neutron scattering by 4he and 3he

Physica 137B (1986) 126-140 North-l'lolland, Amsterdam

NEUTRON SCATI'ERING BY 4He AND 3He

E.C. S V E N S S O N and V.F. S E A R S

Atom/( Enerj O" ~1 Canada l.tmtted. ('halk Rtr('r, Ontario. ('anad¢1 KOJ 1,10

During the last decade, a surprising amount that is nev, and exciting has been learned about liqu,,l .a lie primarily as a rc~,uh of extensive and detailed neutron-scattering measurements of the dynamic and slaUc structure factors The l'irxl neutron studies on liquid ~He have also given us considerable information ahou! the zero-sound and spin-fltlclu;ltion excmttions in this J:ermi fluid. Selected aspects of this ,xork are described and future directions indicated.

I. Introduction

The hel ium isotopes, 4 He (Bose part icles of spin 0) and 3He (Fermi par t ic les of spin 1/2) , in both their solid and l iquid phases cont inue to be of high interest to physicists in general and neutron scat terers in par t icular . To date, neu t ron-sca t te r ing measurements on these quan tum fluids and solids have been repor ted from at least 15 different es tab l i shments with the names of approx ima te ly 85 different au thors appear ing on well over 100 papers . Liquid 4He has been far more extensively s tudied by neutron scat ter ing than any other material . It was only a short while af ter the first sys temat ic app l ica t ions of neu t ron-sca t te r ing tech- niques to the s tudy of p rob lems in condensed- mat te r physics by Wollan, Shull and co l l abora to r s [1] at Oak Ridge Na t iona l Labora to ry that the first neutron measurements on liquid 4He were repor ted [2] from Los Alamos in 1951. These trans- miss ion-measurements were soon followed by dif- f ract ion measurements such as those of Henshav< and Hurst [3] in 1953 and 1955. Then came the ext remely impor t an t inelas t ic-scat ter ing measurements of Palevsky et al. [4], Yarnel l et al. [5] and Henshaw [6] which verified that superf luid 4He d id indeed have a d ispers ion relat ion of the "' phonon- ro ton" form envisaged by Landau [7]. In near ly 3~ decades of extensive and almost con- t inuous s tudy, a p rod igous wealth of in format ion about l iquid ')He has been accumula ted . Vir tual ly every type of neutron spec t romete r has been used

0378-4363/86/$03.50 ' Elsevier Science Publishers B.V (North-Holland Physics Publishing Division)

for these measurements and the neutron energies employed have spanned a nmge of l0 s from ul t ra-cold neutrons of 10 " eV [g] to neutrons of near ly 20 eV [9]. In spite of this history, new and excit ing discoveries are still being made for this fascinat ing material . In fact, a great deal has been learned over the last decade largely as a result of extensive and deta i led studies of the t empera tu re dependence of the dynamic structure factor. S (Q, ~), and the static s tructure factor, S (Q) , (hQ and ho~ being, respectively, the mornentum and energy transfers).

In cont ras t to liquid 4He, neutron studies on liquid ~He are still in their early youth with the first results having been repor ted [10] in 1974. Studies on 3He are extremely' difficult because of the enormous absorp t ion cross section ( - - 11000 barns for 4 ~, neutrons). In spite of this, we now know, as a result of studies carr ied out at the lns t i tu t l ,aue Langevin [10.-12] and at Argonne Nat iona l L a bo ra to ry {13-16], a cons iderab le amoun t about the zero-sound modes (densi ty f luctuat ions) and pa r a ma gnon modes (sp in-dens i ty f luctuat ions) in this unique Fermi fluid.

In this art icle we will concent ra te pr imar i ly on a selection of the most exci t ing results of the last 7 8 years for liquid 4He and liquid 3He which, we feel, reflect the most impor tan t recent advances from neut ron-sca t te r ing studies of quan tum fluids and solids. We will first deal with liquid aHe, then with liquid ~t-le. and in the final section we will call a t tent ion to several areas where further neu-

E.C. Soensson and V.F Sears / Neutron scattering t)v 4He and 3He 127

tron measurements are likely to make major con- tributions. For more complete coverage of neutron studies on quantum fluids and solids, the reader is referred to numerous review articles [17-21].

2. Liquid 4He

Following the early work mentioned above, the inelastic scattering by liquid 4He was studied in great detail through the 1960's particularly at Chalk River Nuclear Laboratories. The situation at the beginning of the 1970's is partially summarized by fig. 1 taken from the 1971 paper by Cowley and Woods [22] which is still one of the most useful and widely quoted papers in the helium literature. By this time there were = 100 points (see fig. 7 of [22]) along the branch of sharp (essentially ~-func- tion) phonon-roton excitations which is shown simply as a solid curve labeled "one phonon" in fig. 1 and which dies out beyond = 3.6 ,~-1. Above this branch there is a broad band of "multiphonon" scattering which is centred at an energy somewhat higher than twice the energy, A of the roton minimum (at Q = 1.9 ,~- 1) for Q values up to ---2.3 A-1 and which then rises rapidly to be

IO0. I

2M

80 . . . . RCS

60..

÷ T~CS ~2- HEIGHT ~ /

/ 40~ , . .

tl~ - " hi Z ta t - / " " i i

2o- ~ / 4

• ~ y • ONE PHONON i

I i , , / . . . . T" . ~

c i 2 3 4 0 ( j~- i)

Fig. 1. The dispersion relation for the sharp one-phonon excitations in liquid 4He at 1.1 K and s.v.p, and, at higher energies, the upper and lower half heights and the mean energy of the broad multiphonon peak in S(Q, ~o). From ref. [22].

centred rather close to the dispersion relation for free 4 He atoms shown by the dashed curve.

Although we already had a great wealth of information by this time, many important questions remained unanswered. For example: (i) What was the shape of the one-phonon curve at low Q; in particular, did it curve upward (anomalous dispersion) or downward (normal dispersion) from the velocity-of-sound line? (ii) How did S(Q, to) vary in detail with pressure and temperature, and why did there appear to be no dramatic change in S(Q, to) in passing through the superfluid transition temperature, T x [2.17 K at saturated vapor pressure (s.v.p.)]? (iii) Why did both the width and the central position of S(Q, to) at large Q (> 3.5 A - ] ) oscillate with Q (see figs. 21 and 22 of [22]) rather than behave as expected from the Impulse Approximation, and why was there no evidence of a sharper central component in S(Q, to) at large Q corresponding to scattering by the zero-momentum atoms in the Bose condensate as proposed by Hohenberg and Platzman [23]? Answers to these and other important questions have been obtained from subsequent neutron-scattering studies as we shall now discuss.

2.1. Anomalous dispersion

The shape of the one-phonon dispersion curve at low Q is extremely important for the understanding of the interactions between the excitations in liquid 4He. If there is anomalous dispersion (i.e., if the one-phonon curve lies above the velocity-of-sound line for some range Q ~< Q~) then 3-phonon decay processes are possible; otherwise, the lowest-order allowed processes are the much less probable 4-phonon processes. Although in- direct information about the form of the dispersion can be inferred from several types of measurements, only an accurate determination of the dispersion relation by neutron scattering can give a direct answer. There was a hint of anomalous dispersion at s.v.p, in the measurements of Cowley and Woods [22] but the accuracy was not sufficient to establish this with any certainty. The first convincing evidence came from the study by Svensson et al. [24] which covered the full range of pressures. The situation was further clarified by

12g I 5. ('. Srens,s'on and V. I-" ,S'('ar~ / ,%'eutron ~( attermff t~v 41h' and 'lh"

I I I I I - T . . . . . . I I

~-- 2 6 0

' ' /*He E 12K > -

~J

o 240

>

< ~ . , - ~ - - . . , z2o fl

z £ / " "

0

z = " ' . . . . 'J '%1 2 O0 I t _ I l ~ .1. ~ . . . .

02 04 06 08

WAVEVECTOR ( ~-1 )

Fig. 2. Phonon phase velocities, ~ / Q , for l iquid 4He at 1.2 K

and s.v.p. The open square shows the sound ,¢ehx:itv and the

insert the devia t ion from this value. F rom ref. [26].

the study of Stirling et al. [25] whose measurements extended up to 20 bars and were particularly good at s.v.p. Recently, Stifling [26] has carried out an even more accurate study at s.v.p. obtaining the results shown in fig. 2. We see ver'v clearly that there is anomalous dispersion up to (..) = 0.55 A ~ substantiating the earlier value. 0.52 A I. of Svensson et al. [241.

2.2. Temperature dependen(e qf S(Q, oa) at h,w Q

Studies of S(Q, ~) carried out prior to 1978 appeared to indicate that nothing dramatic hap- pened on passing through 11\ and suggested that " 'roton-like" excitations persisted well above T~. This was puzzling since rotons had originally been

2 0 0 0

I 000

5OO

2 0 0

iO0 ~ " " " ' "

I @

Z I " Z) 50 , , , , l , , L

D

'°°° E . . . . . . . . . . ,_, f "'-. 1 9 7 K n- 500 " .." I-- "" °°,,

~ ! "'" "" ... o Q

2 0 0 ~ .'. " J- " / ~ " .'' ." ~ o • , ~ - - ~ " ° • i o /

>" i / "~'~ " /

7 ,., oF. ' ~-- i l i i i . . - L . I

Z - - - - T .... r . . . . r . . . . .

2 o o ~ ;;" ": ' - - '< .~

- r . . . . . , I . . . . .

r, ,o • °

lOOK • •

• • ' , " , °

: / I

I O0 -/. I-

I" 50 ~ - _

OI

i , . . . . . , -T -----I

t 38K ! 1 7 7 K

T4_I """ 1

l • " 1

i .t ' ' • , • • • ' , , • • , • •

..... 1 "" .... 2->.i L t . _ ~ f . ~ . . . . 4.

............... ( 2 - 1 2 0 7 K 2 ]

• , • ~ .

, , , / " - -<. /

,/ -/

/

2 27K

• ° • - ° °

• ° " • • ° . • O . °o ° • ° °

° • ° ' ° • o o

°

, i t I 1 . . . . . . . ,

4 2 1 K

• •o . " ° • • , • , °o • • . • • •

. ° ° • ° . • . ° °

i i L . - a . - - i

O'2 0 3 014 0[5 06/01 0'2 0'3 0'4 0 5 06 /0~ 0 2 0'3 0'~,- 0 5

FREQUENCY ( T H z )

t

4 q:

0 6

Fig. 3. The reso lu t ion-broadened dynamic s t ruc ture factor, S ( Q . o~1, for l iquid 4He for Q = 1.13 A ' and s.v.p. The dashed cur',e,, show the normal- f lu id components , i.e.. the last term in e q ( 1 ). From ref. [28].

E.C. Svensson and V.F. Sears / Neutron scattering t~v 4He and ' t ie 129

thought to be associated with the superfluid phase. The dynamics of liquid 4He appeared to be much more complex than had been anticipated and this had a very discouraging effect on many theorists, leading them to abandon liquid 4He. Finally, studies by Svensson et al. [27] and Woods and Svens- son [28] showed that something dramatic did hap- pen on passing through T x - the one-phonon peak in S(Q, o~) disappeared as shown by fig. 3. This peak is a clearly identifiable feature of S(Q, ,,) for all temperatures below T x (2.17 K), but above T x it has disappeared leaving only a single broad component in the scattering. The one-phonon' peak is thus clearly a signature of the superfluid.

Woods and Svensson [28] also found that S(Q, to) could be very well described as the sum of a superfluid component with a weight n.~ = p J p (p~ being the superfluid density) and a normal-fluid component with a weight n , = 1 - n,, namely

S(Q, to)=n.,S~(O, o~)+n,S,(a, ¢o). (1)

Here S~(Q, ~) consists of a one-phonon peak (with a temperature-dependent width) plus a broad multiphonon component while Sn(Q, ~0) consists of only the single broad peak characteristic of non- superfluid 4He. In practice, S.(Q, ~o) is taken to be the distribution observed at a temperature 7"* just above T x (e.g., the 2.27 K distribution in fig. 3) with the leading edge adjusted [28] so as to satisfy detailed balance and preserve the first mo- ment. In the upper part of fig. 4 we see that the normalized one-phonon intensities obtained [28] by application of (1) are indeed in excellent agreement with the quantity n~ = p s / p for all five values of Q that were studied. From fig. 1 we see that these values span the range from below the "maxon" to the roton minimum. We also see from the lower part of fig. 4 that the widths of the one-phonon peaks appear to fall on a universal curve and are in excellent absolute agreement with the widths for rotons (dashed curve) calculated from the theory of Landau and Khalatnikov [29].

The interpretation provided by (1) resolved several previously puzzling problems. First, and most important, it showed that there were no "one-phonon" excitations above T~. It also gave linewidths for rotons (and for other Q values) which were in excellent agreement with the predic-

tions of the Landau-Khalatnikov theory right up to T x whereas the values inferred from earlier neutron measurements were considerably larger especially near T x. In addition, it gave roton-

( L

~Z

r- {3

A

O

N

A

O v

S

A

N

T F- v

OJ

1 . 0

0.8 ~ ~ L ~

o 0 . 8 0

0 . 6 • 1 . 1 3

x I . : 3 0

+ 1.40 o 1 . 9 2 6

o.4 ps/p ---- L - K THEORY

0.2

oo

0 . 0 8

0 . 0 6

I I

# ,/4

4 0.04 /

0.02 7 ~(" x J _

0 . 0 0 - - " " ' ' '

1.0 1.4 1.8 2.2 T(K)

Fig. 4. Upper part: normalized intensities of the one-phonon peaks in S(Q, ¢o) for liquid aHe at s.v.p, for the five Q values indicated and n, = p~/p (solid curve). Lower part: the corresponding intrinsic widths of the one-phonon peaks and the width (dashed curve) for rotons calculated from the theory of Landau and Khalatnikov 129]. Results are based on an analysis in terms of eq. (1). From ref. [28].

130 E. ( ". Sren.~,m and I'. I£ .S'ear~ /' Neutron x('attermy, I~v 4lie and 'lh"

minimum energics, A(T) , which agreed very' well with values inferred from thermodynamic measurements whereas earlier neutron values had been about 25% lower near Ta [30]. [Thc earlier neutron values were taken from the whole distributions and hence were seriously distorted by the presence of the broader normal-fluid component which is centred at a lower energy than the roton peak in

S,(Q. o:).l Mezei [31], using the spin-echo technique, has

determined very accurately the roton linewidths and energy' shifts at temperatures ~< 1.4 K where one would expect the Landau- Khalatnikov theor,,. to be valid. His results confirm this, and extrapola- tion of his curves to higher temperatures gives excellent agreement with the values of Woods and Svensson [28] obtained using (1). but not with the results of Dietrich et al. [32] and Tarvin and Passell [33] who used the whole distributions to determine the roton parameters. The values of these authors also violate a bound placed on the temperature variation of ,.1 by the theory of Bedell ct al. [34] which is, however, satisfied by the values of Woods and Svensson. The work of Tarvin and Passell [33] in fact showed that the linewidths and energies obtained by fitting to the whole distribution depend strongly on the form assumed for the lineshape. This is exactly what one would expect from (1) and fig. 3 since one is then trying to fit a single-peak lincshape to an observed distribution which is actually the sum of two components having different linewidths and centred at different energies. Additional support for ( l ) i s provided by the fact that the analogous relationships

s ( O ) = n S , ( Q ) + n n S n ( O ) (2) and g ( r ) = n g , ( r )+ -n , , g , , ( r ) (3)

implied by (1) are very well satisfied [35,36] by' the best experimental results [36,37] for S(Q) and g(r) .

There is clearly a great deal of evidence m support of ( 1 ), which, originally, was proposed [28] simply as an empirical relationship. Early theoretical studies by Griffin [38] and Griffin and Talbot [39] appeared to give direct support for (1). but, in more recent studies [40]. these authors find that S( Q. w) exhibits a single resonance at all tempera-

lures, contrary to the two-component structure clear],,' visible in the experimental results (e.g.. fig. 3) on which ( I) is based. The region of strict validity of their calculations is. however, restricted to Q values considerably lower than those of the experiments [27,28]. so there may possibly be no serious disagreement. Detailed measurements at lower Q values and further theoretical calculations are required. Although ( I ) will. in the final analysis. almost certainly turn out to be somewhat too simple, it has. we believe, put us back on the right track to understanding the dynamics of liquid 4He. The appealingly. simple picture that it pre- sents has stimulated renewed interest in liquid 4He on the part of theorists, leading to substantial advances in our understanding with, one hopes, more to follow in the near future.

It is important to note that the universality of the widths suggested by the rcsuhs in fig. 4 does not continue to lower O values, as shown b \ the recent measurements of Mezei and Stirling [41] which are summarized in fig. 5. Here wc scc that the widths in the range 0.3~<Q~<0.7 ,& ~ arc strongly Q dependent. One ,,'cry' interesting feature of these results is the drop in width between 0.5 and 0.6 ,& ~ at the lowest temperatures. This is undoubtedly a reflection of the fact that 3-phonon decay processes, which arc dominant in the region of anomalous dispersion, are no longer allowed once one passcs (at Q = 0.55 A i) into the region of normal dispersion (see fig. 2).

, 1 7 K Roton

\ k \

.16K 4

r - - - - - 7

0.8 r Phonon .70

Ok./

o6i 4.eSVP SO 8

= I :40 I-.- C3 0'4 ~- "" * 15K ~

[ • 12K -10 ~ o t ~,~ , 0 9 5 K : 0

03 05 0.7 192 WAVEVECTOR (/~-1)

Fig. 5. Temperature dependence of intrinsic ',~,idth~, of roton~, and Iow-Q phonon.,, in liquid 4tic at .,,.v.p. From ref. [411.

E.C. Soensson and K F Sears / Neutron scattering bY 4He and tHe 131

2.3. The momentum distribution and the condensate fraction

In 1938, Fritz London proposed [42] that the k transition in liquid 4He was the analogue, for a real liquid, of the phenomenon of Bose-Einstein condensation in an ideal Bose gas. Thus began the Condensate Saga, the drama of over four decades of attempts to establish convincingly that a finite fraction of the atoms in superfluid 4He are indeed condensed into the zero-momentum state. Since this saga has recently been reviewed by one of us [43,44] we will be fairly brief in our treatment here.

In 1966, Hohenberg and Platzman [23] pointed out that one ought to be able to determine the condensate fraction, n 0, by neutron-scattering measurements at large Q where S(Q, ~) should directly reflect the momentum distribution, n(p), of the 4He atoms. This stimulated several experimental studies which, however, gave conflicting results with inferred values of n o ranging from 0.02 to 0.17 (for details see [44-47]). With the work of Martel et al. [45] it became clear that these discrepancies were largely attributable to inad- equacies in analysis, in particular to the failure to allow for the fact that the impulse approximation was not strictly valid in the Q range of the measurements because of substantial distortions caused by final-state interactions and inteference effects. By taking account of these effects, Martel et al. were able to obtain a quantitatively correct de- scription of the oscillations in the width of S(Q, ~o) at large Q which had been a puzzle since they were first observed by Cowley and Woods [22]. Martel et al. also proposed a method for obtaining more reliable values of n ( p ) from measurements of S(Q, ~) in the accessible Q range, and this method was subsequently applied by Woods and Sears [46] to the earlier results of Cowley and Woods [22] and by Sears et al. [47] to the results of new measurements undertaken specifically to establish the existence of the condensate and determine its value. The values of n(p) obtained from this latter study are shown in fig. 6. We see that there is essentially no change in n(p) between 4.27 and 2.27 K, but that on further cooling to 2.12 K there is a marked increase in n(p) at

0 . 2 0 , - -

o4 v

v c

o ti:i. i o x °

o i OOox •

io c O.lO~-

O x l l

O ~

0 x

o~O .--I

B, 6

0.05 -

0 .00 0

• 1 . 0 0 K

* 2.12 K

o 2 .27 K

o 4 . 2 7 K

~m 80

o Q

"li i I II1~11 | . . . . . . .

I 2 3 p (A- ' )

Fig. 6. The momentum distribution, n(p), for liquid 4He at four temperatures at s.v.p. The horizontal bar shows the resolution HWHM. From ref. [47].

low p which is a direct consequence of the appearance o f a finite condensate fraction below T x = 2.17 K. The surprisingly large increase at a temperature just 0.05 K below T x reflects an en- hancement caused by a p - 2 singularity in n*(p) , the n(p) for the non-condensate atoms, which is, however, only present if n o is finite and is stron- gest near T x (see [43,44,47,48] for additional details). This observation of the large increase in n(p) at low p just below T x (approximately half the total increase observed at 1.00 K as seen from fig. 6) was thus the long-sought convincing experimental proof that there really was a Bose condensate in superfluid 4He. Forty-four years after London made his intriguing proposal [42], we were finally sure that he was right.

By applying an improved analysis procedure to the results of fig. 6 and to the earlier results of Woods and Sears [46], Sears et al. [47] obtained the values of n o shown as solid circles and a solid square in fig. 7. The solid curve shows the results of a fit [47] to these values and to the values

1 3 2 E . ( 5 " . S r e n v ~ o n a n d ~'. I". Sear~ / Neu t ron .~(attertng I w 4 l ie and ' l h '

indicated by open symbols (see [47] for detailed identification) which were obtained from pair correlation functions via the method of Hyland et al. [49] which will be discussed further in section 2.4. The best theoretical estimates of no(O) are shown by asterisks: upper [50], lower [51]. The difference is characteristic of the differences that can be produced by changes in the interatomic potential that are within the present uncertainties. There is, we believe, no significant discrepancy between the best theoretical values and those from the experiments. This was where we stood [47] in 1982. A great deal more has been added to the saga since then, essentially all of it serving to confirm the 1982 picture. First, Campbell [52] proposed a method for determining n() from the temperature variation of the surface tension. His best values. shown as the dashed curve in fig. 7, are in excellent agreement with the fit to the earlier results. Then Sears [53] pointed out that n.) could also be determined from the temperature variation of the average kinetic energy per atom, (K >, and, from the values of ( K ) available from neutron-inelastic-scattering and neutron and X-ray diffraction measurements, he obtained the values of n(, shown as + "s and × ' s in fig. 7. Since it was assumed that < K*(T)) = <K(Tx) ), where K*(T) refers to the

0 2 0 , . . . . . . l . . . . . . .

O I 5 -

Z 0 F -

c o Z

005) Z 0

0 0 0 "

• - " "f . . . . . l

1

. . . . . . . . I . . . . . . . . 1 . . . . . i15 . . . . . 0 1 _ _

i

O 0 0 5 tO 2 2 5

T E M P E R A T U R E ( K )

Fig. 7. Experimentally-based values for the condensate fraction, n.(T), in superfluid ,I He at s.v.p, and theoretical values (* "s) for n,d0). Eor further details see text. From ref. [44 I.

non-condensate atoms, these values are, strictly speaking, upper limits on n . and there is indeed some indication that they are systematically higher than the other values, especially near T x. Mook [54] has also carried out a neutron-inelastic- scattering study very similar to the one of Sears et al. [47] and obtained the values shown by solid triangles ill fig. 7.

In fig. 7 we thus have theoretical values of n . (0 ) as well as experimental values of nu (T) obtained from many different sets of results (from neutron, X-ray and surface-tension measurements) via five different methods of analysis, and all of these results are in substantial agreement with each other. It seems almost inconceivable that this could be fortuitous, and, hence, in addition to being sure that the condensate exists, we can now be rather confident about its magnitude and it,,, temperature dependence. For T < 1 K. no ~ 0.13 with an uncertainty that is probably no larger than 0.02 or 0.03. Although we have certainly reached a major milestone, the ( 'ondensate Saga is by no means at an end, as will be discussed further m section 4.

Before leaving this section, we want to call at tention to recent work at large O carried out using spallation neutron sources. At Argonne Na- tional Laboratory, measurements [55] on liquid and polycrystalline solid (both hop and bcc) 4He have been carried out at O values up to = 23 A These measurements give distributions that appear to be Gaussian at all temperatures and in all phases, and the inferred values of ( K ) are essentially independent of temperature and phase at a given density. The values of (K > at low density are, however, 15-20% lower than values from the Green 's - funct ion Monte Carlo calculations of Whitlock and collaborators [50,56], while at high densities they are higher than the corresponding calculated values by approximately the same amount . This is very puzzling and clearly indicates that more work is required. For further discussion see Moleko and Glyde [57]. Brugger et al. [58] have carried out measurements on liquid 41-le at Los Alamos National Laboratory at Q = 83 A t. They obtained ( K ) = 13 K in good agreement with accepted values [53], but their resolution was not sufficient to detect any significant change between

E.C. Svensson and V. I:" Sears / Neutron scattering 12v ~He and "tHe 133

4.2 and 1.2 K. Recent measurements [9] on liquid 4He at the KENS facility in Japan have reached Q = 150 ,~- ], the largest value to date. The results show a change in lineshape between 2.5 K and 1.2 K that probably is attributable to the appearance of the Bose condensate below T x.

2.4. Static structure factor and pair correlations

In recent years there has been a renewed interest in the static structure factor, S(Q), of liquid 4He largely because of the tantalizing possibility of determining the condensate fraction from the temperature variation of the pair correlation func-

tion, g(r) , which can be obtained from S(Q) by Fourier inversion. Hyland et al. [49] proposed that this could be done via the relation

g ( r ) - 1 = (1 - no)2[gn(r) - 1], (4)

where gn(r) refers to the non-condensate atoms and, in practice [59,60], is taken to be g(r) for a temperature 7"* just above T x. The first diffraction measurements sufficiently accurate and complete to provide a critical test and application of (4) were those of Svensson et al. [37]. Their very extensive and accurate S(Q) values for 11 temperatures at s.v.p, are shown in fig. 8, and the temperature dependences of various features of the g(r)

1.6

1'4 L 1.2

1.0

I l I I I I I ! I {

4.27 K o 6 ~ a a m •

3.60 K

3.00 K ~ t O iir 011 g. 0 il, • . . . .

2.27 K

2.15 K

2.12 K

(Q) 2.07 K ~ I - o - o = : "- ~ ; -" _.1,7, e o o o ~ , O t ~ . o o o o o l O O _ -

1.97 K

!j //z

0.8 F / , ~

0 . 6 ~ 1 / ' 4

o.4g 0.2 ~- / , , J

0.0 ; / 0

1.77 K

1 . 3 8 K

i p o ~ t o ~ p o ~ o

1.00 K t i t t I I I t I ! ; ~ - ~ : 1 4 i ~ I Q I l l I 1 1 0 1 ~

I 1 L l I I I 4 5 6 7 8 9 I 0

O ( ~ - ' )

I II 12

Fig. 8. The static structure factor, S(Q), of liquid 4He for 11 temperatures at s.v.p. From ref. [37].

134 E ( "..S'ven~wm and 1. I-: £'ears ,," 3,eutr, m vcatterm,v I B" 4tie and "ltc

values obtained by inverting these S(Q) values are shown in fig. 9. Note that the amplitudes of the oscillations about the , g ( r )= 1 line. which are a direct measure of tile correlations between tile atoms, increase (as is normal) with decreasing temperature between 4.27 K and T\. but then there is a dramatic reversal and they decrease cont inuously with further cooling in the superfluid phase. This anomalous behavior, which is unique to superfluid 4He, is, according to Hyland et al. [49], a direct consequence of a toms dropping into the zero-momentum condensate state where they effectively no longer contribute to the correlations between atoms that give rise to the oscillations in g( r ) . Equation (4) is only applicable at sufficiently large r ( > 6 A) that the one-particle densit,,, ma-

I ' j ~ -

I i

I I

I I O; . . . . . . . . .

,] : r I

" . ) 5 -

2 4 ,

6 _14, . . . . . . a¢ 5 . . . . . . ~-£2-L -_- . . . . . . . . . . . .

', . / 5 4

T ! 0 0 K

',.) 0 L____ . _ 2

0 P ' ; 6 8 IO r ' A ' ,

. . . . . . . . 4 0 [ 3 0 '0 . , ~ I

2 " , ." t 138 , ---~-" . gz 96 • ,_ " "

u ? 3 6 ,,

", i 090 .

4 5 6 • 0 8 8 ... , g4 *

r , < ;86 1

. . . . . . . . ; t 6 2 ~ ~ .".r"

(. ~8 " ' I,.).5 , r5 gro

~-,,~ . _ j ,,:;s j

, :, s 4 ~; o ..... ; ;~ ~ T . 5 TEM:SF.RA7 J ~ L ( K I

Fig. 9. The pa i r co r re l a t i on func t i on . ,e, i r ) , f o r hqu id 4 H e at

1.00 K and s.v.p, and the t empe ra tu re v a r i a t i o n o f ~ar ious

f e a t u r e s o f g ( r ) . F r o m ref. [60].

trix has effectively reached its asymptotic value. n . , and. to be physically meaningful, it must give n . values which are independent of r in this region. This implies stringent limitations for the behavior of ,g(r) which in fact appear [35.59,60] to be entirely consistent with the experimental re- ~,uhs. For example, the positions of tile nodes o f

g ( r ) - - 1 must be independent of T for T < 7",,, and from fig. 9 one can see that this is the case. The variation above Tx is attributable to a density change o f - 1 7 q in this region, but below llv where tile density changes by only ~ 0.7q:. tile variations are very small and random, and easily attributable to experimental uncertainty'.

Values of n u obtained by application of (4) are shown by open symbols in fig. 7 (for origins see [47]). They are clearly in excellent agreement with the other values as discussed in section 2.3. There has, however, been considerable controversy (for references see [47]) regarding tile method of Hv- land cta l . [49] and hence, al though (4) has a lot of intuitive appeal and experimental support as an empirical relationship [35.61]. it is still not clear what significance one can attach to values of n,, obtained bv using it. Should this method, which has the great advantage of being far more "'effi- cient'" for obtaining n , values (diffraction measurements being much less time consuming t h a n

inelastic-scattering measurements at large Q I. prove to be reliable, it would enable one to obtain much more detailed information on tile variation of n , with temperature and pressure than is likel 5 ever to he obtained from inelastic-scattering measurements. Further theoretical at tempts to .justify or improve this approach, as well as additional experimental tests, would thus be extremely valuable.

Taking a further step. onto even more shaky ground, one can combine (3) and (4) to obtain [35] an explicit relationship between n , ( T ) and n.(7") for all T < 1~,,, namely,.

n.(T)=n,,(T)[2 n,}(T)] (5) n,,(O)[2 -n,)({})]

Values of n , ( T ) calculated f r o m {5). using the values of no(7") indicated by the same symbols in fig. 7, are shown in fig. 10 together with the

E.C. St,ensson and 1/. F. Sears /Neutron scattering b.v ~He and ¢He 135

v 2

4 L i , 1.2 _ MAYNARD (1976)

0.21_ ooz~ n o FROM g(r ) J. ~/~

/ " " °o FROM °(-~) [If'

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

TEMPERATURE (K)

2.4

Fig. 10. Values of the superfluid fraction, n~(T), calculated from eq. (5) using the values of no(T) shown by the same symbols in fig. 7. The solid curve shows Maynard's [62] experimental values of n, for T > 1.2 K and the values of Brooks and Donnelly [301 for lower T.

experimental n~(T) values of Maynard [62] supp- lemented at T values below 1.2 K by the values of Brooks and Donnelly [30]. The agreement is rather good, indicating that (5), the only existing explicit relationship between no and n, for all T, merits attention by theorists.

In addition to the interest from the point of view of the condensate, accurate values of S ( Q ) and g ( r ) are often essential for the analysis of inelastic-scattering results [63,64], and they are of crucial importance for the testing of the results of theoretical calculations. The results of the beauti- ful Green 's-funct ion Monte Carlo calculations of Kaios et ai. [51] are in fact in very good agreement with the S ( Q ) and g ( r ) values of Svensson et al. [37]. The quality of both the experimental and theoretical results has increased dramatically over the last decade.

3. Liquid 3He

We now turn to 3He. With an absorption cross section of = 11000 barns for 4 A, neutrons, com-

pared with a scattering cross section of --- 6 barns, it is clear that very few of the neutrons that go into the liquid are ever going to come out. The 3He nucleus also has a finite spin (1 /2 ) so that, in addition to the coherent scattering by density fluctuations (zero-sound modes), one would expect [65] to see incoherent scattering by spin-density fluctuations (paramagnon modes). Part icle-hole ( p - h ) excitations in this Fermi system will also provide another decay mechanism (Landau damping) not present in liquid 4He.

The first neutron measurements on liquid 3He [10,11], carried out at the Institut Laue-Langevin , gave no evidence of well-defined collective excitations, but subsequent measurements carried out at Argonne Nat ional Laboratory by Sk~51d and collaborators [13] clearly showed two well-defined peaks in S(Q, o~) attributable to zero-sound modes and, at lower energies, spin-fluctuation excitations. This most impressive feat, especially when one considers that it was achieved using the modest-flux CP-5 reactor, was made possible by the use of a statistical chopper and a very clever design for the sample container. The results for zero-sound modes obtained in the studies at Argonne Nat ional Laboratory [13-16] are summarized in fig. 11. The different energies indicated by different symbols result from different methods of determining the central positions of the zero-sound peaks, there

/ / (a) /

~ /

0.5 1 1.5

q/A- '

Fig. 11. The dispersion reation for zero-sound modes in liquid tHe at T = 40 mK (note that q ~- Q). The solid curve shows the theoretical prediction of ref. [67] and the dashed curve the velocity of sound. For further details see text. From ref. [161.

136 I:'. ( . N+'en.ss+m and 1 '. I~ .S'ear.+ / .Veutr, m ~ attertny+ hv 4 th ' and ' t i e

being considerable ambiguity about how to separate the zero-sound and spin-fluctuaticm contri- butions to the observed scattering. The energies of the spin-fluctuation excitations, observed over the same range of Q as the zero-sound modes, lie within the p -h band (see fig. 4 of [14]). For Q > 1.3 A, ~, only a single broad peak, centred within the p - h band. is observed. The Argonne measurements also showed [15,161 that there wa~, very little if any' change in the zero-sound peaks between 40 mK and 1.2 K (the Fermi temperature, T r, is = 1.5 K for liquid 3He at s.v.p.), but that the spin-fluctuation peaks were considerably broader at the higher temperature. Even at the lowest temperatures, the zero-sound peaks exhibited intrinsic widths that varied [16] from 0.3 meV at Q = 0.4 ~, t to = 1.3 meV just before these modes "'disappear" (undoubtedly because of l.,andau damping) as they enter the p h continuum at Q = l . 3 ~ , 1

In later measurements at the lnstitut l+aue Langevin, Hilton et al. [12] also observed well- defined zero-sound modes in liquid "He for 0.48 < Q < 0.78 A i with energies in good agreement with the results of fig. 11 and intrinsic widths of 0.2 to 0.5 meV. Note that the intrinsic widths observed for ~'.ero-sound modes in liquid 3He at T = 15 40 mK are enormously (1-2 orders of magnitude) larger than the intrinsic widths of rotons in liquid 4He at T = 1 K [31]. In fact, thc • ",ero-sound peaks in liquid ~He are more meaning- fully compared with the broad peaks observed m liquid 4t-le above T x (He I) than with the very sharp one-photon excitations observed in superfluid 4He at 7"= 1 K. The origin of the large widths observed for liquid 3He is not clear at present. In the RPA (Random Phase Approxima- tion) theories that have normally been applied to liquid 3He, the only damping mechanism is single p h decay which cannot account for the large widths in the region where the zero-sound modes arc far from the p - h band. By' allowing for multi- pie p h decay process, Glyde and Khanna [66] have estimated widths of 0.22 and 0.35 meV for Q values of 0.5 and 1.0 A 1, respectively. Sk01d and Pelizzari [16] have suggested that phonon phonon decay processes arc also important, and the fact that their experimental widths are larger than the

theoretical estimates [66] and, further, give evidence for a maximum in the region of (I.6 0.7 A i certainly supports this. The results of fig. 11 indicate that There is anomalous dispersion in liquid 3He, with probably a much larger deviation above the sound velocity (dashed curve) than observed for liquid "+lie (fig. 2), so one would expect 3-phonon decay processes to be important for liquid ~He. Note that the calculations of Aldrich and Pines [67] (solid curve in fig. 111 also indicate anomalous dispersion. Multiple p h and phonon- decay processes are undoubtedl\' both important and the theories have to be modified accordingly. l-or a recent discussion of theoretical work on liquid 3He and the implications of the experiments for these theories, see Glyde [21].

Sokol et al. [68] have recently reported neutron measurements on liquid ~He at large Q, 10 16+& t at 0.2 K and s.v.p. The aim of this study was to determine the momentum distribution, n(p) , and, in particular, to search for the discontinuity at the t'ermi momentum expected for a degenerate Fermi system if the interactions between the particles arc not too strong. The observed scattering was found to be describable by a Gaussian, centred at the recoil energy for free +He aton+s, v,.hose width gave / K } = 8.1 K in reasonably good agreement with several theoretical estimates. There was no evidence for the expected discontinuity, possibly because the experimental resolution and the statistical accuracy were not sufficiently high. Another study of liquid ~l-|e at large Q is underway at Oak Ridge National Laboratory [69]. Polarized neutron diffraction measurements on a single crystal of bcc ~He at T < 1 mK have also just been reported [70]

a phenomenal technological feat.

4. Concluding remarks and future directions

Although the neutron studies of the last decade on liquid 4He have answered most of the important questions that were outstanding in the early 1970's, they have left us with an even greater number of new questions and many obviously important things to do in future. Let's take the condensate as the first example. Now that we know it exists, and a rather lot about it at s.~.p.

E.C. Svensson and 11". F. Sears / Neutron scattering by 4He and ~ He 137

(fig. 7), we would like to have higher accuracy. This will require both better experiments and better analysis procedures since the latter are a major source of the present uncertainties - see, e.g., the very recent work by Griffin [48] on the problem of correcting for the singular behavior of n*(p) in the inelastic-scattering studies. With higher accuracy, we would have a better knowledge of how n o ---, 0 at Tx and this is extremely important since ~/n~ is the order parameter for superfluid 4He. The value of the critical exponent, 2fl, for n o can be inferred from the values of other exponents to be --- 0.70 but it would be highly desirable to make a direct determination. Although the large uncertainty of the present results precludes an accurate determination, it is interesting to note that the dashed curve [52] in fig. 7 gives 2fl = 0.71. We also want to know n o at higher densities (where it should be smaller because stronger interatomic interactions will further deplete the condensate) and whether it falls continuously or discontinu- ously to zero at the superfluid-solid interface (as- suming, as seems almost certain, that there is no condensate in the solid). The results of various theoretical calculations of the density dependence of n o, as well as preliminary results from neutron measurements at Argonne National Laboratory [71], have recently been summarized by Manousakis et al. [72]. The indications are that n o decreases rather slowly with increasing density. Another interesting challenge would be to determine n o for 3He-4He mixtures. One would expect [73] n o to increase, at least initially, as 3He is added to 4 He because the density will be lowered, lessening the interatomic interactions. Ultimately, we want to" see a separate sharper condensate component directly in S(Q, to) as suggested initially by Hohenberg and Platzman [23]. Calcu- lations indicate [44] that this will require very high experimental resolution and excellent statistics at Q values in excess of 100 ~,- 1 (energy transfers > 5 eV). This is the domain of intense spallation sources which, with their high fluxes of epithermal neutrons, will undoubtedly play an increasingly important role in future studies of the condensate.

Going back to lower Q, we need detailed results at higher densities of the type and quality of those for s.v.p, shown in figs. 2-5, and the measure-

ments at low Q .(fig. 5) need to be extended to temperatures well above T x. It will be very interesting to see how well (1) applies at higher densities. The results of Svensson et al. [63] at different densities at low T have shown that, for Q values near the "maxon", S(Q, to) changes dramatically with density while for Q near the roton minimum there is relatively little change. These results have been interpreted [63] as indicating that, like solid 4He, liquid 4He is very "anharmonic" (see also [21]). The concept of anharmonicity in a liquid is, however, not well defined, and in describing the temperature variations by (1) (which is found to work very well over a wide range of Q) no explicit account has been taken of any anharmonicity. Much still remains to be learned about the excitations and their interactions, and this whole area of determining and understanding S(Q, to, P, T) for Q values in the phonon-roton region is clearly a very fertile one for the fruitful interplay of theory and experiment.

Extensive results for S(Q) and g(r) at higher densities such as those for s.v.p. [37] shown in figs. 8 and 9 are also needed for testing theories, for analysing results for S(Q, to) taken under the same thermodynamic conditions, and for further testing the possible validity of (4) for determining n 0 values. Even higher accuracy would be desirable since the changes of S(Q) and g(r) with temperature are very small (e.g., 2-5% [36,37] at the positions of the principal maxima). A further object of such studies is to decide between (or reconcile) the explanation of the anomalous behavior of S(Q) and g( r ) for superfluid aHe (see fig. 9 and section 2.4) as being due to the changing population of the condensate [49] and a competing explanation [74] which ascribes it simply to the changing thermal population of rotons.

The reader will undoubtedly have noticed that all the equations in this paper are essentially empirical relationships, having strong support from experiment but no really firm foundation in theory. The experiments are at present ahead of the theory but hopefully the recent resurgence of interest in liquid 4He on the part of theorists will soon reverse this trend and lead us to a much better fundamental understanding. In the actual application of the relationships (1-4), and other relation-

138 /'.'. ( ". Nt'en,~'son a n d | ~ I i ,S'('ar.~ / ,V('utron ~(attertn.g t)l "~lh" a n d ~t1(,

ships used by Sears et al. [47] and Sears 153] to obtain their no values, it has been necessary to make assumptions about how quantities characteristic of the normal fluid or the non-condensate a toms (e.g., S.,(Q. to), S,(Q) , g,(r), n*(p ) and {K*)) vary below T x, and these assumptions, for which we have little theoretical guidance at present, are often a major source of uncertainty in the analysis. Measurements to determine in detail how these quantities vary above (and especially .just above) T x are needed to provide us with guidance in estimating how they change below T x where they cannot be observed separately. These measurements coupled with fundamental theoretical work on this problem are especially important for the condensate studies.

Even after nearly 35 years of intensive study, there are clearly still a great many' ways in which neutrons can (and will) make further important contr ibut ions to our understanding of liquid ")He. Those of us who have greatly enjoyed studying this fascinating material need have no fear of running out of useful things to do.

Regarding liquid 3 He. we need measurements at higher resolution so that the zero-sound and spin- fluctuation peaks can be better separated, leading to less uncertainty about the energies (fig. 11 ) and intrinsic widths [12,16] of the zero-sound modes. This would give us a clearer picture of the extent and magnitude of the anomalous dispersion and additional guidance to the understanding of the large intrinsic widths of the zero-sound modes even at extremely' low temperatures (15-40 inK). The origin of these widths is of crucial importance for the theoretical understanding of liquid 3He, and it seems quite probable that additional in- sights could be gained by a detailed comparison of the results for liquid 3He with corresponding re- suits for liquid 4He at temperatures above T,\. The measurements should also be extended to lower Q. and better determinations of the lineshape of the spin-fluctuation peak at low Q would have important consequences for theories (see [16]). It would also seem to be valuable to extend the measurements to higher temperatures to determine if and when the zero-sound peaks change signifi- cantly (there is little or no change up to 1.2 K [15]) and whether or not there is any correlation with

T I = 1.5 K. One of the next moves should also be to stud}' the scattering at highcr pressures where the energies of the zero-sound modes are expected [66] to increase by a large amount (a factor of --- 2 at P = 30 atm). Measurements by l t ihon c t a l . [75]. which arc, however, confined to the region Q > 1.1 A i and which show no evidence of well- defined zero-sound modes, suggest a decrease in energy with increasing pressure. Measurements at high Q values at much better resolution than thosc of Sokol et al. [68] are also of high importance and, it is hoped, will eventually reveal the expected discontinuity in n ( p ) at the Fermi momentum. These suggested studies on liquid 3He, ;.is well at, the further studies on solid ~He that undoubtedly will follow the measurements just reported [70]. are going to take enormous effort and timc. Con- sidering what was achieved [13 16] with the ( 'P-5 reactor at Argonne, there is, however, no doubt that these studies are feasible with today 's best reactor sources. One would hope that it will eventually also) prove possible 1o study fully-spin- polarized specimens of ~Hc using polarized ncu- trons. Absorpt ion will then not bc a problem.

References

[11 For an account of the earl.* d a ~ of n e u t r o n scaUerillg ~ee ('.G. Shull. Pr(~,'. of the Conf. on Neulron .~caltcring. R.M. Moon, ed. (('()NI-760601-PI. Nat. Tech. Inf. Service. Sprinfield, VA. 19761. Vol. I. p. 1.

[21 l.. (;oldstein. H.S. Sommcr~..Jr.. I,.I).P. King and C J thfffman, Pr(~:. of the Int. ('onf, on Lo~,~ Tcnlp. Phys., R Box~ers, ed. (1951). p. 8g

[3] I).( i. Hensha~ and l).(,L HtlrM, Phys. b),cx. 91 (1953) 1222: l).(i. I|ursl and I).(L tlcm, ha'~.. Ph,.r,. Re,.. I()(J (1955) 994.

[4] H. Palc.*'sk}.. K. ()tner,, K.E. l,ar~,~,on, R. Pauli and R. Slcdman, Phys. Rex'. 1(18 (1957) 1346; tt. Palc',~,k>, K. ()tnc~, and K.I!. [.arsson. Ph).',. Rex. 112 (195S) 11.

[5] J . L Yarncll, G.P. Arnold. P.J. Bent and I ' . ( . Kerr, Ph',~,. Rex. l.eil. 1 (1958) 9: Phv~,. Rex. 113 (1959) 1379.

[6] I).(i . t lenshaw, Ph',s. Re','. [.eft. I (19581 127. 171 LI ) . Landau. J. Ph\'~,. U.S.S.R 5 11941) 71, 11 (19471 91. [8] R. ( ;olub, ('. Jewell, P. Agcron. W. Mampe, B. tleckcl and

I. Kilvington, Z. Phys. B 51 (1983) 187. [9] N. Watanabe, Neutron Scattering in the "Nineties ( IAI 'A,

Vienna, 19851 p. 279. [10] R. Scherm, W.G. Stifling. A.DB. Wood~,, R.A. ('ox~.le,,

and G.J. ( 'oomhs , J. Phv~,. (" 7 (19741 1.34I.

E.C. Svensson and V.I:. Sears /Neutron scattering by 4He and *He 139

[11] W.G. Stirling, R. Scherm, P.A. Hilton and R.A. Cowley, J. Phys. C 9 (1976) 1643.

[12] P.A. Hilton, R.A. Cowley, R. Seherm and W.G. Stirling, J. Phys. C 13 (1980) L295.

[13] K. SkNd, C.A. Pelizzari, R. Kleb and G.E. Ostrowski, Phys. Rev. Left. 37 (1976) 842.

[14] K. SkNd and C.A. Pelizzari, Quantum Fluids and Solids, S.B. Trickey, E.D. Adams and J.W. Dufty, eds. (Plenum, New York, 1977) p. 195.

[15] K. SkiSId and C.A. Pelizzari, J. Phys. C 11 (1978) L589. [16] K. SkNd and C.A. Pelizzari, Phil. Trans. R. Soc. Lond. B

290 (1980) 605. [17] A.D.B. Woods and R.A. Cowley, Rep. Prog. Phys. 36

(1973) 1135. [18] R.A. Cowley, ref. [1], Vol. II, p. 935; Quantum Liquids, J.

Ruvalds and T. Regge, eds. (North-Holland, Amsterdam, 1978) p. 27.

[191 D.L. Price, The Physics of Liquid and Solid Helium, Part II, K.H. Bennemann and J.B. Ketterson, eds. (Wiley, New York, 1978) p. 675.

[20] W.G. Stirling, J. de Phys. 39 (1978) C6-1334. [21] H.R. Glyde, Condensed Matter Research Using Neutrons,

S.W. Lovesey and R. Scherm, eds. (Plenum, New York, 1984) p. 95.

[22] R.A. Cowley and A.D.B. Woods, Can. J. Phys. 49 (1971) 177.

[23] P.C. Hohenberg and P.M. Platzman, Phys. Rev. 152 (1966) 198.

[24] E.C. Svensson, P. Martel and A.D.B. Wcxxls, Phys. Lett. 55A (1975) 151.

[25] W.G. Stirling, J.R.D. Copley and P.A. Hilton, Neutron Inelastic Scattering 1977 (IAEA, Vienna, 1978) Vol I!, p. 45.

[26] W.G. Stirling, 75-th Jubilee Conf. on Helium-4. J.G.M. Armitage, ed. (World Scientific, Singapore, 1983) p. 109.

[27] E.C. Svensson, R. Scherm and A.D.B. Woods, J. de Phys. 39 (1978) C6-211.

[28] A.D.B. Woods and E.C. Svensson, Phys. Rev. Lett. 41 (1978) 974.

[29] L.D. Landau and I.M. Khalatnikov, Zh. Eksp. Teor. Fiz. 19 (1949) 637.

[30] J.S. Brooks and R.J. Donnelly, J. Phys. Chem. Ref. Data 6 (1977) 51.

[31] F. Mezei, Phys. Rev. Lett. 44 (1980) 1601. [32] O.W. Dietrich, E.H. Graf, C.H. Huang and L. Passell,

Phys. Rev. A 5 (1972) 1377. [33] J.A. Tarvin and L. Passell, Phys. Rev. B 19 (1979) 1458. [34] K. Bedell, D. Pines and I. Fomin, J. Low Temp. Phys. 48

(1982) 417; see also K. Bedell, D. Pines and A. Zawadow- ski. Phys. Rev. B 29 (1984) 102.

[35] E.C. Svensson, V.F. Sears and A. Griffin, Phys. Rev. B 23 (1981) 4493.

[36] E.('. Svensson and A.F. Murray, Physica 108B (1981) 1317.

[37] E.C. Svensson, V.F. Sears, A.D.B. Woods and P. Martel, Phys. Rev. B 21 (1980) 3638.

[38] A. Griffin, Phys. Rev. B 19 (1979) 5946; Phys. Lett. 71A (1979) 237.

[391 A. Griffin and E. Talbot, Phys. Rev. B 24 (1981) 5075. [40] E. Talbot and A. Griffin, Ann. Phys. (N.Y.) 151 (1983) 71;

Phys. Rev. B 29 (1984) 2531. [41] F. Mezei and W.G. Stirling, ref. [26], p. 111. [42] F. London, Nature 141 (1938) 643. [431 E.C. Svensson, ref. [261, p. 10. [441 E.C. Svensson, Prec. of the 1984 Workshop on High-En-

ergy Excitations in Condensed Matter (Los Alamos Na- tional Laboratory publication LA-10227-C, 1984) Vol. II, p. 456.

[45] P. Martel, E.C. Svensson, A.D.B. Woods, V.F. Sears and R.A. Cowley, J. Low Temp. Phys. 23 (1976) 285.

[46] A.D.B. Woods and V.F. Sears, Phys. Rev. Lett. 39 (1977) 415.

[47] V.F. Sears, E.C. Svensson, P. Martel and A.D.B. Woods, Phys. Rev. Lett. 49 (1982) 279.

[48] A. Griffin, Phys. Rev. B 30 (1984) 5057, 32 (1985) 3289. [49] G.J. Hyland, G. Rowlands and F.W. Cummings, Phys.

Lett. 31A (1970) 465. [50] P.A. Whitlock, D.M. Ceperley, G.V. Chester and M.H.

Kalos, Phys. Rev. B 19 (1979) 5598, P.M. Lam and M.L. Ristig, Phys. Rev. B 20 (1979) 1960.

[51] M.H. Kalos, M.A. Lee, P.A. Whitlock and G.V. Chester, Phys. Rev. B 24 (1981) 115.

[521 L.J. Campbell, Phys. Rev. B 27 (1983) 1913; A.I.P. Conf. Prec. No. 103 (1983) 403.

[531 V.F. Sears, Phys. Rev. B 28 (1983) 5109. [54] H.A. Mook, Phys. Rev. Lett. 51 (1983) 1454. [55] R.O. Hilleke, P. Chaddah, R.O. Simmons, D.L. Price and

S.K. Sinha, Phys. Rev. Lett. 52 (1984) 847; P.E. Sokol, R.O. Simons, D.L. Price and R.O. Hilleke, Prec. of the 17-th Int. Conf. on Low Temp. Phys., U. Eckern, A. Schmidt, W. Weber and H. Wiihl, eds. (North-Holland, Amsterdam, 1984) Part II, p. 1213. See also R.O. Sim- mons, ref. [44], p. 416.

[56] P.A. Whitleck and R.M. Panoff, ref. [44], p. 430. [57] L.K. Moleko and H.R. Glyde, Phys. Rev. Lett. 54 (1985)

901; ibid. 55 (1985) 550. [58] R.M. Brugger, A.D. Taylor, C.E. Olson, J.A. Goldstone

and A.K. Soper, Nucl. Instrum. Meth. 221 (1984) 393. [59] V.F. Sears and E.C. Svensson, Phys. Rev. Lett. 43 (1979)

2009. [60] V.F. Sears and E.C. Svensson, Int. J. Quant. Chem.:

Quant. Chem. Symp. 14 (1980) 715. [61] E.C. Svensson, Inst. Phys. Conf. Ser. No. 64 (1983) 261. [62] J. Maynard, Phys. Rev. B 14 (1976) 3868. [63] E.C. Svensson, P. Martel, V.F. Sears and A.D.B. Woods,

Can. J. Phys. 54 (1976) 2178. [64] V.F. Sears, A.D.B. Woods, E.C. Svensson and P. Martel,

ref. [25], p. 23. [65] V.F. Sears, J. Phys. C 9 (1976) 409. [66] H.R. Glyde and F.C. Khanna, Can. J. Phys. 58 (1980) 343. [67] C.H. Aldrich Ill and D. Pines, J. Low Temp. Phys. 32

(1978) 689. [68] P.E. SOkol, K. SkiSId, D.L. Price and R. Kleb, Phys. Rev.

Lett. 54 (1985) 909.

140 t-.(. S¢'ensson and [',f~ Sear~ /.V('utmm scattering hv Qh' attd 'lh,

169} I't.A. Mook. private commun. [70] A. Benoit, J. Bossy, J. Flouquet and J. Schweizer, reported

at the Int. Conf. on Neutron ,'¢¢,:attering, Santa Ft, NM. August 19-23. 1985.

[711 Sccref. 2 in [721 . 172] E. Manousakis, V.R. Pandharipande and Q.N. Usnlani.

Phys. Re'.'. B 31 (1985) 7022.

[73] W.-K. Lee and B. Goodman, Ph,,'s. Re'.. B 24 (1981) 2515. [741 Sec S. Battaini and I,. Rcatto. Ph~,s. Roy. B 28 (1983) 1263

and references therein. [751 P.A. ttilton, R.A. ('owlet., W.(i. Stifling and R. Scherm.

Z. Phys. B 3(I (19781 107.

neutron scattering by 4he and 3he

Documents