Temperature dependence of the phonon and roton excitations ...glyde/Pubs/F1A76D9Ed01.pdf · 4226 W....

PHYSICAL REVIEW B VOLUME 41, NUMBER 7 1 MARCH 1990

Temperature dependence of the phonon and roton excitations in liquid He

W. G. StirlingDepartment of Physics, School of Physical Science and Engineering, University ofKeele, Keele ST5 5BG, United Kingdom

and Science and Engineering Research Counci I, Daresbury Laboratory, Warrington 8'A4 4AD, United Kingdom

H. R. GlydeDepartment of Physics, University of Delaware, Newark, Delaware 19716

(Received 28 August 1989)

We report high-precision neutron scattering measurements of the temperature dependence of thephonon and roton excitations in liquid He at saturated vapor pressure in both the superfluid andnormal-fiuid phases. Two wave vectors were examined in detail: Q =0.4 A in the phonon region

0and Q=1.925 A ' at the roton minimum. The goal is to understand excitations in Bose liquids bydetermining the dynamic structure factor S(Q, tv) accurately in both phases. At Q =0.4 A ' thesharp peak in S(Q, co) broadens with temperature T, but remains sharp and well defined in the nor-

mal phase (T & Ti ) up to T=3 K. The rate of increase of the width of S(Q, tv) shows a break atTi„but otherwise S(Q, tv) is similar in the two phases and a transition could not be identifiedthrough S(Q, co). In contrast at Q=1.925 A ', the sharp component of S(Q, cv) disappears at Tzand S(Q, cv) is very different above and below Ti. This behavior is documented and values of thefrequencies and linewidths at several temperatures are reported.

I. INTRODUCTION

Liquid He is the most readily accessible Bose quantumliquid in nature. It exhibits both a superfluid (Bose bro-ken symmetry) and a normal state. Neutron inelasticscattering studies have provided detailed information' onthe dynamical response of this fundamental Bose liquid.Nevertheless, the character and interpretation of the ex-citations observed in both the superfluid and normalphases, and especially the relation between the excita-tions in the two phases, is not clear. ' In this paper wepresent high-precision measurements of the scattered in-tensity at wave vectors Q =0.4 A ' and Q =1.935 Aover a wide temperature range in both the superfluid andnormal phases. Q =0.4 A ' and Q =1.935 A ' lie inthe "phonon" and "roton" regions, respectively, of thesingle excitation dispersion curve of the superfluid phase.The aim is to display the difference in the dynamicalstructure factor S (Q, co) in the two phases, in an attemptto further understand the nature of the excitations.

In the superfluid phase (temperature T (T&=2. 17 Kat saturated vapor pressure (SVP), S(Q, co) contains a sin-

gle sharp peak superimposed on a broad background. '

The sharp peak may be interpreted as the collectivephonon-roton excitation in the fluid density proposed byLandau and evaluated by Feynman and others.Within the Green's function' ' or dielectric formula-tion" ' of Bose fluids, the sharp peak may also be inter-preted as excitation by the neutron of a single quasispar-ticle, a single, dressed particle' having momentum

p =fiQ. The broad background may be interpreted asmultiphonon or multiquasiparticle excitation.

Indeed, the dielectric theory shows that S (Q, co) in thesuperfluid phase can be represented as the sum of a"singular" and a "regular" part. The singular part is

defined as that part which is proportional to the single-particle Green's function Gi(Q, to). In this way, thesingle-particle excitations can be observed in S(Q, co).The appearance of G, (Q, co) in S(Q, co) is due to a finitecondensate fraction no. As Q~O, the poles of S(Q, co)

and G, (Q, ai) coincide. In the normal phase only theregular part survives in S(Q, co) and single quasiparticleexcitations cannot be observed via S(Q, co). A centralmotivation here is therefore to explore possibledifferences in S(Q, co) in the superfluid and normalphases. This is particularly interesting at lower Q wherethe dielectric theory should be most applicable ' andwhere recent, precise data are not available.

As temperature is increased in the superfluid phase, thesharp peak broadens. ' This broadening may be well de-scribed' by quasiparticle-quasiparticle scattering, as pro-posed initially by Landau and Khalatnikov, ' at least atlow temperature. Precise data 2' at larger wave vectors(Q & 1 A ') suggest that the sharp peak disappears en-tirely from S(Q, to) when T is increased to Tz, leavingonly a broad S(Q, co) in the normal phase. This is partic-ularly clear in the data in the "maxon" region of thephonon-roton dispersion curve ( Q = l. 1 A ' ). ' Thisdisappearance, or abrupt change in S(Q, co), at T& sug-gests that the sharp component in S(Q,ai) is a propertyof the superfluid phase. In the normal phase, S (Q, co) is abroad function of co and largely independent of tempera-ture. As noted, in the dielectric theory, quasiparticle ex-citation can no longer be observed in S(Q, co) in the nor-mal phase where no=0.

On the basis of data at SVP for Q ~0.8 A ', Woodsand Svensson proposed that S(Q, co) could be decom-posed into two parts, a superfluid component Sz(Q, co)

and a normal component Sz(Q, co). In the superfluidphase, each component was weighted by the respective

41 4224 1990 The American Physical Society

41 TEMPERATURE DEPENDENCE OF THE PHONON AND ROTON. . . 4225

superfluid and normal densities ps( T) and p~( T) in theform,

S(g,nI)=[ps(T)/p]Ss(g, co)+[p~(T)/p]S~(g, ro) .

Recently, Talbot et al. made precise measurements ofthe temperature dependence of S(g, ro) over a wide rangeof co in liquid He under 20 bars pressure. Two wave vec-tors were studied, Q =1.13 and 2.03 A ', which corre-spond to the maxon and roton regions of the phonon-roton dispersion curve, respectively. As at SVP, thesharp peak in S ( Q, aI ) either vanished or changed abrupt-ly as T was increased through Tz. However, Talbotet al. found that the Woods-Svensson decomposition,with components proportional to ps(T) and p~(T), didnot describe the data accurately enough to have funda-mental meaning. The important observation by Woodsand Svensson that S(g, co) changed abruptly at TI forQ & 1 A ' was confirmed.

As noted, the temperature dependence of S ( Q, co ) atsmaller wave vectors ( Q (0.5 A '

) is not so preciselydetermined. For 0.2(Q(0.5 A ', Woods and Cow-ley and Woods observed that the one-phonon line shapechanged little on passing from the superfluid to the nor-mal phase. The phonon linewidths increased sharplywith T for T just below Tz, but they continued to in-crease with T for T & Tz. These observations are inmarked contrast to the temperature dependence at largerwave vectors. Is this a real effect in which S(g, nI) forQ 0.5 A is similar in the two phases, while at higher Q,S(g, co) is quite different?

Since the measurements of Cowley and Woods, therehave been significant improvements in signal intensityand instrumental resolution which permit a more precisedetermination of S(g, co), especially of the linewidth.Our purpose here is to present new data at Q =0.4 Aand to explore the temperature dependence of S(Q, ro)and the Woods-Svensson decomposition. We alsopresent new precise data at the roton wave vectorQ= 1.935 A ' for comparison.

The experiment and analysis of data are discussed inSecs. II and III. The results for Q =0.4 and 1.935 Aare presented in Secs. IV and V. The data are discussedand compared with theory in Sec. VI.

Q=04A-~

T=1.35K

N~~CO

E1000—

~~COC4)C

00

~4&004 00+%~ a a a n () g li e e (l g0

I I I I I

0.1 0.2 0.3 0.4 0.5

Energy (THz)

Q=0.4A ~

T=1.35K

1.127 THz (1 THz=4. 135 meV=47. 99 K). Collimationof 60':60':60' was used since the neutron intensity is low,even on a high-flux instrument. With these conditionsthe experitnental (incoherent scattering) resolution [fullwidth at half-maximum (FWHM)] was 0.036 THz (1.73K) and the phonon linewidth (measured FWHM) at thelowest temperature (1.35 K) was 0.044 THz (2.11 K).The order-of-magnitude larger roton structure factor per-mitted higher instrumental resolution; 30' collimationwas used throughout, with a fixed incident energy of0.913 THz. In this configuration the experimental rotonlinewidth (FWHM) at 1.23 K was 0.015 THz (0.72 K).

The high-purity 4He sample (of volume approximately40 cm ) was condensed into an aluminum cell of a diame-ter of 3 cm containing vertically spaced cadmium discs, 1

cm apart, to minimize multiple scattering effects. Thesample cell was cooled in a helium flow cryostat and thesample temperature measured by a calibrated carbonresistor. "Empty-cell" measurements were made at eachwave vector at the lowest temperatures investigated. InFig. 1 we present an example of the "full" and "empty"

II. EXPERIMENTAL DETAILS

The measurements of the dynamic form factor dis-cussed here were made using the IN12 cold-neutrontriple-axis crystal spectrometer at the high-flux reactor ofthe Institute Laue-Langevin (ILL), Grenoble, France.This instrument is situated on a neutron guide tube fromthe ILL vertical cold source and thus has an enhancedflux of long-wavelength neutrons. Pyrolytic graphite(002) was used as a monochromator and analyzer for allexperiments, with a cooled beryllium filter to removehigher-order neutrons. Because of the different energyand wave vector ranges of the two series of measure-ments, two different experimental configurations wereemployed. For the Q =0.4 A ' experiments (and someless complete determinations of Q =0.3 A '), the instru-ment was operated with a

fixed

ana neutron energy of

50-CIC

ez0

Il

I) " I)

al.2 0.60.50.3 0.4Energy (THz)

FIG. l. Upper: empty-cell (0) and liquid He plus cell (Cl)inelastic spectrum for the phonon wave vector (g =0.4 AT =1.35 K). Lower: expanded view of multiphonon peak aftersubtraction of the ce11 scattering. The solid line is a Gaussian fitas described in the text.

4226 W. G. STIRLING AND H. R. GLYDE 41

0

spectra for the phonon wave vector Q =0.4 A '. As isseen in Fig. 1, the inelastic spectrum is dominated by thesharp one-phonon peak at 0.16 THz, with a much weakerbroad multiphonon peak at 0.41 THz; this latter featureis shown with the empty-cell scattering subtracted on anexpanded scale in Fig. 1.

There is, in fact, little published data on the completescattering function (one-phonon peak plus, at higher en-

ergies, the multiphonon continuum} at small wave vec-tors, other than that of Svensson et al. ' for Q =0.3A '. In the present study, therefore, the widest energyrange that was kinematically possible was investigated atsmall wave vectors. The present roton study concentrat-ed on the peak, the dominant feature of the inelastic spec-trum, since a wide range of co has been observed byWoods and Svensson. At this wave vector the empty-cell scattering was very weak and flat over the range ofthe measurement.

III. DATA ANALYSIS

The empty-cell scattering was firstly subtracted fromthe full spectra to give the net He scattering. Examplesof the resulting helium spectra are shown in Figs. 2 and3.

To determine the instrumental resolution width, it wasassumed that the intrinsic width of the one-phonon (ro-ton} line at T =1.35 K (T =1.23 K) was negligible. Forexample, at Q =0.4 A ' and T =1.35 K the intrinsichalf-width at half-maximum (HWHM) of the one-phononline measured by Mezei and Stirling using neutron spinecho techniques is approximately 0.8 X 10 THz. Theobserved width here therefore represents the instrumentresolution width.

The one-phonon (roton) line at T=1.35 K (T = 1.23

Q=0.4A-i

7=1.35K

A t(g, co) =2[co—co(Q, T)] +PQ, T)

PQ, T)

[co+co(g, T)] +I (Q, T)(2)

is the corresponding one-phonon response function.Here ntt(co) is the Bose factor for temperature T and fre-quency co, Z(Q, T) is the one-phonon intensity, and1 (Q, T) is the HWHM. This choice of analytic form forS,(g, co) is discussed in detail in Ref. 22 and is expectedto be valid when P Q, T) is sufficiently small that the fullfrequency-dependent I (Q, co, T) and co(g, co, T) can be

K) was fitted by least squares to a Gaussian function (tak-en to represent the instrumental resolution function} tak-ing account of the charging analyzer resolution, whereappropriate. The fitted functions are shown as the linesof Figs. 2 and 3. This procedure yielded energies and fullwidths of 0.161 (+0.001) and 0.044 (+0.002) THz for

Q =0.4 A ', and 0.179 (+0.0005) and 0.0153 (+0.0013)THz for Q =1.925 A '. These Gaussians were used asthe instrumental resolution functions at all higher tem-peratures.

The dynamic form factor was expressed as a sum of aone-phonon part S ( Q, co) plus a multiphonon part

S (Q, ro) =S&(Q, co)+S~(Q,~),in which SM(g, co) contains any possible interferenceterms ' between the one and multiphonon components.The one-phonon scattering function St(g, co) wasrepresented, as in the work of Talbot et al. , by aLorentzian function at both positive (neutron energy loss)and negative (neutron energy gain) excitation energies.S, ( Q, co ) is then written as

S&(g,co)= [ntt(co)+1]Z(Q, T)A&(g, co),1

2~

where

~~CO

4)

500—Z

4000-C

C

z 2000—

0- 0 nQ

1 I

0.2Energy (THz)

0.4 0.60.14 0.16

I

0.18

Energy (THz)

0.20 0.22

FIG. 2. Net He scattering after subtraction of empty-cellscattering (Q =0.4 A ', T=1.35 K). The solid line is the re-

sult of a least-squares fit to two Gaussians as described in thetext.

FIG. 3. Net He scattering at the roton wave vector(Q =1.925 A ', T=1.23 K). The solid line represents aGaussian fit.


IV. PHONON RESULTS (Q =0.4 A ')

Although the bulk of the phonon data was determinedat a wave vector of 0.4 A ', a few measurements werealso made at 0.3 A '. Results for temperatures of 1.35and 2.96 K at Q =0.3 A ' are displayed in Fig. 4 wherethe large line broadening at the higher temperature isclearly apparent.

Figures 5 and 6 summarize the inelastic scattering re-sults for Q =0.4 A '. As the temperature is raised, thephonon line broadens and its peak intensity is reduced,

Q=0.3A-'

o T=1.35K

o T=2.96K

Ca

8 &000-C

Z

0--

04

I I

0.1 0.2Energy (THz)

oooo o=~

0.3

FIG. 4. Comparison of superfluid (T=1.35 K) and normalfluid ( T =2.96 K) spectra for Q =0.3 A

well approximated by their on energy shell values [e.g.,I (Q, cu, T) = I'(Q, T) at all co]. By combining theLorentzians, A&(Q, ro) can be written in the "harmonicoscillator" function form often used to describe phononresponse functions. Thus the "Lorentzian'* and har-monic oscillator functions are equivalent providedco(Q, T) is interpreted as the phonon frequency in eachcase and not the sum [co (Q, T)+ I (Q, T)]' . The func-tions defined in Eqs. (1) and (2) were convoluted with aGaussian (instrumental resolution) function of unit areaand FWHM defined by the lowest temperature data, asdescribed above, and were fitted to the helium data.

For the phonon data, the multiphonon componentSM(Q, co) was modeled by a normalized Gaussian func-tion Gsr(Q, ro) with peak height scaled by the appropriatetemperature factor of Eq. (1), SM(Q, co) =[ns(co)+1]Gsr(Q, co). Since the width of G~(Q, co) was large,instrument resolution effects were not removed.

In practice, the final fitted one-phonon parameterswere only weakly dependent on the particular form usedto describe the multiphoton continuum. For the rotondata the weak multiphoton scattering was considered as asmall temperature-independent "background"; thissimplification has little effect on the one-phonon parame-ters extracted by the fitting procedure. The fit ofS~(Q, cu) to the multiphonon scattering at Q =0.4 Ais shown in Fig. 1.

ps( ~) px(S(Q, ru) = Ss(Q, ru)+ S~(Q, ru) .

p'

p(3)

In the normal phase where p&=p, S(Q, co) reduces toSz(Q, ro). To explore the WS model for Q =0.4 A ', wedetermined S~(Q, ro) as S~(Q,co)=S(Q,co) for T=2.32K. We denote this as S~(Q, co). In the superfiuid phasewhere both components of S(Q, co) contribute, we deter-mined [ps(T) Ip]Ss(Q, ro) as

while the peak position remains essentially constant.The full lines in Figs. 5 and 6 represent the "best-fit"

convolutions of Gaussian and Lorentzian functions as de-scribed above in Sec. III. The simple Gaussian. model forthe multiphonon continuum peak introduced in Sec. IIIdescribes this part of the spectrum rather well, except atthe highest temperature (3.94 K) where multiphononscattering has largely disappeared as a contribution dis-tinguishable from S, (Q, ro). Taken overall, the fit of thesimple two-component model to the observed spectra isvery satisfactory.

The phonon parameters obtained in this manner arepresented in Fig. 7. The phonon frequency is seen to belargely independent of temperature, although there maybe a slight increase just below T&. At about 4 K there isan apparent "softening" (somewhat exaggerated by thesuppressed zero of the lower frame of Fig. 7); inspectionof Fig. 6 suggests that this is indeed a real effect as thereis a significant shift of intensity to lower frequency be-tween 3 and 4 K. The half-width at half-maximum I ofA, (Q, co) increases smoothly through the A, point with lit-tle sign of any discontinuity, although the rate of increasedecreases above the transition. We note that a small sys-tematic error is introduced in our analysis since we haveused the 1.35-K linewidth as a measure of the instrumen-tal broadening. However, the work of Mezei and Stir-ling, which concentrated on temperatures below 1.7 K,has shown that the HWHM at 1.35 K for the wave vec-tor considered is less than 0.001 THz, producing a negli-gible effect on the phonon widths extracted. Finally, Fig.7 shows that the one-phonon intensity also shows littleevidence for a discontinuity at the A, point. So there isindeed little evidence for a qualitative change in the formof S(Q, co) at or near the A. point. To further support thiscontention, we compare spectra just below (2.04 K) andjust above T (2.32 K) in Fig. 8. The only significantdifference is in the peak intensity, while there is a slightshift to higher frequency just above T& as seen also in

Fig. 7.Above the A, point the spectra at 2.24, 2.32, 2.56, and

2.96, are similar, except for a slight increase in widthwith temperature and some increase in scatter of the datain the peak region with temperature. The similarity isdisplayed in Fig. 9 where a low temperature spectrum isincluded for comparison. As noted, the T =3.94 K spec-trum is significantly displaced toward lower frequency.

In the Woods-Svensson (WS) model, S(Q, co) is ex-pressed as a sum of a superAuid and normal component


[ps( T)/p]Ss(Q m):—S(Q, co) — Sg(Q, ~)

p~( T)=S&(Q,~)— px( T)

S& (Q, co)+ [n&(co)+1]— [ns(co)+1] G~(Q, co),p

(4)

where we have used S(Q,co)=S,(Q, co)+Ssc(Q, co) andSM(Q, co)=[ns(co)+1]GM(Q, co). The Ss(Q, co) deter-mined in this way is dominated by the one-phonon partsof S(Q, co). In the co range where G(Q, co) is significant,

I

ns(co) and ns(co) are negligible so that the second termreduces to [ I —[pz(T)/p]I Gsr(Q, co). This term serves

simply to add a multiphonon component as T decreasesbelow Tz, w'hich may or may not be meaningful.

1500-1.89 K

1500 "2.08 K

1000 . 1000-

500- 500-

0----I

-0.2)

0 0.2I

0.4 0.6

~ ao d ~o~o o~0 ---a -s-e-------- —- --'--'--

-0.2 0 0.2 0.4 0.6

ENERGY (THz) ENERGY (THz)

1500-1.72 K

1500-2.04 K

1000-

zI-UJZ

500-

-0.2 0

0cO

Q ---~p-~—0—--w-—0 0 ) I

0.2I

0.4 0.6

1000-

500-

0%ae i3Q 0=~

0I

-0.2 0 0.2I

0.4 0.6


1500- 0

0 1.35 K1500-

1.95 K

1000— 1000-

500— 500-

-0.2

l00 ~00~ 0+9Q.00k

I

0 0.2 0.4 0.6

0 --~=,-0.2

I

0.2

00 0~0(I

0.4 0.6


0FIG. 5. Temperature dependence of the phonon spectrum (Q =0.4 A '). The solid lines are the fitted Lorentzian convolutions as

described in the text.


The resulting instrument resolution broadened

[ps(T)lp]S&(Q, co) is shown in Fig. 10. For the lowesttemperatures the ratio of normal to superfluid densities isrelatively small (0.06 for T = l. 35 K) so Ss is only slight-

ly different from the total S,. At higher temperatures,however, the superfiuid component Sz consists of a sharppeak which goes slightly negative in the wings, indicatingthat too much intensity has been subtracted. Althoughthis effect is small, of the same order of magnitude as thescatter in the low intensity data points, we believe the net

negative intensity suggests that Ss(Q, c0) is not physicallymeaningful. We have also made the subtraction via theintrinsic S(Q, co) (not resolution broadened) using Eq. (4)and the data in Table I and obtained similar negative in-tensity in the wings. As in our previous study, we findthat (3) is not an accurate enough description to havecompelling fundamental meaning. In addition, thepresent data at Q =0.4 A ' do not suggest a priori adecomposition of the form (3).

1500- 1500-

2.32 K1000-

3.94 K

500- 500-

Q ~00 5 0 0

-0.2 0

000~0~I

0.2 0.4 0.6

0

-0.2I

0.2 0.4 0.6


1500- 1500-

0)Z 1000-LUI-zt- 500-UJZ

2.24 K1000-

500-

2.96 K

0 —=—~o~o oI

-0.2 0.2 0,4 0.6

0 o o-0- —————------I I I

-0.2 0 0.2 04 0.6


1500-

1000 .2.14 K

1500-

1000—

2.56 K

500- 500—

o tk)oQ -- —-~ =~-

o

-0.2

00~06 5Q~

0 0.2 0.4 0.6

~oQ ~ 9

0

-0.2 0.2 0.6

0~o~—--- --0-+—I

0.4


FIG. 6. As Fig. 5.


CO

I-oN ~

0(1

4He Q=O.4 A-' SVrI I I

I I

Q=0.4A-~

o T=2.24K

o T=2.32K

o T=2.56K~ T=2.96K

01

0.08-

N0.06-

-4

0.1 0.2I

0.3 0.4 0.5

0.04-Q4

O.O2 =—

01

0.16-

NO.14-

0.12-0

0.10-

0&~Mezei and Stirling

I I s I

2 3

Ipp 0

I'G&1

I 0

I

-7

-6

1500—~~COC+ ~000-C

500—

0

opera,0.1

Q=0.4A 1

T=1.35K

0.2 0.3 0.4

Energy (THz)0.5

n~~OO On+OnO

I I I I I I I

0.081 3

Temperature (K)

) -44

0

FIG. 9. Comparison of the phonon spectrum (Q =0.4 A ')above Tq, with the lowest temperature ( T = 1.35 K) result.

FIG. 7. Fitting parameters co(Q, T), I (Q, T), and Z(Q, T) ofEq. (2) for the phonon spectra (Q =0.4 A '). The dashed line

shows the Gaussian resolution width I G used in the convolu-tion procedure. The solid lines are previous half-widths at half-

maximum [I (Q, T)] observed by Cowley and Woods (Ref. 24)and Mezei and Stirling (Ref. 25).

I I

g 0 4/-1

o T=2.04K

o T=2.32K

CO

soo—C

Z

ay5i~ey

0I

0.2

Energy (THz)0.4

0FIG. 8. Comparison of the phonon spectrum (Q =0.4 A )

just below Ti (2.04K; 0) and just above T» (2.32 K; (&).

V. ROTON RESULTS (Q =1.925 A ')

The observed scattering intensity at the roton wavevector is displayed in Figs. 11 and 12. The solid lines arefits of Eqs. (I) and (2) to the data convoluted with aGaussian to represent the instrument resolution as dis-cussed in Sec. III.

As observed in previous studies, the roton linewidth in-creases and the roton peak intensity drops rapidly withincreasing T for T ( T&. The increase in the linewidth isparticularly rapid in the temperature range just belowTx. The rapid change in character of the scattered inten-sity just below T& is displayed in Fig. 13 where the scat-tered intensity at T=1.79, 2.02, 2.09, and 2.17 K arecompared. It is clear that there is a qualitative change inthe inelastic spectrum between 2.09 and 2.17 K. In thesuperfluid phase there remains an identifiable peak in thescattered intensity which has disappeared at T&. AboveT& there is relatively little change in the shape of the ob-served scattering intensity with temperature, althoughthe frequency m(Q, T) decreases significantly and thelinewidth I (Q, T) continues to increase as T is increasedabove T&. The similarity of the observed intensity at alltemperatures in the normal phase above Tz is displayedin Fig. 14, where the spectra at 2.33, 2.48, and 2.96 K arecompared. In general, the shape of the scattered intensi-ty is quite different in the superAuid and normal phases,particularly in its temperature dependence.


a=0.4A-~

T=2.24K

&~/~(I =1.00

0.5

T=2.14K

&l„/) =0.88

Co l 1' Fi al5 Pl Pi 5iI q~ t ~ a-

0.1I

0.2 0,3 0.4I

0.5

ChCIc 50Q—

g-0 4g-1

T=2.04K

&~/&=0.83

I'..II Ii, & .'qij~nIH&y

I I I

0.1 0.2 0.3 0.4

As seen from Figs. 11 and 12, the fitted S, (Q, co} ofEqs. (1) and (2) provide a very satisfactory description ofthe temperature dependence of the scattering intensity.A single Lorentzian function was also fitted to the dataand, while providing a reasonable fit at low and high tern-peratures, this form was unsatisfactory between 1.93 and2.33 K. The roton parameters Z(Q, T), co(Q, T), andI (Q, T) obtained using Eqs. (1) and (2) are shown in Fig.15. The co(Q, T) and I (Q, T) reported here agree wellwith previous measurements. For example, the rotonHWHM at 1.37 K found here is 0.1 K, while Mezei re-ports 0.11 K at T =1.4 K. Comparing Fig. 7 and Fig. 15we see that the roton linewidth increases significantlymore rapidly with temperature than does the phononline width. Indeed, at T& and above, the spectral

0linewidth at Q =1.925 A ' is approximately three timesgreater than the linewidth at Q =0.4 A ', whereas atT=1 K both lines are extremely narrow. Below T& theroton co(Q, T) also decreases with increasing T (a fre-quency shift 6=[co(Q, T) co(Q, T—=O K)] spiv(T))(Refs. 19 and 27), while the phonon co(Q, T) is essentiallyindependent of T. In the normal phase the temperaturedependence of co(Q, T}at Q =1.925 and 0.4 A ' is quitedifFerent. In particular, at Q =0.4 A ' the ratio ofI'(Q, T)/co(Q, T) in the normal phase is sufficiently smallthat the scattered intensity can reasonably be attributedto a single collective excitation. This is not the case atthe roton wave vector.

CD (5 i505 5razm VI. DISCUSSION AND INTERPRETATION

I

0.1 0.2I

0.3 0.4I

0.5A. Energies and lifetimes

CD QI Rx ~nn&&n&as~9@g

I

0.1I

0.2I

0.3I

0.4I

0.5

COCIC

ICL

CD

Q=o 4A-

T=1.35K

0.08

Oo ~~~~O mo Pmo~o

0.1 0.2 0.3 0.4

Energy (THz)

i

0.5

FIG. 10. Resolution broadened values of pz(T)lp)S&(Q, co)

obtained by subtracting a "normal fluid component" from thesuperfluid phase phonon spectra (Q =0.4 A '), as described in

the text.

To obtain the energies co(Q) and inverse lifetimes I (Q)we fitted the response function A, (Q, co) in (2) to the in-trinsic one-phonon part of the data. This A, (Q, co) is theexact response function' for a boson (a Bose quasiparti-cle or phonon) in which the self energy X(Q, co)=b,(Q, co) —iI (Q, co) is approximated by its "on-shell"value X(Q, co }=X[Q, co( Q) ]. In this approximation, X be-comes independent of frequency and the full' A, (Q, co)reduces to (2) in which co(Q)=co (Q)+b[Q, co(Q)],I (Q)=I [Q,co(Q)], and coo(Q) is the noninteracting bo-son energy.

By combining the two Lorentzians in (2), A, (Q, co)takes the form

8col(Q)co(Q)

t~' —[~'(Q)+ r'(Q)] I'+4~'r(Q)' (5)

Equation (5) is often referred to as the harmonic oscilla-tor (HO) function. The Lorentzian (2), which has aStokes and anti-Stokes term, is clearly the same as theHO function. However, in using (5) the sumE (Q) =co (Q)+I (Q) is often interpreted as the phononenergy. The relation E (Q) =co (Q)+ I (Q) may be usedto relate the present co(Q) in Table I to the values E(Q)quoted by Tarvin and Passell for the roton energy.Given the fundamental nature of (2), we believe co(Q) isthe most appropriate energy. However, when S(Q,co) isvery broad and A, (Q, co) is needed over a wide range ofco, defining a frequency-independent co(Q) and 1 (Q) may

4232 %'. G. STIRLING AND H. R. GLYDE 41

not be meaningful.The present half-width I (Q) at Q =0.4 A ' is com-

pared with the values quoted by Cowley and Woodsand by Mezei and Stirling in Fig. 7. The values quotedby Cowley and Woods include the instrument resolu-tion and are therefore larger than the present values. Thepresent values at low T are larger than those quoted byMezei and Stirling. We believe this is because thepresent instrument resolution I G is much larger than the

intrinsic 1(Q) at low T, which makes is difficult to ex-tract a sinall I (Q) with precision. At low T the Mezeiand Stirling values are more reliable.

In Fig. 16 we compare the present I (Q) for the rotonwith those obtained by Woods and Svensson At SVPand by Talbot et al. at 20 bar pressure. The presentI (Q) agree with the WS values at low T but are larger athigher T [T near T&]. The present I (Q) are larger be-cause we have fitted (2) to the whole of the one-phonon

1500-

2000 .-

0.14 0.16 0.18 0.20 0.22

ENERGY (THz)

———g- - - ——I- ————I- ———-&- ——00.10 0.15 0.20 0.25

ENERGY (THz)

6000- 00

2000

4000zIXI

zI-

2000z

0.16 0.18 0.20

ENERGY (THz)

0.10 0.15 0.20

ENERGY (THz)

0.25

3000

6000-

4000-

2000-1000-

0 -o- 1—00.16 0.18 0.20

ENERGY (THz)

0.12 0.16 0.20 0.24

ENERGY (THz)

FIG. 11. Temperature dependence of the roton spectrum (Q =0.1925 A '). The solid lines are the fitted Lorentzian couvoutionsas described in the text.


peak rather than that part of it identified by WS as theone-phonon peak in the superfluid component of S ( Q, co).The present analysis is similar to the simple subtractionmethod used by Talbot et al. If the interpretation dis-cussed in Sec. VI that the sharp component of S(Q, co)for the roton arises from single quasiparticle excitation isindeed correct, then the lifetimes extracted by WS may bea better representation of the quasiparticle lifetime.

8. Comparison of normal liquid Hewith other Suids

To interpret the observed scattering from normal Hewe make comparisons with liquid He, heavier fluids, andsolid He. In Fig. 17 we compare our observed intensityfor normal He in the phonon region (Q =0.4 A ') andT=2.32 K with S(Q, co) observed in liquid He at

600

400

0-I———————I ———0-—

-0.2 0.2

ENERGY (TMz)

04 -0.2 0 0.3 0.4

ENERGY (THz)

750

(0500

Z

250

0

-0.1 0 0.1 0.2 0.3

0

0.40-01 0 0.2 04


1000

200

-0.1 0.2

ENERGY (THz)

0.4 -0.2I

0 0.2

ENERGY (THz)

0.4

FIG. 12. As Fig. 11.

4234 %'. G. STIRLING AND H. R. GLYDE

2000—

~~CO

(000

Z

0 pppppoi I I 1 I

%.2 0 0.2

Energy (THz)

Q=1.925A ~

o T=1.79K

O T=2.02K

p T=2.17K

I

OA

g- t AQSl(-1

~ Y=2.02K

a FWHM =21"=0. 1 THz for He and 0.05 THz for He.Since the peak position and width ' in He is quite suc-cessfully interpreted as a collective zero sound mode, it isnatural to interpret the peak in normal He in the sameway. Liquid Ne supports a well-defined collective densityexcitation up to Q=0. 16 A ' but not3 for Q~0. 8A '. Liquid Rb and other liquid metals support awell-defined collective mode at Q =0.3 A ' (the FWHMis approximately one-half the peak frequency), but thepeak is certainly gone by Q =1.0 A '. Historically' andon the basis of these comparisons, the peak in S ( Q, co) innormal 4He at Q =0.4 A ' may be interpreted as a col-lective zero sound mode in the fluid density.

In Fig. 18 we compare the scattering intensity atQ =1.13 A ' (maxon) and at Q =1.93 A ' (roton) fromnormal He at p =20 bar at several temperatures with

S(Q,co) observed in liquid 3He at Q =1.2 and 2.0 A

1000-COC,

C

Z 500—

0 ppopoI

%2I I I

0 02Energy (THz)

O T=2ASK

pT 2.17K

0.4

CO

I'c 40

o D

N a 20-l0

0.30

I I

MLK

p I

I

4He Q=1.925 A ' SVP

-12

FIG. 13. Comparison of roton spectra (Q =1.925 A ')below Tq and at T& (T=2.17 K). For clarity, some data arepresented twice.

Q =0.5 A ' and T =0.12 K by Scherm et al. Thepeak in S (Q, E) in liquid He at v=0. 18 THz is identifiedas scattering from the collective zero sound (phonon)mode with spin Iluctuation scattering at lower v. Thewidth of the one-phonon peak is similar in the two cases;

600

N

I- 0.20-

e~1

0 Q

0.20

OO 0 0oOo0

Nr0.10—l-

0

hC

-8

0-4

I'GI- — - 0

-e

I-4 0

400—~~COCI

200—

Z8&8&

01

I I

Temperature (K)

i

-0.2I 1 i

0 0.2

Energy (THz)

0.4

FIG. 14. Comparison of roton spectra (Q =1.925 A ')above Tz.

FIG. 15. Fitting parameters co(g, T), I (g, T), and Z(g, T) ofEq. (2) for the roton spectra (Q =1.925 A '). The dashed lineshows the Gaussian resolution width used in the deconvolutionprocedure. Results for the superQuid phase are also displayedmultiplied by 10; for comparison results of Mezei (Ref. 27) andLandau and Khalatnikov (Ref. 19) are shown as the dot-dashedline.


0.14—

0.12-

0.10—

NZ 0.08-

t-~ 0.06-

0 Q=1.925A 1, SVP Present

~ Q=2.030A, P=20 bars

o Q=1.926A, SVP WS----- LK Theory, P=SVP

LK Theory, P=20 bars

TA, (SVP)

T), (20 bars)

1000-L

6$

CO

0) 500-

IZ

4He

Q=0.4 A-'

svp2.32 K

0 0 0I

%2 0I

0.2

000 00~

0.4 0.6

0.04—A.2

Energy (THz)

0.2 0.4I

0.6

0.02—

0.00',1.0 1.2 1.4 1.6 1.8

Temperature (K)2.0 2.2

FIG. 16. One-phonon half-widths I'(Q, T) at the roton4

Q =1.926 A ': open circles with error bar present values ob-tained by fitting (2) to the whole of the "one-phonon" peak (seetext); open squares with error bar represent Woods andSvensson (WS) values (Ref. 20) obtained by fitting to the one-phonon peak showing intensity approximately proportional topz(Q, T); closed circles with error bar represent values at p =20bar obtained by Talbot et aI. (Ref. 22) following the WSanalysis. Lines are the Landau-Khalatnikov (Ref. 19) theory.

~He

Q=0.5 A1

SVP2K

~ 0

~ ~ 0.1

s A

I

E(meV)

FIG. 17. S(Q, co) observed in liquid 'He at Q =0.5 A ' (Ref.29) and in liquid He at Q =0.4 A ' (present).

The intensity observed in normal liquid He is largelytemperature independent. In liquid He between T =40mK and 1.2 K, S(g,co) also differs little. Thus in normalHe and liquid He, S(g, co) is largely independent of

temperature. In He there is no collective excitation forQ & 1.2 A '. Rather, S(g, co) represents broad scatter-ing from particle-hole excitations. The scattering ob-served at Q =1.13 A ' in normal liquid He is broaderthan S(g, co) in He (i.e., FWHM=0. 6 THz in He com-pared with FWHM=0. 4 THz in He}. To set the scale,the zero sound mode width at Q = 1 A ' in He is2I =0.2 THz and the width of a typical longitudinalphonon midway in the Brillouin zone of bcc He (p =25bar) is 2I =0.2S THz. ' These comparisons suggeststrongly that normal He does not support a collectiveexcitation at Q = 1. 1 A '. Rather, S(g, co) at Q = 1.13A ' in normal He represents scattering from pairs ofsingle-particle excitations. Also, the data of Woods andSvensson and of Talbot et aI. for T &T&, reproducedhere in Fig. 19, suggest that the sharp peak in S(g, co) inthe superfluid phase disappears at T=Tz leaving onlybroad scattering. The situation is similar but not so clearat the roton wave vector, chiefly because the roton energyis so small. However, there is probably no collective

mode in normal He at the roton wave vector. On thebasis of their data and comparison with other fluids,Pederson and Carneiro also suggest there is no collec-tive excitation in normal He at the roton Q.

C. Some speculative remarks0

The present data at Q =0.4 A ' and previous data forQ &1.1 A ' show that the temperature dependence ofS(g, co) is quite different at these two Q values. AtQ50.4 A ', the peak in S(g,co) broadens graduallywith increasing T, but has the same basic shape in thesuperfluid and normal phases. In the normal phase, thepeak remains well dered as expected for a collectivedensity mode. At Q & 1.1 A ' the sharp peak inS(g,co), seen in the superfluid phase, apparently disap-pears at T&. In the normal phase, S(g,co) is very broad,which is characteristic of scattering from pairs (or highmultiples} of quasiparticles and no collective mode. Wemake some speculative remarks relating this behavior tomicroscopic theories of liquid He.

Landau, and Feynman and Cohen proposed thatsuperfluid He supports a collective excitation in the Quiddensity p(r)= fdr e'~'p(g). This excitation appears asthe sharp component in S ( Q, co },


250V)

200-

I5PCQ

IOO-~~CO

8 5p-C

Ot&

Z-50

-0, 2 0 0.2 0.4 0.6 0.8 I.O I.2

I 4-1I Q*I.I3AI

i T l90K' T *I928Kk

~ T 2.05K-I + T*2.96K

I

I

N~ O T 394K.4 4 %so He

I

+ ~ tc e~gI

af. p

3000

2500-

2000-

I 500-

1000-

I 0Q&2.03A I + T~ I.SOK

T) l.928 K ~ T I.93K+ T ~ I.96K

I v T ~ 2.06K+

o T 2.97K)O 4HeI

aI T

500

0+~i s

-0.2 0 0.2 0.4 0.6 0.8 I.O I .2

v (THz)

v (THz)-0.2 0 0.2 0.4 0.6

I I I I

0.5

0.8 1.0 1.2} I

~He

Q=~.zk'

-0.2 0

0.5—

v (THz)

v (THz)02 04 06 08 10 12

I I I I

~He

Q=2.0R i

0 0I

E(me V) E(me V)

FIG. 18. Upper: scattered intensity from liquid He at Q =1.13 A ' (maxon) and g =2.03 A ' (roton) (Ref. 22) for tempera-tures near T~ and above. Lower: S(g, co) of liquid 'He at Q =1.2 and 2.0 A ', T =40 mK (solid line) and T=1.2 K (open circles)(Ref. 36).

TABLE I. One-phonon parameters appearing in the response function A, (g, co) of Eq. (2) at thephonon (Q =0.4 A ') and roton (Q =1.925 A ') wave vectors.

1.351.721.891.952.042.082.142.242.322.562.953.94

co(Q, T) (THz)

0.161+0.0010.161+0.00150.1600.1590.1590.1600.1630.165+0.0040.1640.1650.1570.105+0.008

I (Q, T) (THz)

Phonon (Q =0.4 A ')

0.007(4)+0.0020.011(1)0.015(0)0.023(6)0.026(9)0.031(4)0.043(8)+0.0060.045(0)0.049(4)0.061(0)0.070(6)+0.011

Z(Q, T) (arb. units)

2.97+0.093.223.264.044.134.004.52+0.305.145.497.217.56+0.50

Roton (Q =1.925 A ')1.231.371.511.721.791.932.022.092.172.332.482.96

0.1789+0.00050.1785+0.00050.17700.17390.17340.16710.16420.16030.1428+0.0040.10480.07580.0249+0.015

0.002 +0.0030.0050.012(5)0.01500.0290.0460.0660.123 +0.0100.1830.2230.26 +0.02

5.70+0. 156.447.607.548.649.90

11.316.4 +1.226.854.0

187.2 +200.0


600

400-

'c 0

z 0

I II

I

I

I

III )I

III ilI

i I

200-I

II

I

I 1o I tL

I

Lal ~ I&I- 'I1g 8

He

20 atm

o -).(zk '

TX I' 28

a macroscopic fraction Xo of the N atoms in the zeromomentum state (k =0) &o and ao may be replacedby numbers (au ~k =0)=au ~k =0)=+No ~k =0) ).The density operator therefore separates into twoparts p(g)=+No(ct&+ct &)+p'(Q), where p'(Q)=g'k al, ctk+& involves particles above the condensate(kAO) only. Substituting p(g) into (7), and the dynamicsusceptibility g separates into two parts

y(g, co)=no g A G tt(g, co)Att+ytt(g co)aP

where

S(g, co) = — [n—tt(co)+1]Imp(g, co) .1

Here nz(co) is the Bose function and no=No/N. Thesingular part y~ =noAGA is proportional to single-particle Green's functions of the form G»(g, t)

i ( T—a&(t)a& (0) ). This comes from the first term ofp(g) and represents excitation of single quasiparticleshaving momentum Q. The regular part is

yIt ( Q, t) = —i ( Tp'(Q, t)p' (Q, O) &,

III

0-

0-

0 ~I-0.2 0 0.2 0.4 0.6 0.8

v (THz)

I.O I.2

FIG. 19. Temperature dependence of the net scattering in-tensity from liquid He at p =20 bar and Q =1.13 A ', fromTalbot et al. (Ref. 22).

S ( Q, co ) = f dt e ' '( p( Q, T)p ( Q, 0) ) .=1

The corresponding dynamic susceptibility is

y(g, t)= i ( pT(g,—t)p (Q, O)) . (7)

Here p (Q)= gk ak+&ak is the density creation opera-+

tor which may be expressed as a coherent sum of particle(ak+Q) and hole (ak) pair excitations. The collectivedensity excitation follows ' from the strong interactionbetween the He atoms and Bose statistics. The tempera-ture dependence of the collective mode was not fully dis-cussed.

Alternatively, Bogolubov, Hugenholtz and Pines,Gavoret and Nozieres, and the subsequent dielectric for-mulation ' present a somewhat different view based onGreen's functions. ' The Bose broken symmetry and thecondensate play a central role. Specifically, when there is

which represents excitation of particle-hole pairs, involv-

ing atoms outside the condensate only.The functions A (Q, co) are given by A (Q, co)

=[1—P (Q, co)] in which the P (Q, co) incorporates theinterference between scattering, which excites a singleparticle (G &) and pairs of particles (yIt). The expres-sion (8) is formally the same form as S ( Q, co) for phononsin solids ' in which scattering from single phonons,pairs of phonons, and one-two phonon interference termsare retained.

The general result (8) has several interesting features'which motivated this experiment. Within this descrip-tion of superfluid He, S(Q, co) contains a singular part,proportional to no, which involves excitation of a singlequasiparticle having wave vector Q. Thus any pole in thesingle quasiparticle Green s function G tt(g, co) will beobserved in S(g, co) when no@0. Since no=No/N -0.1,the intensity in S(g, co) resulting from ys may be small.For T ) Tz, in the normal phase, the poles in G & can nolonger be observed in S ( g, co).

The y'R(g, co) is the dynamic susceptibility involvingquasiparticles above the condensate. As in liquid Heand heavier fiuids, gz ( g, co ) may display a collective(sound) mode, at least at low Q. It is this mode in yz[strictly in the total g(g, co)], which is discussed by Feyn-man and evaluated in detail using correlated basis func-tion methods. Both yz and yz can contain singularstructure arising from long-lived quasiparticles and a col-lective density mode, respectively. Unfortunately, thespectral shape of G tt, A(Q, co), and y„'(Q, co) is notknown for strongly interacting fluids. When both yz and

yz have sharp, singular structure, it may not be useful toseparate y into two parts.

Gavoret and Nozieres show that the total y(g, co) andsingle-particle Green s functions in G&(g, co) have a singlecommon pole as Q~O. This occurs at frequency co=cQwhere c is the macroscopic sound velocity. Thus as


Q~0 it is not meaningful to distinguish between single-particle and collective density excitations —there is onlya single excitation at co=cQ, as shown by Bogolubov in

the dilute gas limit. As Q =0.4 A ' the observedS (Q, co) in Figs. 5 and 6 is concentrated almost entirely ina single peak with a very small tail reaching to higher co

This is true in both the superfluid and normal phases.This single peak is most readily interpreted ' as duechiefly to a collective mode in y„' at T & Tz and a collec-tive mode in the total g for T )Tz. For T & Tz the in-

tensity due to y& must also lie in the same single peak.Indeed, if Q =0.4 A ' may be regarded as low Q, thenwe expect G& and g to peak at the same energy. Thepresent resolution width I G =1 K is certainly not small

enough to resolve two closely lying peaks, if G, and gzpeak at slightly different energies. Also, since no~0. 1

most of the observed intensity in the single peak is prob-ably due chiefly to y„' rather than ys. On the basis ofcomparison with liquid He and other fluids, we expect gto show rather little temperature dependence.

In contrast, at the maxon and roton Q vectors, S ( Q, co)

consists of a sharp peak superimposed on a broad back-ground in the superfluid phase. Particularly at Q =1.13A ', a large fraction of the intensity lies in the broadbackground (see Fig. 19). We might interpret the sharppeak as the single quasiparticle peak of G in ys and thebroad background as due to gz. In this case y~ does notcontain a collective mode in the superfluid phase. As T isincreased, the sharp peak in S(Q, co) decreases in intensi-ty. This is expected from (8) since ys may continue tohave sharp structure for T & T&, but the contribution tog, which is what is observed by neutrons, decreases withno( T) and vanishes at T) . At T & T) in the normalphase we observe only y =yz which, from the discussionin Sec. VI 8, does not appear to show a collective mode.This interpretation suggests that y„' is broad and doesnot have singular structure in both phases. In this casethe separation of y into gz and y„ is very useful andoffers a possible interpretation of the data not open toother formulations. This picture, originally suggested byWoods and Svensson, is especially attractive atQ =1.13 A '. In the roton region the data are not soclear because the roton energy is low, and above Tz it isnot clear whether y, has a collective mode or not. How-

ever, from Fig. 13 the sharp roton peak seen in thesuperfluid phase does seem to disappear at Tz. Also, atthe roton Q we find I (Q, T}& co( Q, T) for T & Ti„when(2} is fitted to the observed intensity. This is uncharac-teristic of a weil-defined single mode suggesting the inten-sity is not due to scattering from a single mode. This alsomeans that fitting (2), a single mode expression, to the ob-served intensity is not meaningful for T ) T& at the roton

In this picture, the phonon and roton regions aresignificantly different. The y contains a collective modeat low Q but not for Q & l. 1 A '. At low Q the peak in

S(Q, co) is predominantly due to the collective mode in y.As Q~0 it is meaningless to distinguish the peak in yand G since there is only one mode having a single ener-gy. At Q&1. 1 A ' the sharp peak in S(Q, co) forT & T& is the quasiparticle peak in G. In this modelliquid He does not support a collective density mode athigher Q. The sharp peak in S(Q, co) disappears at Tibecause G no longer contributes to S ( Q, co ) when

no( T) =0. Thus liquid He apparently behaves like a nor-mal cold liquid in that it supports a collective densitymode at small wave vector and this mode dominatesS(Q,co). In addition, the existence of a finite no in thesuperfluid phase means that single quasiparticle excita-tions can be observed in S(Q, co) and these constitute thesharp component of S(Q, co) in the superfiuid at largerwavevectors. This interpretation will be elaborated fur-ther in a forthcoming paper (Glyde and Griffin).

ACKNOWLEDGMENTS

The authors wish to thank S. Hayden, D. Marcenac,D. Puschner, and C. C. Tang for assistance with the ex-periments and data treatment. Valuable discussions withProfessor A. GrifBn are gratefully acknowledged. Theyare grateful to the Institut Laue-Langevin for makingavailable the experimental facilities used in this study andgratefully acknowledge the award of a North AtlanticTreaty Organization (NATO) collaboration grant. Sup-port from the U.S. Department of Energy, OSce of BasicEnergy Sciences, under Contract No. DE-F602-84ER45082 and from the Science and EngineeringResearch Council is also gratefully acknowledged.

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P. Feynman and M. Cohen, ibid. 102, 1189 (1956).C. W. Woo, in The Physics ofLiquid and Solid Helium (Ref. 1),

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5S. T. Beliaev, Zh. Eksp. Teor. Fiz. 34, 417 (1958) [Sov. Phys. —

JETP 7, 289 11958)].N. Hugenholtz and D. Pines, Phys. Rev. 116,489 (1959).

7J. Gavoret and P. Nozieres, Ann. Phys. (N.Y.) 28, 349 (1964).K. Huang and A. Klein, Ann. Phys. (N.Y.) 30, 203 (1964).P. C. Hohenberg and P. C. Martin, Ann. Phys. (N.Y.) 34, 291

{1965).' V. Ambegoakar, J. M. Conway, and G. Baym, in Lattice Dy-

namics, edited by R. F. Wallis (Pergamon, New York, 1965),p. 261, Eq. (2.15).

"S.K. Ma and C. W. Woo, Phys. Rev. 159, 165 (1967).S. K. Ma, H. Gould, and V. K. Wong, Phys. Rev. A 3, 1453(1971).I. Kondor and P. Szepfalusy, Acta Phys. Hung. 24, 81 (1968);P. Szepfalusy and I. Kondor, Ann. Phys. (N.Y.) 82, 1 (1974).


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Phys. Rev. B 38, 11 229 (1988).A. D. B.Woods, Phys. Rev. Lett. 14, 355 (1965).

24R. A. Cowley and A. D. B. Woods, Can. J. Phys. 49, 177(1971).

25F. Mezei and W. G. Stirling, in 75th Jubilee Conference on

Helium-4, edited by J. G. M. Armitage (World Scientific,Singapore, 1983), p. 111.

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Temperature dependence of the phonon and roton excitations ...glyde/Pubs/F1A76D9Ed01.pdf · 4226 W....

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