Size Effects in Superfluid Helium II - I

Size effects in superfluid helium II - I. Experiments in porous systems E. Guyon, I. Rudnick

To cite this version:

E. Guyon, I. Rudnick. Size effects in superfluid helium II - I. Experiments in porous systems. Journal de Physique, 1968, 29 (11-12), pp.1081-1095. 10.1051/jphys:019680029011-120108100. jpa-00206747

HAL Id: jpa-00206747 https://hal.archives-ouvertes.fr/jpa-00206747 Submitted on 1 Jan 1968

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LE JOURNAL DE PHYSIQUE TOME 29, NOVEMBRE-DÉCEMBRE 1968, 1081.

SIZE EFFECTS IN SUPERFLUID HELIUM II I. EXPERIMENTS IN POROUS SYSTEMS

By E. GUYON (1) and I. RUDNICK (2), Physics Department, University of California, Los Angeles, California, (Reçu le 22 avril 1968.)

Résumé. 2014 Les propriétés de la phase superfluide de l’hélium liquide sont fortement modi- fiées par les effets de taille. Nous discutons dans ce travail une étude experimentale de l’hélium. contenu dans des poudres très fines (~ 100 à 104 Å) : abaissement de la température d’établis- sement de l’état superfluide, T0, et propriétés superfluides pour T To : vitesse d’écoulement critique, chaleur spécifique, densité superfluide. Cette dernière donnée est obtenue à partir de la mesure de la vitesse du son hydrodynamique qui se propage dans un tel système poreux 2014 le quatrième son. Nous discutons en détail les techniques et les propriétés physiques liées à la mesure du 4e son. Une discussion des résultats est présentée par analogie avec l’étude des effets de proximité dans les supraconducteurs.

Abstract. 2014 The properties of the superfluid phase of liquid helium are modified by size effects. We discuss here experiments on helium contained in very fine compressed powders (~ 100 to 104 Å) : lowering of the onset temperature for superfluidity, T0, and superfluid properties for T To : critical flow rate, specific heat, superfluid fraction. These last data are obtained from the measurement of the hydrodynamic sound velocity that propagates in such porous systems : 4th sound. We present in some detail the techniques of the measurements and the physical properties of 4th sound. A discussion of the results is presented in analogy with the study of the proximity effect in superconductors.

IntroduCtion,. - The properties of the superfluid rikse [3] had measured the specific heat of helium phase of helium, He II, are strongly affected by adsorbed on a jeweller’s rouge powder and found the effects of size. These effects occur when the helium following properties : the singularity and discontinuity is in a nearly one or two dimensional container : of specific heat, which is observed in bulk helium II superfluid film of helium (Rollin film) or helium at T À is replaced by a peak having its maximum contained in the small channels of a porous system. at T p ( Tx). This peak becomes broader and This work will present some aspects of the modifica- shifted to lower temperatures for thinner films [4]. tion of superfluidity in this last case; the channels are In order to decide unambiguously the fundamental the tangled and interconnected pores of highly relevance of the temperature broadening from the compressed fine powders. Previous experimental experiments, one should have some information on the studies have described mostly the modification of the thickness distribution in the films. Unfortunately the onset temperature for superfluidity, To. We will also values of dF, obtained from adsorption isotherms present and discuss some properties of such systems measurements, are only of statistical nature and may in the superfluid state at T T0. vary with the substrate properties. The same charac- The problem of size effects in He II has been a teristic results of flow and specific heat measure- subject of considerable experimental interest in the ments [5, 6] were obtained for helium filling porous last 15 years. The first experiments of superfluid systems : To is smaller than Tx and the specific heat flow in unsaturated films [1] by Long and Meyer [2] has a broad maximum at a temperature Tp below Tx. show that To ( Tx) is an increasing function of the The same uncertainty remains in these results as the film thickness, dF. They also showed that the critical effect of the pore size distribution is certainly impor- flow rate goes smoothly to zero at To in contrast with tant in this case. The two sets of experiments were the sharp drop for the bulk sample. In 1949, Frede- done on the same samples of porous Vycor glass (solid perforated by interconnected channels). In (1) Permanent addyess : Laboratoire de Physique des the case of helium completely filling the pores of the Solides, associ6 au Centre National de la Recherche Vycor specimen, it was found on one sample that Faculté des France. Par- Scientifique, Sciences, Orsay, = = 2.06 OK. However in the case of unsa- tially sponsored by NATO. To TP (2) Research sponsored in part by the U.S. Office of turated films, it was observed that TP was larger than Naval Research, Acoustics Branch, Nonr 233-(48). To [2]. This contrasts with the case of bulk helium

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019680029011-120108100 1082

where To = TP = Tx with an accuracy of the order velocity of the unattenuated 4th sound made is given of 10-6 °K. by [10] : A fundamental property of superiluid helium is its superfluid density. This quantity can be obtained from the velocity measurement of the hydrodynamic wave which propagates in helium II filled micropores : ul and U2 are the first and second sound velocity, This 4th sound wave was predicted by Pellam [7] P, is the isobaric isothermal expansion coefficient, and Atkins [8] and first observed by Shapiro and so the entropy of He II. E( T) is a small coefficient Rudnick [9] for coarse samples where the shift of To and is given in figure 1 for bulk helium. The correc- was too small to be detected. Similar experiments were performed in much finer powders by Rudnick et al. [10] and will be discussed and developed here. We will recall in Part I some essential results for 4th sound based on two fluid hydrodynamics. We discuss in Part II the properties of the powders and the experimental techniques. The determination of To, of flow measurement and of superfluid density, will be given in Part III. We present in Part IV a theoretical discussion based essentially on the use of modified Gins- burg-Pitaevski [11] (G. P.) equations to discuss these results. Some analogies will appear with the case of superconductors where the corresponding Ginsburg- Landau [12] (G. L.) equations are known to describe well, within certain limits, the size effects.

FIG. 1. - Correction to the simplified 4th sound velo- I. Summary of 4th sound properties. - The propa- city using (1.1). gation of sound waves in liquid helium II presents a number of which can be descri- interesting properties tion is maximum around 2 OK and decreases with bed accurately in terms of two fluid components, a zero near 1.17 OK where normal fluid and a of and temperature, becoming superfluid density PN pg [13] This correction should be different such that PN + Ps = P where p is the total density of PP changes sign. for a onset as the liquid. The superfluid carries no entropy and samples having depressed temperature the involved in it on The has no The sound wave which parameters depend To. viscosity. corresponds calculated variation is in to the sound is due to the vibration of the temperature of U4( T) given ordinary 2 with that of first and second sound two components in phase and is mainly characterized figure together in the case of "coarse" has to be smaller by density and pressure oscillations. Another mode powders : d then but not so small as to see a shift in the onset corresponds to the oscillation of the two fluids in XN 10-3 OK. The value of opposite phase such that the momentum density j temperature : Tx - T0 was taken from the disk measure- remains constant. This wave corresponds to an en- pg/p(T) oscillatory ments of Dash and We will not tropy (temperature) wave and is called second sound. Taylor [14]. repro- It is absent in bulk liquid helium above Tx. In the duce the calculation [9] leading to (I. 1). However limit where - 1 - 0 is the ratio of the one easily can understand the major contribution to y (y specific the 4th sound from the heats), which is a good approximation not too close expression (1.2) thermohydro- from the lambda point, the temperature (entropy) dynamics equations : changes are not coupled to the pressure (density) 1) The force on the superfluid is due to the variation of changes and there are no temperature changes in first the chemical potential y : sound and no pressure changes in second sound. Atkins [8] considered the case where the normal fluid does not move. This occurs when the viscous wavelength for the normal fluid, X,,, becomes large compa- The thermal fluctuations associated with the 4th sound red to the size of the system. Classically are small. 2) The continuity equation is : at the frequency m ; "1JN is the normal fluid viscosity. This is true if the is Near Tx, PN goes to as the disappears only powder homogeneous along p superfluid its progressively, and AN N 5 X 10-3 (0-112 CM. In’ our length. differentiation and from systems where the pore size varies between d = 100 By eliminating b. (1.3), and 104 A, the condition d ÀN is well satisfied for one gets : the frequencies o = 103 to 105 used. The phase 1083

near the onset temperature as p,,. The expression for u4 is obtained without any detailed knowledge of the pore size and we expect it to describe the Psi P ( T) variation for small powders as well as for "coarse" ones in the limit of applicability of the two fluid hydrodynamics. The grains of the superleak can be considered as immobile scattering centers. They cause a reduction of the effective phase velocity in the case of our experiments where the wavelength of the sound is much larger than the size of the centers. In addition, we expect the scattering correction to be independent of the frequency in this limit. Shapiro and Rudnick [9] experimentally determined a frequency independent analytical scattering correction which depended on the porosity p (percentage of empty volume) of their "coarse" samples and was given by : n = (2 - p) 1-/2 (I .5) where n is the index of refraction. Since the multiple scattering is large for the present samples and because of lack of certainty about the range of validity of this correction, it was decided that the best course would be to arbitrarily "normalize" all 4th sound velocities to give the calculated value of 232 m/s at 1.2oK. However, a reasonable correction for all powders should be one corresponding to a porosity of around 0.5 as estimated from the weight of powder. (The minimum porosity for close packed spheres is close to 0.3.)

- FIG. 2. - The fourth sound velocity is calculated II. Experimental techniques. A. SUPERLEAK using (I.1) and the experimental lst and 2nd sound MATERIAL. - We have used pressed powders of results. There is a good agreement with the experi- different sizes as our porous media. The grain pro- mental variation for coarse powders. perties are described in table I. The sizes of the grains were obtained from electron microscope photographs such as the one given for the Elf powder in which is the wave equation for a density wave of velocity : figure 3. In the case of the smaller carbon black powders, the size distribution is quite uniform. However, the coarser powders (usually polishing The proportionality to p. comes from the continuity compounds such as "rouge" or alumina Linde B equation. In the presence of the driving force, only powder) show a wide size distribution. In this last a fraction p./p can move. One expects U24 to vanish case, the particle size given by the manufacturer

TABLE I

PROPERTIES OF THE PACKED POWDERS PACKING PRESSURES WERE NOT THE SAME FOR ALL SAMPLES BUT WHERE OF THE ORDER OF 104 lb/in.2

(a) We are grateful to TRW Systems for providing us with electron microscope photographs of our samples. 1084

- FIG. 3. - Electron microscope picture of a carbon FIG. 4. The fourth sound cylindrical resonator is black powder (1 cm correspond a environ 500 Å). symmetric with respect to x. In particular, the two end - drive and receiver - transducers are identical. The copper body - B - of the resonator contains corresponds approximately to the size we find for the the packed powder - P. The 6 f1. thick mylar foil M coarse grains. However, the pore size is determined is the moving part of the transducer and has its single the finer For the aluminized side against B. The active capacitance mostly by grains. example, pro- of the transducer is that between the aluminized mylar of the "0.05 Linde B" are perties {JL powders very and the copper block C insulated from the end plate A similar to those of the 300 k carbon powders. by a nylon spacer N. The filling leak is produced by The powder is in a solid open copper a small scratch in the sealing indium ring I. The packed cylin- of the coaxial is small der and a funnel attached passive capacitance pole kept pressed through long rigidly Contact between the inner conductor and to it. The is and several (30 pf/m). powder highly compressible C is insured by a watch band pin W. In flow experi- fillings of the funnel are necessary to fill the resonator. ments only the central part (B + P) is used. The same pressure is applied after each filling. Various pressures up to 103 kg/cm2 were used. Howe- is applied right against the cylinder end surface with ver, after the final when the end and packing, plate the single metallized side against the copper. The are removed, 10 to 20 of the piston % % packed dead space between the mylar and the carbon is powder extrudes out of both ends in a resulting kept as small as possible, typically of the order of 20 [1.. release of the pressure. Thus, while the partial In some cases where this was not satisfied, a large can be the final packing pressure specified accurately, complexity of resonance modes other than 4th sound pressure cannot. The best correlation between the ones was observed. They are probably related to the modification of and the size for a superfluidity pore conversion of 1 st and 2nd sound near the ends into given powder is obtained by weighing the amount of 4th sound. The second plate of the capacitative included. The shift in onset powder largest tempera- transducer is the copper end plate put against the ture is observed for the largest amount of powder. transducer. The capacity of the condenser is typically After the the end areas are cleared of the packing, C = 150 pf [17]. The resonator is sealed with indium extruded carbon. Care is taken to the carbon get rings put between the end plates and the body of the surface flat and aligned with the cylinder end. This cylinder. A small scratch is made in the already turns out to be delicate for the finest packed powders compressed indium rings which acts as a leak to fill which are very friable. The same cylinder is used the capsule. The complete filling with helium II, for fourth heat and flow measurements. sound, specific observed from the 4th sound signal below the onset It is under vacuum between the to kept experiments temperature, took between a fraction of a minute limit the amount of adsorbed materials which would and 1 on the powders and the size reduce the size for Atkins et al. hour, depending pore (see example [16]). of the scratch. With an ordinary fluid a tight but B. FOURTH SOUND. - The resonance technique leaking termination presents a high impedance because used has been described previously. We only point the acoustic resistance is high by virtue of viscosity. out here the new or critical features in the experiment. The superfluid component being viscousless, one must Figures 4 gives a sketch of the fourth sound resonant rely instead on a high mass reactance for the leaks. cavity. The ends of the cylinder are closed by two In the absence of a high impedance termination, capacitative transducers : an aluminized mylar sheet there would be an end correction whose amplitude 1085 is temperature dependent and is not determined by the fourth sound velocity alone. Accordingly, it is essential that all leaks to the resonator have a superfluid mass reactance large compared with the characteristic acoustic impedance of the fourth sound; this is obtained by the scratch sufficiently long and of small cross-section area. The resonator was put directly into the helium bath in a glass cryostat whose temperature could be varied between 1.2 OK and 4.2 OK, and stabilized within 10-3 OK. The output signal of a tracking generator "General Radio wave analyzer type 1900 A" was fed into the drive transducer after amplification. An A.C. voltage of 1 to 50 volts was used. The frequency was varied up to 20 kC to determine the plane wave resonant modes excited by the A.C. input signal (the cavities used were 2.54 cm long and the first plane wave made at low temperature had a frequency about 4 kilocycles). The signal from the other-receiving-transducer was fed into a cathode follower, amplified by 40 dB and detected by the wave analyser. It is possible to produce an acoustic signal at a frequency of2co, double that of the A.C. signal, by not polarizing the drive transducer with D.C. voltage. This allowed a separation of the effect of the acoustic signal from the electric pickup at a frequency w. A characteristic of these transducers is that the oscillating mass of the transducers is very small. Kinetic reaction and the mechanical "crosstalk" are thus small. The signals in the absence of helium II in the cavity (either absence of helium or helium above the lambda point) were at least 30 dB lower than with He II through most of the temperature range. A set of recorded resonance curves at different temperatures is given in figure 5 for rouge. They show the lowest plane wave harmonic modes. Q’s of several hundreds at low temperatures are not unusual. Al- though the amplitude of the detected signal becomes lower for fine pores, the Q, of the signal does not necessarily decrease. On the contrary, in the limit of very coarse systems with d tV XN, the fourth sound is strongly attenuated [18] and becomes progressively FIG. 5. - Frequency dependence of the pickup signal a first sound mode as d becomes larger than XN. (At of the resonator with a drive signal at the same fre- The resonance modes to the six the same time, a second sound mode, which is a diffu- quency. correspond first plane wave harmonics. The large low frequency sion wave in the fine powder, appears [19].) For the signal is present only below To. Its amplitude depen- input powers used, the Q and the position of the reso- dence is used to determine To. nances are essentially independent of the input signal Erratum : On the first pike of the two lower spectra, the figure 1 amplitudes. However, the response amplitude is not has been forgotten. proportional to the input for large input powers. The residual spacing between the transducer and the closest space is larger. In improperly packed resonators it powder grains defines the maximum displacement of can be larger than all the other peaks. Its frequency the drive transducer and may cause saturation effects. range decreases only slightly with temperature. When In addition to the complexity of the resonance T - To (that is when u4 0) all peaks "fall" into modes, a low frequency signal is observed. Its ampli- this signal and it becomes impossible to determine tude increases with decreasing frequency (this last their frequencies. The amplitude of this signal de- feature is not seen on figure 5 as the use of a 60 cycles creases only very close to To. We use this variation rejection filter attenuates the signal amplitude below for an accurate determination of the onset temperature 200 cycles). This signal is larger when the dead end (see I I I . A) . 1086

C. FLOW MEASUREMENTS. - The end plates of the cavity were taken off in order to measure the flow rate through the powder. A Lucite cylinder container was sealed on top of the cylinder. It had a larger cross-section area than the resonator and did not limit the flow. This container was closed except for a small hole near the top to allow for the fillings. A coaxial piston could be moved up and down in the cryostat. It was used to change the outside level. In this way the container could be filled and a level difference created, giving rise to a chemical potential difference between the ends of the powder filled cylinder. The change of level in the inside vessel is a linear function of the flow rate. It is obtained from the change of capacitance of a vertical parallel plate capacitor, using a standard technique. The condenser a resonant is part of oscillating circuit kept below FIG. 6. - Flow rate measurement through carbolac II the helium level with a stable self-inductance and a powder in the immediate vicinity of To. To was tunnel diode biased in the region of negative dynamic measured independently as in figure 7. At lower resistance. The resonant frequency was recorded temperature the increase in flow rate is much slower. using counter and digital to analog converter. A Hz Hz was of 10 in 107 achieved. - frequency stability D. SPECIFIC HEAT. In addition to these data, The effect of vibrations on the helium level in the specific heat measurements are in progress on the limits the capacitance stability. same capsules and will be reported independently. A change of level of 1 mm in the capacitance gave a frequency variation of several kilocycles (this is III. Results. - A. ONSET TEMPERATURE. - A what is expected from the change of level of the helium conventional measurement of To comes from that of of dielectric constant N 1.05). In addition to the the flow rate as a function of temperature. We give flow through the porous medium, the level variation in figure 6 the results obtained on a carbolac II powder is also due to the evaporation. It is measured as a as a function of temperature. In order to minimize background variation of level above To and is nearly the thermal effects due to the flow, we took the initial constant up to T À’ It is generally less than 10 % of the slope of the level versus time curve to measure the total measurement. The film flow through the 1 mm flow rate. We give on the same curve temperature diameter hole is certainly negligible (calculated to be variation of the low frequency signal amplitude mea- of the order of 10-4 times smaller than the flow rate). sured on the same specimen. The flow rate goes During the flow rate measurement, the temperature smoothly to the constant background value above To. in the two baths could be recorded. Their difference This is in contrast with the critical flow rate for helium was consistent with the thermomechanical effect. in coarse systems. Reppy [20] measured the critical When there is superflow from the inside upper bath A flow rate in coarse fibrous foam from angular velocity to the bath B lower by AH, the temperature diffe- measurements. The velocity drops to zero in less rence AT = T A - TB becomes positive. The dri- than 10 y degrees in samples where the average grain ving force on the superfluid is : size is 150 {JL [21]. However, it is also found [21] that the range of fast variation of the critical velocity becomes much larger in smaller pores. This had also The thermal effect compensates partly that of the been observed in previous experiments : after the first level difference. Only a small temperature difference work of Atkins et al. [16], Brewer et al. [6] measured is needed to limit the OH effect. the flow of helium filling porous Vycor glass and found This is most pronounced near T À where the specific a curve going to zero at a depressed To with a nearly entropy is very large. In order to reduce the effect zero slope. Brewer and Mendelsohn [22] measured of A T which increases as the superflow takes place, the heat transport properties in unsaturated films we have measured only the initial flow rate at a adsorbed on glass and found a smooth superfluid constant initial flow rate at a constant initial value transition for thin films of low onset temperature (down of AH = 1 cm. The measurements in a carbolac II to To = 1.8 °K). In this last system, it does not powder are given in figure 6 and will be discussed seem possible to attribute the effect to the distribution in II.C. of sizes as could be the case for the Vycor glass or our Our main concern here is the determination of To powders. We will discuss this shape in connection independently of the 4th sound data. As there is with the 4th sound results. only a small AT associated with a given AH, this is a In fourth sound measurements, To is defined as the favorable factor for an accurate determination of To. temperature above which no acoustic signal is trans- 1087

mitted through the packed powder. To can be deter- to be presented next. Qualitatively, we observe larger mined directly from the decrease of the amplitudes of depressions for smaller grain size powders and in the the sound signal. However, the amplitude of the case of a given powder, for higher mass densities. resonance signals does not vary regularly far from To. A direct study of the measured To as a function of We rather use the sharp decrease of the amplitude of an average pore size is very desirable and is under the low frequency signal observed in the vicinity of To. way. A histogram giving the total pore size area as As T approaches To, the frequency of the resonant a function of pore size (typically per range of 5 A in modes collapse into this low frequency region. For pore size) can be obtained from adsorption measu- example, To = 2.122 ± 0.002 oK on the curve of rements. Such data had already been obtained on the low frequency signal (fig. 7) for a carbolac II carbon powders similar to ours by J. H. Atkins [23] powder. The values of To obtained from this last from the adsorption isotherms of N2 at low tempera- method are shown on figure 8 by arrows for various ture. Some preliminary results [24] on packed CI powders together with the fourth sound measurements powders indicate an average pore diameter of 45 A for an onset temperature T = 2.095 OK. This pore size corresponds to 1/2 of the average grain size and is a reasonable value for a close packed array of spherical carbon grains. It also gives a good agreement with the value of depressed onset temperature of films of comparable thickness and a phenomenological model to be discussed in IV. C.

B. FOURTH SOUND RESULTS. - We give in figure 8 the normalized fourth sound results for the different powders of table I. The ratio between the normalized velocity u4(1.2 OK) = 232 m/s and the uncorrected value determined from the frequency of the first harmonics varied between 1.14 (Linde B) and 2.4 (tightly packed carbolac I). It was the same for different harmonics. In this long wavelength limit the scattering correction is indeed independent of the frequency. It is very likely that a part of the effect is due to reduction of the fourth sound velocity at 1.2 OK with respect to the value for the bulk. If one assumes FIG. 7. - Determination of To from the variation of the refractive index correction (I.5) to remain valid amplitude of the low frequency signal on a carbolac II and a value of P = 0.5 we see that u4(1.2 OK) should powder. decrease with respect to the bulk value when To de-

FIG. 8. - Fourth sound velocity as a function of temperature in powders of progressively lower size from A to G. We have not identified the different harmonics used in the velocity measurements. The properties of the powders are given in table I. The arrows correspond to independent measurements of To as in figure 7. 1088

creases. Direct measurements of porosity together mentally identical at all temperatures. However, the with fourth sound measurements [24] tend to support grain size distribution and probably the pore size this idea. We have observed no appreciable disper- distribution were very different in the two cases. This sion between the temperature variation for different point is in favor of the fundamental significance of the harmonics (up to 20 kc). Figure 8 includes measure- results. We will focus our attention on the immediate ments made on many different harmonics. In parti- neighborhood of To (r = To - T 0.1 °K) for two cular, in the vicinity of To, it is convenient to follow reasons : first the experimental determination of To by the variation of the fourth sound velocity using the an extrapolation of the data requires it. Secondly this higher harmonics. We observe also, in some cases, region has a considerable theoretical interest as will be an added complexity of peaks at fixed frequencies. discussed later. The choice r 0.1 OK is somehow It is not easy to locate precisely the 4th sound reso- arbitrary as the range of validity of the theoretical nance when it crosses these peaks. This can also be models is not accurately known. improved by measurements on several harmonics. In the case of very coarse powders, our results show The following features are observed in figure 8 for that Ps/p( T) varies as (Tx - T) 2’3 over a large tempe- pores of progressively smaller pore size : rature range, typically r N 0.1 OK. We give in fi- a) The negative slope of the normalized fourth gure 9 more recent data [24] on the temperature varia- sound increases with T. Even tion of the fourth sound velocity for an A1203 Linde C velocity monotonically with A size. It is at low temperatures, U4 V) for small powders is different powder 10,000 average grain from that in the bulk. The fact that the negative possible to deduce from a least square fit on this measurement and of the exact form of the relation at low temperature is for small pore size slope larger with the the law : may just mean that the fourth sound velocity is actually superfluid density (1.2) lower in this case. We can conclude that size effects are even far important below To. This result differs with : from other 4th sound data in Vycor [25] where the slope is relatively temperature independent and smaller on a temperature range T = .15 OK. We obtain the than that for bulk helium up to nearly To where u4(T) same 2/3 critical exponent [26] found in the angular decreased very sharply. momentum measurements of Clow and Reppy on b) Near To, U4(T) goes to zero with a smaller slope "coarse" powder and the later ones of Tyson and than in "coarse" powders. To, as obtained in a), Douglas on oscillating disks with a larger separation decreases as the pore size decreases. These indepen- than our pore size. The multiplying factor was 1.438, dent determinations of To agree with a reasonable 1.43 instead of 1.41 found here. The difference may extrapolation of the fourth sound data to zero. be due to a modification of the properties of the helium c ) The fourth sound curves look like family of curves in these powders with respect to the bulk. No shift characterized only by the value of To. We have of To was observed with an accuracy of 10-3 OK, but measured in different runs a powder of Linde B (2) a modification of To of this order for this powder and one of Elf 8 which had the same To within 10-3 OK. is possible. The normalized velocity determinations were experi- Let us consider now the case of powders with a shift of To of several millidegrees. We have plotted in figure 10 the normalized superfluid density on a carbolac II (1) powder calculated from the fourth sound velocity on a carbolac II (1) powder. The variation is approximately linear in a temperature range ,r = 0.2 OK, with a law Ps/p(T) = 3.5 (T0 - T). This value of To agrees well with that determined in figure 6 from flow rate measurements on the same sample. However, at lower temperature, there is departure from the linear variation and one can find a region between T = 2 OK and 2.1 OK where a 2/3 power law applies well. For lower To, the behavior of Ps/p(T) near To indicates an even stronger rounding. We take the example of the sample C II (4) ( fig. 11 ) where :

FIG. 9. - of the corrected Log Log plot superfluid density varies in a variation as a function of for a coarse Ps/p( T) nearly quadratically temperature temperature 2.00 T 2.06 °K. For small the A120a powder (average grain size of 10,000 A). No range pores depression of Tx is observed. A straight line fit of domain of r where it is not possible to locate the resonance is is small. this curve gives Ps/p = 1.41 (Tx - T)2/3 in a tempe- large : first the output amplitude rature interval Tx - T = .15 OK. Also the frequency does not increase as rapidly with r 1089

FzG. 10. - The fourth sound results for the carbolac II powder (the same as fig. 6) indicate a linear variation of Ps/p(T) near To. The results at lower temperature may be fitted with a 2/3 power law down to 2 OK.

C. ATTENUATION OF FOURTH SOUND. - Sanikidze et al. [29] have recently calculated the fourth sound attenuation near the X point using time dependent Ginsburg-Pitaevski equations. In the vicinity of the lambda point, the characteristic time associated with the relaxation of Ps goes to infinity as p. 1. This type of behavior is usual in second order phase transitions where it is associated with some type of critical opa- lescence. The resulting attenuation coefficient is :

where ( is the coefficient of second viscosity. Its measurement should give some interesting information on the relaxation mechanism involved. In addition to this basic source of attenuation, there are other possible instrumental sources. There are probably parts of the resonator which are incompletely filled

- with In such it is that the FIG. 11. The low value of To was obtained with a powder. regions, possible C II (4) powder packed with fine copper wool which partially unlocked normal fluid can undergo sufficient prevented extrusion after the packing and provided motion to lead to significant viscous losses. Further- good heat transfer. Due to the uncertainty in To we more, any radiation of sound through the leaks used cannot decide unambiguously on the value of the to fill the resonator will lead to an apparent attenua- critical exponent. tion. Before one can use the observed Q, of the resonator to get the attenuation of 4th sound, it is necessary as for large pores. One has to be careful in looking to account for the losses due to these causes. We are for a critical exponent determined in the region explo- not yet confident of our ability to do this. In the red by the fourth sound. In the domain 2.06 OK samples having high Q resonances at low temperature T To, the critical exponent could be easily ( N 100), we observed a continuous decrease of the Q varied by a factor of 2 due to the uncertainty in with temperature up to the vicinity of To. Within To [27]. A theoretical estimate of the domain of the last millidegrees, the Q, drops more sharply and the validity of the scaling laws should greatly help in this last determination of frequencies correspond to values case. These results, where it is found that the critical of Qr- 1 to 2. A similar behavior is seen on the exponent n increases when T0 decreases, have been amplitude measurements of the maxima of the confirmed and developed in recent experiments [28]. 4th sound resonances. In some cases this allowed a 1090

determination of To as from the low frequency signal. in the sound field decreases. We have not observed These curves are very similar to the critical flow rate such an increase in Q,when decreasing the A.C. voltage curves. We will discuss briefly the critical velocity of the drive transducer. - Vc - results and the possible connection between these results. IV. Discussion. - The first theoretical approach Direct measurements of Vc in coarse porous systems of Ginsburg and Pitaevski [11] followed the successful were recently presented by Reppy [20]. At low tem- Ginsburg-Landau phenomenological theory of super- perature, v,, is a constant and is larger for smaller pore conductivity. The properties of the superfluid- sizes. (One can get an estimate of v, from the empi- condensed-phase of He II are characterized by a rical relation V4 d - 1 cm5/s. For pores of 50 A, complex wave function similar to the ordinary wave v,, = 38 cm/s.) Near T,,, v, drops sharply as T function of quantum mechanics which obeys the increases. The domain in which v, varies is larger equation : for small powders. An interpretation of this behavior has been proposed by Langer and Fisher [30]. They describe the critical velocity from a fluctuation model where the fluctuations take the forma- place through This equation is obtained from an expansion of the tion of vortices. In the final result, the critical velo- gain of free energy of the system in the superfluid city appears to vary as Pg oc (T À - T) 2/3 in agree- phase, ð.Fs, and is valid is small. In the ment with the recent Clow and when I12 Reppy [21] experiment. original treatment, ) § 12 appears as the superfluid Our flow experiments are not inconsistent with fraction pg/p : those of Clow and Reppy. In the domain near To, the critical flow rate proportional to Ps v,, in the two fluid hydrodynamics picture, should decrease as p;. gives the "healing length" which represents the length In the case of the C II (1) sample, for example, in which the spatial variation of §(r) takes place. varies with and the pg linearly temperature ( fig. 10) Smaller scale variations are prevented from the form flow rate has a variation with r quadratic (fig. 6). of the first (kinetic energy) term. This would cause This as we do not is, however, only qualitative really a large increase in the free energy of the system. measure the critical velocity. Before going on with this equation let us recall its To discuss the 4th sound attenuation in these terms, limited applicability. The healing length is small we have to look for the velocities generated by the not too close from Tx ( N 2 A) because of the short to the in sound field. It is possible estimate velocity range of the interaction which is usually taken as a of the field the of the the main part from knowledge hard sphere interaction in a Bose gas [32]. This electro-acoustic properties of the cavity. Let us consi- allows large scale fluctuations of the order parameter der, for example, the velocity field in the case of and limits applicability of this local model. This is a C I (3) powder, where we measured, at 2.09 OK, in contrast with superconductors where such an the input voltage V, = 50 V and the pickup voltage approach has been very successful, § being large and V2 = 9.5 X 10-4 V. The pressure amplitude is of the order of 102 to 104 A. However, the G. P. ap- given by [31] :: proach has led to a reasonable physical insight to the problem of size effects in helium II. We will use it before going to a new model developed with K. Maki [33] where attention is paid to the crucial role of the fluctuations of the order The impedance is ZT = (wC) -1 where the capaci- parameter. tance of the transducer C = 150 l and s are the pf, A. FORM OF G. P. EQUATION. - We suppose : length and area of the cavity. A velocity amplitude, v, is associated with P by the relation : We express the two fundamental properties of superfluid He II. The superfluid density varies is In this particular example, with a Q, value of 50, one ) Tx - T 12/3. In the G. P. framework, the equili- gets v = .15 cm/s. Values of Q, of this order imply brium value : losses much larger than those predicted by the different calculations of fourth sound attenuation unless one exceeds the critical velocity. However the value of v calculated is much smaller than Vc and cannot explain the attenuation. We are left with the possibility that There is a finite in the the mechanism rise to the attenuation does not discontinuity specific giving heat take place in the central part of the sound field. This [34]. oc - should vary quadratically with T : does not seem to be the case because one would expect ð.Fs OC2/2p to get large increases in Q, as soon as the amplitude 1091

From (IV. 3) and (IV. 4) and the superfluid den- whose exact solution is the Jacobi double periodic sity [26] and specific heat results [34], one gets : function [38] :

This result was also obtained Mamaladze by [35]. This can be checked from the relations The healing length is given by (IV. 2) and varies easily between the Jacobi solutions [39]. Let us note that : as (TÀ - T)2/3. This agrees with the scaling law result of Josephson which says [26] that :

B. WAVE FUNCTION. - We make no EQUILIBRIUM As pointed out by Fulde and Moorman [37], the attempt do deal with a model which would accurately two boundary conditions can be written respectively as : reflect the complicated 4th sound medium. The spi- rit of this calculation is principally to give the qualitative features of the phenomenon. We use the simple model case of helium inside the volume limited by two parallel walls x = 0, x = d, which is also the situation for the helium film. We use the boundary K is the complete elliptic integral. condition on the walls, § = 0 [36]. The equa- The identification of (IV. 6) and (IV. 5) gives : tion (IV .1 ) can be written in reduced units as :

§ = 0 for ( = 0; d/d = 0 for ( = d/2ç; k is defined from the relation : ( = x/ is the reduced length. The equation has the form [37] : C. ONSET TEMPERATURE. - The highest temperature at which the equation has a nonzero solution is

FiG. 12. - Depression of the onset temperature as a function of the film thickness in the two phenomenological models [11, 35] based on the Landau theory. We indicate the distribution of onset temperature in powders as a function of the particle size (rectangles). 0 indicates the result for a sample of carbolac I where the average pore size has been actually measured. We give as a reference the onset temperature for unsaturated helium films. We have taken the data of Fokkens et al. (Physica, 1966, 32, 2129) and we have used the following expression for the number of layers of helium atoms : 0 = 172/2T Log [Ps(T)/PJ (McCormick et al., to be published). 1092

given by the largest value of §( T) and the smallest one We have assumed the - 2/3 power temperature de- of K (k2) (1 + k2)1/2. K(k2) is minimum for k = 0 and : pendence of §(T). The variation of k2 from 0 to 1 is associated with the following change. a) For k2 - 0 (T -> TO) the order parameter varies smoothly on a distance of the order of d (fig. 14, curve 6). In the limit k2 = 0, it is a sine wave. We have taken : This is characteristic of the properties of the linearized

using the value of A in sect. IV-A. We give in figure 12 a plot of the variation of To with d in the case of this modified model [35] and that of the original treatment of G. P. [11]. We see a reasonable quantitative agreement with data taken on unsaturated helium films. Let us note however that it is not possible to deduce unambiguously from these results the important 2/3 power dependence. We also indicate on this figure the measurements of onset temperatures in our powders as a function of particle size. We do not expect indeed any accurate agree- FiG. 14. - Solution of the nonlinearized G. P. ment with the models. The relevant parameter is equation (IV. 6) with the boundary condition § = 0 at the pore size which is, in particular, a function of the boundary of the film and = 0 at the center the If the was d§/dx packing pressure. porosity p equal of the film. Curves 1 to 6 correspond to the following to 0.5 and if the empty spaces were spherical, these values of k2 : 1 ; 1-10-5 ; 0.99 ; 0.8 ; 0.4 ; 0. two parameters (particle and pore size) would be equal. In the case of the C I powder where direct measurement of the average pore size has been made, equation for the non interacting system and is valid for it corresponds to half of the grain size. In this case T -* To (I 13 I I). the experimental onset temperature is found to be in b ) For k2 - 1 ( T T0), the wave function is good agreement with the two models. nearly constant over most of the width of the film and goes to zero rapidly near the boundary on a dis- D. TEMPERATURE DEPENDENCE OF THE ORDER PARA- tance ç(T) (fig. 14, [2]). This variation for the METER. - We have calculated the solution of strongly interacting system is in agreement with the work and the results on 13. Each eq. (IV. 9) give figure of Gross [40]. In this case, the nonlinear term curve is identified a that a by given To, is, by given d/ço. becomes important and the healing length is an important concept. Below To, one can calculate the amplitude of this wave function. Actually, we are inte- rested in an average of § on the width of the film measured by the 4th sound experiment (see the continuity equation (IV. 4)) :

The integral can be calculated using transforma- tions [39] of the elliptic functions [11]. Then one calculates I 12 using the result [26] :

The results are shown in figure 15 and show qualitatively the type of behavior observed from the 4th sound results close to To. The variation of § 12 (T) is linear near To and can be obtained from an expansion FIG. 13. - Solution of the equation (IV. 9). When k2 of (IV. 9) when k2 - 0 : goes from 0 to 1 the system goes from the weakly interacting state (T -*TO) to the interacting one (T To). 1093

result for k2 --* 1 only to get an indication of the low temperature behavior. E. SPECIFIC HEAT. - a ) A similar calculation giving the temperature variation of the specific heat can be performed along the same line. Fulde and Moor- The slope of the § 12(T) curve near To decreases man [37] studied in such a way the specific heat of as 1.27(T"A - TO)-l/3 when the film or the powder superconductors plated with magnetic materials, gets thinner. The experimental slopes of the norma- where the same boundary condition, § = 0, applies. They find that the discontinuity of the specific heat is reduced with respect to that of the bulk : this is due to the fact that right at To the solution of the linearized equation is a sine wave rather than § = constant as for the bulk. Moreover, the maximum of the specific heat is shifted at a temperature lower than To. The same calculation can be performed in our case. The value of the maximum of the specific heat should be in the range of fast variation of the curve of k2( T) (fig. 13). b) However, in order to correctly describe the specific heat one needs a model which describes the strong effect of the fluctuations on the specific heat giving rise to the logarithmic singularity in the bulk. In the case of a finite system, the effect of the singularity should disappear as the logarithmic singularity is related to the very long wavelength fluctuations FIG. 15. - Calculated temperature dependence of which cannot take place in such systems [33]. Psi P (T) near To. The variation is linear in the imme- A complete description would require taking into diate vicinity of T, when To Tx. account these two effects. We will present later a more appropriate model. However, the difficulty in lized superfluid density (taking ps/p (1.2 oK) = .98) the treatment of the specific heat in finite systems is related to the "Fulde and Moorman that are too small by a factor of 3 but qualitatively one Effect", is, observes the expected decrease of slope when To de- to the effect of the change of the form of the equilibrium solution when T varies. creases. We do not develop this comparison as the exact normalisation for the fourth sound - velocity F. NATURE OF THE TRANSITION NEAR T0. We are would be required. not able to decide unambiguously on experimental this model However, predicts that far from TO (kl - 1) grounds if the temperature variation near To is due the order is constant over the width parameter nearly to a progressive disappearance of the superfluidity of the film and one the bulk gets superfluid density. in pores of different sizes. However, this seems the of one By taking asymptotic expression K(k2), unlikely in our description of helium II as a condensed obtains in this limit : system spatially ordered on a distance ç( T). In the vicinity of To, ç( T) becomes of the order of the thickness of the pores. The probability of a sharp boundary between a normal domain and a superfluid one and when T --> 0 : is unlikely in the Landau framework we have used. To illustrate this, let us study a two dimensional model of a tube along Ox with a periodic succession of channels of equal lengths 2b and widths d1 and d2 (d1 d2) (fig. 16). a ) b x 3b : We consider the part of the tube For for = 2.1 the value of the example, To OK, of width d1 when is small we look for the solution superfluid density at T = 0 should be 10 % below of the linearized equation : that of the bulk. Our 4th sound results seem to indicate a lowering of the superfluid density below the bulk value larger than that (using an arbitrary scattering correction n = 2 - 0.5 = 1.22). Let us insist on the fact that in the limit whereI is large the validity of (IV .1 ) based on a free energy expansion is very questionable. We have indicated the 1094

From the continuity of the function and its first derivative for x = b :

The solution of this equation is obtained easily from a graphical solution and has a single solution for To : T01 To To2. In the limit T01 To2, To is given by :

The onset temperature is the same as for a single film of thickness d. Such a result does not depend indeed on the precise geometry used here. However it requires : FIG. 16. - Variation of the order parameter in a two dimensional periodic channel near the onset tempera- i) The change in pore size takes place over a length ture. [ §(x) varies sinusoidally in the domain where comparable to the width. ) §J decreases rapidly in the superfluid state is favored (domain 2) and expo- the unfavorable to the amplitude nentially in the domain where it would not be stable region superfluidity; without proximity (domain 1). associated with the superfluid state becomes exponen- tially small when b becomes large compared to a and the helium will look "normal like". The criterion we have the effect of the variation separated of § along b r- a should hold well for powders. Ox and the width of the tube : along ii ) A regularity in the pore size distribution along the length of the resonator : in our example the distribution is periodic. We believe that this is approxi- mated in the uniformly compressed powders. In the vanishes as it should to the Landau according general case of a sample inhomogeneous along its length, one at the onset if there picture, right Tol, temperature still expects sharp 4th sound resonances, as the 4th was an infinite length channel of width d1. sound wavelength of the first harmonics is of the order x b : In the same we have : b) b way, of the length of the resonator. However, the distribution of the eigenmodes should not be the same as for an ordinary cylindrical cavity.

where : - Acknowledgments. The work reported here was a collective effort involving more persons than those vanishes at To.. At a temperature T01 T T02, listed as authors. The experiments on flow rate were A1 is positive and A2 negative [41]. We look for done with B. K. Jones who is continuing with these a solution : experiments. We have presented fourth sound measurements which were obtained in collaboration with K. A. Shapiro, M. Kriss, S. A. Scott and J. Fraser. We have benefited from discussions with P. Pincus, § is normalized at x = 0. K. Maki, R. Kagiwada and M. Le Ray.

BIBLIOGRAPHIE

[1] At a temperature T, the thickness of the Rollin pressure. Reviews of these properties are given film 2014 dF 2014 in equilibrium with the vapor pres- by Brewer (Proceedings of the 10th Low Tempe- sure p, is an increasing function of p/ps(T) where rature Conference, Moscow) and Manchester Ps(T) is the saturated vapor pressure. dF goes (Rev. Mod. Phys., 1967, 39, 383). from zero to indefinitely large values when P/Ps(T) [5] BREWER (D. F.), Superfluid Helium, J. F. Allen from zero goes to 1. editor (Academic Press, 1966), p. 159. Adv. in [2] LONG (E. A.) and MEYER (L.), Physics, 1953, [6] BREWER (D. F.), CHAMPENEY (D. C.) and MENDEL- 2, 1. SOHN (K.), Cryogenics, 1960, 108. [3] FREDERIKSE P. Physica, 1949, 15, 860. (H. R.), [7] PELLAM (J. R.), Phys. Rev., 1948, 73, 608. [4] In the case of the thinnest films (1 or 2 atomic ATKINS 962. layers), the situation is further complicated as [8] (K. R.), Phys. Rev., 1965, 113, the Van der Waals forces modify the helium pro- [9] RUDNICK (I.) and SHAPIRO (K. A.), Phys. Rev. Let- perties near the solid boundaries. In particular, ters, 1962, 9, 191 ; SHAPIRO (K. A.) and RUD- the first layer behaves like solid helium under NICK (I.), Phys. Rev., 1965, 137 A, 1383. 1095

[10] RUDNICK (I.), GUYON (E.), SHAPIRO (K. A.) and [28] KRISS (M.) and RUDNICK (I.), Bull. Am. Phys. Soc., SCOTT (S.), Phys. Rev. Letters, 1967, 19, 488. 1968, 13, 505. [11] GINSBURG (V. L.) and PITAEVSKI (L. P.), Sov. Phys. [29] SANIKIDZE (I.), Sov. Phys. JETP, 1967, 24, 1045. JETP, 1958, 7, 858. SANIKIDZE (I.), ADAMENKO (I.) and KAGANOV, Sov. [12] For a review of the Ginsburg-Landau equation, Phys. JETP, 1967, 25, 383. see DE GENNES (P. G.), Superconductivity of [30] LANGER (J. S.) and FISHER (M. E.), Phys. Rev. Metals and Alloys (Benjamin, New York, 1966). Letters, 1967, 19, 560. FINCH KAGIWADA BARMATZ and [13] The definition of 03C1n, 03C1s and of the corresponding [31] (R. D.), (R.), (M.) RUDNICK 6 A 1425- velocities vn and vs can be seen as follows : in a (I.), Phys. Rev., 1964, 134, A, A 1428. Galilean frame moving at a velocity vn, the superflow is characterized by a velocity vs 2014 vn a mass [32] LEE (T. D.) and YANG (C. N.), Phys. Rev., 1958, current density j = 03C1s(vs 2014 vn) and a free energy 112, 1419. [33] Published as a second part of this paper. 2014 serves as a definition density 1/2 03C1s(v vn)2. This [34] See BUCKINGHAM (M. J.) and FAIRBANK (W. M.), in Low vol. for 03C1s : in the laboratory frame Progress Temperature Physics, III, D. J. Gorter ed. (North-Holland Co., j = 03C1vn + 03C1s(vs-vn) 03C1svs + 03C1nvn. Publishing Amsterdam), Chapter 3. It is well to keep in [14] DASH (J. G.) and TAYLOR (R. D.), Phys. Rev., 1957, mind the limitations in the model : JOSEPHSON 105, 1. (Phys. Rev. Letters, 1966, 21, 608) has pointed [15] The presence of the pressure and temperature terms out from scaling law arguments that the identifi- is a direct of the thermomechanical consequence cation 03C1s ~ |03C8|2 is generally not good. More- effect. For = 0 a variation is ~03BC, pressure over, we describe the discontinuity in specific associated with a corresponding ~T = - ~p/03C1s. heat at T03BB but not the logarithmic singularity. A of 1 mm He level difference (pressure head) [35] MAMALADZE (Yu. G.), Sov. Phys. JETP, 1967, 25, to ~ 50 10-6 oK at 1.5 °K. corresponds 479. Let us note however that the stability requi- ATKINS and CONDON [16] (K. R.), SEKI (H.) (E. V.), rement of the solution |03C8|= 0 at T = T03BB is Phys. Rev., 1956, 102, 582. not fulfilled with this choice of 03B2(T) . [17] This is nearly five times the stray capacitance due [36] This choice has not been justified microscopically. to the coaxial line to the electric circuit. going It can be deduced from an analogy ([12], p. 232) [18] EDELSON (B. N.), DYUMIN (N. E.), RUDAVSKII with the case of superconductors where a micro- (E. Ya.) and SERBIN (I. A.), JETP Letters, 1967, scopic description of the proximity effect is known 6, 243. This may explain some negative results (DE GENNES (P. G.) and GUYON (E.), Phys. Rev. of experiments where the 4th sound was looked Letters, 1963, 3, 168 ; DE GENNES (P. G.), Rev. for in too coarse samples. Mod. Phys., 1964, 36, 225 ; WERTHAMER (N. R.), [19] POLLACK (G.) and PELLAM (J.), Phys. Rev., 1965, Phys. Rev., 1963, 132, 2440). Not too close to To, 137, 1383. the exact choice of the boundary condition which [20] REPPY (J. D.), Phys. Rev. Letters, 1965, 14, 733. expresses the modification of 03C8 near the boundary [21] CLOW (J. R.) and REPPY (J. D.), Phys. Rev. Letters, is not very crucial (TSUNETO (T.), Progr. Th. 1967, 19, 291. Phys., 1964, 31, 330) because of the small value [22] BREWER (D. F.) and MENDELSOHN (K.), Proc. Roy. of the healing length. Soc., 1961, A 260. [37] FULDE (P.) and MOORMAN (W.), Phys. Con. Matt., [23] ATKINS (J. H.), Carbon, 1965, 3, 299, whom we wish 1967, 6, 403. to thank for information of the carbon powders’ [38] This solution corresponds to that written by Gins- properties. burg-Pitaevski under the form of a Legendre [24] KRISS (M.), private communication. normal integral which is the inverse function of ~. [25] BREWER (D. F.), LEPPELMEIER (G. W.), LIM (C. C.), [39] JAHNKE-EMMDE-LOSCH, Tafeln hoherer Funktionen, EDWARDS (D.O.) and LANDAU (J. D.), Phys. Rev. p. 75-79. Letters, 1967, 19, 491. [40] GROSS (E. P.), J. Math. Phys., 1963, 4, 195. [26] CLOW (J. R.) and REPPY (J. D.), Phys. Rev. Letters, [41] In a similar calculation, KIKNADZE et al. (JETP 1966, 16, 887. Lett., 1966, 3, 197) studied the problem of the TYSON (J. A.) and DOUGLAS (D. H.), Phys. Rev. horizontal boundary separating the superfluid Letters, 1966, 17, 472. above T03BB and the normal fluid (below it) below T03BB. [27] In determining critical exponents, very often one However they write in their calculation that A has freedom in adjusting the value of the critical vanishes in the normal phase. This is not consis- temperature with limits determined by experi- tent with the Landau model. Recently Guenter mental error. It is worth emphasizing that such Ahlers (to be published) has studied experimen- adjustments can have profound effects on the tally the effect of the gravitational field on the magnitude of the exponent (and vice versa) espe- superfluid transition in He II and found a sharp cially if the range of validity of it is unknown. horizontal transition layer. This is not inconsis- This remark may be applied to some recent tent, however, with the proximity effect descrip- angular momentum experiments of CLOW and tion. The predicted transition region between REPPY [21] in five alumina powders (average grain the superfluid He II (above) and the quasinormal size 2,000 Å). The value T0 = T03BB 20143/4,000 °K helium below should extend only about 10-3 cm, is a direct result from the assumption that the which was too small to be detected in this critical exponent for the 03C1s/03C1(T) law is still 2/3. experiment.