CURRENT STATE OF THE PROBLEM OF LIQUID HELIUM

(Conclusion)*

J. G. Daunt and R. S. Smith

CONTENTS

6. Surface films (saturated and unsaturated). 6.1. Film thickness and its heat capacity. 6.2. Theories of film formation. 6.3. Unsaturated helium film—static properties. 6.4. Superfluidity in unsaturated films. 6.5. Theories of the unsaturated film . . . . . . . . . . . . . . . . . . 93

7. Rate of helium transport along the film. 7.1. First experimental results. 7.2. New measurements of transport over the surface of glass. 7.3. Measurements of transport over surfaces of metals and plastics. 7.4. Determination of the linear flow velocity. 7.5. Critical velocity of film flow and its interpretation . . . . 109

8. Hydrodynamics of a two-component liquid. 8.1. Thermohydrodynamic equations. 8.2. Experimental studies of the nonlinearity of the equations of motion . . . . . . . . . . . . . . . . 126

9. First and second sound. 9.1. Velocity and attenuation of first sound. 9.2. Tisza and Landau equations for the velocity of second sound. 9.3. Other equations for the velocity of second sound. 9.4. Experimental study of second sound. 9.5. Effect of pressure on the velocity of second sound. 9.6. Rayleigh disk and Pitot tube in a second-sound field. 9.7. Attenuation of second sound . . . . . . . . . . . . . . . . . . . . . . 133

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6. SURFACE FILMS (SATURATED AND UNSATURATED)

6.1. Film thickness and its heat capacity

It was first shown by Rollin^R.36, and also by Rollin and Simon^R.39, that films are formed on all solid surfaces immersed in liquid helium. These films, which constitute one of the characteristic features of helium II, have been the object of numerous investigations (apparently concerned chiefly with their thickness and rate of transport).

* J. G. Daunt and R. S. Smith, Rev. of Mod. Phys. 26, No. 2, 172–236 (1954). (For the beginning of the translation see UFN, LVI, issue 3, 1955.)

The film thickness was first measured by Daunt and Mendelssohn \(^{\mathrm{D.38b}}\) and by Kikoin and Lazarev \(^{\mathrm{K.38b}}\). The first group of authors used a developed copper surface (area \(103\ \text{m}^2\)), held by a suspension thread near the surface of liquid helium. Along a thin wire attached to the copper surface and immersed in liquid helium, the film flowed onto the copper at the bath temperature. The copper surface was then moved into a region at room temperature, and the amount of helium evaporating from it was observed from the rise of the liquid level in the bath. In this way the authors were able to detect films with thickness \(10^{-7}\ \text{cm}\) and greater. As a result of the measurements the mean thickness of the surface helium film was found to be \(3.5\cdot 10^{-6}\ \text{cm}\). The film thickness depended little on temperature in the temperature interval \(1.59\)—\(2.14^\circ\text{K}\). It was found that above the \(\lambda\)-point \((2.18^\circ\text{K})\) the film thickness is less than \(10^{-7}\ \text{cm}\).

Using another experimental arrangement, also connected with evaporation of the film from a developed surface, Kikoin and Lazarev \(^{\mathrm{K.38b}}\) found for the mean film thickness, measured at an unspecified temperature, the value \(2 \div 3\cdot 10^{-6}\ \text{cm}\). These authors also reported a qualitative observation of a very rapid formation of the film: they speak of a “jump” of the bath level that occurred when the film ran over the large surface of their apparatus.

We must temporarily turn aside from our discussion of the question of film thickness in order to mention one difficulty in the interpretation of data relating to films and the resolution of this difficulty. The thickness measurements that we have just discussed are in fact measurements of volume, namely, of the volume of bulk liquid. In order to pass to the film thickness, one has to assume that the density of the helium film differs little from the density of the bulk liquid. The opinion that this method is justified not merely by considerations of convenience may perhaps be expressed on the basis of Fredericks’s \(^{\mathrm{F.49a}}\) measurements of the heat capacity of adsorbed helium films. The heat capacity was measured in a calorimeter in which known quantities of gaseous helium were adsorbed on the surface of crocus powder \((\mathrm{Fe_2O_3})\) with a total area of \(4000\ \text{m}^2\). The results are presented in Fig. 32. It was found that, for the thinnest of the films subjected to measurement, consisting, according to Fredericks, of approximately four atomic layers, the heat capacity differs very strongly from the heat capacity of the bulk liquid and shows no \(\lambda\)-point; but for films whose thickness is greater than approximately 20 layers, the mean heat capacity almost coincides with the heat capacity of bulk liquid helium. These experiments clearly demonstrate the smoothing of the \(\lambda\)-transition and its disappearance as the film thickness decreases. From these results it is reasonable to conclude that the averaged

the properties of an ordinary film consisting of more than 100 atomic layers are very close to the properties of the liquid in the vessel.

The experiments described above give only the mean value of the film thickness and say nothing about its variation with height above the level in the bath. But intuition, as well as comparison with water wetting glass, suggests that the film thickness may vary with height. The experiments of Jackson and Berg J.49a and B.51d had as their aim the determination of this variation, as well as the testing of theories of the helium film that had already been advanced by that time.

Jackson and Berg carried out a series of elegant experiments on measuring the thickness of a surface film in the undisturbed state. Their method, based on the method used by Rothen R.45 for determining the thickness of barium stearate layers, makes it possible to estimate the thickness of a helium film adhering to a stainless-steel mirror, the lower end of which is immersed in a bath of liquid helium II, from measurements of the state of polarization of the light reflected from the mirror. Plane-polarized light incident on the film, upon reflection from the film-coated mirror, becomes elliptically polarized, the eccentricity and orientation of the ellipse being a measure of the film thickness.

Fig. 32. Heat capacity of adsorbed helium (in cal/deg) according to F. Frederick F.49a. Curve 1—3–4 layers, curve 2—5–6 layers, curve 3—7–9 layers, curve 4—9–12 layers, curve 5—for liquid helium in a vessel. The indicated numbers of layers are given by Frederick. For a relatively new interpretation see Section 6.3.

(Of course, the refractive index of the film must be known. As in the analogous case mentioned above, in the absence of better data one has to use the properties of the bulk liquid.) The chief advantage of this method is that the film thickness can be measured at various heights above the level of the bath. Measurements can be made with equal success on a film at rest and on a moving film. Some of the results of Berg and

Jackson^B.51d for resting films are shown in Figs. 33 and 34. In these figures \(H\) denotes the height above the liquid level in the bath and \(\Delta N\) is a number proportional to the film thickness. It is seen that:

a) at a given height \(H\), the film thickness changes little with temperature in the interval between the extreme values of the temperatures used*), namely between \(1.1^\circ\) K and a temperature much lower than the \(\lambda\)-point; this result is in agreement with the results obtained earlier by Daunt and Mendelssohn^D.39b;

Fig. 33. Results of measurements of the temperature dependence of the film thickness at various heights \(H\) (Burdge and Jackson^B.51d). The angle of rotation of the Nicol prism \(N\) is proportional to the film thickness.

b) the film thickness falls to immeasurably small values at the \(\lambda\)-point \((2.18^\circ\text{ K})\) and at higher temperatures, which had also been observed earlier^D.39b. The behavior of the film in the region near the \(\lambda\)-point could be observed during a slow increase or decrease of the bath temperature. These results, relating to temperatures close to \(T_\lambda\), as we shall see later, have a certain theoretical significance. Subsequent observations by Jackson and Henshaw^J.53 showed, however, that at temperatures above \(T_\lambda\) the helium film has a thickness close to that of 10 atomic layers. In accordance with this observation, to the thicknesses obtained from Figs. 33 and 34 one must add a constant thickness of the order of 10 atomic layers. This surface film can easily be removed by radiation incident upon it.

* Henshaw and Jackson^H.51, using the same method, published a curve showing a slight monotonic change of the film thickness with temperature. It is not clear to us which of the results should be considered more reliable.

Berdzh and Jackson attempted to express the change in the thickness of the film with \(H\) at a given temperature by means of the formula

\[ d=\frac{d_0}{H^{1/z}}, \tag{6.1} \]

where \(d_0\) is the thickness of the film at a height of one centimeter.

A relation of this kind was used because it had already occurred in some theoretical works devoted to film thickness. However, experiments showed that \(z\) varies both with temperature and with the height \(H\). In other words, even at fixed temperature equation (6.1) is only approximate. For example, in measurements with a stationary film it was found that \(z_{\mathrm{avg}}=2.5\) at \(2.1^\circ\mathrm{K}\) and increases to 3.5 at \(1.1^\circ\mathrm{K}\); at the same time, at some unspecified temperature, \(z\) changed from 2.9 to 3.3 when \(H\) was changed from \(0.25\ \mathrm{cm}\) to \(1.2\ \mathrm{cm}\).

Unfortunately, a correction toward increasing them by the amount indicated by Jackson and Henshaw \(^{J.53}\) must be introduced into the results for \(z\) given in the preceding paragraph. Since Jackson and Henshaw did not give definite values of the film thickness

Fig. 34. Temperature dependence of the film thickness at temperatures close to \(T_\lambda\), according to the data of Berdzh and Jackson \(^{B.51d}\). The rotation angle of the Nicol, \(\Delta N\), is proportional to the film thickness. The black points refer to the upper, the light points to the lower temperature scales.

with allowance for the correction, we shall not recalculate \(z\) on the basis of their experiments. Bowers, in two recently published papers, added a number of new data important for the question under discussion. Bowers investigated the thickness of immobile helium films at temperatures above and below \(T_\lambda\) for both saturated and unsaturated films. His method consisted in weighing the film formed on aluminum foil, using microbalances mounted in a cryostat. In experiments with saturated films at temperatures below \(T_\lambda\), Bowers used foil \(7.6\ \mathrm{cm}\) high, which was immersed in liquid helium II and connected to it by a thin wire about \(3\ \mathrm{cm}\) long, so that the measured

the values of the film thickness proved to be averaged and corresponded approximately to \(H_{\mathrm{mean}} = 5\ \mathrm{cm}\) (equation (6.1)). Integrating equation (6.1) for the case of the foil he used, Bowers found that his results can be represented by the relation

\[ n = \frac{295}{H^{1/z}}, \tag{6.2} \]

where \(n\) is the number of layers and \(z = 2.0 \pm 0.3\). Bowers points out, however, that the number 295 does not necessarily lead to such a value of the film thickness at a height of \(1\ \mathrm{cm}\) above the liquid level, since the foil he used was on average raised above the liquid level much more than \(1\ \mathrm{cm}\). (For the method used to establish the relation between \(n\) and the film thickness, see Bowers’ work B.53d.) These results for \(z\) do not agree with the already cited results of Jackson and coworkers, nor with the results of Atkins (see below). Averaging equation (6.2) over the values of \(H\) in his experiment, Bowers obtained agreement with the earlier results of Daunt and Mendelssohn D.39b. At temperatures above \(T_\lambda\), Bowers found the film thickness to be approximately 10 layers, in agreement with Jackson and Henshaw J.53. Bowers also notes that the film thickness is extremely sensitive to radiation. As for the variation of film thickness with temperature, Bowers’ results are in quite satisfactory agreement with those of Burge and Jackson.

The results set forth in the preceding paragraphs refer to films at rest. By placing a heat source at the upper end of their steel mirror, Burge and Jackson were also able to observe films moving over the surface of their mirror. They found that the thickness of the moving film at a given height is as much as 20% greater than the thickness of the film at rest. Moreover, the variation of film thickness with height is also different in these two cases. Describing the data obtained from experiments with moving films by means of equation (6.1), the authors established that in this case \(z\) varies from 2.5 to 9 when \(H\) changes from 0.25 to 1.8 cm. These results were reported by the authors of paper B.51d only as preliminary, since at that time they did not have definite data on the thermal state of the moving film.

Up to now we have said nothing about the value of the parameter \(d_0\) in equation (6.1), obtained by Burge and Jackson. In their first communications J.49a and J.49b Jackson and Henshaw give the value \(d_0 = 1.9 \cdot 10^{-6}\ \mathrm{cm}\) at \(1.5^\circ\mathrm{K}\) as a preliminary value. In a later paper on the moving film, Jackson and Henshaw J.50 indicate a value of \(d_0\) agreeing with that mentioned above, so that the latter may be regarded as established. Both these values are also in approximate agreement with the values found by Atkins, which

will be given later. (The preliminary character of the first of the values named was connected with the difficulty of an absolute measurement of the film thickness as compared with its relative measurement.)

Atkins’ experiments A.50a and A.50b on determining the film thickness were also carried out with moving films. His method consisted in a quantitative investigation of a phenomenon first observed by Allen and Misener A.39a, who found that the level of liquid helium II in a reservoir from which the liquid flows out through a very narrow slit does not immediately assume its equilibrium value (coinciding with the level of liquid helium in the bath into which the reservoir is immersed), but executes oscillations*) (with an amplitude of the order of 1 mm) about the equilibrium position. Typical results of observations of these oscillations are presented in Fig. 35.

Atkins observed similar oscillations of the level of liquid helium II in a reservoir emptied only by transfer through a film when the levels in the vessel and the bath approached one another. In the mathematical treatment of the results of observations of these oscillations he made certain assumptions about the character of the motion, for example that the velocity of the motion is constant over the cross section of the film. As a result Atkins arrived at the following expression for the period of the oscillations:

\[ \tau = 2\pi \left\{ \frac{\rho}{\rho_s}\, \frac{r}{2z}\left(1+\frac{r}{R}\right) \int_0^l \frac{dH}{dl} \right\}^{1/3}. \tag{6.3} \]

The experiment to which this equation applies consists in the fact that the film moves along the inner and outer surfaces of a tube (with radii, respectively, \(r\) and \(R\)), projecting vertically to a height \(l\) above the surface of the helium in the bath. If it is assumed that the variation of \(d\) with \(H\) is given by equation (6.1), then, measuring \(\tau\) as a function of \(l\) and using equation (6.3), one can obtain the values of both parameters \(d_0\) and \(z\) of equation (6.1). In this way Atkins found that \(d_0 = 1.5 \cdot 10^{-6}\) cm at \(1.1^\circ\) K and \(d_0 = 2.4 \cdot 10^{-6}\) cm at \(2.0^\circ\) K and that \(z \cong 7\). These results, both for \(d_0\) and for \(z\), are in satisfactory agreement with the results of Burge and Jackson; however, it is necessary to recall that the data of the two works cannot be subjected to direct comparison, since Atkins obtained an average value of \(z\) for a region in which,

*) These oscillations were isothermal or quasi-isothermal, since the reservoir was connected with the helium bath through the gas phase. If, however, two reservoirs connected by a narrow slit are well insulated from one another, then the oscillations that can arise when the levels in the reservoirs approach one another will be strengthened by the thermomechanical effect. An analysis of such adiabatic oscillations was carried out by Robinson R.51, who came to the conclusion that experimental observations of such oscillations could prove effective from the point of view of measuring the temperature dependence of the entropy of the normal component of helium II.

as Berdge and Jackson showed, \(z\) changes noticeably. (For analogous reasons it remains unclear whether an exact comparison can be made between the results of Jackson and co-workers and those of Bowers.)

Atkins’s results were criticized by Kaganov and EselsonK.51, who, by modifying equation (6.3), obtained a dependence of \(d_0\) on temperature which, in their opinion, is in better agreement with the results of Berdge and Jackson. However, Dingle (R. B. Dingle), in a private communication, pointed out that he had carried out a direct derivation of Atkins’s equation, leaving no doubt that, under the assumptions made, it is correct.

Fig. 35. Graph showing the oscillatory character of the superfluid outflow of helium from a reservoir when the level difference is close to zero. Allen and MeissnerA.39a.

6.2. Theories of film formation

Theories of the formation of thick films on vertical surfaces immersed in a bath of helium II were proposed by FrenkelF.40, SchiffS.41, Bailey, de Boer and MichelsB.41, and TemperleyT.49b. Both Frenkel and Schiff assumed that the film

is formed as a result of the action of van der Waals forces between the atoms of the wall material and the atoms forming the film. Such forces, after integration over all the atoms of the wall, which is practically semi-infinite, lead to an expression for the potential containing the distance from the wall to the minus third power. If such a potential is added to the gravitational potential, one can verify that the equation of the free surface will be precisely equation (6.1) with \(z = 3\). This result is in satisfactory agreement with the experimental material now available; however, the difficulty is that the theory “explains more than is required.” It contains no indication of any distinction between helium I and helium II. Of course, strictly speaking, this theory applies only to absolute zero, since it considers the internal energy rather than the free energy; but if the observed approximate constancy of the film thickness in the temperature interval \(1—2^\circ\) is taken as proof of the validity of the theory in this temperature range, then, from the point of view of the theory itself, there apparently are no reasons that would prevent the film from existing in the helium I region. Schiff assumes that the observed difference in the film thickness for the cases of helium I and helium II can be explained by the difference in the hydrodynamic properties of these two phases: in the superfluidity region, rapid transport of matter leads to the formation of a film when the temperature is lowered and balances any evaporation of the film when the temperature is raised, whereas in the helium I region the viscosity hinders the formation of the film and leads to the evaporation of any film already formed at a temperature below the \(\lambda\)-point, if the temperature, increasing, passes through the \(\lambda\)-point. However, this assumption was to some extent refuted by Burge and Jackson. Jackson and co-workers showed experimentally that at the \(\lambda\)-point the film thickness drops discontinuously to a very small value (\(\sim 10\) atomic layers), and that the temperature of the sharp decrease in film thickness does not depend on the rate at which heat is supplied to the film. This apparently excludes the possibility of explaining the disappearance of the film at \(T_\lambda\) solely by the hydrodynamic properties of the helium film.

Beal, de Boer, and Michels \(^{\text{B.41}}\) approached the film problem from another point of view. They regard all the helium atoms in the film as particles of an ideal Bose–Einstein gas condensed at the lowest energy level. This lowest energy is determined by the thickness \(d\) of the film. The part of the wave function depending on the motion perpendicular to the wall is sinusoidal, and the energy per particle is equal to

\[ \varepsilon = \frac{h^2}{8 m d^2}, \tag{6.4} \]

where \(h\) is Planck’s constant and \(m\) is the mass of a helium atom. If the gravitational potential energy is added to this zero-point energy and it is assumed that the number of particles \(n\) falling on unit area of the film is proportional to its thickness, i.e. \(n=Kd\), then the energy per unit area of the film at constant height \(H\) will be

\[ E=Kd\left(mgH+\frac{h^{2}}{8md^{2}}\right). \tag{6.5} \]

The thickness \(d\) at constant \(H\) is found from the condition

\[ \frac{dE}{dd}=K\left(mgH-\frac{h^{2}}{8md^{2}}\right)=0. \]

Thus, in equilibrium

\[ d=\left(\frac{h^{2}}{8m^{2}g}\right)^{1/2}H^{-1/2}. \tag{6.6} \]

This dependence of \(d\) on \(H\) is in agreement with the results of Bowers and, possibly, is not very far from the experimental data of Burge and Jackson. The constant factor in equation (6.6), which contains no arbitrary parameters, agrees, to within a factor of the order of five, with the observed value \(d_0\), which is certainly a surprising result. In conclusion, we point out that according to this theory the change in film thickness on passing through \(T_{\lambda}\) is qualitatively explained by the initial assumption that the particles forming the film are Bose–Einstein particles in their lowest state. When the temperature rises to values exceeding the \(\lambda\)-point, the film disappears, since the number of particles suitable for forming the film falls to zero.

However, it is necessary to note that equation (6.5) contains only positive terms. In equilibrium \(E=2KdmgH\), so that, apparently, there is no reason at all for the formation of a film. To overcome this difficulty it is necessary to introduce an interaction with the wall, and moreover one sufficiently strong to raise the particles from the level of the bath, where the energy given by equation (6.4) would have to be equal to zero. However, one may expect that any reasonable potential of interaction with the wall would lead to a change in the wave functions and, consequently, in the dependence of the energy on the thickness of the film. Another serious objection was raised by Mott[^49b], who pointed out that the sinusoidal wave functions required for the validity of equation (6.4) imply large changes of density across the section of the film, so that all the particles prove to be concentrated at the center of the film. In the presence of any interaction whatever between the particles, this could not correspond to the state with lowest

energy. A more acceptable wave function would be a wave function with an amplitude constant over the greater part of the film cross section, not leading to an energy dependence of the type \(d^{-2}\).

A more artificial theory of the film was given by Temperley \(^{49b}\). He starts from a one-dimensional wave equation in which the energy of the van der Waals interaction of a given helium atom with the wall and with the other helium atoms in the film is taken into account; in other words, the remaining helium atoms provide a self-consistent field. His wave equation has the form

\[ \frac{\partial^2 \psi}{\partial x^2} + \left[ \frac{8\pi^2 mE}{h^2} + \frac{k^2}{(s+d+x)^3} + \frac{l^2}{(s+x)^3} - \frac{l^2}{(s+d+x)^3} \right]\psi=0. \tag{6.7} \]

In this equation \(s\) is the diameter of the atom, \(d\) is the thickness of the film, \(k^2\) is a measure of the energy of the atom associated with the presence of the wall, and \(l^2\) is a measure of the energy of the atom in the field of the remaining atoms of the film. By variational methods Temperley finds that the lowest energy level in this problem is

\[ E=-\frac{h^2}{32\pi^2 m}\left(a+\frac{b}{d^2}\right)^2, \tag{6.8} \]

where \(a\) is a constant depending on \(s\) and \(l\), and \(b\) is a constant depending on \(k\) and \(l\). The gravitational energy is added to equation (6.8), and the equation of the equipotential surface is taken as the equation of the film surface,

\[ mgH-\left(\frac{h^2}{32\pi^2 m}\right)\left(a+\frac{b}{d^2}\right)^2=\mathrm{const}. \tag{6.9} \]

The experimental data hardly make it possible to estimate the constants accurately; however, Temperley finds that there is agreement, in order of magnitude, between the slope of the film surface determined from equation (6.9), using the values of \(a\) and \(b\) found by him, on the one hand, and the experimental results of Jackson and Burge, on the other. We note in passing that the form of equation (6.9) emphasizes the fact that an equation of the type (6.1) is, evidently, quite unsuitable for describing the experimental results. We have already mentioned the changes in \(z\) over wide limits that arise when the experimental data are substituted into equation (6.1).

Temperley also qualitatively predicted the temperature dependence of the film thickness. According to his expectations, the film thickness, plotted as a function of temperature, should have a shallow maximum at approximately \(1.8^\circ\mathrm{K}\) and fall to zero at the \(\lambda\)-point. This prediction is not confirmed by the experimental results shown in Fig. 33.

J. G. Daunt and R. S. Smith

6.3. Unsaturated helium film—static properties

An unsaturated helium film is a film in equilibrium with its vapor, the pressure of which is less than the saturated-vapor pressure at the given temperature. Its properties have been studied

Fig. 36. Helium adsorption isotherms on crocus powder. Dependence of the adsorbed volume \(v\) (in cubic cm of helium at normal temperature and pressure per \(1\ \mathrm{m}^2\) of adsorbing surface) on the degree of saturation \(p/p_0\) at various temperatures: \(4.21^\circ\mathrm{K}\) (lower curve), \(3.02^\circ\mathrm{K}\), \(2.42^\circ\mathrm{K}\), \(2.25^\circ\mathrm{K}\), \(2.14^\circ\mathrm{K}\), \(2.01^\circ\mathrm{K}\), \(1.80^\circ\mathrm{K}\), and \(1.59^\circ\mathrm{K}\) (upper curve). Data of Stratuss (see Long and Meyer \(^{53a}\)).

by several investigators, and it must be admitted that the results of these studies are extremely contradictory. These results and the problems connected with them were the subject of a review,

published recently by Long and MeyerL. 53a, to whose work we refer the reader for further details; however, new data published after the appearance of their review do not agree with all the results presented in that review.

The principal subject of investigation is adsorption isotherms. These isotherms were obtained for the case of adsorption on glass by Keesom and SchmidtK. 33b, Keesom and ShveersK. 41c, KistemakerK. 47b, and Brewer and MendelssohnB. 53e, on crocus by Long and MeyerL. 49 and StraussL. 53a, on charcoal by Schiffer, Smith, and WundellS. 49, on crocus and steel by Frederiks and GorterF. 50, on TiO₂ by Mastrangelo and AstonM. 51a, and on aluminum foil by BaurB. 53d.

The general picture obtained is illustrated in Fig. 36, which shows the adsorption isotherms on Fe₂O₃ obtained by Strauss; however, there are deviations from this picture. The results of Frederiks and Gorter are similar to those obtained by Strauss, whereas Long and Meyer found that at temperatures below \(T_\lambda\) the isotherms

Fig. 37. Temperature dependence of helium adsorption on \(0.5\ \mathrm{m}^2\) of glass for various values of the degree of saturation \(p/p_0\) (in percent), according to Brewer and MendelssohnB. 53e. The dashed line shows the temperatures at which superfluidity appears for various values of \(p/p_0\).

may be superposed on one another. The most striking, however, is the result of Brewer and Mendelssohn, shown in Fig. 37, from which it follows that, on a graph similar to that depicted in Fig. 36, the isotherms for \(T > T_\lambda\) lie lower.

In addition to the discrepancies noted above, there is considerable uncertainty in the question of the maximum thickness of the adsorbed

films at temperatures below the \(\lambda\)-point for a pressure \(p\) (at which the measurement is made) somewhat less than \(p_0\) (the saturated-vapor pressure). Kistemaker’s data, recalculated by Fredericks and Kistemaker (see Long and Meyer \(^{\mathrm{L}.53a}\), p. 11), led, in the case of a film very close to saturation, to a thickness equal to 150 atomic layers. Long and Meyer \(^{\mathrm{L}.49}\) give a similar value. However, Bowers \(^{\mathrm{B}.53d}\) observed films not exceeding 20 atomic layers in thickness, right up to a saturation of 99.92%. Another discrepant result was recently reported by Brewer and Mendelssohn \(^{\mathrm{B}.53f}\) (see also Brewer and Mendelssohn \(^{\mathrm{B}.53g}\)). In their experiments a helium II film was deposited on a stack of glass plates, the distance between which was about \(10^{-4}\) cm; the amount of gas desorbed upon heating the plates to a temperature above \(T_\lambda\) was then measured. This amount exceeded by a factor of ten the amount expected for the case of a normal helium II film\(^*\). The authors attribute this excess to the formation of thickenings associated with anomalous surface tension.

Despite the already mentioned discrepancies in the results for nearly saturated and saturated films, adsorption isotherms for the case of small values of \(p/p_0\) can quite well be discussed in the language of the Brunauer—Emmett—Teller theory of multilayer adsorption \(^{\mathrm{B}.38}\). This theory predicts the course of the isotherms and, in addition, makes it possible to calculate certain physical constants of the adsorbed substance. In particular, the volume of the first adsorbed layer can be determined from the isotherms. The behavior of helium differs from that predicted theoretically in that the volume of the first adsorbed layer, calculated from the isotherms, is approximately four times greater than the volume of a monolayer possessing the properties of bulk helium. This shows that the first adsorbed layer has a higher density than bulk liquid helium. The factor of four is precisely the ratio of the volume of atoms with the interatomic distances found in the liquid to the volume of densely packed helium atoms, whose diameter is determined in the hard-sphere approximation. The exceptionally large volume of liquid in the vessel is attributed to a high zero-point energy (see Simon \(^{\mathrm{S}.34}\) and Benevis and Simon \(^{\mathrm{B}.23}\)); in the adsorbed film, large attractive forces from the wall overcome the action of zero-point repulsive forces and lead to the formation of close packing, so that the first layer is, in essence, rather a layer of a two-dimensional solid than a layer of liquid. These results lead

* Similar unpublished experiments by Gager (W. B. Gager) and one of us, carried out in 1952 on horizontal stacks of brass, glass, and aluminum plates separated by wire spacers of diameter \(0.005\) cm, also led to anomalous values of the apparent film thickness (\(\sim 50\cdot 10^{-6}\)) in the case of brass and Al. The results for the case of glass were not anomalous.

to the new interpretation of Frederikse’s data on the heat capacity of the adsorbed film (see Fig. 32). The first four layers in Frederikse are in fact one layer, which is under conditions of strong compression and, as the heat capacity shows, has the properties of a two-dimensional solid.

A modification of the Brunauer—Emmett—Teller theory, required by the above-mentioned anomalous packing of the atoms of the first layer, was undertaken by BandB.49b and by Aston and MastrangeloA.51c. Both attempts led to improved agreement between theory and experiment for sufficiently small values of \(p/p_0\). (Deviations from the Brunauer—Emmett—Teller isotherms are observed in a quite general way for values of \(p/p_0\) greater than 0.4.) BandB.51e also gave a possible explanation of adsorption isotherms on the basis of the conception of condensation of an ideal Bose–Einstein gas in a layer of two-dimensional gas bound to the surface with energy \(W(n)\), where \(n\) is the number of the layer counted from the solid surface. The form of the resulting isotherms depends on the relation between \(W\) and \(n\). Unfortunately, the best agreement with the experimental data is not obtained for the most natural choice of this dependence.

Recently, Textra, Hoftman, and Meydenberg (Physica 19, 935 (1953)) used an ingenious differential McLeod manometer to measure adsorption isotherms of helium II on glass. Within the accuracy of their measurements, the isotherms taken at temperatures in the interval from \(1.5^\circ\) K to \(T_\lambda\) cannot be distinguished from one another.

6.4. Superfluidity in Unsaturated Films

Long and Meyer reported experiments undertaken with the aim of detecting superfluidity in adsorbed filmsL.50a and L.54a. In one series of experiments (their “method I”) they found that, when the temperature was lowered, superfluidity set in at a quite definite temperature below \(2.18^\circ\) K, and that this temperature decreased as the film thickness decreased. However, in another series of experiments (their “method II”), in which their system, it may be supposed, was closer to a state of mechanical equilibrium, superfluidity appeared at \(T_\lambda\) (for bulk liquid helium) for films of any thickness. The latter result seems to testify in favor of the conclusion that the position of the \(\lambda\)-point does not depend on the film thickness. However, experiments by Bowers, Brewer, and MendelssohnB.51a on heat transfer in unsaturated films lead to the opposite conclusion. These investigators found that, for a given ratio \(p/p_0\), the heat flux which itself, one may think, is connected with superfluidity, decreases when the temperature is raised. In the case

for a thin film the heat flow fell to zero at a low temperature, whereas for a thick film the cessation of heat flow occurred at a higher temperature. Recently Long and Meyer[^L52c] repeated the experiments of Bowers, Brewer, and Mendelssohn and obtained results consistent with those of these investigators. In addition, the values of the temperatures at which, in these experiments, heat transfer just begins, for a given film thickness, agree with the temperatures found by method I in the work of Long and Meyer[^L52a], and indicate a change of \(T_\lambda\) with film thickness.

Further information on this question was obtained in the work of Brewer and Mendelssohn[^B5e], mentioned in Section 6.3; the results of Brewer and Mendelssohn are shown in Fig. 37. The onset of superfluidity occurred at an increasingly lower temperature as the film thickness decreased; however, this point apparently did not correspond to any anomaly in the observed isotherms. On this basis Brewer and Mendelssohn maintain that, by studying the isotherms, one cannot find any indications of a transition from the superfluid to the nonsuperfluid state in an adsorbed film.

6.5. Theories of the Unsaturated Film

In addition to comparing observed isotherms with theoretical adsorption isotherms, some authors have attempted to give a more far-reaching thermodynamic interpretation of these data. These efforts include calculations aimed at correlating data on adsorption isotherms (Section 6.3) with data on superfluidity (Section 6.4), as well as consideration of isotherm data in order to find indications of a \(\lambda\)-transition. (The beginning of the onset of superfluidity upon lowering the temperature need not necessarily coincide with the appearance of thermodynamic anomalies.)

Conclusions concerning heat capacity, entropy, enthalpy, etc., were drawn from absorption data, for example, by Mastrangelo and Aston[^M51a] and Aston[^A51a], Meyer and Long[^M52], Long and Meyer[^L53a][^L53b], and Rice and Eidon[^R53c]. However, in view of the discrepancies among the available data discussed in Section 6.3, the authors of the present article believe that arguments based on derived thermodynamic quantities can hardly be relied upon.

The discussion of the problem of the helium film on the basis of the properties of an ideal Bose–Einstein ensemble has a somewhat different character. Osborne[^O49b] showed that a two-dimensional Bose–Einstein gas does not undergo condensation comparable with condensation in a three-dimensional gas, but rather is characterized by a gradual “accu-

“...accumulating” into the lowest energy state. This accumulation occurs at a temperature much lower than the temperature of condensation of the gas in the vessel. ZimanZ.53a investigated the properties of ensembles having the shape of a rectangular slab. If \(L_1\) is the finite edge length of this slab and \(L_x \ll L_y = L_z\), then the \(\lambda\)-temperature of such an ensemble is lower than the \(\lambda\)-temperature of an ensemble of cubic form \((L^3)\). Assuming that liquid helium, both in a bath and in films, can be divided into “domains” with edge length of the order of \(10^{-5}\) cm, Ziman was able to obtain the variation of the \(\lambda\)-temperature with the thickness of an ideal Bose–Einstein film, which reproduces very well the data on the onset of superfluidity obtained in the already discussed experiments of Bowers, Brewer, and MendelssohnB.51a.

7. RATE OF TRANSFER OF HELIUM ALONG A FILM

7.1. First experimental results

The remarkable property of helium II—the flow of helium along a surface film—was discovered and investigated by RollinR.36, R.39; subsequently this property was studied in detail by Daunt and MendelssohnD.38b, D.38c, D.39b, D.39c. The basic properties of the flow of a surface film, discovered by these investigators, have repeatedly been presented in reviewsK.42, B.40, D.40, K.48, M.49c, S.53b; however, in order to clarify the relation of the first results to recent investigations, it is useful to consider these results briefly. For details of the experiments the reader is referred to the earlier reviews.

Fig. 38. Dependence of transfer along a film from a glass vessel on time (after Daunt and MendelssohnD.39c). The curve represents the level of liquid helium in the vessel at different moments of time.

The first investigations of Daunt and Mendelssohn were carried out under isothermal conditions and consisted in measuring the transfer of helium II from one vessel to another along the connecting surface. The course of a typical experiment is shown graphically in Fig. 38, which gives the time dependence of the height of the level of liquid

...helium in a glass test tube, from which the helium flowed as a film along the surface of the test tube into the bath. It should be noted that, except for the anomalous region, when the inner level was near the very rim of the test tube, the rate of lowering of the inner level remained constant, and did not change even when, at the 33rd minute, the difference of the levels was suddenly changed. As a result of a whole series of measurements, Daunt and Mendelssohn came to the conclusion that at temperatures below the λ-point:

1) under isothermal conditions, helium II always collected at the lowest possible level,

2) transfer of helium from one reservoir to another took place in the form of a film formed on the solid surface connecting the reservoirs,

3) the rate of transfer was practically independent of the difference in levels*),

4) the rate of transfer did not depend on the material of the wall,

5) the rate of transfer was proportional to the perimeter of the connecting surface,

6) transfer from one vessel to another was limited by the narrowest part of the connecting surface situated above the upper level. At places situated below the upper level, the liquid from the film could collect into drops**),

Fig. 39. Dependence of the rate of transfer through a film on a glass surface (in cm³/sec cm of width) on temperature (Daunt and Mendelssohn D. 39b).

7) the rate of transfer, within the errors of the experiment, did not depend on the length of the connecting surface,

8) numerical values of the rate of transfer of helium (over a glass surface), measured in cubic cm of transported liquid per 1 cm of film perimeter per 1 sec, as a function of temperature, are given in Fig. 39.

In other experiments by Daunt and Mendelssohn D. 39a, D. 39b and Rollin and Simon R. 39 it was shown that under non-isothermal

*) Daunt and Mendelssohn observed a small change in the rate of transfer with height when the observations were made at height differences of more than 6 cm. The change in the rate of transfer was approximately 2% per centimeter of height.

**) An elegant visual confirmation of the latter circumstance was obtained recently by Jackson and Henshaw J. 53 and by Hamm and Jackson H. 53 with the aid of an optical method (see also Section 7.4).

under such conditions the helium film moves to those places where the greatest evolution of heat occurs, and the flow velocity of the film coincides with the value obtained under isothermal conditions.

7.2. New measurements of transport over the surface of glass

The picture of the phenomenon described above, which had been regarded as established, was called into question in 1948 and early 1949 by the experiments of Atkins A.^48 and of de Haas and van der Berg H.^49, who obtained results that differed greatly from those given above. In connection with this, interest in the phenomenon of transport of a helium film was renewed, and during 1949–1952 many works were carried out on this question, invariably confirming the results of the early investigations of Daunt and Mendelssohn.

In view of the great theoretical significance of the data on the flow of the film (see below), it is necessary to give some information on the work of Atkins and of de Haas and van der Berg. Atkins’s experiments consisted in studying the isothermal flow of a helium II film from a glass test tube; his results proved to be different from the results of Daunt and Mendelssohn summarized in items 3, 5, 7, and 8 of Section 7.1. Atkins A.^48 observed transport velocities exceeding by more than a factor of 5 the velocity values at the same temperatures shown in Fig. 39; moreover, he found appreciable changes in the transport velocity with the length of the film connecting the vessels.

Atkins further reported the dependence he had observed of the transport velocity on the difference in levels; at a constant level difference, the transport velocity depended in a complicated way on the width of the connecting surface and on the geometry of the test tube. De Haas and van der Berg, in a short communication, gave the results of their experiments, which also concerned the isothermal outflow of a helium film from a glass test tube. The transport velocity they obtained for \(1.4^\circ \mathrm{K}\) was equal to \(140 \cdot 10^{-5}\ \mathrm{cm^3/(cm \cdot sec)}\) (compare with the value in Fig. 39, equal to \(7.5 \cdot 10^{-5}\ \mathrm{cm^3/(cm \cdot sec)}\)); the transport velocity depended strongly on the difference in levels. De Haas and van der Berg explained their result by the fact that in their experiments radiation from external sources was prevented from reaching the surface with the film.

Subsequent experiments on the study of the flow of a film from glass test tubes, carried out with the use of a varied experimental technique, gave values of the transport velocities very close to the data of the first experiments of Daunt and Mendelssohn. Confirmation of the correctness of the early experiments was obtained, for example, by Lane with collaborators W.^49, F.^49, Esselson and Lazarev E.^51, Burks and Dash W.^50d, and then by Atkins A.^50b himself and by de Haas and van der Berg B.^51f.

The credit for elucidating the physical causes of the anomalous results of Atkins[^48] and of de Haas and van der Berg[^49] belongs to Mendelssohn and his collaborators.

A series of elegant experiments by Bowers and Mendelssohn[^49c][^50e] showed that the anomalous effect observed by Atkins and by de Haas and van der Berg is secondary in character and does not reflect the true properties of film flow. Comparing the experimental methods of various investigators, Bowers and Mendelssohn came to the conclusion that the diversity of the results obtained is explained by the fact that the state of the film substrate—the solid surface over which the film flows—differs in different experiments; in particular, the substrate may be contaminated by condensed gases. They carried out experiments on the isothermal outflow of a helium film from a glass test tube, applying controlled contaminations to the substrate. The results of a series of such experiments are given in Fig. 40,

Fig. 40. Dependence of transport along the film from a glass vessel on time for different states of the glass surface (see text). Bowers and Mendelssohn[^50c].

on which the height of the helium II level inside the test tube is plotted as a function of time. Curve 1 corresponded to film flow over a very clean glass surface. As before, the transport velocity, apart from a small region near the edge of the test tube, is independent of the difference in levels (see Section 7.1) and coincided with the value obtained earlier by Daunt and Mendelssohn[^39c]. Curve 2 reproduces the results of flow over glass on which a thin layer of solid air had previously been deposited by condensation of gaseous helium with a small admixture of air directly in the Dewar containing the test tube. In this case the transport velocity increased slightly at the beginning of the flow, but the general character of the flow remained the same. Curves 3, 4, and 5 reproduce the resul-

tats obtained after three successive depositions of air on the test tube at 2° K. It is seen that the character of the film flow changed completely. The rate of transfer at the beginning of the flow increased many times over; its dependence on the difference of levels coincided with that observed by Atkins A.^48 and de Haas and van der Berg H.^49. Curve 5 corresponded to “saturation” in the sense that subsequent depositions of air did not noticeably change the results.

It should be noted that this deposited film of solid gas was completely transparent and colorless and that similar results were obtained when H₂ and He were used as contaminants. In addition, it should be noted that deposition of the solid film occurred immediately as soon as the contaminated gas was admitted into the system, and irrespective of whether the test tube had been removed from the liquid helium or immersed in it. After warming the apparatus to room temperature and cleaning it by prolonged pumping, it was possible to reproduce the results shown by curve 1.

Fig. 41. Apparatus of Bowers and Mendelssohn V.^50e for studying the influence of radiation on the rate of transfer along a film.

Labels in the figure: suspension \(R\); copper jacket \(C\); screens \(S\); test tube \(B\); liquid helium.

On this basis it was concluded that the large values of the transfer rate, its dependence on the difference of levels, and other anomalous effects observed by Atkins and by de Haas and van der Berg were connected with contaminants of the substrate, whose granular structure and random character greatly complicated the phenomenon. In light of numerous studies of film transfer over the surface of metals, it seems possible that the increase in the transfer rate over contaminated surfaces is connected with an increase in the perimeter of the surface caused by the presence of a deposit.

The possibility of an effect of changing the transfer rate upon illumination of the film surface, proposed by de Haas and van der Berg, was also investigated by Bowers and Mendelssohn V.^50c. In their experiment a glass test tube \(B\) (see Fig. 41) was placed

in the copper jacket \(C\), in which two narrow slits had been cut for observing the liquid level in the test tube. This entire system was externally protected by copper shields \(S\), so that, by rotating the vessel with the test tube on the suspension \(R\), it was possible either to shield the test tube completely from external radiation or to make observations of the level possible. The transfer of helium II from the test tube took place alternately under irradiation and without it; the results are shown in Fig. 42. As can be seen from the curve (Fig. 42), all points

Fig. 42. Typical result of the experiments of Bowers and Mendelssohn\({}^{\mathrm{B}.50e}\) with the apparatus shown in Fig. 41. The curve plots the time dependence of the height of the helium level in the test tube during its filling.

lay on one and the same straight line; consequently, the rate of transfer in the absence of irradiation of the film remained unchanged. The absolute value of the transfer rate agreed with the results of earlier work\({}^{\mathrm{D}.39c}\), as was to be expected for a clean surface. The absence of an effect of irradiation on the transfer rate was subsequently confirmed by Atkins\({}^{\mathrm{A}.50b}\).

The subsequent work of Brown and Mendelssohn\({}^{\mathrm{B}.50f}\) on the outflow of the film from clean glass test tubes of different shapes also confirmed the earlier investigations of Daunt and Mendelssohn and showed that Atkins’s results are of a secondary character and are connected with contamination of the surface.

Mendelssohn and White\({}^{\mathrm{M}.50a}\) investigated in detail the temperature dependence of the transfer rate over clean glass surfaces; their results, together with the results of the earlier work of Daunt and Mendelssohn\({}^{\mathrm{D}.39c}\) and the later work of Weber, Fairbank, and Lane\({}^{\mathrm{W}.49}\), are given in Fig. 43. It is seen that the agreement of all these measurements with one another is satisfactory. Mendelssohn and White proposed the following expression, describing their

results:

\[ R=A\left[1-\left(\frac{T}{T_\lambda}\right)^\alpha\right], \tag{7.1} \]

where \(R\) is the rate of transport in \(\text{cm}^3/\text{sec}\) per cm of width, and \(\alpha\) lies between 6 and 8. Unfortunately, the errors of measurement do not permit a more

Fig. 43. Temperature dependence of the rate of transport over a film on a glass surface (in \(\text{cm}^3/\text{sec}\) per cm of width). (Solid curve according to the data of Mendelssohn and White \(^{\text{M. 50a}}\), dashed curve according to Webber et al. \(^{\text{W. 49}}\), dotted curve according to Daunt and Mendelssohn \(^{\text{D. 39c}}\).)

precise determination of the value of \(\alpha\). The constant \(A\) for glass lies between \(7.3\cdot 10^{-5}\) and \(7.65\cdot 10^{-5}\ \text{cm}^3/\text{sec}\,\text{cm}\).

Further, Mendelssohn and White confirmed the earlier results of Daunt and Mendelssohn, namely that (a) the rate of transport is insensitive to the height of the film and (b) the rate of transport increases when the helium level is very close to the rim of the vessel. They also noted that, when the difference of levels is less than 3 mm, the rate of transport decreases. A similar effect was also observed by Atkinson \(^{\text{A. 50b}}\) in experiments with clean glass test tubes (see Fig. 44).

Fig. 44. Dependence of the rate of transport over a film on glass on the difference of levels (pressure difference). Atkinson \(^{\text{A. 50b}}\).

These deviations of the transport velocity from a constant value have not yet received a satisfactory explanation; one interesting observation by Jackson and co-workers, connected with these effects, will be described below. The question of film flow at very small level differences was again examined experimentally by Pikus. In a preliminary communication P.53a it was stated that the full transport velocity (i.e., the velocity corresponding to large level differences) was observed even for level differences of the order of 1 mm. Further discussion of this question requires more complete information.

Fig. 45. Transport velocity in a film on glass at temperatures below \(1^\circ\) K (Ambler and Kurti A.52b).

Investigating the temperature dependence of the transport velocity, Ambler and Kurti A.52b made measurements at temperatures below \(1^\circ\) K. They observed the outflow of liquid helium from a glass test tube that had been cooled to temperatures down to \(0.15^\circ\) K by direct contact with a paramagnetic salt. Their preliminary results are shown in Fig. 45. A considerable scatter is evident in the numerical values of the transport velocity obtained in different experiments; nevertheless, there is no doubt that these values lie above those obtained in the work of M.50a for glass in the temperature range from \(1^\circ\) K to \(2.18^\circ\) K. In Fig. 45 the low-temperature results are “normalized,” i.e., shifted so that at \(1.2^\circ\) K they coincide with the data of M.50. Of interest in these results is the rapid increase of the transport velocity with decreasing temperature near \(0.5^\circ\) K. Lesen and Burse L.52b measured the flow velocity of the film over a copper surface down to \(0.75^\circ\) K; their measurements also revealed a tendency for the transport velocity to increase as the temperature was lowered below \(1^\circ\) K.

7.3. Measurements of transport over surfaces of metals and plastics

In their early work, Daunt and Mendelssohn noted (see Section 7.1) that the transport velocity of a surface film of helium II is the same over a polished copper surface and over glass. However, they noted that the transport velocity over drawn copper wire

was somewhat higher. The question of the influence of the (uncontaminated) substrate on the rate of transfer has in recent years been thoroughly investigated experimentally. The results of this investigation have proved rather complex.

Observations of the rate of transfer of helium II over copper, stainless steel, lucite, iron, and lead (treated in various ways) were made by Bowers and DashB.50d, B.50g, B.51g, over platinum and nickel by Mendelssohn and WyattM.50a, M.50b, over stainless steel by Jackson and HenshawJ.50, over plastics—lucite and perspex—by ChandrasekharC.52b, and over stainless steel by Chandrasekhar and MendelssohnC.52c. It was found that the general character of the temperature dependence of the rate of transfer over metallic surfaces is the same as in the case of glass, but the absolute values of the transfer rate proved appreciably larger. For example, as reported by Bowers and DashB.51g, the rate of transfer over etched copper at \(1.5^\circ \mathrm{K}\) is \(50 \cdot 10^{-5}\ \mathrm{cm}^3/\mathrm{cm\,sec}\).

In connection with measurements of the outflow of helium from opaque test tubes, which must be carried out in studying transfer over metallic surfaces, it is appropriate to mention the method of Bowers and DashB.50d, B.50g, B.51g, distinct from the visual method used by all the other investigators. In the method of Bowers and Dash the indicator of the total amount of liquid helium in the vessel was the dielectric properties of the liquid helium located between the electrodes of a cylindrical capacitor. A change in the level of liquid helium in the annular gap of the capacitor led to a change in its capacitance, which was measured from the change in the frequency of an oscillatory circuit of which this capacitor formed a part. The capacitor, serving to measure depth, was placed inside a test tube made of the material along whose surface the transfer rate was to be measured.

After ordinary mechanical treatment, the surfaces of metals and plastics were, in general, rougher and more prone to surface defects than the fire-polished surface of glass. It was therefore to be expected that these defects of the surfaces of metals and plastics would lead to increased values of the transfer rate, just as artificial damage to a glass surface by depositing solid air on it (see Sec. 7.2) led to an increase in the transfer rate. This conclusion is confirmed by: (a) the work of Bowers and DashB.50g, B.51g, who established that the transfer rate at \(1.5^\circ \mathrm{K}\) over mechanically treated copper was \(14.8 \cdot 10^{-5}\ \mathrm{cm}^3/\mathrm{cm\,sec}\), and after etching of this same vessel increased to \(49 \cdot 10^{-5}\ \mathrm{cm}^3/\mathrm{cm\,sec}\) at the same temperature, and it turned out that the irregularities of the copper surface after etching were visible under a microscope; (b) the work of ChandrasekharC.52b, which showed that the transfer rate over perspex at \(1.5^\circ \mathrm{K}\) changes from \(16.0 \cdot 10^{-5}\) to \(10.7 \cdot 10^{-5}\ \mathrm{cm}^3/\mathrm{cm\,sec}\) after polishing with crocus; (c) the work

Chandrasekhar and Mendelson^C.52c, who showed that the rate of transfer over the surface of stainless steel, after its final finishing, is the same as for pure glass \((R = 8.0 \cdot 10^{-5}\ \mathrm{cm}^3/\mathrm{cm}\ \mathrm{sec})\), whereas the very same surface, after being heated to red heat, which disturbed the cleanliness of the treatment, and lightly polished, gave the value \(R = 12.8 \cdot 10^{-5}\ \mathrm{cm}^3/\mathrm{cm}\ \mathrm{sec}\).

It may be concluded, especially in connection with the above-mentioned work of Chandrasekhar and Mendelson, that the rate of transfer over different materials does not differ from the rate of transfer over uncontaminated glass, provided only that the surfaces of the materials can be made as smooth as the surface of glass. This result was obtained for copper in an earlier work of Downt and Mendelson. The same conclusion was, in the main, confirmed by the recent work of Smith and Burse^S.53a.

Apparently one may conclude that the increase in the rate of transfer over roughly treated surfaces is connected with an increase in the perimeter of the joining surface, just as occurred in the case of contaminated glass. This question will be considered below, in Section 7.5.

In conclusion one may point to one circumstance illustrating the complexity of the phenomenon of film transfer over uneven surfaces, namely that at about \(1.5^\circ \mathrm{K}\) a maximum was found on the curve of the temperature dependence of the rate of transfer over lucite and perspex^B.50g, B.51g, C.52b. Chandrasekhar^C.52b suggested that there exist two competing transfer processes: one, pure superfluid transfer (as in the case of glass), and the other, a siphon process of a semicapillary nature, inherent only to uneven surfaces. There is, however, no complete certainty as to the correctness of this supposition, in particular because Smith and Burse^S.53a did not observe any maximum. It may be supposed that in subsequent work a detailed study of the microstructure of the surfaces used would be useful.

7.4. Determination of the Linear Flow Velocity

In order to determine the linear flow velocity of a film from the values of the transfer rate, which represents the volume of liquid flowing through \(1\ \mathrm{cm}\) of film width in \(1\ \mathrm{sec}\), it is necessary to have data on the thickness of the film. A series of elegant experiments, in which the film thickness and the transfer rate were measured simultaneously, was carried out by Jackson and Henshaw^J.50. Their apparatus is shown schematically in Fig. 46. It consisted of a cylindrical test tube \(A\) made of stainless steel, on the side surface of which there was, parallel to the axis of the test tube, a flat strip \(2\ \mathrm{mm}\) wide; the strip was polished and constituted a mirror. This mirror was covered with a film of barium stearate; determination of the thickness

helium film, moving along the mirror, was carried out by an optical method developed by Jackson and described in Section 6.1. A glass capillary tube \(B\) was attached to the bottom of the test tube through a Kovar-glass seal \(K\), by means of which it was possible to observe the liquid level in the test tube. In this way it was possible to determine the amount of liquid helium that had flowed out or flowed in and at the same time to measure the film thickness. The results obtained are collected in Table 7.1 (see p. 120); the film thickness \(d\) given in it was measured at a height of \(1\ \mathrm{cm}\) above the external liquid level.

Column 4 in Table 7.1 gives the critical velocity of flow of the film along the inner surface of the test tube at a height of \(1\ \mathrm{cm}\) above the liquid-helium level, obtained by dividing the transfer velocity \(R\) (column 2) by the film thickness \(d\) at a height of \(1\ \mathrm{cm}\)*).

In this calculation it is assumed:

a) that the mean density of helium in the film is equal to the density of liquid helium in the bulk;

b) that the thickness of the film inside the test tube varies with height in accordance with the observed variation outside the test tube;

c) that all the liquid forming the film takes part in the motion.

The assumptions a) and b) are apparently correct. On the contrary, assumption c) gives rise to doubts. If, instead of c), it is assumed that only the superfluid component takes part in the motion of the film and that the density of the superfluid component in the film is the same as in bulk liquid helium (see Section 2), then the mean critical velocity \(v_s\) of the superfluid component in the film can be obtained by multiplying the values in column 4 of Table 7.1 by \(\rho_s/\rho\); it is given in column 5.

Fig. 46. Jackson and Henshaw’s apparatus \(^{1.50}\) for the simultaneous measurement of the transfer velocity and the film thickness. \(A\)—steel test tube with a glass tube \(B\) attached to it by means of a Kovar-glass seal \(K\), \(C\)—copper screen with a window \(W\) for observation, \(E\)—thermal shield with liquid helium, \(S\)—suspension.

*) The data given here differ numerically from the data obtained by Henshaw and Jackson, since these authors calculated the velocity of the film outside the test tube. However, the flow velocity outside is not the critical velocity, since the transfer velocity is determined by the narrowest perimeter of the connecting surface, located above the upper level.

From these results one may draw the following conclusions:

1. The magnitude of the mean critical velocity \(\bar v_s\) of the superconducting component in the film is insensitive to changes in temperature. A theoretical interpretation of this fact will be discussed below in Section 7.5.

2. The values of \(\bar v_s\) are of the order of \(110\ \text{cm/sec}\), i.e., approximately twice as large as the values calculated from equation (7.2). However, it is still

Table 7.1*)

\(T\) \((^\circ\mathrm{K})\)	\(R\) \((\text{cm}^3/\text{cm sec})\)	\(d\) \((\text{cm})\)	\(\bar v_{\mathrm{crit}}\) \((\text{cm/sec})\)	\(\bar v_s\) (crit.) \((\text{cm/sec})\)
1.1	\(16.9\cdot 10^{-5}\)	\(1.63\cdot 10^{-6}\)	104	105
1.3	16.9	1.63	104	110
1.5	16.8	1.66	101	114
1.7	16.1	1.82	88.5	118
1.9	12.8	1.94	66	118

premature to conclude that equation (7.2) is unsuitable, since (a) the transport velocities given in Table 7.1**) are approximately twice as large as the transport-velocity values obtained for glass and for the most carefully polished stainless steel (see above, Section 7.2), and (b) in these experiments the film thickness was measured on the surface of barium stearate, whereas the critical transport velocity was measured for stainless steel.

Further observations by Jackson and Henshaw concerned the peculiarities of the phenomenon of film transport from a test tube in the case when the inner level was only a few millimeters from the rim of the test tube. As was reported earlier by Daunt and Mendelssohn (see Section 7.1), the transport velocity is anomalously high while the level is falling through the first few millimeters if the experiment is begun with a full test tube. After the inner level has dropped below this anomalous region, the transport velocity assumes its characteristic value, independent of the difference in levels, as was described in detail in Section 7.2. Using their optical method for determining the film thickness, Jackson and Henshaw observed that in the anomalous region of the flow, bright specks corresponding to drops appear in the field of view of the microscope,

*) The values given in columns 1, 2, and 3 are taken from the data of Henshaw and Jackson in Proceedings of the Symposium on Low Temperature Physics (Proc. Natl. Bur. Standards, 185, 1952) and from private communications by Jackson.

**) Translator’s note: in the original, “in Section 7.1” is apparently a misprint.

formed from the liquid; these drops have a length from \(\frac{1}{10}\) to \(\frac{1}{3}\) mm and move downward rather slowly, with a velocity of about \(1\) cm/sec. When the level receded from the edge and the transfer ceased to depend on the difference of levels, the drops disappeared.

Simultaneous measurements of the film thickness and the transfer velocity in the case of flow over a glass surface were made by Atkins.^A.50b The transfer velocity was determined directly from the change in the liquid level during the emptying or filling of a glass test tube; the film thickness was calculated from observations of oscillations of the liquid level in the test tube near the equilibrium level (see above, Section 6.1). Atkins’ results are given in Table 7.2 and represent smoothed values taken from the filling curve of the test tube at \(1.47^\circ\) K. In this table, the different values of the film thickness at the same temperature \(1.47^\circ\) K correspond to different heights above the level of the helium bath (from \(0.5\) cm to \(3.5\) cm).

These results indicate, first, that the film thickness on glass does not differ very greatly from the value obtained by Jackson and Henshaw (Table 7.1) for a film on barium stearate, and, second, that the critical velocity \(\bar v_{\text{crit}}\) is approximately equal to \(37\) cm/sec. This critical velocity, obtained by dividing the quantities in the first column of Table 7.2 by the quantities in the second column, is approximately equal to the critical velocity of flow of the superfluid component \(\bar v_s\), since at \(1.47^\circ\) K the density of the superfluid component differs little from the total density. The value \(37\) cm/sec for \(\bar v_s\) is in good agreement with the predictions made by means of equation (7.2), which will be discussed below.

Table 7.2

\(R\) (cm\(^3\)/cm sec)	\(d\) (cm)	\(v_{\text{crit}}\) (cm/sec)
\(7.4\cdot 10^{-5}\)	\(1.9\cdot 10^{-6}\)	39
7.8	2.1	37
8.15	2.3	35

Knudsen and Dillinger^K.53d in a brief communication give the results of experiments on the flow of a film along copper spirals. They obtained a critical velocity of \(37\) cm/sec at \(1.3^\circ\) K and a film thickness of approximately \(2\cdot 10^{-6}\) cm—values in remarkable agreement with Atkins’ results.

7.5. Critical velocity of film flow and its interpretation

The characteristic velocities of motion of the film discussed in the preceding section are, in order of magnitude, \(50\) cm/sec, i.e., they are relatively small, equal, for example, to the velocity obtained in free fall without friction under the action of

gravity from a height of 3 mm. All measurements of isothermal emptying of the test tube, as is evident, for example, from Fig. 38, took place at a difference of levels (i.e., a pressure difference) considerably greater than 3 mm. Consequently, there exists some mechanism that limits the rate of flow of the film to these comparatively “low” velocities, which, as established experimentally, do not depend on the magnitude of the pressure difference that caused this motion.

The experimental fact that the rate of flow of the film is independent of the length of the path, and also the ability of the film to perform only weakly damped oscillations, indicate that at these critical velocities and at lower velocities there is no resistance to the flow of the film*).

Thus, the flow of the film may be regarded as superfluid. Furthermore, it can be shown that at velocities exceeding the critical one the resistance to flow increases sharply to enormous values that are not amenable to measurement. This question was studied in observing the non-isothermal flow of the film, which will now be discussed.

Rollin and Simon^R.39 had already shown that the flow of the film can be produced equally well both by a temperature difference and by a difference of levels, the greatest velocities attainable in the former case being equal to the critical velocities observed in isothermal flow under the action of gravity.

The flow of a film caused by a temperature gradient occurs in the direction toward the heat source. This phenomenon was studied in detail by Daunt and Mendelssohn^D.38b, D.39c, in particular in their work on the thermomechanical effect in films^D.39a, D.50b, by Daunt et al.^D.47a, Atkins^A.48, Brown and Mendelssohn^B.50f, and Chandrasekhar and Mendelssohn^C.51 **). In the work of the latter authors, helium II flowed through a narrow slit situated above the liquid level into a completely closed Dewar when a current was passed through a heater placed inside the Dewar (see Fig. 47). Fig. 47 gives the measured dependence of the volume rate of liquid helium on the supplied power at 2.09° K. It is seen that the volume rate increased linearly with the supplied power until a critical value was reached \((2.4\cdot 10^{-4}\ \text{cm}^3/\text{cm sec})\); after this, increasing the supplied power did not lead to any noticeable increase in the volume rate. The sharp bend of the curve may be interpreted as an indication that, for velocities smaller than the critical value, the motion is free from friction

*) K. Kasuya^K.53f proposed explaining the damping of the oscillations by the equalization of temperature between the reservoirs by means of thermal conduction through the copper wall; in this connection he concluded that the motion of the film is entirely free from friction.

**) The significance of this effect for the construction of cryostats was recently studied experimentally by Ambler and Kurti^A.52d.

and that large frictional forces prevent an increase of the velocity above the critical one.

The observed fact that the flow of a film, caused both by gravity and by a temperature gradient, is superfluid up to a certain critical value of the transfer velocity, which is a function only of temperature, apparently has primary importance for understanding the properties of helium II. A similar effect is also observed in the flow of bulk liquid

Fig. 47. Apparatus and results of measurements by Chandrasekhar and MendelssohnC.51 of the dependence of the volume rate of flow of helium II along a film into a reservoir on the power released in the reservoir.

helium through very narrow slits (see Section 2.5); however, for the flow of bulk liquid helium even through the narrowest slits this effect is not so sharply expressed as for the flow of a film. This effect was compared by Daunt and MendelssohnD.42 with the existence of a limiting density of the superconducting current on the surface of a superconductor, exceeding which, as is known, leads to the destruction of superconductivity (the so-called “critical current”). The authors further suggested that the change in the limiting transfer velocity of the film with temperature may be connected with the change in the density of the superfluid component. Such a dependence should lead to the existence of some maximum, temperature-independent velocity of superfluid motion. The results on the flow of a helium film given in the preceding section do not contradict this hypothesis. With the aid of this hypothesis one can calculate the fraction of the normal component \(\rho_n/\rho\), i.e. the ratio of the density of the normal

components to the total density, using data on the rate of transport of the film. Unfortunately, the scatter of the results for the transport rate is too large (see Fig. 43) for any sufficiently accurate calculations to be made. Nevertheless, it is evident that formula (7.1), proposed for the dependence of the transport rate on temperature with the value of the exponent \(a = 6\), coincides with formula (2.2) for \([1 - (\rho_n/\rho)]\). The values of \(\rho_n/\rho\) calculated in this way agree, within the limits of accuracy, with the data obtained by Andronikashvili and others (see Sec. 2.3), and do not contradict the hypothesis that the maximum velocity of superfluid motion is independent of temperature.

In connection with the flow velocity of the film, one should point out an interesting circumstance noted by Bailey, de Boer, and Michels \(^{B.41}\), and independently of them by F. London \(^{L.45}\), a circumstance implicitly used by Gorter and Ray \(^{G.44}\). Namely, the product of the mean momentum of superfluid atoms moving with the critical velocity and the film thickness is, in order of magnitude, equal to Planck’s constant:

\[ mvd \simeq \hbar . \tag{7.2} \]

London \(^{L.45}\) established that the amount of helium (in grams) transported per second through one centimeter of width of the connecting surface, divided by the number of superfluid atoms in \(1\ \text{cm}^3\), has the dimension of angular momentum and is numerically close to \(\hbar\). Moreover, for cases in which the necessary data were available, London showed that such a quantum relation is valid not only for helium II films, but also for the superconducting body of electrons in superconductors, which may serve as an argument in favor of the analogy proposed by Daunt and Mendelssohn \(^{D.42}\) between the properties of helium II and superconductivity.

Relation (7.2) shows that, independently of the value of the film thickness \(d\), the transport rate \(R\) (in \(\text{cm}^3/\text{cm}\cdot\text{sec}\)) is constant, since \(R\) is proportional to \(v \cdot d\). Therefore, although the film thickness varies with height above the liquid level, \(R\) must be independent of height; this has been proved experimentally with great accuracy. Experimental proof of the independence of \(R\) from height has been obtained for heights up to \(6\ \text{cm}\) (see Sec. 7.1 and, for example, Fig. 44). However, recently Eselson and Lazarev \(^{E.52}\) reported a peculiar dependence of \(R\) on height. Kasuya \(^{K.53e}\) suggested that at great heights the transport rate of the film begins to be determined by Gorter mutual friction. He concluded that the transport rate \(R\) at great heights should be proportional to \((\Delta p)^{1/3}\). There is as yet no experimental confirmation of this supposition.

Another consequence of equation (7.2) is that, although the film thickness may change in passing from one substrate

to another, the total rate of transfer \(R\) should not depend on the material of the substrate. This is in agreement with the most recent experiments on the transfer of a film over stainless steel \(C.^{52c}\), in which \(R\) was found to be equal to the rate of transfer over glass. However, measurements of the film thickness on substrates of different materials continue to be of interest.

Various attempts have been made to prove theoretically the existence of a critical velocity of superfluid motion and to obtain its numerical values.

Landau \(L.^{41a}\) considered the onset of viscous resistance at a flow velocity slightly exceeding the critical one. For this purpose he analyzed the interaction of a liquid at rest with a wall moving through it. He postulated that the motion of the liquid will begin with the excitation in it of a phonon or a roton. For the excitation of a phonon the relative velocity of motion must be greater than the speed of sound \(u_i\) (i.e. \(v_{\mathrm{crit}} > u_i\)); for the excitation of a roton the velocity must be greater than \((2\Delta/\mu)^{1/2}\), where \(\Delta\) and \(\mu\) are the quantities described in Section 3.3*).

However, the absolute value of \(v_{\mathrm{crit}}\) given by both these inequalities was approximately \(10^4\ \mathrm{cm/sec}\), which considerably exceeded the experimental values of the critical velocity. Thermodynamic considerations advanced later by Ginzburg \(G.^{44}\) likewise led to values of \(v_{\mathrm{crit}}\) several orders of magnitude greater than those obtained experimentally.

In 1949 Ginzburg \(G.^{49}\), repeating to some extent the arguments put forward earlier by Daunt and Mendelssohn \(D.^{46b}\), considered the zero-point energy of particles and came to the conclusion that \(v_{\mathrm{crit}}\) should be connected with the zero-point velocity, determined by means of the uncertainty relation for a particle confined in a volume with a linear dimension equal to the film thickness \(d\). This immediately leads to equation (7.2). This point of view was also developed by Mendelssohn \(M.^{45}\), who suggested that in both cases—superfluidity of helium II and superconductivity—mass transfer is a diffusion caused by the zero-point motion of particles. Such a consideration should lead to a temperature-independent mean transfer velocity; relation (7.2) will then be satisfied automatically.

Mott \(M.^{49b}\) proposed another mechanism explaining the existence of a maximum mean velocity of superfluid motion. In his treatment \(v_{\mathrm{crit}}\) varies rather in proportion to \(d^{-1/2}\), and not to \(d^{-1}\), as follows from equation (7.2). A brief description of Mott’s model is given in Section 2.5. The data currently available do not make it possible to choose with confidence between Mott’s model and the model connected with equation (7.2).

*) For a discussion of the question of \(v_{\mathrm{crit}}\), see also Dingell \(D.^{52a}\).

A somewhat different possible explanation for the existence of a critical velocity \(v_{\mathrm{crit}}\) has been proposed by one of us (R. Smith, unpublished). The film is represented as an aggregate of chains of atoms stretched parallel to the \(z\)-axis, i.e. perpendicular to the plane of the substrate; the length of the chains is \(d\), and the distance between them (the lattice constant) is \(a_1\). If now the crystal lattice of the substrate (with lattice constant \(a_0 \ne a_1\)) moves along the \(x\)-axis with velocity \(v\) relative to the film, then the lower ends of each chain are acted upon by a periodic force with fundamental frequency \(\nu = v/a_0\). As a result, waves are formed in the chains; but, for a given \(d\) and sufficiently small \(v\), the reflected waves have such a phase that the chain does not absorb the energy of the moving crystal. However, if \(v\) increases, the chain finally begins to resonate, and a phonon appears in it at the expense of energy absorbed from the crystal. The velocity at which viscosity is first observed is the critical velocity and is determined from the equation

\[ v_{\mathrm{crit}} = a_0 \nu_0 = a_0 \frac{u_1}{\lambda_{\max}} = \frac{a_0 u_1}{4d}, \tag{7.3} \]

where \(u_1\) is the speed of sound in the chain.

It is curious that from formula (7.3) there follows the constancy of the product \(v_{\mathrm{crit}} d\). Further, if one substitutes into (7.3) a reasonable numerical value of \(a_0\) and takes the known value of \(u_1\) for liquid helium, one obtains \(v_{\mathrm{crit}} d \simeq 10^{-4}\ \mathrm{cm}^2/\mathrm{sec}\), which agrees with the experimental data.

It should be noted that in this somewhat idealized model the absence of microscopic roughness of the film is assumed; this was explicitly indicated earlier in Tisza’s theory\({}^{47}\). Otherwise, the tangential forces arising during the motion of the crystalline substrate would lead to the excitation of waves propagating in directions other than perpendicular to the substrate; in this case the fundamental frequencies would be much smaller than \(u_1/4d\), and therefore the interaction would appear at vanishingly small relative velocities.

8. HYDRODYNAMICS OF A TWO-COMPONENT LIQUID

8.1. Thermo-hydrodynamic equations

The ordinary hydrodynamic equations of a one-component liquid must be supplemented and modified in a theory using the two-component model, in such a way as to explain the specific effects arising in helium II. In the ordinary one-component case, the motion of a liquid in the absence of external forces is determined by Euler’s equations, the continuity equation, and the gra-

...with boundary conditions. These equations have the form

\[ \rho \frac{D\mathbf{v}}{Dt} = \rho \frac{\partial \mathbf{v}}{\partial t} + \rho \mathbf{v}\,\operatorname{grad}\mathbf{v} = \]

\[ = -\operatorname{grad} p + \eta\left( \frac{4}{3}\operatorname{grad}\operatorname{div}\mathbf{v} - \operatorname{rot}\operatorname{rot}\mathbf{v} \right), \tag{8.1} \]

\[ \frac{\partial \rho}{\partial t} + \operatorname{div}\rho\mathbf{v} = 0, \tag{8.2} \]

where the second coefficient of viscosity is taken to be zero in accordance with Stokes’ approximation.* This section will be devoted to the question of what equations should replace (8.1) and (8.2) in the hydrodynamics of a two-component liquid.

We have already earlier (in the section on viscosity) discussed the question of the decomposition of the density of helium II. Naturally, in exactly the same way one can decompose into two terms the density of the mass flux (momentum density). Then we obtain:

density

\[ \rho=\rho_s+\rho_n \tag{8.3} \]

and momentum density

\[ \mathbf{j} = \rho\mathbf{v} = \rho_s\mathbf{v}_s + \rho_n\mathbf{v}_n. \tag{8.4} \]

In accordance with the ideas expressed in Sections 2, 4, and 5, \(\mathbf{v}_s\) and \(\mathbf{v}_n\) in (8.4) may be quite different depending on the flow conditions. In particular, any one of the velocities \(\mathbf{v}\), \(\mathbf{v}_s\), and \(\mathbf{v}_n\) may be equal to zero, while the other two are nonzero. Also in the spirit of the results considered in Sections 2, 4, and 5, one may imagine that different viscosities affect the flow of the normal and superfluid components. Namely, we assume

\[ \eta_s=0,\qquad \eta_n\ne 0. \]

On the basis of the above-mentioned results (or as an assumption consistent with the theories considered in Section 3) we can write the entropy-conservation equation (if irreversible processes are absent or negligibly small) in the form

\[ \frac{\partial(\rho S)}{\partial t} + \operatorname{div}(\rho S\mathbf{v}_n) = 0, \tag{8.5} \]

where \(S\) is the entropy of one gram of the whole liquid. This means that only the normal component carries entropy. Some authors prefer to regard the entropy of the superfluid component as very small; we shall take it to be zero, putting

\[ \rho S=\rho_n S_n \tag{8.6} \]

* See Rayleigh, Theory of Sound, Vol. II, Gostekhizdat, 1955.

instead of the possible

\[ \rho S=\rho_s S_s+\rho_n S_n \]

(here \(S_n\) is the entropy of one gram of the normal component).

If equations (8.3) and (8.4) are substituted into the continuity equation (8.2), then we obtain two equations whose sum is equal to zero:

\[ \left. \begin{aligned} \frac{\partial \rho_n}{\partial t}+\operatorname{div}(\rho_n \mathbf v_n)&=\Gamma,\\ \frac{\partial \rho_s}{\partial t}+\operatorname{div}(\rho_s \mathbf v_s)&=-\Gamma. \end{aligned} \right\} \tag{8.7} \]

In (8.7) \(\Gamma\) is the amount of normal component formed in 1 sec. in unit volume. In most cases, when considering the hydrodynamics of a two-component liquid, an approximation is used in which \(\Gamma=0\).

We have not yet written down analogues of equation (8.1) for a two-component liquid. At present, exact equations not giving rise to doubt (one for \(D\mathbf v_n/Dt\), the other for \(D\mathbf v_s/Dt\)) have not yet been obtained. Equations linear in the velocities, suitable in the limit of small velocities, are known; however, the terms quadratic in the velocities proposed by different authors are different. The first-approximation equations (written with irreversible processes neglected), about which there is no disagreement, have the form

\[ \left. \begin{aligned} \rho_n \frac{\partial \mathbf v_n}{\partial t} &=-\frac{\rho_n}{\rho}\operatorname{grad}p-\rho_s S\operatorname{grad}T,\\ \rho_s \frac{\partial \mathbf v_s}{\partial t} &=-\frac{\rho_s}{\rho}\operatorname{grad}p+\rho_s S\operatorname{grad}T. \end{aligned} \right\} \tag{8.8} \]

These equations were first given (in a somewhat different form) by TiszaT.40 in 1940; however, one of the equations had been written by him still earlierT.38b, T.38c. Instead of the terms with \(\operatorname{grad}T\), Tisza wrote terms proportional to the gradient of a certain osmotic pressure \(p_n\); this osmotic pressure was then taken to be equal to the fountain pressure, determined by the equation of F. London (see Section 4.7), and Tisza’s equations were brought to the form (8.8). In deriving (8.8), Tisza assumed that \(\Gamma\) in (8.7) is equal to zero; this assumption is always made in deriving (8.8). (A complete and clear derivation of the hydrodynamic equations in first approximation was given by DingleD.49.)

These same equations were derived by LandauL.41a, who proceeded from other premises. Landau assumed that the thermodynamic Gibbs potential of helium II containing excitations (i.e., the normal component) is equal to the thermodynamic potential of the pure superfluid component plus the kinetic energy of the excitations relative to the superfluid component. If one also takes into account

if one uses the law of conservation of momentum, then, as a linear approximation, one obtains equation (8.8).

In his last paper, where the equations for second sound are derived (see Section 9.2), Tisza^T.47^ obtained equations (8.8) by still another method. Dingle^D.49^ also used this method for deriving (8.8). These equations can also be obtained from Hamilton’s principle, as a linear approximation.

If one becomes interested in approximations of the hydrodynamic equations better than (8.8), then differences of opinion are found. Equations with terms quadratic in the velocities were obtained (or postulated) by a whole series of authors: Landau^L.41a^, Gorter and Mellink^G.49a^, Nakajima, Tomita and Usui^N.50a^, Zilsel^Z.50a^, Temperley^T.51c^, Prigogine and Mazur^P.51b^, Usui^U.51a^, and Dingle^D.52a^. Interest in these equations is mainly determined by the fact that the superfluidity of helium II is limited by a certain critical velocity (see Section 7). A complete system of hydrodynamic equations must explain the existence of a critical velocity (or region of velocities), whereas equations (8.8), of course, cannot provide any information about this.

Here we are not in a position to describe and discuss the theories of all these authors in any detail, especially since Dingle, in his review, devoted sufficient attention to them. We shall merely list, using, as far as possible, the same notation, the variants of the equation of motion of the superfluid component proposed by various authors, and briefly discuss the terms contained in them.

Landau:

$$ \rho_s \frac{D\mathbf{v}_s}{Dt} = -\frac{\rho_s}{\rho}\,\operatorname{grad} p +\rho_s S\,\operatorname{grad} T +\rho_s\,\operatorname{grad}\left[\frac{\rho_n}{2\rho}(\mathbf{v}_n-\mathbf{v}_s)^2\right]; $$

Gorter and Mellink:

$$ \rho_s \frac{D\mathbf{v}_s}{Dt} = -\frac{\rho_s}{\rho}\,\operatorname{grad} p +\rho_s x\left(\frac{\partial S}{\partial x}\right)\operatorname{grad} T - A\rho_s\rho_n(\mathbf{v}_s-\mathbf{v}_n)^3; $$

Nakajima, Tomita and Usui:

$$ \rho_s \frac{D\mathbf{v}_s}{Dt} = -\frac{\rho_s}{\rho}\,\operatorname{grad} p +\rho_s S\,\operatorname{grad} T -\frac{\Gamma}{2}(\mathbf{v}_n-\mathbf{v}_s)+\mathbf{f}; $$

Zilsel:

$$ \rho_s \frac{D\mathbf{v}_s}{Dt} = -\frac{\rho_s}{\rho}\,\operatorname{grad} p +\rho_s S\,\operatorname{grad} T +\frac{\rho_n\rho_s}{2\rho}\,\operatorname{grad}(\mathbf{v}_n-\mathbf{v}_s)^2; $$

$$ \rho_n \frac{D\mathbf{v}_n}{Dt} = -\frac{\rho_n}{\rho}\,\operatorname{grad} p -\rho_s S\,\operatorname{grad} T - \frac{\rho_n\rho_s}{2\rho}\,\operatorname{grad}(\mathbf{v}_n-\mathbf{v}_s)^2 -\Gamma(\mathbf{v}_n-\mathbf{v}_s); $$

Temperley:

\[ \rho_s \frac{D\mathbf v_s}{Dt} = -\frac{\rho_s}{\rho}\operatorname{grad}p +\rho_s \tilde S \operatorname{grad}T -\frac{\Gamma}{2}(\mathbf v_n-\mathbf v_s); \]

Prigogine and Mazur:

\[ \rho_s \frac{D\mathbf v_s}{Dt} = -\frac{\rho_s}{\rho}\operatorname{grad}p +\rho_s x\left(\frac{\partial S}{\partial x}\right)\operatorname{grad}T +\frac{\rho_n\rho_s}{\rho}\operatorname{grad}(\mu_n-\mu_s) - \]

\[ - \frac{\rho_s}{\rho}\Gamma(\mathbf v_n-\mathbf v_s) +\zeta(\mathbf v_n-\mathbf v_s)^2(\mathbf v_n-\mathbf v_s); \]

Dingle:

\[ \rho_s \frac{D\mathbf v_s}{Dt} = -\frac{\rho_s}{\rho}\operatorname{grad}p +\rho_s \tilde S \operatorname{grad}T +\rho_s \operatorname{grad} \left[ \frac{1}{2\rho} \int_{0}^{v=v_n-v_s} \rho_n d(v^2) \right]. \]

As was indicated above, Landau obtained his equation by postulating the form of the thermodynamic potential for a two-component liquid. In deriving the complete system of equations he also made use of the conservation laws (of mass, momentum, energy, and entropy), valid for reversible processes.

Gorter and Mellink obtained their equations by generalizing (8.8). The last term on the right-hand side of their equation takes account of the mutual friction (see above, Sections 2 and 5), acting on the two components as forces equal in magnitude and opposite in direction; this mutual-friction force was postulated by the authors in order to explain heat-transfer experiments*). The second term on the right-hand side of their equation contains the combination \(x\dfrac{\partial S}{\partial x}\) \(\left(x=\dfrac{\rho_n}{\rho}\right)\), obtained as a result of Gorter’s proposed[^49b] generalization of Tisza’s postulate on entropy (see Section 4.7). If equality (8.6) is valid, this combination is equal to the quantity \(S\) entering into (8.6).

Nakajima, Tomita, and Usui derived their equation from the conservation laws. The vector \(\mathbf f\) is a separation parameter, introduced because the conservation law determines the behavior only of the sum

\[ \rho_s \frac{D\mathbf v_s}{Dt}+\rho_n \frac{D\mathbf v_n}{Dt}. \]

The third term of their equation contains \(\Gamma\) from (8.7). An analogous term was introduced by the authors into their equation

*) Kasuya proposed, in addition to the mutual-friction force of Gorter–Mellink, a friction force acting on the superfluid component,

\[ f_s=-B\rho_s(v_s)^3, \]

where \(B=0\) for \(v_s<v_{\mathrm{crit}}\). This additional force makes it possible to describe experimental results better than by mutual friction alone. It is quite obvious how this force should be inserted into the hydrodynamic equations.

for \(\left(\dfrac{D\mathbf v_n}{Dt}\right)\). Discussing their equations in the light of the experimental results, Nakajima et al. set \(\Gamma\) equal to zero, and took \(\mathbf f\) to be equal to the Gorter–Mellink mutual-friction force.

Zilsel obtained his equations using Eckart’s variational principle\({}^{38}\). This principle is formally similar to Hamilton’s principle, but is formulated for quantum mechanics, so that the concept of particle trajectories is absent from it and the variation is performed not with respect to displacements, but with respect to velocities. We have written out both of Zilsel’s equations in order to show that they are not symmetric with respect to \(\Gamma\), in contrast to the equation of Nakajima et al. and to Temperley’s equation. Zilsel showed that this asymmetry is connected with the representation of the superfluid particles as “concentrated” in the sense of Bose–Einstein condensation.

It is curious that Eckart’s principle automatically gives

\[ \operatorname{rot}\mathbf v_s = 0; \]

let us recall that this condition was postulated by Landau.

Let us also note that in Zilsel’s equation there is a term very similar to the last term of Landau’s equation.

Temperley’s equations are derived from the conservation laws. They are also symmetric with respect to \(\Gamma\) (we have slightly changed Temperley’s notation, using the relation between \(\Gamma\) and \(\dfrac{DS_n}{Dt}\), given by (8.5), (8.6), and (8.7)).

Prigogine and Mazur derived their equations by the methods of the thermodynamics of irreversible processes. In addition to using the conservation laws, the authors introduced the idea of two partial pressures, each of which acts on one of the components. The equation written by us is a combination of equations (3.41) and (5.2) from their paper. The quantities \(\mu\) in their equation are the chemical potentials of the components; they are not equal to one another if \(x=\dfrac{\rho_n}{\rho}\) is not equal to its equilibrium value. It can be shown that the term with \(\operatorname{grad}(\mu_n-\mu_s)\) in Prigogine and Mazur is equivalent to the term with

\[ \frac{1}{2}\operatorname{grad}(\mathbf v_n-\mathbf v_s)^2 \]

in Zilsel’s equation. The last term in Prigogine and Mazur is the Gorter–Mellink mutual-friction force; it was postulated by the authors in order to obtain the Gorter–Mellink equation.

We have omitted Usui’s equation, since it differs from the equation of Nakajima, Tomita, and Usui only in that in it, instead of \(x\dfrac{\partial S}{\partial x}\), there stands \(S\).

A general remark should be made concerning all equations containing terms proportional to \(\Gamma(\mathbf v_n-\mathbf v_s)\). In all derivations from the conservation laws there arises a certain arbitrariness,

in how to divide this term between the two components, since this division requires additional assumptions. The nonsymmetric division in Zilsel’s equations follows from the assumption that Eckart’s principle is applicable to the question under consideration.

Dingle’s equation is a generalization of Landau’s equation. Landau restricted his consideration to terms of order \((v_n - v_s)^2\), whereas Dingle, in his calculations, took into account all powers of the relative velocity.

The complete system of thermo-hydrodynamic equations (for a nonviscous, non-heat-conducting liquid), including the definitions (8.3), (8.4), and (8.6), the conservation laws (8.5) and (8.7), and the Euler-type equations (8.8), is usually derived wholly or partly from thermodynamic considerations. Recently, however, Kronig \(^{\mathrm{K.}\,53g}\), using statistical mechanics, derived equations (8.5) and (8.8) in the linear approximation. Kronig’s derivation was based on Landau’s ideas \(^{\mathrm{L.}\,41a}\) and is applicable, strictly speaking, only to the temperature region in which, by assumption, all excitations in the liquid are phonons, i.e., to temperatures below approximately \(0.6^\circ\mathrm{K}\). Kronig’s work is a development and modification of the derivations concerned mainly with the velocity of second sound and obtained by Dingle \(^{\mathrm{D.}\,52c}\) and by Ward and Wilks \(^{\mathrm{W.}\,51b,\mathrm{W.}\,52}\); these results will be considered in Section 9.3 on second sound.

H. A. Kramers \(^{\mathrm{K.}\,52d}\) undertook a treatment of the question similar to that carried out by Kronig. He considered excitations of a general type, with respect to which phonons and rotons were special cases. However, Kramers’s calculations led to an expression for the velocity of second sound (see Section 9) that is valid only in the limit \(T \to 0^\circ\mathrm{K}\).

A statistical treatment of the problem on the basis of the Bose-Einstein gas model was undertaken by Bogolyubov \(^{\mathrm{B.}\,47}\) and Zilsel \(^{\mathrm{Z.}\,58b}\). Bogolyubov considered a nonideal gas and, using the method of second quantization, showed that in the case of weak interaction the excited states of the gas can be regarded as an ideal Bose-Einstein gas of “quasi-particles.” The energy spectrum of the quasi-gas is not known exactly; however, it was shown that for a special form \(E(p)\) (it is essential that the curve \(E(p)\) not be convex toward the \(p\)-axis as \(p \to 0\)) the gas splits into excited and condensed fractions, which can have different mean velocities, provided only that the relative velocity is sufficiently small.

In a similar way, Zilsel discussed the possibility of two velocities in a Bose-Einstein gas and obtained analogous restrictions for the energy spectrum. In particular, he considered the spectrum of Belyaev, de Boer, and Michels \(^{\mathrm{B.}\,41}\) and discussed the possibility of the occurrence of a critical velocity.

8.2. Experimental investigations of the nonlinearity of the equations of motion

Comparatively few experiments have been undertaken to test the nonlinear equations presented above, or to choose the best among them. The first-order equations (8.8) have been tested mainly by means of experiments with second sound (Section 9). It was noted above that the complete equations should describe the phenomenon of critical velocity. However, an explanation of this phenomenon has not yet been found. Fried and Zilsel F.52, using Zilsel’s equations, attempted to obtain criteria for the applicability of the linear equations. They found that the linear approximation ceases to be valid at \(v_s \cdot d \approx 5 \cdot 10^{-4}\ \mathrm{cm}^2/\mathrm{sec}\), which is more or less in agreement with equation (7.2).

The equations of Gorter and Mellink also lead to a rather diffuse effect of the critical velocity, since the mutual-friction force increases smoothly (though rapidly) with the relative velocity. The equations of Gorter and Mellink are the only equations for which a direct comparison with observations has been made. Their equations were constructed in such a way as to explain the results of experiments on thermal conductivity. These and other experiments intended to test the equations were considered in Sections 2 and 5, and were also discussed in considerable detail in a recent review by Atkins A.52a. From their results one may conclude that the equations of Gorter and Mellink do not completely describe the behavior of helium II. Unfortunately, with respect to the other equations considered and presented above, no such experimental verification has been undertaken.

9. FIRST AND SECOND SOUND

9.1. Velocity and attenuation of first sound

Two different velocity fields in helium II lead to the possibility of the existence of two kinds of wave motion in the liquid. This circumstance was first noted and investigated by Tisza T.38b. In the absence of a temperature gradient, both equations (8.8) can be combined with one another; in doing so they yield the hydrodynamic equation for a single liquid, according to which ordinary (first) sound is possible, in which oscillations of density or pressure propagate at constant entropy with velocity

\[ u_1 = \left[\left(\frac{\partial p}{\partial \rho}\right)_s\right]^{1/2}. \tag{9.1} \]

The first measurements of the velocity \(u_1\) were made by Findlay, Pitt, Grayson-Smith, and Wilhelm F.38 by the method of standing ultrasonic waves. The sound was excited by vibrations of quartz immersed in liquid

helium, at a frequency of 1.338 Mc/s. Subsequent measurements were made by Pellam and Squire^P.47, who used a pulse technique at 15 Mc/s, and by Atkins and Chase^A.51b, who also used a pulse technique at 14 Mc/s. All the results are in good agreement with one another; the graph showing the value of the first-sound velocity \(u_1\) as a function of temperature at the saturated-vapor pressure, according to the data of Atkins and Chase, is presented in Fig. 48. This graph reveals an anomalous minimum at the \(\lambda\)-temperature \(2.18^\circ\mathrm{K}\), caused by the transition of liquid helium I into liquid helium II. Also of interest is the maximum near \(2.5^\circ\mathrm{K}\), corresponding to the minimum of the adiabatic compressibility. The agreement of the results of various authors, who used different operating frequencies, shows that, at least above \(1.6^\circ\mathrm{K}\), the dispersion is insignificant (less than 1%).

Fig. 48. First-sound velocity \(u_1\) (m/sec) in liquid helium according to the measurements of Atkins and Chase^A.51b.

One of the distinctive features of the curve is its sharp slope near \(T_\lambda\). Atkins and Chase investigated this region in detail and came to the conclusion that it is impossible to establish the existence of a discontinuity at this point (as is to be expected from Ehrenfest’s relations for a second-order transition). On the other hand, they suggested that both branches of the curve \(u_1\) as a function of \(T\), above and below \(T_\lambda\), may descend to lower values than have so far been observed. Taking into account that the measurements were carried out down to \(1.2^\circ\mathrm{K}\) and that the curve apparently tends to saturation, from Fig. 48 one can obtain by extrapolation the value of \(u_1\) at absolute zero, especially since, according to theory,

\[ \left(\frac{\partial p}{\partial \rho}\right)_S \]

should be practically constant near absolute zero. The extrapolation of Atkins and Chase gives \(u_1 = 237 \pm 2\) m/sec as \(T \to 0^\circ\mathrm{K}\).

The pulse method of measurement, first used by Pellam and Squire^P.47, enabled them to observe the attenuation of first sound. Their results for the attenuation coefficient \(a\) are shown in Fig. 49 by black circles. Subsequent measurements of \(a\) at a frequency of 15 Mc/s were carried out by Atkins and Chase (unpublished, see A.52a and C.53b); the values agree well with the preceding ones.

work of Pellam and Squire. The authors of the later work continued their measurements below the temperature \(1.6^\circ\) K, to which Pellam and Squire had brought their investigations; these results down to \(1.2^\circ\) K are also given in Fig. 49. The very strong attenuation in liquid helium I as the \(\lambda\)-point is approached deserves attention; it occurs along with a monotonic increase of \(\alpha\) in liquid helium II as the temperature is lowered.

The theoretical values of the attenuation coefficient \(\alpha\), shown in Fig. 49 by the dashed curve, were calculated by Pellam and Squire by strictly classical methods, on the basis of an estimate of the separate contributions to absorption caused both by viscous losses and by losses due to thermal conductivity. (See, for example, Bergman B.39b). It is evident that up to \(3^\circ\) K there is good agreement between theory and experiment, due, probably, as Pellam and Squire pointed out, to the monatomic character of helium, which is free from relaxation phenomena associated with internal degrees of freedom of molecules or complexes.

Fig. 49. Ultrasound attenuation coefficient as a function of temperature, according to the measurements of Pellam and Squire. \(^{\mathrm{P}.47}\) In the region from \(1.6\) to \(1.2^\circ\) the curve is drawn from data of the recent work of Atkins and Chase (see A.52a). The dashed curve above \(1.6^\circ\) K represents the theoretical curve for \(\alpha\), calculated by Pellam and Squire \(^{\mathrm{P}.47}\) by the classical method. The dashed curve below \(1.6^\circ\) K represents the coefficient \(\alpha\) according to Khalatnikov’s calculations, obtained by comparing his results with experimental values at high temperature according to the measurements of Atkins \(^{\mathrm{A}.52a}\). \(\bullet\) — measurements of Pellam and Squire \(^{\mathrm{P}.47}\), \(+\) — measurements of Atkins and Chase (see A.52a).

On the other hand, an extraordinarily strong increase in the value of \(\alpha\) below \(3^\circ\) K is observed experimentally, which, apparently,

tends to infinity at \(T_\lambda\). The same authors suggested that this increase is caused by forced transitions of liquid helium I into liquid helium II, occurring locally at temperatures close to \(T_\lambda\), and due to variations of pressure in sound waves. This explanation is consistent with Keesom’s\(^{{K.42}}\) explanation of the specific-heat curve above \(T_\lambda\). Pippard\(^{{P.51a}}\) examined in detail the question of local fluctuation transitions of helium I into helium II and conversely. He obtained an expression for the velocity and attenuation of first sound in an inhomogeneous liquid containing spherical inclusions with a compressibility different from that of the medium in which they are found. Applying these results to liquid helium, Pippard drew attention to the fact that compression of inclusions of helium II will cause their cooling, whereas compression of the basic medium (helium I) will cause its heating. The successive approach to equilibrium must be accompanied by a relaxation process, which will cause anomalous absorption of sound waves. Making plausible assumptions in order to apply the theory to liquid helium, Pippard calculated the curve \(u_1\) as a function of \(T\) near \(T_\lambda\) and found that the anomalous attenuation in liquid helium I near \(T_\lambda\) can be explained by considering inclusions in helium I of volumes of liquid helium II with dimensions of about 850 atoms.

The increase of attenuation in liquid helium II with decreasing temperature was discussed by Pellaam and Squire. They noted that the calculated normal viscous and thermal-conduction losses are completely inconsistent with the observed increase of absorption, and suggested that this discrepancy might be caused by neglecting dissipative phenomena of the relaxation type. In estimating the fraction of ordinary viscous losses entering into \(\alpha\), in order to obtain agreement with the results of Pellaam and Squire one may use the values of \(\eta_n\) given in 2.4. Moreover, according to Landau and Khalatnikov\(^{{L.49}}\), a theoretical calculation of \(\eta_n\) is possible. Concerning the application of their theory to the calculation of the attenuation of first sound in helium II, carried out by Khalatnikov\(^{{L.49}}\), it will be appropriate here to give some explanations.

The theory of the viscosity of liquid helium II, constructed by Landau and Khalatnikov\(^{{L.49}}\), is based on the conception of the liquid arising from Landau’s theory, according to which the normal component is a “gas” of excitations—phonons and rotons—having the energy spectrum described in Section 3. The theory is constructed on the calculation of the scattering of one layer of the gas of excitations by another, adjacent layer, when the two layers have slightly different macroscopic velocities. For this purpose the scattering cross sections of phonons by phonons, phonons by rotons, and rotons by rotons were first calculated; from these the collision integrals were obtained, which determine the change of the distribution function. It turned out that rotons behave as heavy particles, and phonons as light ones. Rotons are scattered

on rotons at all temperatures, and such processes give rise to a component of the viscosity that is independent of temperature. Above \(0.9^\circ\mathrm{K}\) the effect of phonon–phonon collisions is small in comparison with the influence of phonon–roton collisions; however, below this temperature phonon–phonon collisions may play a noticeable role, and below \(0.7^\circ\mathrm{K}\) they make a large contribution. The problem is further complicated by the fact that the rate of formation and decay of phonons is far from negligible in comparison with the rate of the scattering process; in fact, below \(0.9^\circ\mathrm{K}\) the former rate is larger, whereas above \(0.9^\circ\mathrm{K}\) the latter is larger. This leads to two different formulas for calculating the viscosity: one valid below \(0.8^\circ\mathrm{K}\), the other valid above \(1.0^\circ\mathrm{K}\).

The details of these truly heroic calculations cannot be given here. It was found that the phonon part of the viscosity depends strongly on temperature. If this part is subtracted from the experimental values of Andronikashvili\(^{A.48a}\), a constant roton viscosity is obtained. The final expressions for the coefficient of viscosity\(^*\) (in poises) are:

\[ \left. \begin{aligned} \eta \cdot 10^{-5} &= 1 + 8.7 \cdot 10^{-4} T^{1/2} \exp\left(\frac{\Delta}{kT}\right), \qquad T > 1.0^\circ\mathrm{K},\\ \eta \cdot 10^{5} &= 7.8 \cdot 10^{-5} \left[ T^{1/2}\exp\left(\frac{-\Delta}{kT}\right) + 4.8 \cdot 10^{-6}T^{5} \right]^{-1},\\ &\qquad T < 0.8^\circ\mathrm{K}. \end{aligned} \right\} \tag{9.2} \]

The viscosity in the interval between \(0.8\) and \(1.0^\circ\mathrm{K}\) was found by interpolation. It turns out that at extremely low temperatures \(\eta \sim T^{-5}\), which is a consequence of phonon–phonon scattering.

In their work the authors sought to reduce to a minimum the errors due to uncertainty in the values of such parameters as \(\partial^3\Delta/\partial p^3\) and \(\partial p_0/\partial \rho\). Making plausible assumptions about the magnitude of these parameters, the authors obtained excellent agreement with experiment. More important, on the other hand, is that, in the authors’ opinion, the temperature dependence has been obtained with sufficient accuracy. An ingenious application of the theory described above was made by Khalatnikov\(^{X.50}\) for calculating the attenuation of first sound. He showed that in a sound wave the local nonequilibrium phonon and roton densities can assume equilibrium values only over finite intervals of time, which creates a relaxation mechanism that causes attenuation. He considered two relaxation times, characterizing the two most important

\(^*\) Unfortunately, it is not at all obvious whether equations (9.2) really give \(\eta\) or \(\eta_n\). However, Landau and Khalatnikov, in making a numerical estimate, compare their results with the experimental values of \(\eta_n\).

of the collision process by means of which equilibrium is established, namely: phonon–phonon collisions and phonon–roton collisions. From this he was able to estimate the temperature dependence of the attenuation coefficient caused by relaxation effects, and found that \(\alpha\) increases sharply as the temperature is lowered. Combining the absolute value of the attenuation coefficient at some temperature (above \(1.6^\circ\) K) with the experimental results of Pellam and Squire, Khalatnikov obtained, for temperatures below \(1.6^\circ\) K, the curve shown in Fig. 49 by the dotted line.

Recently Khalatnikov\(^{X.52b, X.52c}\) examined these effects in more detail and showed that it is necessary to consider three coefficients of second viscosity along with the coefficient analogous to the coefficient of thermal conductivity in ordinary liquids. He showed that these new results do not alter the conclusions obtained earlier\(^{X.50}\) concerning the attenuation of first sound; however, for second sound they lead to appreciable absorption at frequencies of \(10^4\) cps.

Kronig, Tellung, and Veldring\(^{K.52e}\) also constructed a relaxation theory for calculating the attenuation of first sound. Using a two-component model, previously investigated by Kronig and Tellung\(^{K.50}\), they considered the relaxation of one component into the other and, comparing \(\alpha\) with experimental values, came to the conclusion that below \(T_\lambda\) the relaxation time must be either less than \(10^{-12}\) sec, or greater than \(10^{-4}\) sec.

9.2. Tisza and Landau Equations for the Velocity of Second Sound

In addition to the motion of a liquid corresponding to first sound, in which the density at every point of the liquid changes periodically, there is possible—as Tisza\(^{T.38}\) first discovered—another wave motion, in which the density and pressure (to a first approximation) remain constant, while oscillations of entropy (or temperature) occur. (In such motion the normal and superfluid components move in opposite phase.) In his 1947 paper Tisza considered these two wave motions more fully, showed that the coupling between density waves and entropy waves is in fact very small, and obtained an equation for the velocity of the temperature wave \(u_2\) (the so-called “second sound”),

\[ u_2=\left[-\frac{\partial T}{\partial\left(\dfrac{1}{S'_n}\right)}\cdot\frac{\rho_s}{\rho_n}\right]^{1/2}, \tag{9.3} \]

where \(S'_n\) is the entropy of the normal component per gram of total liquid.

In order to bring equation (9.3) for the velocity of second sound to numerical values, Tisza used a number of assumptions, partially justified experimentally. Namely:

(a) The entropy \(S'_n\) in equation (9.3) was taken to be equal to the observed total entropy of the liquid in the temperature interval \(1^\circ\mathrm{K}—T_\lambda\). It was confirmed experimentally that the entropy of the superfluid component in this temperature region is negligibly small.

(b) The superfluid component, just like the normal one, is capable of containing thermal compressional excitations (Debye waves). As a consequence it follows that at a sufficiently low temperature \((0.4^\circ\mathrm{K} < T < 1.0^\circ\mathrm{K})\) the mechano-caloric effect disappears. This hypothesis, however, was not taken into account in the calculation of \(u_2\) between \(1^\circ\mathrm{K}\) and \(T_\lambda\). Its effect will be that at temperatures below the temperature of the hypothetical second phase transition, second sound should be completely damped.

(c) A quasi-thermodynamic argument suggests that

\[ \frac{\rho_n}{\rho}=\frac{S}{S_\lambda}=\left(\frac{T}{T_\lambda}\right)^{5.5} \tag{9.4} \]

in the temperature region from \(1^\circ\mathrm{K}\) to \(T_\lambda\), where Debye phonon excitations may be neglected. This relation, as shown in Section 4, is an excellent approximation to the experimental results in the temperature region from \(1.6^\circ\) to \(T_\lambda\).

Using points (a) and (c), Tisza obtained the following expression for \(u_2\):

\[ u_2=26\left\{\frac{T}{T_\lambda}\left[1-\left(\frac{T}{T_\lambda}\right)^{5.5}\right]\right\}^{1/2}\ \mathrm{m/sec}. \tag{9.5} \]

The numerical value of the constant was first found by London \(^{L.46a}\), and the exponent 5.5 is obtained from the requirement of the best agreement of equation (9.4) with the experimental results. (Cf. equations (2.2) and (4.3).) It was found that equation (9.5) satisfactorily represents the experimental values of \(u_2\) (see 9.4) in the temperature interval between \(1.2^\circ\mathrm{K}\) and \(T_\lambda\). However, it was noted that at low temperatures \(u_2\), according to equation (9.5), should decrease monotonically with decreasing \(T\), and moreover as \(T \to 0\), \(u_2 \to 0\). This is in complete disagreement with the experimental data.

Landau \(^{L.41a}\), from his hydrodynamic equations, also obtained two velocities of wave propagation, which he called waves of first and second sound. Landau’s terminology continues to exist (perhaps because the idea of “second sound” is intriguing), although the second motion is an almost purely temperature wave and cannot be excited by ordinary mechanical means (such as a microphone), since in the first approximation there are no density or pressure oscillations. Starting from the assumption that the entropy of the superfluid component

is equal to zero, Landau obtained the expression for the velocity of second sound

\[ u_2=\left[\frac{\rho_s}{\rho_n}\cdot\frac{S^2T}{C}\right]^{1/2} =\left[-\frac{\rho_s}{\rho_n}\left\{\frac{dT}{d\left(\frac{1}{S_n'}\right)}\right\}\right]^{1/2}, \tag{9.6} \]

(where \(C\) is the heat capacity).

Although the formulas for the velocity of second sound obtained by Tisza and Landau are almost identical, the authors interpreted the quantities entering them in different ways. The basic ideas put forward by Landau were as follows:

\((a')\) The entropy \(S_n'\) in equation (9.6) was taken to be equal to the observed total entropy \(S\) of the liquid for all temperatures below \(T_\lambda\).

\((b')\) Only the normal component was assumed capable of containing phonon excitations. Hence the mechanocaloric effect and second sound should be observed at the very lowest temperatures.

\((c')\) The value of \(\rho_n/\rho\) was estimated in two different ways*).

1. On the basis of such experiments as those of Andronikashvili\(^{A.46, A.48a}\) (see 2.3);

2. From Landau’s microscopic theory of liquid helium II\(^{L.41a, L.47}\), which gives a quite definite expression for \(\rho_n\) as a function of the parameters \(\Delta\), \(\mu\), and \(p_0\) (see Section 3).

The first calculation of \(u_2\) as a function of temperature on the basis of Landau’s theory was carried out by Lifshitz\(^{L.44b}\), who used early values of the microscopic parameters \(\Delta\) and \(\mu\) to obtain \(\rho_n\). The result agreed qualitatively with experiment (see 9.4) between \(1.2^\circ\text{K}\) and \(T_\lambda\); however, more importantly, the result of the calculation showed that the value of \(u_2\) should increase noticeably near \(1^\circ\text{K}\) and ultimately tend, as was first emphasized by Landau, to the constant value \(u_1/\sqrt{3}\) as \(T\to0\). This remarkable increase at low temperatures, caused by the role of phonons in this region, was fully confirmed by subsequent experiments. The calculation of \(u_2\) carried out by Lifshitz, which gave only qualitative agreement with experiment, was revised by Landau\(^{L.47}\), who proposed new numerical values for \(\Delta\), \(\mu\), and \(p_0\), on the basis of which \(\rho_n\) could be calculated. The revised scheme led to a dependence of \(u_2\) on \(T\) which agrees roughly with Tisza’s equation above approximately \(1.5^\circ\text{K}\), has a minimum near \(1.1^\circ\text{K}\), rises rather sharply below this temperature, and, as \(T\) tends to zero, approaches the constant value \(u_1/\sqrt{3}\).

\[ \overline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]

*) The validity of equation (9.6) is not at present in doubt, and instead of obtaining \(u_2\) by substituting \(\rho_n\) into it, equation (9.6) is usually used together with the observed values of \(u_2\) to calculate \(\rho_n\).

When compared with the experimental data (see 9.4 and 9.6), it turns out, first, that in the temperature region from \(1.2^\circ\) to \(T_\lambda\) both theories agree qualitatively. Secondly, at temperatures below \(1.2^\circ\) K Tisza’s interpretation is incorrect, partly, perhaps, because of the rigidity of relation (c), but mainly because assumption (b) forbids the growth of \(u_2\) at low temperatures. Thirdly, Landau’s expression for \(u_2\) indicates the importance of the phonon contribution to the normal entropy at low temperatures and makes it possible to give a numerical estimate of it. The presence of the phonon contribution indicates that Landau’s hypothesis (b′), namely the existence of mechanocaloric effects down to absolute zero, is correct. In writing equations (9.1) and (9.6) we neglected the coupling between density oscillations and temperature oscillations. This coupling, proportional to the difference between the heat capacities \(c_p - c_v\), divided by \(c_p\), is in fact negligibly small in helium II. This question was considered by Lifshitz,^Л.44b Peshkov^П.48a and Dingle^D.50c.

9.3. Other equations for the velocity of second sound

Besides Tisza and Landau, various authors have proposed several other variants of equations for second sound. We shall dwell only on some of them. For a fuller acquaintance the reader is referred to the previously mentioned review by Dingle^D.52a.

Gogate and Pathak^G.47b gave an elementary derivation of an equation for the velocity of second sound which, although not entirely rigorous, is quite transparent. The equation coincides with Landau’s equation. On the other hand, Gorter, Castelijn, and Mellink^G.50a and Usui^U.51a obtained a somewhat different equation, based on Gorter’s equation^G.49b for the entropy \(S^* = x \dfrac{\partial S}{\partial x}\), where \(x = \dfrac{\rho_n}{\rho}\), mentioned earlier in Section 4.7. Gorter, Castelijn, and Mellink (for high frequencies) obtained

\[ u_2^2 = x(1 - x)G_{xx}, \tag{9.7} \]

and Usui

\[ u_2^2 = x(1 - x)S_xT_x. \tag{9.8} \]

(In this section and in Section 9.7, functions with subscripts denote partial derivatives.)

For low frequencies Gorter et al. obtained another equation for the velocity of second sound, which will be briefly considered in Section 9.7. In the approximation in which (9.7) and (9.8) are valid, they are equivalent. (In one of them the term \(G_{px}\) is neglected, in the other \(G_{pT}\). These quantities are apparently small.)

Nakajima and Shimizu^N.50b,N.51 used Usui’s equation to calculate the velocity of second sound, starting from two different

equations for the entropy of helium II as a function of temperature—one based on Landau’s representation, the other based on a generalization of Tisza’s model, in which the phonon entropy is included in the normal entropy. Applying Landau’s model, they found that the limiting value of \(u_2\) at low temperatures is three quarters of the value predicted by Landau; this value, as will be shown in 9.4, is too small. Dingle \(^{\mathrm{D}.51}\) cast doubt on the validity of Gorter’s equation for the entropy.

Several relations for the velocity of second sound were obtained in an ingenious way, without the aid of any two-component model. In all considerations of this kind the excitations of the liquid were treated as gas particles (following Landau \(^{\mathrm{L}.41a}\)). From this point of view, second-sound waves are density waves in a gas of excitations or “particles.” In order for such waves to propagate, the “particles” must satisfy certain conditions (discussed more fully in the literature cited at the end), among them: (1) only elastic collisions between the particles are possible, (2) the mean free path of the excitations must be small in comparison with the wavelength of second sound, and (3) dissipative processes must not cause too strong an attenuation of second sound.

This consideration was first made by Ward and Wilks \(^{\mathrm{W}.51b,\mathrm{W}.52}\). They constructed an analogy between phonons and photons in order to obtain Landau’s predicted limiting value for the velocity of second sound as \(T \to 0\), namely \(u_2 = u_1/\sqrt{3}\). They briefly discussed the possibility of the existence of second sound in crystals—an investigation first proposed by Peshkov \(^{\mathrm{P}.46a}\). (The analogy constructed by Ward and Wilks also implies the possibility of the existence of second-sound waves in a gas of photons; however, as was pointed out by London \(^{\mathrm{L}.51}\), photons do not satisfy condition (2) formulated above.)

Later Ward and Wilks obtained the same results by applying the Boltzmann equation to the excitations.

Dingle also considered this problem \(^{\mathrm{D}.52a,\mathrm{D}.52c}\) analogously to Ward and Wilks. Dingle introduced thermodynamic functions of the gas of excitations, from which he obtained the expression

\[ u_2^2=\frac{T S^2 \rho}{c_v \rho_n}, \tag{9.9} \]

which differs from equation (9.6) by the factor \((\rho/\rho_s)\), so that agreement with that equation occurs only as \(T \to 0\). The difference between (9.6) and (9.9) is due, in Dingle’s opinion \(^{\mathrm{D}.52a}\), to an incorrect allowance for the ground state in the derivation of equation (9.6). In the derivation of the two-component equations by Kronig \(^{\mathrm{K}.53g}\), which was already mentioned in Section 8, the statistical approach was

applied to the solution of the problem of second sound, and Kronig found it possible to take into account the momentum of the ground state.

As we have already indicated, there exist various assumptions concerning the possibility of propagation of second sound in a medium different from helium II. Dingle[^52a] has recently calculated the velocity \(u_2\) expected in some media. In doing so, of course, the restrictions indicated above must be satisfied. In particular, if one is interested in the possibility of the existence of second sound in \(\mathrm{He}^3\), then an answer to this may be obtained from the fact that \(\mathrm{He}^3\), apparently, is not superfluid (see (3.7)), and that dissipative processes in \(\mathrm{He}^3\) may prevent the propagation of second-sound waves. Nevertheless, the investigation of second sound in pure liquid \(\mathrm{He}^3\) will be extremely interesting and important.

9.4. Experimental investigation of second sound

The first experiments with second sound in liquid helium were carried out by Peshkov. In a series of beautiful experiments, published from 1944 to 1949[^44][^46a][^46b][^48c][^49], Peshkov investigated the temperature region from the \(\lambda\)-point to a temperature of about \(1^\circ\mathrm{K}\). Peshkov used the method of standing waves: oscillations of temperature or entropy were excited at one end of a tube filled with liquid helium II. The oscillations were excited by means of a heater consisting of a flat wire coil through which a sinusoidal current was passed. Another flat coil, representing a resistance thermometer of phosphor bronze and mounted perpendicular to the axis of the tube, served as the detector and could be moved along the tube up and down.

The relation between wavelength and frequency could be studied either by establishing resonance of temperature waves in the tube and measuring the resonance curve, or by moving the detector along the tube and observing the distance between the nodes and antinodes of the temperature oscillations. Peshkov used frequencies from 10 to \(10^4\) cps and found no dispersion in the waves. His results are shown in Fig. 50 by the dotted curve. Let us note that the value of the second-sound velocity \(u_2\) rises sharply from zero at \(T_\lambda\) and reaches a maximum, equal to \(20.3\ \mathrm{m/sec}\) near \(1.65^\circ\mathrm{K}\), a value that remains approximately constant over a wide temperature range. However, before the final rise to substantially higher values below \(1^\circ\mathrm{K}\), \(u_2\) passes through a minimum, equal to approximately \(18.4\ \mathrm{m/sec}\), a little above \(1^\circ\mathrm{K}\).

Peshkov[^48a][^48b] also investigated some additional effects which, for lack of space, we cannot discuss here in detail. One of the interesting experiments consisted in studying second sound with the aid of a porous filter placed in front of an ordinary acoustic emitter. Between the emitter

and the filter, the normal and superfluid components move as a single whole, producing density oscillations, or first sound. In the filter, because of viscosity, the normal liquid, unlike the superfluid, is retarded; this leads to the appearance of relative motion of the liquids and, consequently, to the appearance on the other side of the filter of second sound.

Carrying out experiments at lower temperatures, Peshkov \(^{\text{P.48c}}\) was the first to find that, contrary to Tisza’s equation (9.5), the velocity of second sound passes through a minimum at a temperature somewhat above \(1.0^\circ\) K, thereby experimentally confirming Landau’s assumption that the density and entropy of the phonons enter into the corresponding quantities for the normal liquid.

Fig. 50. Velocity of second sound \(u_2\) as a function of temperature. The dotted curve represents Peshkov’s results \(^{\text{P.44, P.46b, P.48b, P.48c, P.49a}}\). Above approximately \(1.25^\circ\) K this dotted curve also represents the results of Lane and co-workers \(^{\text{L.46c, L.47b}}\), Pellam \(^{\text{P.48, P.49b}}\), and Osborne \(^{\text{O.48b}}\)—all of them agree well. The solid curve, drawn through the experimental results represented by circles, was obtained by Maurer and Herlin \(^{\text{M.49d}}\).

Other measurements of the velocity of second sound were carried out by Lane, Fairbank, Shulz, and Fairbank \(^{\text{L.46c, L.47}}\), using a somewhat different method. In their apparatus, a cylindrical volume

of lucite with a heater at the bottom, which served as the source of periodic temperature oscillations, was partially filled with helium II. In the upper part of the volume a magnetic microphone was mounted, serving as a receiver. According to the assumption put forward by Onsager, the “second sound” excited in the liquid is transformed at its surface into ordinary sound in the vapor, with a high coefficient of energy transformation. The pressure oscillations of ordinary sound in the vapor could act on the microphone. Since the level of the liquid in the volume falls slowly, it was possible to observe resonances and, knowing the length of the liquid column at each resonance, to determine the velocity of the temperature waves with an accuracy of up to 0.5%. The results obtained agreed with Peshkov’s data, as shown in Fig. 50.

Later work on measuring the velocity of second sound was carried out with the pulse technique developed by PellamP. 48, P. 49a and OsborneO. 48b. A short heat pulse (of duration about \(10^{-4}\) sec) was sent into helium and received by a receiver. The two pulses—the transmitted and the received—are displaced relative to one another on the oscilloscope screen. For details of the experimental apparatus one should consult the original papers. The velocity is found from the ratio of the path length to the time interval between the sending and receiving of the signal, as determined directly from the oscilloscope. The method has the advantage of convenience (there is no need to attain resonance conditions) and of the small average power radiated into the helium. This second advantage becomes more and more important as the temperature is lowered and the entropy of helium II decreases.

The results of these experiments gave sufficient confirmation of the accuracy of the measurements previously made by Peshkov and by Lane and co-workers. They also showed that there is no measurable dispersion over a wide range of frequencies, as was also found by Peshkov.

In order to deepen the theoretical understanding of the mechanism of second sound, measurements below a temperature of approximately \(1.1^\circ\) K—the lowest boundary of the region investigated in the works cited above—were of interest. The results of such measurements were published during the last several years, first by Peshkov down to \(1.03^\circ\) KP. 48c, then by Maurer and HerlinM. 49d and PeshkovP. 52 down to \(0.86^\circ\) K, and then by Pellam and ScottP. 49b, Atkins and OsborneA. 50c, and Clerk, Hudson, and PellamK. 53a down to temperatures attained by paramagnetic cooling. The results of the work of Maurer and Herlin, and of Peshkov, are given in Fig. 50, where later data are also included. These results, as has already been noted, first indicated the existence of a minimum of the curve \(u_2\) as a function of \(T\) and the onset of a sharp increase of \(u_2\) below \(1^\circ\) K.

Thus, these results for the first time made it possible to make an unambiguous choice between the fundamental assumptions of Tisza, on the one hand, and Landau, on the other (see 9.2), concerning the interpretation of the equation for second sound in liquid helium II.

The results of measurements of \(u_2\) in the region of temperatures of paramagnetic cooling, obtained by Atkins and Osborne\({}^{A.50c}\) and by de Klerk, Hudson, and Pellam\({}^{K.53a}\), who used pulse techniques, are shown in Fig. 51. Leaving aside the question of the accuracy of the temperature measurements in the work of Atkins and Osborne (a later work cast doubt on the temperature measurements), we shall note their results as a certain milestone in measurements of the velocity of second sound. They observed that the velocity \(u_2\) increases very strongly near \(0.4^\circ\text{K}\), and then becomes constant. Extrapolating their results to \(0^\circ\text{K}\), they obtained for \(u_2(0) \simeq 152\ \text{m/sec}\), which is approximately 10 percent higher than the value expected by Landau.

Fig. 51. Velocity of second sound as a function of temperature at low temperatures (after de Klerk, Hudson, and Pellam\({}^{K.53a}\)). Curve \(A\) and the corresponding points are the data of de Klerk et al. The dashed curve \(B\) and the points on it are the data of Atkins and Osborne\({}^{A.50c}\).

Quite recently, de Klerk et al., working in the same temperature interval, found a somewhat different behavior. First, they established that the steep rise occurs rather at \(0.6^\circ\text{K}\) than at \(0.4^\circ\text{K}\). They explained this by the fact that the Atkins apparatus, cooled by a paramagnetic salt, never had its low temperature. The temperature indicated by Atkins and Osborne is apparently closer to the temperature of the salt than to the temperature of the helium. Some time ago, even before the publication of the results of de Klerk et al., Kramers pointed out that the steep rise of \(u_2\) with temperature should occur at the same temperature at which the dependence of the heat capacity on \(T^3\) begins, i.e. at the beginning of the purely phonon region. Measurements of the heat capacity by Kramers et al.\({}^{K.52a}\) showed that this temperature lies near \(0.6^\circ\text{K}\). Therefore,

the experiments of de Klerk et al. were a convincing confirmation of Kramers’ considerations.

Further, de Klerk et al. found that at the lowest temperatures they attained, namely at \(0.01^\circ\) K, the velocity of second sound is noticeably higher than the value predicted by Landau and, apparently, increases still further as the temperature is lowered. (They obtained velocity values as high as \(192\ \text{m/sec}\).) These results are shown in Fig. 51.

De Klerk, Hudson, and Pellam tentatively gave the following three possible explanations of their results:

(1) It is possible that the high value of the velocity obtained by them is an actual property of the propagation of second sound. This would mean that Landau’s prediction concerning the limiting value of the velocity at low temperatures is erroneous.

(2) The effect may have been caused by the shock-wave phenomenon discovered by Osborne \(^{\mathrm{O}.51}\).

(3) The effect may be due to the large mean free path of phonons.

Of course, very little can be said regarding the first of these suppositions. However, Dingle \(^{\mathrm{D}.52d}\) has recently carried out some calculations concerning the propagation of a rectangular or \(\delta\)-shaped pulse in a viscous or heat-conducting medium. The attenuation effect causes a spreading of the pulse and a distortion of its symmetry, so that the beginning of the received pulse will arrive after a time \(t = x/u_{20}\) (where \(u_{20}\) is the velocity calculated from equation (9.6)). If the velocity \(u_{\mathrm{fr}}\) of the front of the pulse is calculated, it proves to be very large. Dingle obtains, at \(0.2^\circ\) K,

\[ u_{20} \simeq u_{\mathrm{fr}}(1 - 60\sqrt{\eta}). \tag{9.10} \]

Estimating*) this expression, he finds that the observations of Atkins and Osborne (who measured the propagation of the leading front of the pulse) agree with Landau’s prediction. This correction to the observed velocity values is not suitable at the low temperatures used by de Klerk et al. If, however, other points of the received pulse are recorded in the measurements, the velocities obtained may be several times smaller. Therefore Pellam suggested that the use of pulses of high-frequency oscillations to determine the velocity of second sound may lead to the appearance of a dependence of the velocity on the carrier frequency. However, despite the fact that carrier frequencies were also used by de Klerk et al., no dependence on the carrier frequency was found.

*) In order to use (9.10), Dingle had to abandon the value of \(\eta\) used by Landau and Khalatnikov (\(\eta \simeq 1\) poise at \(0.2^\circ\) K), and to use Tisza’s approximation \(^{\mathrm{T}.47}\): \(\eta \sim T^{1/2}\). Even under this assumption, formula (9.10) becomes invalid at \(0.06^\circ\) K.

As for shock waves, it is clear that they begin to play a large role as the temperature is lowered, since any heat release into the system different from zero will cause a noticeable rise of temperature in a medium with vanishingly small entropy. If such shock waves cannot be eliminated, in the limiting case they lead, according to Clerk, Haddon, and Pellam[^K53a], to an apparent velocity of second sound equal to the velocity of first sound. Shock waves were considered by Temperley[^T51c] and Khalatnikov[^Kh51]. Khalatnikov criticized Temperley’s work, which was based on erroneous hydrodynamic equations. Starting from Landau’s equations, which he solves in the second approximation, Khalatnikov obtains:

\[ u_{\phi p}=v_{\infty}+\frac{1}{2}\alpha_2\left(v_{n1}+v_{n2}\right), \tag{9.11} \]

\[ \alpha_2=\frac{ST}{C}\,\frac{\partial}{\partial T}\left[\ln\left(u_{20}^{3}\frac{C}{T}\right)\right], \tag{9.12} \]

where \(v_{n1}\) and \(v_{n2}\) are the velocities of the normal component on one side and the other side of the discontinuity surface of the shock wave. As the temperature changes, \(\alpha_2\) changes sign, so that in some temperature regions discontinuity surfaces arise on the leading front, and in others—on the trailing front of the wave. As \(T\to 0\), if the velocity of the normal component in front of the wave front is equal to zero,

\[ u_{\phi p}\to \frac{u_1}{\sqrt{3}}+\frac{1}{3}v_n; \tag{9.13} \]

and since \(v_n\) becomes very large, yet does not exceed \(u_1\),

\[ u_{\phi p}\to u_1\left(\frac{1+\sqrt{3}}{3}\right). \tag{9.14} \]

Much attention has been devoted to shock waves in recent reviews by Dingwall[^D52a] and Atkins[^A52a].

The third possible explanation of the observed velocity values lies in the large mean free path of phonons, as was proposed by Ward[^W53b], Gorter[^G52], and Atkins[^A53]. This may be explained as follows: the appearance of second sound consists in the establishment in the liquid of a periodically varying “local temperature.” Such a local temperature, corresponding to local equilibrium, can occur only if the mean free path of phonons in the liquid is much smaller than the wavelength of second sound. If the mean free path of phonons becomes macroscopic at low temperatures, as was assumed in the viscosity theory of Landau and Khalatnikov[^L49] (see 9.1), equilibrium may fail to be established during propagation of the pulse. In this case the thermal pulse

will propagate with the velocity of phonons, i.e., with the velocity of first sound.

In a recent study of the influence of the mean free path length, Kramers, van den Berg, and Gorter \(^{\mathrm{K}.53h}\) compared the propagation of pulses in two volumes, \(3\ \mathrm{cm}\) and \(6\ \mathrm{cm}\) long, between \(0.1^\circ\mathrm{K}\) and \(1.0^\circ\mathrm{K}\).

At the lowest temperatures the sharply defined leading front of the pulse had a velocity equal to \(230\ \mathrm{m/sec}\). With increasing temperature the velocity decreased and the front of the received pulse became diffuse. In the \(6\ \mathrm{cm}\)-long volume this was observed at a lower temperature than in the \(3\ \mathrm{cm}\)-long volume and up to \(0.8^\circ\). The velocity measured in the long volume was smaller than in the short one. At \(0.8\ \mathrm{K}\) the observed velocity proved to coincide with the results of measurements by other authors. Consequently, the results confirm the postulated effect of the mean free path length.

9.5. Influence of pressure on the velocity of second sound

Measurements of the influence of pressure on the velocity of second sound were made by Peshkov and Zinov’eva \(^{\mathrm{II}.48c}\) in the temperature range from \(1.3^\circ\mathrm{K}\) to \(T_\lambda\), at pressures from the saturated-vapor pressure to the pressures at which helium solidifies, and by Maurer and Herlin \(^{\mathrm{M}.51b,\mathrm{M}.51c}\), Meyer and Herlin \(^{\mathrm{M}.53}\), who extended the temperature range to far below \(1^\circ\mathrm{K}\) and used pressures up to \(25\ \mathrm{atm}\). Above \(1^\circ\mathrm{K}\), the results taken from the published data of Herlin et al. form a family of isobars shown in Fig. 52. It is seen that the maximum of the curve \(u_2\) as

Fig. 52. Velocity of second sound as a function of temperature and pressure according to Maurer and Herlin \(^{\mathrm{M}.51c}\). The experimental points, about 250 in all, lie well on the curves shown; the maximum deviation of a point from a curve is \(\pm 1\%\). The pressures on the curves are indicated in atmospheres.

function of temperature, shifts toward lower temperatures as the pressure increases. As Peshkov and Zinov’eva indicated, the line of maxima (on the curve \(p\), as a function of \(T_{\max}\)) has

\[ \frac{dT}{dp}<0 \]

and is parallel to the \(\lambda\)-line.

At very low temperatures (about \(0.4^\circ\)K, not shown in Fig. 52) the velocity of second sound increases sharply, in full agreement with the reports of de Klerk et al. \({}^{K.53a}\); however, as is seen from Fig. 52, the higher the pressure, the lower the temperature at which the increase in velocity begins. After the sharp rise in velocity, saturation sets in, as expected, but here the curves intersect, and on approaching absolute zero the limiting velocity is greater the higher the pressure.

These results are important from the theoretical point of view. They confirm the assumption, first made by Landau, that at low temperatures only phonons enter the normal liquid. The sharp increase in the velocity of second sound occurs at the temperature at which the phonon part of the entropy constitutes an appreciable fraction of the total entropy, while the limiting velocity at \(0^\circ\)K is proportional to the velocity of first sound. Since the velocity of first sound increases with increasing pressure, and the phonon part of the entropy varies inversely as the cube of the velocity of first sound, an increase of pressure of helium II lowers the temperature at which the increase in velocity begins and increases its limiting value. Both these effects are observed. Similarly to de Klerk, Hudson, and Pellam \({}^{K.53a}\), Meyer and Herlin observed that the limiting values of the velocity obtained were larger than the values \(u_1/\sqrt{3}\) expected from Landau’s theory.

Kondo, Nakajima, and Shimizu \({}^{K.51}\) compared the results of Maurer and Herlin in the region above \(1^\circ\)K with calculations based on Yushi’s equation for the velocity of second sound (9.8). They obtained qualitative, but not quantitative, agreement.

9.6. Rayleigh disk and Pitot tube in a second-sound field

In addition to measurements in which the velocity of second sound as such was of interest, Pellam and his collaborators carried out elegant experiments with second sound in order to test the hypothesis of the two-component model. They performed an extensive series of experiments with a Pitot heat tube and a Rayleigh heat disk in second-sound fields.

In Pellam’s experiments with a Pitot heat tube, in a volume whose length was equal to half the wavelength, standing waves of second sound were established. The apparatus was ingeniously constructed in such a way that the pressure of helium II could be measured

in the middle of the volume (antinode of the wave) and at the end of the volume (node of the wave), using the liquid helium itself as the manometric liquid. The pressure of the liquid proved to be higher at the node. Pellam explained this by an intuitive generalization of Bernoulli’s equation, into which he introduced a term connected with the relative motion of the normal and superfluid liquids, namely:

\[ \frac{1}{2}\rho v^{2}+\rho gh+p+ \]

\[ +\frac{1}{2}\rho\,\frac{\rho_n}{\rho_s} \left(\frac{\dot H}{\rho ST}\right)^2=\mathrm{const}, \tag{9.15} \]

where \(\dot H=\rho STv_n\) is the heat flux. The introduction of the fourth term into equation (9.15) shows that where the kinetic energy of the relative motion of the two liquids is large, the pressure is small, and conversely—a conclusion confirmed by observations.

In this experiment it was established that the pressure changes proportionally to the square of the particle velocity; as noted earlier, there are no pressure changes proportional to the first power of the velocity. In Pellam’s publication only qualitative experimental results were given.

Liquid helium level

Fig. 53. Diagram of the Pitot–Pellam thermal tube \(^{P.50b}\). The dashed curve inside the horizontal vessel \(A\) represents the distribution of the density of the heat flux (\(B\) is a flat electric heater) at resonance. The resulting difference of levels in the vertical tubes, as well as the rise due to capillary forces, are exaggerated here.

The use of Rayleigh’s thermal disk for studying fields of second sound was first proposed by Pellam and Morse \(^{P.50a}\); the experimental results were recently reported by Pellam and Hanson \(^{P.52b}\). The basic idea of these experiments is that each component of liquid helium II will act on the Rayleigh disk as if the other were absent, since the action on the disk is proportional to the square of the velocity, and not to its first power. It is known that the maximum torque acting on a small disk suspended in a sound field is given by the expression

\[ \tau=\frac{4}{3}a^{3}\rho v^{2}, \tag{9.16} \]

where \(a\) is the radius of the disk, \(\rho\) is the density of the liquid, and \(v\) is its velocity. We

we can write the following expression both for the superfluid and for the normal component of helium II:

\[ \left. \begin{aligned} \tau_n &= \frac{4}{3}\,a^3 \rho_n v_n^2,\\ \tau_s &= \frac{4}{3}\,a^3 \rho_s v_s^2 . \end{aligned} \right\} \tag{9.17} \]

In the field of second sound the total rotational moment is the sum of both terms (9.17). The two velocities are connected by the condition that the density of the mass flux in equation (8.4) is zero, and may be replaced by the heat flux \(\dot H\) according to the equation \(\dot H=\rho S T v_n\). Thus, as was shown in detail by Pellam\({}^{\text{p.52b}}\),

\[ \begin{aligned} \langle \tau\rangle_{\text{mean}} &= \langle \tau_n\rangle_{\text{mean}}+\langle \tau_s\rangle_{\text{mean}}\\ &= \frac{4}{3}\,a^3 \rho\,\frac{\rho_n}{\rho_s}\,v_n^2\\ &= \frac{4}{3}\,a^3 \rho\,\frac{\rho_n}{\rho_s} \left(\frac{\dot H}{\rho S T}\right)^2 . \end{aligned} \tag{9.18} \]

Fig. 54. Schematic of the apparatus used by Pellam and co-workers\({}^{\text{p.52b}}\) for experiments with the Rayleigh disk in liquid helium II (see text).

The experimental technique used to observe the Rayleigh disk in liquid helium II by Pellam and co-workers is shown schematically in Fig. 54. The disk \(D\), representing a galvanometer mirror 12 mm in diameter, was suspended on a thin copper ribbon \(H\) at the center of the cylindrical horizontal vessel \(E\). A beam of light reflected from the disk onto a standard scale indicated the smallest deflections of the mirror caused by the field of second sound. The second sound was excited by means of a flat electric heater \(F\), mounted on the vertical wall of the vessel \(E\); a sinusoidal current was passed through the heater. Standing waves of second sound were established in the vessel by adjusting the frequency of the current feeding the heater approximately to the fundamental frequency. At a given temperature the rotational moment acting on the disk was measured as a function of the frequency of the periodic

heating and from the shape of the resonance curve one could calculate the effective heat flux at the center of the volume. From the frequency corresponding to the maximum, i.e., to the maximum torque, one could calculate the velocity of second sound. It turned out to coincide with the earlier results cited above.

More interesting is the torque itself. From equation (9.17) and from the absence of mass flux it follows that the component which has the smaller density produces the larger torque, and if the density practically vanishes, the torque becomes infinite. Thus, near

Fig. 55. Ratio \(\tau/(\dot H^2)_{\mathrm{avg}}\) as a function of temperature \(T\). The curves correspond to theoretical expectations; circles are observed values.

the \(\lambda\)-point the torque is almost entirely due to the superfluid liquid, whereas below \(1.2^\circ\mathrm{K}\) the torque is produced by the normal liquid. The theory is beautifully confirmed by experiment; each of the components produces its own torque, shown in Fig. 55, taken from the paper of Pellam and Hanson. The experiment gives an elegant confirmation of the two-component theory.

The theoretical curve in Fig. 55 is obtained from equation (9.18), if \(\rho_n\) and \(S\) are known. In doing so, Andronikashvili’s data\(^{A.48a}\) were used for \(\rho_n\), and Kapitza’s data\(^{K.41b}\) for the entropy. The agreement between experiment and theory is a reliable check of all the quantities. This was already pointed out in Section (2.3). A simple transformation of equations (9.6) and (9.18) leads to an expression for the heat capacity as a function of the rotational

moment of the heat flux and \(u_2\). Thus, the Rayleigh disk can be used as a calorimeter. Pellam and Hanson evaluated \(C_v\) by this method and obtained good agreement with other experiments.

The derivation of the equations for the Pitot heat tube and the Rayleigh heat disk was made by Pellam rather intuitively than rigorously (see Pellam’s discussion of the flux of mechanical energy, of radiation pressure, and of the transfer of heat flux at the “classical boundary” for heat pulses\(^{\mathrm{P.49c}}\)). It is therefore very gratifying that Usui\(^{\mathrm{U.51b}}\) derived equations, valid for each experiment, from the equations of motion of Nakajima, Tomita, and Usui. In particular, Usui confirmed that the total torque acting on the Rayleigh disk is indeed equal to the sum of two torques according to equation (9.17). As Usui noted, these experiments with the Pitot tube and the Rayleigh disk characterize the first occurrence of the term \((\mathbf{v}\operatorname{grad})\mathbf{v}\), entering into the equation of hydrodynamics.

9.7. Attenuation of Second Sound

Equations expressing the influence on the propagation of second sound of irreversible processes (caused by viscosity, thermal conductivity, and the finite relaxation time governing the transformation of the components into one another in those places where the concentration of one of them exceeds the equilibrium concentration at the given temperature) were considered partially or completely by Dingle\(^{\mathrm{D.48,\ D.50c}}\), Usui\(^{\mathrm{U.51a}}\), Kronig and Thellung\(^{\mathrm{K.50}}\), Gorter, Castelijn, and Mellink\(^{\mathrm{G.50a}}\), Kronig, Thellung, and Woldringh\(^{\mathrm{K.52e}}\), Khalatnikov\(^{\mathrm{X.50,\ X.52b,\ X.52c}}\), and Band and Meyer\(^{\mathrm{B.48a,\ B.48d,\ M.48d,\ B.49d}}\).

Dingle derived expressions for the attenuation coefficients of second sound propagating in an unbounded volume, due separately to viscosity and to thermal conductivity (\(\alpha\) is equal to the reciprocal of the path over which the amplitude of the disturbance wave decreases by a factor \(e\)). These coefficients are equal:

due to viscosity

\[ \alpha_{\eta}=\frac{\rho_s}{\rho_n}\frac{2\eta\omega^3}{3\rho u_2^3}, \tag{9.19} \]

due to thermal conductivity

\[ \alpha_k=\frac{K\omega^2}{2\rho C_v u_2^3}. \tag{9.20} \]

In these equations \(\eta\) is the coefficient of first (ordinary) viscosity, \(k\) is the coefficient of thermal conductivity, and \(C_v\) is the heat capacity. Usui also considered the effects due to viscosity and arrived at an equation equivalent to (9.19). Kronig and Thellung carried out analogous investigations, starting from more general equations including the coefficients of viscosity, thermal conductivity, and volume-

… expansion. They arrived at a cubic equation for the square of the wave vector \(K\left(K_i=\dfrac{\omega}{u_i}\right)\). The three roots of this equation correspond to three types of wave propagation: first sound, second sound, and “third sound.” The third sound is a surface effect (see 5.7), since the wave vector \(K_3\) is an almost purely imaginary quantity; the motion caused by viscous drag is damped over a distance of about \(10^{-5}\) cm. (These waves are not the ordinary viscous waves observed in all viscous liquids. Ordinary viscous waves, which have also been found in helium II, are characterized by the condition that the square of the wave vector is a purely imaginary quantity.) More interesting here is the fact that the damping coefficient of the second-sound waves of Kronig and Thellung is exactly equal to the sum of the two coefficients given by formulas (9.19) and (9.20), if Stokes’ relation between the first and second (bulk) viscosity coefficients is adopted.

Gorter, Casella, and Mellink also considered the influence of relaxation on the velocity of second sound. They assumed that the characteristic parameter \(\beta\), associated with the relaxation time, is such that in the absence of equilibrium \(\dfrac{dT}{dt}=\beta G_x\). Substituting this into the equation of motion, we find that \(u_2\) depends on the frequency: at high frequencies \((\omega \gg \beta G_{xT})\) equation (9.7) is valid; at low frequencies (the opposite inequality) equation (9.7) must be divided by \(\dfrac{1-G_{xx}G_{TT}}{G_{xT}^{2}}\). Kronig, Thellung, and Woldring considered the influence of relaxation somewhat more crudely, by generalizing the equations of Kronig and Thellung[^50]. They established that their results for second sound differ from the results just cited above only by an amount that lies beyond the accuracy of the present measurements. They did not calculate the dependence of the damping coefficient on the relaxation time.

Khalatnikov considered the relaxation process as the decay and absorption of phonons and rotons, restoring equilibrium. This work is a continuation of the work of Landau and Khalatnikov[^49] on viscosity, noted above (Section 9.1). Its results require the introduction of certain experimental data. With such data the author finds that the relaxation phenomenon leads to additional damping, which (for example, at \(2^\circ\) K) gives

\[ a \simeq 2\cdot 10^{-14}\omega^2 . \]

At the same temperature, equations (9.19) and (9.20) give

\[ a_{\eta} \simeq 7\cdot 10^{-18}\omega^2 \quad\text{and}\quad a_{\kappa} \simeq 4\cdot 10^{-14}\omega^2 . \]

Unfortunately, since the theories of Dingle and of Kronig and Thellung are macroscopic, whereas Khalatnikov’s theory is microscopic, it is not clear whether all the damping coefficients will be additive.

Band and Meyer also considered relaxation effects. For this purpose they introduced two relaxation times. The first relaxation time \(\tau_1\)

characterizes the exchange of momentum between the normal and superfluid liquids and leads to a damping term in the wave equation. Thus, if \(z\) is proportional to the displacement of the normal component from the center of gravity of the fluid element, then they write

\[ \frac{d^{2}z}{dt^{2}}+\frac{1}{\tau_{1}}\frac{dz}{dt}=v_s^{2}\operatorname{grad}\operatorname{div} z. \tag{9.21} \]

The second relaxation time characterizes the rate of heat exchange between the components. \(\tau_2\), apparently, is connected with the characteristic time of Gorter et al. and Kronig et al.; \(\tau_1\) was introduced ad hoc. The assumptions of Band and Meyer led to the result that at very low frequencies the wave equation for second sound passes over into the equation of thermodiffusion.

For the frequencies used in measurements with second sound (\(\sim 10^4\) cps), the damping determined by equations (9.19) and (9.20) is extremely small in the temperature range from \(1^\circ\) K to \(2^\circ\) K. However, near the \(\lambda\)-point, \(\alpha_\eta\) and \(\alpha_k\) become very large, since here \(\rho_s\) and \(u_2\) tend to zero. At low temperatures \(\alpha_k\) also becomes large, since the heat capacity tends to zero.

The experimental data on absorption are still very few and sometimes contradictory. PellamR. 48, R. 49a observed an increase of \(\alpha\) with temperature from \(10^{-2}\ \mathrm{cm}^{-1}\) at \(1.65^\circ\) K to \(0.25\ \mathrm{cm}^{-1}\) at \(2.1^\circ\) K, with a sharp increase at temperatures above the latter, which qualitatively agrees with the predictions of the present paragraph. OsborneO. 51, however, found a considerably smaller absorption, which is practically independent of temperature up to \(2.17^\circ\) K. The difference in the magnitudes of the effect found by the two experimenters is explained, in any case partly, by the difference in the fundamental frequencies: Pellam used rectangular pulses of duration \(150\ \mu\mathrm{sec}\), Osborne—of duration \(600\ \mu\mathrm{sec}\), which should give a difference (roughly) by a factor of 16 in the expected absorption.

CITED LITERATURE *)

I. Works of Soviet Authors

A. 46. E. L. Andronikashvili, J. Phys. USSR, 10, 201 (1946); ZhETF 16, 780 (1946).
A. 48a. E. L. Andronikashvili, ZhETF 18, 424 (1948).
A. 48b. E. L. Andronikashvili, ZhETF 18, 429 (1948).
A. 49. E. L. Andronikashvili, ZhETF 19, 535 (1949).

) The translators considered it advisable to divide the bibliography into two parts (Soviet and foreign works). In the first list, references to J. Phys. USSR* are supplemented by references to Soviet journals in Russian in which the corresponding works were published. It should also be borne in mind that this list covers only about half of the works on liquid helium II carried out in the USSR.

A. 52. E. L. Andronikashvili, ZhETF 22, 62 (1952).

B. 47. N. Bogolyubov, J. Phys. USSR, 11, 23 (1947); Izv. AN SSSR, ser. fiz., 11, 77 (1947).

G. 44. V. L. Ginzburg, ZhETF 14, 134 (1944).

G. 49. V. L. Ginzburg, DAN 69, 161 (1949).

E. 51. B. N. Esel’son and B. G. Lazarev, DAN 81, 537 (1951).

E. 52. B. N. Esel’son and B. G. Lazarev, ZhETF 23, 552 (1952).

K. 38a. P. L. Kapitsa, Nature 141, 74 (1938); DAN 18, 21 (1938).

K. 38b. A. K. Kikoin and B. G. Lazarev, Nature 142, 289 (1938).

K. 41a. P. L. Kapitsa, J. Phys. USSR 5, 59 (1941); ZhETF 11, 581 (1941). (See also K. 44).

K. 41b. P. L. Kapitsa, Phys. Rev. 60, 354 (1941).

K. 41c. P. L. Kapitsa, J. Phys. USSR 4, 181 (1941); ZhETF 11, 1 (1941).

K. 44. P. L. Kapitsa, J. Franklin Inst. 237, 491 (1944). (See also K. 41a).

K. 51. M. I. Kaganov and B. N. Esel’son, ZhETF 21, 658 (1951).

L. 41a. L. Landau, J. Phys. USSR 5, 71 (1941); ZhETF 11, 592 (1941).

L. 41b. L. Landau, Phys. Rev. 60, 356 (1941).

L. 44a. L. Landau, J. Phys. USSR 8, 1 (1944).

L. 44b. E. M. Lifshitz, J. Phys. USSR 8, 110 (1944).

L. 47. L. Landau, J. Phys. USSR 11, 91 (1947).

L. 49. L. Landau and I. M. Khalatnikov, ZhETF 19, 637, 709 (1949).

P. 44. V. P. Peshkov, J. Phys. USSR 8, 381 (1944); DAN 45, 385 (1944).

P. 46a. V. P. Peshkov, Reports of International Conference of the Physica Society (London), 2, 19 (1946).

P. 46b. V. P. Peshkov, J. Phys. USSR 10, 389 (1946); ZhETF 16, 1000 (1946); also Nature 157, 300 (1946).

P. 48a. V. P. Peshkov, ZhETF 18, 857 (1948).

P. 48b. V. P. Peshkov, ZhETF 18, 867 (1948).

P. 48c. V. P. Peshkov, ZhETF 18, 951 (1948).

P. 48d. V. P. Peshkov and K. N. Zinov’eva, ZhETF 18, 438 (1948).

P. 49. V. P. Peshkov, ZhETF 19, 270 (1949).

P. 52. V. P. Peshkov, ZhETF 23, 687 (1952).

S. 40. P. G. Strelkov, J. Phys. USSR 3, 53 (1940); ZhETF 10, 743 (1940).

F. 40. Ya. Frenkel, J. Phys. USSR 2, 365 (1940); ZhETF 10, 650 (1940).

Kh. 50. I. M. Khalatnikov, ZhETF 20, 243 (1950).

Kh. 51. I. M. Khalatnikov, DAN 79, 237 (1951). See also ZhETF 23, 253 (1952).

Kh. 52a. I. M. Khalatnikov, ZhETF 22, 687 (1952).

Kh. 52b. I. M. Khalatnikov, ZhETF 23, 8 (1952).

Kh. 52c. I. M. Khalatnikov, ZhETF 23, 21 (1952).

Ya. 43. I. A. Yakovlev, J. Phys. USSR 7, 307 (1943).

II. Works by foreign authors

A. 37. J. F. Allen, Peierls and Uddin, Nature 140, 62 (1937).

A. 38a. J. F. Allen and A. D. Misener, Nature 141, 75 (1939).

A. 38b. J. F. Allen and A. D. Misener, Nature 142, 643 (1938).

A. 38c. J. F. Allen and H. Jones, Nature 141, 243 (1938).

A. 39a. J. F. Allen and A. D. Misener, Proc. Roy. Soc. (London) A172, 467 (1939).

A. 39b. J. F. Allen and J. Reekie, Proc. Cambridge Phil. Soc. 35, 114 (1939).

A. 39c. J. F. Allen and E. Ganz, Proc. Roy. Soc. (London) A171, 242 (1939).

A. 48. K. R. Atkins, Nature 161, 925 (1948).

A. 50a. K. R. Atkins, Proc. Roy. Soc. (London) A203, 119 (1950).
A. 50b. K. R. Atkins, Proc. Roy. Soc. (London) A203, 240 (1950).
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