THE CURRENT STATE OF THE PROBLEM OF LIQUID HELIUM*)

J. G. Daunt and R. S. Smith

CONTENTS

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

1. Introduction. 1.1. Phase transition in liquid helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

2. Viscosity and “normal” density. 2.1. Viscosity of helium I. 2.2. Viscosity of helium II. 2.3. Determination of the density of the normal component. 2.4. The rotating-disk or cylinder method. 2.5. Flow of helium II through narrow slits. 2.6. Flow of helium II through wide slits. 2.7. The problem of viscous forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

3. Theoretical foundations of the two-component model. 3.1. General remarks. 3.2. London’s theory. 3.3. Landau’s theory. 3.4. Green’s theory. 3.5. Prigogine and Philippot’s theory. 3.6. Combining the Landau and London models according to Temperley. 3.7. Experimental data on the connection between the existence of the λ-transition and the type of statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

4. Thermomechanical effect and entropy of the superfluid and normal components. 4.1. Thermomechanical effect. 4.2. Thermomechanical effect and determination of entropy. 4.3. Heat-capacity measurements. Entropy of the normal and superfluid components. 4.4. Mechanocaloric effect. 4.5. Other calculations of the entropy of the superfluid and normal components. 4.6. Mechanocaloric effect as a method of cooling. 4.7. Theories of the thermomechanical effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

5. Heat transfer. 5.1. Thermal conductivity of helium I. 5.2. Heat transfer in helium II. 5.3. Heat transfer through narrow slits (experiment). 5.4. Heat transfer through narrow slits (theory). 5.5. Heat transfer through wide slits (experiment). 5.6. Heat transfer through wide slits (theory). 5.7. Wall layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

6. Surface films. 7. Rate of transfer of helium along a film. 8. Hydrodynamics of a two-component liquid. 9. First and second sound. Bibliography

) J. G. Daunt, R. S. Smith, Rev. of mod. phys., 26*, No. 2, 172—236 (1954).

J. G. Daunt and R. S. Smith

PREFACE

Twenty-seven years have already passed since the existence was established of two different modifications of liquid helium, which were named by W. Keesom “helium I” and “helium II.” Therefore it is now possible to attempt a review of the present state of the problem of liquid helium, especially in view of the fact that in recent years great attention has been devoted to this problem in both experimental and theoretical investigations. In our review only work on liquid helium II is considered, since superfluid helium II is of the greatest interest. Included in the review are only those properties of helium I which are essential for interpreting the properties of helium II. The review includes neither old nor new data on solid helium and on He\(^3\). The authors are aware that there are many works concerning liquid helium II which are not commented on in the review for the sake of brevity.

I. INTRODUCTION

1.1. Phase transition in liquid helium

The first indications that liquid helium passes at a temperature of \(2.18^\circ\text{K}\)*) into some other modification were obtained by Kamerlingh Onnes\(^{K.11}\)**) in 1911 in measurements of the density of the liquid. Kamerlingh Onnes\(^{K.24}\) found that the coefficient of thermal expansion changes sign at this temperature, i.e. the density has a maximum at \(2.18^\circ\text{K}\) (Fig. 1). However, convincing experimental proof of the existence of two phases of liquid helium was obtained by W. H. Keesom and his collaborators, who undertook measurements of the most diverse physical quantities, such as, for example, the dielectric constant\(^{K.28a,K.28b}\) and the heat of vaporization\(^{K.32c}\), which showed anomalous behavior at \(2.18^\circ\text{K}\). From this Keesom concluded that there exist two states of liquid helium which pass into one another at this temperature. He called the state of the liquid in the temperature interval between \(2.18^\circ\text{K}\) and the boiling point \((4.2^\circ\text{K})\) helium I, and the modification existing at temperatures below \(2.18^\circ\text{K}\), helium II\(^{K.28a,K.27}\).

*) On the presently accepted so-called “agreed scale,” the \(\lambda\)-point of liquid helium lies at \(2.186^\circ\text{K}\).

**) The bibliography will be placed in UFN, LVII, issue 0 with the second part of the review. In references to the literature, the initial letter of the author’s surname and the year of publication are indicated. In references to articles published in Russian, the first letter is given in italics. (Translator’s note.)

Most convincingly, the full depth of the changes taking place was shown by Keesom’s measurements of the heat capacity of liquid heliumK.32a, K.32b, K.33a, K.35 (the curve of the dependence of heat capacity on temperature, constructed from the experimental values, is shown

Fig. 1. Density of liquid helium in \(g/cm^3\) as a function of temperature according to Kamerlingh Onnes and BoksuK.24

Fig. 2. Heat capacity, \(cal/(g\cdot deg)\), of liquid helium. Circles — data of Keesom and ClusiusK.32a; squares and triangles — data of Keesom and KeesomK.32b.

in Fig. 2). It was precisely in connection with the form of this curve, resembling the letter \(\lambda\), that EhrenfestE.33 called the transition point the \(\lambda\)-point.

Subsequent work by Keesom $^{K.33a}$ showed that the transition from helium I to helium II occurs without latent heat and that the two modifications cannot be in equilibrium with one another. Detailed reviews of these questions were compiled by Keesom $^{K.42}$*), Barton, Grayson-Smith and Wilhelm $^{B.40}$, and Squire $^{S.53b}$.

2. VISCOSITY AND “NORMAL” DENSITY

2.1. Viscosity of helium I

The first measurements of the viscosity of liquid helium I were made by Wilhelm and co-workers $^{W.35}$ by the method of torsional oscillations of a cylinder immersed in the liquid. The results, however, may be open to doubt, as was pointed out by Kapitza $^{K.38a}$, because of turbulence which in all probability occurred during the measurements.

Later measurements by the method of torsional oscillations of a disk immersed in the liquid were carried out by Keesom and Macwood $^{K.38a}$, Keesom and Keesom $^{K.41a}$, and Smith $^{S.50a}$, and by de Troyer and co-workers $^{T.51a}$ in Leiden. The results obtained by the last-named authors are apparently the most reliable and are shown graphically in Fig. 3.

Measurements of the viscosity of helium I by the flow method (Poiseuille flow), i.e. determination of the velocity of flow of the liquid through capillaries, were made by Jackson and co-workers $^{J.39a}$, and also by Bowers and Mendelssohn $^{B.49a,\ B.50c}$. The results of these measurements are shown in the same Fig. 3, from which it is clear that the two methods give results in excellent agreement with one another.

The principal conclusions may be formulated as follows:

a) The viscosity $\eta$ of helium I in the temperature range from $2.8^\circ$ to $4.2^\circ\ \mathrm{K}$ is practically independent of temperature and has a value of about 30 micropoise;

b) On passing through the $\lambda$-point $(2.18^\circ\ \mathrm{K})$ as the temperature is lowered, no discontinuity in the value of $\eta$ is observed**). The value of $\eta$, however, decreases markedly on passing through the $\lambda$-point and, for example, at $1^\circ\ \mathrm{K}$ becomes twenty-five times smaller than at $4^\circ\ \mathrm{K}$.

For comparison with the data for the liquid, Fig. 3 also gives values of the viscosity of gaseous helium $^{J.38}$. It should be noted,

) Keesom, Helium*, IL, M., 1949; the Russian edition contains additions by E. M. Lifshitz and E. L. Andronikashvili.

**) In the earlier work of Keesom and co-workers $^{K.38a\ \text{and}\ K.41a}$ a discontinuity of $\eta$ at the $\lambda$-point is assumed. However, this was not confirmed by the data of later works.

that the viscosity of liquid helium I is no greater than the viscosity of gaseous helium at the same temperature, and that its value does not increase with decreasing temperature, as it does in ordinary liquids. The latter suggests that even in the liquid state helium retains many properties of gases, as was to be expected for a liquid of such low density. Here it is interesting to draw attention to the recently carried out measurements by Tjerkstra[^53a] of the viscosity of liquid helium I

Fig. 3. Viscosity of liquid helium in micropoise as a function of temperature. Circles are the data of Bowers and Mendelssohn[^49a] and B.[^50c], crosses are the data of de Troyer et al.[^51a]. The dashed curve is the viscosity of helium gas.

at elevated pressures. Tjerkstra worked with pressures from 5 to 50 kg/cm² and found that, as the pressure was increased, the viscosity approached that of a normal liquid, i.e. it increased with decreasing temperature. The greatest viscosity observed in this work was about \(120 \cdot 10^{-6}\) poise at a liquid density of \(0.186\ \text{g}/\text{cm}^3\).

2. 2. Viscosity of helium II

The results of measurements of the viscosity of liquid helium II by the oscillating-disk method are represented by the curve in Fig. 3. However, attempts to measure the viscosity of liquid helium II by the outflow method led to entirely anomalous results. Such experiments were

were first carried out by Allen and Misener A.38a and by Kapitza K.38a; they showed that the flow of liquid helium II through narrow channels is entirely unlike the flow of ordinary liquids, namely, that it is nonviscous.

It was established that the flow of liquid helium II through the finest capillaries (with diameters of the order of \(10^{-4}\)—\(10^{-5}\) cm) is almost independent of the pressure difference producing the flow, and is independent of the length of the capillary. Moreover, for the narrowest channels the volume flowing out is approximately proportional to the surface area, and not to the cross-sectional area (see Section 2.5). A review of these early works was given, for example, by Keesom K.42, by Jones J.35b, and by Darrow D.40. Later works are discussed below, in Section 2.5.

The most recent estimate of the upper limit of the viscosity of helium II from measurements of the rate of flow through narrow slits, made by Kapitza K.41a, K.41b, states that the viscosity is certainly less than \(10^{-11}\) poise. This value should be compared with the value of the viscosity of liquid helium I above the \(\lambda\)-point, equal to \(2\cdot 10^{-5}\) poise.

The data from flow measurements lead to the conclusion that there exists flow without any losses. This phenomenon, called by Kapitza K.38a “superfluidity,” is the principal characteristic of helium II.

The contradictory results of determinations of the viscosity of helium II, which led to values less than \(10^{-11}\) poise in flow measurements and greater than \(10^{-6}\) poise in measurements with an oscillating disk, were reconciled by Tisza T.38a, T.38b, T.38c, T.40. Tisza proposed a hypothesis according to which, at temperatures different from \(0^\circ\) K, the atoms participating in the “superfluid” motion constitute only part of the total number of atoms; the remaining part of the atoms produces viscous drag in experiments with an oscillating disk. Thus, to explain the experimental data one should assume that liquid helium II consists of two components, which may be called “superfluid” and “normal.” The total density of the liquid may be represented as the sum

\[ \rho=\rho_s+\rho_n, \tag{2.1} \]

where \(\rho_s\) and \(\rho_n\) are the densities of the superfluid and normal components, respectively.

However, in considering the two-fluid model one must exercise caution, since, on the one hand, the two components do not appear in the form of separate phases in space, and, on the other hand, helium II does not exhibit any noticeable spatial ordering in its structure that would distinguish it from helium I. The latter has been confirmed by X-ray investigations carried out by Keesom and Taconis K.35a and by Rikke R.40, R.47 and R.53a.

2. 3. Determination of the Density of the Normal Component

The hypothesis of the two-fluid model of helium II, advanced by Tisza T.38a, T.38b, T.38c, T.40*) to explain the problem of the viscosity of liquid helium II, had been implicitly assumed in an earlier theoretical work by F. London L.38a, L.38b, L.39a, who proposed that the

Figure labels: to the pump; elastic thread; mirror; window; liquid helium; disks.

Fig. 4. Diagram of Andronikashvili’s apparatus A.46 for measuring $\rho_n/\rho$ by the method of oscillations of a stack of plates.

$\lambda$-transition in helium II is analogous to condensation in a Bose–Einstein phase. The hypothesis posed the problem of experimental determination of the normal and superfluid densities. In 1946 Andronikashvili A.46 reported direct measurements of the relative density of the normal component, $\rho_n/\rho$. The experiment, based

*) For a critique of Tisza’s theory see Keesom, Helium, IL, 1949, Supplements by E. M. Lifshitz, pp. 426–429 (translator’s note).

on the assumption, put forward by Landau \(^{L.41a}\), consisted in measuring the moment of inertia of a stack of closely spaced aluminum disks (of diameter \(3.45\ \mathrm{cm}\), with a distance between disks of \(0.21\ \mathrm{mm}\)), suspended in a bath of liquid helium II, as shown in Fig. 4. The superfluid component does not affect the motion of the disks. However, the normal or viscous component will be partially carried along by the disks in their oscillatory motion. By measuring the change in the period of oscillation as a function of temperature, one can calculate the change

Fig. 5. Normal density \(\rho_n\) in \(\mathrm{g/cm^3}\). The solid curve is drawn according to the data of Andronikashvili \(^{A.48a}\); the points are the values of Hollis-Hallett \(^{H.52a}\).

in the total moment of inertia of the oscillating system and hence obtain the relative density of the normal component at the given temperature. For details of these calculations, see Hollis-Hallett \(^{H.52a}\).

Similar experiments were then carried out successively by Andronikashvili \(^{A.48a}\) and later by Hollis-Hallett \(^{H.50a;H.52a}\);

their results are shown in Fig. 5, from which it is seen that, within the limits of error, the results of both investigators agree well.

Approximately, the course of the variation of \(\rho_n/\rho\) with temperature can be expressed by the formula

\[ \frac{\rho_n}{\rho}=\left(\frac{T}{T_\lambda}\right)^\sigma, \tag{2.2} \]

where \(T_\lambda=2.18^\circ\mathrm{K}\) (the \(\lambda\)-point). Andronikashvili’s results show that a single value of \(\sigma\) cannot cover the whole temperature region. For \(1.3^\circ<T<1.7^\circ\mathrm{K}\), \(\sigma=6.8\); for \(1.8^\circ\mathrm{K}<T<2.18^\circ\mathrm{K}\), \(\sigma=5.3\). These results, therefore, may seem partly to contradict the assertions made earlier by London\(^{\mathrm{L}.47}\) and by Tisza\(^{\mathrm{L}.46a}\).

Fig. 6. Dependence of the concentration of the normal liquid \(\rho_n/\rho\) on \(T\); borrowed from the work of de Klerk, Hudson, and Pellam\(^{\mathrm{K}.53b}\): ● — data of de Klerk, Hudson, and Pellam\(^{\mathrm{K}.53a}\); ▲ — data of Peshkov\(^{\mathrm{P}.46b}\), Maurer\(^{\mathrm{M}.49d}\), and Gerlin\(^{\mathrm{G}.48c}\), obtained from second-sound measurements; ■ — data of Andronikashvili\(^{\mathrm{A}.46}\), obtained from measurements with an oscillating disk. To avoid confusion, the overlap of the three series of measurements is not shown.

and Tisza\(^{\mathrm{T}.47}\) regarding a single value of \(\sigma\), equal to 5.5. However, in measurements at low temperatures the percentage content of the normal component is small and, consequently, the possible relative error of the measurements is greater than at high temperatures. In the region of lower temperatures the value of \(\rho_n/\rho\) is determined more accurately from experiments with second sound. Values of \(\rho_n/\rho\) may be

computed from the measured velocity of second sound (see § 9), as was done, for example, by Peshkov^P.46b, P.48c, Bendt and Meyer^B.48a, and recently by de Klerk, Hudson, and Pellam^K.53b. These values agree very well with the results obtained by the direct method described above.

An additional indirect method for determining \(\rho_n\) is provided by experiments with a Rayleigh disk in second sound, as reported by Pellam and co-workers^P.50a and P.52b (see also Section 9.6).

These measurements make it possible to determine the quantity \(\rho_n/S\) at low temperatures (about \(1.3^\circ—1.6^\circ\mathrm{K}\)), where \(S\) is the entropy of liquid helium II. Using the entropy values obtained earlier by Kapitza^K.41a, Pellam and Hudson found good agreement between their values of \(\rho_n\) and the values of \(\rho_n\) found by Andronikashvili^A.48a. Recent measurements of the velocity of second sound below \(1^\circ\mathrm{K}\), carried out by de Klerk, Hudson, and Pellam^K.53a, make it possible to calculate \(\rho_n/\rho\) in this temperature range. The values obtained by de Klerk, Hudson, and Pellam^K.53b with the aid of heat-capacity data found by Kramers^K.52a are shown graphically in Fig. 6. If the interpolation formula (2.2) is retained, it is seen that for \(0.7^\circ<T<1.5^\circ\) \(\sigma\) varies from the highest value 13 to 6, whereas for the temperature interval \(0.3^\circ<T<0.6^\circ\mathrm{K}\) \(\sigma=4\). It follows from this that, even if the region below \(0.6^\circ\mathrm{K}\) is excluded, a formula of the form (2.2) is quite unsuitable (see Sections 4.7 and 9.4).

2.4. Method of the Oscillating Disk or Cylinder

Using the density values of the normal component \(\rho_n\) and carrying out measurements by the method of the oscillating disk, one can draw further conclusions concerning the viscosity of the normal component of helium II. These measurements make it possible to determine only the product of viscosity by density. Both Tisza^T.47 and Landau^L.41a pointed out that this method determines not the quantity \(\eta\rho\), where \(\rho\) is the total density of the liquid, as was assumed in obtaining the viscosity values shown in Fig. 3, but the quantity \(\eta_n\rho_n\), where \(\eta_n\) is the viscosity of the normal component.

The values of \(\eta_n\) in helium II obtained in the manner described from the most recent and reliable measurements by the oscillating-disk method of Andronikashvili^A.48b, de Troyer^T.51a, and Hollis-Hallett^H.52a are given in Fig. 7. In the calculations performed by the last author, the values of \(\rho_n\) were taken from Peshkov’s measurements of second sound. It is evident from the figure that there is good agreement among the results of different authors.

Recently Hollis-Hallett (see the review by Atkins^A.52a *)) carried out measurements of the “normal” viscosity \(\eta_n\) of helium II by the method

) For a fuller report see Proc. Camb. Phil. Soc. 49*, 717 (1953).

of the rotating cylinder; in this case the torque acting on the inner cylinder was measured as a function of the rotational speed of the outer concentric cylinder. With such an arrangement the parameter $\eta_n$ was measured directly instead of the parameter $\eta_n \rho_n$,

Fig. 7. “Normal” viscosity $\eta_n$ in helium II. Circles—data of Hollis-Hallett$^{\mathrm{H.\,52a}}$, squares—Andronikashvili$^{\mathrm{A.\,48a}}$, crosses—de Troyer et al. The dotted curve represents Hollis-Hallett’s results obtained with a viscometer with a rotating cylinder, as given in the review by Atkins$^{\mathrm{A.\,52a}}$.

obtained in a viscometer with an oscillating disk. Unfortunately, the observed torque acting on the inner cylinder was not directly proportional to the rotational speed of the outer cylinder, as would be expected for a normal liquid. However, by extrapolating the results to zero rotational speed, one can make a preliminary estimate of $\eta_n$. The results of such an estimate

are shown by the dashed line in Fig. 7, from which it is seen that the values of \(\eta_n\) obtained in this way agree well with the values of \(\eta_n\) obtained by the oscillating-disk method for temperatures from \(T_\lambda\) down to approximately \(1.6^\circ\) K. Below \(1.6^\circ\) the two series of results diverge: the values found by the rotating-cylinder method are smaller. Such a discrepancy at low temperatures may reflect the unreliability of the values of \(\rho_n\) taken from second-sound data and used for calculating \(\eta_n\) from the oscillating-disk experiments.

It should be noted that the value of the viscosity \(\eta_n\) found by the rotating-cylinder method changes little with temperature in comparison with the values obtained with the oscillating disk and shown in Fig. 3. As is seen from Fig. 7, \(\eta_n\) changes by no more than a factor of 2 between \(T_\lambda\) and \(1.3^\circ\) K. Moreover, from the data of both methods of observation (both the disk method and the cylinder method) it follows that at low temperatures \(\eta_n\) begins to exhibit a negative temperature coefficient of \(\eta_n\), analogous to the coefficients of normal liquids (cf. liquid hydrogen \(K.38b\)), whereas for helium I

\[ \frac{\partial \eta}{\partial T} \]

is positive. This negative temperature coefficient of \(\eta_n\) below \(1.7^\circ\) K is opposite to what would be expected for the behavior of a “gas-like” normal component.

A theory explaining the increase of viscosity with decreasing temperature below \(1.7^\circ\) K was proposed by Landau and Khalatnikov \(L.49a\). The theory is based chiefly on the assumption of a rapid decrease in the number of scattering centers (rotons) as the temperature is lowered. A fuller discussion of this question is given in Section 9.1.

2.5. Flow of helium II through narrow slits

The superfluid motion of liquid helium II through very narrow channels was first observed by Kapitza \(K.38a\) and by Allen and Misener \(A.38a\). Although the early works have already been considered in reviews \(K.42, S.39b\) and \(D.40\), here a brief account is given of these fundamental results, which are necessary for a complete discussion of the phenomenon of helium II flow.

Allen and Misener studied in detail the properties of superfluid motion \(A.38b\) and \(A.39a\); other authors—namely Kapitza \(K.41a\), Daunt, Stout, and Barió \(G.38, G.39\), and Jones, Wilhelm, and Grayson-Smith \(J.39a\)—also reported their measurements of flow through narrow capillaries. Of interest is the method by which Allen and Misener made their thinnest capillaries; it was widely used in other experiments (see Brown and Mendelssohn \(B.47a\)). The authors inserted a bundle of fine stainless-steel wires (about 1000 wires

diameter about \(6\cdot 10^{-2}\) cm) into a tube of new-silver and drew the tube through a successive series of steel dies. In this way capillaries with a width of the order of \(10^{-5}\) cm were made.

The main results obtained by Allen and Misener with the best capillaries may be formulated as follows*):

a) The volume rate of flow at temperatures below \(2^\circ\) K became independent of the pressure drop;

b) the rate of outflow was not proportional to the square of the capillary radius, as would be expected for an ordinary viscous liquid; instead, the rate increased proportionally to the decrease in the width of the capillary, from which it could be concluded that the flow is, in the main, a surface effect;

c) the volume rate was not inversely proportional to the length of the capillary, as would be expected from ordinary hydrodynamics, but tended, with increasing length, to a limiting value independent of the capillary length;

d) the flow velocity (defined as the volume flowing out per second divided by the cross section of the capillary) depended on temperature, as shown in Fig. 8. This temperature dependence is very close both in magnitude and in form to the analogous dependence for the flow of a surface film (see Section 7.1). With such a dependence (or independence) on the various mentioned—

Fig. 8. Critical (or maximum) flow velocity of helium II in cm/sec through narrow capillaries at a pressure drop of \(160\) dyn/cm\(^2\), according to measurements by Allen and Misener\(^{A.39a}\). The upper curve is for a capillary of diameter \(1.2\cdot 10^{-5}\) cm; the lower, for a capillary of diameter \(7.9\cdot 10^{-5}\) cm.

*) Allen and Misener\(^{A.39a}\) also carried out experiments on the flow of helium II through tubes filled with pressed powder with particles about \(10^{-5}\) cm in size. They found a noticeable dependence of the flowing volume on the pressure drop at all temperatures, whence it could be concluded that the flow of helium II through powder differs from the flow through capillaries. Similar conclusions were drawn by Bowers, Chandrasekhar, and Mendelssohn\(^{B.50b}\) and \(^{C.53a}\) from experiments with powder, and by Bowers and White\(^{B.51b}\) and White\(^{W.51a}\) from experiments on flow through porous membranes.

From the parameters given above it is clear that the equations of ordinary hydrodynamics are inapplicable, and that in general one cannot determine the viscosity of helium II in the usual way from experiments on flow through narrow capillaries. For flow through capillaries of intermediate dimensions the results, as might have been expected, revealed a complex combination of viscous hydrodynamic flow with the “superfluid” flow characteristic of narrow capillaries. Therefore their quantitative interpretation is very difficult (see Section 8.1). Moreover, as noted in Section 4.1, it is now known that the superfluid flow depends strongly on small temperature differences established at the ends of the capillaries. Consequently, in the absence of direct observation of the exact temperature values in the experiments of Allen and Misener, a complete quantitative interpretation is impossible, although, by supposition, these experiments were quasi-isothermal. The noticeable similarity of the results obtained with the finest capillaries and with flow through a surface film (see Section 7.1) led Allen and Misener to the conclusion that two flows exist: first, a surface superfluid flow, which may be identified with the flow through a surface film, and, second, a bulk flow which possibly obeys ordinary hydrodynamics. As the width of the capillary decreases, the surface or superfluid flow will begin to predominate, and as a result a flow with a limiting velocity \(v_{\mathrm{cr}}\) will be established, which is a function of temperature (see Fig. 8).

The question of the magnitude of this critical velocity for superfluid flow \(v_{\mathrm{cr}}\) in narrow capillaries, and of its dependence on the width of the capillary, has been studied in detail by a number of experimenters; but, as will be seen, the results obtained are to some extent contradictory, and at present it is impossible to give an exact picture of the phenomenon. It is interesting to compare the values of \(v_{\mathrm{cr}}\) obtained by various authors at the lowest temperatures, where, as follows, for example, from Fig. 8, \(v_{\mathrm{cr}}\) tends to a limit independent of temperature. A large part of the known results is presented in Table I. It follows from the table that agreement among the data of different authors exists only as to order of magnitude.

It was assumed, as is discussed in detail in Section 7.5, that for superfluid flow the following relation holds between \(v_{\mathrm{cr}}\) and the capillary diameter \(d\):

\[ m \cdot v_{\mathrm{cr}} \cdot d \simeq h, \tag{2.3} \]

where \(m\) is the mass of the helium atom. Numerically this gives

\[ v_{\mathrm{cr}} \cdot d \simeq 10^{-4}\ \mathrm{cm}^{2}/\mathrm{sec}. \tag{2.4} \]

In the case of a surface film of helium this relation has been confirmed experimentally (see Section 7.5 and the last line of Table I). However, for superfluid motion through very narrow capillaries,

as is seen from the table, the observed values of the products \(v_{\mathrm{cr}}\cdot d\) are not independent of \(d\), as equation (2.3) requires. Moreover, in all cases, except for the observations of Allen and Misener with the thinnest capillaries \((d=1.2\times 10^{-5}\ \mathrm{cm})\), this product is numerically much larger than follows from equation (2.4). Recent

Table I

Critical velocities of superfluid motion through narrow capillaries

Temperature °K	\(d\times 10^5\) (cm)	\(v_{\mathrm{cr}}\) (cm/sec)	\(v_{\mathrm{cr}}\cdot d\times 10^4\) (cm²/sec)	\(v_{\mathrm{cr}}\cdot d^{1/2}\times 10^3\) (CGS units)	Literature references
1.2	1.2	13	1.6	45	A. 39a
1.2	7.9	8	6.3	71	A. 39a
1.54	3	40	12	220	K. 41b
1.54	30	14	42	240	K. 41b
1.58	2	23	4.6	105	M. 47a
1.52	3	20	6	110	M. 47a
1.66	10	15	15	150	M. 47a
1.35	12	25	30	275	B. 52

Temperature °K	\(d\times 10^5\) (cm)	\(v_{\mathrm{cr}}\) (cm/sec)	\(v_{\mathrm{cr}}\cdot d\times 10^4\) (cm²/sec)	\(v_{\mathrm{cr}}\cdot d^{1/2}\times 10^3\) (CGS units)	Literature references
Surface film	Surface film	Surface film	Surface film	Surface film	Surface film
1.5	0.2	40	0.8	57	see Sec. 7.4

experiments of Bowers, Brewer, and Mendelssohn B. 51a with unsaturated surface films also led the authors to the conclusion that \(v_{\mathrm{cr}}\cdot d\) is not constant. In the absence of more accurate and definite experiments on this question, equation (2.3) can be applied to superfluid motion of a liquid through narrow capillaries only with substantial reservations.

Mott M. 49b proposed another mechanism for calculating the maximum mean velocity of superfluid motion and found that \(v_{\mathrm{cr}}\) is proportional to \(d^{-1/2}\), and not to \(d^{-1}\) as in equation (2.3). Mott’s model assumes the existence of a boundary between the moving superfluid part and the superfluid part remaining at rest; the surface tension at this boundary will balance the Bernoulli pressure difference in these parts and, consequently, counteract the growth of the resting volume at the expense of the volume in motion. The critical or maximum velocity of motion will occur when the surface-tension forces can keep the boundary in equilibrium (for further discussion see Sec. 2.6). A detailed description of the method for measuring \(v_{\mathrm{cr}}\), used by Kapitsa K. 41a and by Meyer and Mellink M. 47a, will be given in Sec. 4.2.

A series of experiments on the flow of helium II through narrow capillaries was carried out comparatively recently by Mendelssohn and coworkers (Bowers and Mendelssohn B.50a, B.52, Bowers, Chandrasekhar and Mendelssohn B.50b, Bowers and Wyatt B.51b), with the aim of investigating the pressure gradients existing inside the capillaries themselves. Although the results obtained are complicated, especially when the capillaries were made by packing with powder, their interpretation raised interesting problems. The principle of the experimental method is shown schematically in Fig. 9, which depicts three reservoirs \(A\), \(B\), and \(C\), connected by narrow capillaries \(P\) and \(Q\). \(A\), \(B\), and \(C\) are placed in one and the same bath of liquid helium and, consequently, are at one and the same temperature. Owing to the fact that the level in \(A\) is higher than in \(C\), a superfluid flow takes place from \(A\) to \(C\) through \(B\).

Fig. 9. Flow scheme in the experiments of Mendelssohn and coworkers B.50a, B.50b and C.53a.

By selecting the geometry of the system one can arrange that, in the steady state, the critical velocity of motion will be reached in capillary \(P\), whereas the velocity \(v_Q\) in capillary \(Q\) will be less than \(v_{\mathrm{cr}}\), or conversely. It was found that if, for example, \(v_P < v_{\mathrm{cr}}\) and \(v_Q = v_{\mathrm{cr}}\), then in the steady state the liquid in vessels \(A\) and \(B\) establishes itself at one and the same level, and there exists a difference of levels between \(B\) and \(C\). This means that the superfluid flow from \(A\) to \(B\), at a velocity smaller than the maximum critical velocity, occurs without the presence of any pressure difference at the ends of the connecting capillary, and, moreover, the entire pressure jump is concentrated in the capillary with the greatest “resistance” to the flow. The experiments leading to these results were performed with capillaries of width about \(10^{-4}\) cm, made either by fusing glass surfaces or in porous membranes. Other experiments B.50b, C.53a of a similar type, using capillaries filled with powder, revealed a more complicated picture. These results require considerable caution in considering the pressure drop in narrow capillaries and suggest that, possibly, the pressure drop exists only at the entrance to or exit from the capillary. A brief discussion of possible consequences was given by Meyer and Band M.48a. Moreover, since superfluid flow must be accompanied by a heat flow (see Section 4), it turns out that a complete interpretation of the phenomenon requires further experiments to study both the temperature distribution and the pressure distribution.

To conclude the consideration of the motion of helium II through narrow capillaries, it seems appropriate to make here several remarks concerning the frictional forces considered earlier by Atkins A.^50b. From the facts listed above, namely that for (a) the narrowest capillaries the maximum or “critical” (mean) velocity of motion does not depend on the pressure difference, and (b) for velocities of motion smaller than \(v_{\mathrm{cr}}\) the pressure difference is zero or, at least, very small, one may conclude that the frictional forces opposing the motion are equal to zero or close to it for all velocities smaller than \(v_{\mathrm{cr}}\), and that for velocities greater than \(v_{\mathrm{cr}}\) the frictional forces are unusually large. Such a change of the frictional force with velocity is shown in Fig. 10 by curve \(A\). Almost all data concerning the properties of surface-film transport also indicate (see Sec. 7) that the frictional forces opposing the motion of the surface film have the same character as those represented by curve \(A\). There is, however, some doubt about the form of the curve for velocities somewhat below the critical velocity. Thus, for example, Mendelssohn and White M.^50a and Atkins A.^50b found that when helium flows out along the surface film from a test tube, the flow velocity begins to decrease continuously in comparison with the critical velocity when the pressure difference decreases from values of about 3 mm of liquid helium to zero, whereas for pressure differences \(\Delta p\) greater than 3 mm the flow velocity remains constant, i.e., independent of any further increase of \(\Delta p\). The observed smoothed dependence of \(\Delta p\) on the velocity as \(\Delta p \to 0\) could be explained by frictional forces represented in Fig. 10 by the dashed portion of curve \(B\). Moreover, because of the observed similarity between flow through narrow capillaries and flow along surface films, it may be assumed that the change of the friction curve from \(B\) to \(A\) (Fig. 10) could also apply to the flow of a bulk liquid through narrow capillaries.

Fig. 10. Dependence of the frictional force on velocity (according to Atkins A.^50b). Curve \(A\) — simplified theory of the critical velocity. Curve \(B\) — modified theory of the critical velocity.

Another explanation for the decrease of velocity with decreasing \(\Delta p\) can be found in the energy balance. For free fall, for example, from a height of 3 mm or less, a velocity exceeding 24 cm/sec cannot be obtained, and the curve of the dependence of velocity on \(\Delta p\) must be a parabola for all velocities smaller than \(v_{\mathrm{cr}}\). This explanation, apparently, was not confirmed by recent observations—

tions of Pikes R.53a, who investigated the motion of a film at small pressure differences, or with the experiments of Mendelssohn and co-workersB.50a, B.52, B.50b, B.51b, described above. Obviously, as was emphasized earlier, further experiments are necessary in order to draw final conclusions on these questions of superfluidity.

2.6. Flow of Helium II through Wide Slits

The flow of helium II through narrow capillaries and along surface films is characterized by a critical velocity and an almost complete independence of the velocity from the pressure difference. This may be interpreted as an indication that, at velocities equal to \(v_{\mathrm{cr}}\), frictional forces come into play which make a further increase of the velocity impossible.

On the other hand, the flow of helium II through wide capillaries (\(d \geq 10^{-3}\ \mathrm{cm}\)) does not reveal the presence of any “critical velocity” (if a critical velocity does exist, it must be very small), and the frictional forces are not so large as to hinder the continuous increase of the flow velocity with increasing pressure difference. Thus flow through wide capillaries does not exhibit the complete superfluidity that is characteristic of flow through narrow capillaries. Consequently, the study of flow in wide capillaries is of interest for estimating the nature of the frictional forces acting in the nonsuperfluid part of the helium II stream (for more detail see Section 8.1).

Detailed experimental studies of the flow of helium II through wide capillaries were carried out several years ago by Allen and MisenerA.39a and by Jones, Wilhelm, and Grayson-SmithJ.39a. Recently this question has again been considered in the light of the latest theoretical ideas in the works of AtkinsA.51a and of Hsang, Hsant, and WinklerH.52c. All except the last authors used essentially one and the same experimental method, namely observation of the isothermal motion of liquid helium II from a cylindrical vessel of known internal diameter through a capillary attached to the bottom of the reservoir. With such a method corrections must be introduced for such secondary effects as the outflow of helium from the reservoir along the surface film and the existence of possible temperature differences between the inner and outer reservoirs. The reader will find details of these corrections in the original papers.

To illustrate the noticeable difference in the character of the flow through wide and narrow capillaries, typical results obtained by AtkinsA.51a at \(1.22^\circ\ \mathrm{K}\) for cylindrical capillaries of different diameters are shown in Figs. 11 and 12. In these graphs the observed flow velocity is plotted as a function of the pressure difference \(\Delta p\). First, although certain assumptions can be made regarding \(v_{\mathrm{cr}}\), it is evident that one cannot conclude with certainty that all

Fig. 11. Average flow velocity as a function of pressure drop according to Atkins\(^{A.51a}\). Temperature \(1.22^\circ\mathrm{K}\).

Plot axes: vertical — average velocity \(\bar V\) \((\mathrm{cm}\,\mathrm{sec}^{-1})\); horizontal — \(\operatorname{grad} p\) \((\mathrm{dyne}\,\mathrm{cm}^{-3})\). Curves: \(I\), \(II\), \(III\), \(IV\).

Capillary	Diameter	Length
\(I\)	\(4.40 \times 10^{-2}\ \mathrm{cm}\)	\(8.04\ \mathrm{cm}\) \((+)\), \(1.75\ \mathrm{cm}\) \((\square)\)
\(II\)	\(2.03 \times 10^{-2}\ \mathrm{cm}\)	\(7.90\ \mathrm{cm}\)
\(III\)	\(0.815 \times 10^{-2}\ \mathrm{cm}\)	\(8.03\ \mathrm{cm}\) \((\bigcirc)\), \(0.83\ \mathrm{cm}\) \((\triangle)\)
\(IV\)	\(0.262 \times 10^{-2}\ \mathrm{cm}\)	\(7.76\ \mathrm{cm}\)

Fig. 12. Average flow velocity as a function of pressure drop according to Atkins\(^{A.51a}\). Temperature \(1.22^\circ\mathrm{K}\).

Plot axes: vertical — average velocity \(\bar V\) \((\mathrm{cm}\,\mathrm{sec}^{-1})\); horizontal — \(\operatorname{grad} p\) \((\mathrm{dyne}\,\mathrm{cm}^{-3})\). Curves: \(I\), \(II\), \(III\), \(IV\).

Capillary	Diameter	Length
\(I\)	\(4.40 \times 10^{-2}\ \mathrm{cm}\)	\(48.6\ \mathrm{cm}\)
\(II\)	\(2.03 \times 10^{-2}\ \mathrm{cm}\)	\(46.6\ \mathrm{cm}\)
\(III\)	\(0.815 \times 10^{-2}\ \mathrm{cm}\)	\(8.03\ \mathrm{cm}\)
\(IV\)	\(0.262 \times 10^{-2}\ \mathrm{cm}\)	\(7.76\ \mathrm{cm}\)

the curves intersect the velocity axis \(\Delta p = 0\) at finite distances from zero, as they should if a “critical” velocity exists for inviscid flow. Secondly, with an increase in the pressure difference the velocity increases continuously, showing not even a hint of “saturation,” as in the case of motion through narrow capillaries. The slope of the curves, however, is in complete disagreement with the slope expected for normal hydrodynamic flow, which indicates the complex character of the frictional forces involved here. It turns out that, in order to explain the results over the entire temperature range, some combination is necessary of a term containing the critical velocity and other dissipative terms. This conclusion was drawn by Jones \(^{J.39b}\) in his detailed analysis of the results of Allen and Misener \(^{A.39a}\).

Atkins \(^{A.51a}\) attempted to interpret his results by means of the equation

\[ v = v_1(d) + v_2(\operatorname{grad} p) + v_3(d,\operatorname{grad} p) \tag{2.5} \]

and found that

\[ v_3 = a(d)\operatorname{grad} p. \tag{2.6} \]

However, this leads to the conclusion that \(a(d)\) has a minimum at \(d = 8\cdot 10^{-3}\), which is surprising and seems somewhat arbitrary if one recalls that only four different capillaries were measured. This difficulty can be eliminated only by discarding the data for the capillary of smallest diameter.

It is customary to try to describe these complex frictional forces by means of a two-component model. Then, for the case of stationary isothermal flow, the accelerations of the superfluid and normal components are zero, and one may write, in general form (see Section 8)\(^*\):

\[ -\frac{\rho_s}{\rho}\operatorname{grad} p = F_r + F_s, \]

\[ -\frac{\rho_n}{\rho}\operatorname{grad} p = -F_r + F_n - \eta_n\nabla^2 V_n, \tag{2.7} \]

where \(F_r\) is the mutual-friction force caused by the relative motion of the superfluid and normal liquids: \(F_r = f(v_s - v_n)\); \(F_s\) and \(F_n\) are indeterminate frictional forces depending respectively only on the superfluid velocity \(v_s\) and the normal velocity \(v_n\); the third term on the right-hand side of the second relation (2.7) describes the ordinary effect of viscosity.

To describe the character of the forces \(F_r\), \(F_s\), and \(F_n\), various assumptions have been put forward, some of which are outlined below, in Section 8. The most widely investigated was the assumption of Gorter and Mellink \(^{G.49a}\), who, in order to explain the data on the thermal conductivity of helium II (see Section 5), supposed that domi-

\(^*\) On this question see also the recent review by Atkins \(^{A.52a}\).

... member is \(F_r\)—the force of mutual friction—and that \(F_r\) is proportional to the third power of the relative velocity \((\mathbf{v}_n-\mathbf{v}_s)\).

According to Gorter and Mellink, for isothermal steady flow under the action of a pressure gradient:

\[ \frac{\rho_s}{\rho}\operatorname{grad} p = A\rho_s\rho_n(\mathbf{v}_n-\mathbf{v}_s)^3, \]

\[ \frac{\rho_n}{\rho}\operatorname{grad} p = -A\rho_s\rho_n(\mathbf{v}_n-\mathbf{v}_s)^3 + \eta_n\nabla^2\mathbf{v}_n, \tag{2.8} \]

where \(A\) is a constant characterizing the mutual friction. As was shown by Atkins\(^{\mathrm{A}.51a}\), these equations lead to the following expression for the mean flow velocity \(\bar v\) through a cylindrical capillary of radius \(r\):

\[ \bar v = \frac{\rho_s}{\rho} \left( \frac{-\operatorname{grad} p}{A\rho\rho_n} \right)^{1/3} + \frac{d^2(-\operatorname{grad} p)}{32\eta_n}. \tag{2.9} \]

Comparing his experimental results with the theory of the motion of helium II through wide capillaries, Atkins\(^{\mathrm{A}.51a}\) established that the observed changes of \(v\) with \(\operatorname{grad} p\) occur in accordance with equation (2.9); this confirms the hypothesis that the mutual-friction force is proportional to the third power of \((\mathbf{v}_s-\mathbf{v}_n)\). Assuming that in capillary III (see the caption to Fig. 11) \(v_3\) (see equation (2.5)) is negligibly small, Atkins estimated the Gorter–Mellink constant and found approximate agreement with the values of Gorter and Mellink. Moreover, the coefficient \(A\) proved to be constant in order of magnitude. However, the observed dependence of \(v\) on \(d\) did not coincide with the theoretical one. Later measurements by Hsang, Hanta, and Winkler\(^{\mathrm{H}.52c}\) (briefly described in Section 5) of flow through wide capillaries also showed that at sufficiently large relative velocities \(F_r\sim(\mathbf{v}_s-\mathbf{v}_n)^3\). However, these new measurements also showed that \(A\) (see equation 2.8) is not constant, but depends strongly on the width of the slit, especially at low temperatures \((T\sim 1^\circ \mathrm{K})\).

Atkins suggested that in flow through wide slits turbulence should be taken into account; however, he recalled that if one accepts the basic postulate of Landau and London, namely that \(\operatorname{rot}\mathbf{v}_s=0\) (see Section 3), turbulence apparently should not arise, since turbulence in a superfluid liquid is altogether impossible and hardly exists in a normal liquid for the observed Reynolds numbers (for a normal liquid

\[ R=\frac{1}{2}\,\bar v d\rho_n/\eta_n\leq 750 \]
).

However, several years ago Jones\(^{\mathrm{J}.39b}\) indicated how the results of Allen and Misener on flow in wide capillaries can be qualitatively interpreted by means of the assumption of the existence of turbulent flow in some capillaries. In our opinion, the inclusion of turbulence also ...

among the postulated or at present unknown mechanisms is needed to explain the behavior of helium II in wide capillaries, for it seems very characteristic that, for their widest capillaries, Allen and Misener found that the flow velocity falls as the temperature is lowered from the \(\lambda\)-point, whereas for laminar flow the second term in equation (2.9) predicts an increase in velocity.

Moreover, no choice of the temperature dependence of the Gorter—Mellink constant \(A\) can compensate the behavior of the laminar-flow term in equation (2.9).

In order to show the possibility of turbulence, we shall compare the data of Allen and Misener and of Atkins at \(1.2^\circ\mathrm{K}\) with the semi-empirical formula

\[ v=\frac{a}{d^{1/2}}+bd^{1/2}(\operatorname{grad}p)^{1/2}, \tag{2.10} \]

where \(a\) and \(b\) are constants. The first term in this equation corresponds to Mott’s expression\(^{\mathrm{M}.49\mathrm{b}}\) for the critical velocity (see (2.5)); the second term gives an approximate relation between velocity, diameter, and pressure for classical turbulent flow.

The constant \(a\) in equation (2.10) was determined, first, from the experiments of Allen and Misener on flow through three tubes filled with wire (it was assumed that in such narrow channels the flow does not depend on pressure at \(1.2^\circ\mathrm{K}\)) and, second, from Atkins’s results, shown in Figs. 11 and 12, under certain reasonable assumptions concerning the critical velocities.

The data used are presented in Table II, and they give the value \(a=0.07\) CGS units. It is also clear from Table II that \(v_{\mathrm{cr}}d^{1/2}\) is closer to a constant than \(v_{\mathrm{cr}}d\), which is not so noticeable in Table I.

Table II

Capillary	Diameter \(d,\ \mathrm{cm}\)	\(v_{\mathrm{cr}}\) at \(\Delta p=0\) \((\mathrm{cm/sec})\)	\(v_{\mathrm{cr}}\cdot d^{1/2}\) (CGS units)	\(v_{\mathrm{cr}}\cdot d\) \((\mathrm{cm^2/sec})\)	Reference
A	\(4\cdot10^{-4}\)	4	0.08	\(16\cdot10^{-4}\)	A.39a
B	\(8\cdot10^{-5}\)	8	0.07		A.39a
C	\(1.2\cdot10^{-5}\)	13	0.04	\(1.6\cdot10^{-4}\)	A.39a
I	\(4.4\cdot10^{-2}\)	0.5	0.10	\(220\cdot10^{-4}\)	A.51a
II	\(2.03\cdot10^{-2}\)	0.7	0.10		A.51a
III	\(0.815\cdot10^{-2}\)	1.0	0.08		A.51a
IV	\(0.262\cdot10^{-2}\)	3.1	0.15	\(81\cdot10^{-4}\)	A.51a

To check the form of the curve corresponding to equation (2.10), we plotted \(\ln(v-v_{\mathrm{cr}})\) as a function of \(\ln(\operatorname{grad} p)\) from the data of Atkins (see Figs. 11 and 12) and found that the slopes of these curves lie between 0.43 and 0.59, so that we could take the exponent of \(\operatorname{grad} p\) in equation (2.10) to be equal to 0.5 with sufficient accuracy.

In estimating \(b\) from Atkins’ data, we found for all curves at \(5 < \operatorname{grad} p < 15\) that \(20 \leq b \leq 40\). If we put \(b=30\), then equation (2.10) may be written in the form

\[ v=\frac{0.07}{d^{1/2}}+30d^{1/2}(\operatorname{grad} p)^{1/2}. \tag{2.11} \]

In comparing this equation with the results of Allen and Misener \(^{\mathrm{A}.39a}\), semiquantitative agreement was found for the dependence

Fig. 13. Dependence of the velocity on capillary size for two different pressure gradients at \(1.2^\circ\)K. The solid curves are the experimental results of Allen and Misener \(^{\mathrm{A}.39a}\) (from Fig. 6). The dashed curves are calculated from equation (2.11), given in the text. The lower pair of curves corresponds to \(\operatorname{grad} p = 0.75\ \mathrm{dyn}\cdot\mathrm{cm}^{-3}\), the upper pair of curves to \(\operatorname{grad} p = 8.0\ \mathrm{dyn}\cdot\mathrm{cm}^{-3}\).

of the velocity on the diameter; the position and value of the minimum were predicted with a small error. A comparison of the experimental results with equation (2.11) is given in Fig. 13.

2.7. The Problem of Viscous Forces

The nature of the friction forces arising, for example, in the motion of helium II through wide capillaries was investigated experimentally by Hollis-Hallett \(^{\mathrm{H}.50a,\mathrm{H}.52a}\). These experiments, together with

more recent experiments by the same author H. 52b, in which a viscometer with a rotating cylinder was used, were considered in sufficient quantitative detail in a recent review by Atkins A. 52a. Therefore we shall confine ourselves here to a qualitative description of the main results.

In his measurements of the viscosity of helium II by means of a viscometer with an oscillating disk (see Section 2.4), Hollis-Hallett extended his observations to larger amplitudes of oscillation than those used earlier. If \(\theta\) is the amplitude of the \(n\)-th oscillation of the disk, then the logarithmic decrement

\[ \delta = -\frac{1}{2\pi}\frac{d\ln\theta_n}{dn} \tag{2.12} \]

is a measure of the retarding forces acting on the disk. For an ordinary viscous liquid \(\delta\) does not depend on the amplitude, as was found by Hollis-Hallett for helium I. In helium II, however, he found that the logarithmic decrement \(\delta\) does not depend on the amplitude for \(\theta < 0.1\) radian; for \(\theta > 0.1\) radian \(\delta\) increases appreciably with amplitude, especially in measurements at low temperatures.

In order to explain the change of \(\delta\) with amplitude, it is necessary to introduce frictional forces more complex than the forces of normal viscosity. The mutual-friction forces caused by the relative motion of the superfluid and normal components of the liquid, introduced by Gorter and Mellink G. 49a and equal to (see Sections 2.6 and 5.5)

\[ F_r = A\rho_s\rho_n(v_s - v_n), \tag{2.13} \]

could explain Hollis-Hallett’s results. The solution of the hydrodynamic equations on the basis of the Gorter–Mellink theory of mutual friction for a disk oscillating in a two-component liquid was given by Zwanniken Z. 50b; the results of these calculations are represented by the solid curve in Fig. 14. In this figure the difference of decrements \((\delta - \delta_0)\) (\(\delta_0\) is the decrement of damping at zero amplitude) is plotted as a function of temperature. Fig. 14 also shows the results obtained by Hollis-Hallett and a dotted curve depicting \(\rho_s\) on a scale corresponding to these results. It is seen that the Gorter–Mellink frictional forces, represented by the solid curve, are unsuitable for a complete explanation of the changes of \(\delta\) with temperature. Moreover, it is interesting to note that the difference of decrements \((\delta - \delta_0)\) turns out to be proportional to the density of the superfluid component—a property which may indicate that the participating frictional forces depend only on \(v_s\).

These interesting results, obtained with an oscillating-disk viscometer for large amplitudes, i.e., for large peripheral velocities, were supplemented by further work of Hollis-Hallett H. 52a on the oscillations of a stack of plates in liquid helium II.

In essence this was a repetition of the experiment of Andronikashvili A.^46 (see Section 2.3), in which, however, oscillations of large amplitudes were studied. For oscillations of a stack of plates, Hollis-Hallett found that not only does the logarithmic decrement $\delta$ increase with increasing amplitude $\theta$, as was the case for a single disk, but that the period of the oscillations also increases with increasing $\theta$.

From the increase of the period it follows that the superfluid component is more and more entrained by the stack of plates at high velocities, as would be expected if there existed either a mutual-friction force $F_r$, or a frictional force $F_s$ between the plate and the superfluid component. Detailed results show that, whereas a mutual-friction force $F_r$, similar to the force postulated by the Gorter–Mellink theory, undoubtedly exists, at the same time there must also exist other frictional forces of an undetermined nature.

Fig. 14. Comparison of experimental values of the difference of decrements $(\delta - \delta_0)$, obtained for a single disk with an oscillation amplitude of 0.215 radians and a period of 3.78 sec, with the Gorter–Mellink theory. The solid curve is theoretical; circles are experimental results; the dashed curve represents $\rho_s$ on a scale corresponding to these points.

A further experiment, indicating the possibility of entrainment of the superfluid component into rotational motion, was carried out by Osborne O.^50*) In this experiment a cylindrical test tube (inner diameter 1.4 cm) was rotated about its vertical axis with angular velocities from eight to seventeen revolutions per second, and the shape of the meniscus was observed.

It was found that throughout the whole temperature range the meniscus was parabolic and corresponded to the meniscus that would be expected if the entire liquid rotated with the velocity of the test tube. Osborne concluded that the peripheral velocities used (from 35 to 70 cm/sec) were higher than the “critical” velocities of superfluid motion and that therefore the superfluid component was entrained by the normal liquid.

This conclusion agrees with the conclusions mentioned above, drawn from the experiments of Hollis-Hallett. Very little, however, can be learned from an experiment with peripheral velocities whose values are greater than the “critical” velocity and, in particular

) Analogous experiments had been carried out earlier by E. L. Andronikashvili (Dissertation, 1948); see also E. L. Andronikashvili, I. P. Kavernin, ZhETF, 28, 126 (1955). (Translator’s note.*)

no conclusions can be drawn regarding the postulate of LandauL. 41a and LondonL. 46a, that \(\operatorname{rot}\mathbf{v}_s=0\) for covering velocities.

Recently AndronikashviliA. 54 also carried out experiments with rotating liquid helium. These experiments are broadly similar to those proposed by LondonL. 46b. A system consisting of a stack of circular plates, placed in a copper cylinder (analogously to what AndronikashviliA. 46 had done in his measurements of \(\rho_n\)), was suspended on an elastic thread in liquid helium II and set into rotation at a constant speed of 0.5 revolutions per second at \(2.10^\circ\mathrm{K}\).

The system was cooled to \(1.5^\circ\mathrm{K}\) and stopped by an electromagnetic damping device. The damping was then removed, and the system was heated to \(1.65^\circ\mathrm{K}\) in 15 sec., so that as a result of the heating \(\rho_n\) increased from 12 to 22%. If the superfluid component was in rotation at the low temperature, then upon heating the fraction of the liquid that had passed into the normal state should transfer its angular momentum to the vessel. No experimental evidence that the vessel acquired such an angular momentum was observed.

3. THEORETICAL FOUNDATIONS OF THE TWO-COMPONENT MODEL

3.1. General remarks

As has already been indicated, the peculiar position with the viscosity of helium II is explained with the aid of the two-component model of the liquid, in which

\[ \rho=\rho_s+\rho_n. \tag{3.1} \]

The macroscopic conclusions from the two-component model, characterized by this and by several further equations (which will be given below), were first widely discussed by TiszaT. 38a, T. 38b, T. 38c, T. 40). These conclusions will be described in Sections 4.7 and 8.1; here we shall briefly consider the microscopic theories that serve as the basis of this model and explain other properties of helium II. Our exposition will be purely qualitative and far from exhaustive, since in a general review of the properties of helium II there would not be room for a detailed presentation; moreover, such a presentation became unnecessary after the publication of Dingle’sD. 52a review of theories of helium II. Four basic theories will be considered: F. London’s theoryL. 38a, L. 38b, Landau’s theoryL. 41a, L. 41b, L. 44a, Green’s theoryG. 48a, G. 48b, and the theory of Prigogine and PhilippotP. 52a, P. 53b, P. 53c*). The earliest was F. London’s theory. It preceded Tisza’s phenomenological theory and gave rise to the latter.

) Independently of Tisza, L. D. LandauL. 41a, L. 41b formulated, substantiated, and investigated in detail the two-component model. (Translator’s note.)*

**) Authors’ additions in proof. Unfortunately, Feynman’s theory was published after this review had been written. See the appendix and the note on p. 376.

3.2. London’s Theory

London^L. 38a, L. 38b pointed out the similarity between the transition of helium I into helium II and the phenomenon of degeneracy that occurs at sufficiently low temperatures in an ideal Bose—Einstein gas. The degeneracy, or “condensation,” of an ideal Bose—Einstein gas consists in the fact that a finite fraction of all particles “descends” to the lowest energy level, where the particles are characterized by a uniform distribution in configuration space, but by one single momentum; one may say that the particles are ordered, or “condensed,” in momentum space. This condensation is a phase transition of the third kind (a discontinuity in the derivative of the heat capacity). With cooling below the degeneracy temperature, which is characterized by a maximum on the heat-capacity curve, an ever larger number of particles descends to the lowest energy level, and the entropy of the gas rapidly decreases, tending to zero.

Since noninteracting helium atoms obey Bose—Einstein statistics, London proposed explaining the properties of helium II on the basis of this model. The uniform distribution of the gas in configuration space was, in his opinion, to reflect the fact that in helium II (as in helium I) no regular structure was observed^K. 36a, R. 40, R. 47; the ordering in momentum space, on the other hand, was to correspond to the superfluid flow in helium II. Of course, liquid helium is in no sense an ideal gas, so that the model is far from perfect; however, the degeneracy temperature for the model was found to be equal to 3.13° K, which is rather close to the experimental value (2.18°) of the $\lambda$-point of helium II.

In other respects, however, there is a quantitative difference between the ideal-gas model and liquid helium. The anomalous behavior of the heat capacity in the gas is much less pronounced; the change of $\rho_n$ and of the entropy with temperature in the ideal-gas model is significantly slower than in liquid helium—$\sigma$ in equality (2.2) for the gas case is only $3/2$. These quantitative discrepancies prompted London^L. 39a to modify the theory by replacing the density of states of free particles with the density of states of particles moving in some average potential field of neighboring atoms. By means of such an arbitrary, ad hoc, alteration of the density-of-states function, London was able to bring the theory into agreement with the experimental values of the heat capacity and to increase the exponent $\sigma$ to 5. However, the following difficulty arose: in order to obtain the desired heat-capacity curve, it was necessary to postulate that even at absolute zero only a fraction (0.136) of all atoms is “condensed.” Such a postulate is not consistent with entropy measurements (Section 4) and was subsequently abandoned by London^L. 45.

London’s conclusion that it is necessary to improve the model of an ideal Bose—Einstein gas by considering intermolecular

forces, was taken into account in the works of Goldstein^G. 41^, Lamb and Nordheim^L. 41^, Becker^B. 50b^, Liffrid^L. 50b^, and Galpern^H. 52e, H. 52^, in which effects occurring in external force fields are considered.

Following London’s first work, a whole series of studies was carried out on the theory of the Bose–Einstein gas and on its application and applicability to the problem of liquid helium. We shall not analyze these works in any detail. Let us note only a few improvements of the theory. Bijl, de Boer, and Michels^B. 41^ considered a model of a Bose–Einstein liquid with a smoothed potential, and assumed the existence of an energy gap between the ground state of the molecules and all excited states. In one modification of their theory, the width of the gap depends linearly on the occupation of the ground state. As a result of these measures, the theoretical heat-capacity curve is brought into better agreement with experiment near \(T_\lambda\). Tisza^T. 51d^ then, in order to justify the existence of an energy gap, proposed considering molecules consisting of several helium atoms, in equilibrium with the ground state. His results are similar to those of Bijl, de Boer, and Michels. Goldstein^G. 46^ investigated the properties of a Bose–Einstein liquid possessing certain volume-averaged deviations from an ideal gas, and considered the question of the influence of these deviations on the order of the \(\lambda\)-transition.

A further, more rigorous investigation of the Bose–Einstein model was carried out by Feynman^F. 53a*^, who showed that the interaction between helium atoms does not preclude the possibility of a \(\lambda\)-transition, and by Friedman and Butler^F. 53b^, who obtained an approximate distribution function for a Bose–Einstein gas possessing interaction. Considering only repulsive forces between molecules, Friedman and Butler obtained for \(T_\lambda\) the value \(2.4^\circ\mathrm{K}\) and a heat-capacity curve in better agreement with the experimental data than the ideal-gas curve.

It should be noted that none of the theories mentioned above explains the behavior of the heat capacity at temperatures below \(0.6^\circ\mathrm{K}\). The observed heat capacity varies in this region proportionally to \(T^3\) (see Sections 3.3 and 4) and, apparently, is connected with phonons in the liquid.

3.3. Landau’s Theory

In Landau’s opinion^L. 41a, L. 44a^, the type of statistics obeyed by helium atoms has nothing to do with the superfluidity of helium II. The fundamental property of helium is rather that it remains liquid down to absolute zero. At absolute zero the liquid must be in the ground state; it is assumed that in this

* A complete exposition of Feynman’s theory of liquid helium appeared in print when the present review had been completed. The reader is referred to the original papers, since it is impossible to present this theory in a footnote.

in this state the motion of the liquid is vortex-free. Landau tried to prove the latter assumption by pointing out that the excitation of vortex motion, which is quantized, is connected with imparting to the system a finite energy, analogously to the way in which, in atomic systems, an increase in energy is connected with an increase in angular momentum. The energy increment \(\Delta\) is taken to be positive, i.e., it is assumed that a system with a unit or larger vortex excitation has a greater energy than in the ground state. Although this assumption does not prove that the ground state is vortex-free, it is natural, however, to suppose that the excitation of vortex motion is connected with an increase in the entropy of the system and, consequently, corresponds to removal from absolute zero.

As Landau indicated, removal from the ground state can also occur upon excitation of one or several quanta of sound waves, or “phonons.” Landau proposed, by analogy, to call a quantum of vortex excitation a “roton.”*) Thus, states close to the ground state are characterized by the numbers and energies of the excited phonons and rotons superposed on the ground state.

So long as such a superposition is valid, the heat capacity and entropy of the system coincide with the heat capacity and entropy of the excitations—quantities that can be calculated if the energy spectrum of the excitations is known. For phonons, as is generally accepted, \(E = p u_1\), where \(E\) is the energy, \(p\) the magnitude of the phonon momentum, and \(u_1\) the velocity of sound. For rotons Landau proposed a spectrum of the form

\[ E = \Delta + \frac{p^2}{2\mu}, \tag{3.2} \]

where \(\Delta\) is the above-mentioned energy increment (or gap), \(p\) is the roton momentum, and \(\mu\) its effective mass. By a choice of the parameters \(\Delta\) and \(\mu\), Landau \(^{Л.41a}\) obtained agreement of the heat-capacity curve below the \(\lambda\)-point with the experimental data. For agreement of the heat capacity in the temperature interval from \(1^\circ\mathrm{K}\) to \(2.18^\circ\mathrm{K}\), the values of the parameters were taken to be

\[ \frac{\Delta}{k} \simeq 8 \div 9^\circ\mathrm{K} \]

and \(\mu \simeq 7 \div 8\) masses of the \(\mathrm{He}^4\) atom.**)

Landau came to the conclusion that the ground state and the excitations play the role, respectively, of the superfluid and normal

*) The term “roton” was first proposed by I. E. Tamm. (Translator’s note.)

**) Landau \(^{Л.41a}\) calculated the phonon heat capacity \(\left(C_{\mathrm{phon}} = 4.4 \cdot 10^{-3} T^3\ \mathrm{cal/g\ deg}\right)\) from compressibility measurements made by Keesom \(^{К.36b}\). This value should be compared with the value \(C_{\mathrm{phon}} = 5.6 \cdot 10^{-3} T^3\ \mathrm{cal/g\ deg}\), obtained in recent experiments by Kramers, Wasscher, and Gorter \(^{К.52a}\). It should be noted that at \(1.0^\circ < T < 2.18^\circ\mathrm{K}\), \(C_{\mathrm{phon}}\) is negligibly small in comparison with \(C_{\mathrm{rot}}\), if the chosen values of \(\Delta\) and \(\mu\) are used.

liquids. The excitations are “normal,” since they can be scattered and reflected and thereby create viscosity. If, for some reason, for example because of the existence of a temperature gradient, a transfer of excitations takes place, then the transported momentum can be calculated as a function of the transport velocity. Landau established that the ratio of the momentum density to the transport velocity is equal to the normal density \(\rho_n\). If \(\rho_n\) is less than \(\rho\), then their difference \(\rho-\rho_n\) is equal to \(\rho_s\), the density of the ground state. The normal density \(\rho_n\) determined in this way*) increases with increasing temperature; the temperature at which \(\rho_n\) is equal to \(\rho\) is the \(\lambda\)-point. The \(T_\lambda\) calculated in this way (under the assumption that the excitations form an ideal gas) turned out to be equal to \(2.3^\circ\mathrm{K}\), in good agreement with the observed value.

In order to show that the liquid associated with the ground state is superfluid, Landau considered the flow of helium II through a tube and concluded that helium cannot absorb a phonon from the walls of the tube so long as it flows relative to the tube with a velocity less than the speed of sound. Analogously he concluded that absorption of a roton is impossible so long as the relative velocity of the helium is less than the “critical velocity,” equal to \((2\Delta/\mu)^{1/2}\). At velocities less than the smaller of these two velocities (they are of the same order of magnitude), the flowing helium will not interact with the walls, i.e., it will be superfluid, unless, as Landau noted, other, still unknown causes limit the superfluid character of the flow.

With the help of such arguments Landau justified the two-component model, in which equations (3.1) and the other equations of Tisza’s macroscopic theory acquire meaning. It should be emphasized, however, that the expression “two components” must not be understood literally. In reality there is one liquid in which, owing to the low temperature, not all types of thermal motion are excited. The existing excitations can carry momentum independently of the whole liquid and behave like a normal liquid. The remaining, unexcited part of the liquid behaves like a superfluid liquid.

In his last paper Landau \(^{L.47}\) changed the form of the energy spectrum of rotons and proposed the dependence

\[ E=\Delta+\frac{(p-p_0)^2}{2\mu}. \tag{3.3} \]

*) Temperley \(^{T.51b}\) suggested that at very low temperatures such a “linear” definition of \(\rho_n\) does not agree with the “rotational” definition obtained from experiments with torsional oscillations of a stack of plates (see Section 2.3). This difference may arise if, in the case of rotation, some excited states correspond to zero angular momentum. Dingle \(^{D.52a}\) did not agree with this opinion.

This formula with three arbitrary parameters made it possible to bring the second-sound velocity calculated by Landau (see Section 9) into better agreement with experiment. Landau took \(\Delta/k = 9.6^\circ\mathrm{K}\), \(\mu = 0.77\) of the mass of the \(\mathrm{He}^4\) atom, \(p_0/h = 1.95\cdot 10^8\ \mathrm{cm}^{-1}\). Unfortunately, the values of these constants cannot be checked independently. One can only note that expression (3.3), with the quoted values of \(\Delta\), \(\mu\), and \(p_0\), agrees with measurements of the heat capacity.

Let us make several remarks concerning this theory. First, it should be noted that the theory says almost nothing about the \(\lambda\)-transition. It can reproduce the experimental heat-capacity curve below the \(\lambda\)-point, but it does not predict the \(\lambda\)-discontinuity on the heat-capacity curve. However, the question of the \(\lambda\)-discontinuity and the related question of the order of the phase transition were considered by Goldstein G. 53, who emphasized the importance of the role of phonons even at the temperature of the \(\lambda\)-transition. Secondly, although the theoretically calculated entropy and \(\rho_n\) depend essentially on the sign and numerical value of the energy gap \(\Delta\), this quantity cannot be obtained from any microscopic model. For example, one cannot say whether the sign and magnitude of \(\Delta\) for liquid \(\mathrm{He}^3\) will be such as to lead to superfluidity. Finally, the exact nature of the “roton” is unclear. Perhaps only one thing can be said about the roton: that it is an excitation with an energy determined by equality (3.3).

In making Landau’s theory more rigorous, notable successes have been achieved. Kronig and Thellung K. 52b gave a quantum theory of a non-vortical hydrodynamic field, in which the concept of phonons is refined. Recently Thellung T. 53b succeeded in quantizing a hydrodynamic field containing vortices. The Hamiltonian obtained by him contains a phonon term, a roton term, and an interaction term. Unfortunately, it has not been possible to determine the energy levels of the roton excitation.

3.4. Green’s Theory

Green’s theory G. 48a, G. 48b is based on the quantum formulation of the kinetic theory of liquids by Born and Green B. 47b, B. 47c.

The main feature of the quantum treatment, according to Green, is the difference between the intensity factors determined, first, thermodynamically and, second, from kinetic theory. Especially essential is the difference between the thermodynamic pressure \(p\), determined from the thermodynamic formula \(p\cdot dV = dW\) (the work performed in displacing the boundary), and the pressure \(p_1\)—the kinetic pressure, the gradient of which determines the mean acceleration of molecules. The thermodynamic pressure is given in the form of a series \(p = p_1 + p_3 + p_5 + \ldots\), where \(p_i\) vary differently with temperature. When \(p_3 + p_5 + \ldots\) becomes comparable with \(p_1\), the liquid ceases to satisfy the classical thermo-

dynamic formulas; in particular, the first law of thermodynamics takes different forms depending on whether the liquid is at rest or whether it is undergoing translational or oscillatory motion.

On this distinction between classical and quantum liquids, Green builds an explanation of the properties of He II. Calculation showed that \(\pi=p_1-p\) becomes significant when only energies of order \(2h^2/mr_1^2\) remain excited, i.e., when states with angular momenta \(l=2,4\), etc. cease to play a role. (\(r_1\) is the distance between atoms at which the potential energy has a minimum, equal for helium to three atomic units.) This corresponds to the characteristic temperature

\[ T_0=\frac{2h^2}{mkr_1^2}\simeq 2.3^\circ\mathrm{K}, \]

which Green identified with the temperature of the \(\lambda\)-transition. He showed that below this temperature thermal waves arise; these waves carry heat and mass in opposite directions. The anomalous heat capacity and density of helium II are connected with the presence of these waves; thermomechanical effects (see Section 4) are explained with their help as well.

Green further indicated that at the characteristic temperature \(T_0\) the radial distribution function of the liquid must undergo significant changes; namely, at \(T<T_0\) the maxima in the

Fig. 15. Radial distribution function. The upper curve corresponds to a normal liquid, the lower to liquid helium II at \(1^\circ\mathrm{K}\), according to Green\(^{48\mathrm{b}}\). (The curves are taken from the article by Mendelssohn\(^{49\mathrm{a}}\).)

radial distribution function that are characteristic of a normal liquid disappear (Fig. 15). Since it is known that inclusion of states with large angular momenta strengthens the normal maxima of the radial distribution function, Green concluded that the unusual properties of liquid helium should disappear above \(T_0\).

According to Green, the decomposition of the density in accordance with formula (3.1) is artificial. Green observed that pressure fluctuations may be interpreted as density fluctuations according to the formula
\(\pi_v=\pi+\Delta\pi=\pi_v\frac{\rho_s}{\rho}+\pi_v\frac{\rho_n}{\rho}\), but warned that ascribing real meaning to these densities may lead to unnatural and absurd conclusions. In his theory the two-component model is, in essence, rejected.

A more detailed discussion of this theory within the limits of the present review is impossible because of its great mathematical complexity.

Turning to criticism of this theory, it should be noted that some authors disagree with Green on the very foundations of the theory; they regard \(p_1\) and \(p\) as identically equal. The bibliography on this question and its further discussion are given in the review by Dingle\(^{D.52a}\). Another observation concerns a quite appreciable change in the radial distribution function which, according to Green’s prediction, should occur at the \(\lambda\)-transition (see Fig. 15). It is significant that the x-ray structural investigations of Keesom and Taconis\(^{K.36a}\) and Riqui\(^{R.40, R.47, R.53a}\) did not reveal such a change.

3.5. The theory of Prigogine and Philippot

In the recently published theory of Prigogine and Philippot\(^{P.52a, P.53b, P.53c}\), a model is given which explains only the thermal properties of helium II.

No attempt is made to discuss the dynamic properties of liquid helium. The proposed model is a generalization of the free-volume model of an ordinary liquid, developed by Lennard-Jones and Devonshire. In the free-volume theory the distribution function for a liquid is derived from the distribution function for a single molecule which moves in a cell or free volume bounded by neighboring molecules. Prigogine and Philippot generalized this model by considering fluctuations in the number of particles occupying a cell. In the case of ordinary liquids far from their critical points the cell dimensions are such that it contains one molecule; in the case of helium, however, even at low temperatures, because of the large zero-point energy of the atoms, the cell volume (equal to \(V_{\mathrm{mol}}/N_{\mathrm{mol}}\)) is much greater (approximately three times) than the volume of the “hard sphere” of a molecule. Therefore only in helium can it be expected that a cell may be occupied by more than one molecule. The authors obtained a distribution function for the case in which a cell may be occupied by 0, 1, or 2 particles, and found that the heat capacity of this model is greater than the heat capacity of an ordinary liquid; moreover, with a reasonable choice of the energy constants the heat capacity had a maximum of about \(1\ \mathrm{cal}/\mathrm{mole}\cdot\mathrm{degree}\) at \(1.5^\circ\mathrm{K}\). A second model, in which a cell of volume \(2V/N\) could be occupied by 1, 2, or 3 particles, led approximately to the same re-

results in the case of Bose–Einstein particles; in the case of Fermi particles, however, the hump on the heat-capacity curve was smeared out and the point of the maximum shifted to a much higher temperature (close to the critical temperature of helium). Thus this simple model, which showed in general terms that a $\lambda$-anomaly of the heat capacity is possible in He$^4$ and impossible in He$^3$, is in this respect equivalent to the hypothesis of Bose–Einstein condensation.

In two subsequent papers Prigogine and Philippot showed, first, that the model leads to a negative coefficient of thermal expansion below the “$\lambda$-point,” i.e. the density maximum, and, second, that fluctuations transform the smooth hump of the heat capacity at $T_\lambda$ into an infinite jump. This fluctuation effect, in the opinion of Prigogine and Philippot, occurs because the “excitation energy” associated with the appearance of two atoms in a cell decreases with increasing number of fluctuations, i.e. with increasing temperature.

3.6. Combination of the Landau and London Models According to Temperley

London’s theory of a Bose–Einstein liquid rather successfully explains the properties of liquid helium near $T_\lambda$. In addition to explaining the anomalous behavior of the specific heat, the theory provides a logical basis for explaining the viscosity of liquid helium, which in this temperature region is more like that of a gas than of a liquid (see Section 2.1). In addition, it has been established (see Section 3.7) that particle statistics have a strong influence on their behavior: the properties of liquid He$^3$ differ in many respects from those of He$^4$.

On the other hand, Landau’s theory is in better agreement with experiment at temperatures below $0.6^\circ\mathrm{K}$. As has already been noted, the heat capacity at low temperatures is evidently associated with longitudinal phonons. Further, experiments with second sound (Section 9.4) also indicate that the normal density $\rho_n$ includes the effective density of phonons, just as Landau predicted. Therefore, in order for a theory based on Bose–Einstein condensation to be valid at temperatures below $1^\circ\mathrm{K}$, it is necessary to add the phonon heat capacity and mass to the heat capacity and mass of the Bose gas.

Taking these results into account, Temperley$^{52a}$ concluded that Landau’s method, suitable at the very lowest temperatures, ceases to be valid at temperatures above approximately $0.6^\circ\mathrm{K}$, whereas London’s model is suitable at temperatures above approximately $1.5^\circ\mathrm{K}$. Between these temperatures lies a transition region in which neither model—neither the phonon model nor the model of an almost ideal gas—is a sufficiently good approximation. In attempting to reconcile these two models, Temperley recently proposed a theory$^{52b}$ in which large clusters of particles (with their

…of Debye lattices) in coordinate space are in statistical equilibrium with small accumulations of particles in momentum space, which play the role of a superfluid component*). The $\lambda$-transition corresponds to the appearance of accumulations in momentum space. Temperley explains the possibility of the formation of such accumulations of two types, in equilibrium with one another, by the known fact that the attractive forces in helium are almost sufficient for the formation of stable molecules. Temperley believes that in He$^3$ the formation of accumulations in momentum space is impossible, owing to the different symmetry of the atoms He$^3$ and He$^4$.

3.7. Experimental data on the connection between the existence of the $\lambda$-transition and the type of statistics

In clarifying the merits of these theories, the question of the importance of Bose—Einstein statistics for superfluidity is of great interest. In this connection, much attention is being given to the rare isotope of helium with mass 3. Since its atoms obey Fermi—Dirac statistics and, because of the large molar volume of liquid He$^3$, it can be likened to an ideal Fermi—Dirac gas (in analogy with the way London likened liquid He$^4$ to an ideal Bose—Einstein gas), one may expect that liquid He$^3$ should not possess the property of superfluidity. This criterion was proposed independently by Pollar and Davidson$^{P.\,42}$, Frank$^{F.\,46}$, and Onsager$^{O.\,48a}$.

The first experimental indication of the possible importance of the type of statistics was made by Daunt and collaborators$^{D.\,47a,D.\,47b}$, who studied the superfluidity of dilute solutions of He$^3$ in He$^4$. They found that He$^3$ atoms do not take part in superfluidity either in films or in the bulk liquid; this result was confirmed by Lane and collaborators$^{L.\,48**)}$. More convincing experiments on the flow of pure He$^3$ were carried out by Osborne, Weinstock, and Abraham$^{O.\,49a}$, who established the absence of superfluidity in the temperature interval $1.05—3.2^\circ$ K. The subsequent work of Daunt and Heer$^{D.\,50a}$ on determining the $\lambda$-points of solutions of He$^3$ in He$^4$ (the most concentrated solution contained 89%; its $T_\lambda$ was equal—

*) Thermodynamic conclusions from the assumption of the existence of molecular associations below $T_\lambda$ were considered by Rice$^{R.\,49,R.\,50}$. However, from the existing experimental facts it is difficult to say with complete certainty whether such molecular complexes are actually formed in configuration space; in particular, measurements of the electrical susceptibility$^{K.\,28a,K.\,28b,G.\,50c}$, the refractive index$^{J.\,38}$, and light scattering$^{Я.\,43}$ do not indicate any noticeable difference in these properties in helium I and helium II.

**) L. D. Landau and I. Ya. Pomeranchuk showed (DAN 59, 669 (1948)) that any extraneous particles present in a small concentration in helium II should not participate in the superfluid motion, irrespective of their statistics.

... (0.38° K) showed that in pure He³ superfluidity is hardly to be found at temperatures above 0.25° K.

There is, however, one experimental fact that complicates the picture outlined in the preceding paragraph and has not yet received a satisfactory explanation. Recent experiments by Hammel and Schuch[^52d] with solutions of He³ in He⁴ indicated the participation of the isotope He³ in anomalous transport in a film. It was observed that He³ is transported by an unsaturated film, like the isotope He⁴, i.e. almost in the absence of a pressure difference, and only the rate of transport is smaller. However, the authors did not discover any superfluidity in pure He³, so that their observations do not contradict the experiments of Osborne, Weinstock, and Abraham. Since the presence of He⁴ is evidently necessary for the anomalous transport of He³ in an unsaturated film, and since no anomalous flow of He⁴ was observed in saturated films, no conclusion can be drawn from these experiments that liquid He³ is a liquid of the same kind as He⁴.

There is another, indirect indication that liquid He³ is to some extent similar to liquid He⁴. A whole series of properties of liquid He³ was excellently predicted by de Boer and Lunbeck[^48b] on the basis of de Boer’s “quantum law of corresponding states”[^48c] without any allowance for the type of statistics. In particular, the equation of state of the vapor was predicted with surprising accuracy. Of course, this is not a very weighty objection to F. London’s “Bose–Einstein hypothesis”; rather it serves as evidence that, at very low temperatures, many properties of liquids cease to depend on the nature of the particles, or, in other words, that many properties of liquid He⁴ (and He³) can be successfully interpreted without using all the information about their atoms. As an example of such insensitivity of macroscopic properties to the chosen microscopic model, one may point to the discussion of the change of the λ-point in He³—He⁴ mixtures, given in Daunt’s review[^52b] on the isotope He³ at low temperatures. In this discussion, a whole series of models, both with and without allowance for statistics, led to predictions so close that they could scarcely be distinguished experimentally.

The experimental data considered do not testify in favor of liquid He³ possessing superfluidity; rather, they indicate the opposite: that there is no λ-transition in He³. Of course, measurements of the heat capacity of He³ or of the elasticity of its vapor at lower temperatures are needed in order to make the latter assertion more convincing*).

) Authors’ addition in proof. Reports on measurements of the heat capacity of liquid He³ down to 0.5° K have been made by Brewer and Daunt (Phys. Rev. 91, 631 (1954)), Roberts and Sydoriak (Phys. Rev. 93, 1418 (1954)), and Osborne, Abraham, and Weinstock (Phys. Rev. 94*, 202 (1954)). No λ-transition was found.

4. THE THERMOMECHANICAL EFFECT AND ENTROPY OF THE SUPERFLUID AND NORMAL COMPONENTS

4.1. The thermomechanical effect

The basic ideas about the thermal state in which the superfluid and normal components of helium II are found were obtained from experiments on the study of thermomechanical effects in this liquid. All experiments devoted to these effects were carried out no less than ten years ago (apart from quite recent measurements below \(1^\circ\) K; see the brief communications of Bots and Gorter\(^{\mathrm{B}.53a}\) and of Rodgers and Härlin\(^{\mathrm{R}.53b}\)), and therefore it seems advisable to consider here the results of the earlier experiments, especially since the numerous theoretical works that have appeared in recent times are based on the interpretation of these experiments.

Fig. 16. Fig. 17.
Apparatus used to observe the fountain effect by Allen and Jones (and by Meissner)\(^{\mathrm{A}.38c}\).

For the first time the thermomechanical effect was observed experimentally by Allen, Peierls, and Uddin. These authors, attempting to measure the thermal conductivity of helium II, noticed that their observations were distorted by a surprising flow of liquid occurring in the direction opposite to that of the heat flow. These observations, first published by Allen and Jones\(^{\mathrm{A}.38c}\), were made with an apparatus (Fig. 16) consisting of a closed vessel with helium II, connected with the surrounding liquid helium by means of a narrow capillary. Under isothermal conditions, as was to be expected, the levels of the liquid in the vessel and outside it coincided. Upon the release of heat

inside the vessel it was found that the liquid began to flow into the vessel, and the level in the vessel rose. This unusual behavior of helium II was demonstrated especially vividly by the following experiment of Allen and Jones (and Meissner)A. 38c. The apparatus used by them consisted of a long, narrow glass tube, placed vertically (Fig. 17). The lower part of the tube, immersed in helium II, was tightly packed with fine emery powder. Heat was supplied by illuminating the powder with a lamp placed outside the cryostat; it was found that, upon illumination, liquid helium flowed so rapidly through the emery plug into the tube that it was expelled through the top of the tube in the form of a fountain. Only one superfluid component could so easily pass through the narrow channels between the particles of emery powder; therefore, evidently, the superfluid component had to be ascribed the property of moving in the direction toward the heat source.

Fig. 18. Photograph of a helium II fountain.

A photograph of such a helium “fountain” is shown in Fig. 18. This thermomechanical effect in liquid helium II later received the name “fountain effect.”

Following these first qualitative observations by Allen and his collaborators came quantitative investigations by Allen and ReekieA. 39b, who used, as the narrow channel between the reservoir and the helium bath, a tube with powder; investigations by KapitzaK. 41a, in whose experiments the narrow channel was the gap between optically polished glass disks; and investigations at Leiden University, namely the work of DaikartD. 43, Keesom and DaikartK. 47a, MellinkM. 47b, and Meyer and MellinkM. 47a, who used the same apparatus as Kapitza. In addition, Daunt and MendelssohnD. 39a showed that the fountain effect can also be obtained when the reservoir with helium is connected to the helium bath by only a surface film. This result was confirmed by StrelkovC. 40, and then by many other authors (see Section 7), and was interpreted by Daunt and MendelssohnD. 50b. Finally, it was shown by Long and MeyerL. 50a and L. 52a that the fountain effect occurs when two vessels are connected to one another only by means of an adsorbed unsaturated helium film at temperatures below \(T_\lambda\) (see Section 6.4).

4.2. Thermomechanical Effect and Determination of Entropy

Kapitza \(K.^{41a}\) and Meyer and Mellink \(M.^{47a}\) used the thermomechanical effect in helium II to measure the difference between the entropy of the superfluid component and the entropy of the whole liquid. Kapitza’s apparatus is shown schematically in Fig. 19. It consisted of a Dewar vessel connected with the surrounding helium bath by means of a narrow slit; the slit was a gap between two optically polished glass plates. The mean width of the gap could be varied from \(3\cdot 10^{-5}\) to \(30\cdot 10^{-5}\) cm. Slits of such width were sufficiently narrow to assume that during the experiment only one superfluid component flowed through them. Resistance thermometers were placed inside and outside the Dewar; an electric heater was also placed inside the vessel. When heat was released inside the Dewar, the superfluid component flowed into it from the bath through the narrow channel; the quantity of liquid that had flowed in was measured directly from the level of the liquid inside the Dewar. Fig. 20 shows a typical dependence of the volumetric velocity \(\dot V\), in \(\mathrm{cm^3/sec}\), and of the simultaneously measured temperature difference \(\Delta T\) inside and outside on the power \(\dot Q\) supplied to the Dewar. For small supplied powers, the volumetric velocity \(\dot V\)

Fig. 19. Diagram of Kapitza’s apparatus for observing the thermomechanical effect \(K.^{41a}\). \(H\) — electric heater, \(T_1\) and \(T_3\) — resistance thermometers.

Fig. 20. Graph of the dependence of the volumetric velocity \(\dot V\) of helium II (in \(\mathrm{cm^3/sec}\)) and of the temperature difference \(\Delta T\) at the ends of the narrow slit on the supplied power \(\dot Q\), according to Kapitza’s observations \(K.^{41a}\).

was directly proportional to \(\dot Q\), which can be written in the form

\[ \dot Q=\rho \dot V \cdot T \cdot \Delta S, \tag{4.1} \]

where \(\rho\) is the density of the liquid, \(\Delta S\) is the difference between the specific entropies of the superfluid component and of the whole liquid; \(T \cdot \Delta S\) is equal to the amount of heat absorbed in the transformation of one gram of superfluid liquid into helium II at temperature \(T\). The values of \(\Delta S\) obtained by Kapitza in this way lie well on the smoothed curve of dependence on temperature, constructed from the data of Table III*).

Table III

Entropy, calculated from the results of measurements of the mechano-caloric effect (Kapitza \(K^{41a}\))

\(T\ ^\circ\mathrm{K}\)	2.105	2.04	1.965	1.880	1.746	1.580	1.470	1.345	1.33
\(\Delta S\) \(\mathrm{J}/\mathrm{g}\cdot\mathrm{deg}\)	1.33	1.10	0.91	0.735	0.49	0.284	0.18	0.11	0.10

Meyer and Mellink M. \(^{47a}\) obtained analogous results. Relation (4.1) is valid when the following conditions are fulfilled**): first, the supplied power must not, even in part, be dissipated in the liquid inside the Dewar, i.e. lead to a rise of temperature in the Dewar. As can be seen from Fig. 20, Kapitza observed that the temperature in the Dewar did not rise (\(\Delta T=0\)) for all velocities below a certain critical value \(\dot V_{\mathrm{crit}}\); this meant that the above-mentioned condition is indeed satisfied for small supplied powers. Secondly, no fraction of the supplied power should be dissipated as a result of heat conduction through the slit. As will be shown in Section 5, the true thermal conductivity of helium II is small; heat transfer through the slit is effected by counterflow convection of the superfluid and normal components. In the absence of any appreciable flow of the normal component, as was the case in the experiments described, heat dissipation due to thermal conductivity through the narrow slit is small. Confirmation of this was the fact that the values of \(\Delta S\) did not depend on the width of the slit when it was varied from \(3\cdot10^{-5}\) to \(30\cdot10^{-5}\ \mathrm{cm}\).

As follows from Fig. 20, Kapitza’s investigations showed that at a certain, rather definite critical velocity of flow—

* The values of \(\Delta S\) given in Table III were obtained experimentally by Kapitza, but by another method (see Section 4.5).

** It seems that the heat released in the condensation of vapor in the Dewar vessel, due to the fact that when the liquid level in the Dewar rises the volume of vapor in it decreases, should be taken into account in the above-mentioned Meyer and Mellink M. \(^{47a}\) correction.

... the linear dependence between \(\dot V\) and \(Q\) is violated, and, simultaneously, a temperature difference \(\Delta T\) appears between the liquid helium inside and outside the Dewar. Apparently, at the critical value of the flow velocity \(\dot V_{\rm crit}\), superfluidity is destroyed and a viscous irreversible flow arises, in consequence of which excess heat must be evolved. By means of such observations it was possible to calculate the critical flow velocities. The results of measurements of these critical velocities by Kapitza, and also by Meyer and Mellink, were given in Section 2.5.

Meyer and Mellink\({}^{\mathrm{M}.47a}\) reported that the amounts of heat absorbed by one gram of liquid, when it flows into a reservoir through a narrow slit, in two cases—when the flow along the channel occurs at a velocity smaller than the critical one and when the flow velocity exceeds the critical one—proved to be markedly different. The exact nature of the irreversible processes occurring during flow with a velocity greater than the critical one is not yet clear; likewise unknown is the exact location of the temperature drop that then arises. A possible mechanism of this process, based on the assumption that the temperature gradients are located at the very entrance and exit of the capillary, was briefly discussed by Meyer and Band\({}^{\mathrm{M}.48a}\).

4.3. Heat-capacity measurements.

Entropy of the normal and superfluid components

From measurements of the thermomechanical effect, which were described in the preceding section, one can calculate the difference of the specific entropies of the superfluid component and of all helium, \(\Delta S\). To determine these quantities separately, it is necessary to use data on the total entropy of all helium II obtained by the direct calorimetric method. Such measurements have been made repeatedly during the last 20 years.

Early measurements of the heat capacity of liquid helium in the temperature interval between 1 and \(3^\circ\mathrm{K}\) were carried out by Keesom and Clusius\({}^{\mathrm{K}.32a}\), and also by Keesom and Keesom\({}^{\mathrm{K}.32b,\mathrm{K}.35}\); their results are shown in Fig. 2. Measurements in the temperature interval \(0.6\)–\(1.5^\circ\mathrm{K}\), in which cooling was carried out with the aid of paramagnetic salts, were performed by Keesom and Westmijze\({}^{\mathrm{K}.41b}\) and by Hill, Wilkinson and Wilks\({}^{\mathrm{H}.50b}\); it was found that in this temperature region the heat capacity is approximately proportional to the sixth power of the absolute temperature. Recent measurements by Kramers, Wasscher and Gorter\({}^{\mathrm{K}.52a}\) were carried down to \(0.25^\circ\mathrm{K}\); it was established that, for

\[ 0.25^\circ\mathrm{K} < T < 0.6^\circ\mathrm{K} \qquad C = 0.0223\cdot T^3\ \text{cal}/(\text{mol}\cdot\text{deg}), \]

and for

\[ 0.7^\circ\mathrm{K} < T < 1.4^\circ\mathrm{K} \qquad C = 0.106\cdot T^{6.7}\ \text{cal}/(\text{mol}\cdot\text{deg}). \]

On the basis of these measurements it was noted that, at temperatures above the region in which the \(T^3\) dependence holds, the heat capacity is apparently not a power-law, but an exponential function of temperature.

Clearly, the proportionality of the heat capacity to the cube of the temperature indicates that the excitations in the liquid are longitudinal compression waves (i.e., phonons). This is also indicated by comparison of the theoretical and experimental values of the heat capacity. The absolute magnitude of the phonon specific heat is given by the expression, a discussion of which is given by Kronig and Thellung1

\[ C_{\mathrm{phon}}=\frac{16}{15}\pi^5\frac{k^4}{h^3}\frac{1}{\rho}\frac{T^3}{u_1^3}, \tag{4.2} \]

where \(u_1\) is the longitudinal speed of sound. Using the value \(u_1 = 237\ \mathrm{m/sec}\), obtained by extrapolation toward low temperatures of the results of recent measurements by Atkins and Chase,[^2] we obtain

\[ C_{\mathrm{phon}}=0.020\,T^3\ \mathrm{cal}/\mathrm{mole}\cdot\mathrm{deg}, \]

in excellent agreement with the experimental value of the heat capacity below \(0.6^\circ\mathrm{K}\).

Fig. 21. Entropy of helium II according to Kramers’ data.[^3] Along the ordinate is plotted \(\lg_{10} S\) (\(S\) in joules/2 deg).

The circumstance that the experimental heat capacity at temperatures above \(0.7^\circ\mathrm{K}\) considerably exceeds the phonon heat capacity, obvi—

as is evident, means the appearance in the liquid of other excitations, the possible nature of which was briefly discussed in Section 3.

Using the results of earlier measurements of the heat capacity by Gorter, Keesom, and MellinkG.50a, and also by Band and MeyerB.48a, the total entropy \(S\) was calculated. Since these calculations were made before the recent experiments of Kramers and othersK.52a, they were based on an incorrect extrapolation of the \(T^6\)-type heat-capacity dependence to the region \(1—0^\circ\mathrm{K}\). The calculations rested on the firmly established conclusion that helium remains liquid down to \(0^\circ\mathrm{K}\).*)

These calculations have now been repeated on the basis of the measurements of Kramers et al.K.52a; the heat capacity at low temperatures \((0.6 \div 0^\circ\mathrm{K})\) was taken to be proportional to \(T^3\). The new results are given in Table IV and in Fig. 21. The curve is plotted on a logarithmic scale; it is evident that even above \(0.9^\circ\mathrm{K}\) it does not become rectilinear. This means that if one tries to express the dependence in the form

Table IV

Smoothed values of the heat capacity and total entropy of helium II
according to Kramers, Wasscher, and GorterK.52a

\(T\,^\circ\mathrm{K}\)	\(C\), joule/(g·degree)	\(S\), joule/(g·degree)	\(T\,^\circ\mathrm{K}\)	\(C\), joule/(g·degree)	\(S\), joule/(g·degree)
0,60	0,0051	0,00169	1,45	0,944	0,162
0,65	0,0068	0,00215	1,50	1,127	0,197
0,70	0,0098	0,00276	1,55	1,330	0,238
0,75	0,0146	0,00358	1,60	1,572	0,284
0,80	0,0332	0,00475	1,65	1,83	0,336
0,85	0,0343	0,00644	1,70	2,11	0,395
0,90	0,0510	0,00885	1,75	2,46	0,461
0,95	0,0743	0,0122	1,80	2,80	0,535
1,00	0,1042	0,0168	1,85	3,19	0,617
1,05	0,142	0,0227	1,90	3,63	0,709
1,10	0,191	0,0304	1,95	4,27	0,812
1,15	0,250	0,0402	2,00	4,95	0,929
1,20	0,322	0,0523	2,05	5,82	1,061
1,25	0,410	0,0672	2,10	6,92	1,215
1,30	0,516	0,0853	2,15	8,61	1,40
1,35	0,634	0,1059	2,18	11,6	1,53
1,40	0,780	0,132	2,186	14,3	1,57

\[ \frac{S}{S_\lambda}=\left(\frac{T}{T_\lambda}\right)^{\xi}, \tag{4.3} \]

*) A recent work by QuilongC.52a on the measurement of the melting curve seemed to cast doubt on this conclusion, but this turned out to be a misunderstandingC.52d.

as was proposed by Tisza T.^47 and London L.^45, then a single value of \(\xi\) will be insufficient for a correct description of the results even in the temperature interval \(1^\circ\) K—\(2.1^\circ\) K. From Fig. 21 it is seen that, very approximately, one may take \(\xi = 6.7\) for the interval \(0.9^\circ\text{K} < T < 1.45^\circ\text{K}\) and \(\xi = 5.4\) for \(1.45^\circ < T < 2.1^\circ\text{K}\).

If we now compare the values of \(\Delta S\) obtained from Kapitza’s experiments K.^41a on the thermomechanical effect (Table III) and \(S\)—the total entropy determined calorimetrically (Table IV), it turns out that \(S\) and \(\Delta S\) are equal to one another within the accuracy of the experiment. Since \(\Delta S\) is the difference between the entropies of all the helium and of the component taking part in the superfluid motion, it follows, as Kapitza K.^41a indicated, that the part of the liquid taking part in the superfluid motion has an entropy equal to zero. This fundamental conclusion, as will be shown in Section 4.5, is also obtained from other experimental data and may be regarded as firmly established. It can be expressed otherwise by saying that the normal component of helium II carries all the entropy of the liquid.

It should be noted here that the suggestion that a temperature dependence of the entropy of the liquid at temperatures below \(0.7^\circ\) K of the type \(T^3\) might exist was made in 1938 by Pickard and Simon, who measured the heat capacity of helium II (unpublished, but see Blinney and Simon B.^39a); only in 1952 was this dependence firmly proved by the experiment of Kramers, Wasscher, and Gorter K.^52a. The appearance of a \(T^3\) dependence at low temperatures was interpreted by the earlier authors as indicating that, on lowering the temperature below \(0.7^\circ\) K, liquid helium enters a certain “anomalous region,” in which the entropy is the entropy of a system of the “lattice” type. Such views, in particular the suggestion of a “lower transition temperature” at about \(0.7^\circ\) K, below which the thermomechanical and other effects vanish, were supported by Tisza T.^47 and T.^49a and by Kurti and Simon K.^38c. The latter authors used this hypothesis to explain the very low thermal conductivity of liquid helium observed by them in the region \(0.2\)—\(0.5^\circ\) K. However, recent measurements of the velocity of second sound (see Section 9.4), as well as the results presented in this section and in Section 4.5, clearly show that the \(T^3\) dependence is connected with the “phonon entropy” possessed by the normal component of helium II, and that the entropy of the superfluid component of helium II at all temperatures from \(0^\circ\) K to \(T_\lambda\) is equal to zero.

It is appropriate here to recall superconductors, which are another example of systems in which superfluid motion takes place. Measuring Thomson heat in superconductors, Daunt and Mendelssohn D.^38a, D.^46a established that the electrons participating in the supercon-

conducting current, energetically lie at the lowest level, i.e., are at absolute zero. In another paper M. Mendelssohn again emphasized that in both cases—helium II and superconductors—the superfluid particles remain in the same energy state in which they were at absolute zero, although the rest of the substance may have a finite temperature. The obvious similarity between the phenomena of superfluid motion in helium II and in superconductors, in each of which the entropy of the superfluid component is zero, has been the subject of consideration and discussion by many authors, for example, Landau, Daunt and Mendelssohn, Mendelssohn, and F. London*).

4.4 Mechanocaloric Effect

The essence of the thermomechanical effect, or fountain effect, is that when heat is released in helium II a flow of the superfluid component arises in the direction toward the heat source. If this effect is reversible, then one may expect, as Tisza first noted, that a flow of the superfluid component should be accompanied by the release of heat (or cold). This reverse “mechanocaloric” effect was first observed experimentally by Daunt and Mendelssohn using the apparatus shown in Fig. 22. Helium II could flow out of a closed Dewar vessel through a plug \(P\), made of compressed fine emery powder; evidently, because of the narrowness of the channels, the flow was mainly superfluid. Measurement of the temperature in the Dewar by means of the resistance thermometer \(T\) from the very beginning of the outflow revealed an increase in temperature; this proved that the liquid carried away by the superfluid flow was in a lower thermal state than the liquid as a whole. In another experiment the Dewar was immersed in a helium bath and the liquid flowed into the Dewar; it was found that the temperature in the Dewar was thereby lowered. The quantitative results approximately agreed with the assertion that the entropy of the superfluid component is zero.

Similar experiments were carried out by Kapitza, who observed the inflow of helium II into a thermally insulated vessel through a narrow gap between two optically polished glass disks. The apparatus is shown in Fig. 19; its description is given in Section 4.2. Kapitza found that if the level of helium in the Dewar is above the level of the helium bath, then a superfluid outflow of helium from the vessel is created and the temperature in it rises. Thermal insulation

*) See also V. L. Ginzburg, Superconductivity, Publishing House of the Academy of Sciences of the USSR, 1946; UFN 48, 26–118. (Translator’s note.)

the dewar was sufficiently good to maintain constant level differences and temperature differences for a considerable time. Therefore it was possible to make measurements of the equilibrium values of \(\Delta T\)—the temperature difference of helium II in the dewar and in the bath—and \(\Delta p\)—the corresponding pressure difference. The results obtained

Fig. 22. Daunt and Mendelssohn\({}^{D.39a}\) apparatus for observing the mechanocaloric effect. \(P\)—tube of compressed powder, \(T\)—resistance thermometer.

Fig. 23. Relation between the temperature difference \(\Delta T\) and the fountain pressure \(\Delta p\) in the mechanocaloric effect. According to Kapitza’s data\({}^{K.41a}\), \(\Delta T\) is plotted in millidegrees, \(\Delta p\)—in cm of liquid helium.

are given in Fig. 23; it is evident from them that, at least for small temperature differences, a linear dependence between \(\Delta p\) and \(\Delta T\) is preserved for all the mean temperatures at which the measurements were carried out.

Similar measurements were performed by Meyer and Mellink\({}^{M.47a}\). In their experiments the flow took place through a narrow slit \((d \leq 10^{-4}\ \text{cm})\). The values they obtained of \(\Delta p/\Delta T\) as a function of \(T\) are given in Fig. 24; they are in good agreement with Kapitza’s results and reproduce quite satisfactorily the dependence between the so-called “fountain pressure” \(\Delta p\) and the corresponding

by the temperature differences that arise between two vessels with helium II connected by a narrow slit.

If the slit width is increased, as was investigated for slits with widths from 1 micron to 100 microns by Keesom and Dijkar K.^47a and Melink M.^47, then the relation between the fountain pressure $\Delta p$ and the temperature difference $\Delta T$ becomes more complicated than in the case of very narrow slits. This occurs because in wider slits a flow of the normal (viscous) component of helium II becomes possible. Naturally, this flow proceeds from the vessel with the greater pressure into the vessel with the smaller pressure, i.e., in the direction opposite to the motion of the superfluid component. Such a circulation of helium leads to an increase in the transfer of heat through the slit from one vessel to the other; as a result, the fountain pressure cannot remain constant without a continuous supply of heat to the vessel at the higher temperature. This process will be considered in more detail, together with other observations of the thermal conductivity of helium II, in Section 5.

Fig. 24. Temperature dependence of the fountain pressure (in cm of liquid helium), referred to a temperature difference of $0.001^\circ\mathrm{K}$. Points are the experimental data of Meyer and Melink M.^47a. Solid curves are theoretical values according to equation (4.4). For the upper curve the scale is enlarged by a factor of 5 (the numbers on the ordinate axis in brackets).

4.5. Other calculations of the entropy of the superfluid and normal components

In Section 4.7 the following formula will be considered, first proposed by G. London L.^38c and L.^39b, and relating the fountain pressure to the accompanying temperature difference in the flow of helium II through very narrow channels:

\[ \left(\frac{dp}{dT}\right)_{\Delta T\to 0}=\rho\cdot\Delta S. \tag{4.4} \]

Here $\rho$ is the density of the liquid, and $\Delta S$ is the difference between the specific entropies of all the helium $S$ and of its superfluid component $S_s$*). If the values of $\Delta S$ obtained from formula (4.4) are compared with the aid of observations—

\[ \text{*) Gorter attributes to the quantity }\Delta S\text{ a more complex meaning (see Section 4.7).} \]

given values of $\dfrac{dp}{dT}$ (Table III gives the values of $\Delta S$ obtained by Kapitza$^{K.41a}$ in just this way), with the values of the total entropy $S$, measured calorimetrically (see Table IV), it turns out that $\Delta S$ and $S$ coincide within the limits of experimental error. It follows from this, as was also implied by the measurements set forth in Section 4.2, that the liquid taking part in superfluid motion has entropy equal to zero. This conclusion has recently been confirmed again by measurements of Rogers and Fairbank$^{R.53b}$ *).

It may be noted that the values of $\Delta S$, calculated from the data of Meyer and Mellink on the thermomechanical effect in the range from $1.0^\circ\mathrm{K}$ to $1.4^\circ\mathrm{K}$, proved to be 10% higher than the total entropy measured by Gorter, Kasteleijn, and Mellink$^{G.55a}$. However, subsequent measurements by S. Kramers and others$^{K.52a}$ almost completely removed this discrepancy.

In a recently published brief report on measurements of the fountain effect below $1^\circ\mathrm{K}$, Bots and Gorter$^{B.53a}$ point to deviations from equation (4.4) occurring at these temperatures. However, more detailed information is needed for discussion of these deviations.

4.6. The mechanocaloric effect as a method of cooling

The circumstance that the mechanocaloric effect leads to heating and cooling of liquid helium in two vessels between which a superfluid flow takes place makes it possible to consider this process as a possible method for obtaining low temperatures. Kapitza$^{K.41a,K.41b}$ reported preliminary experiments of this kind, in which it was established that a lowering of the temperature by $0.4^\circ\mathrm{K}$ could readily be obtained. Details of the experimental apparatus were not given.

This method was examined in detail and criticized by Simon$^{S.50b}$. Simon does not dispute the possibility of using the mechanocaloric effect to obtain low temperatures, but notes the unfortunate choice of liquid helium as the working substance, in connection with its small entropy, which decreases monotonically with decreasing temperature down to $0^\circ\mathrm{K}$.

*) De Klerk, Masuda, and Usui$^{K.52c}$, carrying out a numerical analysis of measurements of the velocity of second sound (see Section 9.4) on the basis of Usui’s thermodynamic theory$^{U.51a}$, made an attempt to remove the discrepancies between experiment and theory by assigning a small finite value to the entropy of the superfluid component. However, these discrepancies do not exceed 3% and, in our opinion, can be explained by experimental errors.

4.7. Theories of the thermomechanical effect

A qualitative explanation of the experiments described above was given by Tisza (T.38a and following). Using London’s conception of helium II as a partially condensed Bose–Einstein liquid, Tisza qualitatively explained the effect. As was already said, Tisza divided helium II into two liquids: a superfluid, whose density is a function only of temperature, and a normal one. The superfluid component has no viscosity, and there is no pressure in it; in the normal liquid there is both the one and the other. When helium II flows through a narrow capillary, the normal component, owing to viscosity, lags behind the superfluid one, and the increase in the concentration of the superfluid component leads to the appearance of a temperature gradient (and a pressure difference). If the capillary is so narrow that the normal component in it is completely immobile, then the flow is reversible. Tisza pointed out that the process inverse to that described above is the fountain effect: a temperature gradient creates a gradient of concentration of the superfluid component and a pressure gradient. Taking into account the property of the Bose–Einstein model that the superfluid component is devoid of entropy, Tisza T.38c observed that when the liquid flows through a narrow capillary the entropy will remain in the vessel being emptied, which will lead to heating; in the vessel being filled, cooling will occur. This is the mechanocaloric effect.

A quasi-thermodynamic theory of these effects was given by G. London L.38c, L.39b. London applied a method analogous to that used by Kelvin also in deriving relations for thermoelectric effects, in which it was assumed that reversible processes could be considered separately from irreversible ones. London considered a closed helium circuit consisting of two reservoirs connected by two parallel paths: a narrow capillary and a tube with a turbine. The reservoirs have different temperatures and pressures. The analogue of the thermoe.m.f. in the case of helium is \(\dfrac{dp}{dT}\), the analogue of Peltier heat is the heat left by the superfluid helium in the reservoir when it flows out through a narrow capillary. An analogue of Thomson heat was also introduced for the case of helium flow along a capillary in the direction of and against the temperature gradient. London accepted Tisza’s assumption that the superfluid particles are at the lowest energy level and therefore their flow is reversible, i.e. occurs without interaction; hence it followed that, when helium flows through a capillary, the “Thomson heat” is equal to zero. Further, London obtained equation (4.4) for \(\left(\dfrac{dp}{dT}\right)\)—the helium “thermo-e.m.f.”—and equation (4.1) for the coefficient of the mechanocaloric effect (the helium “Peltier heat”).

Before continuing the discussion of the theories of these effects, let us give a simple derivation of equation (4.4), based on Tisza’s arguments[^47]. Since the experiments considered at the beginning of Section 4 indicate that, in sufficiently narrow capillaries and at sufficiently low velocities, the flow of superfluid helium is reversible, and since the theories mentioned in Section 3 support this point of view, a purely thermodynamic treatment may be applied to the effect under consideration. Let us imagine two isolated vessels connected by such a thin capillary that only the superfluid component can flow through it (the vessels may also be connected by a superfluid film). Let one vessel be characterized by the values \((p, T)\), and the other by \((p + \Delta p, T + \Delta T)\). In the case of equilibrium, the specific thermodynamic potentials of the superfluid component in the two vessels must be equal. Since, in addition, the thermodynamic potential of one gram of the superfluid component must be equal to the thermodynamic potential of one gram of the normal component, we obtain:

\[ G(p, T) = G(p + \Delta p, T + \Delta T) \]

or, in the limit of small \(\Delta T\),

\[ \operatorname{grad} G = 0 = \frac{1}{\rho}\operatorname{grad} p - S \operatorname{grad} T, \]

where the thermodynamic quantities \(G\) and \(S\) refer to one gram of the entire liquid. Hence we obtain F. London’s equation

\[ \frac{\operatorname{grad} p}{\operatorname{grad} T} = \rho S. \tag{4.5} \]

What is unusual in this equilibrium condition is the inequality of the temperatures of the two volumes. Reversible flow of liquid from one vessel to another under conditions in which the temperatures of the vessels are different is possible, of course, only if the entropy of the liquid participating in the flow is equal to zero. If the flowing liquid possesses entropy, then heat conduction arises and, consequently, an increase of entropy; therefore, a rigorous calculation of the fountain effect must be carried out with the aid of the thermodynamics of irreversible processes*).

It should be noted that a derivation of equation (4.5), equivalent to that given above, was presented by Temperley[^52a] in an article devoted to the consideration of the “neutral strip” between the theories of London and Landau. His derivation, however, is based on more special arguments; here, making use of the observed properties of the superfluid component, we have relied on the simplest thermodynamic considerations.

) For another simple derivation of relation (4.4), see, for example, E. M. Lifshitz’s supplement to W. Keesom’s monograph, Helium, IL, 1949, pp. 405–406. (Translator’s note.*)

A possible interpretation of equation (4.5) is that, in the absence of a hydrostatic pressure difference, a “fountain pressure” acts in the superfluid liquid if there is a temperature gradient.

Further, it is natural to postulate that, in order to obtain the correct equations of motion, the term with $\operatorname{grad} p$ in the equations of ordinary hydrodynamics should, in the case of helium II, be supplemented by a term containing $\operatorname{grad} T$. The same result can be reached by generalizing the arguments that led to equation (4.5), i.e., by asserting that, in the absence of equilibrium, the superfluid component connecting two reservoirs will be accelerated in the direction of decreasing thermodynamic potential. Hence, since $\operatorname{grad} G$ is equal to the force acting on unit mass,

\[ \frac{d v_s}{d t}=-\operatorname{grad} G \]

or, in the linear approximation,

\[ \rho_s\left(\frac{\partial v_s}{\partial t}\right) =-\left(\frac{\rho_s}{\rho}\right)\operatorname{grad} p +\rho_s S \operatorname{grad} T. \tag{4.6} \]

This equation coincides with the linear approximation of Landau’s hydrodynamic equation Л.41a for the superfluid component; the justification of this equation here essentially coincides with Landau’s justification.

Equation (4.6), which is a generalization of equation (4.5), may also be regarded as a generalization of ordinary hydrodynamics, in which the acceleration is equal to the gradient of the thermodynamic potential with the opposite sign and in which temperature effects are completely neglected. (These properties of the hydrodynamic equations will be discussed in more detail in section 8.1.)

Equation (4.6), while describing sufficiently well the behavior of the liquid in an oscillatory process such as second sound (see section 9), gives no more than a qualitative description of the flow of the superfluid component through a capillary or a surface film from a high level to a low one. First, such a flow is characterized by a critical velocity, allowance for which requires the inclusion of an additional term in (4.6). Second, in the case of a steady flow through a capillary, $\operatorname{grad} G$ inside the capillary must, by considerations of continuity, be equal to zero, and the gradients of $T$ or $p$ must, at least, exist in the vessel before the entrance into the capillary. Finally, any flow from one level to another and from one temperature to another, higher one, is associated with dissipation of free energy; equation (4.6) says nothing either about this or about the manner in which the superfluid component loses its momentum and mixes with the whole liquid in the “lower” reservoir.

Let us return to equation (4.4) and to its history. First of all, it is clear that such an equation for the fountain pressure can be derived from the equations of hydrodynamics under special statistical conditions (cf. Section 8.1) (if one disregards the fact that the fountain pressure had already been used in deriving the equations of hydrodynamics). Similar relations are derived from the equations of hydrodynamics when all irreversible effects are neglected. From Dingle’s hydrodynamic equations\(^{\mathrm{D.49}}\), in which it is assumed that the superfluid component also carries entropy, one obtains the relation

\[ \frac{\operatorname{grad} p}{\operatorname{grad} T} = \rho (S-S_s), \tag{4.7} \]

where \(S_s\) is the entropy per unit mass of the superfluid component.

Gorter\(^{\mathrm{G.49b}}\) proposed a modification of F. London’s equation, based on a quasi-thermodynamic consideration of the reversible flow of both components in a wide capillary. He obtained an analogue of equation (4.5):

\[ \frac{\operatorname{grad} p}{\operatorname{grad} T} = \rho x \frac{\partial S}{\partial x}; \qquad x=\frac{\rho_n}{\rho}. \tag{4.8} \]

This equation reduces to F. London’s equation only under the condition

\[ S=x\left(\frac{\partial S}{\partial x}\right)_{pT}. \tag{4.9} \]

Equation (4.9) leads to the relation

\[ S=xS_0(p,T), \tag{4.10} \]

which is a generalization of Tisza’s proposal\(^{\mathrm{T.47}}\) that \(S=x\cdot f(p)\). Let us note that, according to (4.10), \(S\) and \(x\) need not vary with temperature according to one and the same law (although in fact this is approximately the case over most of the superfluid region).

Usui\(^{\mathrm{U.51a}}\) also derived equations (4.8), (4.9), and (4.10), and emphasized the difference between \(\left(\dfrac{\partial S}{\partial x}\right)_{p,T}\) and \(\left(\dfrac{\partial S}{\partial x}\right)_{p}\).

Dingle\(^{\mathrm{D.51}}\) subjected Gorter’s work to sharp criticism. He does not consider either Gorter’s results or his interpretation to be correct. Dingle asserts that Gorter implicitly took the concentration of the normal component \(x\) to be equal to its effective mass, whereas, if these quantities are defined and derived according to Landau, it turns out that they are not equal. In this connection it is appropriate to mention F. London’s remark that, if the entropy is derived from the thermodynamic potential in the usual way, then equation (4.9) is obtained immediately, provided only that the entropy of the superfluid component \(S_s\) is taken to be zero; this result can be obtained without [[unclear: continuation after “не при-”]]

resorting to some microscopic model or thought experiment.

Some authors objected to the use of thermodynamics or quasi-thermodynamics in deriving equations describing the fountain effect or the mechanocaloric effect. For example, Meissner M. 38, in an interesting article in which the analogy between the fountain effect in helium and thermoelectric effects in conductors is emphasized, raises the question of the legitimacy of G. London’s use, in his derivations, of the second law of thermodynamics. Meissner calculates the increase of entropy in a helium cycle consisting of two reservoirs at different temperatures connected by two capillaries. He assumes that no improvements in the experiment can make the effect of irreversible processes (thermal conductivity and viscosity) small in comparison with the “Peltier heat,” and that therefore thermodynamic methods in principle cannot give correct results.

As applied to ordinary systems these objections are undoubtedly well founded: a rigorous treatment must be based on the methods of the thermodynamics of irreversible processes. This was carried out for the case of helium II by de Groot and co-workers G. 47a, G. 50b, G. 51a, G. 51b (see especially G. 50b and G. 51b).

De Groot obtained an expression for the fountain pressure and the mechanocaloric effect in the general case of two liquids which can transform into one another by the chemical reaction \(1 \rightleftarrows 2\). He made the special assumptions usual in considering helium II: 1) that both liquids react infinitely rapidly, 2) that only the superfluid component moves through the capillary, and 3) that the superfluid component does not interact with the walls of the capillary, i.e., moves as a whole. In this case equations were obtained which coincide with Gorter’s equations, namely, equation (4.8) and the corresponding analogue of equation (4.1)

\[ \dot{Q} = \rho \dot{V} T x \frac{\partial S}{\partial x}. \tag{4.11} \]

There is, of course, no contradiction between the thermodynamic and the irreversible-thermodynamic approaches to the problem of helium II. (For example, the irreversible processes indicated by Meissner apparently do not arise in the flow of helium II through narrow capillaries, so that G. London’s equations are fully applicable.) Later authors preferred to consider the problems in their most general form, applying their results to helium II as a special case. Earlier authors, on the contrary, preferred to emphasize the remarkable properties of helium II—the disappearance of entropy and viscosity in the superfluid component—and confidently used simple thermodynamic considerations.

5. HEAT TRANSFER

5.1 Thermal Conductivity of Helium I

The thermal conductivity of liquid helium I was measured in the temperature interval between \(T_\lambda\) and \(4.2^\circ\mathrm{K}\) by Grenier G.\(^{51c}\) and by Bowers and Mendelssohn B.\(^{51c}\); separate observations relating to the temperature \(3.3^\circ\mathrm{K}\) were also made by Keesom and Keesom K.\(^{36c}\). The results of all the authors mentioned are in good agreement with one another. They show that helium I behaves in the usual way, having a comparatively low value of thermal conductivity. The results obtained

Fig. 25. Thermal conductivity of helium I according to Grenier G.\(^{51c}\).

by Grenier, who used a modified Lee disk method, are shown in Fig. 25, from consideration of which it is seen that the thermal conductivity increases approximately linearly from the value
\(4.5\cdot10^{-5}\ \mathrm{cal}/\mathrm{deg}\cdot\mathrm{cm}\cdot\mathrm{sec}\) at a temperature of \(2.24^\circ\mathrm{K}\) to
\(6.3\cdot10^{-5}\ \mathrm{cal}/\mathrm{deg}\cdot\mathrm{cm}\cdot\mathrm{sec}\) at \(4.2^\circ\mathrm{K}\).

It is interesting to note that no signs are observed of anomalously high values of thermal conductivity in the region of temperatures slightly exceeding \(T_\lambda\). The absence of any clear anticipation of the \(\lambda\)-transition when approaching \(T_\lambda\) from the side of higher temperatures in measurements of thermal conductivity is in marked contrast with the definite signs of such anticipation that appear in measurements of viscosity (see Section 2.1), in the attenuation of first sound (see Section 9.1), and in observations of heat capacity (see Fig. 2).

It should also be noted that if, for the theoretical calculation of the thermal conductivity, one uses the simple formula of the kinetic-

... of the kinetic theory of gases, namely:

\[ K=2.5\eta C_v, \tag{5.1} \]

one obtains a value of the same order of magnitude as the observed one. At \(4.2^\circ\mathrm{K}\) the gas-kinetic value is \(4.4\cdot 10^{-5}\ \mathrm{cal}/\mathrm{deg}\cdot\mathrm{cm}\cdot\mathrm{sec}\), and the temperature dependence in broad outline coincides with the observed temperature dependence of the thermal conductivity, once again showing that helium I may, to a first approximation, be successfully regarded as a gas.

5.2. Heat transfer in helium II

It was noted long ago by Keesom^K.35 and by Kurti, Rollin, and Simon^K.36d that the thermal conductivity of liquid helium II is noticeably greater than that of helium I; and Keesom and Keesom^K.36c published the first direct observations of an apparent thermal conductivity. The latter found that, for capillaries of diameter \(0.6\ \mathrm{mm}\), the thermal conductivity of helium II reached values approximately 200 times greater than the thermal conductivity of copper at room temperature and approximately \(5\cdot 10^6\) times greater than the thermal conductivity of helium I.

Early measurements of the thermal conductivity of helium II, performed with wide capillaries (diameter from \(0.2\) to \(1.6\ \mathrm{mm}\)), were published by Allen, Peierls, and Uddin^A.37, Keesom, Keesom, and Saris^K.38b, Allen and Ganz^A.39c, Kapitza^K.41c, Keesom and Saris^K.40a, and Keesom, Saris, and Meyer^K.40b; a review of part of these results was given by Keesom^K.42 and Darrow^D.40. All these measurements showed the impossibility of describing heat transfer in capillaries containing helium II by means of the ordinary heat-conduction equation, namely the equation

\[ \dot q=\frac{\dot Q}{A}=K\cdot\frac{\Delta T}{l}, \tag{5.2} \]

where \(\dot q\) is the heat-flux density, \(\dot Q\) the total heat flux, \(A\) the cross section of the capillary, \(K\) the thermal conductivity, \(\Delta T\) the temperature difference between the ends of the capillary, and \(l\) the length of the capillary. Instead, the following general regularities were found:

a) the heat flux \(\dot Q\) is proportional to the total cross section of the capillary, as should be expected according to equation (5.2),

b) the thermal conductivity \(K\) does not depend on the length of the capillary, as should be expected according to equation (5.2),

c) the heat-flux density \(\dot q\) is proportional to the cube root of the temperature gradient, i.e. \(\dot q\sim\left(\dfrac{dT}{dl}\right)^{1/3}\)—a result completely unexpected from the point of view of equation (5.2) and one that makes meaningless the numerical values of the “thermal conductivity” in general

understanding. For the temperature interval \(1\)—\(1.6^\circ\) K,

\[ \dot q=0.623\,T^3\left(\frac{dT}{dl}\right)^{1/3}\ \mathrm{W\cdot cm^{-2}}. \tag{5.3} \]

The results of some measurements are presented in Fig. 26. These results clearly show that heat transfer in helium II is an unusual process, which cannot be explained within the framework of the ordinary theory of thermal conductivity. On the contrary, the theory based on Tisza’s proposed two-component model\({}^{T.38a}\) made it possible to penetrate into the essence of the mechanism with which the heat-transfer process is connected. Considering two volumes of helium II connected by a capillary (the system usually used in measurements of heat transfer), Tisza assumed that under these conditions a mass circulation takes place between the vessels, occurring in such a way that the superfluid part of the liquid moves without viscous losses toward the heat source (as in the thermomechanical effect), while the “normal” component moves in the opposite direction with a velocity determined by frictional forces. Such mass circulation, as was first clearly formulated by London\({}^{L.38c}\), would explain the existence of very large heat flows, analogous in character to the Peltier effect. In such transfer the superfluid component, entering a volume at a higher temperature, is “excited” and absorbs energy in the amount determined by equation (4.1), since for conversion of the superfluid component into a liquid of ordinary “composition” an amount of heat equal to \(T\cdot\Delta S\) per \(1\) g is required; similarly, upon entering the capillary in the colder vessel the corresponding amount of heat is liberated. The total heat transfer \(\dot Q\) for this process would therefore be

\[ \dot Q=\dot M\cdot T\Delta S, \tag{5.4} \]

where \(\dot M\) is the mass flux for any of the components. In addition to this, there exists heat carried along with the normal component and equal to \(\dot M C_v \Delta T\), but this amount of heat is, obviously, negligibly small in comparison with the value determined from equation (5.4)\(^*\). This qualitative explanation was then further developed and quantitatively worked out by many authors (see Sections 5.4 and 5.6). At present it is generally accepted\(^ {**}\).

It is of interest to describe, in passing, some of Kapitza’s early experiments\(^{K.41c}\), in which he elegantly demonstrated the existence of circulating mass flows accompanying heat transfer. In one series of experiments he connected a closed vessel containing a heater with a bath of liquid helium II by means of a horizontal capillary 0.55 mm in diameter, and suspended directly opposite the outlet of the capillary, on a torsion balance (as shown in Fig. 27), a flat vertical disk (a vane). It was found that when the liquid in the vessel was heated, the disk experienced a force acting in the direction away from the capillary outlet, as though a jet of liquid were flowing out of the latter. By means of relative displacement of the disk, Kapitza studied the geometry of the jet and found that its diameter was close to the diameter of the capillary. In addition, by measuring the reaction force, he found that the jet velocity was of the order of \(5\ \mathrm{cm/sec}\). It is interesting to note that Kapitza found the torsion-balance method more sensitive for

Fig. 27. Diagram of Kapitza’s apparatus\(^{K.41c}\) for studying a helium jet. The vessel and the system with the vane are immersed in helium II. \(H\) — heater, \(T\) — thermometer.

\[ \text{Labels in the figure:} \]
\[ \begin{gathered} \text{Stationary Dewar vessel},\quad \text{To a rotating head},\quad \text{Vacuum},\\ \text{Vane},\quad \text{Vane} \end{gathered} \]

\(^*\) The heat flux determined by the expression \(\dot M C_v \Delta T\) is a flux associated with ordinary convection. Unless one postulates fantastically large flow velocities (\(10^4\)–\(10^5\ \mathrm{cm/sec}\)) through the capillaries, it is impossible to explain the observed heat flows by this process alone. In reality, another convective process must occur, the process associated with equation (5.4), i.e. a kind of Peltier process; to distinguish it from ordinary convection, following London we shall call it here “internal convection.”

\(^ {**}\) This fundamental conclusion was obtained by P. L. Kapitza\(^{K.41c}\). (Note added in proof.)

…detecting the inflow of heat to the vessel than direct measurement of changes in the temperature inside the vessel by means of a phosphor-bronze resistance thermometer.

In another series of experiments Kapitza mounted the vessel and capillary on the suspension of a torsion balance and directly measured the reaction force of the jet. He found, as was to be expected, that the helium jet emerging from the capillary has the same properties as those found earlier. On the basis of these experiments it was concluded that the “jet” is associated with the motion of the normal component along the capillary away from the heat source. No superfluid motion along the capillary toward the heat source is observed.

5.3. Heat transfer through narrow slits (experiment)

In considering the process of heat transfer in liquid helium II in greater detail, it is necessary, as in the case of the mass-flow problem, to distinguish between (a) effects occurring in heat transfer through narrow slits and (b) effects occurring in heat transfer through wide slits. The process of “internal convection” considered above in Section 5.2 necessarily presupposes the existence of a “normal” mass flow directed away from the heat source and equal in magnitude to the superfluid mass flow toward the heat source. Therefore, in the case of extremely narrow slits it may be assumed that the flow of the normal component will be (owing to its finite viscosity) negligibly small, and together with it the heat transfer associated with the process of internal convection will also be small. As the width of the slit increases, an ever greater amount of the normal component will be able to flow through it, and one should expect that this flow and the heat transfer associated with it will be limited by the boundary conditions imposed on the normal flow. These boundary conditions for the case of so-called narrow slits (of width \(0.1\)—\(10\,\mu\)), apparently, with a few exceptions, are conditions associated with the Poiseuille laminar flow of the normal component through the slit, i.e. conditions determined mainly by surface viscous forces. In case (b), i.e. in the case of so-called wide slits (from 10 microns to 2 mm), the boundary conditions are less clear. However, it appears, as will be considered in Section 5.6, that the main forces limiting the magnitude of the normal flow are connected in this case with the mutual friction between the superfluid and normal flows over the entire cross-section of the slit. The practically sharp distinction between case (a) and case (b) consists in the fact that in the first case the heat flux is exactly proportional to the temperature difference at the ends of the capillary, whereas in case (b) the heat flux is approximately proportional to the cube root of the gra…

dient of temperature (this peculiarity has already been mentioned in Section 5.2).

The first observations of heat flow through narrow slits were made by Allen and Reekie^A.39b, who used channels formed between particles of a tightly compressed fine powder. They noticed that at the very lowest temperature at which the measurements were made (1.24° K), the heat flux \(\dot Q\) was proportional to the temperature difference \(\Delta T\) between the ends of the tube filled with powder. All later measurements with narrow slits, carried out by Keesom and Dijkart^K.47a, Mellink^M.47b, Meyer and Mellink^M.47a, and by Hwang, Hunt, and Winkel^H.52c, using instruments essentially identical with Kapitza’s instrument^K.41a shown in Fig. 19, also showed that for sufficiently low temperatures and sufficiently narrow slits \(\dot Q\) varies directly proportionally to \(\Delta T\). However, it also followed from them that the transition from the region in which \(\dot Q \sim \Delta T\) (case (a)) to the region in which \(\dot Q \sim (\Delta T)^{1/3}\) (case (b)) also depends on temperature. In Table V, borrowed from the work of Meyer and Mellink, are given the tempera-

Table V

Temperatures above which the linear relation between the heat flux \(Q\) and the temperature difference \(\Delta T\) is not valid: (according to Meyer and Mellink^M.47a)

Slit width in microns	Temperature °K
10	1.3
5	1.6
2	1.9
1	2.1
0.3	2.16

Fig. 28. Experimental dependence (solid curve) of the heat flux \(\dot Q\) on the temperature difference \(\Delta T\) at the ends of a slit of width \(5 \mu\), according to Hwang, Hunt, and Winkel^H.52c. The upper dashed curve corresponds to a linear dependence of \(\dot Q\) on \(\Delta T\), the lower—to a dependence of the type

\[ \dot Q \sim (\Delta T)^{1/3}. \]

tures above which a linear dependence between \(\dot Q\) and \(\Delta T\) is no longer observed for slits of various widths. The transition from one mechanism of heat transfer to another is illustrated by Fig. 28, borrowed from the work of Hwang, Hunt, and Winkel^H.52c.

The works considered above, devoted to the observation of heat flow through narrow slits, led to the following conclusions:

(1) At sufficiently low temperatures (see Table V) the density of the heat flux \(\dot q\) is proportional to the temperature difference \(\Delta T\).

(2) Up to now, however, there are not enough data to formulate a law for the variation of the heat flux with change in slit length.

(3) The heat-flux density $\dot q$ depends on the slit width. Apparently, $\dot q$ varies with the slit width $d$ according to a power law with an exponent lying between 1 and 2. (Unfortunately, exact data on the slit width are lacking.) Moreover, the magnitude of the variation of $\dot q$ with $d$ apparently itself is a function of temperature.

(4) The heat-flux density depends strongly on the temperature at a constant temperature difference. For a slit of width $5\,\mu$, according to the data of Mellink M.47b, $\dot q$ is proportional to $T^{13}$; as Meyer and Mellink M.47a report, for a slit of width $1\,\mu$, $\dot q \sim T^9$, and the dependence on $T$ weakens as the slit width decreases.

(5) The numerical values of the observed “thermal conductivity,” calculated using equation (5.2), are given in Table VI,

Table VI

Experimental and theoretical values $K_{\rm exp}$ and $K_{\rm theor}$ of the apparent thermal conductivity of helium II in narrow slits. Slit width — $d$, slit length — $l$. Data taken from the works of London and Zilsel L.48b

$d$, microns	$l$, cm	$T$ °K	$K_{\rm exp}$	$K_{\rm theor}$, $\dfrac{\mathrm{cal}}{\mathrm{deg}\cdot\mathrm{cm}\cdot\mathrm{sec}}$	$\dfrac{K_{\rm exp}}{K_{\rm theor}}$	Authors
1.75	0.275	1.960	17.2	10.2	1.69	K. 47a
1.75	0.275	1.705	3.5	1.86	1.88	K. 47a
1.75	0.275	1.476	0.61	0.32	1.92	K. 47a
1.75	0.275	1.223	0.066	0.032	2.05	K. 47a
1.15	0.275	2.170	24	15.5	1.55	K. 47a
1.15	0.275	1.989	17.1	5.28	3.25	K. 47a
1.15	0.275	1.799	5.35	1.55	3.45	K. 47a
1.0	0.1	2.159	31	10.8	2.9	M. 47a
1.0	0.1	1.948	21.4	3.06	7.0	M. 47a
1.0	0.1	1.802	12.4	1.19	10.4	M. 47a
0.75	0.275	1.411	1.05	0.060	17.5	K. 47a
0.75	0.275	2.097	30.2	4.3	7.0	K. 47a
0.75	0.275	1.600	1.85	0.156	11.8	K. 47a
0.75	0.275	1.403	0.39	0.031	12.5	K. 47a
0.5	0.1	1.659	2.48	0.108	23	M. 47a
0.5	0.1	1.315	0.354	0.0064	55	M. 47a
0.5	0.1	1.274	0.277	0.0043	64	M. 47a
0.5	0.1	1.086	0.124	0.0006	202	M. 47a
0.3	0.1	1.652	1.92	0.0365	54	M. 47a
0.3	0.1	1.558	1.35	0.018	75	M. 47a
0.3	0.1	1.358	0.48	0.0034	140	M. 47a
0.3	0.1	1.226	0.25	0.00097	260	M. 47a

borrowed from the work of London and Zilsel\(^{\mathrm{L}.48\mathrm{b}}\). The data of Table VI show that even in the case of the narrowest slits the apparent thermal conductivity is many orders of magnitude greater than the thermal conductivity of helium I.

5.4. Heat transfer through narrow slits (theory)

The theory of heat transfer by helium II through narrow slits was considered in detail by London and Zilsel\(^{\mathrm{L}.48\mathrm{b}}\), who used the mechanism of “internal convection” and took into account the limitation of mass transfer during circulation caused by the laminar (Poiseuille) flow of the normal component in the slit. Subsequently this problem was discussed by Andronikashvili\(^{\mathrm{A}.49}\), Gorter and Mellink\(^{\mathrm{G}.49\mathrm{a}}\), and Atkins\(^{\mathrm{A}.52\mathrm{a}}\).

The formula for heat transfer is obtained as follows: first, if the only limitation on the flow is the viscosity \(\eta_n\) of the normal component, flowing laminarly along the capillary, then a pressure gradient must arise along the channel

\[ \operatorname{grad} p = \eta_n \nabla^2 \mathbf{v}_n, \tag{5.5} \]

where \(\mathbf{v}_n\) is the velocity of the normal component. Further, it should be noted that this pressure is, in essence, the “fountain” pressure and is connected with the temperature gradient along the channel. The relation between the pressure and temperature gradients was discussed earlier in Section 4.5; it is given by formula (4.4). Using the equation for the total heat flux (5.4) and combining (4.4) and (5.5), we obtain, for a plane-parallel slit with gap \(d\) per unit width,

\[ \dot{Q} = -K_{\mathrm{theor}} d \cdot \operatorname{grad} T = -\frac{\rho^2 S^2 T \cdot d^3}{12 \eta_n}\operatorname{grad} T, \tag{5.6} \]

where \(S\) is the entropy per unit mass of the liquid, coinciding with \(\Delta S\), and \(K_{\mathrm{theor}}\) is the coefficient of thermal conductivity.

The values of \(K_{\mathrm{theor}}\) were calculated from equation (5.6) by London and Zilsel\(^{\mathrm{L}.48\mathrm{b}}\), using experimentally determined values of \(S\) and \(\eta_n\). For comparison they used the values of \(d\) and \(\operatorname{grad} T\) from the measurements of Keesom and Duykaarts\(^{\mathrm{K}.47\mathrm{a}}\) and of Meyer and Mellink\(^{\mathrm{M}.47\mathrm{a}}\); the results of their calculations are given in Table VI, from which it is seen that \(K_{\mathrm{theor}}\) is of the same order of magnitude as \(K_{\mathrm{expt}}\) for slits with gaps greater than \(1\,\mu\). For slits narrower than one micron the theoretical thermal conductivity turns out to be too small. For one of the narrowest slits of Meyer and Mellink (with a gap of \(0.3\,\mu\)) the ratio \(K_{\mathrm{expt}}/K_{\mathrm{theor}}\) at \(1.23^\circ\mathrm{K}\) is equal to 260.

If the agreement between \(K_{\mathrm{theor}}\) and \(K_{\mathrm{expt}}\) for slits from 1 to \(10\,\mu\) is satisfactory, then the enormous discrepancy for narrower slits is disturbing. Gorter and Mellink\(^{\mathrm{G}.49\mathrm{a}}\), who drew attention to this discrepancy, attribute it to the fact that for the narrowest slits the gap becomes comparable with the mean free path determining the viscosity, and that in this case the flow of the normal

components must be regarded as analogous to Knudsen flow of rarefied gases. However, no satisfactory quantitative explanation has been obtained for the anomalously large thermal conductivity in slits with a gap of less than \(1\mu\), and it may be assumed that much further work will have to be expended on clarifying this question.

Since the dependence of the entropy of helium II on temperature is very strong (see Sec. 4.3), and since \(\eta_n\) does not change appreciably in this temperature range (see Fig. 7), it follows from equation (5.6) that the thermal conductivity \(K_{\text{theor}}\) should be approximately proportional to \(T^n\), where \(n\), in accordance with the experimental data, lies between 10 and 15. London and Zilsel\(^{\text{L. 48b}}\) assumed that \(n = 12.2\), but in the light of later measurements of \(S\) and \(\eta_n\) this value should be revised. The decrease in the observed value of \(n\), noted by Meher and Mellink\(^{\text{M. 47a}}\), is apparently connected with the anomalously large thermal conductivity in the narrowest slits and still requires explanation.

5.5. Heat transfer through wide slits (experiment)

Early measurements of heat transfer through wide slits (with diameters from \(0.2\) to \(1.6\) mm) and the results of these measurements have already been discussed in Sec. 5.2. Later work on heat transfer in the region where the heat flux \(\dot Q\) is not directly proportional to the temperature difference \(\Delta T\) was carried out by Keesom and Dijkart\(^{\text{K. 47a}}\), Mellink\(^{\text{M. 47b}}\), and by Hangu, Hant and Winkler\(^{\text{H. 52c}}\). The value of these works lies in the information contained in them concerning the resistance forces acting over the entire cross section of the slit, i.e., throughout the entire volume of the liquid.

The apparatus used in these works was the same as in the investigations by the named authors of heat transfer through narrow slits (see Fig. 19). The width of the slits varied from 1 to \(15\mu\); for these slits a nonlinear dependence of the heat flow \(\dot Q\) on \(\Delta T\) was observed at sufficiently high temperatures (see Table V).

The results obtained were, in their character, consistent with the results for the case of much wider slits (see Sec. 5.2), namely: (a) the total heat flow \(\dot Q\) proved to be approximately proportional to the cross section \(A\) of the slit, and (b) the density of the heat flux

\[ \dot q \left(\dot q = \frac{\dot Q}{A}\right) \]

varied with \(\Delta T\) as \((\Delta T)^{1/3}\). The measurements, however, were not carried out over a sufficiently wide range to make it possible to determine the influence of the length of the slit on the results obtained.

The latest results of Hangu, Hant and Winkler\(^{\text{H. 52c}}\) show that the region of nonlinearity, i.e., the region in which \(\dot Q \sim (\Delta T)^{1/3}\), may not extend all the way down to zero heat flow. For slits 1–5 \(\mu\) wide, used by the named authors, a linear relation between \(\dot Q\) and \(\Delta T\) is observed even at temperatures exceeding

...critical values given in Table V. Such a transition from the \((\Delta T)^{1/3}\) region to the linear region has already been illustrated in Fig. 28.

In many measurements of heat transport published by the above-mentioned authors, and also in the early measurements of Allen and Reekie\(^{A.39b}\), along with the parameters \(\dot Q\) and \(\Delta T\), the pressure difference at the ends of the slit connecting two volumes of liquid helium II was measured. This pressure difference is the so-called fountain pressure, which arises as a result of the liberation of heat in the warmer volume of helium II and, according to the two-component model, is fundamentally connected with the motion of the superfluid component in the direction of the heat source. However, when wide slits are used, the fountain pressure cannot be maintained without a continuous supply of heat to the warmer vessel, owing to the fact that a “normal” flow arises along the direction of the fountain-pressure gradient and heat is carried into the colder volume by the process of “internal convection.” In the case of extremely narrow slits, however, it proves possible, as was discussed in Section 4.4, to maintain a once-created fountain pressure for considerable intervals of time without an additional supply of heat. It may be supposed that in the ideal case these time intervals could be made indefinitely large if one considered slits wholly impermeable to the flow of the normal component.

The principal results of these measurements of heat flow and fountain pressure in the case of wide slits are as follows:

a) At constant temperature there is observed a strict proportionality between the heat flow \(\dot Q\) and the observed fountain pressure \(\Delta p_{\text{obs}}\), regardless of whether a direct proportionality between \(\dot Q\) and \(\Delta T\) exists or not. This fact was first noticed by Allen and Reekie\(^{A.39b}\); the data collected

Fig. 29. Dependence of the observed fountain pressure \(\Delta p_{\text{obs}}\) (in cm of liquid helium) on the heat flow \(\dot Q\) for a slit of width 10.5 microns at various constant temperatures. Data of Mellink\(^{M.48b}\).

MellinkM. 48b and clearly illustrating this result are shown in Fig. 29.

b) The ratio of the observed fountain pressure to the temperature difference \(\left(\dfrac{\Delta p_{\mathrm{obs}}}{\Delta T}\right)\) is, in general, not equal to \(\rho \Delta S\), as was found in the case of narrow slits (see Section 4.5). It has been established that \(\dfrac{\Delta p_{\mathrm{obs}}}{\Delta T}\) is a complicated function of temperature, slit width, and \(\Delta T\); a typical curve, showing, according to MellinkM. 48b, the dependence of \(\Delta p_{\mathrm{obs}}\) on \(T\) for various constant values of \(\Delta T\) in the case of a slit 5 μ wide, is given in Fig. 30. In this figure the dotted curves show the calculated values of \(\rho \Delta S \cdot \Delta T\), or rather, \(\rho S \cdot \Delta T\).

Fig. 30. Temperature dependence of the fountain pressure (in cm of liquid helium) for various constant values of \(\Delta T\), according to MellinkM. 48b. The measurements were made on a slit 5 microns wide. The dotted lines correspond to the maximum theoretical values of the fountain pressure according to equation (4.4).

5.6. Heat transfer through wide slits (theory)

A possible explanation of the nonlinearity of the change in the heat flux \(Q\) through wide capillaries in helium II with changing temperature difference \(\Delta T\) was proposed by GorterG. 48c and was discussed in detail by Gorter and MellinkG. 49a and by Gorter, Castelijn, and MellinkG. 50a. The explanation is based on the assumption that there exists a mutual friction force between the superfluid and normal components when they move relative to one another, equal to

\[ F_r = A\rho_s \rho_n (v_s - v_n)^3, \tag{5.7} \]

where \(F_r\) is the force calculated per unit volume, and the coefficient \(A\) depends on temperature. The influence of the mutual friction force, proportional to the cube of the relative velocity, on the isothermal flow of helium II has already been considered in Sections 2.6 and 2.7.

Taking into account such a friction force, acting on both the superfluid and the normal components, the simple expression (4.4) for the fountain pressure ceases to be valid, since it was derived under the assumption that irreversible forces such as \(F_r\),

are absent. In this case, during the flow of the superfluid component the theoretical pressure gradient \(\rho S\,\operatorname{grad}T\) is greater than the observed \(\operatorname{grad}p_{\mathrm{obs}}\), owing to the presence of mutual friction. Consequently, in the steady case we have:

\[ \rho S\,\operatorname{grad}T=\operatorname{grad}p_{\mathrm{obs}}+A\rho_s\rho_n(\mathbf{v}_s-\mathbf{v}_n)^3. \tag{5.8} \]

From measurements of \(\operatorname{grad}T\) and \(\operatorname{grad}p_{\mathrm{obs}}\), carried out by Hunt, Hunt and Winkel H. \(^{52c}\) for steady flow or flow with negligible acceleration, the mutual-friction force \(F_r\) can be calculated by means of (5.8).

To explain heat transfer by “internal convection,” taking mutual friction into account, it is necessary to combine equation (5.8) with the equation relating the heat-flux density \(\dot q\) to \(\operatorname{grad}p\). Looking ahead (see Section 8.1), let us combine the hydrodynamic equations for the motion of the superfluid and normal components, as Gorter and Mellink did, and obtain

\[ \operatorname{grad}p_{\mathrm{obs}}=\eta_n\nabla^2\mathbf{v}_n, \tag{5.9} \]

which, together with the basic equation for the process of internal convection (5.4), gives:

\[ \dot q=-\frac{\rho STd^2}{12\eta_n}\operatorname{grad}p_{\mathrm{obs}} \tag{5.10} \]

for flow in a plane-parallel slit with distance \(d\) between the walls. Equation (5.10) points to the interesting fact that the motion of the normal component is completely determined by the boundary conditions on the walls of the slit even in the presence of mutual friction. Moreover, it shows that there exists an exact proportionality between \(\dot q\) and the observed pressure gradient along the channel, \(\operatorname{grad}p_{\mathrm{obs}}\), irrespective of whether \(\operatorname{grad}p_{\mathrm{obs}}\) satisfies the simple relation (4.4) or the more complicated one (5.8). The latter result is in good agreement with the observations first made by Allen and Reekie A. \(^{39b}\), according to which \(\dot q=\mathrm{const}\cdot\Delta p_{\mathrm{obs}}\) (see Fig. 29). It should be noted that the linear dependence between \(\dot q\) and \(\operatorname{grad}p_{\mathrm{obs}}\), which exists according to equation (5.10), will be violated if irreversible forces other than mutual friction act on the superfluid motion.

Combining (5.8) and (5.10), we obtain:

\[ \operatorname{grad}T=-\frac{12\eta_n}{\rho^2 S^2Td^2}\dot q-\frac{A\rho_n}{S(\rho ST)^3}\dot q^{\,3}. \tag{5.11} \]

For narrow slits the first term on the right-hand side of (5.11) is predominant and determines the heat flux; it is easy to see that neglecting the second term again leads to the linear equation (5.6). As the slit width \(d\) is increased, the first term on the right-hand side of (5.11) will decrease, and \(\operatorname{grad}T\) will become proportional to \(\dot q^{\,3}\). Proportional-

the dependence of \(q\) on the cube root of the temperature gradient was indeed observed experimentally. In essence, on the basis of these observations, Gorter and Mellink assumed that the mutual-friction force \(F_r\) is proportional to the cube of the relative velocity.

As was explained in greater detail in Sections 2.6 and 2.7, it is still not clear how well the mutual-friction force introduced by Gorter and Mellink can explain the normal viscosity, thermal conductivity, etc., despite the initial success of this hypothesis, also noted by Nakajima, Tomita, and Usui\(^{\mathrm{N.50a}}\), who carried out extensive calculations including mutual friction. Moreover, Hang, Hunt, and Winkel\(^{\mathrm{N.52c}}\) clearly showed (see, in particular, Fig. 28) that mutual friction does not exist until the flow velocity reaches a certain critical value. This conclusion—that motion at a velocity below the critical one is devoid of friction—had been made earlier by Chandrasekhar and Mendelssohn\(^{\mathrm{C.51}}\) and by Atkins\(^{\mathrm{A.51a}}\) from observations of flow in a surface film and in wide slits. It was suggested that

\[ F_r = A\rho_s\rho_n \bigl[(v_s - v_n) - v_{\mathrm{crit}}\bigr]^3 . \tag{5.12} \]

Recently Kasuya\(^{\mathrm{K.53e}}\) published a short note in which the results of Hang, Hunt, and Winkel\(^{\mathrm{N.52c}}\) are interpreted with allowance for a mutual-friction force of the type (5.12), as well as with allowance for a friction force acting on the superfluid component and proportional to \(v_s^3\). He calculated the critical velocities for both variants. Further communications are needed for a more detailed discussion.

In conclusion one may say (see also Section 8.1) that there is a whole series of possible irreversible terms in the equations of hydrodynamics, and their theoretical and experimental study is only beginning.

5.7. Wall Layers

Along with the problem of heat transfer in liquid helium II itself, investigations of heat transfer across the boundary between a solid and liquid helium II are of interest. The first studies of this question were carried out by Kapitza\(^{\mathrm{K.41c}}\). His apparatus consisted of a metallic parallelepiped inside which a resistance thermometer and a heater were placed, freely suspended on thin threads in a thermostated bath of liquid helium II. The release of heat in such a parallelepiped led to an increase in its temperature relative to the temperature of the helium bath because of the finite thermal resistance of the solid–liquid boundary. Kapitza found that this boundary thermal resistance, to a first approximation, does not depend on the nature of the solid and that the temperature jump at the surface is localized in a liquid layer near the surface with a thickness of less than \(10^{-3}\,\mathrm{cm}\). Moreover, he measured the heat-transfer coefficient, i.e., the power in watts dissipated by \(1\,\mathrm{cm}^2\) of surface, divided by

temperature difference:

\[ n=\frac{w}{\Delta T}, \tag{5.13} \]

and found that for small temperature differences \((\Delta T<0.01^\circ K)\) \(n\) is approximately proportional to \(T^3\). Kapitsa’s results are shown in Fig. 31.

Similar measurements, though by a less direct method, were subsequently carried out by Wyatt, Gonzalez, and Johnston W.\(^{53a}\), who confirmed the values and the temperature dependence obtained by Kapitsa. Osborne O.\(^{51}\), carrying out experiments with second sound, came to similar conclusions.

Gorter, Taconis, and Beenakker G.\(^{51d}\) gave a formal theory of the phenomenon of heat transfer across the boundary solid—helium II. They considered the motion of the superfluid and normal components perpendicular to the solid surface (along the \(z\)-axis), assuming that the temperature changes discontinuously. In this case, near the boundary the normal component is transformed into the superfluid component, or conversely. It is assumed that this transformation occurs at a finite temperature difference; therefore the temperature \(T\) near the boundary differs from the temperature \(T_0\) in the depth of the liquid.

Fig. 31. Temperature dependence of the heat-transfer coefficient \(n\) across the boundary solid—helium II; Kapitsa K.\(^{41f}\).

It is further assumed that the increase of \(x\), i.e. the fraction of the normal component \(\left(x=\frac{\rho_n}{\rho}\right)\), is directly proportional to \((T-T_0)\). Then the continuity equation has the form

\[ \operatorname{div} x v_n + a(T-T_0)=0, \tag{5.13′} \]

and the heat that has flowed through a unit surface is given by the formula*)

\[ w=v_n \rho T S^*-\lambda \operatorname{grad} T=\text{const}, \tag{5.14} \]

where \(\lambda\) is the normal classical coefficient of thermal conductivity of the liquid, while the first term describes heat transport by internal convection.

*) Here \(S^*\), equal to \(x\left(\frac{d^2G}{dx\,dT}\right)\) or \(x\left(\frac{dS}{dx}\right)\), is a quantity introduced by Gorter G.\(^{49b}\). Everywhere in the present review \(S^*\) is taken to be equal to \(S\).

Equations (5.13′) and (5.14) lead to an exponential dependence of \(T\) on \(z\)

\[ T - T_0 = \Delta T \exp\left(-\frac{z}{\delta}\right), \tag{5.15} \]

whence

\[ n=\frac{w}{\Delta T}=\frac{\lambda}{\delta} =\left[\frac{\lambda \rho T a S^*}{x}\right]^{1/2}. \tag{5.16} \]

Here \(\delta\) is the effective thickness of the layer of liquid in which the temperature jump occurs:

\[ \delta^2=\frac{x\lambda}{\rho T a S^*}. \tag{5.17} \]

It should be noted that the coefficient \(a\) is related to the relaxation time for the transformation of the superfluid component into the normal component, introduced by Gorter et al. \(^{\mathrm{G}.50a}\) into the theory of second-sound absorption (see Section 9.7). Using this relation between \(a\) and the results of measurements of second sound, Gorter and co-workers \(^{\mathrm{G}.51d}\) concluded that formula (5.16) does not contradict experiment.

Further investigation of this problem was carried out by Kronig and co-workers. Developing the theory of the propagation and attenuation of second sound in helium II (see Section 9.1), Kronig and Tellung \(^{\mathrm{K}.50}\) concluded that a third type of wave motion may exist, possessing an almost purely imaginary wave vector and, consequently, rapidly attenuated in space. They associated this motion with surface effects near the liquid–solid boundary and came to the conclusion that the thickness of the liquid layer in which these effects are appreciable is less than or equal to \(10^{-5}\) cm. Kronig, Tellung, and Woldring \(^{\mathrm{K}.52e}\), developing this theory, wrote the effective thickness \(\delta\) in the form

\[ \delta=\delta' + \delta'', \tag{5.18} \]

where \(\delta'\) is associated with the wave motion of the third type introduced by them earlier (for the limit \(\omega=0\)), while \(\delta''\) is due to relaxation according to the ideas of Gorter, Taconis, and Beenakker. For \(\delta'\) they gave the expression

\[ \delta'^2=\frac{\eta\lambda}{\rho^2 S^{*2} T}. \tag{5.19} \]

(Here \(S^*\) has the meaning indicated above.)

Recently Khalatnikov \(^{\mathrm{Kh}.52a}\) has suggested that the thermal resistance of the boundary is mainly explained by the emission of sound caused by thermal vibrations of the solid. At the time of writing the present review, the details of this work were not known.

(To be continued in Vol. LVII, issue 1, September 1955.)

1. This expression was first obtained by L. D. Landau. ↩