PHYSICAL REVIE% 8 VOLUME 22, NUMBER 12

Phonons in solid argon

15 DECEMBER 1980

H. R. Glyde and M. G. SmoesDepartment ofPhysics, University ofOttawa, Ottawa, Canada K1N 6N5

(Received 3 April 1980)

The phonon energies and the dynamic form factor, S(Q,co), in solid "Ar at temperatures from 10 K to themelting point are calculated using the self-consistent-phonon (SCP) theory with the pair interaction representedby the recent realistic potential developed by Aziz and Chen. The temperature dependence of the phononenergies agree well with the observed values of Fujii et al. This is a substantial improvement over the previouslyavailable calculations obtained using anharmonic perturbation theory. The temperature broadening of S(Q,co) alsoagrees well with the observed phonon groups for low-energy phonons where data are available and with themolecular dynamics (MD) calculations of Hansen and K1ein. The S(Q,co) for high-energy phonons near melting arefound to be extremely broad, broader than in solid helium, and S(Q,co) becomes very sensitive to intermediatephonon energies used in its calculation, making the present SCP group shapes not reliable for high-energy phononsnear melting. Further experimental data or MD simulation of $(Q,co) near melting for high-energy phonons wouldbe most useful as a test of anharmonic theory, for example, of whether an explicit description of short-rangecorrelations in the atomic motion is important.

I. INTRODUCTION

The dynamical properties of solid argon have along and rich history of study. This is becausethe potential of the solid can be well representedby a sum of pair interactions between argon atomswith the addition of the Axilrod-Teller-Muto (ATM)triple-dipole interaction. Comparison of theorywith experiment then represents a reliable test ofour understanding of the dynamics.

Prior to the pioneering measurements of phononenergies and lifetimes using inelastic neutron scat-tering by Daniels et a/. ' in solid krypton and by Eg-ger et a/. , Batchelder et ak. , and Fujii et al. insolid 36Ar, the early work on solid argon focusedmainly on the elastic and thermodynamic proper-ties. This work, both theory and properties, iselegantly reviewed by Dobbs and Jones, Pollack,Boato, Guggenheim, '0 and Hor ton. ' The calcula-tions of Horton and Leech' and the review by Hor-ton' stand out as the early detailed discussion ofthe phonon frequency dispersion curves. In allthese studies the pair interaction was representedby a model pair potential, generally of the Mie andLennard- Jones type.

The first. detailed calculation of the phonon fre-quencies and lifetimes in solid argon over a widetemperature range was by Bohlin and Hogberg'who used quasiharmonic theory with a perturbationtreatment of the quartic and cubic anharmonicterms (QHPT). Niklasson' also used the QHPTto calculate phonon properties in argon and both heand Bohlin and Hogberg employed the Lennard-Jones potential.

The first calculations in solid argon using real-istic pair potentials appeared in a series of papersby Bobetic and Barker. (BB),ts Barker ef al. ,

's and

Klein eI; a/. ' They employed a, potential developedby Bobetic and Barker to represent the pair argoninteraction and the ATM potential to represent thethree-body interactions. For calculating phononfrequencies this potential is indistinguishablefrom the more widely known and accurate Barker-Fisher-Watts' potential although some differencecan be seen in the expansivity. With this potentialand using the QHPT Bobetic and Barker's and Bar-ker et al. ' obtained elastic constants and phononfrequencies in good agreement with the observedvalues near 0 K. However, Klein et al. found thetemperature dependence of these QHPT frequenciesagreed poorly with the temperature dependence ob-served by Batchelder et al. This was not improv-ed when the QHPT was replaced by the self-consis-ten t harmonic approximation.

Goldman et al. calculated the phonon dispersioncurves at 0 K using a more complete self-consist-ent theory and the Lennard-Jones potential and alsofound good agreement with the observed phonon en-ergies at 0 K available at that time. Using thepowerful molecular dynamics (MD) method, Hansenand Klein ' evaluated directly the dynamic formfactor S(Q, &o} in a "Lennard-Jones solid. " Theyfound significant differences between their "exact"S(Q, o&} and that calculated using the self-consist-

' ent-phonon (SCP) theory for selected phonons athigh temperature. An exhaustive review of all thedynamic and thermodynamic properties of all therare-gas crystals has been made by Klein andKoe hler. '8

In this context we have evaluated the phonon fre-quencies and S(Q, o&) in solid s'Ar at all tempera-tures using the self-consistent-phonon theory andthe recent realistic potential developed by Aziz andChen. The Aziz-Chen potential is shown in Fig. 1.

22 6391 1980 The American Physical Society

6392 H. R. GLYDE AND M. G. SMOES 22

rI

II

li

liI!Il

—50I—I

Ii

—100—

—1505.0 4.0

l

5.0 6.0

s(q ~) fg( I Q kQ

N

x (ezQ. uz(t & -zu. uzi (0))

Here Q is the wave vector of the excitation createdby the neutron and I'v is the energy transferredfrom the neutron to the solid. The S(Q, e) is thenexpanded in powers of the displacements u& ofatoms l from their lattice points R, and, ignoringthe elastic Bragg scattering component,

I

S(Q, ~)=Sz(Q, ~)+Sz2(Q, ~)+S2(Q, ~)+ " . (2)

The Sz(Q, ar), proportional to ([Q u] ), describes thescattering in which the excitation is a single phononhaving wave vector zl =Q. The S2(Q, &o), proportion-al to ((Q u)4), describes the scattering process inwhich two phonons are created (or destroyed) and

Sz2(Q, &o) represents the interference contributionbe'.ween the scattering from one- and two-phononprocesses. The higher-order processes we hopeshould be small or at least uniform, uninterestingfunctions of v.

FIG. l. Argon pair potential in the well region:Aziz-Chen tHef, 22), - ~ ~ Barker-Fisher-Watts (Ref. 19),—- Parson-Siska-Lee (Ref. 35), and -'- Lennard-Jones.

The results are compared with the more recent andaccurate neutron scattering data of Fujii et al. andthe MD studies of Hansen and Klein. The aim is totest whether the temperature dependence of thephonon frequencies and the temperature broadeningof S(Q, &o) can be correctly predicted. This willhelp answer (1) whether at high temperature theSCP theory is a substantial improvement over theQHPT and (2} whether the SCP theory itself is re-liable at high temperatures. Since the role of theATM forces has been clearly established by Bar-ker et al. , we do not include them in the calcula-tion of the phonon frequencies here. Their contri-bution to the frequencies is discussed in Sec. III.

The technical details of the present calculationsare outlined in Sec. II and the low-temperature re-sults are presented in Sec. III. The temperaturedependence of the phonon frequencies and the dy-namic form factors are presented in Secs. IV andV. The results are discussed in Sec. VI.

H. THE SELF-CONSISTENT THEORY

A. Dynamic form factor

The cross section for the coherent scattering ofneutrons from a solid is proportional to the dynam-ic form factor,

B. Phonon dynamics

The one-phonon Sz(Q, &o) is 3

S1(Q~ zzz) = [n(zzz) + l]zf'(Q) Z (Q'&,~)'

Xf,',A(zf y, &o)a(Q —zf }.

Here n(&o) is the Bose function, d(Q) is the Debye-Waller factor, f,z, =(h/2m&v, „), t, z, and e,„are thepolarization and frequency of the phonon havingbranch X, and nz is the atomic mass.

In the self-consistent harmonic (SCH) approxima-tion, the ~,~ are given by the usual harmonic ex-pression, 26

(o,'z, eq, ——(e—"'"«—I)(V(0) V(l)v(roz))e, z,qX qX

in which the force constants (V(0) V(l)v(roz)) areevaluated as derivatives of the interatomic poten-tial v(ro, ) averaged over the vibrational distribu-tion of the atoms about their lattice points. The vi-brational distribution is described by a Gaussianfunction of rms amplitude (u, ) consistent with theSCH frequencies,

(u'z) =Q f,'z [2n(zzz, z,) + I].In the SCH theory in which the cubic anharmonicterm is added as a perturbation2 (SCH+C) theone-phonon response function in Sz(Q, &o) takes theusual anharmonic form,

PHONONS IN SOLID ARGON

8&v, iF(qX, pd)

[ (-o'+ pd,',+2pd„a(qz, (o)p+[2(o„F(qz, (o)]' '

The 4 and 1" are the phonon frequency shift and in-verse lifetime due to the cubic anharmonic term

1t 2

[Ioo ]) 8OZILI

LtjCL4|L

{iso]I l I I

{iii ]

Z20O

0 0.2 0.4 0.6 0.8 I.O 0.8 0.6 0.4 0.2 0 0.1 0.2 0,5 0,4 0.5WAVE VECTOR Q(0/2 Ii)

F(q~, ~}=-(II ) ~ II'(q~ I 2) I'&(I 2 ~)102

(6) FIG. 2. Phonon energies calculated in the SCH+ C ap-proximation (solid line) and observed by Fujii et al.(Ref. 6) (points).

where1 1

M(1, 2, ~) =(n, +n, +1)(d + (O~ + (d2 (d —((d~ + (dp)

-(nt —n()1 1

(0 + G02 —601 (d —602 + (d1

J(1,2, pd)

= (nq +n2 + I)s~[&(pd —((u, + (u,)) —6 ((u+ (u, + (u, )]

+ (n 2 n f )r[6((0 + 47$ (0 $ ) '5 ((0 (0$ + (0f )]where v, = pd«„& and n~ ——n(or&) and so on. In theseexpressions V(qX, 1,2} is essentially the Fouriertransform of (V(0) V(0) V(l)v(p»)), the third de-rivative of v(rp, ) averaged over the same vibration-al distribution as appearing in (4). If 4(qX, ur) andF(qX, v) depend little on frequency, then A(qX, &o) isa simple Lorentzian function peaking at v =co,„+ 2a&,ih(qX, pd) with a full width at half maximumFWHM of W = 2F(qX, &o). The position of this peakis usually identified as the SCH+ C phonon fre-quency.

For harmoniclike phonons,

S2(Q, &o}= [n(ru) + 1Jd'(Q)

X 1 '112

x~(1, 2, ~)~(Q —(q, +q, )) . (6)

Here J'(1, 2, pd) given in (7) is usually referred to asthe two-phonon density of states. Since in S&(Q, &o)

there is a sum over all phonons having wave vectorq~ [with qp fixed by 4(Q —(q, +qt))] we expectS2(Q, &d) to be a reasonably uniform function of &o.

The interference term, S,p(Q, u&}, has been dis-cussed extensively ' and we do not discuss ithere.

SCH + C QHPTAC BB

potential potential Obs.

QH QHAC I J

potential potential

ably confident of the dynamics, we may comparethe calculated phonon frequencies and elastic con-stants with experiment as a test of the interatomicpotential. In Fig. 2 we compare the phonon fre-quencies calculated using the SCH + C approxima™tion and the Aziz-Chen potential with the observedvalues of Fujii et a/. There we see the overallagreement is excellent, especially for the trans-verse branches. The SCH+ C longitudinal frequen-cies do, however, lie definitely above the observedvalues for wave vectors at the Brillouin-zone edge.The SCH + C longitudinal frequencies lie below theobserved values at small wave vector.

Barker et al. ' and Klein et aL. ' have calculatedthe phonon frequencies at 0 K and higher tempera-tures, respectively, using quasiharmonic pertur-bation theory (QHPT). They described the pair po-tential using the Bobetic-Barker form (a slightlyextended Barker-Pompe potential) and included thethree-body, triple-dipole Axilrod- Teller-Muto(ATM) forces. Their transverse phonon frequen-cies at 0 and 4 K are indistinguishable from thepresent results and experiment. This is demon-strated in Table I where some selected transversephonon energies are compared. Their longitudinalfrequencies lie above the present SCH+ C results,due to the addition of the three-body forces, andtherefore agree better with experiment at low qbut less well at high q.

F

TABLE I. The phonon energies (meV) for selectedtransverse phonons along the [t'00] direction in ppAr at&=10 K (AC means Aziz-Chen potential). The QHPT BBare the values calculated by Klein et al. using quasihar-monic perturbation theory and the Bobetic-Barker poten-tial at 7=4 K. The observed values are from Fujii et al.(aef. 6).

HI. PHONONS AT LOW TEMPERATURE

In PPAr at low temperature (T ~ 10 K), where an-harmonic effects are small and we may be reason-

0.4 3.470.7 5.241,0 5.90

3.475.255.91

3.46 + 0.015.25 + 0.015.94 + 0.03

3.355.075.69

3.294.975.58

6394 H. R. GLYDE AND M. G. SMOES 22

TABLE II. Elastic constants c]g of Ar at T =10 K in 10 dynes/cm, & = (c44- ci2)/&f2.

Ci2 &44

Theory

PresentBarker et a$. (T = 0 K).~

(a) with ATM forces(b) without ATM forces

395

416387

205

232200

220

228228

190

186187

0.07

-0.010.014

Experiment

NeutronFujii et al.(a) Model 1(b) Sound speedBatchelder et al.(x=4K)

Dorner and Eggerd(T=4K)

425+ 5421+ 8

240+ 5227+ 9

224+ 1217+ 6

185+ 10194+17

411 190 210 221

367 + 14 174+ 17 234+ 11 193+ 31

-0.07 + 0.02-0.04 + 0.05

0.11

0.34 + 0.12

~ Reference 16. I

Force-constant model and sound speed obtained from fits to the same neutron data, Ref. 6.Reference 5.

d Reference 4.

The impact of including the three-body Axilrod-Teller-Muto forces can be seen clearly from theelastic constants listed in Table II. If we comparethe calculations of Barker et al. with and withoutthe ATM forces, we see the transverse branchescontrolled by c44 and cii-ci2 are unaffected by in-cluding the ATM forces while cii and ci2 are in-creased by -8/q and -14/q, respectively This.means that while transverse phonons are unaffect-ed by the ATM forces, the longitudinal frequenciesare increased by -4%. At high q this increase isreduced to 1-2% (M. L. Klein, private communi-cation}. The phonon frequencies calculated byBarker et al. and Klein et al. are essentially un-changed if the Bobetic-Barker pair potential isreplaced by the Barker-Fisher-Watts pair poten-tial. ' The present elastic constants shown inTable II were calculated from the slope of theSCH+C phonon dispersion curves and have an es-timated uncertainty of e 5 (10~ dynes/cm ).

From this we may conclude (a) that the phononfrequencies calculated using any of the modernrealistic pair potentials and either quasiharmonicperturbation theory or self-consistent theory areessentially the same in Ar at low temperature, (b)that three-body forces increase the longitudinalfrequencies by -4/0 and are needed to get goodagreement with experiment at low q (see Table II),and (c) there remains a, small but definite dis-agreement at high q, with the calculated longitud-inal frequencies lying 1-3% above the observedvalues. Since, as we shall see in Sec. V, the long-itudinal groups at high q are already quite broad at

T =10K, it is not clear whether the disagreementarises from problems in establishing the one-pho-non frequency accurately or whether there remainssome physics in this disagreement. Otherwise,agreement with experiment verifies the pair poten-tials and suggests that the QHPT and SCH+C the-ories predict the frequencies equally well at T &10K.

IV. TEMPERATURE DEPENDENCE OF THEPHONON ENERGIES

The SCH+ C phonon energy is defined in thissection, as in the previous section, as the mid-point of the peak in S(Q, &o) of (2) at one-half thepeak height. The SCH frequencies ai, z are used inall the response functions that enter S(Q, &o). Forthe lower-energy phonons considered in this sec-tion this definition coincides with the definition asthe position of the maximum in S,(Q, &o}. However,for some of the high-energy phonons discussed inthe following section having a broad and somewhatasymmetric S(Q, ~), the midpoint at one-half thepeak height can differ significantly from the posi-tion' of the maximum in S&(Q, &o). In addition, thetwo-phonon contribution has a significant energydependence at higher energy making it difficult toseparate the one-phonon component from the totalS(Q, uy) if it were not calculated separately. In this

case only adore ct comparison of the total calculatedS(Q, ~}with the observed scattering intensity ismeaningful. To discuss phonon energies here werestrict ourselves, as did the experiments of Fujii

22 PHONONS IN SOLID ARGON

g5.4-

& 3.0-0O

2.6

4.0-

-.SCH

3.6—

QH'. ISCH+C

32-

[IOO]T i [IOO) T5 =0.4 h=O.3

I I I I 28 I I I I

20 40 60 80 20 40 60 80TEMPERATURE (K)

6.0-(D

E

&- 5.6-(3CCLIJZ.'LLj

5.2—

OZ0CL 4.8

4.0-

3.6-

h = 0.5I I I I 28 I I I I

20 40 60 80 20 40 60 80TEMPERATURE ( K)

FIG. 3. Temperature dependence of the phonon ener-gies. The solid and open points are from Fujii et al.(Ref. 6) and Batchelder et al. (Ref. 5), respectively.The crosses are the QHPT calculations of Klein et al.(Ref. 17) using the Bobetic-Barker (Ref. 15) pair poten-tial.

IO

[Ioo] L

IO

[Ioo] L8- 75 K

2E

0 I

0.6 0.8 I.O0.2 0.4CLLLJ

LLI

IO

[Ioo] Tz 8- IP0

00.2 0.4 0.6 0.8 I.O

IO

[loo] T8 — 75 K

0.2 0.4 0.6 0.8 I.o 0.2 0.4 0.6 0.8 l.oWAVE VECTOR Q {a/2II )

FIG. 4. The QH (~ ~ ~ ~ ), the SCH (—--), and theSCH+ C ( ) phonon dispersion curves along the h00ldirection at &=10 and 75 K.

et al. , to lower-energy phonons where phonon en-ergies can be meaningfully defined and where theone-phonon S&(Q, &u) can be readily distinguishedwithin the total S(Q, &u).

In Fig. 3 the SCH+C phonon energy for twotransverse phonons is shown as a function of temp-erature. There we see that the temperature de-pendence of the SCH + C phonon energies agreesvery well with the more recently observed valuesof Fujii et al. Also shown on the left-hand side ofFig. 3, as crosses, are the phonon energies cal-culated by Klein et al. ' using the QHPT and the

FIG. 5. Temperature dependence of longitudinal phononenergies along the [100) direction.

Bobetic-Barker potential which do not agree sowell with experiment. We believe the differencebetween their energies at higher temperatures andthe present SCH+ C values represent the differencebetween the SCH+C and QHPT rather than a dif-ference in the potential for the following reasons.Firstly, as shown in Fig. 1, the model realisticpotentials do not differ substantially in the well re-gion. Secondly, the phonon energies at low temp-erature are insensitive to these small differencesas noted in Sec. III, particularly the transversebranch along the [100]direction (see Table I).Thirdly, we have calculated the cubic anharmonicshift in the QHPT and found it to be substantiallydifferent at high temperature from the same shiftin the SCH+C theory. For these reasons we be-lieve the SCH+C theory is a substantial improve-ment over the QHPT at high temperatures. Thedifferences between the QH, the SCH, and the SCH+C frequencies at 10 and 75 K are shown in Fig. 4.

The temperature dependence of two longitudinalphonon energies is shown in Fig. 5. There we seethat, while the SCH+ C energies lie below the ob-served values at all temperatures, owing to theneglect of ATM forces, the temperature dependence

2.4-CP

E

I I I I

SCH

2 2(3CCLLI

~2.O-

OZ'.

CL& I.8-

[III] T

h =O.2

SCH+C

—[I I I ] Ts=o4

3.2-20 40 60 80 20 40 60 80

TEMPERATURE (K)

FIG. 6. Temperature dependence of transverse phononenergies along the [111jdirection.

6396 H. R. GLYDE AND M. G. SMOES 22

TABLE III. Elastic constants c;~ of Ar at T =82 K in 10 dynes/cm, 6=(c44-ci2)/ci2.

Ci2 ~44 ii i2

Theory

PresentKlein and Murphy

(T =80 K)'Fisher and Watts

(Z' =80 K)'Gibbons et al.

T =80 K)c

220

250

237

135

163

161

157

119

112

112

83

80

-0.015

-0.027

-0.030

-0.029

ExperimentNeutrondBrillouin

248+ 6238+ 4

153+ 5156+3

124+ 4112+3

95+ 1182+ 7

-0.019+ 0.04-0.028+ 0.04

Bobetic-Barker potential, Ref. 15;Barker-Fisher-Watts potential, Ref. 34;Parson-Siska-Lee potential, Ref. 35, all including Axilrod-Teller-Muto three-body

forces.Fujii et a/. , Ref. 6.

e Gewurtz and Stoicheff, Ref. 36.

agrees will with experiment. A final exampI. e isshown in- Fig. 6. These comparisons show thetemperature dependence of the phonon energies canbe calculated quite accurately using the SCH+Capproximation.

In Table III are listed the elastic constants of Arat T =82 K. There we see the present SCH+C c44and cii-ci2 agree well with previous values and ex-periment, but the cii and ci2 are too small owing toneglect of the three-body ATM forces. The neutrondata represent zero-sound elastic constants and lieslightly above the first-sound Brillouin values.Clearly, ATM forces are needed to get agreementwith experiment. The good agreement between thetheory including these forces and experiment rep-resents a triumph of both theory and experiment.

/

monic term. The cubic anharmonic force constantwas not, however, reevaluated, and to keep theone-phonon response peak position consistent withthe SCH+ C frequencies defined above, the v~),

+2m, ~h(q A, u} was set at u&scenic in A(qX, a&). Theaim of the second approximation is the use of theSCH+ C frequencies which are the best estimatehere of the true argon-crystal. frequencies in I',Sq2(Q. , ~), and S2(Q, &o) to get a more realistic valuefor the width of the phonon group and the contribu-tion of S~2(Q, &o} and S,(Q, &o) to S(Q, &o) in (2). Set-ting v,'~+ 2&@,~h(qX, &o) = use„+c will be valid ifb (qX, &o) is not a very sensitive function of v. TheS(Q, e) was folded with a Gaussian of FWHM of 0.4meV (0.5 meV) for transverse (longitudinal) pho-nons to simulate the neutron scattering instrument-

V. DYNAMIC FORM FACTOR

In this section we present calculations of thedynamic form factor S(Q, &o) of Eq. (2) using theSCP theory. This is done in two approximations.In the first the cubic anharmonic shift b(qX, &u) andone-phonon damping I'(qX, e) are calculated simplyas perturbations to the SCH frequencies. That isthe averaged cubic force constant, 4, I', S»(Q, u&),

and S~(Q, &o) are all evaluated using the SCH fre-quencies. The SCH phonons are the "intermediatepropagator" phonons in S(Q, &u). It was this ap-proximation that was used to obtain the SCH+ Cfrequencies of the previous section.

In the second approximation the SCH+C frequen-cies are used as the intermediate frequencies inI'(q&, (o), Sq2(Q, &o), and S2(Q, &u). The second ap-proximation simulates the first iteration in a fullyself-consistent theory including the cubic anhar-

2.0 I

[Ioo]l.5- r=r(sI.O-

«D

2.0

I.5—3~ Io-

I I

I IOO] L (5,0,0)

0.5—

5.0 6.0 7.0 8.0 9.0 IO.O

ENERGY TRANSFER %~(meVj

FIG. 7. Dynamic form factor, S(Q, ~), for the longi-tudinal phonon of wave vector Q= (27t'/a) (3.0, 0,0). Theupper (lower) curve uses the SCH (SCH+ C) frequenciesin I'(qX, u} and S2(Q, a).

22 PHONONS IN SOLID ARGON 6397

5.0 I I

[ I I I I L {2.4,2.4,2.4)4.0

(

2.0-cn I.0-

CV

O5.0

I: I I I ] L (2.4,2.4,24)4.0-

I

a 5.0-M

2.0—

I.O-

—2.04PCll

NrC)

LO

3CfV)

0.0I.o

Q = —0 (3,0,0)T=75 K

S~+ S2

S2

5.0 6.0 7.0 8.0 9.0 lo,oENERGY TRANSFER h~(meV)

FIG. 10. Contribution of S2(Q, ~) to St(Q, &u)+ S&(Q, u)for the longitudinal phonon at Q= (2~/a) (3,0, 0) at T= 75K. For this phonon S&2(Q, +)= 0.

5.0 6.0 7.0 8.0 9,0 Io.oENERGY TRANSFER h~(meVj

FIG. 8. As Fig. 7 for Q= (27r/a) (2.4, 2.4, 2.4).

0 Q- 2II

al resolution width.The S(Q, u&) for two high-frequency, longitudinal

phonons having reduced wave vectors near theBrillouin-zone edge are displayed in Figs. 7 and 8.The one-phonon groups for these phonons are al-readybroad atr =10K. AtT =75Kwe seetheone-phonon peak is dispersed over a very wide frequencyrange. It also depends very sensitively on whetherthe SCH or SCH+ C frequencies are used in the one-phonon width I'(q1., v) andS, (Q, &u). Since the SCH+ C

frequencies are lower than the SCH frequencies theI'(SCH+ C) and S,(SCH+ C) lie in a. lower-frequen-cy range than I'( SCH) and S2(SCH). It turns outthat this lowering means that I'(SCH + C) and St(SCH+ C) become substantially larger in the one-phononpeak energy range than the I'(SCH) and St(SCH).This is true only for the high-energy phonons. Thelower-energy phonons lie in an energy range belowwhere either I'(SCH+C) or I'(SCH) are substantial.

This is displayed in Fig. 9 which shows I"(SCH) andI'(SCH+ C) as well as 4(SCH) for the longitudinalphonon at Q =(2wja) (3, 0, 0).

The contribution of S,(Q, ~) to the total S(Q, ~) forthe longitudinal phonon at Q = (2mja) (3.0, 0, 0) isshown in Fig. 10. There we see that S2(Q, &u) islargely responsible for the large shoulder on thehigh-energy side of S(Q, &o). Since this Q lies mid-way between the two reciprocal-lattice vectors theinterference term S»(Q, u&) vanishes by symmetryfor this phonon. The individual contributions of

S~t(Q, &o) and S2(Q, &u) to S(Q, ur) for the longitudinalphonon at Q =(2sja) (2.4, 2.4, 2.4) are shown in Fig.11. There we see the interference term is import-ant for these high-energy phonons. However, ingeneral the interference terms were significantlysmaller in argon than in neon and heliumwhere they are very substantial.

The S(Q, ~) for a lower-energy longitudinal pho-non is shown in Fig. 12. There we see the S(Q, &a)

is relatively insensitive to whether the SCH or SCH+C frequencies are used as the intermediate prop-agator frequencies. These S(Q, m) may be com-pared directly with the observed groups of Fujiiet al. shown in Fig. 13. There we see the calcu-lated S(Q, v) agrees well with the observed inten-sity except at T =75 K. The observed groups show

CDV)

"o

O(D

P (A

O

I L

3CU

OP

O

Q = —"(2.4,2.4 2.4)T=75 K

4 6her (meV)

10 04.0 5.0 S.O 8.0

ENERGY TRANSFER 5~ ( meV )9.0 IO.O

FIG. 9. Functions 2~,),&(q&, ~) and 2~,), ~(q~, ~) calcu-lated with the SCH phonons as the intermediate propa-gator frequencies and 2~,), ~(q~, co) calculated with SCH+ C phonons as the propagator frequencies.

FIG. 11. Contributions of S2(g, u) and Stt(g, cu) toSt (Q, ~)+ ~gp (4, ~)+ S2 (Q, &u) for the longitudinal phonon atg= (2m/a) (2.4, 2.4, 2.4) at T= 75 K.

4.0

H. R t LYDK AND M G. SMOKS

5.0

22

[ IO

50 (2.5CH) 4.0

2.0- 2.0

~ I.O-CDCh

04

'~ 4.0[IOO] L

50 (2.5,0,0)

I I

I (SCH+C )

I.O-

0~ 4.0

Ol

C 50—

2.0

[1001 T

4.0l

5.0

~ 2.0-V)

I.O-

Ol3.5 4.5 5.5 65

ENERGY TRANSFER

. 12. As Fig. 7 for Q=(2x/a) (2.5,0, 0).

2.03(3

I.OCO

03.0

2.0

I.O

4.0I

5.0I

6.0

a very abrupt broadening betwod db y the theory. F' ll

not

tu d d of thtransverse pho

o e phonon rgroups for someonons are shown

Generally, th'

zn Fig. 14.ge groups for bono ner-

symrnetric land insensitive to h

a, well behaved "o w ether the SC

7

e used as intermediare-

e u d' e iate propagator

I

4.0 S.o 70SFER h~(meq)

FIG. 14. c ors for transvernamic form facte SCH+C fr

I'irection using th frfrequencies

S o

500

36AArt

~100) L (2.5, 0,0)Eo= l5.7 meV

E~ 200—

V)

o l50—

Z'DK

100—

50—

05.5

I

4.0I I I

5.5NEUTRON ENE RGY TRANSFER (meV)

I

6.5 7.0

FIG . ].3. Phonon rgroups observed b Fu"'y jii et al. (Ref. 6) for cor comparison vr th F'i ig. 12.

PHONONS IN SOLID ARGON 6899

frequencies. For higher-energy phonons @&a ~ 6meV, S,(Q, &o) is broad and peaks in the energyrange where I'(qX, &o) and S2(Q, &o) are rapidly in-creasing (see Figs. 9 and 10) leading to a plateauon the high-energy side of S(Q, &o). For these pho-nons S(Q, &o) is very sensitive to whetner the SCHor SCH+C frequencies are used as intermediatepropagator frequencies. In a fully self-consistent,second-order theory the frequencies given by themean position of the phonon group should be used.For the higher-energy phonons having a high-fre-quency tail this mean frequency will be larger thanthe SCH+ C frequency. Thus, using the SCH+Cfrequencies will tend to overestimate the effect ofchanging the intermediate frequencies. The pur-pose here is to show that for 5'(d ~ 6 meV the groupwidths are sensitive to the intermediate frequen-cies. A definitive group shape will have to await afully self-consistent theory which would have toinclude shor t-range correlations.

VI. COMPARISON WITH MOLECULAR DYNAMICS

Using the method of molecular dynamics (MD) itis now possible to simulate classical liquids andsolids directly. In this method Newton's equationsof motion are solved numerically for N atoms in avolume V to evaluate the atomic positions r, (t)=R, +u, (t) as a function of time. With this infor-mation properties such as the dynamic form factorS(Q, ur) in (1) can be calculated directly. Withinthe accuracy of the method, the resulting S(Q, &o)

should be exact, including all anharmonic contrib-utions. Thus a comparison of S(Q, v) calculatedusing the $CP theory with MD should provide a testof the theory.

In an interesting study, Hansen and Klein havesimulated the rare-gas solids using MD, assumingthe rare-gas atoms interact via the Lennard- Jonespotential

v(r) =4&I (o/r)" —(o/r) ],where for Ar o =3.405 A and &=119,8 K. Thispotential is compared with the more realistic po-tentials in Fig. 1. To make direct comparison withtheir simulation we have calculated S(Q, &o) usingthe Lennard-Jones potential in Ar at densities andtemperatures considered by them.

In Fig. 15, the S(Q, &o) for the longitudinal phononhaving wave vector Q = (2v/a) (0.5, 0, 0) in 36Ar atT =39.5 K (a = 5.3521) and T = 87.5 K (a =5.4603)calculated here using the SCH+C theory and byHansen and Klein using the MD method are com-pared. Both S(Q, &o} are folded with a Gaussian, ofFWHM=0. 22 in the MD case and of FWHM=0. 4meV in the SCH+C case. In this calculation theLennard- Jones potential was truncated after the

0.05 Q =2ii (0.5,0,0)/Q

0.04

0.03

~ 002

0.0 I

llI /slI jI/ =875KI.ilI.li

\

/

j Il ' )'v'

f(3

t

3

0.03 ). l T =395KjIi

I'i l~

il('i

Ij'

IjI

~1 I/.

l.5 2.0 30

0.02—

O.OI

2.5To

I

4.0I I

5.0'hu (meV)

FIG. 15. S(Q, co) for the longitudinal phononhalfway.to the zone boundary in the I100] direction (Lennard-Jones potential) ——SCH+ C, — MD.

I

7.0

first-neighbor shell. For more direct comparisonthe SCH+ C theory S(Q, &v) was multiplied by asingle scale factor of 1.18 at both temperatures sothat at T =39.5 K the two S(Q, &o) lie directly on topof one another. Without this factor the SCH+ CS(Q, &o) would lie somewhat below the MD S(Q, &o),

perhaps due to multiphonon contribution not includ-ed in the SCH+C S(Q, &u). The temperature T=87.5 K lies slightly above the melting point (T=84 K).

From Fig. 15 we see that the peak positions ofthe S(Q, u&} agree well, particularly the change ofthis peak position with temperature. The increasein the width and intensity of S(Q, &o) with tempera-ture also agrees well. Thus except for the humpson the MD S(Q, &u), which may be due to statisticalnoise, the SCH+ C S(Q, u&) reproduces the MD re-sults well. This phonon group, calculated usingthe Aziz-Chen potential and observed by Fujii eta/. , is also shown in Figs. 12 and 13, respective-ly. Since the SCH+CS(Q, v) in Fig. 12 peaks atenergies slightly below but in a good agreementwith the observed peaks in Fig. 13, the MD andexperiment phonon group peaks are consistent.The observed group at T =75 K in Fig. 13 is sub-stantially broader than would be predicted by MD

H. R. GLYDE AND M. G. SMOES

v) pg—Q =277-(2, l, )/c

o 08875 K

0.7—3

&C' pe-

o 05—O

0.4—I

LL|

o~ 0.3

0.2—(/) I(/) /+ p. l—

........gvO"' . .~ .eS)

—l.6

—l.2

C3—I.OCO

-0.8 O

-0.6 p

—0.4 g)

-0.2

2.0 2.5 3.0

4.0o

5.0'hio (meV)

6.0

FIG. 16. The one phonon, S~, and full S for the trans-verse phonon on the zone boundary in the f100] directionat &= 87.5 K. The dots and solid line are S~ by MD,dashed line S by MD. The dashed (dotted) lines are S(S~) calculated here using the Lennard- Jones potentialand the SCH+ C theory. The peaked line is an SCPS(Q, ~}calculated without folding by Goldman (as inRef. 39).

simulation.In Fig. 16 a similar comparison is made for the

transverse phonon having wave vector Q =(2s/a)(2, 1, 1) [equivalent to a reduced q = (2v/a} (1, 0, 0)at T =87.5 K]. In this case the Lennard-Jones po-tential was truncated after three neighbor shells.The two S(Q, e) are plotted in the unadjusted unitsnoted in the figure. While the intensity and widthof the SCH+ C theory S(Q, ar) and Si(Q, &o) compareswell with the MD results, the SCH+ C S(Q, v) peaksat much higher energy. This disagreement in peakposition is surprising since the peak position of theSCH+ C theory S(Q, &o) for the T[100Jbranch calcu-lated using the Aziz-Chen potential agreed extreme-ly well with experiment at all temperatures, asshown in Fig. 3. Regarding the SCH+ C theory asan interpolation scheme to connect the MD simula-tion and experiment suggests the MD peak positionwould not agree with experiment in this case.

From Figs. 15 and 16 we conclude that the widthand shape of the SCH +C theory S(Q, ic) agree wellwith the MD simulation near melting in 36Ar, al-though the peak position does not always agreewell.

VII. DISCUSSION

In the previous sections we saw that the SCH+Ctheory predicts phonon frequencies in good agree-ment with experiment right up to 82 K. The valuesobtained are much improved over those predictedby QHPT. For low-frequency phonons, where com-paxison with experiment is possible, the phononlifetimes and shape of S(Q, &u) are also quite wellpredicted (see Figs. 12 and 13). These shapes arealso reasonably insensitive to how S(Q, &o} is calcu-lated. In addition, the Aziz-Chen potential, whichdescribes gas and liquid-argon properties pre-cisely, clearly predicts the phonon properties wellalthough the phonon data are not really preciseenough to distinguish between the Aziz-Chen andBarker-Fisher-Watts potentials, for example.

With the above positive points we recall that the

S(Q, &o) for the high-frequency, longitudinal phononspredicted by the SCH+ C theory are very broad andvery sensitive to the values used for the intermedi-ate propagator frequencies at T ~ 55 K (e.g. , see

.Figs. 7 and 8). This is both because S,(Q, ur) itselfis very broad and because S,(Q, ic) falls in a fre-quency range where S,(Q, &o) is rapidly increasingwith ar. For these reasons a fully self-consistenttheory including the cubic anharmonic term is re-quired to establish S(Q, &o} for high-frequency pho-nons precisely. This fully self-consistent theoryshould also include recalculating the averagedharmonic and cubic force constants at each stage ofthe iteration.

It is also clear both from experiment and theorythat S(Q, &o) for high-frequency phonons in 36Ar atT ~ 55 K is indeed very broad. A similarly broadS(Q, &o), having a plateau on the high-frequency sidedue to a rapidly rising S,(Q, &c), was found forhigh-frequency phonons in fcc He and Ne. How-ever, the Si(Q, a&) appears to be even broader in Arnear melting than in Ne, fcc He or even in themore quantum hcp and bcc phases of helium. TheSi(Q, &u) is broad for high-energy phonons in Ar be-cause S,(Q, &u) lies in a frequency range whereI'(q, ~) is large (see Fig. 9). In bcc He, for ex-ample, S,(Q, &e) always lies in a frequency rangebelow which the corresponding I'(q, &c) becomeslarge (see Fig. 4, Ref. 40). Thus the phonons arebroader in Ar due to the closer relative positionsof S,(Q, &o) and F(q, &c) in Ar rather than to largerintrinsic anharmonic effects. Thus, although thequantum phases of helium are the most anharmon-ic, the phonons have long lifetimes because thephonon energies are small and lie in a regionwhere F(q, &u) is small. The rms vibrational amp-litude in bcc He is -30-35 /o of the interatomicspacing compared to 15-18 /o in Ar at the triplepoint.

PHONONS IN SOLID ARGON 6401

Finally, would a theory including an explicitdescription of short-range correlations improveagreement with experiment 7 Certainly, includingshort-range correlations changes the force con-stants and in that sense they are important.Also, the SCH+C theory is an improvement overthe QHPT largely because the initial SCH frequen-cies are closer to the observed values and the sub-sequent cubic frequency shift is smaller; that is,the SCH frequencies are a better basis. If includ-ing short-range correlations provides a still bet-ter basis, then improvement can be expected. Inthis case the cubic anharmonic term would be fur-ther reduced. Experience in solid helium showsthat this reduction in the cubic coefficient(V(0)V(0)V(l)v(ro, )), when short-range correlations

are included, substantially lengthens the phononlifetime. Since experiment and MD tend to con-firm that the phonon lifetimes in Ar are indeedshort, incorporating short-range correlationsshould not change the averaged force constants toodramatically. An experimental determination ofS(Q, u&) for the high-energy phonons in Ar wouldtherefore be most interesting as a critical test offuture theories.

ACKNOWLEDGMENTS

It is a pleasure to acknowledge valuable discus-sions with Dr. G. K. Horton, Dr. M. I . Klein, and

Dr. J. Eckert.

G. K. Horton, in Rare Gas Solids, edited by M. L. Kleinand J.A. Venables (Academic, New York, 1976), Vol.I, Chap. 1.J. A. Barker, in Raze Gas Solids, edited by M. L. Kleinand J.A. Venables (Academic, New York, 1976), Vol.I, Chap. 4; B. M. AxQrod and E. Teller, J. Chem.Phys. 11, 299 (1943); Y. Muto, Proc. Phys. Math.Soc. Jpn. 17, 629 (1943).

3W. B. Daniels, G. Shirane, B. C. Frazer, H. Umeba-yashi, and J. A. Leahe, Phys. Rev. Lett. 19, 1307(1967).

H. Egger, M. Gsanger, E. Luscher, and B.Dorner,Phys. Lett. A28, 433 (1968).

D. ¹Batchelder, M. F. Collins, B.C. G. Haywood, and

G. R. Sidey, J.Phys. C 3, 249 (1970).Y. Fujii, N. A. Lurie, R. Pynn, and G. Shirane, Phys.Rev. B 10, 364V (19V4).E. R. Dobbs and G. O. Jones, Rep. Prog. Phys. 20, 516(1957).

G. K. Pollack, Rev. Mod. Phys. 36, 748 (1964).G. Boato, Cryogenics 4, 65 (1964).E.A. Guggenheim, Applications of Statistical Me-chanics (Clarendon, Oxford, 1966).G. K. Horton, Am. J. Phys. 36, 93 (1968).G. K. Horton and J. W. Leech, Proc. Phys. Soc. London

82, 816 (1963).L. Bohlin and T. Hogberg, J.Phys. Chem. Solids 29,1805 (1968).

4G. ¹iklasson, Phys. Condens. Matter 14, 138 (1972).M. V. Bobetic and J.A. Barker, Phys. Rev. B 2, 4169(1970).

~6J. A. Barker, M. L. Klein, and M. V. Bobetic, Phys.Rev. B 2, 4176 (1970).M. L. Klein, J. A. Barker, and T. R. Koehler, Phys.Rev. B 4, 1983 (1971).M. L. Klein and T. R, Koehler, in Rave Gas Solids,edited by M. L. Klein and J. A. Venables (Academic,New York, 1976), Vol. I, Chap. 6.J. A. Barker, R. A. Fisher, and R. O. Watts, Mol.Phys. 21, 657 (1971).V. V. Goldman, G. K. Horton, T. H. Keil, and M. L.Klein, J.Phys. C 3, L33 (1970).

2~J. P, Hansen and M. L. Klein, Phys. Rev. B 13, 878

0.9v6).R. A. Aziz and H. H. Chen, J. Chem. Phys. 67, 5719(19vv).

3L. Van Hove, Phys. Rev. 95, 249 (1954); W. Marshalland S. W. Lovesey, Theory of Thermal Neutron Scatter-ing (Oxford University, Oxford, 1971).R. A. Cowley, Adv, . Phys. 12, 42 (1963); Rep. Prog.Phys. 31, 123 (1968).H. R. Glyde, Can. J. Phys. 52, 2281 (1974).

6P. F. Choquard, The &nhaxmonic Crystal (Benjamin,New York, 1967); T. R. Koehler, in Dynamical I'~op-eHies of Solids, edited by G. K. Horton and A. A.Maradudin (North-Holland, Amsterdam, 1975), Vol. 2.T.R. Koehler, Phys. Rev. Lett. 22, 777 (1969); H. R.Glyde and M. L. Klein, Crit. Rev. Solid State Sci. 2,181 (19V1).V. Ambegaokar, J. Conway, and G. Baym, in LatticeDynamics, edited by R. F. Wallis (Pergamon, New

York, 1965), p. 261.R. A. Cowley and W. J. L, Buyers, J. Phys. C 2, 2262(1969).L. Bohlin, Solid State Commun. 10, 1219 (1972).H. Horner, Phys. Rev. Lett. 29, 556 (1972).J. Meyer, G. Dolling, R. Schern, and H. R. Glyde, J.Phys. F 6, 943 (1976).

33W. M. Collins and H. R. Glyde, Phys. Rev. B 18, 1132(19V8).

+R. A. Fisher and R. O. Watts, Mol. Phys. 23, 1051(19V2).

3~J. M. Parson, P. K. Siska, and Y. T. Lee, J. Chem.Phys. 56, 1511 (1972).S. Gewurtz and B.P. Stoicheff, Phys. Rev. B 10, 3487(19V4).

3~W. M. Collins and H. R. Glyde, Phys. Rev. B 17, 2766(1978).

3 V. J.Minkiewicz, J.A. Kitchens, G. Shirane, and E. B.Osgood, Phys. Rev. A 8, 1513 (1973); J. Eckert,W. Thomlinson, and G. Shirane, Phys. Rev. B 16, 1057(1977), W. Thomlinson, J.Eckert, and G. Shirane,ibid. 18, 1120 t1978); C. Stassis, G. Kline, W. A.Kamithakara, and S. K. Sinha, ibid. 17, 1130 (1978).

3 V. V. GoMman, G. K. Horton, and M. L. Klein, Phys.Rev. Lett. 24, 1424 (1970).

64Q2 H. R. GLYDK AND M. G. SMOKS 22

H. R. Glyde and F. C. Ehanna, Can. J.Phys. 49, 2997(1971).

~ H. Horner, in Dynamical Properties of Solids, editedby A. A. Maradudin and G. K. Horton (North-Holland,Amsterdam, 1974), p. 450; and in Proceedings of the

L4EA Conference on Neutron Inelastic Scattexin g,Vienna, 1972 (lAEA, Vienna, 1972), p. 119.

~~L. B.K~n~ey and G. K. Horton, Phys. Rev. Lett. 34,1565 (1975).

~3J. Leese and G. K. Horton (unpublished),