THEORY OF FOURTH ORDER STRUCTURE AND RAMAN SCATTERING IN LIQUID 4He

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THEORY OF FOURTH ORDER STRUCTURE AND

RAMAN SCATTERING IN LIQUID 4He

C. Campbell, F. Pinski

To cite this version:

JOURNAL DE PHYSIQUE

Colloque

C6,

supplement au n°

8,

Tome

39,

aout

1978,

page

C6-233

THEORY OF FOURTH ORDER STRUCTURE AND RAMAN SCATTERING IN LIQUID 4

H e+

C.E. Campbell and F.J. Pinski°

School of Physios and Astronomy, University of Minnesota, Minneapolis, Minnesota SS4SS, U.S.A.

Résumé.- On montre que l'intensité IR de la diffusion Raman de la lumière de ''He liquide est

sensi-ble en même temps à la structure à courte portée du liquide et à la partie à courte portée de l'in-teraction entre la lumière et deux atomes d'hélium. L ' I R calculée en utilisant le modèle théorique le plus réaliste est de 40 % plus grande que la valeur donnée par l'expérience.

Abstract.- It is shown that the Raman intensity IR form liquid ""He is sensitive to both the

short-range structure of the fluid and the short-short-range part of the interaction of the light with two He atoms. The calculated I R for the most realistic model is 40 % larger than experiment.

The second order Raman scattering intensity from simple fluids is an integral of the fourth or-der dynamic structure factor of the fluid weighted by the coupling of the fluid to the light/1-2/ : Io(l0) = K / dk dq Y 0 0 Y* ,(q)B. B„ S„(k,q,w) (1)

R 2>2 ^> k q

Where K is a proportionality constant, 6, is the fourier transform of the anisotropic pair-polariza-bility 6(r), and the fourth order dynamic structure function Si,(to) is the frequency transform of S^(t) :

S„(k,q;t) = <{p+(t)p_£(t) - <P£P_£>}

{p+(0)p -KO)-<p+p +>}>/N2 (2)

q -q q -q

where p+ is the density fluctuation operator :

N -»• •+ ~ .

p£ = E. exp (l k.r.)- The function Si, is a probe of the two-excitation states of the systems, since a major part of the two-excitation eigenstate with total momentum zero is given by LPJP.i <P~*'P_">"3'''n

where f is the ground state wave function/1/. There have been several calculations of ID(u)

R using equation (1) as a starting point and introdu-cing approximations for Si, /2-3/. For example, Ste-phen studied the two roton and two maxon region of the spectrum by using the decoupling approximation

HI

S„(k,q;t)

-»

{N'^pgCOp^CO)^}2

<.&Z*&%

t

_+)

(3)

where N_1<p^;(t)p_^(0)> is the time fourier

trans-form of the single quasi-particle contribution to the dynamic structure factor S(k,u) measured in neu-tron scattering. He then used the dipole-induced-dipole approximation for $(r) : 6nTn(r)=6a2/r3,

(where a is the atomic polarizability of the helium atom) ; Bn T n was cut off at small r at some value a_.

While these results are in fair qualitative agree-ment with experiagree-ment/4/, showing a maximum at twice the roton frequency and the suppression of the maxi-mum at twice the maxon frequency, I„(u)) is overesti-mated by an order of magnitude.

Since I_(u) is an integral of the product of R

the anisotropic pair polarizability and the correla-tion funccorrela-tion, it is not clear whether this overes-timate comes from small r approximation for B(r) or the decoupling approximation for S\. Indeed, the two approximations are intertwined by the fact that the-re is a delicate interplay between short-range cor-relations in Si, and the short range structure of B(r). For example, the fact that the wave functions for the liquid become vanishingly small upon the clo-se approach of two particles has the conclo-sequence that a cutoff in (3(r) should not be necessary. Bae-riswyl, however, pointed out that these results are very sensitive to the location of the cutoff/5/, an indication that the decoupling approximation sacri-fices important short-range correlations. Similarly Kleban showed that noninteracting quasi-particle ap-proximations such as Stephen's don't produce corre-lation functions which vanish upon close approach of the particles when fourier transformed back to coordinate space/6/. Finally, one should note that the 8n T T ) form for 6(r) is valid only at large r ; important modifications of 8(r) occur at short and intermediate wavelengths/7/. Corrections to all of these approximations involve wavelengths near the roton wavelength, and thus should be expected to ha-ve significant effects in the two roton spectrum.

To investigate the sensitivity of the theory Supported by U.S. National Science Foundation

Grant DMR 76-14777

Present address : Physics, Suny, Stony Brook, N.Y. 11796, USA.

t o t h e s e c o n s i d e r a t i o n s , we have t a c k l e d a somewhat a n o n - i n t e r a c t i n g q u a s i - p a r t i c l e model, t h e n t h e l e s s ambitious problem, t h e t o t a l Raman i n t e n s i t y

IR =

/

I (U) dw. Then e q u a t i o n ( I ) i s r e p l a c e d by R t h e s t a t i c v e r s i o n -+ -+ 1, = K ' dk dq

Y

2 , 2 ( P ) Y : , ~ ( c ) B ~ B ~ s ~ ( ~ , ~ ) ( 4 )

-+

+

where S ~ , ( k , q ) i s t h e f o u r t h o r d e r s t a t i c s t r u c t u r e

-

f a c t o r SI, (k,q;O)

.

-+ -f

A decoupling scheme f o r Sb(k,q) comes r a t h e r n a t u r a l l y o u t of Wu's c l u s t e r a n a l y s i s of t h e densi- t y f l u c t u a t i o n o p e r a t o r s , where i t i s found t h a t /8/

s t r u c t u r e of U,, g i v e s some s i g n of t h e important in- t e r a c t i o n r e g i o n s . Contour p l o t s of t h e S and D wave p a r t s of UI, a r e i n t h e f i g u r e . The most n o t a b l e fea- t u r e of t h e s e r e s u l t s i s t h e l a r g e , n e g a t i v e e x t r e - mum a t k = q

-

2 A-', t h e r o t o n wave number. This i s p r i m a r i l y r e s p o n s i b l e f o r lowering t h e c a l c u l a t e d i n t e n s i t y t o a f a c t o r of 2-25 g r e a t e r t h a n experimea u s i n g

BDID

; s h o r t range c o r r e c t i o n t o

B

reduce i t t o 1-5 times experiment.

+ - P

SI+(L*,;) = ~(k)'{6~,76$-;;3 + N - ~ U I , ( k , q ) ( 5 ) where S(k) = ~ - ' < p - + p -+> i s measured by X-ray o r in-

k -k

t e g r a t e d n e u t r o n s c a t t e r i n g . Although UI, i s O(N) b o t h terms i n e q u a t i o n (5) c o n t r i b u t e t h e same o r d e r i n N t o IR because of t h e sums i n t h e e x p r e s s i o n f o r

IR. The n a t u r a l decoupling approximation h e r e would b e t o s e t UL,=O. While t h e dynamic v e r s i o n of t h i s decoupling d i f f e r s from Stephen's i n t h a t a l l of < P ~ ( + ) P - ~ ( O ) > would b e used i n e q u a t i o n (3) ( i n s t e a d

-

of <pk(+) ~ ~ ~>I) n e a r twice t h e r o t o n frequency ( 0 ) t h e r e s u l t i n g approximations f o r S4 would be n e a r l y t h e same.

To i n v e s t i g a t e c o r r e c t i o n s t o t h i s decoupling,

+

-f

we have c a l c u l a t e d U4(k,q) and used i t a l o n g w i t h s e v e r a l choices of B(r) t o c a l c u l a t e t h e t o t a l Ra- man i n t e n s i t y IR,. U 4 i s c a l c u l a t e d u s i n g a J a s t r o w wavefunction,

Y

= exp u ( r i j ) , f o r t h e ground

0 2 i < j

s t a t e , where u ( r ) i s chosen t o minimize t h e ground s t a t e energy/8/. We used b o t h a Monte-Carlo c a l c u l a - t i o n and a h y p e r n e t t e d c h a i n based i n t e g r a l equa- t i o n t o c a l c u l a t e U I , / ~ / . The r e s u l t s f o r t h e t o t a l Raman i n t e n s i t y i n t h e two c a l c u l a t i o n s d i f f e r by l e s s t h a n 5 %, s o t h a t t h e key approximation i s t h e c h o i c e o f wavefunction. We i n v e s t i g a t e d both

B

D I D and t h e form proposed by Oxtoby and G e l b a r t / 7 / t o account t h e s h o r t range s t r u c t u r e ' :

BoG(r) = BDID(r)

-

6.22

i3

exp (-r/0.477

i)

The r a t i o of t h e Raman i n t e n s i t y (summed over f i n a l p o l a r i z a t i o n s ) t o t h e B r i l l o u i n i n t e n s i t y was c a l - c u l a t e d a t e q u i l i b r i u m d e n s i t y t o b e 0.0037 u s i n g

BDID

and 0.0025 with

BOG,

compared t o 0.0016 i n t h e e x p e r i n k n t of Greytak and Yan/4/. There i s a r a t h e r weak d e n s i t y dependence i n IR, i n c r e a s i n g by 4 %

u s i n g

BDID

b u t d e c r e a s i n g by l e s s t h a n 3 % u s i n g

BOG

over t h e e x p e r i m e n t a l ' d e n s i t y range ( p

maxlf'o =

1.19).

I f t h e decoupling approximation i s viewed a s

F i g . 1 : Contour p l o t s of t h e S- and D-wave Legendre p r o j e c t i o n s of Us(k,q)/N a t p = 0.0218

A-3.

Shaded a r e a s i n d i c a t e 0 ( U,/N

-

<

0.5:

I n summary, o u r c a l c u l a t i o n s of t h e t o t a l Raman i n t e n s i t y from ' ~ e have shown t h a t important q u a l i t a t i v e and q u a n t i t a t i v e c o n t r i b u t i o n s come from s h o r t range s t r u c t u r e , w i t h t h e roton-roton r e g i o n c o n t r i b u t i n g a major p o r t i o n . We have a l s o shown t h a t p h y s i c a l m o d i f i c a t i o n s of t h e dipole-induced- d i p o l e p a i r p o l a r i z a b i l i t y make c o r r e c t i o n s of t h e o r d e r of 40 % t o t h e t o t a l i n t e n s i t y .

Re£ e r e n c e

/ 2 / Stephen M., Chapt.4 of P h y s i c s of Liquid and So- l i d ~ e l l u m , ed. K.H.Bennemann and J.H.Ketterson, Wiley I n t e r s c i e n c e (1977).

/ 3 / Fetter,A.L., J.Low Temp:Phys.? (1972) 487. / 4 / Greytak,T.J. and Yan,S., Phys.Rev.Lett. z ( 1 9 6 9 )

987.

/5/ Baeriswyl,D., Phys. L e t t . (1972) 297.

161 Kleban,P., Phys.Lett. (1974) 19.

/7/ Oxtoby,D.W. and Gelbart,W.M., Molec. Phys.

30

(1975) 535.

/8/ Wu,F.Y., J.Math.Phys.

12

(1971) 1923. 191 Chang,C.C. and Campbell,C.E.,Phys.Rev.

B15

(1977) 4238.