an amorphous state model for liquid helium

Journal of Low Temperature Physics, Vol. 18, Nos. 1/2, 1975

An Amorphous State Model for Liquid Helium

B. Castaing

Groupe de Physique des Solides* de l'Ecole Normale Sup~rieure, Paris, France

and Y. Sawada

Electrical Communication Laboratory, Tohoku University, Sendai, Japan

(Rece ived Ju ly 19, 1974)

In order to achieve a comprehensive model for the elementary excitations in liquid helium, we modify the original cell model of Matsubara and Matsuda. We avoid the lattice anisotropy, taking an amorphous arrangement for the sites, with f cc short-range order. We compare our results to the experimental data and examine the limitations of the model.

1. I N T R O D U C T I O N

Numerous models of liquid helium-4 have been developed in the last twenty years. The properties of the ground state were studied by McMillan 2 and the elementary excitations were described by Bogoliubov 5 and Feynman. 4'5 Such calculations have been improved by Sunakawa et al. 6 Introducing the phonon-phonon interaction, they obtain a very good fit with the experimental dispersion curve. In their model a pseudopotential between atoms is introduced,

At this level the real problem is to develop a physical interpretation and try to answer questions such as : Are the rotons a simple compressional wave or a more complex entity like the limit of a vortex ring? Is it possible to have a wavelength shorter than the average distance between atoms (3.5 A)?

Our main effort in this paper is to try to give a simpler model, hoping thus to achieve a more intuitive understanding of the problem of helium.

In 1954, Matsubara and Matsuda (MM) 1 proposed a lattice model for superfluid helium. A good agreement with the experimental results was obtained for the pressure dependence of the transition temperature. They

* L a b o r a t o i r e associ6 au C e n t r e N a t i o n a l de la Reche rche Scientif ique.

159

�9 1975 Plenum Publishing Corporation, 227 'West 17th Street, New York, N.Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher.

160 B. Castaing and Y. Sawada

also calculated the sound velocity, the value being anomalously low.* The calculation of the dispersion law of the elementary excitations for large wave vector was impossible, due to the nonphysical lattice anisotropy.

In this paper, starting from a slightly different M M model, the elementary excitation dispersion law and the ground state are calculated and compared to the experimental results.

2. A M O R P H O U S STATE M M M O D E L

In order to avoid the difficulties due to lattice anisotropy we have taken an amorphous model: The helium atoms, of mass m, can only be on sites .A r without periodic arrangement but with a distance d between nearest neighbors. The distance between second neighbors is d' = dx/2 if the local arrangement is fcc. We take the M M Hamil tonian

H = (3h2/md 2) ~ a+a~ - (3h2/2Zmd 2) ~, (a+aj + afar) i <i,j>

+ lw t Z a~-alafaj + �89 Z a{aiafaj ( i , j> <i , j ) '

where <i,j> (<i,j>') means that i and j are first (second) neighbors, Z is the number of first neighbors, and a + is the creation operator on the point i. The hard core of the helium atoms is taken into account by a +2 = 0. The first term in the Hamil tonian is the localization energy when the a tom is surrounded by occupied sites. The second term allows the a tom to go from one site to another. The last two terms are first and second neighbor potential energies.

3. G R O U N D STATE AND E L E M E N T A R Y E X C I T A T I O N S

In this section we calculate the ground state and the excited states. Our calculation will be conducted in three stages: First, restricting ourselves to a subspace of states, a type of "mean field" approximation, we obtain the best possible approximat ion of the real ground state. Second, we construct a Hamil tonian valid for states not too different from the approximate ground state obtained previously. Finally, we diagonalize this Hamiltonian.

In a mean field approximation, one considers each site as independent of the others. The state of a site is thus (ui + via~-)]Oi> where ]0i> is the state of the site i when it is empty (u~ + v~ = 1). The state of the Y points in this approximat ion is I:~b> = ~i (ui + viai~-)lO> where 10> = Hi ]0i> is the empty state (in the following, [~b> always denotes Such a state).

*The numerical value for the velocity of sound, Eq. (4.8) of Ref. (1), was incorrectly enumerated as 2 x 10 a m/s in (1). The true value should be 60m/s.

An Amorphous State Model for Liquid Helium 161

It is useful to define here the operators 7 + :

7~[t/,> = (--vi + uia~-) [ ] (u i + vja+)r0>

~,,10> = o , ~/?2 = o

?/+[~,> is o r thogona l to t~b>. We shall use linear combina t ions of the 7 [ such as (Rj is the posi t ion of the site j)

72 = ( 1 / x / ~ ) ~ exp (ik. R)7 + )

The signification of ~,~ is clear in two extreme cases, the empty state or gaseous state and the total ly occupied state or solid state. In the gaseous state 7~ represents the creat ion of an a tom with the impulse hk. In the solid state 7~ is the creation of a vacancy wave of wave vector k.

The most general state for the system is then a combina t ion of [0>,

�9 " , T u , k ' V ) , ' ' ' ' We return to our p rob lem of finding the g round state for N a toms in

the system. We have thus to minimize <qS[H[4> for any general state I~b> keeping <~b[Xopl~b> = g = cste (Nop = ~ i a[ai is the opera tor " n u m b e r of atoms") . It is equivalent to minimizing (~blg -/~govj~b> with # chosen so that (qSo[goplqSo> = g ([r is the g round state).

A first approx imat ion of IqSo> is the state [~/~o> that makes ( 0 o I H -pNopl0o> extremal in the mean field a p p r o x i m a t i o n Matsuda and Tsuneto 7 show that [0o> is unifortn in the liquid state (ui = u, v~ = v):

@ o l H - pgopl0o> = . [_lind2 + 2 + Z ' v4 - pv 2 (1)

The min imum is for v 2 = ju/A with A = 2[(3h2/md 2) + �89 + �89 Here Z' is the number of second neighbors (in our model Z = 12, Z ' = 6). So

@'o[H - /~gop[0o> = Fo = - N/a~2

We can now define the ?~" and develop H - pNop'

A ~ + + . . . H - - ] /Nop = r 0 -1- Z [~k() ' ; 71~ -]- ~2 _+ k S - k) -~- kt~)k ~)- k -~ ~)k~- k)] -1- (2) k

where

~'k. = < 0 0 1 7 k ( ~ / - , - N o p ~ r 10o>

Ak (~ol (H N + + = - ~ op)~ 7 - ~ [ 0 o >

We neglect terms such as 7s (k # k') or 7s (k # - k ' ) . They are of the order 1 /v / -Y because of the a m o r p h o u s structure (a periodic structure

162 B. C a s t a i n g and Y. S a w a d a

gives terms like 7~-?k+G where G is a reciprocal lattice vector). The other terms in the deve lopment of the Hami l ton ian are of the order 1 /w/~ : and are thus also neglected. We obtain

3h 2 [6h 2 ) e k = ~ 5 ( 1 - (cos k . d}) + imd2 + ZWa b/2vZ(cos k . d)

~-" Z'~'~72/A2/)2(COS k . d ' )

Ak i~d~ + zw1 u2v2@os k . d} + Z'W2u2vZ@os k . d ' )

(cos k . d) = sin (kd)/kd

(the angular brackets denote the mean value when the direction of R~ - Rj = d or R k - R,, = d' varies" i, j (k, m) are first (second) neighbors).

The reason for the appearance of terms like +6 + AkTk Y-k is simple. In the liquid state (uZv z # 0), the ? excitat ions are neither a toms nor holes. The Hamil tonian , which conserves the number of atoms, cannot conserve the number of 7 excitat ions and pair creat ion and destruct ion of 7 excitations O c c u r .

The Hami l ton ian (2) is of Bogol iubov type, and its diagonal izat ion is classical. However , we remark that the ground state of the Hami l ton ian (1) is not free of ?+ excitations, and this modifies [?k,?{;']: We have 17~,?j-I = 6o.(1 - 27/7i); thus

['/k, 7k'] = 6k'k'( l - 2 y Y"i 7+7i) a ~ k . k ,

We look for the e lementary excitat ions ~ - of the form c~- = uu7 ~- - vk?,_u with [ek, e+] = a(u~ - v~) = 1. We replace ?+ by its value

+ + vanish for in (2). The terms % c~_ k

2 1/2 2 2 [eu + (e 2 - Au) 3/2a(eu A2) 1/2 U k ~

and then

A 2 ~ I / 2 N + N H - / ~ N o p = F o - a Z [ e k - ( ~ [ - A ~ ) l / 2 ? + a ~ ( a 2 - - ' - ' k , ~k~k k k

The ground state IqSo) can now be defined by ~kl~o) = 0 for any k. Thus

/Q g + + 2 + n- 14~o) = I~ Xk[1 + ( k/Uk);k 7-k + (Vk/Uu) (Tk 7 - 0 + " ";L00) k


The normal iza t ion gives 2~/[1 - (Vk/uk) 2] = 1 ; thus 2~ = l/au~ and

I, o) = 17 {(1lUke) (3) k n

Suppose for a time that y~- is a fermion operator . The terms with n >_ 2 disappear in (3). I f 100) described the electron gas of a metal in the normal state, 10o) would be the BCS superconduct ing g round state. In our case, we can say that the 4He a toms are "associated by pairs" in the g round state. This p robab ly cor responds to the fact tha t a p roduc t of pair functions is a g o o d approx imat ion for the wave function of the g round state. 1 The size of these pairs is abou t 2n/ko, where k 0 maximizes (Vk/U~)k 2. The value of k o is about 2 ~ - 1 and the size of the pairs is 3 A.

4. D I S P E R S I O N CURVE

The elementary excitation energy is

E k = a(e~ - A~) 1/2

(3h 2 1 sin kd~ [-3h 2 / sin kd] HZu2h 2 sin kd

2 2 ; s i nkd 2 Z ' W d ' 2 2 s inkd '3 ) I /2 +

At low k, E k = hck, where k = [kl and

c = a ( + W(d) + W(d')

is the sound velocity. We take the de Boers-Michiels s Lenna rd - Jones potential for helium :

W(d) = 4el(a/d) 12 _ (a/d)63

where e = 10.22K, a = 2.556A, Z = 12, Z ' = 6, and d' = ,,//2d. If we fit to the experimental rat io between the m a x i m u m and the min imum of the curve, we obtain d -~ 2.5 A. Then the posi t ion of the m a x i m u m is at 1.2 A-- 1 and the m i n i m u m is at 2 A - 1, in g o o d agreement with experiment (Fig. 1). But the ampli tudes are in p o o r agreement, giving 6.3 K for the m a x i m u m of the curve, while the experimental value is 13.9 K. The sound velocity is 80 m/sec (experiment yields 237 m/sec). (We take a = 1 because it is not very different f rom 1 in a lattice model. 9)


('KI

10

5

IEk

J /

/ /

I 1

I

Fig. 1. Excitation spectrum when Eq. (1) is taken for the kinetic energy. The upper curve is the experimental one following the neutron experi- ments of Henshaw and W o o d s ) 0

5. MODIFICATION OF THE MODEL

We thus have poor agreement with the experimental results. It is possible to improve this by changing the expression of the kinetic energy in (1). Matsuda and Tsuneto 7 introduce contributions from second, third, etc. neighbors and the kinetic energy becomes

K ~ a~ a i - ~ uij(a~-a j + a+ ai) (4) i i; jV:i

('K)' ~ E k

, 7 ,/// v

I I 1 2 k (~,-17

Fig. 2. Excitation spectrum with the improved form for the kinetic energy, Eq. (4). The upper curve is the experimental one.


with K = Z j ~ i blij" The aim of this approximat ion is to replace (3h2/md 2) x [1 - sin (kd)/kd] by h2k2/2m for any k, which is evidently a better approxi-

mation. In a simple cubic lattice the value of K is rcZh2/2md 2. We can take this as the approximate value.

The agreement with experiment is now better (Fig. 2). The positions of the max imum (1.1 A -1) and the min imum (1.8A -*) and the velocity of sound (190m/sec) are close to the experimental values. The maximum energy (10.4 K), al though low, has a better value.

6. C O N C L U D I N G R E M A R K S

Although the agreement is good, we have some puzzling problems. First, in our calculation, for a given intersite distance d the chemical potential /~ and the sound velocity squared c 2 are of the same sign :

/,t = 2 v 2 / 2 ~ 2 + ~-W a + T W z

C2 2U21)2a[ rc2h 2 Z Z' ) - ~ l ~ + ~ w l +~-w2

But the pressure P at zero temperature is given by (V is the volume)

P V = - (4o[H - ] ~ / ~ r = �89 + a ~ [e k - - ( • 2 - - A2)1/2] k

Thus the chemical potential must be negative to give a nonvanishing density at zero pressure and cannot be of the same sign as c 2. A possible way to avoid this paradox will be considered later.

Second, we note that our excitations are labeled by a wave vector k. What is the max imum value of k, kin? The excitations are linear combinations of the y+. Thus there are Y modes of elementary excitations and k3~ = 67r2(~M/V). With our numerical values this gives k,, = 1.5 A - 1, which is less than the position of the roton minimum. In reality, for any k and k', (~b0l~kT~[~b0) is of order l / J ; , which we take as negligible. But the part of 7~ [~o) that is or thogonal to all 7~, [~) with k' < k gets smaller as k increases So the development (2) is not valid and E k loses its significance for very high k. If we compare this with an electron in a crystal, we obtain translational states with k as large as we want by combining several atomic electronic states s, p, d , . . . . So, in our model we need another state, like a p state on each point to get a better description of the roton par t of the dispersion curve.

We come back now to 1Lhe chemical potential paradox. A possible way to avoid it is to vary the intersite distance d from place to place. Since/~


C'Ki

10

IEk ///Z j, I I 1 2 kC -l]"

Fig. 3. Excitat ion spectrum when d varies from place to place. Here/~ = - 7.3 K and the density p = 0.145 g/cm 3. The upper curve is the experimental one.

and c 2 do no t have the same dependence on d, it is possible to obtain/~ < 0 and r > 0 (we assume a cons tant density : d3v 2 = Cte).

The calculat ion is identical to that in the preceding section. The average is now to be taken on the modulus and the directions of d. We must choose the limits of the var ia t ion of d. The upper b o u n d a r y is the value of d, d l , for which we must fill all the points in order to agree with the true density of helium. The lower b o u n d a r y d o will be chosen to give a negative/~:

# = ((v2[(Tr2h2/md 2) + ZW(d) + Z'W(d')]))

((()) defines the average on direct ion and modu lus of d) close to the experimental value (# = - 7 . 3 K for p = 0).

The elementary excitation energy is now

( h 2 k 2 [-hak 2 rr2ha //u2l)2\\

sin kd 2v 2 + 2 Z + ( d ) ~ d - U ~ ' / / , 2 2sinkd'\\~} U2 + 2Z ~(,,xW(d)uv ~ - 2 ~

We now compare with experimental results. Fo r p = 0, the density of hel ium is 0.145 g/cm3 ; thus d 1 = 4.02/~. We take do = 2.182 A such that /~ = - 7.3 K (Fig. 3). The value of the m a x i m u m energy is reasonable. But the sound velocity (100 m/sec) and the posi t ion of the m a x i m u m (1.3 A - 1) then are not in good agreement with experiment.


A C K N O W L E D G M E N T

The authors are grateful to Dr. Libchaber for his constant interest in this work and many discussions. One of the authors (Y. Sawada) would like to express his hearty thanks to Professor Bok and his group for the warm hospitality extended 1:o him during his stay in Paris.

REFERENCES

1. T. Matsubara and H. Matsuda, Prog, Theor. Phys. 16, 569 (1954). 2. W. L. McMiUan, Phys. Rev. 138A, 442 (1965). 3. R. P. Feynman, Phys. Rev. 94, 262 (1954). 4. R. P. Feynman and M. Cohen, Phys. Rev. 102, 1189 (1956). 5. N. N. Bogoliubov, J. Phys. (USSR) 11, 23 (1947). 6. S. Sunakawa, S. Yamasaki, and T. Kebukawa, Prog. Theor. Phys. 41, 919 (1968). 7. H. Matsuda and T. Tsuneto, Prog. Theor. Phys. Suppl. 46, 411 (1970). 8. J. De Boer and A. Michiels, Physica 5, 945 (1938). 9. R. T. Whitlock and P. R. Zilsel, Phys. Rev. 131, 2409 (1963).

10. D. G. Henshaw and A. D. B. Woods, Phys. Rev. 121, 1266 (1961).

an amorphous state model for liquid helium

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