Electron scattering and transport in liquid argonG. J. Boyle, R. P. McEachran, D. G. Cocks, and R. D. White Citation: The Journal of Chemical Physics 142, 154507 (2015); doi: 10.1063/1.4917258 View online: http://dx.doi.org/10.1063/1.4917258 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Deviational simulation of phonon transport in graphene ribbons with ab initio scattering J. Appl. Phys. 116, 163502 (2014); 10.1063/1.4898090 Contribution of d -band electrons to ballistic transport and scattering during electron-phonon nonequilibrium innanoscale Au films using an ab initio density of states J. Appl. Phys. 106, 053512 (2009); 10.1063/1.3211310 Leading-order relativistic effects on nuclear magnetic resonance shielding tensors J. Chem. Phys. 122, 114107 (2005); 10.1063/1.1861872 Comment on “Calculation of electron transport in Ar–N 2 and He–Kr gas mixtures—implications for validity ofBlanc’s law method” [Phys. Plasmas 4, 551 (1997)] Phys. Plasmas 6, 4794 (1999); 10.1063/1.873771 Isotropic and anisotropic interaction induced scattering in liquid argon J. Chem. Phys. 107, 10415 (1997); 10.1063/1.474205

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THE JOURNAL OF CHEMICAL PHYSICS 142, 154507 (2015)

Electron scattering and transport in liquid argonG. J. Boyle,1 R. P. McEachran,2 D. G. Cocks,1 and R. D. White11College of Science, Technology & Engineering, James Cook University, Townsville 4810, Australia2Research School of Physical Sciences and Engineering, Australian National University,Canberra ACT 0200, Australia

(Received 27 February 2015; accepted 30 March 2015; published online 20 April 2015)

The transport of excess electrons in liquid argon driven out of equilibrium by an applied elec-tric field is revisited using a multi-term solution of Boltzmann’s equation together with ab initioliquid phase cross-sections calculated using the Dirac-Fock scattering equations. The calculationof liquid phase cross-sections extends previous treatments to consider multipole polarisabilitiesand a non-local treatment of exchange, while the accuracy of the electron-argon potential is vali-dated through comparison of the calculated gas phase cross-sections with experiment. The resultspresented highlight the inadequacy of local treatments of exchange that are commonly used inliquid and cluster phase cross-section calculations. The multi-term Boltzmann equation frameworkaccounting for coherent scattering enables the inclusion of the full anisotropy in the differentialcross-section arising from the interaction and the structure factor, without an a priori assumptionof quasi-isotropy in the velocity distribution function. The model, which contains no free param-eters and accounts for both coherent scattering and liquid phase screening effects, was found toreproduce well the experimental drift velocities and characteristic energies. C 2015 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4917258]

I. INTRODUCTION

The study of electron transport in non-polar liquidsis of fundamental interest for understanding the dynamicsof electronic processes in liquids and disordered systems,including dynamic and scattering processes. More recently,attention has focussed on applications including liquid stateelectronics, driven by use in high-energy particle detectorssuch as the liquid argon time projection chamber (LArTPC).Advances in the fields of plasma discharges in liquids andassociated electrical breakdowns (see, e.g., the review ofBruggeman1) are dependent on a fundamental knowledgeof charged particle transport in liquids. Furthermore, therapidly developing interdisciplinary field of plasma medicinerequires2–5 a detailed knowledge of electron transport throughliquid water and other biostructures, typically under non-equilibrium conditions.

The study of excess electrons in dense gases and fluids isa complex problem, requiring the inclusion of many effectsthat are not present in dilute gaseous systems. The majorcontributions to these effects arise from the small interparticlespacings and their highly correlated separations. For thermalenergies, the de Broglie wavelengths of the excess electronsare often orders of magnitude larger than the interatomicspacing, which leads to significant quantum-like effects. Evenwithin a semi-classical picture, where the excess electrons areassumed to act as point-like particles, no particular volume is“owned” by a single atom. This means the typical picture fortransport in a gas, i.e., a series of individual collision eventsseparated by the mean-free path, is no longer valid, making itimportant to consider multiple scattering effects of the electronfrom many atoms simultaneously. Of particular note is the

effect of “coherent scattering” and the pair correlations of theliquid play a very important role in this and other effects.

Many previous calculations for electrons in dense systemshave neglected these liquid effects for simplicity, modellingdense fluids by applying a theory for dilute gases with only anappropriate increase of the density. However, a few alternativetheories exist that have explored liquids in different ways.Borghesani et al.6 have heuristically combined the liquideffects identified above to obtain an effective cross-section.When used in the standard equations from kinetic theory formobility in a non-zero field, their results have been shown tobe remarkably accurate. Braglia and Dallacasa7 have deriveda theory that addresses both enhancements and reductions tothe zero-field mobility through a Green’s function approachwith appropriate approximations to the self-energy but do notgo beyond linear response theory and hence do not explainnon-equilibrium behaviour at high fields.

In contrast to the above approaches, the seminal articlesby Lekner and Cohen8,9 outline a method to address effectsof a dense fluid from an ab initio approach by appropriatemodifications of the microscopic processes. The article byLekner9 describes how an effective potential for a singlecollision event can be built up from knowledge of only thesingle-atom/electron potential and the pair correlator of thefluid as well as prescribing a method for obtaining effectivecross-sections from this potential. The article by Cohen andLekner8 then describes how the effects of coherent scatteringcan be included with these effective cross-sections in aBoltzmann equation solution for the calculation of transportproperties. Sakai et al.10 have been able to improve agreementwith experiment by empirically modifying the resultant cross-sections of the Cohen and Lekner formalism and by including

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154507-2 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

inelastic processes. Atrazhev et al.11 were able to simplify thearguments of Lekner9 to argue that, for small energies, theeffective cross-section becomes dependent on the density onlyand obtained good agreement with experiment. However, thedistance at which to enforce this new behaviour of the effectivecross-section remains a free parameter in the theory and thisconstant effective cross-section must be found empirically.Atrazhev and co-workers went on to consider the interactionas a muffin tin potential, with each cell being a Wigner-Seitz sphere surrounding each atom in the liquid. They useda variable phase function method which could describe theabsence of a Ramsauer minimum in the liquid cross-sectionalong with density fluctuations of the liquid.12–14

The calculations we present in this paper are basedon a generalization of the Cohen and Lekner formalism,overcoming several approximations which are no longernecessary in modern day transport and scattering theory.With regard to the scattering potential, Lekner9 used theBuckingham polarization potential15 as input, which we willshow is completely inadequate due to its omission of theexchange interaction. This is not noticeable for gas phasemeasurements, due to the fitting parameter of the Buckinghampotential, but shows significant differences after the liquidmodifications are applied. By performing a detailed analysis ofthe partial phase shifts, Atrazhev and co-workers13 were ableto isolate the important properties of the potential which arerequired for accurate determination of the transport properties.Our calculations instead avoid these difficulties by usingaccurate forms for the electron-atom interaction.

With regard to the transport theory itself, we employa previously derived extension of the Cohen and Leknerformalism for the Boltzmann equation from a two-term to afull multiterm treatment of the velocity distribution function.16

This theory utilises the full anisotropic detail of the cross-sections that is available in our calculations. For dilute gaseoussystems, the two-term approximation can be in serious error,17

and in this study, we consider contributions to the errorarising from the neglect of the full anisotropy in both thevelocity distribution function and the differential scatteringcross-section for liquid systems. We perform calculationsspecifically for the noble gas of argon, which is an excellenttest bed for new theories due to the good availability ofexperimental data and the high degree of accuracy to whichab initio calculations can model the gaseous phase. Availableexperimental data include drift velocities and characteristicenergies in both the gas and liquid phases, as well asprecise single-atom cross-sections. We emphasize that weare interested in the full non-equilibrium description of thetransport properties and not only that of zero-field mobilities,and so we must consider the full range of the static structurefactor S(K) instead of S(0) which is fixed by the isothermalcompressibility.

In Secs. II–VI, we consider the calculation of themacroscopic swarm transport properties in the gaseous andliquid environments from the microscopic cross-sections,modified by the screening and coherent scattering effectsdiscussed above. We first detail the calculation of the gasphase cross-sections in Sec. II, using accurate potentials in theDirac-Fock scattering equations and then address, in Sec. III,

effects of screening in the liquid. The transition from a gasto liquid requires a modification of the scattering to includean effective scattering potential and an effective non-localexchange term which we describe in Sec. IV. The applicationof these cross-sections in structured media is outlined in Sec. Vand we present the results of our transport calculations inSec. VI. Initially, in Sec. VI A, we consider only the gasphase, understanding the importance of an accurate treatmentof exchange and polarization and thereby establishing thecredibility of the initial gas-phase potential subsequently usedas input for the calculation of cross-sections for the liquidphase environment. Transport coefficients calculated usingthe screened cross-sections and associated coherent scatteringeffects are considered in Sec. VI B, where they are comparedwith the available measured transport data. Throughout thispaper, we will make use of atomic units (m = e = a0 = ~ = 1)unless otherwise specified.

II. SCATTERING OF ELECTRONS BY ARGON GAS

The core of a transport calculation is based on an accuratedescription of the scattering of the electron off a particle in thebulk. Effective interaction potentials are often used to deter-mine various measurable properties, such as scattering lengthsor polarizabilities. These effective potentials are successful solong as they correctly reproduce these quantities for input inother simulations. However, as mentioned above, there aremany additional effects due to a dense gas or liquid whichcan modify the details of the scattering processes. Hence,we require a potential that does not only produce the correctscattering properties in the dilute limit but also well describesthe scattering properties under a perturbation of the potential.

In the pure elastic energy region, in addition to the staticpotential, there are only two interactions which need to betaken into account in electron-atom collisions, namely, polari-zation and exchange. The polarization can be accounted for bymeans of long-range multipole polarization potentials, whilethe exchange interaction is represented most accurately by ashort-range non-local potential formed by antisymmetrizingthe total scattering wavefunction.

In the present work, the scattering of the incidentelectrons, with wavenumber k, by argon atoms is describedin the gaseous phase by the integral formulation of the partialwave Dirac-Fock scattering equations (see Ref. 18 for details).In matrix form, these equations can be written as

*,

fκ(r)gκ(r)

+-= *,

v1(kr)v2(kr)

+-+

1k

r

0dx G(r, x)

×U(x) *

,

fκ(x)gκ(x)

+-− *,

WQ(κ; x)WP(κ; x)

+-

, (1)

where the local potential U(r) is given by the sum of the staticand local polarization potentials, i.e.,

U(r) = Us(r) +Up(r), (2)

and WP(κ; r) and WQ(κ; r) represent the large and smallcomponents of the exchange interaction. In Eq. (1), fκ(r) andgκ(r) are the large and small components of the scattering

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154507-3 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

wavefunction where the quantum number κ can be expressedin terms of the total and orbital angular momentum quantumnumbers j and l according to

j = |κ | − 12

with l =

κ, if κ > 0−κ − 1, if κ < 0

. (3)

The free particle Green’s function G(r, x) in Eq. (1)is defined in terms of Riccati-Bessel and Riccati-Neumannfunctions (see Eqs. (23) and (24a,b) of Ref. 18). The kineticenergy ϵ of the incident electron and its wavenumber k arerelated by

k2 =1~2c2 ϵ

�ϵ + 2mc2�. (4)

We note that if we ignore ϵ with respect to 2mc2, we obtain theusual non-relativistic relationship between the wavenumberand the energy of the incident electron.

The static potential Us(r) in Eq. (2) is determined in theusual manner from the Dirac-Fock orbitals of the atom.18 Thepolarization potentialUp(r)was determined using the polarizedorbital method19 and contained several static multipole termsas well as the corresponding dynamic polarization terms.20,21 Intotal, the potential Up(r) contained all terms up to and includingthose that behave as r−14 asymptotically.

Finally, the exchange terms WP(κ; r) and WQ(κ; r) in Eq.(1) are given by

WP or Q(κ2; r) = (1 + γ)n′κ′

�Pn′κ′(r) or Qn′κ′(r)

×�−�ϵn′κ′ + ϵ

�∆n′κ′ δ(κ, κ′) + e2

ν

qn′κ′

× 12ν + 1

C2 � j j ′ν;− 12

12

� 1ryν(n′κ′, κ; r).

(5)

Here, C�j j ′ν;− 1

212

�is the usual Clebsch-Gordan coefficient

and the sum over n′κ′ in Eq. (5) is over the radial part ofthe atomic orbitals (Pn′κ′(r) and Qn′κ′(r)) of the ground state,while qn′κ′ = 2 j ′ + 1 is the occupation number of these closedsub-shells where the ϵn′κ′ are the eigenvalues of these sub-shells. The exact form of the definite integral ∆n′κ′ and theindefinite integral r−1 yν(n′κ′, κ; r) is given in Eqs. (11) and(12) of Ref. 22.

We note that the dependence of exchange terms (5) on thewavefunction requires an iterative solution for Eq. (1).

In the integral equation formulation, the scattering phaseshifts can be determined from the asymptotic form of the largecomponent of the scattering wavefunction, i.e.,

fκ(r) −→r→∞

Aκ l(kr) − Bκ nl(kr), (6)

where

Aκ = 1 − 1k

∞

0dr

v1(kr) �U(r) fκ(r) −WP(κ; r)�

+ v2(kr) �U(r) gκ(r) −WQ(κ; r)� (7)

and

Bκ = −1k

∞

0dr

v1(kr) �U(r) fκ(r) −WP(κ; r)�

+ v2(kr) �U(r) gκ(r) −WQ(κ; r)�. (8)

The partial wave phase shifts are then given by

tan δ±l (k) =Bκ

Aκ, (9)

where the δ±l

are the spin-up (+) and spin-down (−) phaseshifts.

The total elastic and momentum transfer cross-sectionsare given, in terms of these phase shifts, by

σel(k2) = 4πk2

∞l=0

(l + 1) sin2δ+l (k) + l sin2δ−l (k)

(10)

and

σmt(k2) = 4πk2

∞l=0

(l + 1)(l + 2)2l + 3

sin2�δ+l (k) − δ+l+1(k)�

+l(l + 1)2l + 1

sin2�δ−l (k) − δ−l+1(k)�

+(l + 1)

(2l + 1)(2l + 3) sin2�δ+l (k) − δ−l+1(k)�

(11)

which can be shown to reduce to the non-relativistic results ifwe set δ+

l(k) = δ−

l(k) = δl(k).

As can be seen in Figure 1, neither the polarization nor theexchange interaction alone is capable of reproducing the trueRamsauer minimum in the argon momentum transfer cross-section; it is only when we combine these two interactions thatthere is an agreement with experiment. This is also true forthe Ramsauer minimum in the elastic cross-section.

In the original work of Ref. 9, Lekner described the elasticscattering of electrons by argon atoms by just the local dipolepolarization potential of Buckingham15 which is given by

Up(r) = − αd

2 (r2 + r2a)2

, (12)

where αd is the static dipole polarizability of argon and ra isan adjustable parameter. Lekner chose this parameter so asto obtain the experimental scattering length a0 = −1.5 a.u. ofRef. 23. This value is very close to the current recommended

FIG. 1. Cross-sections for electron scattering from argon. The calculationsdescribed in this paper, which use the non-local exchange interaction (solidline) are in good agreement with the recommended set of Buckman et al.24 Acomparison with a calculation similar to that of Lekner9 using a Buckinghampotential (dotted line) shows loose qualitative agreement at the Ramsauerminimum, but quantitatively is incorrect. Also shown are the results fromusing two different effective models of a local exchange potential (thickdashed25 and dashed-dotted lines26) which do not agree with experimentalmeasurement at all, as well as the cross-section when exchange is includedbut polarization is neglected (thin dashed line).

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154507-4 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

value of a0 = −1.45 a.u. of Ref. 24. The value obtained in thecurrent work is a0 = −1.46 a.u.

As a consequence of Lekner’s choice for the adjustableparameter ra, his simple polarization potential in Eq. (12) wasable to mimic the effects of both the polarization and exchangeinteractions at low energies of the incident electron andhis calculation was able to produce a low-energy Ramsauerminimum in the momentum transfer cross-section. At higherenergies, his momentum transfer cross-section deviates fromthe experimental cross-section.

We show our cross-sections calculated using (11) inFigure 1. We obtain very good agreement with the rec-ommended set of cross-sections of Ref. 24 which combinemany different experimental measurements and theoreticalcalculations. In order to demonstrate the importance ofincluding the non-local exchange interaction, we have alsorepeated the calculation using two different model potentialsthat replace the non-local exchange with an effective localterm in the potential.25,26 One of these local approximations25

is qualitatively wrong, showing the same behaviour as thatwithout exchange. The other approximation26 is qualitativelysimilar but differs in the scattering length and position of theRamsauer minimum by an order of magnitude. It is clear to seethat there is a significant difference between the results. Whenwe compare our results to those of the Buckingham potential,where we set rα = 1.087 a.u. such that the scattering length isa0 = −1.50 a.u., we find that it does follow the general shapeof the Ramsauer minimum. However, we emphasize that thisis a result of the fitting parameter rα and this potential doesnot accurately describe the details of the scattering.

III. SCREENING OF THE POLARIZATIONINTERACTION

The effects of the high density of the liquid are included inour calculations by several modifications of the gas scatteringproperties. The first of these is to account for the screeningof a single induced atomic dipole by the induced dipoles ofall other atoms. Our procedure outlined in this section closelyfollows that of Lekner.9

In the dilute gas limit, the mobile electron undergoes acollision with a single atom of the gas effectively in isolationfrom all other atoms in the gas. During this collision, theelectron induces a set of multipole moments in the atom,which in turn interact with the electron through a charge-multipole potential, resulting in the polarization potential,Up(r), of Sec. II above. For a dilute gas, the range of thepotential produced by these induced multipole moments isrelatively small compared with the large interatomic spacingand so it is a good approximation to neglect their effect on otheratoms. However, with higher densities of the gas or liquid,many atoms can have a non-negligible induced set of multipolemoments originating from both the mobile electron and fromall other atoms in the bulk. The effective charge-multipolepolarization potential felt by the electron at any particularlocation re is then the sum of the polarization potentials fromall atoms.

We consider effects originating from the induced dipolesof the atoms only and determine the effective polarization

of an individual atom self-consistently. We first assume thatthe induced dipole strength for every atom in the bulk can bewritten as f (r)αd(r)e/r2, where r is the distance of the electronfrom the atom, αd(r) is the exterior dipole polarizability(see Ref. 27, Eq. (1)) for a single atom that results fromthe interaction with the electron, and f (r) accounts forpolarization screening which must be determined. This simplemultiplicative factor is valid, so long as we average overthe atomic distribution. In the dilute-gas limit, we can safelyapproximate f (r) = 1, and in the dense case, we must obtaina self-consistent expression for f (r). By choosing a particular“focus atom” i at location ri such that r = re − ri, and assumingthat the coefficient f (r) is known for all other atoms, whichwe denote by fbulk(r), we can calculate9 the dipole strengthfor atom i from

f i(r) = 1 − πN ∞

0ds

g(s)s2

r+s

|r−s |dtΘ(r, s, t)αd(t) fbulk(t)

t2

(13)

which has been obtained using bipolar coordinates, s and t,where N is the density of the bulk, g(s) is the isotropic paircorrelator of the bulk, and the factor

Θ(r, s, t) = 32(s2 + t2 − r2)(s2 + r2 − t2)

s2 + (r2 + t2 − s2) (14)

arises due to the form of the electric field of a dipole. Theintegrations over s and t represent the contribution from anatom located at a distance s from atom i and a distance t fromthe electron. The likelihood of finding an atom is determinedby g(s) and so, it can be seen that Eq. (13) approximates theexact polarization by that resulting from the ensemble averageof all atomic configurations, given that one atom is locatedat ri. In this approximation, the polarization itself is alwaysaligned along the vector r between the focus atom and theelectron.

The self-consistent solution to Eq. (13) is obtained bysetting f i(r) = fbulk(r) and solving for f i(r), which we doby iteration. The most important quantity in Eq. (13) is thepair correlator, which represents the next order in the particledistribution in the bulk beyond the average density. In thecalculation of Lekner, the pair correlator was taken to bethe analytical solution of the Percus-Yevick model for easeof calculation. In our calculation, we go beyond this byusing the experimental measurements of Yarnell28 to moreaccurately describe the correlations. The data we use, whichwere obtained for a bulk density of N = 0.0213 Å−3, are shownin Figure 2 and compared with the Percus-Yevick model at thesame density.

Using the pair correlator of argon, we have self-consistently calculated the screening function f (r) andshow the result in Figure 3. Although this screening factortechnically applies to the dipole term only, we work with arather more complicated form of the polarization term thanLekner had originally considered. However, as the largestcontribution to the polarization does indeed come from thedipole term, we have decided to apply the screening factorf (r) to the entire polarization potential. Hence, with thescreening of the polarization taken into account, the screenedpolarization potential, Up(r), of an electron with one atom in

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154507-5 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

FIG. 2. Pair correlator for argon, as reported in Yarnell,28 measured inneutron scattering experiments. Also plotted is the pair correlator calculatedin the analytical Percus-Yevick approximation as used by Lekner.9

a dense fluid is given by

Up(r) = f (r)Up(r). (15)

We note that, in contrast to Lekner, who used only the staticdipole polarizability αd, the more accurate representation ofthe atom-electron interaction as described in Sec. II has alreadyled to a radial dependence of the polarization potential Up(r)beyond that of a potential whose asymptotic behaviour isr−4. The effect of the screening has hence led to a furthermodification of Up(r) which is density dependent.

IV. EFFECTIVE POTENTIAL IN LIQUID

For input into the kinetic theory, we require appropriatecross-sections for the scattering of the electron from a single“focus atom” in the bulk. As discussed above, the presence ofthe other atoms screens the polarization interaction betweenthe electron and the focus atom. However, there is anothermore obvious effect resulting from the other atoms in the bulk:their interaction with the electron itself remains significanteven when the electron is very close to the focus atom. Hence,as outlined in Lekner,9 we build up an effective potential thatis experienced by the electron throughout a single scatteringevent, as well as define what is meant by “a single scattering

FIG. 3. The screening function f (r ) of the polarization interaction potentialfor scattering of an electron from a single argon atom in a bulk of densityN = 0.0213 Å−3.

event.” Although we follow the general principles of Ref. 9,we calculate the cross-sections in a distinctly different fashion.

The effective potential that we consider Ueff = U1 +U2is made of two parts: U1(r), which corresponds to the directinteractions with the focus atom and U2(r), which correspondsto the interaction of the electron with the rest of the bulk. Asit is prohibitively expensive to treat exact configurations ofatoms in the bulk, we build the external potential U2 by againtaking the ensemble average,

U2(r) = 2πNr

∞

0dtU1(t)

r+t

|r−t |ds sg(s), (16)

where the order of integration has been reversed in comparisonto (13) for numerical convenience.29 We note that taking theensemble average has the advantage of enforcing sphericalsymmetry of the total effective potential Ueff. In calculating(13) and (16), we make use of the quantity σcore, whichcorresponds to the hard-core exclusion diameter for thedistribution of atoms in the bulk, i.e., the probability fortwo atoms to approach within a distance σcore is vanishinglysmall. For argon, σcore ≈ 6 a0 and we take advantage of thisby explicitly setting g(s) = 0 for s < σcore and adjusting thelimits of Eqs. (13) and (16) accordingly.

In addition, we go beyond Lekner’s calculation byincluding the effects of the exchange terms in the bulk. We dothis by performing the same ensemble average as in (16) butover the quantities WP and WQ instead of U1, obtaining bulkaverages WP,2 and WQ,2. These are then included as effectiveexchange terms, W (P or Q),eff = WP or Q +W (P or Q),2 in theDirac-Fock scattering equations (1). In contrast to U2, theseexchange terms are dependent on the wavefunction itself, sothe ensemble averages must be recalculated at every iterationin the solution of (1).

A plot of the functions Ueff, U1, and U2 is shown inFigure 4. It can be seen that there is a turning point thatoccurs at a distance we denote by rm. In the dense gaslimit, that we are investigating, this value is rm ≈ 4.3 a0. Theturning point at rm provides a natural distinction between thevolume that is under the influence of the focus atom, i.e.,

FIG. 4. Plots of the total effective potential Ueff felt by an electron whencolliding with one atom in the liquid. Also shown are the components,U1 andU2, which represent the direct potential of the atom and the contribution of theremaining atoms in the bulk, respectively. The dashed vertical lines at σcore/2and rm indicate the hard-core exclusion radius and the proposed collisionalsphere, respectively. Note that effects of exchange are not represented in thisfigure.

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154507-6 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

the sphere of radius rm, and that of the rest of the bulk.Hence, we can say that a single collision event takes placewhen an electron enters and leaves the radius rm of a singleatom. We note that rm ≈ 2

3σcore > σcore/2, i.e., rm is largerthan half of the minimal interatomic separation, which couldbe considered to define the volume “owned by” the focusatom and hence a logical choice for the “collision eventradius.” rm is also different from the Wigner-Seitz diameterdWS = 2(4πN/3)−1/3 ≈ 4.2 a0,14 although it is very similar.

We would now like to solve for the scattering properties,in particular the cross-sections, from such a collision process.We assume that it remains reasonable to extract the cross-sections through the phase shifts in a partial wave expansion.In order to determine these, Lekner chose to shift the effectivepotential by an amount U0 such that Ueff(rm) +U0 = 0 and toset the potential Ueff(r > rm) = 0, and finally matched to theasymptotic form of each partial wave in the usual fashion.In contrast, we choose to leave the potential unaltered, butcalculate the phase shift at the point rm instead, effectivelysetting the upper limits of Eqs. (7) and (8) to be rminstead of infinity. We note that this is also known ascalculating the “phase function”13 at the point rm, which isequivalent to setting Ueff(r > rm) = 0 and matching to theasymptotic form of the wavefunction. We believe that thismore accurately represents the available energy states inthe bulk. We denote this cross section, including externalcontributions and screening effects, by σscr(ϵ, χ).

As we may assume g(s) = 0 for s < σcore and because wecalculate the potential only up to a distance of rm ≈ 2

3σcore,we can see that the integral over t in (16) is non-zero only fort & 1

3σcore ≈ 2 a0. At these ranges, the dominant contributionto the potential comes from the polarization component. Wealso note that the values of WP,2 and WQ,2 are not wellbehaved for larger distances and so we set them to be zero forr > σcore/2. We have performed calculations that neglect thecontribution of WP,2 and WQ,2 to the bulk and compared theseto the full calculations, which showed very little differencein the high energy regime of the resultant cross-sectionsand a small difference of up to 5% otherwise. The effectof this change on the transport properties was a small butnon-negligible deviation.

A. Cross-sections and variation of rmThe choice of the value for rm is a crucial part of our

calculation. It is worth mentioning that the choice we makeabove is consistent in the limit of N → 0; in this case, U2 isso weak that it is only after U1 has significantly decayed forvery large values of r that d(U1 +U2)/dr = 0. Hence, rm → ∞as N → 0 and our calculation reduces to the usual scatteringcalculation from a single atom. However, in the dense case, itis not known whether d(U1 +U2)/dr |rm = 0 is the best choiceto model the scattering in the liquid. Hence, we have alsoperformed a sensitivity analysis on the parameter rm. Wedenote the distance at which we calculate the phase shiftsby r∗ and allow it to vary from our initial choice of r∗ = rm.The resultant cross-sections from a variation of ± 1

16 a0 areshown in Figure 5 as well as the more straightforward choiceof r∗ = σcore/2. We note that Atrazhev and Timoshkin14 have

FIG. 5. Screened elastic total and momentum-transfer cross-sections forargon calculated from the phase shifts determined at a distance r ∗. Ourpreferred choice for transport calculations in this paper, r ∗= rm, correspondsto the solid line, the dashed lines are those corresponding to a variationsr ∗= rm± 1

16a0 , and the dotted line corresponds to a variation of r ∗=σcore/2.

implicitly investigated this variation previously, in order todescribe the effect of density fluctuations on the effectivecross-sections. In their case, the value of r∗ was set to be theWigner-Seitz cell radius, which itself depends on the densityof the liquid. In contrast, we keep the density fixed whilevarying r∗.

It can easily be seen that the largest modification to thecross-sections due to the variation in rm occurs at low energies.Importantly, the more obvious choice of r∗ = σcore/2 yields adramatically different behaviour. As will be shown later, theeffect that these variations have on the transport measurementsis significant and shifts the peak observed in various transportproperties.

We note that we neglect the effect of density fluctuations,which would modify the effective cross-section for the liquid.This was investigated by Atrazhev and Timoshkin14 andshown to have a significant contribution to the cross-sections.However, their article focused on a density for which theeffective liquid cross-section vanishes, causing the densityfluctuations to be the largest contribution for small electronenergies. In our case, we can expect density fluctuations tocause both enhancements and reductions of the cross-sections,which would cancel out on average.

V. KINETIC THEORY AND TRANSPORT PROPERTIES

A. Multi-term solution of Boltzmann’s equation

The behaviour of electrons in gaseous and liquid argon,driven out of equilibrium via an electric field E, can be

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154507-7 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

described by the solution of the Boltzmann’s equation forthe phase-space distribution function f (r,v, t),30

∂ f∂t+ v · ∇ f +

eEme· ∂ f∂v= −J( f ), (17)

where r, v, and e denote the position, velocity, and charge ofthe electron, respectively. The collision operator J( f ) accountsfor interactions between the electrons of mass me and thebackground material. We restrict our considerations in thisstudy to those applied reduced electric fields E/N (where Nis the number density of the background material) such thatno internal states of the individual argon atoms are excited.

To calculate the drift and diffusion coefficients, werepresent the spatial dependence of the distribution functionsas31,32

f (r,v, t) = F(v, t)n(r, t) − F(L)(v)∂n∂z

− F(T )(v)cos φ

∂n∂x+ sin φ

∂n∂ y

+ · · ·, (18)

where the superscripts L and T define quantities that areparallel and transverse to the electric field (defined to bein the z direction), respectively. Solution of Boltzmann’sequation (17) requires decomposition of the coefficients invelocity space through an expansion in (associated) Legendrepolynomials,

F(v) =∞l=0

Fl(v)Pl(cos θ) ,

F(T )(v) =∞l=0

F(T )l

(v)P1l (cos θ),

(19)

where θ denotes the angle relative to the electric field direction(taken to be the z-axis).

This is a true multi-term solution of Boltzmann’s equa-tion, whereby the upper bound in each of the l-summationsis truncated at a value lmax, and this value is incrementeduntil some convergence criteria is met on the distributionfunction or its velocity moments. By setting lmax = 1 weobtain the two-term approximation commonly used in allelectron transport theories in liquids,6,8,10 which enforces aquasi-isotropic distribution. The current theory does not makethe quasi-isotropic assumption for the velocity distributionfunction a priori. By using the orthogonality of (associated)Legendre polynomials, the following hierarchy of equationsmust be solved to calculate the drift velocity and diffusiontensor:32

J lFl +l + 1

2l + 3a

(∂

∂v+

l + 2v

)Fl+1

+l

2l − 1a

(∂

∂v− l − 1

v

)Fl−1 = 0, (20)

J lF(T )l+

l + 22l + 3

a(∂

∂v+

l + 2v

)F(T )l+1

+l − 1

2l − 1a

(∂

∂v− l − 1

v

)F(T )l−1 = v

(Fl−1

2l − 1− Fl+1

2l + 3

),

(21)

where the J l represent the Legendre projections of the collisionoperator detailed below and a = eE/me. The solution of the

system of Eqs. (20) and (21) provides sufficient informationto calculate the drift velocity W via

W =4π3

∞0

v3F1dv, (22)

and the characteristic energy, defined as the ratio of thetransverse diffusion coefficient DT to the electron mobilityµ(= W/E), via calculation of the transverse diffusion coeffi-cient,

DT =4π3

∞0

v3F(T )1 dv. (23)

B. Collision operator for interactions in structuredmatter

The collision operator appearing in (17) describes therate of change of the distribution function due to interactionswith the background material. At low electron energies, wherethe de Broglie wavelength of the electron is of the order ofthe average inter-particle spacing ∼N−1/3, the charged particleis best viewed as a wave that is coherently scattered fromthe various scattering centres that comprise the medium. Athigher energies, the de Broglie wavelength becomes muchless than the inter-particle spacing and the effects of coherentscattering are no longer important. In this limit, the binaryscattering approximation is recovered, although the interactionpotential is modified as discussed above. For liquid argon, theaverage interparticle spacing is approximately 4.5 Å, implyingthat “low” energies are those less than ∼7.4 eV, which isseveral orders of magnitude larger than the thermal energy of∼0.01 eV.

Recently, the two-term approximation of Cohen andLekner8 was extended to a multi-term regime,16,33 where theLegendre projections of the collision operator in the smallmass ratio limit were shown to be

J0 (Φl) = me

Mv2

ddv

vνmt(v)

vΦl +

kTme

ddvΦl

, (24)

J lΦl = νl(v)Φl for l ≥ 1, (25)

where M is the mass of an argon atom, Φl =�Fl,F(L),F(T ),

and

νmt(v) = Nv2π π

0σscr(v, χ) [1 − P1(cos χ)] sin χdχ

= Nvσscrmt(v) (26)

is the binary momentum transfer collision frequency in theabsence of coherent scattering effects, while

νl(v) = Nv

(2π

π

0Σ(v, χ) [1 − Pl(cos χ)] sin χdχ

)(27)

are the structure-modified higher-order collision frequencies.The effects of the structure medium are encapsulated in thestatic structure factor S(K), which is the Fourier transform ofthe pair correlator, g(r), used in Secs. III and IV. The structure

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154507-8 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

factor is included via the term

Σ(v, χ) = σscr(v, χ) S(

2mev

~sin

χ

2

), (28)

which represents an effective differential cross-section. If werepresent Σ(v, χ) through an expansion in terms of Legendrepolynomials

Σ(v, χ) =∞λ=0

2λ + 12Σλ(v)Pλ (cos χ) , (29)

then one can make connection with the previous calculationsof the collision matrix elements for dilute gaseous systems.The effective partial cross-sections Σl(c) are defined by16,33

Σl(v) = 2π 1

−1Σ(v, χ)Pl(cos χ)d(cos χ)

=1

4π

λ′λ′′

(2λ′ + 1)(2λ′′ + 1)2l + 1

×C2(λ′ λ′′ l; 0 0)σλ′(v)sλ′′(v), (30)

where

σl(v) = 2π 1

−1σscr(v, χ)Pl(cos χ)d(cos χ) (31)

and

sl(v) = 12

1

−1S

(2mev

~sin

(χ

2

))Pl(cos χ)d(cos χ). (32)

It then follows that

νl(v) = Nv [Σ0(v) − Σl(v)] . (33)

It is sufficient for this study to consider only lowenergy coherent elastic scattering processes. At higher fields,incoherent inelastic scattering effects including excitation andionization would need to be considered.16,33

VI. RESULTS

Swarm experiments are a test of the particle, momentumand energy balance in the cross-section set, and the associatedtransport theory or simulation. In the low-field regimeconsidered in this manuscript, only conservative quasi-elasticprocesses are operative, and hence, the ability of the calculatedvalues of drift velocity and characteristic energy to match themeasured coefficients provides this test on momentum andenergy balance.

In Secs. VI A–VI C, we consider the calculation ofthe macroscopic swarm transport properties in the gaseousand liquid environments from the microscopic cross-sections,including screening and coherent scattering effects as dis-cussed above. Initially, in Sec. VI A, we consider onlythe gas phase, focussing on understanding the importanceof an accurate treatment of exchange and polarization andestablishing the credibility of the initial gas-phase potentialsubsequently used as input for the calculation of cross-sectionsfor the liquid phase environment. Transport coefficientscalculated using the screened cross-sections and associatedcoherent scattering effects are considered in Sec. VI B, where

FIG. 6. The drift velocity (top) and characteristic energy (bottom) of elec-trons in gaseous argon, calculated using the potentials and associated cross-sections detailed in Sec. II and compared with available experimental data(Robertson34,36 at 90 K, Warren and Parker35,37 at 77 K, and Townsend andBailey37,38 at 288 K). The full non-local treatment of exchange consideredhere is compared to two forms of local exchange potentials (LocExA25

and LocExB26) and to the case when the exchange interaction is neglectedaltogether. The background argon gas for the calculations was fixed at 90 Kfor determination of the drift velocity and 77 K for the characteristic energy.

they are compared with the available measured transport datain the liquid phase.

A. Electrons in gaseous argon — benchmarkingthe potential and exchange treatment

The calculated drift velocity and characteristic en-ergy transport properties using the gas-phase cross-sectionsdetailed in Sec. II are presented in Figure 6. They are comparedagainst various experimental data for this gas.34,35 We restrictourselves to the reduced electric fields of less than 3 Td, toensure we are in the regime where only elastic scattering isoperative.

Our current potential, with a non-local treatment ofexchange, generally reproduces drift velocities to within theexperimental errors. There are small regions of E/N (wherethe properties vary rapidly with E/N) where errors can be aslarge as 3% in the drift velocity and 10% in the characteristicenergy. For the characteristic energy, this can be outside theexperimental errors in this region. If the exchange interactionis neglected in the calculation of the cross-section, we observethat the calculated values of the transport properties departfrom the measured by an order of magnitude or more,reflecting the qualitative disagreement in the form of the cross-sections predicted in Figure 1. Given the similarities in the

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154507-9 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

cross-sections calculated using the local exchange potentialB26 to those neglecting exchange, the calculated transportcoefficients are quite similar between the two techniques.Using the local treatment of exchange A,25 which reproducesthe Ramsauer minimum in the cross-section (although itsdepth, location, and width disagree quantitatively), the trans-port coefficients have a similar qualitative form; however,they are displaced to significantly higher fields relative tothe measured values. As expected, implementation of theBuckingham potential as in Lekner,9 which was tuned toreproduce the zero-energy gas-phase cross-section, producesdrift velocities that are accurate to within 10%; however, thecharacteristic energies are significantly worse than those usingthe current potential.

The results shown here for argon demonstrate the validityof the electron-argon interaction potential developed for thecurrent study and the necessity for a strict non-local treatmentof exchange and an accurate treatment of polarization, inorder to generate the accurate microscopic differential cross-sections. The small disagreement for the characteristic energyover a small range of E/N may reflect some minor limitationsin the cross section data base.

B. Electrons in liquid argon

In Figure 7, we compare the drift velocity and charac-teristic energies in both the gaseous and liquid phases. Thetransport coefficients are presented as a function of the reducedelectric fields, so that the explicit density dependence has beenscaled out and we have a true comparison of the gaseous andliquid phases. For a given reduced field, we observe that thedrift velocity in the liquid phase is enhanced by as much asan order of magnitude over the gaseous phase in the reducedfield range considered. Contrarily, the characteristic energy inthe liquid phase is reduced relative to the gaseous phase byas much as 500% over the range of fields for which the dataexist. Importantly, the measured data emphasize that transportof electrons in liquids cannot be treated by using only thegas phase cross-sections and scaling of the density to those ofliquids.

We now assess the importance of including variousphysical processes present in liquids in reproducing themeasured transport coefficients.

First, we assess the importance of coherent scatteringeffects, by implementing the gas-phase interaction potentialand associated cross-sections into the coherent scatteringframework detailed in Sec. V B. The resulting cross-sectionsare displayed in Figure 8. We observe in Figure 7 that theinclusion of only coherent scattering effects acts to enhanceboth the drift velocity and the characteristic energy. This isa reflection of the reduced momentum transfer cross-sectionin Figure 8 in the regime where coherent scattering effectsare operative.16 Interestingly, coherent scattering produces thephysical process of negative differential conductivity (i.e.,the fall of the drift velocity with increasing electric field)which is absent from the gas-phase calculations, as discussedelsewhere.16 While the inclusion of coherent scattering effectsresults in a calculated drift velocity of the same order ofmagnitude as the experimental data, it does not reproduce

FIG. 7. Comparison of the measured drift velocities W and characteristicenergies DT/µ in gaseous and liquid argon, with those calculated fromthe various approximations to the cross-sections. Experimental data (Robert-son34,36 at 90 K, Miller et al.39 at 85 K, Halpern et al.40 at 85 K, Warrenand Parker35,37 at 77 K, Townsend and Bailey37,38 at 288 K, and Shibamuraet al.41 at an unmeasured liquid temperature). The various approximationsused are: gas-phase only cross-sections (Gas), gas-phase cross-sections withcoherent scattering (Gas + Coh), and liquid phase cross-sections with coher-ent scattering effects (Liq + Coh). The results have been calculated usingthe full differential cross-section and results are converged multi-term values.Experimental uncertainties are estimated at 2% for Robertson and less than15% for Shibamura et al..

the correct shape in the profiles, with errors as large as250%. Further, the calculated characteristic energy producedby including coherent scattering enhances the characteristicenergies relative to the gas phase which is inconsistent withthe experimental data.

Second, in addition to the coherent scattering, we nowinclude the full liquid induced effects on the potential asdetailed in Secs. III and IV. The resulting cross-sectionsare displayed in Figure 8, where we emphasize that sucheffects act to essentially remove the Ramsauer minimum inthe cross-section. This produces an enhanced and relativelyconstant cross-section in that energy regime. This is verysimilar to that predicted by Atrazhev and Iakubov,11 in theirreduction of the Cohen and Lekner theory, which suggestedthat a cross-section that is only density dependent would occurfor low impact energies. In Figure 7, we demonstrate thatthe inclusion of both scattering potential modification andcoherent scattering produces drift velocities and characteristicenergies that are both qualitatively and quantitatively inagreement with the experimental data. Errors in the driftvelocities and characteristic energies are significantly reduced.

In Figure 5, we highlighted the sensitivity of the calculatedcross-sections in the liquid phase to the value of r∗ at which the

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154507-10 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

FIG. 8. The momentum transfer cross-sections in the gas-phase (Gas) andliquid-phase (Liq) and their modifications when coherent scattering effectsare included (+Coh). The recommended transfer cross-section of Ref. 24 fora dilute gas is a combination of experimental measurements and theoreticalcalculations.

phase shifts are determined. The macroscopic manifestationsof this sensitivity on both the drift velocity and characteristicenergy is presented in Figure 9. Slight modifications of r∗

by a0/16 from the preferred value of r∗ = rm emphasize thesensitivity of the transport coefficients to this value. Thechoice of r∗ = σcore/2 produces results that are essentiallytranslated to higher reduced electric fields. Importantly, theseresults indicate that the value of rm may be energy dependent.One could possibly tune the value of rm to match theexperimental data; however, we have strived to eliminateadjustable parameters in our formalism. One may also look

FIG. 9. Comparison of the calculated drift and characteristic energy withvariation in the distance r ∗ at which the phase shifts are determined. Experi-mental data are as detailed in Figure 7.

FIG. 10. Top: Differential cross-sections in square angstroms per steradianfor electrons in Ar for (a) dilute gas phase, σ(ϵ, χ), (b) effective liquidphase including screening effects, σscr(ϵ, χ), and (c) liquid phase cross-section including coherent scattering effects Σ(ϵ, χ). Bottom: Differentialcross-sections taken at 10 eV for the same cases. The experimental data forthe gas phase are taken from Gibson et al.42

at using an alternative scheme that is energy-dependent forchoosing the value of rm, e.g., including contributions fromthe exchange terms WP,eff and WQ,eff.

C. Impact of scattering anisotropy and the two-termapproximation

We conclude this study by considering the impact ofthe anisotropy in both the scattering cross-sections and thevelocity distribution function on the calculated transportproperties.

In Figure 10, we display the differential cross-sectionsfor the gas phase and for the liquid modified differentialcross-sections, highlighting the impact of coherent scatteringeffects. For the dilute gas phase, we observe at low energiesthat the differential cross-sections are small and essentiallyisotropic. As we move to higher energies, the differentialcross-section begins to demonstrate an increased magnitudeand also enhanced anisotropy, with peaks in the forward andbackscattering directions. This is confirmed by agreementwith the experimental data of Gibson et al.42 When weaccount for liquid effects in the scattering potential, we observethat similar qualitative structures are present in the resultingdifferential cross-section, with slightly more structure thanfor the dilute gas phase. When the liquid phase differentialcross-section is combined with the structural factor accounting

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154507-11 Boyle et al. J. Chem. Phys. 142, 154507 (2015)

FIG. 11. Percentage differences between the two-term and multi-term valuesof the characteristic energy for the gas and liquid phases using the fulldifferential cross-sections (solid lines) and percentage differences between themulti-term results with using only the momentum transfer cross-section andthe full differential cross-section (dashed lines). All percentages are relativeto the converged multi-term result using the full differential cross-section.

for coherent scattering effects, the resulting differential cross-section Σ(ϵ, χ) takes on a completely different qualitativestructure. The forward peak in the differential cross-section isremoved, with suppression of the cross-section at low energiesand low scattering angles. The backscattering peak in thedifferential cross-section at high energies remains unaffected,while subpeaks in the differential cross-section are enhancedby the coherent scattering effects.

The degree of anisotropy in the distribution function isevidenced by an enhanced value of lmax required in sphericalharmonic expansions (19) to achieve convergence in thevelocity distribution or transport properties. In Figure 11, wedisplay the error in the two-term approximation (lmax = 1)and the converged multi-term result. In the gas and liquidphases, we see that the two-term approximation is sufficient toensure accuracy to within 0.5% in the drift velocity; however,errors as large at 10% are present in the characteristic energy.This indicates a failure of the two-term approximation forthe evaluation of the characteristic energy. Similar findings inthe gas-phase were found by Brennan and Ness.43 Theoriesthat have used the two-term approximation to iterativelyadjust cross-sections may produce cross-sections that areinconsistent with a multi-term framework.

In Figure 11, we also consider the impact of anisotropicscattering on the validity of the two-term approximation.The two-term approximation can only sample the mo-mentum transfer cross-section. Higher-order spherical har-monic components of the distribution function in expansions(19) are coupled to, and hence sample, higher-order coeffi-cients in the expansion of the differential cross-section (see,e.g., Eq. (31)). In Figure 11, we highlight the differences,using dashed lines, between the multi-term approximationsusing only the momentum transfer cross-section (i.e., weassume σl≥2 = σ1) and those where the full differential cross-section is considered. The differences are less than 1% (usuallyless than 0.1%) indicating the distribution function is notsufficiently anisotropic to couple in higher-order partial cross-sections. Equivalently, anisotropy in the differential cross-sections has only a minimal impact on the anisotropy in thevelocity distribution function.

VII. CONCLUSIONS

We have extended the approach of Lekner and Cohen,8,9

overcoming some of its limitations, to calculate the effectivecross-sections and transport properties of electrons in liquidargon. For the first time, an accurate multipole polarizabilityin the electron-atom potential and a fully non-local treatmentof exchange were included in the calculation of liquidphase cross-sections using the full machinery of the Dirac-Fock scattering equations. The accuracy of the potentialimplemented and associated cross-sections calculated wasconfirmed by comparison with experiment in the gas-phase,and the importance of a fully non-local treatment of exchangewas demonstrated. The result calls into question cross-sections(gas, liquid, or clusters) which assume a local treatment ofexchange. Sensitivity to the radial cutoff for the electron-atom potential was presented, and while the maximum in thepotential was shown to be a suitable choice; enhanced accuracymay be achieved with an energy dependent choice of thecutoff.

The calculation of the drift velocity and characteristicenergies was performed for the first time using a multi-termsolution of Boltzmann’s equation accounting for coherentscattering. The full anisotropy of the liquid phase differentialcross-section was considered including anisotropy arisingfrom both the interaction and the structural factor. The multi-term framework enabled an assessment of the sensitivity to thisanisotropy in the differential cross-section and in the velocitydistribution function. While the two-term approximation wasfound to be sufficient for accuracies to within 1% for thedrift velocity, errors of the order of 10% or more were foundin the characteristic energy. The latter was found to be thedominant contribution to the differences in the two- andmulti-term results. It was found that both coherent scatteringand screening of the electron-atom potential are requiredto reproduce the measured transport coefficient values. Weemphasize that there are no free parameters in the currenttheory and its implementation, and hence, the high level ofagreement between the calculated and measured transportcoefficients yields confidence that the essential physics hasbeen captured in the theory.

ACKNOWLEDGMENTS

The authors acknowledge the financial assistance of theAustralian Research Council (ARC) through its Discovery andCentres of Excellence programs.

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