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First principles theoretical modeling of the isomer shift of Mossbauer spectraKurian, Reshmi

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Chapter 1

Introduction and Objective

Synopsis

The basic ideas behind Mossbauer Spectroscopic technique is outlined in the

present chapter. The recoilless emission and absorption of γ-rays, which is

the key feature of this spectroscopy is explained with illustrations. The un-

derstanding of the electronic structure of the compounds under study, from

the Mossbauer spectra parameters are discussed in detail. The chapter also

includes the objective and scope of this dissertation.

1.1 Introduction

Fifty years ago Rudolf L. Mossbauer discovered the recoilless nu-

clear resonance absorption of γ-rays while working on his doc-

toral thesis. This phenomenon, which rapidly developed into a new

spectroscopic technique is known as Mossbauer effect [1–4]. Over

the last couple of decades, Mossbauer spectroscopy has become one

of the most captivating tools in chemical physics providing informa-

tion about the chemical environment of the resonating nucleus on an

atomic scale [1–4]. The most well-known application is the determi-

nation of iron 57Fe in crystalline and in disordered solid samples. Be-

sides iron, there are many elements in the periodic table which have

4 1. Introduction and Objective

Mossbauer active nuclei [5–8]. The Mossbauer effect has been ob-

served for the elements which are dotted in the periodic table shown

(Figure 1.1).

Figure 1.1: The periodic table showing the Mossbauer active nuclei, thedotted elements.

The phenomenon of recoilless resonance absorption/emission of

γ-rays by nuclei is the basic characteristic of Mossbauer Spectoscopy

[5, 9]. When a nucleus in an excited state of energy Ee undergoes a

transition to the ground state with energy Eg by emitting a γ quan-

tum, E0 = Ee - Eg, the γ quantum energy E0 may be totally absorbed

by a nucleus of the same kind in its ground state. This phenomenon is

called nuclear resonance absorption of γ-rays and is shown schemati-

cally in Figure 1.2. The resonance absorption is observable only if the

emission and absorption lines overlap sufficiently. The mean lifetime

of the excited state τN , determines the width of the resonance lines

(Γ) according to time-energy uncertainity relation, Γ.τN ≥ ~. There-

1.1. Introduction 5

fore, longer lifetimes produce too narrow transition lines and shorter

lifetimes produce too broad transition lines, and the resonance over-

lap between the emission and absorber lines is not possible. The suit-

able lifetimes of the excited nuclear states for the resonance absorption

range from 10−6 to 10−11s. The spectral line shape may be described

by the Lorentzian or Breit-Wigner form [5,9]. For the 14.4 keV level of57Fe, the natural line width Γ is determined using the mean life time τ

= t1/2ln2

= 1.43×10−7 s , which is Γ = ~/2π = 4.55 × 10−9eV.

Figure 1.2: Schematic representation of nuclear resonance absorption of γ-rays.

When a γ-ray is emitted from an excited nucleus of mass M, there

occurs a recoil of the nucleus according to which the nucleus moves

with velocity v in the opposite direction of γ-ray emission. According

to the conservation of momentum, the linear momenta of the nucleus

and the γ quantum are equal and the energy of the emitted γ quantum

are reduced as, Eγ = E0 - ER. The recoil energy can be written as,

ER =p2

2M=

E2γ

2Mc2(1.1)

The range of transition energies Eγ for the occurrence of resonance

absorption is 5-180 keV. However, the difficulties of high-energy nu-

clear transitions can get around by the use of synchrotron radiation

1.2. Applications of Mossbauer Spectroscopy 15

Figure 1.8: The Magnetic Splitting of the nuclear energy levels and corre-sponding Mossbauer spectrum.

in a molecule, study the magnetic ordering, etc.

1.2 Applications of Mossbauer Spectroscopy

The fields of applications of Mossbauer spectroscopy includes solid-

state physics / chemistry, bio-chemistry / physics, catalysis, nano-

science, materials science, metallurgy etc. This technique has even

become established for the planetary exploration on the surface of

Mars, the presence of water on Mars is confirmed by Mossbauer spec-

troscopy [15]. Even though there are numerous studies on Mossbauer

active nuclei like Sn, Au, Hg, I, etc., the best studied Mossbauer active

nucleus is 57Fe and we will survey a few applications of Mossbauer

spectroscopy to this element. For a complete review of applications of

6 1. Introduction and Objective

sources [10]. For the Mossbauer transition of 57Fe from the excited to

the ground state (E0 = Ee - Eg = 14.4 keV), the recoil energy ER is eval-

uated to be 1.95 × 10−3 eV according to Eqn. 1.1. This value is about

six orders of magnitude larger than the natural width of the spectral

transition under consideration (Γ = 4.55 × 10−9 eV). Figure 1.3 shows

intensity I(E) as a function of the transition energy for emission and

absorption of γ transition. The recoil effect reduces the transition en-

ergy by ER for the emission process and increases the transition en-

ergy by ER for the absorption process. Therefore, the transition lines

for the emission and absorption are separated by a distance 2ER on the

energy scale as shown in Figure 1.3, which is about 10−6 times larger

than the natural line width Γ. Hence, the overlap between two transi-

tion lines and nuclear resonance absorption is not possible in isolated

atoms or molecules in the gaseous or liquid state.

If the γ-ray emission takes place while both the emitter and absorber

Figure 1.3: The transition lines for emission and absorption in an isolatednuclei, which is separated by 2ER.

nucleus are moving relative to each other, then the γ-photon of energy

Eγ receives a Doppler energy ED, ie, Eγ = E0 - ER + ED. For 57Fe, mov-

1.1. Introduction 7

ing the source at a velocity of 1mm/s towards the sample increases the

energy of the emitted photons by about ten natural line widths. Thus,

the Doppler shift of the emission and absorption lines allows for the

fine tuning of the resonances in Mossbauer experiments [5,9]. The rel-

ativistic Doppler formula for an emitter and absorber with a relative

velocity of υ is,

v = v0

√

1− υ/c

1 + υ/c(1.2)

where c is the speed of light. Expanding Eqn 1.2 into Taylor series

gives the velocity shift, i.e.,

v = v0

(

1− υ/c+1

2υ2/c2 − .......

)

(1.3)

where the first order part vanishes because of the random nuclear dis-

placements and the second order term remains. However, the second

order Doppler shift is quite small, as an example, 0.07 mm/s for 57Fe

for a decrease of 100 K [11].

Figure 1.4: Recoil-free emission or absorption of γ-rays when the nuclei arein solid matrix.

Rudolf L. Mossbauer observed that the recoilless emission and ab-

sorption of γ rays is possible if nuclei are embedded into a solid envi-

ronment (like shown in Figure 1.4) and the transition lines can over-

lap, thus resulting in the resonance absorption. This can be explained

from the classical point of view, when the γ-ray is emitted by a nu-

cleus bound in a lattice, the entire crystal lattice will absorb the recoil.

8 1. Introduction and Objective

In this case, the mass M in the denominator of Eqn. 1.1 should be the

mass of the whole crystal, not the individual nucleus. This reduces

the recoil energy to a negligible amount. Therefore, Eγ = E0 and the

entire process becomes a recoilless resonant absorption [5, 9].

Figure 1.5: Energy levels separated by ~ω of a Debye solid.

The quantum mechanical description of the recoil effect is some-

what more complicated. Of course, the lattice vibrations are quan-

tised and the energy can be absorbed or emitted by the crystal lattice

in quanta of certain energy (phonons). For a Debye solid with one vi-

brational frequency ω, the lattice can only receive or release energies

in integral multiples of ~ω. The energy distribution defined by the

population of the levels spaced by ~ω is shown schematically in Fig-

ure 1.5. If ER <~ω, the lattice cannot absorb the recoil energy, i.e., the

zero phonon process occurs, and the γ-ray is emitted without a recoil.

It suggests that there must be zero-phonon transitions ie, emission

process without excitation of phonons in the lattice. There is a cer-

tain probability f (known as Debye-Waller factor or Lamb-Mossbauer

factor) that no lattice excitation (zero-phonon processes) takes place

during γ-emission or absorption whereas f denotes the fraction of nu-

clear transitions which occur without recoil and only for this fraction

1.1. Introduction 9

is the Mossbauer effect observable. This recoil-free fraction can be ex-

pressed as, f = exp (-ER/~ω). Within the Debye model for solids, f

increases with the decreasing transition energy Eγ , with decreasing

temperature, and with increasing Debye temperature θ. Thus, θ is a

measure of the strength of the bond between the Mossbauer atom and

the lattice.

Figure 1.6: The partial overlap of emission and absorption lines due to thedifferent electron-nuclear interactions in the source and absorber.

The recoilless resonant absorption is necessary for the maximum

overlap of the emission line and absorption line. In-order to make the

nuclear resonance absorption of γ-rays successful, the emission and

absorption lines should coincide or at least partially overlap. How-

ever, the complete overlap between the emission and absorption lines

is possible only if identical materials are used as source and absorber.

If the source and the absorber nuclei are in different chemical environ-

ment, which is usually the case, they have slightly different absorp-

tion/emission frequencies due to the interactions with the surround-

ing electrons [5, 9]. This is schematically shown in Figure 1.6, where

the partial overlap of the absorption and emission lines is because of

the different electron nuclear interactions in the absorber and emit-

ter. Because the Mossbauer lines are very sharp, even small energy

10 1. Introduction and Objective

differences will destroy the resonance. However, with the use of the

Doppler effect, ie, by moving the source and absorber relative to each

other, the perfect overlap can be obtained. For 57Fe, Doppler veloci-

ties up to a few mm/s are sufficient to achieve good overlap between

the emission and absorption lines. A Mossbauer spectrum, which is a

plot of the relative transmission of the gamma radiation as a function

of the Doppler velocity, reflects the nature and strength of the hyper-

fine interactions between the Mossbauer nucleus and the surrounding

electrons. The Mossbauer effect makes it possible to resolve the hyper-

fine interactions and provide information on the electronic structure.

The three main hyperfine interactions corresponding to the nuclear

moments are,

1) Isomer Shift

2) Quadrupole Splitting

3) Magnetic Hyperfine Splitting

1.1.1 Isomer Shift

Mossbauer Isomer Shift (MIS) is defined as a displacement of the fre-

quency of the nuclear γ-transition in the target (absorber) nucleus ∆Eaγ

with respect to the reference (source) nucleus ∆Esγ [5,6,12]. The varia-

tion of the nuclear volume, ie, the nuclear charge radius, during the γ

transition is responsible for the occurrence of Mossbauer isomer shift,

because the atomic nucleus is not a point-like object but an object of

a finite spatial extent. The different nuclear charge radius in the ex-

cited and ground state induce different electron-nuclear interactions

therein, hence the frequency of the γ transition in the nucleus im-

mersed in a specific electronic environment is different than in the

bare nucleus. In a Mossbauer experiment one measures the change

1.1. Introduction 11

of the energy of the resonance γ quantum between the source (s) and

the absorber (a) nuclei, thus there appears a dependence of the en-

ergy of resonance γ quantum on the electronic environment in which

the given nucleus is immersed. The MIS, δ measured in terms of the

Doppler velocity necessary to achieve resonance is given in Eqn. 1.4,

δ =c

Eγ

(∆Eaγ −∆Es

γ) (1.4)

where c is the velocity of light and Eγ is the energy of the γ quantum.

Figure 1.7: The isomer Shift and Quadrupole Splitting of the nuclear energylevels and corresponding Mossbauer spectra.

This shift appears in the spectrum as the difference between the

position of the baricenter of the resonance signal and zero Doppler

velocity as shown in Figure 1.7 [5, 6, 12]. Traditionally, the energy dif-

ferences ∆Ea/sγ are calculated within the framework of perturbation

12 1. Introduction and Objective

theory, whereby the variation of the electron-nuclear interaction po-

tential during the γ-transition is treated as a weak perturbation of the

nuclear energy levels [5, 6, 12–14]. This approach leads to the well

known expression for the isomer shift of Mossbauer spectra as a lin-

ear function of the so-called contact electron density (electron density

at the nucleus) in the absorber ρae and source ρse compounds, see Eqn.

1.5,

δ = α(ρ(a) − ρ(s)) (1.5)

where α is a calibration constant, which depends on the parameters

of the nuclear γ-transition. The most valuable information derived

from isomer shift data refers to the oxidation state and spin state of a

Mossbauer-active atom, its bond properties etc.

1.1.2 Quadrupole splitting

Quadrupole splitting in the Mossbauer spectrum occurs when a nu-

cleus with an electric quadrupole moment experiences a non-uniform

electric field [5, 9]. The nuclear charge distribution deviates from the

spherical symmetry for a nucleus that has spin quantum number I >

1/2 and thus has a non zero electric quadrupole moment. The mag-

nitude of the quadrupole moment may change in going from one

state of excitation to another. The sign of the electric quadrupole

moment, Q indicates the shape of the deformation. Q is negative

for a flattened (pancake-shaped) nucleus and positive for an elon-

gated nucleus (cigar-shaped). Q is constant for a given Mossbauer

nucleus, ie, changes in the quadrupole interaction energy observed

in different compounds of a given Mossbauer nuclide under constant

experimental conditions can only arise from the changes in the elec-

tric field gradient (EFG) generated by the surrounding electrons and

1.1. Introduction 13

other nuclei. Therefore, the interpretation of quadrupole splittings

requires the knowledge of the EFG. The interaction between the elec-

tric quadrupole moment of the nucleus and EFG at the nuclear po-

sition give rise to a splitting in the nuclear energy levels into sub

states, which are characterised by the absolute magnitude of the nu-

clear magnetic spin quantum number |mI |.

As the Mossbauer spectroscopy involves the absorption of the γ-

rays to promote a nucleus from the ground state to an excited state,

the quadrupole Hamiltonian has to be solved for each energy level if

both levels have nuclear spin greater than 1/2. For 57Fe, the ground

state has nuclear spin I = 1/2 and the lowest excited state has I = 3/2.

The second part of Figure 1.7 shows the quadrupole splitting of the

nuclear energy levels of 57Fe, where the absorption line is split due to

the interaction of the nuclear quadrupole moment with non-zero EFG

at the nucleus. The separation between the lines , ∆EQ, is known as

the quadrupole splitting and is written as,

∆EQ =1

2qQVzz

(

1 + η2

3

)1/2

(1.6)

where e is the electrical charge, Q is the nuclear quadrupole moment,

and V is the electric field gradient due to the total electron density

plus all nuclear charges. V can be decomposed into three principal

components, Vzz, Vyy, and Vxx, in descending order of magnitude, and

η is the asymmetry parameter defined as (Vxx - Vyy)/Vzz. For the

substates with axially symmetric EFG (η = 0), the energy separation

∆EQ is,

∆EQ =1

2qQVzz (1.7)

The quadrupole splitting provides information on the symmetry of

the coordination sphere of the resonating atom.

14 1. Introduction and Objective

1.1.3 Magnetic Hyperfine Splitting

The dipole interaction between the nuclear spin moment and the mag-

netic field is called the Magnetic Hyperfine Splitting (Nuclear Zeeman

effect) [5, 9]. A nuclear state with spin I > 1/2 possesses a magnetic

dipole moment µ. The magnetic field splits the nuclear level of spin

I into (2I + 1) equispaced non-degenerate substates characterised by

the magnetic spin quantum numbers mI . Therefore for 57Fe, the ex-

cited state with I = 3/2 is split into four, and the ground state with I

= 1/2 into two substates as shown in Figure 1.8. The energies of the

sublevels are given from first-order perturbation theory by,

EM (mI) = −µHmI/I = −gNβNHmI (1.8)

where βN is the nuclear Bohr magneton, µ is the nuclear magnetic mo-

ment, mI is the magnetic spin quantum number and gN is the nuclear

g-factor.

The magnetic hyperfine splitting enables one to determine the ef-

fective magnetic field acting at the nucleus. The total effective mag-

netic field is the vector sum of externally applied magnetic filed and

the internal magnetic field, ~Heff = ~Hext + ~Hint. The latter consist of

three parts, ~Hint = ~HL + ~HD + ~HC . ~HL is the contribution from the

orbital motion of the electrons, ~HD is the contribution of the magnetic

moment of the spin of the electrons outside the nucleus (spin-dipolar

term) and ~HC is the contribution of the spin-density at the nucleus

(Fermi contact term).

The magnetic hyperfine interaction gives a clear understanding of

the magnetic properties of materials. In compounds with unpaired

electrons the Mossbauer spectroscopy enables one to distinguish be-

tween the high-spin and low-spin states, spin density at various nuclei

16 1. Introduction and Objective

Mossbauer spectroscopy, see Refs. [16–18].

Soon after its discovery, Mossbauer spectroscopy was used to solve

problems in solid state research. Fluck et. al [19] successfully used

this technique is to distinguish between Prussian Blue (PB) and Turn-

bull’s Blue (TB). Both these are compounds with the general molecular

formula AxMa[Mb(CN)6]zH2O (where, A = alkali cation and Ma/Mb

= metal ion). PB is prepared by adding FeIII salt to a solution of

[FeII(CN)6]4−, and TB by adding FeII salt to a solution of [FeIII(CN)6]

3−.

It was believed for a long time that these were chemically different

compounds, Prussian Blue with [FeII(CN)6]4− anions and Turnbulls

Blue with [FeIII(CN)6]3− anions. However, the Mossbauer spectra reco-

rded by Fluck et. al [19] were nearly identical for both PB and TB

showing only the presence of [FeII(CN)6]4− and FeIII in the high spin

state. The explanation is that during the preparation of TB, imme-

diately after adding a solution of FeII to a solution of [FeIII(CN)6]3−, a

rapid electron transfer takes place from FeII to the anion [FeIII(CN)6]3−

with subsequent precipitation of the same material as of PB. It is now

agreed that TB and PB are the same because of the rapidity of electron

exchange through a Fe-CN-Fe linkage.

Thermal spin transition (spin crossover) and electron transfer re-

sulting in valence tautomerism are another aspects which attracted

increasing attention by chemists and physicists because of the promis-

ing potential for practical applications in sensors and display devices

[20]. Prussian Blue Analogues are demanding among those compounds

which show pressure and temperature induced electron transfer [21].

For instance, the CoII- FeIII cyano complex when doped with potas-

sium ions, K0.1Co4[Fe(CN)6]2.718 H2O, may undergo a thermally in-

duced electron transfer around 20 K from the CoII with spin S=3/2 to

the FeIII with spin S=1/2, turning the CoII-HS to CoIII-LS and the FeIII-

1.2. Applications of Mossbauer Spectroscopy 17

LS to FeII- LS [21]. The diamagnetic pair CoIII-FeII (total spin S = 0) can

be converted back to the paramagnetic pair with the application of

light. It has been observed that the formation of the diamagnetic pairs

can be enhanced by increasing the potassium concentration or doping

with the alkali ions like rubidium or caesium and with the application

of pressure. Ksenofontov et. al. [21] proved this by measuring the

Mossbauer spectra under applied pressure on a sample which con-

tained a small fraction of potassium. There is no thermally induced

electron transfer at ambient pressure, at 4.2 K, and the spectrum is

magnetically split into a sextet with a local magnetic field of 165 kOe

arising from the S=1/2 Fermi contact field. At a pressure of 3 kbar,

most of the sextet intensity disappears and a singlet arising from the

FeII-LS sites emerges. At 4 kbar, the pressure-induced electron trans-

fer from CoII-HS to FeIII-LS is complete and the spectrum shows only

the typical FeII-LS singlet [21].

Recent discovery of high-temperature superconductivity (HTSC)

in iron-based compounds has initiated a considerable research activ-

ity comparable or even in excess to the discovery of HTCS in cuprates

[22]. Iron-based superconductors belong to pnictide (compounds of

group V elements) [23–25] or chalcogenide (compounds of group VI

elements) type of compounds [26]. In resolving the origin of supercon-

ductivity in these compounds, the knowledge of their local geometry

and local electronic structure is extremely important. Although the

X-ray diffraction methods can provide the reliable crystalline struc-

tures and the long range magnetic order can be studied by the neu-

tron diffraction method, the local geometric and electronic structure is

accessible via the use of Mossbauer spectroscopy, which is capable of

providing information on an atomic scale [5, 27].

Mossbauer spectroscopy technique finds numerous applications in

18 1. Introduction and Objective

biological systems. For example, Schunemann et. al [28] have stud-

ied the spin distribution and iron oxidation states of the intermediate

states in the reaction cycle of cytochrome P450. The enzyme super-

family cytochrome P450 are found in many living organisms, and play

an important role in many physiological processes for example in the

biotransformation of xenobiotics and synthesis of steroid hormones.

The active site of cytochrome P450 contains a substrate binding site

next to the heme iron centre. They catalyse a variety of reactions by

the transfer of an active oxygen from its heme unit to the substrates.

When a substrate binds to the active site of the enzyme, intermediate

states are formed, however little is known about these intermediate

states. Mossbauer spectroscopy can give clear evidence about the iron

oxidation states of these intermediate states [28].

The advent of third-generation synchrotron radiation sources ex-

tends the applicability of the Mossbauer technique. The use of syn-

chrotron source is an alternative to the conventional Mossbauer tech-

nique [1, 2]. The radiation from the synchrotron source is intense,

tuneable in energy, and is available in the form of short pulses. There

are several ways of Mossbauer filtration of synchrotron radiation (MFSR),

such as pure nuclear reflection, total external reflection (TER), forward

scattering (nuclear resonant forward scattering (NFS) and nuclear in-

elastic scattering (NIS)) etc. The use of synchrotron radiation over-

comes some of the limitations of the conventional technique. For in-

stance, NFS allows the direct determination of the Lamb-Mossbauer

factor [10]. In addition, the high brilliance and the extremely colli-

mated beam lead to a large flux of photons through the very small

size of the sample (0.1-1 mm2), makes possible to measure extremely

small samples, and also samples under unusual conditions like high

pressure [1, 2].

1.3. Significance of the present work 19

1.3 Significance of the present work

The parameters of Mossbauer spectra, such as the isomer shift, quadru-

pole splitting, magnetic hyperfine splitting, are sensitive characteris-

tics of the electronic structure and carry important information on the

spin- and oxidation-state of the resonating atom as well as on its lo-

cal chemical environment. The theoretical estimation of Mossbauer

isomer shifts and the evaluation of the nuclear structure parameters

therein, is the main goal of this thesis.

The Mossbauer isomer shift is the measure of the energy differ-

ence between the energies of γ-transitions occurring in the absorber

and source. Because the electronic environments in which the sample

and the reference nuclei are immersed are different, the isomer shift

probes this difference. However, the relationship between the isomer

shift and the local electronic structure is not straightforward. The iso-

mer shift depends on the nuclear structure parameters, such as the

charge radius variation during the γ-transition, as well as on the lo-

cal electronic structure parameters, such as the electron density ρa/se

in the vicinity of nucleus [6, 12]. Although measurable in principle,

these characteristics are not directly accessible from the experiment.

In such a situation, the first principles quantum chemical calculations

of the local electronic structure and of the isomer shift are of the ut-

most importance. Thus, the theoretical calculation of the electron con-

tact density at the target nucleus in chemical compounds, and calibra-

tion against the experimental isomer shifts remains the most reliable

way of determining the nuclear structure parameters. In these calcu-

lations, all relevant effects, such as the effects of relativity, the effects

of electron correlation and of the solid-state environment, have to be

included, which makes such calculations very demanding.

20 1. Introduction and Objective

The success of the traditional approach for MIS calculation relies

on the availability of the contact densities from the theoretical cal-

culations. The contact density is easily available from the calcula-

tions in which the theoretical methods fulfilling Hellmann-Feynman

theorem, such as the self-consistent field (SCF) method or the Kohn-

Sham (KS) method of density functional theory (DFT), are employed

[29–31]. However, the use of the most sophisticated methods of the

ab initio wave function theory, the coupled-cluster method or single-

reference and multi-reference Møller-Plesset perturbation theory, re-

quires the calculation of the so-called relaxed density matrix which

considerably increases the amount of computational work necessary

to obtain the density. Hence relatively low-level computational meth-

ods are commonly employed in the calibration of the Mossbauer iso-

mer shift.

The standard model of MIS calculations is based on certain as-

sumptions, such as constant density inside the nucleus and point char-

ge nuclear model [6, 12]. The former is valid only within the non-

relativistic formalism, where ρ is replaced with the electron density at

the nuclear position ρ(0). However, the electronic wavefunction in the

vicinity of the nucleus is considerably modified by relativity [12, 32],

hence the relativistic density is divergent near the nucleus and cannot

be defined as ρ(0). Within the standard approach, the non relativistic

contact densities ρ(0) calculated are scaled using an element specific

constant S(Z), in order to account for the effect of relativity. Gen-

erally, this scaling factor is obtained through the comparison of the

four-component relativistic electron density for a one-electron atom

with the corresponding non relativistic density [6]. From the values

of the scaling factor for various elements it is clear that the effect of

relativity is significant in the contact density [6]. Another limitation

1.4. Scope of the thesis 21

is the assumption of point-charge nuclear model. The effect of the

finite size of the nucleus is considered as a perturbation and the per-

turbation Hamiltonian is obtained as the difference between the po-

tential of a uniformly charged sphere and the usual Coulomb poten-

tial [6, 12–14, 18].

Taking all these limitations into consideration, it is desirable there-

fore to employ a more direct theoretical approach for the calculation

of MIS, which allows the straightforward inclusion of the most impor-

tant effects such as relativity and electron correlation. This thesis ex-

plains the development and calculations based on such an approach,

which is based on the non-perturbative inclusion of the finite size nu-

cleus into theoretical calculations. Within this new approach, the most

accurate ab initio methods can be employed, which offers a possibility

for a systematic improvement of the results of theoretical modelling

of the Mossbauer isomer shift.

1.4 Scope of the thesis

In chapter 2, all the quantum chemical formalisms within the scope of

this thesis are introduced. The inclusion of relativity via Normalized

Elimination of Small the Component method in the calculations are

explained. In this chapter, the traditional approach for the MIS calcu-

lations and its limitations are explained, describing the necessity of a

more accurate theoretical framework. The new approach, according

to which the isomer shift is treated as the linear response of the elec-

tronic density with respect to the nuclear radius is described in detail.

Chapter 3 validates density functional theory for the calculation

of MIS within the context of the new theoretical formalism. The val-

22 1. Introduction and Objective

idation of pure and hybrid density functionals for the modelling of

Mossbauer Isomer shift is illustrated. The investigation of the effect

of basis set truncation on the calculated values of isomer shifts are

carried out and described in detail.

In Chapter 4, the importance of the proper account of relativity and

electron correlation is demonstrated in a series of atomic calculations

for various oxidation states of iron atom. From the Mossbauer exper-

iments the parameters of the nuclear γ-transition are not known with

sufficiently high accuracy, hence the calibration of the theoretically ob-

tained contact densities with the experimental isomer shifts is impor-

tant (see Eqn. 1.5). The calibration constant α in Eqn. 1.5 gives infor-

mation on the internal parameters of the nuclear γ-transition, such as

the variation of the nuclear charge radius. Chapter 4 describes the cal-

ibration of 119Sn contact densities with the experimental isomer shifts

to obtain a reliable value of the calibration constant α. The obtained

value of calibration constant α is validated by the comparison of cal-

culated vs. experimental isomer shifts. An independent test of the

calibration constant is carried out by studying the isomer shift varia-

tion under pressure for CaSnO3 perovskite.

Chapter 5 details the calculation of 57Fe calibration constant α us-

ing a series of iron compounds. The reliability of the obtained cali-

bration constant, α is tested via calculating the relative isomer shifts

∆δx = δx − δref , choosing various reference compounds.

Chapter 6 describes the investigation on the possible origin of two

different signals in the Mossbauer spectra of Prussian blue analogue

RbMnFe(CN)6H2O, in which a switching of magnetism occurs with

light and temperature. The analysis is based on two parameters, i)

the distribution of Rb+ around iron and ii) the difference in oxidation

states of iron.

1.4. Scope of the thesis 23

Chapter 7 details the study on the recently discovered iron based

superconductors [22–26]. The aim is to analyse the influence of the

structural variation near the 57Fe nucleus and chemical substitutions

in the coordination sphere of 57Fe, on the isomer shift and magnetic

hyperfine coupling constants in these compounds. This will give us an

idea on the applicability of Mossbauer parameters to understand the

structure of these iron based superconductors under phase transitions

and upon chemical substitutions.

Chapter 8 give a conclusion and outlook of the dissertation.