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Journal of Low Temperature Physics, VoL 59, Nos. 5/6, 1985

Impulse Approximation in Solid Helium

Henry R. Glyde

Department o f Physics, University o f Delaware, Newark, Delaware 19716

(Rece ived Oc tober 31, 1984; revised D e c e m b e r 18, 1984)

The incoherent dynamic form factor S/Q , to) is evaluated in solid helium for comparison with the impulse approximation (IA). The purpose is to determine the Q values for which the IA is valid for systems such as helium where the atoms interact via a potential having a steeply repulsive but not infinite hard core. For 3He, Si(Q, to) is evaluated from first principles, beginning with the pair potential. The density of states g (to) is evaluated using the self-consistent phonon theory and Si(Q, o~) is expressed in terms of g(to). For solid 4He reasonable models of g(to) using observed input parameters are used to evaluate Si(Q, to). In both cases Si(Q, to) is found to approach the impulse approximation SIA(Q, to) closely for wave vector transfers Q ~> 20 ~ -1. The difference between S~ and SIA, which is due to final state interactions of the scattering atom with the remainder of the atoms in the solid, is also predominantly antisymmetric in (to -ton), where toR is the recoil frequency. This suggests that the symmetrization procedure proposed by Sears to eliminate final state contributions should work well in solid helium.

1. INTRODUCTION

The measurement of atomic momentum distributions in condensed matter by deep-inelastic neutron scattering measurements is a field of growing interest. ~-9 For example, the momentum distribution has now been observed in liquid 3-7 and solid 4He,8 in liquid neon, 9 and in atomic hydrogen in solids. ~° The present calculations particularly address deep-inelastic scattering in solid helium.

These experiments are carried out at large momentum (hQ) and energy transfer (E ~ E R = h2Q2/2M) from the neutron to the sample of atomic mass M. Under these conditions the scattering is predominantly from a single nucleus, so that the observed coherent S(Q, to) can be approximated by the single-atom, incoherent Si(Q, to). In addition, if the momentum transferred to the atom is sufficiently large compared to the forces on the scattering atom due to other atoms in the system, then these forces may be

561

0022-2291 / 85/0600-0561 $04.50/0 © 1985 Plcn um Publishing Corporation

562 Henry R. Glyde

neglected. In this case the scattering atom may be approximated by a free particle and Si(Q, to) reduces to the impulse approximation (IA),

Si(Q'to)~SIA(Q'to)=f n(P)8( to hQ22M QMP) dp (1)

These other forces on the scattering atom due to the surrounding atoms are denoted final state interactions, since they affect the final state into which the atom is scattered. If the IA holds, we can readily extract the momentum distribution n(p) from the observed S(Q, to).

The precise values of Q at which the IA becomes valid enough to use for determining n(p) has been a subject of debateJ 'H-14 This debate is exhaustively reviewed by Sears, ~4 who draws analogy with deep inelastic scattering in other areas of physics. He discusses particularly scaling methods and symmetrization procedures that can be used to identify and eliminate the contributions from final state interactions to S(Q, to) at finite Q. For systems having finite forces the IA is asymptotically e x a c t 2'14 at Q ~ . Weinstein and Negele ~ and Reiter ~2 have presented interesting counter examples of systems having infinite forces or infinite impulses where the IA is not reached even at Q-~ oo.

The aim of the present paper is to test the values of Q at which the full Si(Q, to) approaches the IA for a realistic model of solid helium. We consider two cases. First, for solid bcc 3He, we begin with the bare H e -H e interatomic potential, which has ahard core. From this potential we evaluate the dynamics of solid helium from first principles using the self-consistent phonon theory, including a Brueckner T-matrix treatment of the hard core of the potential. From the dynamics, a density of phonon frequencies is generated and used to calculate Si(Q, to) directly. This Si(Q, to) is compared with the IA. Second, having demonstrated that we can derive a g(to) from first principles, we construct model g(to) to represent solid 4He at volumes V=21.1 and 16.0cma/mole. These model g(to) are used to calculate Si(Q, to) for comparison with the IA. These comparisons show that in solid helium, in which the atoms do interact via a potential having a hard but not infinite core, SI(Q, to) approaches the IA in the region of Q>~20/~-1 for practical purposes.

If the momentum distribution is assumed to be Gaussian

n(p~,) = (27r(p])) -1/2 exp (-pZ/2(p])) (2)

then the IA in (1) is

SIA(Q , to) = (277"0"2) -1/2 exp [ -- (to - t o R ) / 2 0 "2] (3)

where or 2= QE(p2)/M2, the z axis is chosen along Q, and toR = ER/h is the recoil frequency. For an isotropic system herr 2= 4ER(K)/3, where (K) is

Impulse Approximation in Solid Helium 563

the kinetic energy per atom. In this form it is clear that the kinetic energy or (pZz) can be extracted from the width of SIA(Q, to). As noted by Hilleke et al., 8 (K) can be extracted directly from the observed SI(Q, to) without assuming the IA, since the second moment of Si(Q, w) [(to -toR) 2 averaged over SdQ, to)] is also equal 14'~5 to o "2. Thus, to obtain the kinetic energy from Si(Q, to) we do not need to assume the IA.

In Section 2 we discuss the dynamic form factors and how Si(Q, t 0) is evaluated here. The results for Si(Q, to) and SIA(Q, to)are presented in Section 3 and discussed in Section 4.

2. THE DYNAMIC FORM FACTORS

2.1. Introduction

Observed 16 is the coherent, inelastic dynamic form factor,

So(Q, to) = - ~ dt exp ( itot )

XN~(exp[ - iQ . r ( l , t ) ] exp[ i Q . r ( l ' ,O ) ] } (4)

Here we evaluate the incoherent dynamic form factor Si(O, to) obtained from So(Q, to) by retaining only the l '= l terms in (4):

Si(Q, to) = ~ dt exp (itot) (exp [ - i Q . r(t)] exp [ iQ- r(0)]) (5)

The terms l ~ l' in Sc(Q, to) are often referred to as the "atomic interference" terms. They represent the interference among the scattering from different atoms in the crystal. The Si(Q, to) evaluates the scattering from individual atoms one by one. The atomic interference terms are believed to be small, of order Q >> 21r/R, where R is the interatomic spacing) 4 They lead to oscillations in the peak position and width of So(Q, to). When these oscillations die out at higher Q, So(O) settles down to So(Q) = 1. Since Si(Q) --- 1, the Q value at which the observed S~(Q) reduces to unity is generally taken as a signal that the atomic interference terms are negligible (at Q ~ 8-10 N-1)4 in liquid 4He. The remaining difference between S~(O, to) and SIA(Q, to) is a measure of the magnitude of the final state interactions between a single atom and the remainder of the atoms in the system.

Writing r = R + u, where u is the displacement from the lattice point R, we have

(exp [ - i Q . r(t)] exp [ iQ . r(0)]) = (exp [ - i Q - u(t)] exp [ iQ . u(0)]) (6)

= exp ( -2 w) exp ([Q. u( t ) ] [Q, u(0)])

564 Henry R. Glyde

In the coherent case, the last equality holds only for a harmonic crystal. 17:8 We believe it is a much better approximation in the incoherent case, since the anharmonic interference terms involving a single phonon do not contribute to Si(Q, to). In the coherent case there is a wave vector selection in the one-phonon SI(Q, to), which requires the reduced wave vector q of the phonon to be equal to Q ( A ( Q - q)). The anharmonic interference terms are odd functions of q. In Si(Q, to) there is always a free sum over q having equal positive and negative q values. The interference terms involving one phonon therefore vanish in S~(Q, to). We have not investigated higher order interference terms, but a similar argument should be possible. In the Debye- Waller factor we retain only the leading term, 2 W = ([Q • u]2).

We now expand the u(t) in terms of phonon normal coordinates AqA (t),

h (Q.u(t)][Q.u(O)])=l ~ 2--~wq (Q.eqA)E(aqx(t)a~x(O)) (7)

where A is the phonon branch index, eqx is the polarization vector, and toqx is the phonon frequency. The expectation value can be Fourier-analyzed in terms of the anharmonic response function A(qlt, to) and the Bose function n(to) as 18-z°

(Aq. ( t)Aq. (O)) = J-~ 2~r e-i~'[n(to) + l]A(q;t, to) (8)

Using the detailed balance 21 result

[n(- to) + 1]A(q;t, -to) = n(to)A(qA, to), (9)

we can write (8) in terms of positive to only. The expectation value in (7) can then be written, using (8) and (9), as

([Q. u(t)][Q" u(0)])

=wR dtog(to) l {[n(to)+l]e-i°~t+n(to)e ~"~'} (10) to

and

2 W = ( [ Q • u]2}= toR dtog(to) [2n(to) + 1] (11)

where

1 A(qA, to) 12) g(to) = to ~--/~ Y~ 3 (0" eqx) 2 (

qA , Z'TTtoqX

Here toR= hQ2/2M and I~ = Q/IQI. This density of states g(to) is a slight

Impulse Approximation in Solid Helium 5 6 5

generalization of the usual harmonic density of states; the "number" of phonons having frequency in the range to to to + dw is obtained by summing over all qA and determining the weight of each A(qh, to) in the frequency range to to to +dto. The anharmonic 19'z° A(qA, to) is

8to2ar(qx, w) A(qh, co)=[_toz+to2q~ +2toq;~A(qh ' to)]z+[2toqAF(q& to)]2 (13)

where toq~ are the "unperturbed" frequencies, say the self-consistent harmonic (SCH) frequencies, z°'2z and A(qA, to) and F(qA, w) are the frequency shift and inverse lifetime due to anharmonic effects, say the cubic anharmonic term. In an approximation in which the phonons have infinite lifetime, such as the SCH approximation,

A(qA, to) = 2zr[6(to - to**) - 8(to + wqa)]

In this case and for a cubic system in which 3((~-eqx) 2 averages to unity, then on writing (3N) -1 ~qx "-> ~o ~ dto g(to), we find that (12) reduces to the usual harmonic-like density of states.

Using (10) and ( l l ) , we obtain for the Si(Q, to)

Si(Q' to) = 2--l~"rr I : dtexp(itot)exp(-2W)

( I o o xexp toR dto g(to) 1 {[n(to) + 1] e -i'°' to

+ n(to) ei~°'}) (14) f

Si(Q, to) depends solely on g(to). Provided we use the same g(to) in 2W and in the last exponential of Si(Q, to), then Si(Q) = 1.

It is convenient to define three moments of g(to):

( to- l )_ f dto g(to)(1/to)[2n(to)+ 1]

(too) =_ f dto g(to) = 1 (15)

(co I) =- f dto g(to)to[2n(to) + 1]

The inverse first moment, from (11), is

2 W = toR(to-') = (hQ2/2M)(3h/2kODw) (16a)

where the last equality is the Debye expression for 2 W at T = 0 K. The observed value of 2 W is often expressed in terms of an empirical value of

566 Henry R. Glyde

0Dw. The zeroth moment is just a normalization of g(to). The first moment is related to (K) and o- by

(K) ---- ( 3 / 4 ) h ( t o 1) and tr 2 = tOa(tO') (16b)

2.2. Impulse Approximation

The IA may be obtained as a short-time approximation by expanding the two exponentials e ~°' ' in the last term of (14) and keeping terms up to t z only, i.e.

e ±i'°t = 1 + ioJt -½~o2t 2

The last exponent in (14) is then

w R dwg(~o) {'''}=2W--iooRt--½o't 2 (17)

and

Si(Q, w)-* S,A(Q, w) = (1/21r) f ~ dt exp (i~ot) exp (-iooRt) exp ( -½crt 2)

= (2~'o'2) -1/2 exp [ - (to - WR)2/2O -2]

Several comments on SIA(Q, o)) are worthwhile. First, the difference between Si(Q, ~o), which contains the final state interactions, and SIA(Q, to) is due entirely to the t 3 and higher order terms in (14). From (17) we expect first that the higher order terms in t will be negligible when 0 is large. In this case both WR and o- are large. A large WR will lead to rapid oscillations in the integrand of (14) [see (17)] with t, causing the contribution to the integral to average to zero except near t = 0. A large or means that the integrand of (14) damps rapidly to zero as t increases. Both these factors mean that s(O, w) is dominated by the t = 0 region, where higher powers of t are small. Second, if the higher moments of g(w) ((w"), beyond n = 1) are small relative to the lower moments n = - 1 to 1, then the coefficients of the higher terms in t in (14) will be small. This requires a g(w) that peaks at low w. Low w means weak forces, so that the IA will hold best in systems having weak forces. In particular, we cannot have any "infinite" frequencies leading to infinite moments. We find that the IA holds at the lowest Q, where 0DW and g(w) are lowest. S(Q, o)) satisfies the f - sum rule

I ~ d o o w s ( o , w ) = h O 2 / 2 M

In the IA case, this follows from the presence of the itoRt term in (17),


which in turn follows from requiring g(to) to be normalized. This demon- strates explicitly in the IA case that the f-sum rule is a particle-number- conserving rule.

3. RESULTS

3.1. BCC 3He

In this section the aim is to evaluate Si(Q, to) from a microscopic model ofbcc 3He beginning from the pair He-He potential. The input for Si(Q, to) is the potential, the crystal structure and volume, and T = 0 K. The para- metrization of the He-He potential developed by Beck 23 is used, which has a hard core of height ~106K. Although not quite so accurate 24 as the HFDHE2 potential of Aziz et al., 2a the Beck potential provides a convenient description adequate for our purpose here. The microscopic model 25 is the self-consistent phonon (SCP) theory with a T-matrix treatment of the short-range correlations induced by the hard core. This latter component in the model is most important. With it the treatment of the close approach between two atoms is nonperturbative. It means that wherever the interatomic potential becomes extremely large, the atoms will avoid each other and the atoms do not sample the regions in space of extremely large potential.

The phonon frequencies are evaluated first in the self-consistent harmonic (SCH) approximation. These SCH frequencies, denoted here as toqA, with the cubic term added as a perturbation are used to evaluate the anharmonic response function A(qA, to) in (13) and the anharmonic, one- phonon coherent dynamic form factor S~(Q, to). From SI(Q, to) we define representative, infinite lifetime (real) frequencies by

I ° O3qx= dtotoSl(Q, to) dtoS~(Q,w) (18) - - c o

Using the ACB sum rule 18

I d to toSs(Q, w) = e-ZWh(Q • eqa)2/2M (19)

and

1 SI(Q, to) [F(Q, qA )]2 ~ [n(to) + 1]A(qA, to)A(Q - q)

where F(Q, qh ) = e-2W(h/2Mtoq~)l/2(Q • e~q~), we find that this definition reduces to

~-~ [n(to) + 1]A(qA, to) (20) toqA

568 Henry R. G l y d e

Ld

c U')

2.0

I.O

I I I

bcc 3He

V = 24 (onVmole)

Q = (-g-~) (0.7, 0,0) LONG.

I I

SI(Q'E) ,

/SGH

I 2

SGH

O0 :3 4 5

E (meV) Fig. 1. The self-consistent harmonic (SCH) frequency, the peak position (SCH+C) of the one-phonon dynamic form factor [SI(Q, E)], and the o3 frequency defined by (18) for the longitudinal phonon at Q = (2~/a)(0.7, 0, 0). S,(Q, E) is in arbitrary units.

As shown in Fig. 1, the o3 frequency lies below the SCH frequency but above the position of the main peak of A(qA, to), due to the long, high- frequency tails of the A(qA, to). Using these o3 frequencies, we constructed the density of states ~(to) shown in Fig. 2. Essentially this ~(to) is a reasonable representation of a microscopically evaluated g(to) for bcc 3He beginning from a pair potential having a hard core. For the present purposes, we could equally well have used the SCH frequencies, or the S C H + C frequencies defined as the position of the main peak of A(qA, to).

In Fig. 3 we show the Si(Q, to) calculated from (14) using ~(to) and the corresponding SIA( Q, to) with 0 -2 = WR<to 1>, where (to 1) is the first moment of if(to). From Fig. 3 we see that Si(Q, w) rapidly approaches SIA(Q, w) as Q increases. The deviation DS = Si - S~A from the IA in Fig. 3 due to the final state interactions in Si is chiefly antisymmetric in ( to - toR). Sears2'26 has shown that the leading contribution to DS from final state interactions (~Q-1) is indeed antisymmetric in (to-toR). The next order (_Q-Z) is symmetric in (to - toR). In Fig. 3, the symmetric component of DS is clearly very small by Q = 15 A-1 in bcc 3He. The kinetic energy computed from

3 ^ if(to) in Fig. 1 is (K)=xh(to~)= 15.5 K. Since the O3q~ frequencies given by (18) have infinite lifetime, the full

effect of the high-frequency tails of A(qA, to) is not included in if(to). Their effect is included only in an averaged sense. To simulate the full g(to)


C (meV) 0 I 2 5 4 5

I I I I

bcc 5He

V=24 cm 5/mole

I I I

0 0.2 0.4- 0.6 Q8 1.0 1.2 FREQUENCY (THz)

Fig. 2. The density of states in bcc 3He at V= 24 cm3/mole calculated from the 03 frequencies defined by (18). The 03 frequencies are calculated from the Beck pair potential via the self-consistent phonon theory.

, : t:~n

obtained from (12) using the A(qA, to) having high-frequency tails, we extend g(to) up to high to and round off the spikes of g(to) in an arbitrary, reasonable way. The resulting g(to) is shown in Fig. 4. This g(to) gives a higher kinetic energy (K) = 18.0 K, due to the larger first moment. In Fig. 5 we show the Si(Q, to) at Q = 10/~lk--1 using the g(to) of Fig. 4. Clearly this Si differs little from that in Fig. 3 at Q = 10/~-1 except for a broader peak due to the increased kinetic energy. By Q = 10/~-1, the Si is not greatly sensitive to the details of g(to).

3.2. Solid 4He

Having demonstrated that Si(Q, to) can be calculated from first principles from v(r) and that Si(Q, w) is not greatly sensitive to g(to), we now evaluate Si in solid 4He using model g(w). We choose volumes V=21.1 and 16.0 cm3/mole, for which neutron scattering measurements 28'29 provide empirical values of 0r)w. At both these volumes 4He is in the hcp phase. We choose a model g(w) of the form

g(w) = (w-1)w2[A exp ( - o~to 2) + B exp ( -~w2/4) + C exp ( -a to2/9)] (21)

which has four free parameters A, B, C, and a. The A, B, and C were

570 Henry R. Glyde

×

I o

o ~ d - o

o

h o

o

' " ~0

E

~ )

o

o

( i .Aa w)

i i i ; i

• ~ 0 0 - -

I i 0 ..'

o o d

0

i

li_ ~

)

C3'0) S ~ Oi

o o

o

o o r - -

o

o o

o

oo

o

o

o o 0 J

o

o

~..0-- 0

~o~

II

~4


E: (meV)

0 I 2 5 4 5

/

i i

bcc 5He

V=24 cm 5/mole

0 0.2 0.4 0.6 0.8 1.0 1.2

FREQUENCY (THz)

Fig. 4. Density of states in bee 3He at V = 24 cma/mole calculated from the 03 frequencies with the finite widths and high- frequency tails of the response function A(q)t, to) added.

4.0

--'-', 5 2 t

2.4 E

o2 ( 1 3

~ 0.8 _o

, i , i i

- - ZA

Q=IO A -I

0 ... ; - . , , .... .

. . . . . . . . . . . . . . . . . . . . . . . . 0.150 ........

- 0 . 8 I " I I I

55 50 65 80 95 110 125

E (meV)

Fig. 5. As Fig. 3 at Q = 10 .~.-~ calculated usng the density of states shown in Fig. 4.

572 Henry R. Glyde

TABLE I

Data U s e d i n ( 2 2 ) to Fix g(w) ~ r S o l i d 4He

V 0DW , (K)Mc, cm3/mole K K

21.1 25 ~ 23.0 c 16 50 b 36.6 c

~Ref. 29 bRef. 28 CRef. 27 and extrapolation.

determined by requiring that g(~o) be normalized, that (w-~) reproduce the observed Debye-Waller factor, and that (w) reproduce the calculated kinetic energy of Whitlock et aL 27 That is, from (15) and (16)

(w -1) = 3h/2kOow

(w °) = 1 (22)

(w 1) = (4/3h)(K)Mc

The values of 0DW and (K)Mc used are listed in Table I. The (K)Mc values were obtained by interpolating the Monte Carlo values of Whitlock et al. 27

The g(w) in (21) has the flexibility to display two peaks and a high- frequency tail. The two peaks approximately represent contributions from transverse (or acoustic) and from longitudinal (or optic) phonons. The weight of g(w) in the two peaks can be adjusted by varying a, although a must be kept within a narrow range to keep g(w) positive everywhere. The position of the first peak in g(o)) is given by ~Oo = 21rvo = (1/2or) 1/2.

In Fig. 6 we show the model g(v) for hcp 4He at V= 16 cm3/mole. The a is chosen so that g(v) has a large weight in the high-frequency, "optic" branch range. The Si(Q, w) calculated from this g(v) via (14) is shown in Fig. 7 and compared with the IA. The IA, assuming a Gaussian n(p), is characterized solely by ~r 2 = O)R(W 1) obtained using the g(v) in Fig. 6. Again the difference DS = Si-SIA is chiefly antisymmetric in ( w - o)R). Beyond Q = 15 ~-1 we and that DS scales as Q-~, The DS in Fig. 7 for 4He at V= 16.0 cm3/mole is somewhat larger than the corresponding DS for 3He in Fig. 3. This is because the frequencies are larger in 4He than in 3He, making the higher moments of g(v) more important.

In Fig. 8 we show the model g(v) for hcp 4He at V=21.1 cm3/mole with a chosen to weight the low-frequency "acoustic" branch range. The corresponding SI(Q, w) and IA for this g(v) are shown in Fig. 9. Comparing Fig. 9 and Q = 10 ~-1 of Fig. 7, we see again that DS = S i - SIA depends

Impulse Approximation in Solid Helium

C (meV) 2 4 6 8

I I I I

4H e

' ~ V=I6 cm3/mole

/ \

I I I I I

0 0.4 0,8 1.2 1.6 2.0 2.4 FREQUENCY (THz)

Fig. 6. Model density of states (21) in solid 4He at V= 16 cm3/mole designed to reproduce the observed 0 D = 50 K, 28 the f-sum rule, and the Monte Carlo calculated KE = 36.6, 27 with Uo = (2ct)-1/2/27r = 0.20 THz. The origin ( u = 0) is shifted upward by one unit.

573

little on the precise shape of g(v). The chief difference is that Si(Q, o9) is narrower in Fig. 9 due to the smaller kinetic energy at V= 21.1 cm3/mole. Also, DS in Fig. 9 is smaller than DS for Q = 10 A-~ in Fig. 7 since the frequencies are lower at 21.1 than at 16.0 cm3/mole.

We also made similar calculations for 4He at 9.04 cm3/mole, where 3° 0DW = 164 K. The results are similar, except that the IA approximation is not reached until substantially higher Q, about twice the Q values shown in Fig. 7.

4. DISCUSSION

The results of this paper show that, beginning with a realistic representation of the dynamics of solid helium, the incoherent Si(Q, to) approaches the IA at Q values in the range Q ~> 20 .~-~. This was demonstrated for bcc 3He by using a density of phonon frequencies calculated directly from the pair interatomic potential, which has a hard core. Essential in treating the dynamics adequately is a nonperturbation treatment 3~ of the hard core, here via a Brueckner T-matrix method. Values of Q ~ 2 0 A- I are readily access- ible using present pulsed neutron sources, as demonstrated most recently by Hilleke et al. s

574 Henry R. Glyde

i#)

. . . . ! _~

.__~) 00 G _ b co

i : ~ ' < • ® i : , o ........ o

~" '~"" ,~i" i i / ,~- CO C,] ~0 O eJ " d (~

' ' ~ ' ~ t

/ ~ H ~ e S - ~ E

~ ; ...... o '2 o 2 °

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i i ~ i i _ ° °

= o 0 d 0

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o i , , , - - N

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I :: Lo I iT,

g

Js

2 - - o o O

(I_A aw) (3'b)$ gOl

II

L~

S

L~

L~


6 (mew I 2 5 4 5 6 7 I I I I I I r

4He

cm3/mole - e ~ = 2 5 K

i t 2 t 0 0.3 O J6 0.19 I. 1.5 1.8

FREQUENCY (THz)

Fig. 8. Model density of states (21) in solid 4He at V=21.1 cm3/mole designed to reproduce the observed 0D = 25 K, z9 the f-sum rule, and the Monte Carlo KE = 23 K, with 9 o = 0.15 Thz.

( . f )

('4 0

:t

-I I0

J i i f 7

rA

- - - - - S i ................. DS x4 Q:r0b

'

" , . . , . . . , , i I 1 I I

25 40 55 70 85 I00

E (meV)

Fig. 9. As Fig. 3 for solid 4He at V= 21.1 cm3/mole using g(v) of Fig. 8 for Q = I O A -l.

576 Henry R. Glyde

The Si(Q, to) is also not greatly sensitive to reasonable variation of the density of states g(to). As noted in Section 2, the solid cannot have any infinite frequencies (implying infinite forces), otherwise Si(Q, to) will not approach the impulse approximation. The results show that the lower the phonon frequencies (the weaker the forces), the more rapidly Si(Q, to) approaches the IA. Thus, the IA approximation is reached more rapidly in solid 4He at lower density than at high density and more rapidly in bcc 3He than in 4He. We also checked that the kinetic energy extracted from the width (0 -2) of the calculated Si(Q, to) agreed with that used as an input to construct g(to). This agreement held down to Q = 5 A-1 and is a demonstra- tion that the second moment of (to - toR) is equal 14'15 to 0 -2 independent of whether S~(Q, to) is in the region in which the IA holds or not.

The difference between Si(Q, to) and SIA(Q, to), due to final state interactions, is almost entirely antisymmetric in ( t o - tOR). This is demonstrated in Figs. 3 and 5 for Q = 10 A-~, where the value of the maximum in DS on each side of the recoil energy ER is given. The DS is larger in Fig. 5 because the higher moments of g(to) are larger when g(to) has a high-frequency tail. Sears ~4 has proposed a method of eliminating the lowest order contributions from final state interactions by symmetrizing the observed S(Q, to) about toP.- The present calculations suggest that this method should be very effective in solid helium. The remaining symmetric deviation of Si(O, to) from S~A(Q, to) will then be negligible at lower Q, perhaps as low as 15 ~-~, depending upon the accuracy required.

The IA in Eq. (1) can be obtained from Si(Q, to) by (1) neglecting the potential energy relative to the kinetic energy (i.e., treating the atom as free) or (2) making a short-time (impulse) approximation to SI(Q, t) as was done here in Section 2. These two conditions become strictly true only in the limit as Q ~ ~ . Also, two examples have been presented in which the IA does not hold at Q ~ . Weinstein and Negele ~ have shown that the IA is not reached at Q ~ oo in fluid of hard spheres treated perturbatively. We believe here that the key is a perturbative treatment of the hard core interaction, which leaves the scattering atom exposed to infinite potential changes and therefore infinite forces. A particle described by a plane wave function is able to unrealistically "penetrate" regions of infinite potential when the potential is added as a perturbation. In this case the potential energy of interaction leading to final state interactions can never be negligible compared to the energy transfer ER in the scattering. If the hard core were not quite perfectly hard and could be treated using a Brueckner theory or by a Jastrow method, so that the atom does not actually see regions of infinite change in potential, then the IA would again hold as Q ~ oo. This is the case here for solid helium.


Reiter ~2 has shown that the IA is never reached for a particle subject to instantaneous random impulses. In this example, since the impulse is instantaneous, there is a finite momentum transfer to the particle, however short a time scale we consider. Thus, in this example, a short-time approximation in which forces can be neglected is never reached. Reiter also examined the hard sphere fluid and a particle (HO) bound in a perfectly harmonic well. In the latter case he shows that, since the HO has a discrete set of frequencies, the Si(Q, to) is a set of delta functions. The envelope of this set of delta functions does, however, approach S~A(Q, to). Given a finite resolution width, only the envelope would be observed. Here we have effectively made this averaging by treating g(to) as a continuous function of to.

In summary, provided the potential can be neglected either relative to the kinetic energy or on a short time scale, the IA holds in the limit Q ~ ~ . For solid helium at V ~ < 15 cm3/mole, this limit is reached, for practical purposes, at Q ~ 20 A -1. Also, the best value of the kinetic energy of bcc 3He at 24 cm3/mole will be obtained from Eq. (12). Using the O3qx frequencies to calculate A(qA, to) explicitly gives, at T = 0 K, a kinetic energy of 27 K. This is more than twice that expected from a Debye model using the observed 32 Debye temperature 0D=20K, (K)=(9/16)kOD=llK. The difference between 27 K and the Debye estimate is a direct manifestation of the extreme anharmonic nature of solid helium. This anharmonic charac- ter leads to high-frequency tails in A(qA, to) and therefore to g(to) in (12). The existence of a pronounced high-frequency tail in g(to) due to anharmonic effects in solid helium was inferred by Sears 33 from an analysis of neutron scattering data. The high-frequency tails will contribute strongly to the first moment of g(to), making (K) large, but little to 0DOC(to-1).

ACKNOWLEDGMENTS

It is a pleasure to acknowledge valuable discussions with Dr. W. B. Daniels, Dr. D. L. Price, Dr. R. O. Hilleke, and Dr. S. K. Sinha, and the valuable assistance of J. Lefever with computations and developing the model of g(w). This work was supported by the U.S. Department of Energy, Division of Materials Science under contract DE-FG02-84ER45082.

REFERENCES

1. P. C. Hohenberg and P. M. Platzman, Phys. Rev. 152, 198 (1966). 2. V. F. Sears, Phys. Rev. 185, 200 (1969). 3. H. A. Mook, Phys. Rev. Lett. 32, 1167 (1974). 4. P. Martel, E. C. Svensson, A. D. B. Woods, V. F. Sears, and R. A. Cowley, J. Low Temp.

Phys. 23, 285 (1976).

578 Henry R. Glyde

5. A. D. B. Woods and V. F. Sears, Phys. Rev. Lett. 39, 415 (1977). 6. V. F. Sears, E. C. Svensson, P. Martel, and A. DI B. Woods, Phys. Rev. Lett. 49, 279 (1982). 7. H. A. Mook, Phys. Rev. Lett. 51, 1454 (1983). 8. R. O. Hilleke, P. Chaddah, R. O, Simmons, D. L. Price, and S. K. Sinha, Phys. Rev. Lett.

52, 847 (1984). 9. V. F. Sears, Can. J. Phys. 59, 555 (1981).

10. R. M. Bragger, A. D. Taylor, C. E. Olsen, J. A. Goldstone, and A. K. Soper, Nucl. Instrum. Meth. 221, 393 (1984).

11. J.J. Weinstein and J. W. Negele, Phys. Rev. Lett. 49, 1016 (1982) ; see also T. R. Kirkpatrick, Phys. Rev. B 30, 1266 (1984).

12. G. Reiter, in Workshop on High Energy Excitations in Condensed Matter, Los Alamos National Laboratory, February 1984.

13. J. M. F. Gunn and M. Warner, Z. Phys. B 56, 13 (1984). 14. V. F. Sears, Phys. Rev. B 30, 44 (1984). 15. A. Rahman, K. S. Singwi, and A. Sj61ander, Phys. Rev. 126, 986 (1962). 16. L. Van Hove, Phys. Rev. 95, 249 (1954). 17. A. A. Maradudin and A. E. Fein, Phys. Rev. 128, 2589 (1962). 18. V. Ambegaokar, J. Conway, and G. Baym, in Lattice Dynamics, R. F. Wallis, ed. (Pergamon

Press, New York, 1965), p. 261. 19. R. A. Cowley, Rep. Progr. Phys. 31, 123 (1968). 20. H. R. Glyde and S. I. Hernadi, Phys. Rev. B 25, 4787 (1983). 21. P. Schofield, Phys. Rev. Lett. 4, 239 (1960). 22. H. R. Glyde, in Rare Gas Solids, M. L. Klein and J. A. Venables, eds. (Academic Press,

New York, 1976), Vol. 1. 23. D. E. Beck, Mol. Phys. 14, 311 (1968). 24. R. A. Aziz, in Inert Gases, M. L. Klein, ed. (Springer, Berlin, 1984), p. 5; R. A. Aziz,

V. P. S, Nain, J. S. Carley, W. L. Taylor, and G. T. McConville, J. Chem. Phys. 70, 4330 (1979).

25. H. R. Glyde and F. C. Khanna, Can. J. Phys. 49, 2997 (1971). 26. V. F. Sears, Phys. Rev. A 5, 452 (1972); 7, 340 (1973). 27. P. A. Whitlock, D. M. Ceperley, G. V. Chester, and M. H. Kalos, Phys. Rev. B 19, 5598

(1979). 28. R. A. Reese, S. K. Sinha, T. O. Bran, and C. R. Tilford, Phys. Rev. A 3, 1688 (1971). 29. V.J. Minkiewicz, T. A. Kitchens, F. P. Lipschultz, R. Nathans, and G. Shirane, Phys. Rev.

174, 267 (1968). 30. W. Thomlinson, J. Eckert, and G. Shirane, Phys. Rev. B 18, 1120 (1978). 31. L. H. Nosanow, Phys. Rev. 146, 120 (1966). 32. D. S. Greywall, Phys. Rev. B 15, 2604 (1977); 16, 5129 (1977). 33. V. F. Sears, Solid State Commun. 11, 1307 (1972).

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