PH Y SICAL RE VIE% B VOLUME 18, NUMBER 6 15 SEP'jI'EMBER 1978

Anharmonic properties of Li

S. H. Taole and H. R. GlydePhysics Department, University of Ottawa, Ottawa, It" lN 6N5 Ontario, Canada

Roger TaylorDivision of Physics, National Research Council of Canada. Ottawa, KlA OR6 Ontario, Canada

(Received 24 March 1978)

The energies and lifetimes of phonons in metallic Li between 110 and 424 K are calculated using the self-

consistent phonon theory and the e6ective ion-ion interaction developed by Dagens, Rasolt, and Taylor(DRT). The results are compared with the recent neutron scattering measurements of Beg and Nielson. Liturns out to be surprisingly harmonic, less anharmonic than Na or K, due to the strong Li ion-ioninteraction which has a relatively soft. repulsive core. The anharmonic corrections do improve agreement with

the observed dispersion curves at 110 K, and they provide a good description of the phonon frequency

changes with temperature, but the calculated phonon lifetimes are often two or three times the observedvalues. The remaining discrepancies with experiment suggest (i) the presence of three- or four-body

interactions in Li which are not included in the DRT pseudopotential method, and (ii) that the phonons

decay significantly via processes higher than those accounted for by cubic anharmonicity.

I. INTRODUCTION

Starting with the early work of Toya, theoreticalcalculations of the lattice dynamics of alkali metalshave progressed, during the past two decades, tothe point where first-principles' calculations canbe expected to yield reliable results. This is es-pecially true in Na and K where it is possible toobtain good agreement with experiment at all tem-peratures, even those where anharmonic effectsare comparatively large."With their sphericalFermi surfaces, free-electron behavior, and fairlytightly bound ionic cores these two metals are,from the theoretical point of view, the easiest ofthe alkali metals to deal with. Rb and Cs are ra-ther more difficult for two reasons. Firstly, thedensity of electrons is so low that the compressi-bility of the electron gas is negative for Cs and

virtually so for Rb, rendering a calculation basedon the electron gas as a zeroth-order approxima-tion, highly questionable. Secondly, the ratherlarge ionic cores of these materials are moreloosely bound than in the lighter alkalis giving riseto appreciable core polarizabilities. This lattereffect has yet to be included in a lattice-dynamiccalculation for Rb and Cs, although Kukkonen' hasincorporated it into the electron-electron scatter-ing contribution to the thermal resistivity. Kuk-konen suggested that the core-polarization effectcould be included by renormalizing the density ofthe electron gas to an effective higher density.This has the additional benefit of removing thecompressibility problem. Hence this type of ap-proach applied to the lattice-dynamic problemmight very well be the answer to the Rb and Csproblem, although this remains to be investigated.

This brings us to Li, the lightest of the alkalimetals with the highest density of electrons and nosignificant core polarization. However Li has itsown special problem which arises from the factthat the ionic core consists of only two 1s elec-trons. This means that the ionic potential seen bythe p component of the conduction band is notshielded by the core, resulting in a strong scatter-ing phase shift. The pronounced deviations fromsphericity of the Fermi surface"' are probablymanifestations of this fact. This means that aproperly constructed Li pseudopotential is highlynonlocal and relatively strong, which in turn meansthat the application of perturbation theory poses aserious problem. Hence although, superficially,the lattice dynamics of the alkali metals appearsto be reasonably well understood, only Na and Kseem to have no significant remaining difficultiesin interpretation.

In this paper we address ourselves to the prob-lem of anharmonic lattice dynamics in Li. Al-though there exists a number of calculations in theharmonic approximation (e.g. ,Refs. 8-13), therehave been no attempts to evaluate the anharmoniccontributions. In view of the fact that Li' is a verylight ion it is conceivable that significant anhar-monic effects could be present even at the lowesttemperatures (liquid nitrogen) for which neutronscattering measurements have been made. " Thispoint alone is worthy of investigation since manyof the aforementioned calculations were adjustedto fit the low-temperature experimental data. Inaddition, the recent neutron measurements of Begand Nielsen" of the phonon dispersion curves up toI' = 424 K provide an interesting test of theory tosee how well these can be reproduced from a first-

2643 1978 The American Physical Society

S. H. TAOLK, H. R. GLYDE, AND ROGER TAYLOR 18

principles point of view. In Secs. II-V we discussthe interionic potential used, the techniques tohandle it, the lattice dynamics, and the comparisonof our results with experiment over the entire tem-perature range. In our treatment there are no ad-justable parameters determined from experimentand it can be regarded as a first-principles calcu-lation.

II. THE INTERIONIC POTENTIAL

In the theory of simple metals, it is normal toreplace the ionic potential seen by a conductionelectron by a pseudopotential. This is then as-sumed to be a sufficiently weak perturbation onthe system, that second-order perturbation theorycan be used to generate the interionic potential.In view of our remarks in Sec. I concerning thestrong scattering properties of Li, this assump-tion is highly questionable. But to go beyond sec-ond-order perturbation theory when calculatingion-ion interactions is a very difficult and unre-warding computation. " Qne way out of this di-lemma is to follow the prescription provided byDagens, Rasolt, and Taylor" (DRT). These au-thors first calculated the charge density inducedby an isolated ion placed in an infinite electrongas of the same density as the metal. For thisthey used the self -consistent Hartree-Fock-Slaterscheme with Kohn and Sham" exchange. They thengenerated the linear response (first-order) resultusing a pseudopotential screened by a dielectricfunction, formally equivalent to the Kohn and Shamexchange approximation, and they adjusted theparameters of the pseudopotential to make thissecond calculation agree with the first. By thisprocedure all the higher-order terms representingmultiple-scattering events at a single ion site werefolded into the pseudopotential, which could thenbe used in low-order perturbation theory. Usingthe DRT pseudopotential to second order in theinterionic potential problem provides an effectivesum of all higher contributions to the two-bodypair potential. Hence it is entirely appropriateto use the DRT pseudopotential to second orderfor a calculation of a Li pair potential and thisis what we have done. We should point out that noeffects of N-body forces (N ~3) are folded into thepair potential and we do not include these forcesin our calculation. It is quite probable that atleast three-body forces play a significant role inLi and we shall comment on the possible influenceof these forces in the discussion.

For our calculations we used the Geldart andTaylor" dielectric function to screen the pseudo-potential. This dielectric function includes bothexchange and correlation in the electron gas and

is probably the most appropriate one to use forthese problems. " Using the same combination ofpseudopotential and screening, DRT obatined pho-non dispersion curves approximately 5% higherthan the experimental results of Smith et al."at90 K. The formulas used to calculate the inter-ionic potential are given by Rasolt and Taylor. "

A particularly important feature of the DRTpseudopotential which is common to all realisticdescriptions of Li, is the strong degree of nonlo-cality. This comes from the fact that the absenceof p-electrons in the Li' core means that the Pcomponent of the conduction band is scatteredmuch more strongly than the s component. As aresult of this feature the corresponding interionicpotential exhibits asymptotic Friedel oscillatorybehavior of the form

(r) = 2k+(Ze)'[A, cos2kzr/(2kzr)'

+4, sin2ih~r/(2kzr)4+ . ],where the amplitude of oscillation is very large.This severely complicates the problem of anhar-monicity since sums of this function in real spacebarely converge at infinite distance. This is be-cause the number of neighbors summed over is r'.Hence truncation of sums at any finite distancecan lead to serious errors. To display this pointwe have plotted r'V(r) in Fig. l, where it can beseen that the long-range oscillations are indeedvery large, particularly compared to the DRT Kinterionic potential. Because of this, special tech-niques bad to be developed to treat the dynamicsof Li which were not necessary for Na (Ref. 8) andK (Ref. 4). These are described in the followingparagraphs and Sec. ID.

Although Eq. (l) contains only the first two termsof an infinite series, it gives a good descriptionof the long-range oscillations of Li in the range ofinterest, and we shall truncate the series at thatpoint. Strictly speaking the coefficient A4 displaysa weak logarithmic dependence on r, but for ourpurposes it is sufficient to regard it as a constant.Thus the problem of summing V„(r) over all neigh-bors can be solved by evaluating sums of functionsof the form e''"/r". These can be computed easilyto any desired accuracy by using a modified Ewaldtechnique. We follow Duesbery and Taylor" andwrite, for the interionic potential at all values of

(2)

The short-range part V„(r) is of course left overwhen V„(r) is subtracted from VI(r).

To evaluate the asymptotic part V„(r), in (1) we

ANHARMONIC PROPERTIES OF Li

O.I2—

0.08—Li: 0 = 5.5l05 A

0, 0.04—4)

—0 04—

0.04—

0.08—0.6

I

I.OI

l.4I

I.8I

2.2I

2.6I

3.0I

5.8I

4 2 4.6

PFIG. l. Effective ion-ion interactions p Vyy{p) in Li and K, where p =&/a and a is the lattice constant.

use the Ewald-Fuchs" method to evaluate x ' andx ', where x= 2k'. This employs the identity

and V,",(r) is the corresponding expression withJ" replacing J. The total potential is then

V. (r)=[V (r)+V,",(r)]+ V,', (r)

= v„, (r)+ ve(r). (6)( a

qltl le ('ggPdqI'(m/2) ~,

qtll lg -(le)-2

a )

2[Zo(x) +J"„(x)], (3)

v„(r) = v.",(r)+ v,', (r), (4)

where

Vo(r) A, [2/I"~(—,')]Be(x)cosx

+A, [2/I'(2)]84e(x)sinx (5)

where I'(~~) is the gamma function. The secondterm J"(x) of this representation of 1/x convergesrapidly as x increases (much more rapidly than1/x") while the first term J@(x) can be evaluatedrapidly in Fourier space. Using (3) we may write(1) as

This representation raises the possibility of re-placing Vn (r) by V,«(r) = V„(r)+ V,",(r), which ig-nores Ve(r) completely This .can sometimes bedone by suitable choice of u. This is regularlydone in the reciprocal space formulation of dyna-mics, "where the Ewald transformation is used toseparate the entire potential into two parts, oneevaluated in reciprocal space and the second eval-uated in configuration space. With suitable choiceof n the part in configuration space may be neglec-ted. However, we found no value of u which madeV «(r) suitably convergent and at the same timemade V~~(r) negligible. The long-range part ofthe potential, which is now collected in V o(r), par-ticularly affects the low-frequency T, branch alongthe [110] direction. For example, for a choice ofo. = 0 (V o = 0), this branch is still unstable when theforce constants are summed to 19 shells. For u=0.17 the V,«(r) converges after about 20 shellsand the harmonic T, frequency at the zone bound-ary ignoring Ve„ is 0.54 Mev (the complete valueincluding Vo is 0.43 MeV). We chose o, =0.23

2646 H. TAOLE, H. R. GLYDE, AND ROGER TAYLOR 18

which made V,«negligible beyond 15 shells andevaluated Va(r) in reciprocal space.

method, we can evaluate D 8(q) exactly in spiteof the long-range character of V» (r).

III. DYNAMICS B. Self-consistent harmonic approximation

In the pseudopotential description of Li, wewrite the metallic energy as the sum of terms de-pending only upon the volume plus the term

~V (r, -r, ),g &g& II

/

which depends upon the positions r, of the ions.The dynamics of ionic motion at a given volumethen depends solely on V» (r, —r, ).

The lowest-order self-consistent phonon theo-ry" " is the self-consistent-harmonic (SCH)approximation in which the phonon frequenciesare again given by expressions (8) and (9). Now,however, the derivative of V«(r» ) in (9) is aver-aged over the relative vibrational amplitudes,u(ll') =u(l) -u(l'), of the ions about their latticepoints

( V„(l)V (l')V„) = [(2&)'IA I ]-'~')

A. Quasiharmonic approximation

~(q~) =g~ (q~)D ()(q)~8(q~),a8

where e(q&) is the phonon polarization vector and

(8)

D„()(q) = —Q(e ' '" ' —1}V (1) V(8l') V«(r„) (9)g

t

In the quasiharmonic approximation the frequencyof a phonon having wave vector q and branch ~ is

x)"d3u exp(--,'u A ' u)

xv (l)v&(l')V„.

The width of this Gaussian distribution is

A„B(fl ') = (u (ll')u() (ll' ))

(1 sk ((' ROE )) (qZ) (qZ)g X

(12)

is the dynamical matrix. " The derivative of V»(r„i) is evaluated at r» ——R»~, the separation be-tween the lattice points. Since 8, —R, varies with

7 as the lattice expands, the quasiharmonic har-monic frequencies are temperature dependent.

With V,, (r) written as the sum in (6), the D 8(q)has two parts

D 8(q)=D»8(q)+D 3(r). (10}

The first part D„()(q) is obtained by substitutingV,«(r) directly into (9). The second part is ob-tained by substituting Vu' (r) in (9) in the form

~ d3V'(r) = c*"&(Q)„(2»)'

which reduces DiO8(q) to

ID.s(q) =

~ F [(q+q).(q+7),&(lq+q()

The Fourier transform E(Q) needed in this ex-pression was evaluated directly from (5) and theintegrals involved reduce to exponential integralsand Dawson's integral. A sum over v of three re-ciprocal-lattice points was sufficient to obtainD o

() (q) with o. = 0.23 in (3). By writing D~»(q) asa sum of two parts following the Ewald-Fuchs

coth[-3'Pk (o(qA)]»)(@)

(13)

where S is Planck's constant, m is the mass,p = (kT) ', and N is the number of atoms in thecrystal. Since the (q)(qA. ) in (13) are the SCH fre-quencies, Eqs. (9), (12), and (13) must be evalu-ated interactively until self -consistent.

The TP & (q) part of D„()(q) can be evaluated straight-forwardly in the SCH approximation since for thispart V.«(r) replaces V,, (r» ) in (9). However, D„8(q}cannot be evaluated in the same way since the average(12) introduces correlations between the atomic mo-tions and the reduction of Do8(q) to the reciprocalspace expression (11)requires no correlation betweenthe atoms. Since V,«(r) continues out to 15 shellsand Vo (r) accounts for the remainder, we thoughtit would be a good approximation to ignore theaveraging for atoms separated more than 15 shellsand use the quasiharmonic value of Do()(q) for theseatoms [A„8(ll') «R'(ll')]. This turned out to bethe case for all branches except the T, [110]branch, which depends heavily on the long-rangepart of the potential. To improve this branch, weapproximated the outer shell averaging by an un-correlated Einstein model like Gaussian averagingof the same width in Do& (q). This improved theT, [110] branch but the frequencies of this branchstill showed unrealistic variation with wave vector.This point is discussed further in Sec. V.

18 ANHARMONIC PROPERTIES OF Li

8(d'((IZ)F((IX, (d)

[-&o'+ ((P((IX)+2~(Q)d(Q, ~)J'+ [2&v('ig}F(Q&, &0)J'

(14)which appears in the one-phonon part of the dyna-mic form factor (see Sec. IIID). The expressionsfor 4(tl&, (l)) and F((I&, co) are given in E(ls. (5) and

(6), Ref. 3 and two approximations were made intheir evaluation. Firstly, the cubic anharmoniccoefficient

s'&rr (vi( )sr()ar )(,)a8r((l")) (15)

was summed out'to five shells only. Tests by com-paring phonon frequencies obtained with three- andfive-shell summations showed this gave all fre-(Iuencies within 2% for q& 0.1(2&/a). Secondly,the self-consistent harmonic fre(luencies (l)((I~)used in 6 and 1 mere those given by the first partof D ()(g) in E(I. (10) only. Tests of the dependenceof 4 and F on (l)((I&) showed this could lead to 10%error in 6 and F or -2% in the final SCH+ cubicfrequencies.

D. Dynamic form factor

To complete this section, we note that the co-herent inelastic cross-section for neutrons scat-tering from crystals is proportional to the dyna-mic form factor, 8(Q, re). This 8(Q, &a) is usuallyexpanded in powers of scattering from single pho-nons, pairs of phonons,

8(Q, ~) =8,(Q, ~)+ 8.(Q, ~)+8,.(Q, ~)

=8~(Q, (())+8,(Q, e)+ ~ (16)

and so on. The term 8»Q, (l)) represents the in-terference contribution between the one- and two-phonon cases. For all phonons considered herethe 8~(Q, (l))+8,(Q, &l)) in E(I. (16) was calculated

C. SCH + cubic approximation

The SCH approximation includes all the evenanharmonic terms that would appear in a first-order perturbation evaluation of the anharmonicshift to the phonon frequencies. " This includesnotably the quartic anharmonic term. The leadingcorrection to the SCH approximation in perturba-tion theory is the cubic anharmonic term. Thisterm is also the leading term in a second-orderself-consistent theory. Here we simply add thecubic term as a perturbation to the SCH frequen-cies.

The phonon frequencies with the cubic term in-cluded are identified with the peak in the one-pho-non response function, "A(jk, v}, of form

and this was usually dominated by the one phononpart

8„(Q, (d) =[n(&0)+ lj~E(Q, (lx)~'A((I&, ~)n(Q- j),(I'I)

where &(Q, (I&) is the structure factor" and n(&())

is the Bose function.

IV. RESULTS

A. Low-temperature phonons

The representation of the Li ion-ion interactionused in the present calculations is shown in Fig. 1.The phonon energy dispersion curves for Li at j.loK calculated in the self-consistent harmonic ap-proximation with the cubic term added (SCH+ cubic)using this ion-ion interaction are shown in Fig. 2.A comparison of the calculation and observed pho-non energies shows good overall agreement withsome clear discrepancies.

In assessing the meaning of these discrepancies,it is interesting to compare first the quasiharmonii. c(QH) and SCH+ cubic phonon energies. In a pre-vious QH calculation using the same ion-ion inter-action as in Fig. 1, DRT noted significant dis-crepancies with experiment around the Q~ = Q(a/2v)= (1,0, 0) point and along the I.[qqOJ branch. Herein Fig. 2 we see that the discrepancy around Q'=(1,0, 0) has effectively been removed. This ispartly because the SCH+ cubic energy is lowerthan the QH value (see Table I) and partly becausethe additional single point due to Beg and Nielson"(see Table I) lies higher than the original valuesof Smith et a/. '4 However, the discrepancy along[qq0] L at higher q remains. We shall see that theSCH+ cubic theory does not predict a large enoughdownward shift due to anharmonicity with temper-ature for this branch (for example, the discrepancyalong I, [qq0j is greater at 293 K). Thus some ofthe discrepancy here might be due to additionalanharmonic contributions. However, these areunlikely to reduce the phonon energy by more thanI/p-Y/p on the basis of the temperature shifts dis-cussed below. Most of the discrepancy thu' resultsfrom the representation of the ion-ion interaction.Since this ion-ion interaction simulates the non-linear electron screening between a pair of ions toall orders in perturbation theory, the discrepancyprobably reflects the role of three- and four-ioninteractions in Li which are not represented in thepresent pair ion interaction.

B. Temperature shifts

The SCH+ cubic phonon energies for Li at 293 Kare compared with the observed values of Beg and

S. H. TAOLK, H. R. GLYDE, AND ROGER TAYLOR

[F.,o,oj

I

LdCtLL IIO 'K

a =5.49

0,J—=.— I

0.8 I.O O.e O.~ 0REDUCED WAVE VECTOR

0.2

FIG. 2. Phonon energy dispersion curves for Li. The solid lines are the SCH cubic calculations at 110 K, theopentr iangles (circles) are the longitudinal-(transverse) phonon energies observed by Smith et al. (Ref. 14 and unpublisheddata) at 98 K, and the sobd triangles and circles are the corresponding values observed by Beg and Nielsen (Ref. 15) at110 K.

Nielsen in. Fig. 3. There we see that the discrep-ancy along [qq0]L, noted at 110 K has increasedand, in addition„ the calculated phonon energiesalong [qqO]T, and along [qqq]I. clearly lie abovethe observed values. These discrepancies showthat the theory does not predict a large enoughdownward shift in phonon energy with temperature.

The [qqO] f', branch in Fig. 3 is interesting sincethe intrinsic anharmonic shifts in the phonon ener-gy from the QH to SCH, and on adding the cubicterm, are both negative in Na and K. Here theQH-SCH shift is negative but very small (see Table I).Also the cubic shift is very small (see Table I andFig. 4). Generally the cubic shifts and the totalshifts are smaller in Li as a percentage of thephonon energy than in K and Na. For example, forthe longitudinal phonon at Q* = (0.5, 0.5, 0) the per-centage shift in energy between 110 and 424 K(0.94 of the melting point) is 5.5% (calculated 3.9%)in Li. In K, this phonon ha. s an energy shift of 8%(ca,lculated 9%) between 90 and 311 K (0.93 of themelting point). ' The corresponding shifts for theT,, phonon at Q* = (0.5, 0.5, 0) between the abovetemperatures are 10.2% (calculated 3.4%%uo) in I,iand 9% (calculated 8.8%) in K. The percentageshift in the calculated QH frequency with tempera-ture is also smaller in I i than in K. The magni-tudes of the temperature shifts in I i are shown inFig. 5.

As a. sumxnary statement, the calculated shiftsw&th temperature are too small Thxs probably

means ultimately that the representation of theion-ion interaction in the replusive region is toosoft given the better agreement of the SCH+ cubictheory with experiment in K and Na.

C. Phonon groups

The dynamic form factor, S(Q, &u), calculatedwith SCH+ cubic theory for the longitudinal polari-zation at Q+ = (2 —$, 2 —$, 2 —g) with g =0.677 iscompared with the corresponding observed pho-non groups in Figs. 6 and V. The calculated S(Q, a&) includes the one-phonon scattering, the two-phonoo scattering, and the interference betweenthese two [ see Eq. (16)]. The S(Q, ao) is also con-voluted with a Gaussian function of full width athalf-maximum equal to the instrument resolutionwidth (-1.3 meV) shown by the horizontal bar withthe observed groups to make the comparison withexperiment more direct. Thi.s comparison showsthat although the calculated phonon group has ap-proximately the correct shape, its overall widthis too small (i.e., =2.5 meV at 424 K comparedwith =4.0 me7, both including the resolutionwidth). Also, the calculated S(Q, a&) does not re-produce the large intensity on the low-energy sideof the observed phonon group. This would have tocome from the two phonon or interference contri-bution to S(Q, &u) in the theory. Large intensitieson the low-frequency sides of some phonons groupshave been observed in K,"and these could be re-

18 ANHARMONIC PROPERTIES OF I i

TABLE I. Selected phonon energies (meV) and intrinsic phonon group widths 2I' (meV)(phonon lifetime v'= 1 ).

Phonon

(1,0, O)r.

(0.5, 0.5, O)L

(0.5, 0.5, 0) 2",

(0.5, 0.5, O)r,

2I

2I

Model

QHSCHSCH+ cubicObs.

Gale.Obs.

QHSCHSCH+ cubicObs.

Calc.Obs.

QHSCHSCH+ cubicObs.

Calc.Obs.

QHSCHSCH+ cubicObs.

Calc.Obs.

110 K

40.039.838.637.5 + 1.00.3

41.242.341.0

0.6

24.124.023.5

0.3

7.3(5.6)(5.1)

0.3

Temperature293 K

39.338.936.935.8 + 1.00.3

. 1.94 +0.6

40.241.740.636.5 ~1.00.80.0 + 0.3

23.923.923.1

0.82.20 + 0.5

7.3(7.6)(6.6)8.6 + 0.6

1.6

38.538.036.033.1 + 3 .0

0.73.85 + 0.6

39.040.839.4

1.03.12 + 0.45

23.823.622.7

1.82.77 +0.64

7.3(8.6)(7.5)

(0.677, 0.677, 0.677)L

2I"

(0.5, 0.5, 0.5)L

QHSCHSCH+ cubicObs.

Calc.Obs.

QHSCHSCH+ cubicObs.

Calc.Obs.

17.217.516.816.4 +0.2

0.3

32.833.232.4

0.3

16.917.816.315.1 +0.5

1 23.23 + 0.22

32.232.831.8

0.6

16.718.116.013.0 + 0 ~ 5

1,73.69&0.5

31 232 230.8

Observed values are those of Beg and Nielsen. 2l for (0.435, 0.435, 0)L phono+.

produced by including the interference terms.To investigate this intensity further, we show

the interference contribution for this phonon atthree different wave vectors Q which have theequivalent reduced wave vector $ in Fig. 8. Therewe see the interference contribution is large butdoes not contribute much scattering intensity atlow energy. Apparently the observed intensity atlow energy arises from scattering from three ormore phonons.

In Figs. 9 and 10 we compare the calculatedS (Q, &u) with the observed phonon groups of Begand Nielsen for some second transverse phonons

along the [110]direction. Again the calculatedS (Q, &u) is convoluted with a Gaussian of width givenby the experimental resolution width shown by thehorizontal bars under the observed groups. As inFig. 8 the calculated S(Q, &u) reproduces the observed group shapes well but the calculated line-width is too small at high temperature. %e findthat the calculated width of the [$(0]T, phonon with)= 0.079 is given entirely by the resolution widthas in the observed case shown and we have notshown the calculation of this phonon group.

From Fig. 5 and the phonon groups shown inFigs. 6-10 we see that the calculated phonon lite-

S. H. TAOLK, H. R. GLYBK, AND ROGER TAYLOR 18

I I

[q o o]I I

l~ qq]

g) 30E

K 20

10

0.2 I.O 0.8 0.6 0.4 0.2%AYE VE'CTOR q (2w/g )

0 0.2

FIG. 3. Phonon energy dispersion curves for Li at 293 K. The solid lines are the SCII+cubic calculations and theopen (solid) circles are the longitudinal (transverse) energies observed by Beg and Nielsen.

[qqo]

@0

50

FIG. 4. elf-consistentharmonic (SCH) and the self-consistent harmonic with thecubic term included (SCH+cubic)dispersion curves along the[110]direction.

i0

0.1 0.2 0.5 0.4 0.5NAV E VECTOR q(2w/a)

ANHARMONIC PROPERTIES OF Li 2651

4-[(((]T

2I' 2-

0

"26QJ

—4

4

~ 21'2Z

0

-4 [(((]L

I

[((o jr& ~

II

Q

- [((o]T,

n

0.5

Q 293-----& 385o424---

- [((o)T2 g [(oo]L

&

0- [((o]L —[(oo]L

gI.O 0 0.5 0

REDUCED WAVE VECTOR

pI.O

comparison of the SCH+ cubic lifetimes with amolecular dynamics determination" using thesame effective ion-ion interaction showed thata factor of two discrepancy could be attributedto the SCH+ cubic theory rather than the poten-tial. In the SCH+ cubic theory, a single phononcan decay to two phonons via the coupling throughthe cubic anharmonic term. Clearly, higher orderprocesses are important. The shortest-lived pho-non compared to the frequency is the longitudinal

Q*=((, g, g) with )=0.68 phonon which vibratesapproximately seven times before it decays.

FIG. 5. Phonon energy shifts 54co between 110 K andthe temperatures indicated, and the intrinsic full widthat half-maximum 2I' of the phonon groups at the temper-atures indicated. The lines are the SCH+ cubic calcula-tions and the points are the observed values of Beg andNielsen.

times are often three or four times the observedlifetimes. A discrepancy of a factor of two ischaracteristic of calculations made using theSCH+ cubic theory, for example in K. There a

V. DISCUSSION

The aim of this study has been to examine theanharmonic properties of lithium, both for its in-trinsic interest and as a means of testing the ef-fective ion-ion interaction. Perhaps the most sur-prising single result of Sec. IV is that lithium isnot very anharmonic. To put the results in per-spective we compare in Table II the percentageanharmonic frequency shift of three phonons with

[f E'i]Lf= 0.677 550—

[i C f]L('= 0.677

400-500

500-(h

Z200-

K

K+ ioo-CO

K

3 500—CO

250

400—

FIG. 6. Calculated S(Q, )= ~$ (g, ) + ~2 (g ) + ~)g (g &)for the longitudinal phonon atQ + = Q a/2x = (1.323, 1.323, 1.323).

250 350—

200

300—

l4 l6 l8 20 l2

ENERGY (meV)

l4 l6 IS 20

2652 S. H. TAOLE, H. R. GLYDE, AXD ROGER TAYLOR

300

the usual expansion of the interionic potential P inpowers of the atomic displacements &. Schemati-cally, this is

Q= g + —'P u2+ —Q u'+ —

Q u'+ (18)~ 200OC3

100OCL'

J

~ 300

200

A traditional"'" measure of the importance ofthe anharmonic terms is the ratio of the BMS vi-brational amplitude to the interionic spacing&=(u') '~'/A. Table III shows estimates of 6 atthe melting point of Li, Na, and K. The (u') isestimated from

1008 12 16 20 10 14 18 22

ENERGY (rneV)

FIG. 7. Phonon groups observed by Beg and Nielsen(Ref. 16) for the same phonon shown in Fig. 6, Q*=(2-h, 2-k, 2-k).

(u') = (lf /0)' (9kT/Mesa), (19)

which is the harmonic result in the high-tempera-ture limit using a Debye approximation to the fre-Iluency spectrum. (An accurate value of 6 obtainedusing the exact frequency spectrum calculated

temperature in Li with those in Na and K whichshow that Li is the least anharmonic of the three.We might expect lithium to be the most anharmonicof the alkali metals, and possibly of all metals,since its mass is small. However, if the inter-ionic forces are strong these forces can counter-act the effects of a light mass by providing astrong restoring force as an ion vibrates awayfrom its lattice point. This will keep the vibra-tional amplitudes small. The large phonon fre-quencies in Li do indeed suggest strong restoringforces. For example, the maximum phonon fre-quency in Li is 2.5 times that in Na and 4.0 timesthat in K, wbile on the basis of the mass differencealone (arccm '~') we expect ratios of only 1.8 and2.4, respectively.

To make these points more precise, we consider

I50- IIOK

IOO-

200—

I50-z

IrIOO

I—

03

250

[f 4 o]T2$ = 0.3I6200-

200-

150-

I 00—

200—

150—

E(.' C o] T2

(.' ~ 0.5

IIO K

293 K

385 K

4

KI—303

R((jL

I I I I I I I I

Pq

,'

', /=2. 677

3

CA 200-

l50-

300 "

250

250— 350—l4 16 IS l2 l4 l6 IS IO l2 l4 l6 IS

ENERGY (meV)

FIG. 8. Calculated 8& (Q, ~) + 82(Q, w) (dashed line) andthe S&(Q, &o)+S&(Q, ~) (solid line) for three phonons hav-ing identical reduced wave vector $ = 0.677 and Q~= (0.677, 0.677, 0.677), Q*= (1.323, 1.323, 1.323) andQ~= (1.677, 1.677, 1.677). The difference between thedashed and solid lines, is the interference contribution.

424 K

200— 300

1 I

l6I I

I 8 20

250I I I I

20 22 24 26ENERGY (mV)

FIG. 9. As Fig. 6 for the phonons indicated.

ANHARMONIC PH, OPKRTIKS OF Li

f5Ã)T. &3 (&)9

300- K

g*0.3&6(c)

gn 0.5110K

TABLE III. Estimates of the expansion parameter ~=(+2)t/2/R fusing Eq. (19)) at the melting temperature T„,.

200-

100

500

-160

- 120

'- 80LiNaK

M T~(amu) (K)

453371337

3.544.255.34

430160100

12.714.213.3

(~) (K) (&o)

~ 300

cu 100xr

a 500

~ 300

600

400

200

0 4 8 12 16 20 16 20 24 28ENERGY {meV)

FIG. 10. As Fig. 7 for the phonons indicated.

TABLE II. Percentage anharmonic shift &~/~ in pho-non frequency calculated using the SCH+ cubic theory forLi between 90 and 424 K (T~=451 K), for Na between 5and 361 K (T~= 371 K), and for K between 5 and 311 K{T = 335 K). The observed values in the brackets are:Li IBeg and Nielsen (Ref. 15)l and K IBuyers and Cowley(Ref. 33)l.

Phonon Li KNa

(1.0, 0, 0)

(o.5, o.5, 0)1.

(0.5, 0.5, o)T2

-7.2 %(-12+ 6%)

4 0%(-6 +3%)03.5%

(-11+3Wo)

-12.8%

-10.0%

-10.8 %%d

-17.5%(-14%)-11.5 $

(-1o%)

(-14%)

from the SCH+ cubic frequencies gives" 5=17%for Na at T =361 K). Clearly the larger value of

e~ for Li has counteracted the smaller mass togive comparable values of & in Li, Na, and K. Onthis basis we expect Li to have comparable ormarginally smaller anharmonic shifts as noted inTable II. The larger e~ for Li reflects the sub-stantially stronger ion-ion interaction (see Fig. 1).

As noted in Sec. I, the stronger ion-ion inter-action in Li can be readily understood in terms ofthe ion cores. The ions interact via the conductionelectrons through an ion-electron-ion indirect in-teraction. The lithium ion has only two core elec-trons, both in an s state. This means that a con-

duction electron in a state having p-wave charactersees directly the strong Coulomb potential of thenucleus unshielded by the core electrons. As aresult the indirect ion-electron-ion interactionwill be strong in Li, much stronger than in Na orK where the larger ions shield the Coulomb poten-tial more effectively. It is interesting that a smal-ler mass (fewer nucleons) is balanced by a stron-ger interaction (fewer core electrons) in such away that the Lindemann" ratio, 5, is roughly heldconstant.

To compare the relative sizes of the cubic andquartic anharmonic terms, we turn to the moreprecise expansion parameter for (18) pointed out

by Horner. " He noted that the expansion para-meter should include a measure of the size of thederivatives of p in (18) as well as noting that eachderivative brings a factor of 1/R. The size ofthese derivatives is fixed largely by the steepnessof the repulsive part of g which we could repre-sent crudely by

y(r) = e.(c/r)",where the index n measures the steepness of P(r)The derivatives of p(r) are

A more appropriate expansion parameter is then)t= nb =n(n') '/'/R, which is approximately theratio of the kinetic to the potential energy. If n

and hence & is small, the Taylor-series expansionconverges rapidly.

In the rare-gas crystals, where n is large (n= 12)we find a large shift in frequency 4, due to thequartic anharmonic term. This shift is approxi-mately given by the difference in the SCH and QH

frequencies. This 44 is about twice the shift 6, dueto the cubic anharmonic term, the difference be-tween the SCH and SCH+ cubic frequencies. In theAlkali halides, the quartic (E,) and cubic (A, ) an-harmonic frequency shifts are comparable in sizesuggesting that ~ is smaller there than in the rare-gas crystals. From Table I we see that the dif-ference in the SCH and QH frequencies is verysmall in Li, while the cubic anharmonic shift 6,is relatively much larger. This means that in Lithe quartic term is small compared to the cubic

S. H. TAOLE, H. R. GLYOE, AND ROGER TAYLOR 18

IO—

I I

[q, q, o] T~

SCH-G

O. I 0.2 0.3 0.4WAVE VECTOR q(2m'/a)

0.5

term so that the core of the interionic potential isquite soft (n small). A small b,, is also requiredto get agreement with experiment since 6, is gen-erally positive and in general, lower frequenciesare required to improve agreement at high tern-perature (see Fig. 3). The A, is larger in Na and

K, suggesting that the core of the ion-ion interac-tion is steeper in these larger ion metals. Thesofter core in Li is consistent with weak shieldingof the Coulomb interaction which leaves the netion-ion interaction closer to 1lr.

Finally, we note that the agreement of the calcu-lated frequencies and anharmonic shifts with ex-perirnent in Li is not as good as in Na or K. Low-er frequencies at high T (see Fig. 3) or larger

FIG. 11. Transverse Tg [110]phonon energy dispersioncurve in (a) the quasiharmonic approximation (QH), (b)the self-consistent harmonic approximation (SCH) withthe force constants for the fifteenth shell and beyond ap-proximated by the nonaveraged QH force constants(SCH-N), and (c) the SCH approximation with the fifteenthshell force constant and beyond obtained by averagingwith an uncorrelated spherical Gaussian function(SCH-G).

negative shifts (see Table II) are required to im-prove the agreement with experiment. Since 4, isalways negative, a generally larger cubic term iscalled for in Li. This is not the case in K and Nawhere agreement is much better. This error inLi probably arises from the breakdown of thepseudopotential method limited to second order inthe ion-e&ectron interaction when this interactionis as strong as in the Li case. As discussed inSec. II, the DRT ion-ion interaction used here isconstructed to simulate all the nonlinear higher-order contributions between pairs of ions. Thusthe present discrepancy probably reflects the pre-sence of three- or four-body forces. A three-bodyforce leads to an additional class of cubic termsin A, which could substantially increase its size.

To conclude, we note that the long-range natureof the Li ion-ion interaction shown in Fig. 1 madethe calculations difficult. This affected particu-larly the [qq0]T, branch, and the frequencies ofthis branch did not stabilize even for force con-stant summations out to 30 shells of neighbors.The QH frequencies could be obtained exactly byexpressing Vn (r) as the sum of a V,ff (r), whichbecame negligible after 15 shells of neighbors,and an asymptotic part which was evaluated in re-ciprocal space. .

In the SCH approximation, averaging of the forceconstants involving the asymptotic part turned outto be important for the [qq0] T, branch and couldnot be done exactly in reciprocal space. Figure 11shows the dispersion curves of this branch withno averaging of the asymptotic part and for an in-dependent Einstein model averaging using a Gaus-sian distribution of appropriate width. This branchis clearly sensitive to the averaging of the distantneighbor force constants. It is also likely that thedip in the [qq0] & branch in the region q=0.7 issensitive to the distant neighbor force constants.In Fig. 3 this [qOOJ I. branch does fall as far belowthe transverse branch as suggested by experimentand this may be due to our approximate treatmentof the distant neighbor force constants.

Note addedin proof: We thank Dr. H. G. Smithfor permission to quote the previously unpublisheddata shown in Fig. 2.

~T. Toya, J. Res. Inst. Catal. Hokkaido Univ. 6, 183(1958).

By "first principles" we mean calculations based on po-tentials or pseudopotentials whose parameters have beer.determined from fundamental considerations such asatomic spectra or charge densities. We do not includein this category force-constant models obtained by fit-ting to metallic properties.

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J.J. Paciga and D. L. Williams, Can. J. Phys. 49, 3227(1971).

7D. L. Bandies and M. Springford, J. Phys. F 6, 1827(1976).

ANHARMONIC PROPERTIES OF Li 2655

V. Bortolani and G. Pizzichini, Phys. Rev. Lett. 22, 840(1969).

ST. Schneider and E. Stoll, in ComPutational Solid StatePhysics, edited by F. Herman, N. W. Dalton, and T. R.Koehler (Plenum, New York, 1972), p. 99.D. L. Price, K. 8. Singwi, and M. P. Tosi, Phys. Rev.B 8, 2983 (1970).P. L. Srivastava, N. B.Mitra, and N. Mishra, J. Phys.F 3, 1388 (1973).

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