Journal of Low Temperature Physics, Vol. 101, Nos. 5/6, 1995

Density and Temperature Dependence of the Momentum Distribution in Liquid Helium 4 W. M. Snow*

Intense Pulsed Neutron Source, Argonne National Laboratory, Argonne, IL 60439 and P. E. Sokol

Department of Physics, The Pennsylvania State University, University Park, PA 61802 (Received May 11, 1995; revised July 18, 1995)

Deep inelastic neutron scattering measurements' on liquid 4He have been carried out for temperatures from 0.35 K to 4.2 K and densities from 0.125 to 0.200 g/cm 3 at a momentum transfer of 23 ~ 1. These measurements are at large enough momentum transfer that deviations from the Impulse Approximation are accurately described by current theories and information on the single particle momentum distribution may be extracted from the measured scattering. The scattering exhibits non-Gausian behavior in both the normal liquid and superfluid phases. A distinct change in the scattering, marked by a reduction in the width and increased deviations from the classical Gaussian shape, occurs at the suerfluid transition. We present a comparison of our experimental results with recent calculations at a variety of temperatures and densities and show that theory and experiment are in excellent agreement. We also present model scattering functions, obtained by correcting for instrumental resolution and final state effects, that represent the scattering in the IA limit. Finally, we present values for the average kinetic energy and the Bose condensate fraction over a broad range densities and temperatures. *Current Address: Indiana University Cyclotron Facility, 2401 Milo B. Sampson Lane, Bloomington, IN 47408.

881 0022-2291/95/1200-0881507.50/0 9 1995 Plenum Publishing Corporation

882

W . M . Snow and P. E. Sokoi

1. I N T R O D U C T I O N The superfluid properties of liquid 4He provide one of the most prominent examples of the macroscopic consequences of microscopic quantum effects. At the macroscopic level, superfluidity can be described in terms of a macroscopic wave function. 1 A successful description of many of the properties of the superfluid phase, such as the succes of the two fluid model, quantized vortex rings, and characteristics of the phonon-roton dispersion relation can be obtained in terms of this macroscopic wave function. At the microscopic level, our understanding of superfluidity is based on the concept of Bose condensation. 2 Both the microscopic and macroscopic descriptions appear in the form of a macroscopic wave function. The magnitude of the macroscopic wave function is proportional to the square root of the condensate fraction, no, and its phase is coherent on a macroscopic scale reflecting the long range correlations introduced by the presence of a condensate. According to our present understanding of Bose liquids, the unique properties of the superfluid phase of liquid helium depend on the existence of the condensate, but not (so far as we know~ directly on its magnitude) Thus, direct information on the condensate has been difficult to obtain. The only observable property of the liquid which can directly reveal the presence of a Bose condensate, independent of theoretical interpretation, is the single particle momentum distribution, n i p ) . If Bose condensation occurs, the momentum distribution must develop a c%function singularity with an intensity proportional to the condensate fraction.4"5 Thus measurements of the momentum distribution are one of the few experimental techniques capable of providing direct information on the condensate. The most direct method for measuring the momentum distribution in liquid 4He is through the use of deep inelastic neutron scattering (DINS). 6 In a deep inelastic scattering event, the energy and momentum transferred from the scattering probe to a particle in the target are very high compared to the characteristic energies and momenta of the particles in the system. In this limit, the dynamic structure factor, which describes the scattering, may be related to the momentum distribution through the Impulse Approximation (IA). this direct relation between the observed scattering and n ( p ) , coupled with the interest in the Bose condensate, has been a major source of inspiration for numerous inelastic neutron scattering experiments performed on helium]" 8 Recently, a number of studies of liquid helium at high momentum transfer have been reported. 9-17 These studies have been motivated by two recent developments. First, the availability of high fluxes of epithermal neutrons at spallation neutron sources 18 has made possible high reso!ution.

Density and Temperature Dependence in Liquid Helium 4

883

high statistical accuracy measurements at momentum transfers large enough that the scattering is approximately described by the Impulse Approximation. Second, quantitative theoretical descriptions of the deviations from the Impulse Approximation that occur at finite momentum transfer, known as final state effects (FSE), have become available. 19-27The availability of these theories allow FSE to be taken into account and information on the scattering in the IA limit, and hence the momentum distribution, to be extracted. In this paper we report new measurements of the observed scattering from bulk liquid 4He at variety of densities above the saturated vapor pressure (SVP) density in the superfluid and normal liquid phases. These measurements complement previous studies that have examined the temperature dependence in both the superfluid and normal liquid at low density 14 and the density dependence of the normal liquid at high temperature. 1~ When combined with these previous measurements a rather complete coverage of the phase diagram as a function of density and temperature results. A brief review of the fundamental properties of Deep Inelastic Neutron Scattering is presented in Sec. II. Section IlI describes the details of the experimental setup and data reduction. Section IV presents the measured scattering and discusses, qualitatively, the behavior of the scattering as a function of temperature and density. The results presented in this section are uncorrected for instrumental resolution or FSE. Therefore, they do not depend on the particular form for the instrumental resolution or, more importantly, FSE that have been used. Thus, the trends that are observed are intrinsic features of the momentum distribution. Section V presents a comparison to theoretical calculations and further analysis of the data. These include comparison to microscopic theories, extraction of model scattering functions, moments of the scattering and the condensate fraction. 2. D E E P INELASTIC N E U T R O N S C A T T E R I N G

The dynamic structure factor, S(Q, co), which describes the scattering of neutrons has, in the limit of large momentum transfers, the simple form2S,29: lira S(Q, co)=SrA(Q, co)=

Q~

o9

n(p) fi co-cor

MHejdp

(1)

where Q and co are the momentum and energy transfer, n(p) is the atomic momentum distribution, cot =h2Q2/2Mne is the recoil energy and Mne is the mass of the helium atom. In this limit, which is known as the Impulse

884

W . M . Snow and P. E. Sokol

Approximation (IA), the typical duration of the scattering event is much shorter than the helium-helium interaction time. The scattering atom then acts as a free particle during the collision. The Doppler broadening term (Q .p) couples the momentum distribution to the observed scattering. The scattering from an isotropic system, such as a liquid, can be expressed as a function of a single scaling variable 3~

y M(co_co~) Q

(2)

when the IA is valid. S(Q, co)can then be written as a function of Y only: Q J~A(Y) = ~ SIA(Q, co)

(3)

where JIA(Y) is the well-known Compton profile. The Compton profile is given by 1

:o

JIA(Y) =x--'trzr~p~ pn(p) dp

(4)

which is simply the longitudinal momentum distribution. JIA( I0 exhibits several features which are characteristic of the IA: it is symmetric about Y = 0 and depends on Q only through the scaling variable Y. This behavior J(I1) is the Y-space manifestation of the well-known characteristics of the IA: the scattering function is centered at the recoil energy and, at constant Q, is symmetric with a width proportional to Q. For helium, the observed S(Q, co) at Q's greater than 15 A -~ is qualitatively in agreement with the IA. 3l The IA only approximately describes the scattering from liquid 4He for currently accessible momentum transfers. Deviations from the IA, generically known as final state effects (FSE), result from the interaction of the recoiling helium atom with its neighbours during the scattering process. 2a These interactions alter the ideal freeparticle behavior of the final state of the recoiling atom required for the validity of the IA. At high Q's, where our measurements are carried out, the observed scattering approximately scales with Y. In this case it is useful to define a momentum dependent Compton profile

J(Y, Q)=QS(G co)

(51

Density and Temperature Dependence in Liquid Helium 4

885

The momentum dependent Compton profile may be thought of as the true Compton profile at infinite Q with corrections due to deviations introduced by FSE. The deviations introduced by FSE have been theoretically studied using several different techniques. They have been commonly expressed as either additive corrections 13"23, 24, 3~36 to the true Compton profile or as a convolution of a final state broadening function 19 22,26,27,29 with the Compton profile. Both techniques provide an adaquate description of FSE for systems with a nonsingular n(p), such as in the normal liquid. However, the appearance of a sharp singularity due to the Bose condensate in the superfluid phase presents difficulties for theories which treat FSE as additive corrections.ll Therefore, we shall treat FSE as a convolution of a final state broadening function R( Y, Q) with the IA result

J(Y, Q)=f R(Y- Y', Q) JIA(Y')dY'

(6)

The final state broadening function has been calculated in several recent theories and the form of the broadening has been experimentally confirmed in the high Q limit. 11-13

3. E X P E R I M E N T A L

DETAILS

Inelastic neutron scattering measurements of liquid helium were carried out using the PHOENIX spectrometer at the Intense Pulsed Neutron Source (IPNS) at Argonne National Laboratory. The experimental setup and analysis procedures have been decribed in detail elsewhere 14 and we present only a brief review of the salient features. PHOENIX is a time-of-flight spectrometer using a Fermi chopper for incindent energy selection and time-of-flight measurement for final energy analysis. A single high-angle detector bank (135 < O < 144) containing 25 detectors is used for observation of the scattered neutrons. The incident energy used for these measurements was approximately 500 meV which corresponds to an average momentum transfer at the recoil peak of 23 ~-1 The helium sample was contained in a cylindrical sample cell made of 6061-T6 aluminium. The cell was 10 cm high with an inner diameter of 4 cm and a wall thickness of 0.16 cm. The cell was attached to either the mixing chamber of a dilution refrigerator or a pumped 3He pot in a specially designed crystat with thin A1 windows and no cyrogens in the neutron beam. The cell temperature was monitored using germanium resistance thermometers attached to the top and bottom of the cell.

886

W . M . Snow and P. E. Sokol

The scattered neutrons were histogramed as a function of time of flight for each detector individually. The data from each detector is transformed to J(Y) using the mean incident energy and time al sample obtained by comparing a Monte Carlo simulation of the incident beam with the observed monitor spectra. Since the statistical accuracy of the results from an individual detector is low, the data from the 25 detectors are added together after being converted to a common Y scale. The data are also placed on an absolute intensity scale by comparison to measurements of low density (0.0073 g/cm 3) helium gas at 5.6 K where the momentum distribution is known. Finally, in order to take into account any systematic errors in the Y scale, the scattered data are shifted by ~-0.04 A -1 rabout half a channel width) so that the first moment sum rule is satisfied. The effects of instrumental resolution, which cannot be expressed as a simple analytic function, must be taken into account in order to determine the true scattering from the liquid. However. when the scattering is localized in a small region of Q - E space, as is the case for helium, an effective resolution function which is a simple one-dimensional convolution can be defined. This effective resolution function, [( Y, Q), is calculated by a Monte Carlo simulation of the spectrometer. In terms o f / ( K Q), the observed resolution-broadened Compton profile Jobs( K Q) is JoBs( Y, Q) = j

J( Y', Q) I( Y - Y', Q) dY'

(7)

--o0

where J(Y, Q) is the unbroadened Compton profile. The instrumental broadening has a full width at half maximum (FWHM) of 0.6 A-1 and is much narrower than the total observed scattering. The final state broadening must also be taken into account in order to extract information on the true Compton profile at infinite Q. In this work we make use of the final state broadening function calculated by Silver ~9-22 at a momentum transfer of 23 A -1 and a density of 0.147 g/cm 3. The FSE broadening function, while only weakly dependent on temperature, does depend on density. This calculation may be extended to other densities using a simple scaling procedure. 37 In Silver's theory, the final state broadening function depends on density primarily through the density dependence of the pair correlation function g(r). The fact that, to a first approximation, g(r) scales with density implies that R(Y, Q) also approximately scales with density. In this case, the FSE broadening function at a density P2 may be obtained from the broadening function at a different density p~ by

Density and Temperature Dependence in Liquid Helium 4

887

where RI(Y) and R2(I1) are the broadening functions at the densities pl and P2 respectively. This procedure has been used previously to take into account the density dependence of F S E in measurements of the normal liquid. ~~ 4. R E S U L T S In this paper we report the scattering from bulk liquid 4He at variety of densities in the superfluid and n o r m a l liquid phases. 38 New measurements of the temperature dependence of the observed scattering in both the n o r m a l liquid and the superfluid phase at densities above SVP are reported, as well as a measurement in the solid phase at low temperatures. These measurements complement previous studies that have examined the temperature dependence at low density 14 and the density dependence at high temperature, l~ W h e n combined with these previous measurements a rather complete coverage of the phase diagram as a function of density and temperature results. The location of the current measurements on the phase diagram, along with previous measurements, are shown in Fig. 1.

10080hcpS'/ 120

E 60 "-"r, 40

bcc~, /

20

-'9

.

.

9 -

.

.

.

- .... *""

o-o- o _ . - - - o - - - - -

_--O

,S,uperf!u,id,, Normal Liquid 0

1

2

3

4

5

T (K) Fig. 1. The phase diagram of liquid 4He as a function of temperature and pressure. The solid circles indicate the locations of the current DINS measurements. The solid diamonds are from Ref. 10 and the solid triangles are from Ref. 14. The dashed lines connect measurements at constant density. The lines correspond to, in order of increasing pressure, densities of 0.147, 0.162, and 0.172g/cm 3. The open squares the location of Green Function Monte Carlo 5~176 calculations of the ground state properties. The open circles indicate the location of Path Integral Monte Carlo sS,57,s9 calculations.

888

W . M . Snow and P. E. Sokol

The measurements presented here, when combined with previous studies, can be classified as either constant density or constant temperature slices through the phase diagram. This allows the density and temperature dependent contributions to the momentum distribution to be separated. We will use this classification to simplify the presentation of the results. In this section we present the raw experimental results, without reference to particular theories or models, In particular, corrections for final state effects and instrumental resolution have not been included. Therefore, the general features evident in the scattering are unaffected by the particular forms or procedures used to take these effects into account. First, consider the scattering data, denoted as Job~( K Q), taken at a constant density of 0.147 g/cm 3 as a uniform of temperature as previously reported by Sosnick e t al. 14 This is the density of liquid helium under saturated vapor pressure near the superfluid transition, which occurs at a temperature of 2.17 K for this density. The scattering, which is shown m Fig. 2, is broad and featureless at all temperatures studied. At high temperatures, near 4 K, Jobs(Y, Q) has a nearly Gaussian shape with a width determined by the zero point motion of the 4He atoms. While no major

2.5

,~" 1.5 Cr >.-

1

0.5

0 -4

-2

0

y(A -1)

2

Fig. 2. Observed J( K Q), uncorrected for instrumental resolution or final state effects, at a density of 0.147 g/cm 3 as a function of temperature. The results have been shifted vertical!y for clarity.

Density and Temperature Dependence in Liquid Helium 4

889

changes occur in the normal liquid phase, the scattering does becomes slightly narrower and more peaked as the temperature is lowered. The scattering becomes visibly more peaked around Y= 0 as the temperature passes from above to below T~. As the temperature is lowered further in the superfluid phase the intensity at small Y continues to increase with decreasing temperature until 1.0 K, after which the scattering is relatively temperature independent down to the lowest temperature measured. No sharp feature indicative of a condensate peak is observed in the measurements below T~. This is similar to the behavior observed in previous measurements at lower Q's. 39ml The changes in Jobs(Y,Q) with temperature can be more clearly observed if the scattering at 4 K is removed. Figure 3 shows AJobs( Y, Q), the difference between Jobs( Y, Q) at a temperature T and a Gaussian fit to Jobs( Y, Q) at 4 K. Removing the nearly Gaussian scattering at 4 K highlights the changes with temperature. Changes in the scattering at temperatures above Tz are relatively small and represent a narrowing of the scattering and increased deviations of the scattering from a pure Gaussian shape with decreasing temperature. However, for temperatures close to Tx a large increase in the scattering at small Y occurs. This is coupled with a significant decrease in the scattering at larger Y due to the sum rules for incoherent scattering. 42 At lower temperatures, there is little variation with temperature. - 0.08

i -

0.04

J(Y,Q)

~

- 0.00

o :,'~ q.o

7.0

~o.~

Fig. 3. AJ( II, Q), the difference between the observed J( 11, Q) and a Gaussian fit to the uncorrected results at 4.25 K, as a function of teperature at a density of 0.147 g/cm 3.

890

W . M . Snow and P. E. Sokol

The changes the shape of the scattering function can be characterized, 43 without reference to any models, using the standard deviation, defined as

(Y=~

~ (Jobs(Yi,Q)- J)~

(9)

and the Kurtosis, defined as

:]4} 3 where J( Yi, Q) is the observed scattering at the discrete points Yi, ] is the average value of J( Y, Q) with respect to Y, and N is the number of data points. The standard deviation provides a measure of the width of the observed scattering. The Kurtosis provides a dimensionless measure of the deviations of the observed scattering from a pure normal distribution (Gaussian form). If the scattering is more peaked than a Gaussian then the Kurtosis is positive and the distribution is designated leptokurtic. Alternately, if the scattering is flatter than a Gaussian distribution then the Kurtosis is negative and the distribution is designated playkurtic. It should be noted that the Kurtosis provides a measure of the deviations of the distribution from Gaussian behavior without regard to the width of the deviations. Changes in a will not affect K: only changes in the form of the distribution will affect K. The values for a and K extracted directly from the experimental data have the effects of instrumental resolution and FSE included. Thus, the absolute values of these quanties cannot be directly compared to theoretical calculations. However, since the instrumental resolution does not depend on temperature and FSE are only weakly temperature dependent, the relative changes in these quantities are meaningful. Thus, these quantities are useful since they provide a characterization of the results without reference to any theory or model. Figure 4 shows a and K as a function of temperature for the scattering at a constant density of 0.147g/cm 3. In the normal liquid o- decreases slowly from its high temperature value indicating that there is some temperature dependence in n(p) even in the normal liquid. A rapid decrease in occurs in the vicinity of the 2 transition with little temperature dependence thereafter. This is in agreement with the qualitative observations above and with previous measurements.44

Density and TemperatureDependencein LiquidHelium4 1.15-

891

0.80

1.10-

/ lx

,ff0.147g/cm 3

Ir I 'ID,.~\

0.60 0.40

~

1.05" 0.20

1.00- ~

0.0

\ ",

0.95-

\\

-0.20

ID

0.90 . . . . . . . . . . . . . . . . . . . . . . . . 0

1

2

3

4

_~.

-0.40 5

r (K) Fig. 4. The standard deviation, ~r, and the Kurtosis, K, as a function of temperature at a constant density of 0.147 g/cm3. The results have been obtained from the measured scattering without correcting for either instrumental resolution or final state effects.

The Kurtosis, which provides a measure of the shape of the scattering, independent of its width, is also shown in Fig. 4. At high temperatures K is platykurtic (negative). However, as we will show later, the true scattering is leptokurtic and the negative value of K is due to the effects of instrumental resolution and FSE which have not been removed at this point. As the temperature is decreased, in the normal liquid phase, K becomes more positive indicating the scattering is becoming more peaked than a normal distribution. Thus, deviations from a Gaussian form are present even in the normal liquid and these deviations become more pronounced with decreasing temperature. Recent measurements at higher resolution have confirmed this result. 13 The non-Gaussian nature of the scattering in the normal liquid deserves comment. In classical systems, n(p) can be expressed as a coherent superposition of statistically independent normal modes (not necessarily harmonic). 36 In this case, n(p) must be Gaussian as a consequence of the central limit theorem. Measurements of several classical and quasi-classical systems, such as neon, 13.45.46 krypton,4S and hydrogen, 47' 48 have confirmed the Gaussian form. Thus, the observation of non-Gaussian behavior implies that such a decomposition has limited validity, even in the normal liquid, due to the extreme quantum nature of liquid helium.

892

W . M . Snow and P. E. Sokol

Upon cooling through the 2 transition the Kurtosis increases rapidly indicating that the shape of the scattering is changing significantly. We note that the Kurtosis is sensitive only to changes in the shape of the scattering and not the overall width. Thus, if the width of the scattering changed, without changing the shape, the Kurtosis would remain constant. As shown in Fig. 4, both the width and shape of the scattering change significantly at the superfluid transition. Below the 2 transition the Kurtosis relatively constant, even though a is still varying with temperature indicating that the shape of the scattering is relatively independent of temperature in the superfluid phase. The behavior of the scattering is in qualitative agreement with our expectations for the momentum distribution when a Bose condensate is present. However, it is important to note that the data do not provide a direct, theory-independent verification of the existence of a condense. The distinct change observed in the scattering data on eitehr side of the superfluid transition is almost certainly associated with a distinct change in the momentum distribution and a large change in n(p) at small p is required

1.6

cY >:

1.2

0.8

0.4

6 -5 -2.5 0

2.5

y(A-I)

Fig. 5. Observed J( K Q), uncorrected for instrumental resolution or final state effects, at a density of 0.162 g/cm3 as a function of temperature. The results have been shiftedverticallyfor clarity.

Density and Temperature Dependence in Liquid Helium 4

893

to produce even a small change in J(Y) at small y.49 Thus, the general increase in the scattering at small Y is consistent with the development of a condensate peak broadened by instrumental resolution and final state effects. However, the scattering is also consistent with forms for n(p) which do not include a condensate peak. 14 Therefore, we wish to emphasize that we can draw no conclusions regarding the existence or magnitude of the condensate from the observed scattering alone without recourse to either theory ol- models for the scattering. The temperature dependence of the scattering at higher densities exhibits behavior similar to the low density measurements. Figure 5 shows the scattering at 0.162 gm/cm 3 and Fig. 6 shows a and K obtained from this data. Figure 7 shows the measured scattering and Fig. 8 shows a and K at 0.172g/cm 3, which is at a density just below the superfluid-solid phase boundary at low temperatures. At high temperatures the observed scattering at both densities has a nearly Gaussian shape. Again, as the temperature is decreased, in the normal liquid, the scattering becomes narrower and changes in the shape of the scattering become more apparent, as in the low density measurements. The variations of a an K explicitly illustrate this behavior. However, the changes in the scattering, both in the width and the shape, are not as prominent as in the lower density measurements.

1.osy-T7 t

o.5o

\~

~. 1.o6t

0.162g/cm3

To / ~

0.40

0.20 ~" 0.10 0.98 ~ 0

0.0 1

2

3

4

5

T (K) Fig. 6. The standard deviation, cr, and the Kurtosis, K, as a function of temperature at a constant density of 0.162 g/cm3. The results have been obtained from the measured scattering without correcting for either instrumental resolution or final state effects.

W.M. Snow and P. E. Sokol

894

1.6 0.172 g/cm 3

;08pi/A\' -

y~~~:~ ~2.01 :/~

0.4

0

s-a~..,,~

3.00

| I I i i i I I r T ~

-5-3-1

.,.v.-a--I

1 3 5

Y ( A -1 )

Fig. 7. Observed J( Y, Q), uncorrected for instrumental resolution or final state effects at a d:ensity of 0.172 g/cm 3 as a function of temperature. The results have been shifted vertically for clarity.

The behavior of Jobs( Y, Q) at the superfluid transition in the higher density studies, which occurs at T = 2 . 0 5 K for 0.162gm/cm 3 and T = 1.95 K for 0.172 gm/cm 3, are also similar to the low density results. The scattering becomes narrower and noticeably more peaked near Y= 0 upon entering the superfluid phase. The standard deviation and Kurtosis both increase in the region of the 2 transition. However, the rapid change in these quantities in the region of the ~ transition is not as apparent as in the lower density data. This is consistent with the expectation that the Condensate fraction decreases with increasing density due to increased importance of inter-particle interactions. We now turn to the constant temperature measurements which allow us to focus on the effects of density alone. Measurements were carried out at 4.25 K in the normal fluid phase and 0.75 K in the superfluid phase over a broad range of densities. A measurement was also carried out at 0.55 K in the solid phase at a density of 0.192 g/cm 3, just above the liquid-solid phase boundary. This measurement allows us to examine the effect of solidification on the momentum distribution,

Density and Temperature Dependence in Liquid Helium 4

895

0.10

1.13" t

O.172 g/cm 3

0.0

1.11'

-0.10

:<

1.09'

-o.2oR

v

0

-0.30~_.

1.07'

-0.40 1.05'

0 ~-~

\\t['//

, l l l l l l l , l l l l l l l l

1.03 0

1

2

-0.50 i i i i l l l

3

4

-0.60

5

T (K) Fig. 8. The standard deviation, ~r, and the Kurtosis, K, as a function of temperature at a constant density of 0.172 g/cm3. The results have been obtained from the measured scattering without correcting for either instrumental resolution or final state effects.

Consider the scattering in the normal liquid at a constant temperature of 4.25 K which is shown in Fig. 9. Figure 10 shows o- and K for these measurements. The general trend evident in the scattering data is the increase in the width of JobJ I7, Q) with increasing density. This increase is consistent with the decrease in average volume per atom as the density increases. Therefore, at higher densities, the resulting zero point motion has a higher average momentum distribution should increase. Furthermore, the scattering approaches a Gaussian form at high density. As the density is decreased deviations from the Gaussian form, as shown by the variation in K, become evident. The density dependence of K clearly illustrates the importance of quantum effects, even in the normal liquid. At high densities, the atoms are well localized and the behavior will approach that of a classical system (i.e. a Gaussian n(p)). As the density is decreased the atoms are less well localized leading to overlap of the wavefunctions and the appearance of deviations from the classical result. Measurements in the superfluid phase at 0.75 K are shown in Fig. 12. The width of the scattering, cr, increases with increasing density, similar to the results at higher temperatures. However, unlike the high temperature results, and additional scattering near Y = 0 is present in the superfluid

896

W, M. Snow and P. E. Sokol

2.5

~,1.5

"~

1

0.5

-4-3-2-1 0 1 2. 3 4 y(A 1)

Fig. 9. Observed J( Y, Q), uncorrected for instrumental resolution or final state effects, at a teperature of 4.25 K as a function of density. The results have been shifted vertically for clarity.

0.20

1.50-

T:4.25 K

1~ 1.40-

\

"'~ \I

I.~O-

/

0.0

-0.20=

~"--

-o.44

t3 1.20"

~'

-0.60

1.10-

-0.80

...................

1.00 0.12

0.14

0.16

0.18

T (K)

0.2

-I .0 0.22

Fig. 10. The standard deviation, a, and the Kurtosis, K, as a function of density at a constant temperature of 4.25 K. The results have been obtained from the measured scattering without correcting for either instrumental resolution or final state effects.

Density and TemperatureDependencein Liquid Helium 4

897

phase. The presence of this additional scattering is evident in the larger value of K at low densities. As the density is increased the additional scattering near Y= 0 decreases, consistent with the expectation that the condensate fraction decreases with density, and K decreases. We have also measured the scattering in the sotid phase at low temperatures near the superfiuid-solid phase boundary which is shown in Fig. i 1. The width of the scattering is broader than in the superftuid, consistent with the higher density of the solid phase. Somewhat surprisingly, the width of the scattering in the tow density solid, as shown in Fig. 12, is a smooth extrapolation, as a function of density, of the behavior in the superfluid phase. The shape of the scattering in the solid phase is very nearly Gaussian, as we will demostrate later. Examination of K indicates that the shape of the scattering is also a smooth extrapolation of the behavior in the superflui& Thus, when the difference in density between the

2

T=0.75 K 1.6

>-

N"

Fig. 11. Observed J( Y,, Q), uncorrected for ins~umental resolution o r final state effects, at a temperature of 0.75 K as a function o f density. The results have been shifted vertically for clarity.

[

t

g=%

~

--1

~o.16Zl

-4-3-2-10 1 2 3 4 y (i-1)

898

W . M . Snow and P. E. Sokol

1.20

It

/

"o

1.15-

0.50

~0.75

K

- o -o,

0.40 F

~

,--, 1.10-

I x\\/~

~'~

/ \\\\\\

1.05

-~

0.30 7~

\

1.00-

0.95-

0.20 0.100

/

~x\o

0.0

0.90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.10 0.14 0.16 0.18 0.2

T (K) Fig. 12. The standard deviation, or, and the Kurtosis, K, as a function of density at a constant temperature of 0.75 K. The results have been obtained from the measured scattering without correcting for either instrumental resolution or final state effects.

superfluid (just below the liquid-solid boundary) and the solid (just above the liquid-solid boundary) is taken into account, the scattering appears to change smoothly between these two phases.

5. ANALYSIS The observed scattering, presented in the previous section, provides valuable insight into the behavior of the momentum distribution as a function of density and temperature. However, these results are broadened by the effects of instrumental resolution and final state effects. To extract the maximum information about the scattering in the IA limit from the measurements these effects must be taken into account. In this section we present various forms for analyzing the data. These include comparison to theories for the momentum distribution, deconvolution of instrumental and final state broadening via model functions, moments analysis (kinetic energy), and extraction of the condensate fraction via a model momentum distribution. This analysis depends on the particular form of the instrumental resolution and final state broadening that have been used. Different choices for these broadening function, particularly the final state broadening derived from theory, will affect the results. Current theories 19-24'26'27 for final state effects differ in their

Density and TemperatureDependence in Liquid Helium 4

899

theoretical approaches. However, we note that most produce only small differences in the scattering which cannot be resolved given the current statistical accuracy of the measurements. Therefore, at the present statistical accuracy our results should be viewed as independent of any particular theory for final state effects. 5.1. Comparison to Theory

The most direct test of theoretical predictions for the momentum distribution is a comparison to the experimental data. However, the theoretical predictions for n(p) and the experimental results for Jobs( Y, Q) can not be directly compared. To avoid the complications associated with deconvolution and the associated loss of information on the experimental uncertainties, we convert the theoretical predictions to J( Y, Q) so that they may be directly compared to the experimental results. The IA is used to convert the theoretical n(p) to JIA(Y), and JIA(Y) is then converted to J( Y, Q) using theoretical predictions for the final state effects. 19 22 Finally, the results are broadened by the instrumental resolution so that they may be directly compared to the experimental results. Theoretical calculations of the momentum distribution of liquid 4He exist at several points in the phase diagram, 5~ 6o as indicated in Fig. 1. In this section, we will compare these theoretical calculations to our scattering data. In most of the cases, the temperatures and densities of the experimental measurements and the theoretical calculations do not coincide precisely. In these cases, we have taken the liberty of performing comparisons between the theoretical calculations and the measurement which is closest to it in the phase diagram. In all of these cases, the temperature and density dependance of the scattering data in the neighborhood of the theoretical calculation, as determined by the changes in the observed scattering in the region, is small enough for a valid comparison to be made. The constant density slice at 0.147 g/cm 3 contains several points which may be compared to theoretical calculations as reported previously.9' 14 Figure 13 shows comparisons with Path Integral Monte Carlo (PIMC) calculations.55.59 The data were taken at temperatures of 3.50 K, 2.80 K, 2.30 K, 1.80 K, 1.50 K, and 1.00 K, and the theoretical calculations were performed at temperatures of 3.33 K, 2.50 K, 2.22 K, 1.82 K, 1.54 K, and 1.18 K. In the superfluid phase, the PIMC calculations predict a condensate fraction of 0.063, 0.087, and 0.069 respectively as a function of decreasing temperature. The densities of these theoretical calculations all lie between 0.1449 and 0.1462 g/cm 3 except for the 3.33 K result, which is at a density of 0.138 g/cm 3. The agreement between theory and experiment is quite good. There is a small systematic difference between the theoretical

900

W . M . Snow and P. E. Sokol

CY

>=

0.:

b -4-3-2-10 1 2 3 4

y(A-I)

Fig. 13. Comparison of the observed Compton profile with theoretical prediction obtained from PIMC calculations55.57,59at a density of 0.t47 g / c m 3 as a function of temperature. The IA and the instrumental and final state broadening functions have been used to compare the theoretical prediction to the experimental results. The details and conditions of the measurements and calculations are described in the text. The results have been shifted vertically for clarity.

and experimental results in the form of a small excess scattering near Y = +2 ,~-1 which we will discuss in detail later. Except for this difference, the theoretical calculations give a good account of the temperature dependence of the scattering data. Comparison of the predicted and experimental temperature dependence have also been carried out at a density of 0.172 g/cm 3 as shown in Fig. 14. The experimental data were taken at temperatures of 0.75, 1.50, 2.01 and 4.25 K. P I M C calculations 5m59 exist at temperatures of 1.18, 2.00, and 4.25 K at a density of 0.1728 g/cm 3. The theoretical condensate fraction for these temperatures are 0.06 and 0.012, respectively. G F M C ealculationsSO. 6o at T = 0 and a density of 0.17 g/cm 3 are also available and yield a condensate fraction of 0.032. The G F M C results may be compared to the T = 0 . 7 5 K measurements, as discussed in detail later. The theoretical calculations are in good agreement with the observed scattering at both temperatures with the exception of the small excess scattering at Y= +2/~-1. We now discuss the measurements carried out at constant temperature. P I M C calculations ss" s9 are available at 4.00 K for densities of

Density and Temperature Dependence in Liquid Helium 4

901

1.4 0.I 72 glcm 3

1

C/ >= .-)

Fig. 14. Comparison of the observed Compton profile with theoretical prediction obtained from PIMC55's7.s9 and GFMC 5~60calculations at a density of 0.172 g/cm3 as a function of temperature. The IA and the instrumental and final state broadening functions have been used to compare the theoretical prediction to the experimental results. The details and conditions of the measurements and calculations are described in the text. The results have been shifted vertically for clarity.

-4-3-2-1 0 1 2 3 4 y (~-1)

0.138, 0.173, and 0.191g/cm 3. These are close to the experimental measurements at 4.25 K for densities of 0.138, 0.173, and 0.186g/cm 3. Figure 15 shows a comparison of the P I M C predictions and the observed scattering. The theoretical calculations give a good account of the density dependence of the width of the scattering data, with the exception of a small additional scattering near Y = 2 g.-1, for these densities. The m o m e n t u m distribution in the ground state, at T = 0, has been studied using a variety of techniques. The most extensive calculations are G F M C studies. 5~ 6o These have been carried out at densities of 0.147, 0.159 and 0.174 g/cm 3. These calculations predict a condensate fraction of 0.092, 0.052 and 0.038 respectively" at these densities. We have compared the results to the scattering data measured at densities of 0.147, 0.162 and 0.172 g/cm 3 at a temperature of 0.75 K as shown in Fig. 16. As before, the agreement between theory and experiment is quite good with the exception of the additional scattering near Y = 2 A -1. The G F M C calculations provide an excellent description of the scattering in the superfluid phase over a range of densities. We note that the theoretical results are in the ground state and the experimental results are at low, but finite, temperature. However, finite

902

W . M . Snow and P. E. Sokol

-4-3-2-10

1 2 3 4

y(A-I)

Fig. 15. Comparison of the observed Compton profile with theoretical prediction obtained from PIMC 55's7, 59 calculations at a temperature of 4.25 K as a function of density. The IA and the instrumental and final state broadening functions have been used to compare the theoretical prediction to the experimental results. The details and conditions of the measurementsand calculations are described in the text. The results have been shifted vertically for clarity.

temperature extensions of variational calculations (using the H N C / S technique, 53'54 for example) indicate that the m o m e n t u m distribution of superfluid helium is changed only slightly for temperatures below 1 K. In addition, thermodynamic properties of superfluid helium such as the entropy, specific heat, isothermal compressibility, first sound velocity, and free energy are known to be temperature independant to an excellent approximation below 1 K. We therefore feel that it is likely that the underlying m o m e n t u m distribution in the scattering data at 0.75 K is very close to that at T = 0 K and that this comparison is valid. We now address the systematic difference between the theoretical predictions and the experimentally observed scattering that appears on the positive Y half of the data for all the temperatures and densities studied. The scattering data is systematically lower than the theoretical predictions at low positive Y and is systematically higher than theory at high K despite the good agreement of the negative half of the data with theory. Because the scattering data are slightly asymmetric in I1, the deviations from the theoretical predictions cannot be described as simply a difference in the width. Thus, the differences observed cannot be solely due to the fact

Density and Temperature Dependence in Liquid Helium 4

903

c~ >-.-1

Fig. 16. Comparison of the observed Compton profile with theoretical prediction obtained from GFMC 5~176 calculations at a temperature of 0.75 K as a function of density. The IA and the instrumental and final state broadening functions have been used to compare the theoretical prediction to the experimental results. The details and conditions of the measurements and calculations are described in the text. The results have been shifted vertically for clarity.

-4-3-2-1 0 1 2 3 4

y(A1)

that the theoretical calculations and the experimental data are at slightly different densities and temperatures. The asymmetry on the positive Y side of the scattering function appears to be a universal feature since it is present at all densities and temperatures studied. It does not depend on either the phase (superfluid or normal liquid), temperature, or density of the sample. These differences are also present in lower Q measurements using the same instrument 61 and in other independent studies. 16 Thus, it appears that the differences between the theoretical predications and the observed scattering are not simply an instrumental effect. Alternately, it might be possible to attribute the asymmetric feature to the effects of multiple scattering which could produce an asymmetry in the observed scattering in the same direction. However, the relative size of this asymmetric feature appears to be the same for several different measurements using a variety of different sampmle geometries. Multiple scattering effects, on the other hand, would be expected to be strongly dependent on sample geometry. In addition, direct calculations of the intensity of multiple scattering for these studies indicate that the asymmetric feature is too large to ascribe to multiple scattering.

904

W . M . Snow and P. E. Sokol

The asymmetry in the observed scattering, therefore, appears to be a real effect. However, this asymmetry cannot be a property of the underlying momentum distribution because the sample container constrains the total momentum of the liquid to vanish. Therefore, we believe that the most likely explanation is that the asymmetry is due to an aspect of FSE which is not included in the Silver theory 19-22 that we have employed to account for final state effects in these comparisons. The recent theory of Carraro and Koonin, 26'27 which includes some physical effects of order 1/Q not present in Silver's theory, is in better agreement with the superfluid at a density of 0.147 g/cm 3 at 0.75 K in the region Y = 2 A -~. (Unfortunately, calculations are not available at other densities or temperatures for the Carraro and Koonin formalism.) It appears that final state effects may account for most of the present differences between theory and experiment. 5.2. Model Momentum Distribution

Comparison of the experimental results to theoretical predictions, as carried out in the previous section, provides the most direct and stringent test of theories. However, such comparisons test only a particular theory and are possible only if theoretical and experimental results are available at the same temperature and density. Therefore, it is desirable to obtain a theory-independant representation of the results that can be used to extract model independent values for quantities such as the kinetic energy, the condensate fraction, and the shape of the momentum distribution. This requires that the effects of instrumental resolution and final state broadening either be taken into account or removed. Several approaches are possible for eliminating the effects of instumen, tal and final state broadening. The most direct method is simple deconvolution of the broadening functions from the experimental data. In the absence of statistical noise, such a procedure would indeed provide details on the true scattering function in the IA limit. However, when statistical noise is present such a procedure is ill-posed and can lead to unphysical results, such as negative scattering or spurious sharp features. 62 An alternate approach, 14 which we follow, is to define a model scattering function, Jmoael(Y)" This model is a parameterized function which, when broadened by instrumental resolution and final state effects, provides a good description of the observed scattering for the appropriate choice of parameters. The model scattering function can be viewed as the scattering in the IA limit without the complications of instrumental and final state broadening. As such, it can be used to extract information of quantities such as the kinetic energy or condensate fraction without reference to a particular theory.

Density and TemperatureDependence in Liquid Helium 4

905

The model scattering function that we have found most convenient for describing the observed scattering is a sum of Gaussians ai

JModel(Y) =

(12 9/ 6 2~)

e_{( Y rc)2/2 2}

i=1

( 11 )

whose amplitudes ai, widths o-i, and common center Yc may be varied. In addition, we constrain the amplitudes

~ai= 1

(12)

i

to ensure that JMod~l(I0 is properly normalized. The model we have used is a phenomenological choice that is simple and convenient. Other choices are also possible. Nevertheless this form, with the restrictions that the amplitudes are always positive and the centers are locked together, does provide a physically realistic model scattering function. The model function is symmetric, positive definite, and free of spurious oscillations and other obviously unphysical features which would certainly appear if a direct deconvolution were attempted. In practice the center of the scattering function Yc, which relaxes to a value close to the IA result of zero in the course of the fit, is allowed to vary to take into account uncertainties in the center of the FSE broadening function and any errors in the definition of the Y scale, through our determination of the incident energy. The model scattering function obtained by convoluting JModel(Y)with the instrumental and final state broadening provides the best fit to the observed scattering. However, thes fits are not unique due to the finite statistical accuracy of the data. In practice, a range of parameters may yield model scattering functions which describe the observed scattering nearly as well as the best-fit values. In addition, the two parameters for the two Gaussians used to describe the data can be highly correlated. Thus, while the model scattering will describe the overall behavior of J~A(I1) accurately, care must be exercised in attributing significance to small differences in the detailed features. The model scattering function can be directly related to the underlying momentum distribution. The scattering, in the IA limit, and the momentum distribution are related by

n(p)=

1 dJ(Y) 2~Y d Y

(13)

906

W . M . Snow and P. E. Sokol

0.7

C~ ~0

0.2

-4-3-2-1 0 1 2 3 4

y(A

Fig. 17. Typical fit of the mode! scattering function discussed in the text to the observed scattering at T = 0.75 K, in the superflnid phase, and T = 35 K, in the normal liquid, at a density of 0.147 g/cm 3.

Thus, our model scattering function is equivalent to using a model momentum distribution of the form " ai {p2/2o-~} nmodel(p) = ~ (2~Za2)3/2e -

(14)

which can be viewed as the experimentally determined momentum distribution. However, we wish to note that, as discussed above, there are a whole family of model functions which adequately describe the experimental results. The 1/Y factor in the conversion between J(Y) and n(p) will further amplify these small differences. 14,49 Therefore, we will present our results in terms of JModel(~0, rather than nmodd(p), to maintain closer contact with the actual experimental results. Typical fits of JModel(Y) to the observed scattering in both the normal and superfluid phases are shown in Fig. 17. Similar quality fits were obtained at all temperatures and densities studied. As can be seen, JModel(Y) provides an excellent description of the observed scattering. As with the comparisons to theory, a systematic difference at Y = +2 ~ - 1 is present. This difference appears, as it did in the comparisons to theoretica!

Density and Temperature Dependence in Liquid Helium 4

907

calculations, since the model is constrained to be symmetric and normalized. Thuns, it will not be able to fit the assymmetric feature which we believe is due to the approximate treatment of final state effects. Aside from this feature the models provide an excellent description of the scattering. The parameters obtained for the model scattering function at the temperatures and densities studied are given in Tables I through V. Two Gaussians are always required to adequately fit the experimental data, even in the normal liquid at high temperature and density. This implies that the underlying momentum distribution has a shape that is non-Gaussian. The addition of further Gaussians did not significantly improve the quality of the fit. In general, a broad Gaussian, with a width on the order of 1 ~ - i, and a narrow Gaussian, with a width on the order of 0.3 to 0.5 A -1, are required to fit the data. The weight of the broad component is largest at high temperatures and decreases, with a corresponding increase in the narrow component, as the temperature is lowered. It is tempting to identify the narrow component with the condensate and the broad component with the uncondensed atoms. However, we wish to emphasize that such an identification is not valid. For example, a narrow component is present even in the normal liquid phase at high temperature where we believe there is no condensate. Therefore, the parameters in the model n(p) must be viewed as simply fitting parameters and no physical significance can be attached to them. The temperature dependence of the model momentum distribution at a density of 0.147 g/cm 3 is shown in Fig. 18. At temperatures above the superfluid transition, which occurs at 2.17 K, the model n(p)'s are broad, featureless, and approximately Gaussian. The behavior is most nearly Gaussian at high temperatures (i.e. 4 K). As the temperature is decreased towards the 2 transition deviations from a pure Gaussian form become apparent. These deviations from Gaussian behavior reflect the importance of quantum effects even in the normal liquid. Upon cooling through the 2 transition Jmodel(Y) exhibits a distinct change in the shape of the scattering. The scattering becomes visibly more peaked around Y= 0 due to the rapid increase in the weigh of the narrow component in the model. As the temperature is lowered further in the superfluid phase the intensity at small Y continues to increase with decreasing temperature down to approximately 1 K, after which the scattering is relatively temperature independent down to the lowest temperature measured. The changes in the shape of the scattering can be illustrated by comparing the model scattering function to a pure Gaussian scattering. Figure 19 shows a comparison of the model scatttering function to single Gaussian with a with chosen to provide the best fit, in terms of the mean

908

W . M . Snow and P. E. Sokol TABLE I

Parameters for Jmod~L(Y) as a Function of Temperature at 0.147 g/cm ~ as Described in the Text. The Standard Deviation of the Single Gaussian Fits to J,,oa~l(Y) Are also Listed

Temp. (K)

at

( h -~)

a2

( h -~)

f.)c (h -I)

G~g ( h -~)

0.35 1.0 1.5 1.8 2.0 2.3 2.8 3.5 4.2

0.785 0.757 0.763 0.765 0.779 0.843 0.592 0.875 0.925

0.95 0.99 1.00 0.98 1.00 1.01 1.11 1.00 1.01

0.215 0.242 0.237 0235 0.22l 0.158 0.408 0.125 0.075

0.29 0.34 0.34 0.39 0.40 0.42 0.67 0.45 0.33

- 0.03 - 0,02 - 0.01 - 0~03 -0,.04 0.00 + 0.0 t - 0.04 - 0.03

0.737 0.767 01768 0.781 0,810 01876 0~890 0.903 0.1935

0-1

0"2

TABLE II Parameters for Jmoaet(Y) as a Function of Temperature at 0.162 g/cm 3 as Described in ti~e Text. The Standard Deviation of the Single Gaussian Fits to Jmodel(Y) Are also Listed

Temp. (K)

aa

~t ( h -~)

a2

0-2 (A - I )

Pc (A - l )

0-sg ( h -1)

0.75 1.50 1.89 2.01 3.0 4.25

0.881 0.898 0.9t8 0.954 0.838 0.774

0.98 1.02 1.04 1.03 1.09 1.11

0.119 0.102 0.082 0.046 0.162 0.226

0.30 0.30 0.30 0.30 0.71 0.69

0.01 0.01 0.03 0.03 0.00 0.00

0.861 0.912 0.952 0.98 I 1.0t3 0.99I

TABLE Ill Parameters for Jmodel(Y) as a Function of Temperature at 0.172 g/cm 3 as Described in the Text. The Standard Deviation of the Single Ganssian Fits to Jmodel(1I) Are also Listed

Temp. (K) 0.75 1.50 1.80 1.90 2.2 3.0 4.25

al 0.894 0.875 0.873 0.873 0.955 0.979 0.772

c~i, ( h -1) 1.03 1.01 1.00 1.03 0.98 0.98 1.20

a2 0.106 0.125 0.127 0,127 0.045 0.019 0.228

0-2 (A-b 0.31 0.30 0.35 0.35 0.30 0.31 0.66

Pc ( h ~i) 0.01 - 0.03 - 0.05 -0.03 0.00 -0.01 0.00

~s~ ( h -~) 0,878 0.879 0.903 0,935 0.962 1.035

Density and Temperature Dependence in Liquid Helium 4

909

TABLE IV

Parameters for J,~oa~l(Y) as a Function of Density at a Temperature of 4.25 K as Described in the Text. The Standard Deviation of the Single Gaussian Fits to J~od~(Y) Are also Listed

/9

(71

(72

Pc

(Tsg

(g/cm~)

(/1

(A-l)

a2

(A-l)

(A-l)

(A-l)

0.125 0.130 0.138 0.140 0.147 0.160 0.173 0.181 0.186 0.195 0.200

0.924 0.700 0.580 0.527 0.461 0.774 0.772 0.397 0.648 0.596 0.396

0.84 1.02 1.08 1.14 1.29 1.11 1.20 1.54 1.36 1.52 1.67

0.076 0.300 0.420 0.473 0.539 0.226 0.228 0.603 0.352 0.404 0.604

1.63 0.60 0.70 0.74 0.76 0.69 0,66 0.94 0.86 0.95 1.07

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.872 0.860 0.890 0.917 0.949 0.989 1.035 1.118 1.140 1.233 1.251

square error, to Jmoa~l(Y). Figure 20 shows the temperature dependence of O-sG, which will be discussed more fully later. As illustrated in Fig. 19, the scattering is most nearly Gaussian at high temperatures. Deviations from the pure Gaussian form are, however, apparent even in the normal liquid. The deviations from a pure Gaussian form increase as the temperature is decreased, as was observed in the raw data. The scattering becomes clearly non-Gaussian in the vicinity of T~ and the deviations are largest at low temperature. The changes in the shape of the scattering can also be illustrated through the temperature dependence of the standard deviation and the Kurtosis which can be obtained directly from J~odel( Y)' The behavior of TABLE V

Parameters for Jmode~(Y) as a Function of Density at a Temperature of 0.75 K as Described in the Text. The Standard Deviation of the Single Gaussian Fits to Jmodel(Y) Are also Listed

(72

P~

(Tsg

(g/era 3)

al

(A 1)

(/2

(A -1)

(A 1)

(A-l)

0.149 0.156 0.162 0.167 0.172 0.192

0.851 0.872 0.881 0.882 0.894 0.573

0.93 0.96 0.98 1.01 1.03 1.07

0.149 0.128 0.119 0.118 0.106 0.427

0.25 0.25 0.25 0.30 0.31 1.13

0.02 0.04 0.01 0.02 0.01 ~0.01

0.778 0.824 0.851 0.886 0.917 1.090

,0

G1

910

W . M . Snow and P. E. Sokol

2.5

1.5

>-r O

E

0.5

13 0

1

2

3

Fig. 18. Jmodel(Y) as a function of temperature at 0.147 g/cm 3. The results have been shifted vertically for clarity.

y(A-~) 1.5k

0.147 g/cm 3

1 >-o O

E

0.5

.... ,'~,-,~ ?,2,5,

0 O

1

2

3

y(A -I)

4

Fig. 19. Comparison of Jmodel(Y) (solid lines) and a single Gaussian fit to Jmoael(}r) as described in the text at a density of 0.147 g/cm 3 at temperatures of 0.35, 2.0, 2.3, and 4.2 K. The results have been shifted vertically for clarity,

Density and Temperature Dependence in Liquid Helium 4

911

1.15 1.10

%I1) (3 gl

oO

1.05

1.00 0.95 0.90 0.85 0.0

0.5

1.0

1.5

2.0

2.5

T/T X Fig. 20. The standard deviation of the single Gaussian fits to Jmodel(I7) as a function of temperature. The temperature has been scaled by T;~ and the magnitude of a has been scaled by the value at Tz. The results are at densities of 0.147 g/cm 3 (O), 0.162 g/cm 3 ( 9 and 0.172 g/cm 3 (fi).

0.8

1.05"

0.I 47 glcm 3

/1_o l.O0' ar

0.7

\

~

0.6

/

'7

~~ - e__ d

0.95'

0.5 c2~

/~ t9

0.90

.< 0 /

~"

e-I-

0,4 cnO

~ /

--'

9

e

\

0,3 o~

0.85" d

9 -.

"o

0.2

080. . . . . . . . . . . . . . . . . . . . . . . . . 0

1

2

3

01 4

5

T (K) Fig. 21. The standard deviation, ~r, and the Kurtosis, K, as a function of temperature at a density of 0.147 g/cm 3. The results have been obtained from the model scattering functions discussed in the text.

912

W . M . Snow and P. E. Sokol

and K, as shown in Fig. 21, is similar to that observed for the raw data. However, the absolute values are different and the temperature dependence is more pronounced than for the corresponding quantities obtained from the observed scattering. These changes result from the correction for instrumental resolution, which affects both cr and K, and FSE, which only affect K, in Jmodel(Y). As observed in the uncorrected data, a (K) slowly decreases (increases) in the normal liquid from the high temperature value at 4 K. The most prominent change from the results obtained from the observed scattering is that K is now positive at all temperatures. The negative values of K obtained directly from the observed scattering are simply an artifact of instrumental and final state broadening which had not been removed. The positive values of K obtained from Jmodel(}1) indicate that the scattering, and thus n(p), is more peaked than a simple Gaussian even at the highest temperature studied (4 K/. Thus, quantum statistical effects are important even in the normal liquid relatively far away from the superfluid transition. The increase in K and decrease in a with decreasing temperature indicates that the momentum distribution is becoming both narrower and more non-Gaussian as the temperature is decreased in the normal liquid. This indicates the increasing importance of the Bose statistics even before a condensate develops. The standard deviation and Kurtosis both change rapidly near T~, with little change in the superfluid phase thereafter. This behavior is consistent with the appearance of a condensate marked by a drastic increase in the scattering at small Y. The decrease in intensity at large K which contributes strongly to the standard deviation, leads to the rapid decrease in a. Likewise, the increase in intensity at small Y makes the scattering more peaked and leads to the rapid increase in K. The small changes at lower temperature indicate that once the condensate has developed neither the width or the shape of the scattering change dramatically. We note that, while the weight at small Y shows a dramatic increase. no 6-function singularity appears in Jmodel(Y). The lack of such a singular behavior, which we would associate with a condensate, is due to the finite statistical accuracy of the data which allows a family of curves to adequately describe the observed scattering. The Jmodel(Y) we have chosen does not explicitly contain a singular component and therefore one does not appear. If such a component were explicitlyincluded in the fit then, depending on the choice of parameters, it would appear in Jmodel(Y]' Jmoael(Y) at densities of 0.162 g/cm 3 and 0.172 g / c m 3 a r e shown as a function of temperature in Figs. 22 and 23. The superfluid transition occurs at temperatures of 2.05 K and 1.95 K. respectively, for these densities. As with the lower density measurements, Jmoaet(I1) is nearly Gaussian at high

Density and Temperature Dependence in Liquid Helium 4

913

temperatures and shows a narrowing and a deviation from Gaussian behavior with decreasing temperature in the normal liquid phase. A rapid change in both the width and shape of the scattering are observed in the vicinity of Ta, consistent with the development of a condensate. However, the changes are much smoother and not as prominent as in the lower density case. This is consistent with a decrease of the condensate fraction with increasing density. Again, below T~ the scattering exhibits very little variation with temperature. The behavior of ~ and K is quite similar to that in the lower density measurement, but with a smoother and smaller variation as a function of temperature. The single Gaussian fits to ,]model(Y), which were introduced to highlight the non-Gaussian behavior of the scattering, provide a useful characterization of the changes in the scattering. They are characterized by a single parameter Crsg, but, unlike the standard deviation, they are not unduly influenced by the behavior at either large or small I1. The temperature dependence of O-sgshows an interesting behavior. Figure 20 shows o-sg, scaled so that c~g = 1 at T;~, as a function of T/T~ for the three densities studied. Above T~, the scaled width appears to follow a universal

1.6

1.2 CY >-

0.8

0.4

Fig. 22. Jmodel(Y) as a function of temperature at 0.162 g/cm 3. The dashed lines are the single Gaussian fits to Jmodel(}z) as discussed in the text. The results have been shifted vertically for clarity.

0 0

1

2

y(A1)

3

4

W . M . Snow and P. E. Sokol

914

behavior, independent of density, even though the values of asg and Ta are quite dependent on density. The scaled width also exhibits a discontinuity in the slope at Tx. The magnitude of the discontinuity in the slope is similar for the different densities studied. Finally, below T;~ the scaled widths for the different densities show differing behavior. The scaled width appears to saturate, as a function of temperature, with the higher densities saturating first. This correlates well with the behavior of the condensate fraction where no decreases with density. The behavior of Jmoael(Y) may also be examined as a function of density at constant temperature. In this case, the changes in the scattering, and hence the momentum distribution, are entirely due to changes in the interparticle separation. Figure 24 shows Jmoa~(I1) as a function of density at a temperature of 4.25 K. The width of the scattering increases with increasing density reflecting the increased localization of the atoms. The scattering is also nearly Gaussian as can be seen by comparison to the single Gaussian fits which are also shown in the figure. Figure 25 shows Jmodel(Y) and Fig. 26 shows o- and K as a function of density at a temperature of 0.75 K. As in the high temperature data, the

1.5

0.5

0 0

1

2

y (,~-1)

3

4

Fig. 23. Jmodel(Y) as a function of temperature at 0.172 g/cm 3. The dashed lines are the single Gaussian fits to Ymodel(I1) as discussed in the text. The results have been shifted vertically for clarity.

Density and Temperature Dependence in Liquid Hefium 4

915

with of the scattering, as reflected in a, increases with increasing density reflecting the increased localization of the atoms. The values of o- are lower than for the high temperature studies reflecting the presence of the condensate at low temperatures. The shape of the scattering is definitely nonGaussian, as shown by the finite values of K. The deviations from Gaussian behavior are largest at low density, where the condensate is largest, and decrease smoothly as the density increases and the condensate fraction decreases. The results for the single measurement in the solid phase at low temperature are also shown in Figs. 25 and 26. As was seen in the observed scattering, o- is larger for the solid measurements and appears to be a smooth extrapolation of the lower density liquid measurements. The shape of the scattering in the solid phase is very nearly Gaussian, as shown by the very small value of K, which again appears to be a smooth extrapolation of the liquid results. Thus, when the difference in density between the superfluid (just below the liquid-solid boundary) and the solid (just above the liquid-solid boundary) is taken into account, the scattering appears to change smoothly between these two phases.

1.5

1

>"D O

,-) E

0.5

Fig. 24. Jmod~l(Y) as a function of density at a temperature of 4.25 K. The dashed lines are the single Gaussian fits to Jmodel(Y) as discussed in the text. The results have been shifted vertically for clarity.

0 0

1

2

y(A

3

4

916

W . M . Snow and P. E. Sokol

1.2 >-

0.8

0.4

0 0

1

2

3

Fig. 25. J m o d e l ( } / ) as a function of density at a [emperature of 0.75 K. The dashed !ines are the single Gauss\an fits to Jmod~l(Y) as discussed in the texL The results have been shifted vertically for clarity.

4

y(A 1.2

0.7

e

T=0.75 K

\ \ \

1.1

\

7~

/

0\

1.0

0.5

/

\

v

P

/

\

0.6

0,4

/

o=lk

/

O 0.3 u~ 0.2

0.9=

\

/

\

0.1 \

tlili1111111iJiailaI|aiilIll

0.8 0.14

0.16

0.18

'0.0 0.2

13(g/cm 3) Fig. 26. The standard deviation, or, and the Kurtosis, K, as a function of density at a constant temperature of 0.75 K. The results have been obtained from the model scattering functions discussed in the text.

Density and Temperature Dependence in Liquid Helium 4

917

5.3. Kinetic Energy The average kinetic energy per atom, (Ek}, is proportional to the second moment of n(p) and, as such, has traditionally been used to characterize changes in the momentum distribution. (E~} may also be compared to theoretical calculations. However,we note that the lineshape comparisons presented earlier provide a much more detailed test. The average kinetic energy can be obtained from the model scattering functions presented previously. Alternately, (Ek} may be obtained directly from the second moment of the observed scattering when instrumental resolution is taken into account. FSE, which do not change the second moment of the scattering, do not need to be taken into account. The former method has the advantage that both instrumental resolution and final state effects are implicitly taken into account and that the tails of the Theoretical predictions for (Ek) are also shown in Fig. 27. A detailed comparison between the experimentall~ 14,40.44,63 67 and theoretical 5~176 results at 0.147 g/cm 3 has been reported previously 14 and will not be repeated here. We note, however, that in general there is good scattering, where the statistics are poor, do not unduly influence the values of (Ek} since they are assumed to be Gaussian. However, the values of (Ek} obtained are biased towards the model scattering function used and no direct estimate of the experimental uncertainty can be obtained. The latter method has the advantage that a direct estimate of the uncertainty in (Ek} can be obtained. However, as pointed out previously, the tails of the scattering will contribute greatly to the experimental uncertainties and a cutoff must be imposed in the calculation of the second moment. Therefore, while both methods give comparable results, we have used the second moment obtained directly from the observed scattering for the values of (Ek} reported here since realistic estimates of the errors are obtained. The model functions yield essentially the same values but with errors, obtained from the variance of the fitting parameters, which are significantly smaller. Figure 27 shows (Ek} as a function of temperature at densities of 0.147, 0.162, and 0.172 g/cm 3. (Ek} for the lowest density measurement, at 0.147g/cm 3, is relatively temperature independent in the normal liquid, decreases rapidly in the region of T~, and is again relatively temperature independent in the superfluid. This behavior is consistent with the development of a finite condensate fraction in the superfluid phase. The results at densities of 0.162 g/cm 3 and 0.172 g/cm 3 exhibit a general decrease in (Ek} with temperature. However, unlike the lower density result, a distinct decrease in (Ek} is not observed at these higher densities at Tx. This behavior is consistent with the presence of a smaller condensate in the superfluid at higher densities.

918

W.M. Snow and P. E. Sokol

agreement at low densities. Reasonable agreement with theory 5~176 is also obtained for the higher density measurements. However, the theoretical values for (E~) are systematically higher than the experimental values, unlike the lower density results. The theoretical and experimental values are in agreement with the experimental uncertainties, but just within the limits of the experimental errors. The origin of these systematic differences is not clear at present. Figure28 shows ( E k ) as a function of density at constant temperatures of 4.25 K and 0.75 K. The value obtained in the solid phase at 0.50 K just above the liquid-solid phase boundary is also shown. At both temperatures ( E k ) increase smoothly with increasing density; This is similar to the behavior observed for a discussed previously. The increase in (Ek} is primarily, though not wholly, due to the increased localization of the atoms as the density increases. In addition, the values for ( E k ) in the superfluid are systematically lower than those in the normal liquid, reflecting the presence of the condensate. T h e o r e t i c a l results s~ for (Ek} are also shown in Fig. 28. The agreement between theory and experiment for the high temperature measurements is quite good, both in terms of the absolute value and the density dependence. At low temperatures, both in the superfluid and in the solid, the theoretical results are systematically higher than the experimental measurements. However, the theories do predict the density dependence quite well.

25

25 a)

0.147 g/crn 3

25r b)

o.162 g/crn 3

C)

0.172 glcrn 3

201 t+t ii

20

201-

15

15I t t i t

A

-i

11~

0

,i,,,,i

1

....

2 T (K)

i ....

iiiii

L

3

4

0

1

2

t

3

T (K)

4

0

1

2

3

4

5

T (K)

Fig. 27. Temperature dependence of the measured (EK> at densities of (a) 0.147g/cm3, (b) 0.162 g/cm3, and (c) 0.172 g/cm3. The solid symbols are the theoretical calculations described in the text.

Density and Temperature Dependence in Liquid Helium 4

919

35

4.25 K 30

A

25

T

~4

0.75 K 20 15 10 0.12

i

i

i

0.14

0.16

0.18

F

i

0.2

0.22

Density (g/cm 3) Fig. 28. Density dependence of the measured ( E k ) at temperatures of 4.25 K ((2)) and 0.75 K (oo). The solid symbols are the GFMC and P I M C calculations at the corresponding temperatures.

Systematic differences between theory and experiment are present in both the normal and superfluid phases. However, they appear to be largest in the superfluid phase and to increase slightly with density. We note that these differences exist even though the predicted and observed scattering are in good agreement. However, these comparisons are sensitive primarily TABLE VI The Average Kinetic Energy per A t o m as a Function of Temperature at a Constant Densities of 0.147, 0.162, and 0.172 g/cm 3. The Temperature, Kinetic Energies, and Experimental Uncertainties Are All in Units of K p = 0.147 g/cm 3

T

(Ex)

AfEK)

0.35 0.75 1.0 1.5 1.8 1.9 2.0 2.2 2.3 2.8 3.0 3.5 4.25

13.3

1.3

14.1 14.5 14.0

1.4 1.4 1.4

14.8

1.5

16.1 16.6

1.6 1.7

16.2 17.1

1.6 1.7

p = 0.162 g/cm 3

p = 0.172 g/cm 3

(EK)

A{EK)

(EK)

A{EK)

15.5

1.6

17.5

1.8

16.5 16.3 17.1

1.7 1.6 1.7

17.3

1.8

18.3 18.2

1.8 1.8

16.9

1.7

17.1

1.7

19.7

2.0

19.2

1.7

19.6

2.0

920

W . M . Snow and P. E. Sokol TABLE VII

The Average Kinetic Energy per Atom as a Function of Density at a Constant Temperature of 4.25 K. The Temperature, Kinetic Energies, and Experimental Uncertainties Are All in Units of K T

(E,,)

A(E~)

0.125 0.130 0.138 0.140 0.147 0.160 0.173 0.181 0.186 0.195 0.200

15.5 15.2 16.1 17.1 19.6 19.2 21.8 26.5 26.5 31.6 32.7

1.6 1.5 1.6 1.7 2.0 t:9 2.2 2.7 2:7 3.2 3.3

to the central p o r t i o n of J ( Y ) where the intensity is large. T h e y are least sensitive to the wings where the finite statistical accuracy of the experimental m e a s u r e m e n t s m i g h t m a s k a n y differences. This suggests t h a t the b e h a v i o r in the high-p tails of the m o m e n t u m d i s t r i b u t i o n m a y be r e s p o n sable for the discrepancy. G F M C calculations 5~ have p r e d i c t e d a n(p) which has an e x p o n e n t i a l d e c a y at large p. O u r m e a s u r e m e n t s m a y u n d e r estimate the c o n t r i b u t i o n of these tails to ( E k ) a n d explain the systematic differences between t h e o r y a n d experiment. A similar a r g u m e n t h a s been p r o p o s e d 68'69 to describe systematic differences between theoretical a n d e x p e r i m e n t a l values for ( E k ) in 3He. TABLE VIII

The Average Kinetic Energy per Atom as a Function of Density at a Constant Temperature of 0.75 K. The Temperature, Kinetic Energies, and Experimental Uncertainties Are All in Units of K T

(EK)

A(EK)

0.149 0.156 0.162 0.167 0.172 0.192

13.6 14.7 15.5 16.7 17.5 21.8

1.4 1.5 1.6 1.8 1.9 2.3

Density

and TemperatureDependence in Liquid Helium 4

921

5.4. Condensate Fraction The condensate fraction is of fundamental interest since it underlies our microscopic understanding of the behavior of liquid helium. However, the condensate, which appears as a c~-function in J(Y), is not directly observable in these measurements due to the finite statistics and the presence of instrumental and final state broadening. The model scattering functions are consistent with the development of a finite condensate fraction. Furthermore, the comparison of theoretical calculations of J(Y) with the experimental results provides strong evidence for a condensate fraction with a magnitude in agreement with the theoretical predictions. Despite this good agreement between theory and experiment it is stil desirable to directly extract the condensate fraction from the experimental measurements without reference to a particular numerical calculation of

n(p). Several different procedures for extracting the condensate fraction from measurement of J(Y) have been proposedfl4' 39,70-74 These methods all, in one form or another, rely on models for the behavior of n(p) when a condensate is present. These different methods will be reviewed in detail elsewhere. In this work, we make use of the model developed by Snow, Sokol, and Wang. v3' 74 Their model n(p) explicitly incorporates the correct limiting behavior of n(p) known from microscopic theory while making as few assumptions about the shape of n(p) as possible. In addition, the procedure explicitly includes the deviations from the IA by making use of current many body calculations of FSE broadening. A similar model has recently been proposed by Glyde 24 which, while differing in several details, gives equivalent results. We have applied this method to the neutron scattering data over a wide range of temperatures and densities to extract the condensate fraction. T A B L E IX The Condensate Fraction Versus Temperature at a Constant Density of 0.147 g/cm 3

T (K)

no (%)

An 0 (%)

0.35 1.00 1.50 1.80 2.00 2.30

10.0 10.0 8.9 5.8 4.0 0.0

1.5 1.5 1.5 1.5 1.5 1.5

922

W . M . Snow and P. E. Sokol

Before presenting the results there is a unique aspect of the model that is worthy of comment. This model makes no explicit reference to the thermodynamic phase (normal liquid, superfluid, or solid). The form has sufficient flexibility to describe either the superfluid, with no finite, or the normal liquid or solid, where n o is expected to be small I or non-existent }. When we fit the data with this model we find that the fit forces no to zero both in the solid and in the normal liquid. This result is determined only by the shape of the scattering; no reference to the location of the normalsuperfluid or liquid-solid transitions or the phase of the sample is used. Figure29 shows the values of no extracted from the scattering measurements and the theoretical predictions at a density of 0.147 g/cm 3 as a function of temperature. At low temperatures, no is 10 + 1.25 %, in good agreement with the theoretical result 5~ 52-54 of 9.2 % at T = 0. Experimentally, no changes little below 1 K. This is in agreement with the results of many body theory and the finite temperature extensions of variational calculations.53, 54, 75 Above 1 K, the values for no decrease and tend to zero at the superfluid transition, in qualitative agreement with the theoretical values obtained from PIMC calculations. 57-59 The temperature dependence of no is also in agreement with a recent calculation which attributes the depletion of the condensate to the effect of thermal excitations of the r o t o n s . 76 In this interesting approach the scale for the size of no at T = 0 is set by the ratio of the thermal energy at the 2 transition (3k~T,J2) to the energy corresponding to the roton minimum (p~/2m). The temperature dependence of no near T~ can also be obtained from the normalization group theory of second order phase transitions. The temperature dependence of no near Tz is expected to take the form na = At2Q where t = ( 1 - TITs) and 2fl -- 0.70. The errors in no are too large to extract TABLE X The Condensate Fraction Versus Density at a Fixed Temperature of 0.75 K

T

no

A!'t o

(K)

(%}

(%)

0.149 0.156 0.162 0,167 0.172 0.192

10.0 9,0 8.5 6.5 55 0.0

1.5 1,5 1.5 1.5 1.5 1.5

Density and Temperature Dependence in Liquid Helium 4

923

b o t h t and A from our results. However, we m a y c o m p a r e this prediction with our experimental results using the accepted value of ft. Such a comparison, with A = 20.1, is shown in Fig. 29. As can be seen, the renorrealization group prediction is in g o o d agreement (at the 10 % level) with the experimental results down to a t e m p e r a t u r e of 1.5 K. This result is consistent with the m u c h m o r e precise m e a s u r e m e n t s of the critical behavior of the superfluid fraction 77 and the specific heat, 78 which also indicate that the width of the critical region in liquid helium is relatively broad. The density dependence of n o at T = 0.75 K is shown in Fig. 30. The values for no decrease with increasing density. This behavior is precisely what one would expect on physical grounds, since the higher rate of interparticle scattering at higher densities tends to lower the probability for a given a t o m to remain in the condensate. Nevertheless, the values for no remain finite close to the superfluid-solid phase boundary. This is also a physically reasonable result since there is not reason for the condensate to vanish continuously near the critical density for a first order phase transition. In addition, while a condensate m a y exist in the solid 79 81 we see no evidence for the presence of a condensate in the solid phase, in agreement with previous studies. 82-86 The values for n o are systematically s o m e w h a t higher than the theoretical results, but show the same density dependence within the errors. 15

IO c~

0

1

2 T (K)

Fig. 29. no as a function of temperature at a constant density of 0.147 g/cm3 (crosses). Also shown are the theoretical calculations of n o at T= 0 from GFMCSO.6o (square) and at finite temperatures from PIMC 55'ST.59 calculations (diamonds). The solid line is a plot of the renormalization group theory prediction no A t ~176with A = 20.1. The dashed line is the prediction of Giorgini e t al. 76 =

924

W . M . Snow and P. E. Sokol

15

10

!ii - / j i

5

r.)

< z_ i_

0

0.14

,

0.t5

0.16

.

0.17

it,

0.18

0:19

0.2

Density (g/crn 3) Fig. 30. no as a function of density at a constant temperature of 0.75 K (crosses). The G F M C s~ 60 results for n o are the solid symbols. The dashed line is the prediction of Giorgini e t al. 76

The G F M C calculations s~ 6o at T = 0 are also shown in Fig. 30. Both the magnitude and density dependence of the calculations are in reasonable agreement with the theoretical results. However, the theoretical values do appear to be systematically lower than the experimental values and to exhibit a stronger dependence on density. The origin of these discrepancies is not clear at present. The figure also shows the theoretical predictions of the density dependence of n o from Giorgani e t al. 76 assuming that the parameter m in their theory is independent of density. These results are in reasonable agreement with the experimental values. However, the errors on the estimates of no are too large to distinguish among the different density dependencies predicted by theory.

6. CONCLUSIONS Based on our analysis of the data using recent calculations for final state effects in deep inelastic neutron scattering, we summarize our conclusions on the momentum distribution, the kinetic energy, and the condensate fraction in liquid 4He as follows: (1) The observed scattering, and therefore the momentum distribution, of liquid 4He in the normal liquid phase is approximately (though not exactly) Gaussian in shape at high temperatures ( ~ 4 K). The width of the

Density and Temperature Dependence in Liquid Helium 4

925

scattering, and therefore the kinetic energy, increases rapidly with increasing density, due to the effect of confinement on the zero point motion, and decreases slowly with decreasing temperature. (2) The observed scattering in the superfluid phase develops a narrow component centered at Y= 0 whose relative intensity increases with decreasing temperature and decreases with increasing density, as one would expect if the narrow component is due to the presence of a Bose condensate. The remainder of the momentum distribution has a width which shows the same trends as a function of temperature and density as for the normal liquid. (3) The observed scattering is in excellent agreement with a number of recent theoretical calculations for n(p) at all temperatures and densities in both phases, including the superfluid phase for which the calculations predict Bose condensation. A small anomaly near Y= +2 A 1 is present in the liquid, superfluid, and solid phases in our data as well as other recent high resolution data. (4) The average kinetic energy is in agreement with theoretical calculations at all measured points in the phase diagram. The temperature and density dependence of the kinetic energy behaves as one would expect in a dense system dominated by zero-point motion. The density and temperature dependence are in better agreement with theory than are the absolute values. The uncertainties in determining the average kinetic energy from the non-Gaussian scattering are large due to the sensitivity of the second moment to the high-Y tails of the distribution. (5) Using a phenomenological model for the momentum distribution which incorporates the known behavior in the presence of a condensate, the magnitude of the condensate fraction has been extracted from the observed scattering. The condensate fraction increases with decreasing temperature, decreases with increasing density, and vanishes in the normal liquid and solid phases. Near the phase transitions, the condensate approaches zero continuously at the superfluid-normal liquid transition but remains finite near the superfluid-solid phase boundary. The values are in agreement with theoretical calculations at all temperatures and densities in the superfluid phase. Because we have not observed a resolution limited spike in our scattering data, we cannot claim to have demonstrated, beyond a shadow of a doubt, the existence of a delta-function singularity in the momentum distribution of superfluid 4He. Nevertheless, on the basis of the distinct change in the scattering upon entering the superfluid phase, the detailed agreement between the scattering data and the theoretical calculations, and the consistency of the trends in the scattering data with all the expectations based

926

W.M. Snow and P. E. Sokol

o n t h e o r e t i c a l g r o u n d s , we c o n c l u d e t h a t the s c a t t e r i n g d a t a p r o v i d e s t r o n g evidence for the existence of a Bose c o n d e n s a t e i n b u l k l i q u i d 4He w i t h a m a g n i t u d e w h i c h is in g o o d a g r e e m e n t w i t h t h e o r e t i c a l p r e d i c t i o n s .

ACKNOWLEDGMENTS

W e w o u l d like to a c k n o w l e d g e useful c o n v e r s a t i o n s w i t h Profs. H. R. G l y d e , A. Griffin, a n d W. G. Stirling. T h i s w o r k was s u p p o r t e d b y the N a t i o n a l Science F o u n d a t i o n t h r o u g h g r a n t D M R - 8 7 0 4 2 8 8 a n d the D e p a r t m e n t of E n e r g y t h r o u g h O B E S / D M S s u p p o r t o f the I n t e n s e P u l s e d N e u t r o n S o u r c e at A r g o n n e N a t i o n a l L a b o r a t o r y u n d e r D O E g r a n t W - 3 1 109-ENG-38.

REFERENCES

1. M. E. Fisher, Rep. Prog. Phys. 30, 615 (1967). 2. P. Nosieres and D. Pines, The Theory of Quantum Liquids, Volume 2 (Addison-Wesley, New York, 1989). 3. A. Griffin, Can. J. Phys. 65, 1368 (1987). 4. F. London, Nature 141, 643 (1938). 5. F. London, Superfluids, VoI. 2 (Wiley, New York, 1954). 6. P. E. Sokol, R. N. Silver, and J. Clark, in Momentum Distributions, R. N. Silver and P. E. Sokol, Eds. (Plenum, New York, 1989), p. 1. 7. P. E. Sokol, in Bose-Einstein Condensation, A. Griffin, Ed. (Cambridge, N.Y., 1995), p. 51. 8. A. Griffin, Excitations in Bose-Condensed Liquids (Cambridge, N.Y., 1993). 9. T. R. Sosnick, W. M. Snow, P. E. Sokol, and R. N. Silver, Europhys. Lett. 9, 707 (1989). 10. K. W. Herwig, P. E. Sokol, T. R. Sosnick, W. M. Snow, and R. C. Blasdell, Phys. Rev, B 41, 103 (1990). 11. T. R. Sosnick, W. M. Snow, R. N. Silver, and P. E. Sokol, Phys. Rev. B 44, 308 (1991). 12. K. W. Herwig, P. E. Sokol, W. M. Snow, and R. C. Blasdell, Phys. Rev. B 44, 308 (1991). 13. R.T. Azuah, W. G. Stirling, H. R. Glyde, P. E. Sokol, and S. M. Bennington, Phys. Rev. B 51, 605 (1995). 14. T. R. Sosnick, W. M. Snow, and P. E. Sokol, Phys. Rev. B 41, 11185 (1990). 15. K. H. Anderson, W. G. Stirling, H. R. Glyde, R. T. Azuah, and S. M. Bennington et aL, Physics B 197, 198 (1994). I6. W. G. Stirling, L. K. H. Anderson, A. D. Taylor, I. Bailey, and Z. A. Bowden, ISIS Experimental Report No. ULS 85/1RB610 (1989)., unpublis 17. R. S. Holt, L. M. Needham, and M. P. Paoli, Phys. Lett. A 126, 373 (1988). 18. J. M. Carpenter and W. B. Yelon, in Methods of Experimental Physics, Vol. 23A, D. L. Price and K. Skold, Eds. (Academic Press, New York, 1986), p. 99. 19. R. N. Silver, Phys. Rev. B 38, 2283 (1988). 20. R. N. Silver, Phys. Rev. B 39, 4022 (1989). 21. R. N. Silver, Phys. Rev. B 37, 3794 (1988). 22. R. N. Silver, in Momentum Distributions, R. N. Silver and P. E. Sokol, Eds. (Plenum, New York, 1989). 23. H. R. Glyde, Phys. Rev. B 50, 6726 (1994). 24. H. R. Glyde, Physiea B 194-196, 505 (1994). 25. S. Stringari, Phys. Rev. B 35, 2038 (1987). 26. C. Carraro and S. E. Koonin, Phys. Rev. B 41, 6741 (1990). 27. C. Carraro and S. E. Koonin, Phys. Rev. Lett. 65, 2792 (1990).

Density and Temperature Dependence in Liquid Helium 4

927

A. Miller, D. Pines, and P. Nozieres, Phys. Rev. 127, 1452 (1962). P. C. Hohenburg and P. M. Platzman, Phys. Rev. 152, 198 (1966). G. B. West, Physics Reports 18C, 263 (1975). P. E. Sokol, Can. J. Phys. 65, 1393 (1987). B. Tanatar, G. C. Lefever, and H. R. Glyde, Physica B 136B, 187 (1986). B. Tanatar, G. C. Lefever, and H. R. Glyde, J. Low Temp. Phys. 62, 489 (1986). H. R. Glyde, J. Low Temp. Phys. 59, 561 (1985). V. F. Sears, Phys. Rev. B 30, 44 (1984). V. F. Sears, Can. J. Phys. 63, 68 (1984). R. N. Silver, (1995), unpublis The raw data and the instrumental resolution and final state broadeing functions presented in this paper are available in electronic format by contacting the authors at [email protected]. 39. V. F. Sears, E. C. Svensson, P. Martel, and A. D. B. Woods, Phys. Rev. Lett. 49, 279 (1982). 40, H. A. Mook, Phys. Rev. Lett. 32, 1167 (1974). 41. H. A. Mook, Phys. Rev. Lett. 51, 1454 (1983). 42. A. Rahman, K. S. Singwi, and A. Sjolander, Phys. Rev. 126, 986 (1962). 43. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Nurnerical Recipies (Cambridge University Press, Cambridge, 1986). 44. V. F. Sears, Phys. Rev. B 28, 5109 (1983). 45. R. O. Simmons, ZeitschriJt fiir Naturforschung A. Journal of Physical Sciences 48a, 415 (1993). 46. M, A. Fradkin, S. -X. Zeng, and R. O. Simmons, Zeitsehrift fuer Naturforschung A: Journal of Physical Sciences 48a, 438 (1993). 47. W. Langel, R. O. Simmons, D. Price, and P. E. Sokol, Phys. Rev. B 38, 11275 (1988). 48. K. W. Herwig and R. O. Simmons, in Momentum Distributions, R. N. Silver and P. E. Sokol, Eds. (Plenum, New York, 1989), p. 203. 49. P. E. Sokol, in Momentum Distributions, R. N. Silver and P. E. Sokol, Eds. (Plenum, New York, 1989), p. 139. 50. R. M. Panoff and P. A. Whitlock, Can. J. Phys. 65, 1409 (1987). 51. P. A. Whitlock, D. M. Ceperley, G. V. Chester, and M. H. Kalos, Phys. Rev. B 19, 5598 (1979). 52. E. Manousakis and V. R. Pandharipande, Phys. Rev. B 30, 5062 (1984). 53. E. Manousakis and V. R. Pandharipande, Phys. Rev. B 31, 7029 (1985). 54. E. Manousakis, V. R. Pandharipande, and Q. N. Usmani, Phys. Rev. B 31, 7022 (1985). 55. D. M. Ceperley and E. L. Pollock, Phys. Rev. Lett. 56, 351 (1986). 56. E. Manousakis, S. Fantoni, V. R. Pandharipande, and Q. N. Usmani, Phys. Rev. B 28, 3770 (1983). 57. D. M. Ceperley and E. L. Pollock, Can. J. Phys. 65, 1416 (1987). 58. D. M. Ceperley and E. L. Pollock, Phys. Rev. B 36, 8343 (1987). 59. D. M. Ceperley, in Momentum Distributions, R. N. Silver and P. E. Sokol, Eds. (Plenum, New York, 1989). 60. R. Panoff and P. Whitlock, in Momentum Distributions, R. N. Silver and P. E. Sokol, Eds. (Plenum, New York, 1989). 61. K. W. Herwig, (1990)., unpublis 62. D. S. Sivia and R. N. Silver, in Momentum Distributions, R. N. Silver and P. E. Sokol, Eds. (Plenum, New York, 1989), p. 377. 63. O. K. Harling, Phys. Rev. A 3, 1073 (1971). 64. A. G. Gibbs and O. K. Harling, Phys. Rev. A 3, 1713 (1971). 65. A. D. B. Woods and V. F. Sears, J. Phys. C: Solid State Phys. 10, L341 (1977). 66. H. N. Robkoff, D. A. Ewen, and R. B. Hallock, Phys. Rev. Lett. 43, 2006 (1979). 67. F. W. Wirth, D. A. Ewen, and R. B. Hallock, Phys. Rev. B 27, 5530 (1983).

28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

928

W . M . Snow and P. E. Sokol

68. J. Carlson, R.M. Panoff, K.E. Schmidt, P. A. Whitlock, and M. H. Kalos, Phys. Rev. Lett. 55, 2367 (1986). 69. P. E. Sokol, K. Skold, D. L. Price, and R. Kleb, Phys. Rev. Lett. 55, 2368 (1986). 70. G. J. Hyland, G. Rowlands, and F. W. Cummings, Phys. Lett~ 31A, 465 (1970). 71. F. W. Cummings, G. J. Hyland, and G. Rowlands, Phys: Lett. 86A, 370 (!981). 72. A. Griffin, Phys. Rev. B 32, 3289 (1985). 73. W. M. Snow~ Y. Wang, and P. E. Sokol, Europhysics Letters 19, 403 (1992). 74. P. E. Sokol, in Excitations in 2D and 3D Quantum Fluids, A. J. Wyatt and H. J. Lauter, Eds. (Plenum, New York, 1991). 75. K. Kehr, Z. Phys. 221, 291 (1969). 76. S. Giorgini, L. Pitaevskii, and S, Stringari, J. Low Temp. Phys, 89, 449 (1992). 77. J. A. Lipa and T. C. P. Chui, Phys. Rev. Lett. 51, 2291 (1983). 78. D. S. Greywall and G. Ahlers, Phys. Rev. A7, 2145 (1973). 79. G. V. Chester, Phys. Rev. A2, 256 (1970). 80. A. Andreev and I. M. Lifshitz, Soy. Phys JETP 29, 1107 (1969). 81. A. J. Legget, Phys. Rev. Lett. 25, 1543 (1970). 82. A. Andreev, K. Keshishev, L. Mezhov-Deglin, and A, Shalnikov, JETP Lett: 9, 306 (1969). 83. J. Suzuki, J. Phys. Soc. Jpn. 35, 1473 (1973). 84. V. L, Tsymbalenko, JETP Lett. 23, 653 (1976). 85. D. S. Greywall, Phys. Rev. B 16, 1291 (1977). 86. D. J. Bishop, M. A. Paalanen, and J. D. Reppy, Phys. Rev. B 24, 2844 (1981).