoLow-temperaturespecific heat of glasses temperature and time dependencerdquoPhysical Review B vol 38 no 3 pp 2084ndash2088 1988

[58] V G Karpov M I Klinger and F N Ignatrsquoev ldquoTheory ofthe low-temperature anomalies in the thermal properties ofamorphous structuresrdquo Soviet PhysicsmdashJETP vol 57 no 2 pp439ndash448 1983

[59] M Paliienko S Bonfanti and G Jug to be published[60] J L Black ldquoRelationship between the time-dependent specific

heat and the ultrasonic properties of glasses at low tempera-turesrdquo Physical Review B vol 17 no 6 pp 2740ndash2791 1978

[61] M SahimiApplications of PercolationTheory Taylor amp FrancisLondon UK 1994

[62] R Pynn andA Skjeltorp Eds Scaling Phenomena inDisorderedSystems (NATO-ASI ldquoGeilo School 1985rdquo) Plenum Press NewYork NY USA 1985

[63] D Stauffer and A Aharony Introduction to Percolation TheoryTaylor amp Francis London UK 2nd edition 2003

[64] A Borisenko ldquoHole-compensated Fe3+ impurities in quartzglasses a contribution to subkelvin thermodynamicsrdquo Journalof Physics Condensed Matter vol 19 no 41 Article ID 4161022007

[65] G X Mack and A C Anderson ldquoLow-temperature behavior ofvitreous silica containing neon soluterdquo Physical Review B vol31 no 2 pp 1102ndash1106 1985

[66] W M MacDonald A C Anderson and J Schroeder ldquoLow-temperature behavior of potassium borate glassesrdquo PhysicalReview B vol 32 no 2 pp 1208ndash1211 1985

[67] A Wurger ldquoPersistent tunneling currents and magnetic-fieldeffects in glassesrdquo Physical Review Letters vol 88 no 7 ArticleID 075502 4 pages 2002

[68] M W Klein B Fischer A C Anderson and P J AnthonyldquoStrain interactions and the low-temperature properties ofglassesrdquo Physical Review B vol 18 no 10 pp 5887ndash5891 1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



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OpticsInternational Journal of



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Biophysics


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It is believed that other metallic glasses should presentsimilar structural features and thusmdashon general groundsmdashone would expect that nonmetallic window glasses too likepure SiO

2and all the more so the commercial multisilicates

of complex chemical composition should present a mul-tiphased structure with the size and concentration of thenear-crystalline regions or RER depending for exampleon composition quench rate and the presence of impuritiesacting as nucleation centres for the RER Indeed materials ofthe general composition (MgO)

119909(Al

2O

3)119910(SiO

2)1minus119909minus119910

(MASin short) are termed ceramic glasses (one of the best knowncommercial examples being Schottrsquos Ceran where Li

2O

replaces MgO and of course CaO or BaO can also replaceor be added to MgO and still yield a ceramic glass) Thesematerials are known to contain microcrystals embedded inan otherwise homogeneously amorphous matrix [3] This isnot surprising for materials made up of a good glass-former(eg SiO

2 Al

2O

3 etc) and good crystal-formers (eg BaO

K2O ) are known to be multiphased [4] with the good

crystal-formers generating their own pockets and channelscarved out within the otherwise homogeneously amorphousnetwork of the good glass-formerrsquos ions [5] Within thesepockets and channels incipient nano- or even microcrystalsmay form but the point of view will be taken in this workthat on general grounds even the purest single-component(eg As SiO

2) glass-former will be rich in RER unless the

quench-rate from themelt is so large as to avoid the formationof crystalline regions or RER

These refined structural details of glasses are evidentlyhard to reveal in all and especially the near-ideal cases(no good crystal-formers no impurities added and rapidquenches) with the available spectroscopic techniques Forexample X-ray spectroscopy does not reveal nano-crystalsbelow the nanometer size However at low and very lowtemperaturesmdashwhere all said structural features remain basi-cally unalteredmdashsome recent experimental findings mightnow improve perspectives with what would appear setto become a new spectroscopy tool Indeed a series ofremarkable magnetic effects have recently been discovered innonmagnetic glasses (multisilicates and organic glasses) [6ndash13] with in the opinion of the present authors a most likelyexplanation for the newphenomena stemming precisely fromthe multiphase nature of real glasses and the presence ofthe RER or microcrystalline regions in their microscopicstructure In turn when the multiphase theory shall befully developed the magnetic effects could represent a validnew spectroscopic tool capable of characterizing micro- ornanocrystals or even incipient crystals and RER in the realglasses The key to this possible development is some newexciting physics of the cold glasses in the presence (and evenin the absence as shown in the present paper) of a magneticfield The magnetic effects in the cold glasses could becomeeventually the amorphous counterpart of the de Haas-vanAlphen and Shubnikov-de Haas effects in crystalline solidsin determining the real structure of amorphous solids

Systematic research on the low-temperature propertiesof glasses has been ongoing for more than 40 years andsome significant theoretical and experimental progress has

been made in the understanding of the unusual behaviourof glasses and of their low-temperature anomalies [14ndash16]This temperature range (119879 lt 1K) is deemed importantfor the appearance of universal behaviour (independent ofcomposition) as well as for the effects of quantummechanicsin the physics of glasses However to make progress in theunderstanding of the low-temperature physics of glassesthere remains a wide range of important questions that arestill open or only partially answered particularly in thelight of some still poorly understood recent and even olderexperiments in cold composite glasses

It is well known that cold glasses show somewhat univer-sal thermal acoustic and dielectric properties which are verydifferent from those of crystalline solids at low temperatures(below 1K) [17 18] The heat capacity 119862

119901of dielectric glasses

is much larger and the thermal conductivity 120581 is orders ofmagnitude lower than the corresponding values found intheir crystalline counterparts119862

119901depends approximately lin-

early and 120581 almost quadratically on temperature119879 (in crystalsone can observe a cubic dependence for both properties)The dielectric constant (real part) 1205981015840 and sound velocity atlow frequencies display in glasses a universal logarithmicdependence in 119879 These ldquoanomalousrdquo and yet universalthermal dielectric and acoustic properties of glasses arewell explained (at least for 119879 lt 1K) since 1972 whenPhillips [19] and also Anderson et al [20] independentlyintroduced the tunnelling model (TM) the fundamentalpostulate of which was the general existence of atoms orsmall groups of atoms in cold amorphous solids which cantunnel like a single quantum-mechanical particle betweentwo configurations of very similar energy (two-level systems(2LS)) The 2LS-TM is widely used in the investigation ofthe low-temperature properties of glasses mostly becauseof its technical simplicity In fact it will be argued in thispaper that tunneling takes place in more complicated localpotential scenarios (multiwelled potentials) and a situationwill be discussed where the use of a number of ldquostatesrdquogreater than two is essential Moreover new insight will begiven on the role of percolation and fractal theory in theTM of multicomponent glasses We present in this paper thejustification and details of the construction of an extendedTM that has been successfully employed to explain theunusual properties of the cold glasses in a magnetic field [21]as well as in zero field when systematic changes in the glassrsquocomposition are involved [22]

The linear dependence in ln(119879) of the real part ofthe dielectric constant 1205981015840(119879) makes the cold glasses usefulin low-temperature thermometry and normally structuralwindow-type glasses are expected to be isotropic insulatorsthat do not present any remarkable magnetic-field responsephenomena (other than a weak response in 119862

119901to the

trace paramagnetic impurities) For some multicomponentsilicate glass it has become possible to measure observablemuch larger than expected changes in 1205981015840(119879 119861) (12057512059810158401205981015840 sim

10minus4) already in a magnetic field as weak as a few Oe

[6 7] A typical glass giving such strong response has thecomposition Al

2O

3-BaO-SiO

2thus a MAS ceramic-glass

herewith termed AlBaSiO The measurements were made





2+119909C119910H




2(Suprasil



0(120596) relax-





2O)

119909and (SiO

2)1minus119909(Na

2O)



119901(119879) for these




















119867(2)

0= minus(12)(Δ120590

119911+ Δ





119867(2)

0= minus

1

2(Δ Δ

0

Δ0minusΔ) (1)




2

0



constant)

119875 (Δ Δ0) =

119875

Δ0

(2)








119901(and




119901(119879 119861) and

1205981015840(119879 119861) [6 7 21] or in 119862

119901(119879 119909) and 1205981015840(119879 119909) [22 24]) are








119908= 2-


119894









minus084

minus086

minus088

minus09

minus119864119864

crys

t




0(suitably defined)




119908= 3 (3LS)

119867(3)

0= (

1198641119863

0119863

0

11986301198642119863

0

1198630119863

01198643

) (3)

where 1198641 119864

2 119864



119894=1119864119894= 0 and taken from



119908= 4

(4LS)

119867(4)

0= (

1198641119863

1119863

2119863

1

11986311198642119863

1119863

2

1198632119863

11198643119863

1

1198631119863

2119863

11198644

) (4)

where 1198641 119864

2 and 119864

3 119864




1198632(nnn hopping |119863

2| ≪ |119863






4or (AlO



2Osdot3(SiO

2) (or (Na

2O)

025(SiO

2)075

) at 2100K (above 119879119892





50Cu

45Al

5metallic



50Cu

45Al













119908-welled 3D



119867(3)

0=

3

sum

119894=1

119864119894119888dagger

119894119888119894+ sum

119894 = 119895

1198630119888dagger

119894119888119895+ hc (5)


0|119895⟩





0≃ 119886ℏΩ119890


0≳ ℏΩ119886 gt 0 and 119887 =




3-symmetric local



1= 119864

2= 119864



119908= 3)









(a) (b)

OSi

Ba Al

119881(119909 119910)

(c)







0gt 0 is the



1 119864

2 119864

3 and 119863

0of a tunneling


119908= 4 model


1= 119864

2= 119864



119875ATS (1198641 1198642 1198643 1198630) =

119875lowast

(1198642

1+ 119864

2

2+ 119864

2

3)119863

0

(6)




0119890119894120601119894119895




E119896

1198630

= 2radic1 minus

sum119894 = 119895119864119894119864119895

61198632

0

cos(13120579 + 120579

119896)

cos 120579 = (cos120601 +119864111986421198643

21198633

0

)(1 minus

sum119894 = 119895119864119894119864119895

31198632

0

)

minus32

(7)



2 119864

3

and 1198630≫ radic119864

2

1+ 119864

2

2+ 119864

2



0 Φ(B) = S




(2)

0(119864

1 119864

2 119863


1 119864

2 119863

0) =

119875lowast119863

0radic119864

2

1+ 119864

2






0 05 1 15 2 25 3

0

05

1

15

2

Dim

ensio

nles

s ene

rgy

leve

ls

minus05

minus1

minus15

minus2

Aharonov-Bohm phase

Energy levels 1198641 + 1198642 + 1198643 = 0



1 119864

2 and 119864

3





in the best glasses






119901





0 We can


2119891(119865)120597119865

2|119865=0

where 119891(119865) = minus(1119896119861119879) ln119885(119865)




12 119885 = 119890minusE1119896119861119879 + 119890minusE2119896119861119879


1015840(119879) minus 120598

1015840(119879

0)]120598

1015840(119879

0) (where


2


0and + ln119879 for 119879 gt 119879





Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162LS=Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES+Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES=2119875119901

2

0

31205980120598119903

int

119911max

119911min

119889119911

119911

radic1 minus (Δ

0min2119896

119861119879119911)

2

tanh 119911

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL=119875119901

2

0

31205980120598119903

times int

119911max

119911min

119889119911int

120591max

120591min

119889120591

120591radic1 minus

120591min120591


1 + 12059621205912

(8)


160

140

120

100

80

60

40

20

0

60

40

20

0

0001

0001

001

001

01

01

1

1

Temperature (K)

Temperature (K)

105middotΔ

120598998400120598

998400

105middotΔ

120598998400120598

998400

Al2O3-BaO-SiO2

Glass (AlBaSiO)

AlBaSiO

ATS

2LSKFS2LS + ATS

SiO2 (Spectrosil)







119861119879119911) [17 18]

120591minus1=119864Δ

2

0

120574tanh( 119864

2119896119861119879) (9)




minus1

min = 1198643120574 tanh(1198642119896

119861119879) 120591minus1max =

119864Δ2



0


120598 (119879 120596) = 1205981015840

RES (119879) + 1205981015840

REL (119879)1

1 + 119894120596120591 (10)



Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES≃

minus2

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0max) if 119879 lt

Δ0max2119896

119861

0 if 119879 gtΔ

0max2119896

119861

(11)


Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL≃

0 if 120596120591min ≫ 1

1

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0min) if 120596120591min ≪ 1

(12)



1198961198611198790(120596) ≃ ((12)120574120596)



0slope given

by 119878 = 119875119901203120598

0120598119903



2)1minus119909(K

2O)









0lt E

1≪ E

2


2

1+ 119864

2

2+ 119864

2

3≪ 119863

0limits




1minusE

0


1+ 119864

2

2+ 119864

2

3equiv 119863 (13)


2+ 119864

3= 0



140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400


119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS


2O)

119909glasses as



Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910


2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)


120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)




lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630


119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)




Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)


Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)



0and an


0with 119878ATS =

7120587lowast1199012

16120598




2)1minus119909(K

2O)








lowast1199012

1120598



2LS = 1198751199012

0120598







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







2+119909C119910H




2(Suprasil



0(120596) relax-





2O)

119909and (SiO

2)1minus119909(Na

2O)



119901(119879) for these




















119867(2)

0= minus(12)(Δ120590

119911+ Δ





119867(2)

0= minus

1

2(Δ Δ

0

Δ0minusΔ) (1)




2

0



constant)

119875 (Δ Δ0) =

119875

Δ0

(2)








119901(and




119901(119879 119861) and

1205981015840(119879 119861) [6 7 21] or in 119862

119901(119879 119909) and 1205981015840(119879 119909) [22 24]) are








119908= 2-


119894









minus084

minus086

minus088

minus09

minus119864119864

crys

t




0(suitably defined)




119908= 3 (3LS)

119867(3)

0= (

1198641119863

0119863

0

11986301198642119863

0

1198630119863

01198643

) (3)

where 1198641 119864

2 119864



119894=1119864119894= 0 and taken from



119908= 4

(4LS)

119867(4)

0= (

1198641119863

1119863

2119863

1

11986311198642119863

1119863

2

1198632119863

11198643119863

1

1198631119863

2119863

11198644

) (4)

where 1198641 119864

2 and 119864

3 119864




1198632(nnn hopping |119863

2| ≪ |119863






4or (AlO



2Osdot3(SiO

2) (or (Na

2O)

025(SiO

2)075

) at 2100K (above 119879119892





50Cu

45Al

5metallic



50Cu

45Al













119908-welled 3D



119867(3)

0=

3

sum

119894=1

119864119894119888dagger

119894119888119894+ sum

119894 = 119895

1198630119888dagger

119894119888119895+ hc (5)


0|119895⟩





0≃ 119886ℏΩ119890


0≳ ℏΩ119886 gt 0 and 119887 =




3-symmetric local



1= 119864

2= 119864



119908= 3)









(a) (b)

OSi

Ba Al

119881(119909 119910)

(c)







0gt 0 is the



1 119864

2 119864

3 and 119863

0of a tunneling


119908= 4 model


1= 119864

2= 119864



119875ATS (1198641 1198642 1198643 1198630) =

119875lowast

(1198642

1+ 119864

2

2+ 119864

2

3)119863

0

(6)




0119890119894120601119894119895




E119896

1198630

= 2radic1 minus

sum119894 = 119895119864119894119864119895

61198632

0

cos(13120579 + 120579

119896)

cos 120579 = (cos120601 +119864111986421198643

21198633

0

)(1 minus

sum119894 = 119895119864119894119864119895

31198632

0

)

minus32

(7)



2 119864

3

and 1198630≫ radic119864

2

1+ 119864

2

2+ 119864

2



0 Φ(B) = S




(2)

0(119864

1 119864

2 119863


1 119864

2 119863

0) =

119875lowast119863

0radic119864

2

1+ 119864

2






0 05 1 15 2 25 3

0

05

1

15

2

Dim

ensio

nles

s ene

rgy

leve

ls

minus05

minus1

minus15

minus2

Aharonov-Bohm phase

Energy levels 1198641 + 1198642 + 1198643 = 0



1 119864

2 and 119864

3





in the best glasses






119901





0 We can


2119891(119865)120597119865

2|119865=0

where 119891(119865) = minus(1119896119861119879) ln119885(119865)




12 119885 = 119890minusE1119896119861119879 + 119890minusE2119896119861119879


1015840(119879) minus 120598

1015840(119879

0)]120598

1015840(119879

0) (where


2


0and + ln119879 for 119879 gt 119879





Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162LS=Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES+Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES=2119875119901

2

0

31205980120598119903

int

119911max

119911min

119889119911

119911

radic1 minus (Δ

0min2119896

119861119879119911)

2

tanh 119911

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL=119875119901

2

0

31205980120598119903

times int

119911max

119911min

119889119911int

120591max

120591min

119889120591

120591radic1 minus

120591min120591


1 + 12059621205912

(8)


160

140

120

100

80

60

40

20

0

60

40

20

0

0001

0001

001

001

01

01

1

1

Temperature (K)

Temperature (K)

105middotΔ

120598998400120598

998400

105middotΔ

120598998400120598

998400

Al2O3-BaO-SiO2

Glass (AlBaSiO)

AlBaSiO

ATS

2LSKFS2LS + ATS

SiO2 (Spectrosil)







119861119879119911) [17 18]

120591minus1=119864Δ

2

0

120574tanh( 119864

2119896119861119879) (9)




minus1

min = 1198643120574 tanh(1198642119896

119861119879) 120591minus1max =

119864Δ2



0


120598 (119879 120596) = 1205981015840

RES (119879) + 1205981015840

REL (119879)1

1 + 119894120596120591 (10)



Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES≃

minus2

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0max) if 119879 lt

Δ0max2119896

119861

0 if 119879 gtΔ

0max2119896

119861

(11)


Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL≃

0 if 120596120591min ≫ 1

1

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0min) if 120596120591min ≪ 1

(12)



1198961198611198790(120596) ≃ ((12)120574120596)



0slope given

by 119878 = 119875119901203120598

0120598119903



2)1minus119909(K

2O)









0lt E

1≪ E

2


2

1+ 119864

2

2+ 119864

2

3≪ 119863

0limits




1minusE

0


1+ 119864

2

2+ 119864

2

3equiv 119863 (13)


2+ 119864

3= 0



140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400


119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS


2O)

119909glasses as



Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910


2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)


120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)




lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630


119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)




Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)


Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)



0and an


0with 119878ATS =

7120587lowast1199012

16120598




2)1minus119909(K

2O)








lowast1199012

1120598



2LS = 1198751199012

0120598







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119867(2)

0= minus(12)(Δ120590

119911+ Δ





119867(2)

0= minus

1

2(Δ Δ

0

Δ0minusΔ) (1)




2

0



constant)

119875 (Δ Δ0) =

119875

Δ0

(2)








119901(and




119901(119879 119861) and

1205981015840(119879 119861) [6 7 21] or in 119862

119901(119879 119909) and 1205981015840(119879 119909) [22 24]) are








119908= 2-


119894









minus084

minus086

minus088

minus09

minus119864119864

crys

t




0(suitably defined)




119908= 3 (3LS)

119867(3)

0= (

1198641119863

0119863

0

11986301198642119863

0

1198630119863

01198643

) (3)

where 1198641 119864

2 119864



119894=1119864119894= 0 and taken from



119908= 4

(4LS)

119867(4)

0= (

1198641119863

1119863

2119863

1

11986311198642119863

1119863

2

1198632119863

11198643119863

1

1198631119863

2119863

11198644

) (4)

where 1198641 119864

2 and 119864

3 119864




1198632(nnn hopping |119863

2| ≪ |119863






4or (AlO



2Osdot3(SiO

2) (or (Na

2O)

025(SiO

2)075

) at 2100K (above 119879119892





50Cu

45Al

5metallic



50Cu

45Al













119908-welled 3D



119867(3)

0=

3

sum

119894=1

119864119894119888dagger

119894119888119894+ sum

119894 = 119895

1198630119888dagger

119894119888119895+ hc (5)


0|119895⟩





0≃ 119886ℏΩ119890


0≳ ℏΩ119886 gt 0 and 119887 =




3-symmetric local



1= 119864

2= 119864



119908= 3)









(a) (b)

OSi

Ba Al

119881(119909 119910)

(c)







0gt 0 is the



1 119864

2 119864

3 and 119863

0of a tunneling


119908= 4 model


1= 119864

2= 119864



119875ATS (1198641 1198642 1198643 1198630) =

119875lowast

(1198642

1+ 119864

2

2+ 119864

2

3)119863

0

(6)




0119890119894120601119894119895




E119896

1198630

= 2radic1 minus

sum119894 = 119895119864119894119864119895

61198632

0

cos(13120579 + 120579

119896)

cos 120579 = (cos120601 +119864111986421198643

21198633

0

)(1 minus

sum119894 = 119895119864119894119864119895

31198632

0

)

minus32

(7)



2 119864

3

and 1198630≫ radic119864

2

1+ 119864

2

2+ 119864

2



0 Φ(B) = S




(2)

0(119864

1 119864

2 119863


1 119864

2 119863

0) =

119875lowast119863

0radic119864

2

1+ 119864

2






0 05 1 15 2 25 3

0

05

1

15

2

Dim

ensio

nles

s ene

rgy

leve

ls

minus05

minus1

minus15

minus2

Aharonov-Bohm phase

Energy levels 1198641 + 1198642 + 1198643 = 0



1 119864

2 and 119864

3





in the best glasses






119901





0 We can


2119891(119865)120597119865

2|119865=0

where 119891(119865) = minus(1119896119861119879) ln119885(119865)




12 119885 = 119890minusE1119896119861119879 + 119890minusE2119896119861119879


1015840(119879) minus 120598

1015840(119879

0)]120598

1015840(119879

0) (where


2


0and + ln119879 for 119879 gt 119879





Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162LS=Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES+Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES=2119875119901

2

0

31205980120598119903

int

119911max

119911min

119889119911

119911

radic1 minus (Δ

0min2119896

119861119879119911)

2

tanh 119911

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL=119875119901

2

0

31205980120598119903

times int

119911max

119911min

119889119911int

120591max

120591min

119889120591

120591radic1 minus

120591min120591


1 + 12059621205912

(8)


160

140

120

100

80

60

40

20

0

60

40

20

0

0001

0001

001

001

01

01

1

1

Temperature (K)

Temperature (K)

105middotΔ

120598998400120598

998400

105middotΔ

120598998400120598

998400

Al2O3-BaO-SiO2

Glass (AlBaSiO)

AlBaSiO

ATS

2LSKFS2LS + ATS

SiO2 (Spectrosil)







119861119879119911) [17 18]

120591minus1=119864Δ

2

0

120574tanh( 119864

2119896119861119879) (9)




minus1

min = 1198643120574 tanh(1198642119896

119861119879) 120591minus1max =

119864Δ2



0


120598 (119879 120596) = 1205981015840

RES (119879) + 1205981015840

REL (119879)1

1 + 119894120596120591 (10)



Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES≃

minus2

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0max) if 119879 lt

Δ0max2119896

119861

0 if 119879 gtΔ

0max2119896

119861

(11)


Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL≃

0 if 120596120591min ≫ 1

1

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0min) if 120596120591min ≪ 1

(12)



1198961198611198790(120596) ≃ ((12)120574120596)



0slope given

by 119878 = 119875119901203120598

0120598119903



2)1minus119909(K

2O)









0lt E

1≪ E

2


2

1+ 119864

2

2+ 119864

2

3≪ 119863

0limits




1minusE

0


1+ 119864

2

2+ 119864

2

3equiv 119863 (13)


2+ 119864

3= 0



140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400


119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS


2O)

119909glasses as



Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910


2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)


120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)




lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630


119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)




Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)


Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)



0and an


0with 119878ATS =

7120587lowast1199012

16120598




2)1minus119909(K

2O)








lowast1199012

1120598



2LS = 1198751199012

0120598







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus084

minus086

minus088

minus09

minus119864119864

crys

t




0(suitably defined)




119908= 3 (3LS)

119867(3)

0= (

1198641119863

0119863

0

11986301198642119863

0

1198630119863

01198643

) (3)

where 1198641 119864

2 119864



119894=1119864119894= 0 and taken from



119908= 4

(4LS)

119867(4)

0= (

1198641119863

1119863

2119863

1

11986311198642119863

1119863

2

1198632119863

11198643119863

1

1198631119863

2119863

11198644

) (4)

where 1198641 119864

2 and 119864

3 119864




1198632(nnn hopping |119863

2| ≪ |119863






4or (AlO



2Osdot3(SiO

2) (or (Na

2O)

025(SiO

2)075

) at 2100K (above 119879119892





50Cu

45Al

5metallic



50Cu

45Al













119908-welled 3D



119867(3)

0=

3

sum

119894=1

119864119894119888dagger

119894119888119894+ sum

119894 = 119895

1198630119888dagger

119894119888119895+ hc (5)


0|119895⟩





0≃ 119886ℏΩ119890


0≳ ℏΩ119886 gt 0 and 119887 =




3-symmetric local



1= 119864

2= 119864



119908= 3)









(a) (b)

OSi

Ba Al

119881(119909 119910)

(c)







0gt 0 is the



1 119864

2 119864

3 and 119863

0of a tunneling


119908= 4 model


1= 119864

2= 119864



119875ATS (1198641 1198642 1198643 1198630) =

119875lowast

(1198642

1+ 119864

2

2+ 119864

2

3)119863

0

(6)




0119890119894120601119894119895




E119896

1198630

= 2radic1 minus

sum119894 = 119895119864119894119864119895

61198632

0

cos(13120579 + 120579

119896)

cos 120579 = (cos120601 +119864111986421198643

21198633

0

)(1 minus

sum119894 = 119895119864119894119864119895

31198632

0

)

minus32

(7)



2 119864

3

and 1198630≫ radic119864

2

1+ 119864

2

2+ 119864

2



0 Φ(B) = S




(2)

0(119864

1 119864

2 119863


1 119864

2 119863

0) =

119875lowast119863

0radic119864

2

1+ 119864

2






0 05 1 15 2 25 3

0

05

1

15

2

Dim

ensio

nles

s ene

rgy

leve

ls

minus05

minus1

minus15

minus2

Aharonov-Bohm phase

Energy levels 1198641 + 1198642 + 1198643 = 0



1 119864

2 and 119864

3





in the best glasses






119901





0 We can


2119891(119865)120597119865

2|119865=0

where 119891(119865) = minus(1119896119861119879) ln119885(119865)




12 119885 = 119890minusE1119896119861119879 + 119890minusE2119896119861119879


1015840(119879) minus 120598

1015840(119879

0)]120598

1015840(119879

0) (where


2


0and + ln119879 for 119879 gt 119879





Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162LS=Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES+Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES=2119875119901

2

0

31205980120598119903

int

119911max

119911min

119889119911

119911

radic1 minus (Δ

0min2119896

119861119879119911)

2

tanh 119911

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL=119875119901

2

0

31205980120598119903

times int

119911max

119911min

119889119911int

120591max

120591min

119889120591

120591radic1 minus

120591min120591


1 + 12059621205912

(8)


160

140

120

100

80

60

40

20

0

60

40

20

0

0001

0001

001

001

01

01

1

1

Temperature (K)

Temperature (K)

105middotΔ

120598998400120598

998400

105middotΔ

120598998400120598

998400

Al2O3-BaO-SiO2

Glass (AlBaSiO)

AlBaSiO

ATS

2LSKFS2LS + ATS

SiO2 (Spectrosil)







119861119879119911) [17 18]

120591minus1=119864Δ

2

0

120574tanh( 119864

2119896119861119879) (9)




minus1

min = 1198643120574 tanh(1198642119896

119861119879) 120591minus1max =

119864Δ2



0


120598 (119879 120596) = 1205981015840

RES (119879) + 1205981015840

REL (119879)1

1 + 119894120596120591 (10)



Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES≃

minus2

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0max) if 119879 lt

Δ0max2119896

119861

0 if 119879 gtΔ

0max2119896

119861

(11)


Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL≃

0 if 120596120591min ≫ 1

1

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0min) if 120596120591min ≪ 1

(12)



1198961198611198790(120596) ≃ ((12)120574120596)



0slope given

by 119878 = 119875119901203120598

0120598119903



2)1minus119909(K

2O)









0lt E

1≪ E

2


2

1+ 119864

2

2+ 119864

2

3≪ 119863

0limits




1minusE

0


1+ 119864

2

2+ 119864

2

3equiv 119863 (13)


2+ 119864

3= 0



140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400


119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS


2O)

119909glasses as



Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910


2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)


120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)




lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630


119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)




Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)


Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)



0and an


0with 119878ATS =

7120587lowast1199012

16120598




2)1minus119909(K

2O)








lowast1199012

1120598



2LS = 1198751199012

0120598







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






50Cu

45Al

5metallic



50Cu

45Al













119908-welled 3D



119867(3)

0=

3

sum

119894=1

119864119894119888dagger

119894119888119894+ sum

119894 = 119895

1198630119888dagger

119894119888119895+ hc (5)


0|119895⟩





0≃ 119886ℏΩ119890


0≳ ℏΩ119886 gt 0 and 119887 =




3-symmetric local



1= 119864

2= 119864



119908= 3)









(a) (b)

OSi

Ba Al

119881(119909 119910)

(c)







0gt 0 is the



1 119864

2 119864

3 and 119863

0of a tunneling


119908= 4 model


1= 119864

2= 119864



119875ATS (1198641 1198642 1198643 1198630) =

119875lowast

(1198642

1+ 119864

2

2+ 119864

2

3)119863

0

(6)




0119890119894120601119894119895




E119896

1198630

= 2radic1 minus

sum119894 = 119895119864119894119864119895

61198632

0

cos(13120579 + 120579

119896)

cos 120579 = (cos120601 +119864111986421198643

21198633

0

)(1 minus

sum119894 = 119895119864119894119864119895

31198632

0

)

minus32

(7)



2 119864

3

and 1198630≫ radic119864

2

1+ 119864

2

2+ 119864

2



0 Φ(B) = S




(2)

0(119864

1 119864

2 119863


1 119864

2 119863

0) =

119875lowast119863

0radic119864

2

1+ 119864

2






0 05 1 15 2 25 3

0

05

1

15

2

Dim

ensio

nles

s ene

rgy

leve

ls

minus05

minus1

minus15

minus2

Aharonov-Bohm phase

Energy levels 1198641 + 1198642 + 1198643 = 0



1 119864

2 and 119864

3





in the best glasses






119901





0 We can


2119891(119865)120597119865

2|119865=0

where 119891(119865) = minus(1119896119861119879) ln119885(119865)




12 119885 = 119890minusE1119896119861119879 + 119890minusE2119896119861119879


1015840(119879) minus 120598

1015840(119879

0)]120598

1015840(119879

0) (where


2


0and + ln119879 for 119879 gt 119879





Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162LS=Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES+Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES=2119875119901

2

0

31205980120598119903

int

119911max

119911min

119889119911

119911

radic1 minus (Δ

0min2119896

119861119879119911)

2

tanh 119911

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL=119875119901

2

0

31205980120598119903

times int

119911max

119911min

119889119911int

120591max

120591min

119889120591

120591radic1 minus

120591min120591


1 + 12059621205912

(8)


160

140

120

100

80

60

40

20

0

60

40

20

0

0001

0001

001

001

01

01

1

1

Temperature (K)

Temperature (K)

105middotΔ

120598998400120598

998400

105middotΔ

120598998400120598

998400

Al2O3-BaO-SiO2

Glass (AlBaSiO)

AlBaSiO

ATS

2LSKFS2LS + ATS

SiO2 (Spectrosil)







119861119879119911) [17 18]

120591minus1=119864Δ

2

0

120574tanh( 119864

2119896119861119879) (9)




minus1

min = 1198643120574 tanh(1198642119896

119861119879) 120591minus1max =

119864Δ2



0


120598 (119879 120596) = 1205981015840

RES (119879) + 1205981015840

REL (119879)1

1 + 119894120596120591 (10)



Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES≃

minus2

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0max) if 119879 lt

Δ0max2119896

119861

0 if 119879 gtΔ

0max2119896

119861

(11)


Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL≃

0 if 120596120591min ≫ 1

1

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0min) if 120596120591min ≪ 1

(12)



1198961198611198790(120596) ≃ ((12)120574120596)



0slope given

by 119878 = 119875119901203120598

0120598119903



2)1minus119909(K

2O)









0lt E

1≪ E

2


2

1+ 119864

2

2+ 119864

2

3≪ 119863

0limits




1minusE

0


1+ 119864

2

2+ 119864

2

3equiv 119863 (13)


2+ 119864

3= 0



140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400


119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS


2O)

119909glasses as



Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910


2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)


120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)




lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630


119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)




Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)


Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)



0and an


0with 119878ATS =

7120587lowast1199012

16120598




2)1minus119909(K

2O)








lowast1199012

1120598



2LS = 1198751199012

0120598







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




(a) (b)

OSi

Ba Al

119881(119909 119910)

(c)







0gt 0 is the



1 119864

2 119864

3 and 119863

0of a tunneling


119908= 4 model


1= 119864

2= 119864



119875ATS (1198641 1198642 1198643 1198630) =

119875lowast

(1198642

1+ 119864

2

2+ 119864

2

3)119863

0

(6)




0119890119894120601119894119895




E119896

1198630

= 2radic1 minus

sum119894 = 119895119864119894119864119895

61198632

0

cos(13120579 + 120579

119896)

cos 120579 = (cos120601 +119864111986421198643

21198633

0

)(1 minus

sum119894 = 119895119864119894119864119895

31198632

0

)

minus32

(7)



2 119864

3

and 1198630≫ radic119864

2

1+ 119864

2

2+ 119864

2



0 Φ(B) = S




(2)

0(119864

1 119864

2 119863


1 119864

2 119863

0) =

119875lowast119863

0radic119864

2

1+ 119864

2






0 05 1 15 2 25 3

0

05

1

15

2

Dim

ensio

nles

s ene

rgy

leve

ls

minus05

minus1

minus15

minus2

Aharonov-Bohm phase

Energy levels 1198641 + 1198642 + 1198643 = 0



1 119864

2 and 119864

3





in the best glasses






119901





0 We can


2119891(119865)120597119865

2|119865=0

where 119891(119865) = minus(1119896119861119879) ln119885(119865)




12 119885 = 119890minusE1119896119861119879 + 119890minusE2119896119861119879


1015840(119879) minus 120598

1015840(119879

0)]120598

1015840(119879

0) (where


2


0and + ln119879 for 119879 gt 119879





Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162LS=Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES+Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES=2119875119901

2

0

31205980120598119903

int

119911max

119911min

119889119911

119911

radic1 minus (Δ

0min2119896

119861119879119911)

2

tanh 119911

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL=119875119901

2

0

31205980120598119903

times int

119911max

119911min

119889119911int

120591max

120591min

119889120591

120591radic1 minus

120591min120591


1 + 12059621205912

(8)


160

140

120

100

80

60

40

20

0

60

40

20

0

0001

0001

001

001

01

01

1

1

Temperature (K)

Temperature (K)

105middotΔ

120598998400120598

998400

105middotΔ

120598998400120598

998400

Al2O3-BaO-SiO2

Glass (AlBaSiO)

AlBaSiO

ATS

2LSKFS2LS + ATS

SiO2 (Spectrosil)







119861119879119911) [17 18]

120591minus1=119864Δ

2

0

120574tanh( 119864

2119896119861119879) (9)




minus1

min = 1198643120574 tanh(1198642119896

119861119879) 120591minus1max =

119864Δ2



0


120598 (119879 120596) = 1205981015840

RES (119879) + 1205981015840

REL (119879)1

1 + 119894120596120591 (10)



Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES≃

minus2

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0max) if 119879 lt

Δ0max2119896

119861

0 if 119879 gtΔ

0max2119896

119861

(11)


Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL≃

0 if 120596120591min ≫ 1

1

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0min) if 120596120591min ≪ 1

(12)



1198961198611198790(120596) ≃ ((12)120574120596)



0slope given

by 119878 = 119875119901203120598

0120598119903



2)1minus119909(K

2O)









0lt E

1≪ E

2


2

1+ 119864

2

2+ 119864

2

3≪ 119863

0limits




1minusE

0


1+ 119864

2

2+ 119864

2

3equiv 119863 (13)


2+ 119864

3= 0



140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400


119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS


2O)

119909glasses as



Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910


2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)


120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)




lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630


119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)




Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)


Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)



0and an


0with 119878ATS =

7120587lowast1199012

16120598




2)1minus119909(K

2O)








lowast1199012

1120598



2LS = 1198751199012

0120598







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 05 1 15 2 25 3

0

05

1

15

2

Dim

ensio

nles

s ene

rgy

leve

ls

minus05

minus1

minus15

minus2

Aharonov-Bohm phase

Energy levels 1198641 + 1198642 + 1198643 = 0



1 119864

2 and 119864

3





in the best glasses






119901





0 We can


2119891(119865)120597119865

2|119865=0

where 119891(119865) = minus(1119896119861119879) ln119885(119865)




12 119885 = 119890minusE1119896119861119879 + 119890minusE2119896119861119879


1015840(119879) minus 120598

1015840(119879

0)]120598

1015840(119879

0) (where


2


0and + ln119879 for 119879 gt 119879





Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162LS=Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES+Δ120598

1015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES=2119875119901

2

0

31205980120598119903

int

119911max

119911min

119889119911

119911

radic1 minus (Δ

0min2119896

119861119879119911)

2

tanh 119911

Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL=119875119901

2

0

31205980120598119903

times int

119911max

119911min

119889119911int

120591max

120591min

119889120591

120591radic1 minus

120591min120591


1 + 12059621205912

(8)


160

140

120

100

80

60

40

20

0

60

40

20

0

0001

0001

001

001

01

01

1

1

Temperature (K)

Temperature (K)

105middotΔ

120598998400120598

998400

105middotΔ

120598998400120598

998400

Al2O3-BaO-SiO2

Glass (AlBaSiO)

AlBaSiO

ATS

2LSKFS2LS + ATS

SiO2 (Spectrosil)







119861119879119911) [17 18]

120591minus1=119864Δ

2

0

120574tanh( 119864

2119896119861119879) (9)




minus1

min = 1198643120574 tanh(1198642119896

119861119879) 120591minus1max =

119864Δ2



0


120598 (119879 120596) = 1205981015840

RES (119879) + 1205981015840

REL (119879)1

1 + 119894120596120591 (10)



Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES≃

minus2

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0max) if 119879 lt

Δ0max2119896

119861

0 if 119879 gtΔ

0max2119896

119861

(11)


Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL≃

0 if 120596120591min ≫ 1

1

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0min) if 120596120591min ≪ 1

(12)



1198961198611198790(120596) ≃ ((12)120574120596)



0slope given

by 119878 = 119875119901203120598

0120598119903



2)1minus119909(K

2O)









0lt E

1≪ E

2


2

1+ 119864

2

2+ 119864

2

3≪ 119863

0limits




1minusE

0


1+ 119864

2

2+ 119864

2

3equiv 119863 (13)


2+ 119864

3= 0



140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400


119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS


2O)

119909glasses as



Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910


2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)


120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)




lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630


119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)




Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)


Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)



0and an


0with 119878ATS =

7120587lowast1199012

16120598




2)1minus119909(K

2O)








lowast1199012

1120598



2LS = 1198751199012

0120598







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




160

140

120

100

80

60

40

20

0

60

40

20

0

0001

0001

001

001

01

01

1

1

Temperature (K)

Temperature (K)

105middotΔ

120598998400120598

998400

105middotΔ

120598998400120598

998400

Al2O3-BaO-SiO2

Glass (AlBaSiO)

AlBaSiO

ATS

2LSKFS2LS + ATS

SiO2 (Spectrosil)







119861119879119911) [17 18]

120591minus1=119864Δ

2

0

120574tanh( 119864

2119896119861119879) (9)




minus1

min = 1198643120574 tanh(1198642119896

119861119879) 120591minus1max =

119864Δ2



0


120598 (119879 120596) = 1205981015840

RES (119879) + 1205981015840

REL (119879)1

1 + 119894120596120591 (10)



Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162RES≃

minus2

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0max) if 119879 lt

Δ0max2119896

119861

0 if 119879 gtΔ

0max2119896

119861

(11)


Δ1205981015840

1205981015840

1003816100381610038161003816100381610038161003816100381610038162REL≃

0 if 120596120591min ≫ 1

1

3

1198751199012

0

1205980120598119903

ln( 2119896119861119879Δ

0min) if 120596120591min ≪ 1

(12)



1198961198611198790(120596) ≃ ((12)120574120596)



0slope given

by 119878 = 119875119901203120598

0120598119903



2)1minus119909(K

2O)









0lt E

1≪ E

2


2

1+ 119864

2

2+ 119864

2

3≪ 119863

0limits




1minusE

0


1+ 119864

2

2+ 119864

2

3equiv 119863 (13)


2+ 119864

3= 0



140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400


119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS


2O)

119909glasses as



Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910


2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)


120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)




lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630


119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)




Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)


Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)



0and an


0with 119878ATS =

7120587lowast1199012

16120598




2)1minus119909(K

2O)








lowast1199012

1120598



2LS = 1198751199012

0120598







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




140

120

100

80

60

40

20

0002 01 1

Temperature (K)

105middotΔ

120598998400120598

998400


119909 = 02

119909 = 01119909 = 008119909 = 005

2LS 119909 = 005

ATS 119909 = 005

ATS 119909 = 02

2LS + ATS


2O)

119909glasses as



Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ATS=Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES+Δ120598

1015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860RES=

120587lowast1199012

1

31205980120598119903119863min

int

infin

1

119889119910

1199102tanh(119863min

2119896119861119879119910)

Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL=

120587lowast1199012

1

21205980120598119903119863min

(119863min2119896

119861119879)

times int

infin

1

119889119910


2119896119861119879119910)

1

1 + 12059621205912

119860max

(14)


120591minus1

119860max =119863

5

Γtanh( 119863

2119896119861119879) (15)




lowast= 119875

lowast ln(119863

0max119863

0min) (16)

1198630min and1198630max being1198630


119892 (119864) = 1198922LS + 119892ATS (119864) ≃ 2119875 +

2120587lowast

119864120579 (119864 minus 119863min) (17)




Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816ARES≃

0 if 119879 lt 119863min2119896

119861

120587lowast1199012

1

61205980120598119903119896119861119879ln(2119896119861119879

119863min) if 119879 gt 119863min

2119896119861

(18)


Δ1205981015840

1205981015840

100381610038161003816100381610038161003816100381610038161003816119860REL≃

0 if 120596120591119860max ≫ 1

120587lowast1199012

1

1205980120598119903119896119861119879ln( 119896119861119879

119863min) if 120596120591

119860max ≪ 1(19)



0and an


0with 119878ATS =

7120587lowast1199012

16120598




2)1minus119909(K

2O)








lowast1199012

1120598



2LS = 1198751199012

0120598







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119891 = 1 minus (119863 minus 119863119904) ] (20)


119904with 119863





119904le 119863



2LS sim 119909ℓ119863119904


2LSV sim 119909119909(119863119904minus119863119891)] = 119909


119904= 119863



119904= 119863



119888 Our attitude


119888and





119904= 119863




2)1minus119909

(K2O)


120

80

40

0


105middotΔ

120598998400120598

998400

minus40

minus80

minus120

minus160

2LS + ATS


10kHz

30kHz


2O)

119909glasses as


300

250

200

150

100

50

0

105middotp

refa

ctor

0 005 01 015 02Concentration 119909

ATS2LS

119860 = 119860119909

119860 = 146952

119860 = 119860119861(1 minus 119909) + 119860119878119909119891

119860119861 = 4715 119860119878 = 74807 119891 = 081

ATS

2LS



119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 (21)







mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








mol 10minus5

10minus8 sJ3 10


SiO2


SiO2

119909 = 0

119871 119871

119897

Our case

K2O

(119909 ≪ 119909119888)

Extra2LS

119909 ≲ 119909119888119909 ≪ 1



2)1minus119909(K

2O)

119909for 119909 gt 0 [24]



119862ATS119901

(119879) = minus1198791205972119865ATS (119879)

1205971198792 (22)



= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(23)




119862ATS119901

(119879) = 119896119861int

infin

0

119889119864119892ATS (119864) (119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)

(24)


119892ATS (119864) = int119889119863int1198891198630119875 (119863119863

0) 120575 (119864 minus 119863)

≃

2119875lowast

119864if 119864 gt 119863min

0 if 119864 lt Dmin

(25)


119862ATS119901

(119879) = 119861ATS [ln(2 cosh(119863min2119896

119861119879))

minus119863min2119896

119861119879tanh(119863min

2119896119861119879)]

(26)












2O)














119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics











119904minus119878= 119881

119911119904119911119878119911


matrix and 119904119911and 119878



12= plusmn(54)|119881

119911|



0 The


119866 (119881119911) =

41198812

119911

1198813

0

exp(minus2119881

119911

1198810


(27)



119862Fe3+ (119879) = minus1198791205972119865Fe3+

1205971198792 (28)


one evaluates as


= minus119896119861119879 ln(2 cosh( 119864

2119896119861119879))

(29)






119862Fe3+119901

(119879) = 120588 (119909) 119899119895119896119861

times int

infin

0

119889119881119911(

119864

2119896119861119879)

2

coshminus2 ( 119864

2119896119861119879)119866 (119881

119911)

= 120588 (119909) 119899119895119896119861int

infin

0

119889119881119911

251198814

119911

161198792

1

1198813

0

119890(minus21198811199111198810)


119911

8119896119861119879)

(30)

where 119899119895= 119909




119862119901(119879) = 119861ph119879

3+ 119861

2LS119879 + 119862ATS119901

(119879) + 119862Fe3+119901

(119879) (31)


2)1minus119909(K

2O)

119909glasses from



119901(119879 119909) data [24] for the


2O)



2)1minus119909(K

2O)














2O)

119909






(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




(SiO2)1minus119909 (K2O)119909

Temperature (K)

119909 = 02

119909 = 01

119909 = 008119909 = 005

119909 = 0

103

102

101

100

10minus1

10010minus1

119862119901

(Jm

3K)



2)1minus119909(K

2O)



0 005 01 015 02Concentration 119909

(SiO2)1minus119909 (K2O)119909400

300

200

100

0

119861 = 24555 119861 asymp 0 = 31295

119861119861 = 70 119861119878 = 104729

= 119861119909

119861 = 45555

Phonons2LSATS

105middotp

refa

ctor

(uni

ts in

Tab

le 2

)

= 119861119861(1 minus x) + 1198611198781199091198911198612LS

119861ATS

119862ph

119861ph = 119861SiO2

SiO2

(1 minus 119909) + 119861K2O

K2O

119909 + 119862ph119909119891







119892




2O)

119909 we


119860ATS119861ATS

=120588 (119909)

21205980120598119903119896119861119863min1199011

2 (32)



0= 254D these small


119873


0(the order of






2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2)1minus119909(K

2O)


0= minus042K as

fixed


1198612LS times 108

119861ATS times 108

119909119895



K-Si 005 26092 15523 2277 2986K-Si 008 26636 19611 3644 1815K-Si 010 26946 22162 4555 1054K-Si 020 28142 33719 9111 300





2)1minus119909(MgO)

119909 which







Appendix




119867(4)

0= (

1198641

11986311198901198941206014

11986321198901198941206012

1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

11986321198901198941206014

1198632119890minus1198941206014

1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

1198632119890minus1198941206012

1198631119890minus1198941206014

1198644

) (A1)

where sum4

119894=1119864119894= 0 is imposed 119863

1and 119863

2are the nn and


120601 = 2120587Φ (B)Φ

0

Φ (B) = Ssdot B Φ

0=ℎ119888

119902 (A2)


) closed


2119890plusmn1198941206012 with 119863




E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1minus 119863

2

2(1 + cos120601))

minusE( sum

119894lt119895lt119896

119864119894119864119895119864119896+ 4119863

2

1119863

2(1 + cos120601))

+ 1198641119864211986431198644minus 119863

2

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

41198641)

minus1

2119863

2

2(119864

11198643+ 119864

21198644) (1 + cos120601) minus 21198632

1119863

2

2(1 + cos120601)

+ 21198634

1(1 minus cos120601) + 1

8119863

4

2(3 + 4 cos120601 + cos (2120601))

= 0

(A3)


0123(with E

0lt E

1lt E

2lt E

3) is for us the


1| ≪ 1 and

|1198632119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 4119863

2

1)

minus 1198632

1(119864

11198642+ 119864

21198643+ 119864

31198644+ 119864

11198644)

+ 21198634


(A4)


119867(4)

0 red = (

1198641

11986311198901198941206014

0 1198631119890minus1198941206014

1198631119890minus1198941206014

1198642

11986311198901198941206014

0

0 1198631119890minus1198941206014

1198643

11986311198901198941206014

11986311198901198941206014

0 1198631119890minus1198941206014

1198644

)

(A5)

always for |119864119894119863


solutions

E0123

1198631

= plusmn1

radic2

4 minussum

119894lt119895

119864119894119864119895

1198632

1

plusmn [

[

(4 minussum

119894lt119895

119864119894119864119895

1198632

1

)

2

+ 411986411198642+ 119864

21198643+ 119864

31198644+ 119864

41198641

1198632

1

+ 8 (cos120601 minus 1)]

]

12

12

(A6)


1 E









1119863

2| ≪ 1 and always

in the limit case |119864119894119863



E4+E

2(sum

119894lt119895

119864119894119864119895minus 2119863

2

2cos2

120601

2)

minus 1198632

2(119864

11198643+ 119864

21198644) + 119863

4

2cos4

120601

2asymp 0

(A7)


E0123

1198632

= plusmn1

radic2

2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

plusmn [

[

(2cos2120601

2minussum

119894lt119895

119864119894119864119895

1198632

2

)

2

+ 411986411198643+ 119864

21198644

1198632

2

minus 4cos4120601

2

]

]

12

12

(A8)




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

1

2

3

Dim

ensio

nles

s ene

rgy

leve

ls 0

2

Energy levels 1198641 1198642 1198643 1198644 = 0

minus1

minus2

0

0

1

1

2

2

3

3

Aharonov-Bohm phase

Dim

ensio

nles

sen

ergy

leve

ls



1= 0


2 119864

3 119864

4with radic1198642

1+ 119864

2

2+ 119864

2

3+ 119864

2

4119863

2= 001


2= 0 and a

choice of 1198641 119864

2 119864

3 119864

4with119863






0ℏ asymp Ω(Δ

0ℏΩ)



1198630asymp 119873Δ

0 119863min asymp 119873Δmin (A9)



119902 = 1198731199020 119878

Δasymp 4119873119886

2

0 (A10)







0|119902119890| 119878



1198630

1003816100381610038161003816100381610038161003816

119902

119890

1003816100381610038161003816100381610038161003816119878Δasymp 8119873

3Δ

01198862

0 (A11)



0are unknown



3




119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119894119877= Δ

119894and dropping

factors of minus12)

1198672= sum

119886=119871119877

Δ1119886119888dagger

11198861198881119886+ Δ

01sum

119886 = 1198861015840

119888dagger

111988611988811198861015840 + hc

+sum

119886

Δ2119886119888dagger

21198861198882119886+ Δ

02sum

119886 = 1198861015840

119888dagger

211988611988821198861015840 + hc


11198711198881119871119888dagger

21198711198882119871+ 119888

dagger

11198771198881119877119888dagger

21198771198882119877)

(A12)


1015840⟩ =


2|119871119871⟩ ⟨119877119877|119867

2|119877119877⟩ ⟨119877119877|119867

2|119871119871⟩ and ⟨119871119871|119867

2|119877119877⟩ of

Hamiltonian (A12)

⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = Δ 1+ Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = minusΔ 1minus Δ

2minus 119880

⟨11987711987710038161003816100381610038161198672

1003816100381610038161003816 119871119871⟩ = ⟨11987111987110038161003816100381610038161198672

1003816100381610038161003816 119877119877⟩ = Δ 01+ Δ

02

(A13)



01+ Δ

02


1198671015840

1= (

Δ1+ Δ

2minus 119880 Δ

01+ Δ

02

Δ01+ Δ

02minusΔ

1minus Δ

2minus 119880

) (A14)


1+Δ

1015840



1198673=

3

sum

119894=1

sum

119886=119871119877

Δ119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(Δ0119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888119894119886119888dagger

11989410158401198861198881198941015840119886

(A15)


⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = Δ 1+ Δ

2+ Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = minusΔ 1minus Δ

2minus Δ

3minus 3119880

⟨11987711987711987710038161003816100381610038161198673

1003816100381610038161003816 119871119871119871⟩ = ⟨11987111987111987110038161003816100381610038161198673

1003816100381610038161003816 119877119877119877⟩ = Δ 01+ Δ

02+ Δ

03

(A16)


3|119871119871119877⟩ and

⟨119871119871119877|1198673|119877119877119871⟩ ⟨119877119871119877|119867

3|119871119877119871⟩ and ⟨119871119877119871|119867

3|119877119871119877⟩ and


⟨119871119877119877|1198673|119877119871119871⟩ and ⟨119877119871119871|119867






119873

119894=1Δ



0= sum

119894Δ

0119894


119867119873=

119873

sum

119894=1

3

sum

119886=1

119864119894119886119888dagger

119894119886119888119894119886+ sum

119886 = 1198861015840

(1198630119894119888dagger

1198941198861198881198941198861015840 + hc)

minus 119880sum

119894lt1198941015840

sum

119886

119888dagger

119894119886119888i119886119888

dagger

11989410158401198861198881198941015840119886

(A17)





1198671015840

1=

3

sum

119860=1

119864119860119888dagger

119860119888119860+ sum

119860 =1198601015840

1198630119888dagger

1198601198881198601015840 + hc (A18)





119864119860= ⟨119886119886 sdot sdot sdot 119886

1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩ 119860 = 119886 = 1 2 3

1198630= ⟨119886

10158401198861015840sdot sdot sdot 119886

1015840 1003816100381610038161003816119867119873

1003816100381610038161003816 119886119886 sdot sdot sdot 119886⟩

= ⟨119886119886 sdot sdot sdot 1198861003816100381610038161003816119867119873

1003816100381610038161003816 11988610158401198861015840sdot sdot sdot 119886

1015840⟩ 119886 = 119886

1015840

(A19)



radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





radic1198642

1+ 119864

2

2+ 119864

2






0119894


119894and 119863



0minand 119863


Acknowledgments


References



50Cu

45Al

5bulkmetallic































1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1minus119909(KCN)

119909 a model










2O-SiO

2and K

2O-SiO



SiO2-B

2O

3-Na





2O




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Multilevel Tunnelling Systems and Fractal ... · systems (ATS) proposed by one of...

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Transcript of Research Article Multilevel Tunnelling Systems and Fractal ... · systems (ATS) proposed by one of...