Properties of the Light Helium Isotope He$^3$ at Low Temperatures. II*)

R. A. Chentsov

5. Theories of Liquid He$^3$ and Liquid Mixtures He$^3$—He$^4$

In the present part of the review we have sought to give only a general idea of the concepts underlying the principal theories of pure liquid He$^3$ and of solutions of He$^3$ in He$^4$. Of course, the theory of He$^3$ cannot be presented in complete isolation from questions of the theory of liquid He$^4$: the very difference between the properties of liquid He$^3$ and He$^4$ is of outstanding interest for the theory of the superfluidity of He$^4$—one of the fundamental phenomena of low-temperature physics. Therefore we shall at times have to touch, in passing, on certain questions of the theory of helium II. Comparatively great attention will be paid to the works of the Soviet scientist I. Ya. Pomeranchuk$^{1\text{–}2,\ \mathrm{I},55}$, since a number of important predictions contained in these works concerning the behavior both of liquid He$^3$ and of solutions of He$^3$ in He$^4$ have received convincing experimental confirmation. As in the presentation of the experimental material, we have not set ourselves the goal of giving an exhaustive survey of the literature, and have compensated for this incompleteness by providing a detailed bibliography. Let us also recall that a number of theoretical questions (on the liquefaction of He$^3$, on the preservation of the character of the $\lambda$-transition in solutions of He$^3$ in He$^4$, on the viscosity of gaseous He$^3$, and others) have already been briefly touched upon in the presentation of the corresponding experimental data.

As was already mentioned in I (pp. 50–51), the main properties of the ordinary isotope of helium He$^4$ in the liquid state, including superfluidity, are well explained by the theory of Acad. L. D. Landau$^{1\text{–}4}$. In Landau’s theory, liquid He$^4$ is regarded as a weakly excited quantum system whose thermodynamic properties are determined by the state of a “gas” of elementary thermal excitations (an approach,

) Continuation (see UFN 55*, No. 1, pp. 49–80, 1955). References to the first part will be accompanied by the numeral I.

basically similar to the approach to the theory of the heat capacity of solids (Debye). The brilliant quantitative agreement with experiment in the overwhelming majority of cases (see, for example, O—5, O—7), as well as the most recent theoretical developments (Tellung and Kronig P—1, Zaiman P—2, Feynman P—3, and others*) convincingly prove the validity of such an approach. However, Landau’s theory does not cover the entire important region of the \(\lambda\)-transition, referring only to temperatures below the \(\lambda\)-point. A discrepancy of \(10^3—10^5\) times is observed with the experimental values of the “critical velocities” of flow at which superfluidity is destroyed. A weak side of the theory is its certain incompleteness, its failure to take into account the microscopic properties of the atoms that form the system, and, in connection with this, the impossibility of answering the question of whether, for example, liquid \(\mathrm{He}^3\) and \(\mathrm{He}^6\) will also possess the same properties as \(\mathrm{He}^4\) (in particular, a \(\lambda\)-transition and superfluidity). It is significant that in this theory (at least in its modern version) roton-type excitations are considered only as a concept describing the state and collective motion of the system of all atoms of a given specimen of liquid \(\mathrm{He}^4\) as a whole; they are not connected with any concrete notion of specific motions of \(\mathrm{He}^4\) atoms. Therefore the quantitative characteristics of rotons are parameters which in essence do not admit of theoretical estimation and are to be determined only from comparison with experiment.

In this connection, the fact that liquid \(\mathrm{He}^3\) (as may be judged from the experimental data set forth in I, pp. 65—66) is devoid of the property of superfluidity is of fundamental significance.** Most theorists concerned with liquid helium believe that the element determining the sharply different behavior of the liquid isotopes \(\mathrm{He}^3\) and \(\mathrm{He}^4\) is the different statistics of the atoms.

The idea of the role of statistics as a factor determining the presence of a \(\lambda\)-transition in liquid \(\mathrm{He}^4\) belongs to F. London P—4. Considering the phenomenon of the jump of the heat capacity at the \(\lambda\)-point, London compared this phenomenon with the kink on the curve of the temperature dependence of the heat capacity (Fig. 1) of an ideal gas whose particles obey the quantum Bose–Einstein statistics. The calculation of the temperature at which the maximum of the heat capacity should occur in such a “gas,” which would consist of \(\mathrm{He}^4\) atoms and possess a dens-

*) References to literature placed in the bibliography are accompanied by the letters O, Э, T, or P, according to the sections of the bibliography (reviews, experimental papers, theoretical papers, miscellaneous).

**) Incidentally, it must be noted that in one of the recent works de Boer T—174 advances the suggestion that at \(T \sim 0.2^\circ\mathrm{K}\) liquid \(\mathrm{He}^3\) may have a \(\lambda\)-transition. If this suggestion were justified, the question would arise whether \(\mathrm{He}^3\) below the \(\lambda\)-point possesses superfluid properties.

ness of liquid He\(^4\), led to the value \(T_0 = 3.09^\circ K\), close in order of magnitude to the temperature of the \(\lambda\)-transition He I to He II (\(2.19^\circ K\)). The mechanism causing a change in the character of the temperature dependence of the heat capacity of an ideal Bose gas upon passing through \(T_0\) is called “Bose–Einstein condensation”

Fig. 1.

and consists in an ordering of the distribution of the momenta of the gas molecules: with decreasing temperature down to \(T < T_0\), some of the molecules begin to “resettle” into states of minimum energy; at \(T = 0\) all gas molecules are in these states. The process of “condensation” should lead to a fall in the “measure of disorder”—the entropy—with decreasing temperature, which may be compared with the observed sharp fall of the entropy of liquid He II upon cooling below \(T_\lambda\) (a more detailed account of the question of Bose–Einstein condensation may be found by the reader, for example, in \(^{8-15}\)). London identified atoms “in the condensed state” with the superfluid component, and the remaining ones with the normal component of He II.

London’s hypothesis encounters a number of very serious difficulties. For example, the transition in a Bose gas is not, unlike the \(\lambda\)-transition in liquid He\(^4\), a second-order transition: the heat capacity does not undergo a jump at \(T_0\); a discontinuity exists only in the derivative of the heat capacity with respect to temperature. Further, measurements of the velocity of second sound in liquid He II at very low temperatures have unambiguously shown that the role of the normal component at these temperatures is played not by atoms in a special state, but by excitations of the phonon type. Experiment \(^{P-5}\) does not at all confirm the predictions of the “condensation” theory concerning enhanced scattering of light \(^{P-6}\), as well as of X-rays and slow neutrons at the \(\lambda\)-point of He\(^4\). However, the main objection is not even this: doubt is raised by the very comparison of a real liquid with an ideal gas. In the latter case the molecules are noninteracting; meanwhile, in liquid helium the interaction is not a correction, but the principal factor determining the prop-

properties of the liquid and its very existence. In connection with this, the question arose: what change might the theoretical conclusions undergo if the interaction of helium atoms in liquid He\(^4\) were properly taken into account? The proponents of London’s theory expressed the hope that taking the interaction into account would improve the agreement with experiment; others voiced the legitimate concern that such an allowance might have the opposite effect and render false the entire theoretical conclusion about the existence of a transition of any kind in liquid He\(^4\).

A substantial shortcoming of London’s theory was also the absence of an explanation for the phenomenon of superfluidity (the disappearance of viscosity). However, N. N. Bogoliubov\(^{5-7}\) showed that under certain conditions a Bose gas exhibits superfluidity, whereas a Fermi gas under analogous conditions does not possess superfluidity.

The production of liquid He\(^3\) in quantities sufficient for experimentation made it possible, to some extent, to test the influence of atomic statistics on the properties of liquid helium without resorting to complicated calculations. As is known, the quantum statistics to which gas atoms are subject is determined by the parity of the spin of the particles. Therefore gaseous He\(^4\) (the particle spin is zero) must obey Bose–Einstein statistics, whereas gaseous He\(^3\) (spin \(1/2\)) must obey Fermi–Dirac statistics. At present the difference between the statistics of atoms in gaseous He\(^3\) and He\(^4\) has been firmly established experimentally both by spectroscopic measurements and by the study of various properties, in particular by measurements of viscosity (see I, pp. 64–65) at low temperatures. But then it is appropriate to recall that in the case of an ideal gas whose molecules are distributed according to Fermi–Dirac statistics, the phenomenon of “condensation” in momentum space described above does not occur. Accordingly, the maximum in the heat capacity shown above in Fig. 1 is absent.

The heat capacity at low temperatures turns out to increase monotonically with temperature according to a linear law (compare the heat capacity of metals at low temperatures, determined by the contribution of electrons, which also have spin \(1/2\)). Consequently, if London’s parallel is legitimate, liquid He\(^3\) should not undergo—in contrast to liquid He\(^4\)—a phase transformation of the \(\lambda\)-point type. This, as we have seen, is indeed confirmed by experiment (see I, pp. 61–64), which considerably increases confidence in London’s hypothesis. Nevertheless, for a final solution of the question it would not be superfluous to carry out an experimental investigation of the properties of He\(^3\) at temperatures much closer to absolute zero than the region studied, and to determine the thermodynamic and hydrodynamic properties of pure He\(^3\), which, as may be expected (see, for example, I\(^{-44}\) or T\(^{-150}\)), is superfluid.

Thus, the available experimental material compels us, while regarding the validity of L. D. Landau’s theory as proven, at the same time to acknowledge the presence, also in London’s theory, of a valuable kernel, consisting in the assumption of an essential influence of atomic statistics on the properties of the corresponding substances at low temperatures.

We now turn to an account of specific works devoted to the theory of pure He\(^3\) in the liquid state, and first of all to the work of I. Ya. Pomeranchuk \(^{1—29}\).

Let us note at the outset that the problem of the theory of liquid He\(^3\) includes, first of all, the explanation and prediction of the general character of the temperature dependences of the most important properties: heat capacity, entropy, vapor pressure, transport coefficients, etc., as well as the elucidation of specific features of the \(p,T\)-diagram of state. It should be emphasized that the experimental material available at present is still far from sufficient for a complete test of any theory of this kind. In particular, information on the properties of He\(^3\) at \(T < 0.5^\circ\)K, where essentially new results may be expected, is almost entirely lacking. However, all the more valuable is every success of a theory constructed on such comparatively limited experimental material.

I. Ya. Pomeranchuk, in accordance with Landau’s general theory of quantum liquids, assumes that in the region of sufficiently low temperatures the energy of the excited states of liquid He\(^3\) is a sum of the energies of elementary excitations. The latter may be regarded as quasiparticles, the system of which in the present case is assumed to obey Fermi–Dirac statistics. At temperature \(T\), excitations with energy \(\varepsilon \sim kT\) play the principal role. The number of such excitations per unit volume is

\[ n \sim p_0^2 \Delta p / \pi^2 \hbar^3 \sim p_0^2 \varepsilon / v \hbar^3 \sim p_0^2 kT / \hbar^3 v, \]

where \(p_0 \sim (3\pi^2)^{1/3}\hbar N^{1/3}\) is the radius of the Fermi sphere (\(N\) is the number of He\(^3\) atoms in \(1\ \mathrm{cm}^3\)). Hence a quadratic dependence on temperature is obtained for the energy of liquid He\(^3\), and a linear law for its heat capacity: \(C = aT\), where \(a \sim \pi^{-2} p_0^2 \hbar^3 v\). In considering kinetic effects it is necessary to take into account that the number of excitations \(n \sim N (T/T_0) \ll N\) for \(T \ll T_0\); consequently, pair collisions must play the principal role under these conditions. A simple calculation leads to the following temperature dependences of the transport coefficients: viscosity \(\eta = Nmlv = A/T^2\), thermal conductivity \(\chi = Clv = B/T\) (\(A\) and \(B\) are constants). These equations make possible an experimental test of the theory if the behavior of \(C\), \(\eta\), and \(\chi\) is measured experimentally at sufficiently low temperatures.

I. Ya. Pomeranchuk made important conclusions about the state of the system of spins of atomic nuclei in liquid He\(^3\). Since, in the latter, the de Broglie wavelength of the atoms is of the same order as the distance between them, an exchange interaction of a special type must play a large role. Considering a system of two atoms of liquid He\(^3\), I. Ya. Pomeranchuk establishes that, for the interaction of these atoms, the parallelism or antiparallelism of the nuclear spins of both atoms is essential. In the first case the coordinate part of the wave function must be antisymmetric with respect to interchange of the two atoms; this leads to the appearance of a repulsive force (i.e., a weakening of the total attractive force) when the atoms approach each other. On the contrary, in the case of antiparallel spins the coordinate part of the total wave function of the diatomic system must be symmetric; in this case the change of the potential with distance must be more rapid, the attractive forces (prevailing over the repulsive forces by virtue of the zero-point energy of the large molecular volume of liquid He\(^3\)) increased, and the state as a whole energetically more favorable than the state with parallel-oriented spins. (This conclusion, apparently, can also be checked independently of the study of the properties of He\(^3\) at low temperatures: the influence of the mutual orientation of the spins on the interaction of He\(^3\) atoms should lead to a splitting of the lines of the optical—chiefly electronic and vibrational—and microwave spectra of the metastable diatomic molecule He\(^3_2\), and to a shift of these lines relative to their positions calculated from the positions of the lines of He\(^4_2\), without taking the indicated effect into account.) Thus, one may say that in liquid He\(^3\) a large role must be played by an exchange effect connected with the exchange of atoms. This effect must lead to a correlation of the orientations of the nuclear spins, expressed in a predominantly antiparallel orientation of the nuclear spins of neighboring atoms. This should manifest itself above all in the magnetic properties of liquid He\(^3\), which should be regarded not as a nuclear ferromagnet, but as a nuclear exchange paramagnet, analogous to electronic exchange paramagnets of the type of solid oxygen. Beginning at the temperature at which the free orientation of spins ceases, deviations from Curie’s law should be observed for the magnetic susceptibility of liquid He\(^3\); the latest measurements of this quantity \(^{3-8}\) at temperatures below \(1^\circ\) K show that such deviations are in fact observed at \(T < 0.5 \div 0.7^\circ\) K. But perhaps still more interesting are the thermodynamic consequences of the indicated correlation of nuclear spins.

A random distribution of nuclear spins in liquid He\(^3\) corresponds to a definite contribution to the entropy of the liquid, equal to \(R \ln 2\) per 1 mole. If, however, in liquid He\(^3\) there occurs a corre-

tion of the directions of the nuclear spins, this contribution disappears. As a result, upon sufficient cooling (down to \(T \lesssim 1^\circ\mathrm{K}\)), when the contribution to the entropy associated with thermal motion becomes small, the total entropy of liquid \(\mathrm{He}^3\) falls to values smaller than \(R\ln 2\). The data presented in I (pp. 62–63) indicate that this prediction of Pomeranchuk’s theory apparently begins to receive experimental confirmation. However, in order to obtain a convincing answer to this question, it is highly desirable to carry out careful measurements of the heat capacity at \(T<0.4^\circ\mathrm{K}\).

The fall of the entropy of liquid \(\mathrm{He}^3\) to values smaller than \(R\ln 2\) should, as I. Ya. Pomeranchuk noted, be reflected in the phase diagram of \(\mathrm{He}^3\), namely in the solid—liquid \(\mathrm{He}^3\) phase equilibrium. The point is that, unlike liquid helium, in solid helium (as was pointed out by L. D. Landau) the amplitude of zero-point oscillations is considerably smaller than the interatomic distances. This makes impossible, or very unlikely, the exchange of atoms responsible for the correlation of spins in liquid \(\mathrm{He}^3\). As a result, in solid \(\mathrm{He}^3\) the orientations of the nuclear spins should have a chaotic distribution; correlation should set in only at \(T\sim 10^{-7}\) degree, when the magnetic interaction of the nuclear spins causes a decrease of the entropy from \(R\ln 2\) to zero. Thus, in a certain fairly broad temperature range \((10^{-7}<T<1)\) the entropy of liquid \(\mathrm{He}^3\) proves to be smaller than the entropy of solid \(\mathrm{He}^3\). This should have as a consequence a negative heat of fusion

Fig. 2.

\(Q_{\mathrm{m}}=T(S_{\ell}-S_{\mathrm{s}})\simeq -RT\ln 2\); in other words, under isothermal melting heat is not absorbed but released. On the other hand, from the Clausius–Clapeyron equation it follows that in this region the derivative with respect to temperature of the melting pressure is also negative:

\[ \frac{dp_{\mathrm{m}}}{dT} = \frac{S_{\ell}-S_{\mathrm{s}}}{V_{\ell}-V_{\mathrm{s}}} <0 \]

and only at the same temperatures \(\lesssim 10^{-7}\) degree should \(\lvert dp_{\mathrm{m}}/dT\rvert\) begin to decrease to 0, in accordance with Nernst’s theorem. The expected general character of the dependence of \(p_{\mathrm{m}}\) on temperature is shown in Fig. 2 (not to scale). The solidification pressure has a minimum at the temperature at which the entropies of solid and liquid \(\mathrm{He}^3\), in equilibrium with each other, become equal. As measurements of the curve \(p_{\mathrm{m}}(T)\) show (see I, pp. 59–60),

It is very likely that precisely the dependence predicted by I. Ya. Pomeranchuk’s theory actually takes place, and that the temperature of the minimum of \(\rho_{пл}\) lies near \(0.5^\circ\) K. For a final resolution of this question it is necessary to measure \(\rho_{пл}\) at \(T < 0.5^\circ\) K by a method free of the drawback, mentioned in 1, of the blocked-capillary method. As I. Ya. Pomeranchuk notes in his paper, the behavior of the entropy curves of solid and liquid He\(^3\) discussed above makes it possible to propose a new method for obtaining record low temperatures: by carrying out adiabatic solidification of He\(^3\) (by compression), one can, in principle, obtain solid He\(^3\) at temperatures \(10^{-6}—10^{-7^\circ}\)K. In the solidification of each \(\mathrm{cm}^3\) of liquid He\(^3\), about \(\sim 10^{-2}\) joule of thermal energy would be removed from bodies in thermal contact with it.

On the basis of a model of liquid He\(^3\) in the form of an ideal Fermi gas of excitations with particle mass equal to the atomic mass and density equal to the density of the liquid (the entropy of the latter takes account of the spin correlation of the nuclei), E. M. Lifshitz \({}^{1–27}\) calculated the chemical potential, the heat of vaporization, and the heat capacity of liquid He\(^3\) and obtained values close to the experimental ones. A number of other authors (including Buckingham and Temperley \({}^{1–125}\), Sinha \({}^{1–153}\), Tkesh-Dalitz and de Boer \({}^{1–156}\)) also used modified Fermi-gas models for the theoretical consideration of the properties of liquid He\(^3\) and derived expressions for thermodynamic quantities (vapor pressure, viscosity, etc.) approximately agreeing with values determined experimentally. Nevertheless, comparison with experiment shows the inadequacy of such simple models for a quantitative description of the properties of liquid He\(^3\).

Prigogine \({}^{1–181}\) applies to He\(^4\) a model analogous to the Lennard-Jones and Devonshire model of a classical liquid, in which the starting point for calculating the properties of the liquid is the distribution function of an individual molecule moving in a “cell” formed by the surrounding other molecules. In ordinary liquids the cell volume approximately coincides with the volume of the molecule itself; however, in the case of liquid helium the former volume is significantly larger than the latter, and more than one molecule can fit in a single cell. It turns out that such a model leads to a peak of the heat capacity in the case of He\(^4\); however, in the case of He\(^3\) (Fermi particles!) the maximum of the heat capacity is strongly smeared out and shifted almost to the critical point, which, in the author’s opinion, gives a qualitative explanation of the difference between the behavior of the two isotopes. It is not hard to see that Prigogine’s theory is to some extent a theory of association of atoms in liquid He\(^4\) and He\(^3\).

We now turn to the consideration of theories of liquid mixtures He\(^3\)–He\(^4\), which are of interest above all because their study casts additional light on the properties of the pure helium isotopes at low temperatures. It is necessary here to note that the most

Of greatest interest at the present stage are the theories of dilute solutions of He³ in He⁴ (as well as He⁴ in He³), since the theory of concentrated solutions, even in the case of ordinary solutions, whose properties have been studied experimentally in detail and are not changed by the presence of quantum effects, is still far from complete. Therefore we shall begin the exposition with a description of the theoretical works devoted to dilute solutions of He³ in He⁴, which (in contrast to dilute solutions of He⁴ in He³ and concentrated He³—He⁴ solutions) have been studied experimentally to a sufficient extent.

Since in solutions of He³ in He⁴ the λ-transition and the property of superfluidity below the λ-point are preserved, the most natural approach in constructing the theory of these solutions is, in this case as well, the application of the “two-component model,” which proved so successful in the case of liquid He⁴. But in that case the first question that arises is: to which component—normal or superfluid—does the He³ impurity contribute in He II. L. D. Landau and I. Ya. Pomeranchuk1–44 put forward the assertion that any foreign particles dissolved in He II (at low concentration) belong to the normal component, since transfer of energy from them to the superfluid component, with the appearance in the latter of new excitations, is impossible. A justification of an analogous statement, but proceeding from the kinetic equation for impurity particles dissolved in a weakly nonideal Bose gas possessing superfluidity, is also given in the work of S. V. Tyablikov7–106. As was indicated in I (pp. 71–73), the nonparticipation in the superfluidity of He II of He³ impurities and, apparently, He⁶ is proven experimentally.

On the basis of the assumption that additional thermal excitations are associated with the impurity atoms He³ in He II, I. Ya. Pomeranchuk1–53 constructed a theory of dilute solutions of He³ in He⁴. He assumes that these excitations are characterized by an energy dependence on momentum of the form \(E = E_0 + (p - p_0)^2/2\mu\) (\(\mu\) is the effective mass of the elementary excitation; \(p_0\) may be either equal or not equal to zero) and that their totality forms a statistically independent ensemble obeying classical statistics. We shall confine ourselves to the case \(p_0 = 0\), since comparison of experimental and theoretical data for the second-sound velocity in solutions of He³ in He⁴ (I. M. Khalatnikov1–51) excludes the possibility \(p_0 \ne 0\). I. Ya. Pomeranchuk points out that for dilute solutions (concentration of the light component \(c < 10^{-1}\)) the degeneracy temperature \(T_0 = \mathrm{const}\cdot c^{2/3} < 0.2^\circ\) K and, consequently, at ordinary temperatures (\(\sim 0.5–1^\circ\)K and higher) one may use classical statistics for the impurity particles. Thus one obtains expressions for the entropy \(S = S_0 + (Rc/3)\ln[(\ldots)/c]\) and the heat capacity \(C = C_0 + (3/8)Rc\) of the solution (quantities with subscript …).

“0” pertain to pure liquid He\(^4\); upon dilution \(\mu=m_{\mathrm{He^4}}\).

In the equations of hydrodynamics there also appear changes relative to the hydrodynamics of pure He II. In particular, in the equation of motion for the velocity \(v_s\) of the superfluid component it is necessary to introduce an additional term that takes into account the osmotic pressure of the dissolved particles. There also arises a new continuity equation for the He\(^3\) atoms:

\[ \partial(\rho c)/\partial t+\operatorname{div}(\rho c v_n)=0, \]

where \(\rho\) is the density of He II, and \(v_n\) is the velocity of the normal component, to which the impurity atoms belong. At equilibrium \(v_n=v_s=0\), and the thermodynamic potential \(\Phi=\text{const}\); hence one obtains one of the equilibrium conditions

\[ -S_0 \Delta T - V \Delta p=(k/\mu)\Delta(c/T). \]

From consideration of this equation (at \(p=\text{const}\)) it is evident that if in some part of a vessel containing He II (\(T\sim1\text{--}2^\circ\) K) with an admixture \(c\sim10^{-6}\) He\(^3\) atoms the temperature is lowered by \(10^{-4}\text{--}10^{-5}\) degrees, then practically the entire admixture will collect at that place. The effects described by the equation given play an important role in the operation of low-temperature installations for separating helium isotopes (see I, pp. 52–53).

Since the number of impurity excitations is, numerically, much smaller than the absolute concentration of He\(^3\) in the solution, it is quite obvious that in any case, at small concentrations, superfluidity must be preserved in the mixture as well (see also \(^{1-46}\)). In particular, it manifests itself in the propagation of second sound. The speed \(u_2\) was subsequently also calculated in the work of I. Ya. Pomeranchuk and was found to be equal to

\[ u_2^2=(\rho_s/\rho_n)T\cdot[(S_0+kc/\mu)^2/C+kc/\mu], \]

where \(\rho_s/\rho_n\) is the ratio of the densities of the superfluid and normal components of pure He II. The formula, in agreement with experiment, shows that at temperatures not much below the \(\lambda\)-point \(u_2(c)>u_2(0)\): the presence of the He\(^3\) impurity increases the speed of second sound by an amount proportional to the concentration. However, upon further cooling \(u_2(c)\) reaches a maximum (the temperature of which decreases as \(c\) increases), and at temperatures close to absolute zero (when practically the entire normal component consists of excitations of the impurity type) the speed of second sound in the mixture falls according to the law

\[ u_2=\sqrt{(5/3)(k/\mu)}\,T^{1/2}. \]

At the lowest temperatures (\(\sim0.1^\circ\) K) the situation is complicated by the quantum degeneracy of the impurity gas. These theoretical conclusions were then brilliantly confirmed by experiment; in processing its data, I. M. Khalatnikov\(^{1-54}\) established that the effective mass of the impurity excitation is about 3 times greater than the mass of the He\(^3\) atom. At the same time it turned out\(^{1-46}\) that, in order to obtain complete quantitative agreement over a broad temperature range, the theory requires some refinement, since the parameter \(\mu\) proves to depend rather strongly on the tempera-

true. At lower temperatures \(p \simeq 2m_3 v_s\), which agrees with the theoretical value of this quantity recently calculated by Khalatnikov \(^{\text{г—176}}\).

A number of works by I. M. Khalatnikov are devoted to the theory of rich mixtures of liquid \(\mathrm{He}^3\) and \(\mathrm{He}^4\). He derived \(^{\text{г—150}}\) the equations of hydrodynamics for concentrated solutions of foreign particles in He II and showed that impurity particles take part in normal motion at high concentrations only in the case when the dissolved substance itself, in pure form, does not possess superfluidity. In a subsequent work \(^{\text{г—151}}\) the hydrodynamic equations obtained are applied to the calculation of the velocities of first and second sound in concentrated solutions. The question of the form of the dissipative function in solutions is also considered, and the conclusion is drawn that the solution is characterized by two kinetic coefficients in addition to those of pure He II: a diffusion coefficient and a thermodiffusion coefficient. A comparison with experimental data for dilute solutions of \(\mathrm{He}^3\) in \(\mathrm{He}^4\) was carried out in the work of V. N. Zharkov and I. M. Khalatnikov \(^{\text{г—152}}\). There are as yet no experimental data on the velocities of first and second sound, diffusion, and thermodiffusion in concentrated solutions of \(\mathrm{He}^3\) in \(\mathrm{He}^4\).

A large number of other theoretical works devoted to rich \(\mathrm{He}^3\)—\(\mathrm{He}^4\) mixtures have as their object the study of the concentration dependence of the temperature of the \(\lambda\)-transition, as well as the liquid–vapor phase equilibrium and the change in the thermodynamic functions of liquid \(\mathrm{He}^4\) and \(\mathrm{He}^3\) upon mixing. Let us recall that on the second question—which includes, in particular, the vapor pressure over solutions and the distribution coefficient of the components between the two phases—up to the most recent time \(^{\text{г—39}}\) there were no reliable experimental results. Therefore we consider it premature to dwell specially on this question, and shall only briefly mention that the theory of equilibrium of \(\mathrm{He}^3\)—\(\mathrm{He}^4\) usually represents an extension to the case of \(\mathrm{He}^3\)—\(\mathrm{He}^4\) of the thermodynamics of classical solutions, with corrections for quantum effects (allowance for zero-point energy, atomic statistics, etc.). In particular, we shall not dwell on the so-called “Taconis hypothesis” \(^{\text{э—56}}\), often invoked for the consideration of this range of phenomena and consisting in the formal assumption of the solubility of \(\mathrm{He}^3\) only in the normal component of He II. Since at the same time the curve of \(T_\lambda\) as a function of concentration is known with fairly good accuracy over almost the entire concentration interval (with the exception of the region \(c_{\mathrm{He}^3}=0 \div 11\%\)), it seems advisable to mention briefly some of the relevant theories and to compare them with the experimental data. The very fact of the decrease of \(T_\lambda\) in passing from pure \(\mathrm{He}^4\) to a solution can be qualitatively understood from the following very simple considerations:

An admixture of a small amount of He\(^3\) increases the density of the normal component of He II by no more than the amount \(m\mu n_k\) (where \(n\) is the number of impurity atoms in \(1\ \text{cm}^3\)), since the effective mass \(\mu\) of the impurity is several times greater than \(m_{\mathrm{He}^4}\) (cf. the experiment on second sound in solutions). It is therefore natural to suppose that the admixture of He\(^3\) leads to a redistribution of mass between \(\rho_s\) and \(\rho_n=\rho-\rho_s\) in He II in favor of the normal component. If we take into account that the \(\lambda\)-point corresponds to the temperature up to which, upon heating, the condition \(\rho_n=\rho\) is reached, then the effect of the lowering of \(T_\lambda\) at \(c\ne0\) becomes quite understandable. At the same time, as we see from this reasoning, essential for the occurrence of the lowering of the temperature of the \(\lambda\)-point is the interaction of He\(^3\) atoms with He\(^4\) atoms, which manifests itself in the inequality of \(\mu\) and \(m_{\mathrm{He}^3}\).

To explain the curve \(T_\lambda(c)\) and other properties of liquid He\(^3\) and He\(^4\) mixtures over a wide range of concentrations, the authors of many recent works use a model of liquid mixtures of helium isotopes due to Heer and Daunt \(^{7-143}\), with one or another modification. These authors regard the collections of atoms of both components of the solution as independent systems. In this, the two components—He\(^4\) and He\(^3\)—are treated as an ideal Bose gas and a nondegenerate (at “ordinary” temperatures) Fermi gas, respectively, enclosed in a rectangular potential well of depth

\[ \chi=\sum_{i=3,4} N_i \chi_i / \sum N_i \]

(\(N_i\) is the number of atoms; \(\chi_i\) is the potential well for the pure component in the same model). The volume of the solution is assumed equal to

\[ V=\sum_{i=3,4} N_i v_i, \]

where \(v_i\) is the volume per 1 atom in pure liquids He\(^3\) and He\(^4\). The temperature of the \(\lambda\)-transition of the solution is calculated as the degeneracy temperature of the Bose gas (\(N_4\) atoms of He\(^4\), placed in the volume of the solution \(V\)) and turns out to be equal to

\[ T_\lambda(c)=T_\lambda(0)\left\{(1-c)/[1+c(v_3/v_4-1)]\right\}^{2/3}. \]

This dependence, in general features, agrees with the experimental data. By allowing additional assumptions, for example about the existence of an energy gap between the lowest and the remaining energy states in the Bose gas or about the association of atoms into groups, it is possible to bring the theory still closer to experiment. However, the theory of Heer and Daunt has the same weak points discussed above as London’s theory of liquid He\(^4\). In addition, one may think that an exact theory of liquid He\(^3\)—He\(^4\) mixtures must explicitly take into account the interaction of both components of the solution: after all, the presence of such an interaction is sharply manifested already in the change of the effective mass of He\(^3\) atoms when He\(^3\) enters a solution with He II.

In conclusion, we may say that, despite the first successes, the theory of both pure liquid He\(^{3}\) and liquid solutions of He\(^{3}\)—He\(^{4}\) is still very far from completion. The principal reason for this situation lies in the youth of the He\(^{3}\) problem and in the insufficiency of experimental material. This insufficiency became especially noticeable when the first indications appeared of the emergence of certain new processes occurring in liquid He\(^{3}\) at temperatures \(T < 0.5^\circ\) K. The most necessary among the data lacking at present are: the heat capacity of liquid He\(^{3}\) at \(T < 0.4^\circ\) K; the vapor pressure of pure He\(^{3}\) and of liquid mixtures He\(^{3}\)—He\(^{4}\) at \(T < 1^\circ\) K; the viscosity of liquid He\(^{3}\), and especially a direct test of the absence of its superfluidity at \(T > 0.1^\circ\)—\(0.2^\circ\) K; the speed and attenuation of sound in liquid He\(^{3}\); the magnetic properties of liquid solutions He\(^{3}\)—He\(^{4}\), especially at \(T < 1^\circ\) K; the heat of vaporization of liquid He\(^{3}\); the change of volume and entropy upon mixing liquid He\(^{3}\) and He\(^{4}\) over a wide range of temperatures and concentrations; the velocity of second sound in mixtures at \(T \sim 0.5^\circ\) K and below; the course of the melting curve \(p_{\text{melt.}}(T)\) at \(T < 0.5^\circ\) K; and the thermodynamic and elastic properties of solid He\(^{3}\).

However, it is already clearly evident that the study of the properties of He\(^{3}\) at low temperatures considerably broadens the range of physical knowledge. A comparison of the properties of the ordinary and light isotopes of helium is a remarkable demonstration of how subtle differences in atomic structure (differences in nuclear spin) lead—owing to quantum effects—to a sharp difference in the macroscopic behavior of the two substances. The study of He\(^{3}\) will provide rich additional material for investigating the unique physical object represented by quantum liquids, and will certainly confirm the validity and fruitfulness of Landau’s theory of quantum liquids. The results of investigations of He\(^{3}\) also shed additional light on the problem of the superfluidity of He\(^{4}\). They show that, apparently, an essential condition for the occurrence of superfluidity in a quantum liquid is the absence of spin in the particles composing it, which therefore obey Bose–Einstein statistics. These results have, in our opinion, great value also for the interpretation of existing theories of superfluidity. In particular, the proximity—established (I–54 and others)—of the effective masses of impurity excitations in solutions of He\(^{3}\) in He\(^{4}\) to the actual masses of the atoms (this was previously known for the final parameters—the effective mass, momentum, and energy of roton-type excitations and of He\(^{4}\) atoms in pure He II), as well as the observed “imposition” on an ensemble of excitations of the statistics of the corresponding atoms (cf. I–29, I–27, and others), suggest that excitations of the deformation type, introduced into the theory of the quantum liquid as a means of describing

liquid as a whole are in fact associated with a definite spatial structure (of the order of atomic dimensions) and a characteristic internal motion (for example—but not necessarily—of the type of the motion^{175–176} analogous to the motion of a classical liquid around a solid body moving through it, which leads to a change in the effective inertial mass of the latter). The study of these subtle questions of the microscopic theory of quantum liquids is an attractive subject for future investigations.

BIBLIOGRAPHY

The bibliographic list includes: review articles on the properties of He³ and mixtures of He³–He⁴, as well as reviews and monographs on He⁴ published in Russian; works on the vapor pressure of He³ and on the methods for enriching He³; and works on the isotope effect He³; experimental investigations of the properties of He³ and of solutions; theoretical works. If the theoretical treatment is carried out in connection with the presentation by the authors of original experimental data, the corresponding work is placed in the section of experimental works. The theoretical works include some works devoted mainly to the description of He⁴, but also containing essential material relating to He³ and He³–He⁴ mixtures. At the end are given works on He⁴ cited in the second part of the review in the course of the exposition. For works not published in Russian, the original language is indicated (with the exception of works published in Physical Review and several other well-known journals).

The bibliography basically covers the period approximately to the middle of 1954. The author intends soon to publish a bibliographic supplement in order to make the bibliography for 1954 exhaustive.

1. REVIEW ARTICLES. MONOGRAPHS.

1. 1. P. L. Kapitsa, Problems of liquid helium. Soviet Science, No. 1, 33–50.
2. 1. P. L. Kapitsa, On the superfluidity of liquid helium, UFN, 26, No. 2, 133–143.
3. 1. E. L. Andronikashvili, K. A. Tumanov, Development in the Soviet Union of the theory of superfluidity and superconductivity, UFN, 33, No. 4, 469–532.
4. 1. E. M. Lifshitz, Theory of the superfluidity of helium II, UFN, 34, No. 4, 512–559.
5. 1. E. L. Andronikashvili, Superfluidity (experimental data). In: V. Keesom, Helium, IL, Moscow, pp. 430–533.

6. V. Keesom, Helium, IL, Moscow.

7. E. M. Lifshitz, Superfluidity (theory). In: V. Keesom, Helium, IL, Moscow, pp. 385–429.

8. K. A. Tumanov, Isotope He³, UFN, 37, No. 4, 403–413.

9. K. A. Tumanov, New data on He³, UFN, 38, No. 3, 439–441.

1. F. London, Rare helium isotope He³, key to the unusual properties of ordinary liquid He⁴, Nature, 163, No. 4148, 694–695.

2. Liquid helium 3, Nature, 163, No. 4143, 477–478.

3. 1. J. G. Daunt, Properties of helium 3 at low temperatures, Adv. in phys., 1, No. 2, 209–268.

4. R. B. Dingle, Theory of He II, Adv. in phys., 1, No. 2, 111–168.

5. 1. B. M. Abraham, D. W. Osborne, B. Weinstock, Properties of liquid He³, Science, 117, No. 3032, 121–125 (Engl.).

6. C. F. Squire, Physics of low temperatures, McGraw-Hill, New York, 1953, 70–73, 95–96 (Engl.).

7. J. Wilks, Recent Soviet work on liquid helium, Nuovo Cimento, 10, No. 4, Suppl. 569–538 (Engl.).

2. ABUNDANCE OF He³. METHODS OF ENRICHING HELIUM WITH THE He³ ISOTOPE AND OF OBTAINING PURE He³.

1. 1. L. W. Alvarez, R. Cornog, He³ in helium, Phys. Rev., 56, No. 4, 379.

2. L. W. Alvarez, R. Cornog, Helium and hydrogen of mass 3, Phys. Rev., 56, No. 6, 613.

3. 1. L. T. Aldrich, A. O. C. Nier, Abundance of He³ in atmospheric helium and helium from sources, Phys. Rev., 70, Nos. 11–12, 983–984.

4. J. Franck, A proposed method for separating He³ from He⁴, Phys. Rev., 70, Nos. 7–8, 561.

5. R. C. Johns, W. H. Furry, Separation by a combined thermodiffusion method, Rev. Mod. Phys., 18, No. 2, 151–234 (see pp. 210–214).

6. 1. J. G. Daunt, R. E. Probst, H. L. Johnston, L. T. Aldrich, A. O. C. Nier, New method for separating helium isotopes He³ and He⁴, Phys. Rev., 72, No. 6, 502–403.

7. B. B. McInteer, L. T. Aldrich, A. O. C. Nier, Coefficient of thermodiffusion of helium and separation of He³ by thermodiffusion, Phys. Rev., 72, No. 6, 510–511.

8. 1. L. T. Aldrich, A. O. C. Nier, Variations in the relative amounts of He³ and He⁴ in natural helium sources, Phys. Rev., 74, No. 10, 1225.

9. L. T. Aldrich, A. O. C. Nier, He³ content in natural helium sources, Phys. Rev., 74, No. 11, 1590–1591.

10. A. Andrew, W. R. Smith, Increase of the concentration of He³, Phys. Rev., 74, No. 4, 493–497.

11. C. T. Lane, H. A. Fairbank, L. T. Aldrich, A. O. C. Nier, Separation of He³ in the flow of helium gas through a long and narrow tube, Phys. Rev., 73, No. 3, 256–257.

12. B. B. McInteer, L. T. Aldrich, A. O. Nier, Increase of the concentration of He³ by the thermodiffusion method, Phys. Rev., 74, No. 8, 946–949.

13. R. V. Rollin, J. Hatton, Increase of the concentration of He³ from ordinary liquid helium at temperatures below 1° Kelvin, Phys. Rev., 74, No. 4, 508.

14. 1. J. H. Coon, Abundance of the helium isotope He³, Phys. Rev., 76, No. 9, 1355–1357.

15. C. A. Reynolds, H. Fairbank, C. T. Lane, B. B. McInteer, A. O. Nier, Thermal shock in liquid He³, Phys. Rev., 76, No. 1, 64–66.

1. 1. O. F. Schuette, Jr, A. Zucker, W. W. Watson, A thermodiffusion apparatus for increasing the concentration of He³, Rev. Sci. Instrum., 21, No. 12, 1016—1018.

2. K. W. Taconis, Nederl. tijdschr. natuurkunde, 16, 101 (Dutch).

3. 1. F. A. Paneth, P. Reasbeck, K. I. Mayne, The content of He³ and the age of meteorites, Geochim. et cosmochim. acta, 2, No. 5/6, 300—303 (Engl.).
4. 1. K. R. Atkins, J. C. Findlay, D. R. Lovejoy, W. H. Watson, Method for enriching mixtures of He³—He⁴, Canad. J. Phys., 31, No. 4, 679.

5. T. Soller, W. M. Fairbank, A. D. Crowell, Rapid separation of He³ from He⁴ by the “thermal-shock” method, Phys. Rev., 91, No. 5, 1058—1060.

6. 1. B. V. Rollin, On the processes of formation of H³ and He³ in nature, UFN, 52, No. 3, 472—477.

3. EXPERIMENTAL INVESTIGATIONS OF THE PROPERTIES OF He³ AND MIXTURES He³—He⁴ AT LOW TEMPERATURES.

1. 1. J. G. Daunt, R. E. Probst, H. L. Johnston, Distribution of the isotopes He³ and He⁴ in the liquid phase, J. Chem. Phys., 15, No. 10, 759—760.

2. H. A. Fairbank, C. T. Lane, L. T. Aldrich, A. O. Nier, Concentrations of He³ in the liquid and vapor phases of He⁴, Phys. Rev., 71, No. 12, 911—913.

3. 1. J. G. Daunt, R. E. Probst, H. L. Johnston, A new isotopic effect in liquid helium, Phys. Rev., 73, No. 6, 638.

4. J. G. Daunt, R. E. Probst, S. R. Smith, Experiments with He³ at low temperatures, Phys. Rev., 74, No. 4, 495—496.

5. H. A. Fairbank, C. T. Lane, L. T. Aldrich, A. O. Nier, Distribution of He³ between liquid He⁴ and vapor, Phys. Rev., 73, No. 7, 729—732.

6. H. A. Fairbank, C. A. Reynolds, C. T. Lane, B. B. Mc-Inter, L. T. Aldrich, A. O. Nier, Increase in the vapor pressure of liquid helium associated with the presence of dissolved He³, Phys. Rev., 74, No. 3, 345—346.

7. 1. B. M. Abraham, B. Weinstock, D. W. Osborne, Temperatures of the λ-points of mixtures He³—He⁴, Phys. Rev., 76, No. 6, 864.

8. B. M. Abraham, B. Weinstock, D. W. Osborne, Temperatures of the λ-points of mixtures He³—He⁴, Proc. Internat. conf. on phys. of very low temps, M. I. T., p. 45.

9. J. G. Daunt, Note on the temperatures of the λ-points of mixtures He³—He⁴, Proc. Internat. conf. on phys. of very low temps, M. I. T., p. 46.

10. L. Goldstein, E. F. Hammel, J. E. Kilpatrick, L. D. P. King, On the investigation of mixtures He³—He⁴ by means of slow neutrons, Proc. Internat. conf. on phys. of very low temps, M. I. T., p. 53.

11. E. R. Grilly, E. F. Hammel, S. G. Sydoriak, Approximate values of the density of liquid He³ between 1.27 and 2.79° K, Phys. Rev., 75, No. 7, 1103—1104.

12. C. T. Lane, Equilibrium He³—He⁴ at temperatures below the λ-point, Proc. Internat. conf. on phys. of very low temps, M. I. T., p. 51.

Bibliography

1. C. T. Lane, H. A. Fairbank, L. T. Aldrich, A. O. Nier, On the equilibrium He³–He⁴ at temperatures below the λ-point, Phys. Rev., 75, No. 1, 46–50.

2. D. W. Osborne, B. M. Abraham, B. Weinstock, Vapor pressure over He³ and mixtures of He³–He⁴, Proc. Internat. conf. on phys. of very low temps, M. I. T., p. 48.

3. D. W. Osborne, B. Weinstock, B. M. Abraham, Comparison of the flows of isotopically pure liquid He³ and He⁴, Phys. Rev., 75, No. 6, 988.

4. R. B. Probst, H. L. Johnston, J. J. Fritz, Thermal conductivity of liquid He³ and He³–He⁴, Proc. Internat. conf. on phys. of very low temps, M. I. T., p. 52.

5. S. G. Sydoriak, E. R. Grilly, E. F. Hammel, Condensation of pure He³ and its vapor pressure between 1.2 K and its critical point, Phys. Rev., 75, No. 2, 303–305.

6. S. G. Sydoriak, E. F. Hammel, New experiments with liquid He³, Proc. Internat. conf. on phys. of very low temps, M. I. T., pp. 42–44.

7. K. W. Taconis, J. J. M. Beenakker, A. O. C. Nier, L. T. Aldrich, Measurements relating to vapor–liquid equilibrium for solutions of He³ in He⁴ at temperatures below 2.19 K, Physica, 15, Nos. 8–9, 733–739.

8. K. W. Taconis, J. J. M. Beenakker, A. O. C. Nier, L. T. Aldrich, Measurements relating to vapor–liquid equilibrium for solutions of He³ in He⁴ at temperatures below 2.19 K, Phys. Rev., 75, No. 12, 1966.

9. K. W. Taconis, J. J. M. Beenakker, A. O. C. Nier, L. T. Aldrich, Measurements relating to vapor–liquid equilibrium for solutions of He³ in He⁴ at temperatures below 2.19 K, Proc. Internat. conf. on phys. of very low temps, M. I. T., pp. 49–50.

10. B. Weinstock, D. W. Osborne, B. M. Abraham, Viscosity of liquid He³, Proc. Internat. conf. on phys. of very low temps, M. I. T., p. 47.

1950.

1. B. N. Eselson, B. G. Lazarev, Certain properties of solutions of He³ in He⁴, DAN, 72, No. 2, 265–267.

2. B. N. Eselson, B. G. Lazarev, Certain properties of solutions of He³ in He⁴. I. Separation of helium isotopes, ZhETF, 20, No. 8, 742–747.

3. B. N. Eselson, B. G. Lazarev, N. E. Alekseevskii, Measurement of the vapor pressure over solutions of He³ in He⁴, ZhETF, 20, No. 11, 1055–1056.

4. B. N. Eselson, B. G. Lazarev, I. M. Lifshitz, Certain properties of solutions of He³ in He⁴. II. Displacement of the λ-point and features of the transfer effect, ZhETF, 20, No. 8, 748–759.

5. B. M. Abraham, D. W. Osborne, B. Weinstock, Vapor pressure, critical point, heat of vaporization, and entropy of liquid He³, Phys. Rev., 80, No. 3, 366–371.

6. J. G. Daunt, C. V. Heer, λ-temperatures of solutions of He³ in He⁴ at temperatures below 1°K, Phys. Rev., 79, No. 1, 46–51.

7. E. A. Lynton, H. A. Fairbank, Velocity of second sound in mixtures of He³ and He⁴, Phys. Rev., 79, No. 4, 735–736.

8. E. A. Lynton, H. A. Fairbank, Second sound in mixtures of He³ and He⁴, Phys. Rev., 80, No. 6, 1043–1046.

9. K. W. Taconis, J. J. M. Beenakker, Z. Dokoupil, Measurements of the influence of He³ dissolved in He⁴ on the fountain effect, Phys. Rev., 78, 171.

BIBLIOGRAPHY

1. B. Weinstock, D. W. Osborne, B. M. Abraham, Phase diagram of He³—He⁴ solutions, Phys. Rev., 77, No. 3, 400—401.

2. 1. B. M. Abraham, D. W. Osborne, B. Weinstock, Melting curve of He³, Proc. Internat. conf. on low temp. phys., Oxford, pp. 29—30.

3. J. J. M. Beenakker, Influence of He³ on the fountain effect in He⁴, Proc. Internat. conf. on low temp. phys., Oxford, pp. 80—81.

4. E. A. Lynton, H. A. Fairbank, Velocity of second sound in He³—He⁴ mixtures, Phys. Rev., 81, No. 2, 295.

5. E. A. Lynton, H. A. Fairbank, Effective mass of He³ in dilute solutions of He³ in He⁴, Proc. Internat. conf. on low temp. phys., Oxford, pp. 82—83.

6. D. W. Osborne, B. M. Abraham, B. Weinstock, Solidification of He³, Phys. Rev., 82, No. 2, 263—264.

7. H. S. Sommers, Jr., Phase diagram of dilute solutions of He³ in He⁴ at temperatures below the $\lambda$-point, Proc. Internat. conf. on low temp. phys., Oxford, pp. 78—80.

8. K. W. Taconis, Thermal conductivity of liquid helium II containing a certain amount of He³, and heat transfer to the walls of a vessel, Proc. Internat. conf. on low temp. phys., Oxford, pp. 29—30.

9. 1. J. J. M. Beenakker, K. W. Taconis, E. A. Lynton, Z. Dokoupil, G. Sees van, Heat conductivity of He II containing a certain amount of He³, Physica, 18, Nos. 6—7, 433—448.

10. J. G. Daunt, C. V. Heer, $\lambda$-point of He³ solutions in He⁴ at temperatures below 1 K, Phys. Rev., 78, No. 3, 342.

11. J. G. Daunt, C. V. Heer, Solubility of He³ in liquid He⁴, Phys. Rev., 86, No. 2, 205—208.

12. J. G. Daunt, C. V. Heer, Solubility of He³ in liquid He⁴, Phys. Rev., 86, No. 4, 627.

13. E. F. Hammel, H. L. Laquer, S. G. Sydoriak, W. E. McGee, Investigation of the magnetic properties of liquid He³, Phys. Rev., 86, No. 3, 432.

14. E. F. Hammel, A. F. Schuch, Anomalous type of flow of He³, Phys. Rev., 87, No. 1, 154—155.

15. H. S. Sommers, Jr., Values of the vapor elasticity of He³—He⁴ mixtures at temperatures below the $\lambda$-point, Phys. Rev., 83, No. 1, 113—127.

16. B. Weinstock, B. M. Abraham, D. W. Osborne, Vapor pressure of liquid He³, Phys. Rev., 85, No. 1, 158—159.

17. 1. W. M. Fairbank, W. B. Ard, H. G. Dehmelt, W. Gordy, S. R. Williams, Temperature dependence of the nuclear susceptibility of He³ between 1.2° K and 4.2° K, Phys. Rev., 92, No. 1, 268—269.

18. J. C. King, H. A. Fairbank, Velocity of second sound in He³—He⁴ mixtures, Phys. Rev., 90, No. 2, 347.

19. J. C. King, H. A. Fairbank, Velocity of second sound in He³—He⁴ mixtures at temperatures below 1° K, Phys. Rev., 90, No. 5, 989—990.

20. J. C. King, H. A. Fairbank, Shift of the $\lambda$-point in He³—He⁴ mixtures, Phys. Rev., 91, No. 2, 489.

21. H. S. Sommers, Jr., W. E. Keller, J. G. Dash, Heat of mixing of He³ with He⁴, Phys. Rev., 91, No. 2, 489.

22. H. S. Sommers, Jr., W. E. Keller, J. G. Dash, Heat of mixing of He³ with He⁴, Phys. Rev., 92, No. 6, 1345—1346.

23. G. Vries de, J. G. Daunt, Heat capacity of He³ in the temperature interval 1.3° and 2.3° K, Phys. Rev., 92, No. 6, 1572—1573.

1. B. Weinstock, J. R. Pellam, Measurements with a thermal disk of Rayleigh in mixtures of He³—He⁴, Phys. Rev., 83, No. 2, 521.

2. 1. B. N. Esel’son, Some properties of solutions of He³ in He⁴. II. Ultrasonic dispersion, ZhETF, 26, No. 6, 744—750.
3. B. N. Esel’son, B. G. Lazarev, Solidification of mixtures of helium isotopes, DAN, 97, No. 1, 61—64.
4. B. N. Esel’son, P. G. Borzunyak, Surface tension of solutions of helium isotopes, DAN, 98, No. 4, 569—571.
   95a. B. N. Esel’son, P. G. Borzunyak, Surface tension of liquid helium isotopes, DAN, 99, No. 3, 365—367.
5. E. W. Becker, R. Misenta, F. Schmeissner, Viscosity of gaseous He³ and He⁴ in the temperature range 1.3—4.2° K, Phys. Rev., 93, No. 1, 241.
6. E. W. Becker, R. Misenta, F. Schmeissner, Viscosity of gaseous He³ and He⁴ in the temperature range 1.2° K. Toward a quantum-statistical gas-kinetic theory of collisions at low temperatures, Zeits. f. Phys., 137, No. 1, 126—136.
7. W. M. Fairbank, W. B. Ard, G. K. Walters, Fermi—Dirac expression in liquid He³ at temperatures below 1° K, Phys. Rev., 95, No. 2, 566—568.
8. J. C. King, H. A. Fairbank, Second sound in mixtures of He³—He⁴ at temperatures below 1° K, Phys. Rev., 93, No. 1, 21—27.
9. D. W. Osborne, B. M. Abraham, B. Weinstock, Heat capacity and entropy of liquid He³ in the temperature interval 0.42°—1.06° K, Phys. Rev., 94, No. 1, 202—203.
10. T. R. Roberts, S. G. Sydoriak, Heat capacity of liquid He³, Phys. Rev., 93, No. 6, 1418.
11. C. F. Squire, Third International Conference on Low-Temperature Physics and Chemistry, Science, 119, No. 3091, 400—402.
12. G. Vries, de, J. G. Daunt, Heat capacity of 9.3%-liquid He³ in the temperature interval 1.15°—2.3° K, Phys. Rev., 93, No. 2, 361—362.
13. G. Vries, de, J. G. Daunt, Heat capacity of 9.3%-He³ at temperatures below 1° K, Phys. Rev., 93, No. 3, 631—632.

4. THEORIES OF LIQUID He³ AND SOLUTIONS He³—He⁴.

1. 1. L. D. Landau, I. Ya. Pomeranchuk, On the motion of extraneous particles in helium II, DAN, 59, No. 4, 669—670.
2. S. V. Tyablikov, Behavior of impurities in a weakly nonideal gas obeying Bose statistics, ZhETF, 18, No. 11, 1023—1029.
3. J. Boer, de, R. J. Lunbeck, Properties of condensed He³, Physica, 14, No. 5, 318.
4. J. Boer, de, R. J. Lunbeck, Properties of the condensed phase of the light isotope of helium, Physica, 14, No. 8, 510—519.
5. F. London, O. K. Rice, On solutions of He³ in He⁴, Phys. Rev., 73, No. 10, 1188—1193.

6. J. W. Stout, Second-order transitions in two-component systems. Application to solutions of He³ in He⁴, Phys. Rev., 74, No. 5, 605—609.

7. 1. I. Ya. Pomeranchuk, Influence of impurities on the thermodynamic properties and speed of second sound in He II, ZhETF, 19, No. 1, 42—53.
8. J. Boer, de, Statistical mechanics of mixtures of He³ and He⁴, Phys. Rev., 76, No. 6, 852—853.

Bibliography

1. L. Goldstein, M. Goldstein, On exchange energy in an ensemble of He³ atoms, Phys. Rev., 76, No. 3, 464.

2. D. Haarter, Properties of He³, Amer. J. of Phys., 17, No. 7, 399–403.

3. D. Haarter, H. Wergeland, Viscosity of mixtures of He³ and He⁴, Phys. Rev., 75, No. 5, 886–887.

4. J. Kramendonk, van, K. Compaan, J. Boer, de, Equation of state of gaseous He³, Phys. Rev., 76, No. 7, 998–999; 76, 1728 (correction).

5. A. R. Miller, Possibility of a phase transition in the pure isotope of helium with mass 3, Nature, 163, No. 4138, 283–285.

6. O. K. Rice, Thermodynamics of liquid helium on the basis of a two-component model, Phys. Rev., 76, No. 11, 1701–1708.

7. K. S. Singwi, L. S. Kothari, Viscosity and thermal conductivity of mixtures of He³ and He⁴, Phys. Rev., 76, No. 2, 305–306.

8. J. W. Stout, Solutions of He³ in He⁴, Phys. Rev., 76, No. 6, 864–865.

9. 1. I. Ya. Pomeranchuk, On the theory of liquid He³, ZhETF, 20, No. 10, 919–924.

10. J. Boer, de, C. J. Gorter, Statistical mechanics of liquid mixtures of He³ and He⁴, Physica, 16, No. 3, 225–228; 667–668 (correction).

11. J. Boer, de, C. J. Gorter, Statistical mechanics of liquid mixtures of He³ and He⁴, Phys. Rev., 77, No. 4, 549.

12. J. Boer, de, J. Kramendonk, van, K. Compaan, Equation of state of gaseous He³, Physica, 16, No. 6, 545–554.

13. R. A. Buckingham, H. N. V. Temperley, Viscosity of liquid He³, Phys. Rev., 78, No. 4, 482.

14. O. G. Engel, O. K. Rice, Temperatures of the λ-points of solutions of He³ in He⁴, Phys. Rev., 78, No. 1, 55–57.

15. L. Goldstein, M. Goldstein, On the magnetic properties of liquid helium 3, J. chem. phys., 18, No. 4, 538–543.

16. C. J. Gorter, J. Boer, de, Phase transition and vapor pressure of liquid mixtures of He³ and He⁴, Phys. Rev., 77, No. 4, 569–570.

17. J. E. Kilpatrick, On the elasticity of vapor of solutions He³—He⁴, Phys. Rev., 79, No. 3, 529–530.

18. O. K. Rice, Influence of He³ on the λ-point of He⁴, Phys. Rev., 77, No. 1, 142–143.

19. O. K. Rice, O. G. Engel, Thermodynamics of solutions He³—He⁴, Phys. Rev., 78, No. 2, 183.

20. O. K. Rice, Thermodynamics of solutions of liquid helium He³—He⁴, Phys. Rev., 79, No. 6, 1024–1025.

21. 1. E. M. Lifshits, On the heat capacity of liquid He³, ZhETF, 21, No. 5, 659–660.

22. I. M. Khalatnikov, Effective mass of He³ and the speed of second sound in solutions of He³ in He⁴, DAN, 79, No. 1, 57–59.

23. J. Boer, de, Viscosity of He³ and He⁴, Proc. Internat. conf. on low temp. phys., Oxford, pp. 94–96.

24. J. Boer, de, E. G. D. Cohen, Viscosity of gaseous He³ at low temperatures, Physica, 17, No. 11—12, 993–1000.

25. R. B. Dingle, Remarks on the two-component model of helium II, Philos. Mag., 42, No. 333, 1080–1088.

26. J. G. Daunt, C. V. Heer, Solubility of He³ in liquid He⁴, Proc. Internat. conf. on low temp. phys., Oxford, pp. 76–78.

27. C. J. Gorter, K. W. Taconis, J. J. M. Beenakker, Some considerations on temperature gradients in helium II, Physica, 17, No. 9, 841–846.

Bibliography

1. O. Halpern, On the viscosity of gaseous He³, Phys. Rev., 82, No. 4, 561.

2. A. Hatasi [[unclear: name]], On the λ-point of He⁴—He³ mixtures, J. Phys. Soc. Japan, 6, No. 1, 271—274 (in English).

3. C. V. Heer, J. G. Daunt, On the theory of Bose–Einstein liquids, Phys. Rev., 81, No. 2, 295.

4. C. V. Heer, J. G. Daunt, On the theory of Bose–Einstein liquids, Phys. Rev., 81, No. 3, 447.

5. S. Koide, T. Usui, Influence of the He³ impurity on propagation in liquid helium II, Progr. Theor. Phys., 6, No. 4, 506—523.

6. S. Koide, T. Usui, Influence of the He³ impurity on the final amplification of the thermal disk of Rhodes, Progr. Theor. Phys., 6, No. 4, 622—623.

7. S. Koide, T. Usui, Influence of the He³ impurity on attenuation of waves in liquid helium II, Proc. Internal. conf. on low temp. phys., Oxford, pp. 106—108.

8. P. Mazur, I. Prigogine, On the hydrodynamics of liquid mixtures He³ and He⁴, Physica, 17, No. 7, 680—693 (in French).

9. J. G. Morrow, Thermodynamic properties of a nonideal liquid solution He³—He⁴, Phys. Rev., 84, No. 3, 502—505.

10. M. Toda, A. Isihara, On liquid He³ and its mixtures with He⁴, Progr. theoret. phys., 6, No. 4, 480—485.

11. 1. I. M. Khalatnikov, Hydrodynamics of solutions of foreign particles in He II, ZhETF, 23, No. 2, 169—181.

12. I. M. Khalatnikov, Sound in solutions of foreign particles in He II and the dissipative function of the solution, ZhETF, 23, No. 3, 265—274.

13. J. Boer, de, C. J. Gorter, Thermodynamics of the He I—He II transition in mixtures of He³ with He⁴, Physica, 18, No. 8/9, 565—568.

14. R. A. Buckingham, K. A. Scriven, Diffusion in gaseous helium at low temperatures, Proc. Phys. Soc., A 65, No. 5, 376—377.

15. J. G. Daunt, T. S. Tseng, C. V. Heer, Further consideration of questions of the theory of Bose–Einstein liquids, Phys. Rev., 86, No. 4, 627.

16. S. Koide, N. Matsudaira, T. Usui, Description of liquid helium II by means of a two-component model, Sci. Pap. Coll. Gen. Educ. Univ. Tokyo, 2, 129—144 (in English).

17. R. A. Nelson, W. Band, Elasticity of the vapor of He³—He⁴ mixtures, Phys. Rev., 88, No. 6, 1431.

18. P. J. Price, Second sound in He³—He⁴ mixtures, Phys. Rev., 87, No. 2, 381.

19. K. S. Singwi, Entropy and heat capacity of liquid He³, Phys. Rev., 87, No. 3, 540—541.

20. H. N. V. Temperley, A new theory of liquid helium, Proc. Phys. Soc., A 65, No. 8, 619—628.

21. 1. A. I. Akhiezer, V. Aleksin, Time of magnetization of a weak solution of He³ in He⁴, DAN, 92, No. 2, 259—262.

22. V. N. Zharkov, I. M. Khalatnikov, Theory of diffusion and thermodiffusion in weak solutions of He³ in helium II, DAN, 93, No. 6, 1007—1009.

23. B. M. Abraham, D. W. Osborne, B. Weinstock, Orientation of nuclei and entropy of liquid He³, Phys. Rev., 90, No. 2, 347.

Bibliography

1. W. Band, R. A. Nelson, Theory of the elasticity of helium isotope mixture vapors based on the model of molecular groups, Phys. Rev., 90, No. 5, 744—747.

2. K. Buckthought, Quantum-statistical theory of liquid helium, Canad. J. Phys., 31, No. 6, 932—941.

3. Chen Tien Chi, F. London, Entropy of liquid He³, Phys. Rev., 89, No. 5, 1038—1040.

4. L. Goldstein, On the theory of liquid He³—He⁴ mixtures, Phys. Rev., 92, No. 3, 820.

5. D. Haarter, Elasticity of vapor and entropy of liquid He³, Phys. Rev., 91, No. 4, 1018—1019.

6. J. G. Morrow, Vapor–liquid equilibrium in He³—He⁴ mixtures, Phys. Rev., 89, No. 5, 1034—1037.

7. J. G. Morrow, Temperature λ-point of dilute solutions of He³ in He⁴, Phys. Rev., 92, No. 1, 1—3.

8. P. J. Price, Macroscopic theory of superfluid mixtures He³—He⁴, Phys. Rev., 89, No. 6, 1209—1215; 90, 1131 (correction).

9. I. Prigogine, J. Philippot, On the molecular theory of liquid helium. III. Solutions of He³ and He⁴, Physica, 19, No. 3, 235—240 (French).

10. B. Weinstock, B. M. Abraham, D. W. Osborne, Nuclear orientation and entropy of liquid He³, Phys. Rev., 89, No. 4, 767—769.

1954.

1. S. M. Bhagat, P. K. Katti, On the elasticity of vapor of liquid He II and liquid He³, Sci. and culture, 19, No. 9, 461—462 (English).

2. J. Boer, He³ viscosity at low temperature; anomalous transition? Nederl. tijdschr. natuurkunde, 20, No. 7, 178—182 (Dutch).

3. E. G. D. Cohen, M. J. Offerhaus, J. Boer, d₂, Transport phenomena and equation of state of gaseous mixtures of helium isotopes, Physica, 20, No. 8, 501—515.

4. R. P. Feynman, Atomic theory of the two-component model of liquid helium, Phys. Rev., 94, No. 2, 262—277.

5. L. Goldstein, On the theory of liquid He³—He⁴ mixtures, Phys. Rev., 95, No. 4, 869—884.

6. J. E. Kilpatrick, W. E. Keller, E. F. Hammel, N. Metropolis, Second virial coefficients of He³ and He⁴, Phys. Rev., 94, No. 5, 1103—1110.

7. Z. Mikura, On a model of liquid helium considered as a Bose–Einstein liquid. Part I. He³—He⁴ mixtures. Progr. theoret. Phys., 11, No. 1, 25—52.

8. V. S. Nanda, Heat of mixing of He³ and He⁴ solutions, Phys. Rev., 94, No. 2, 241—243.

9. I. Prigogine, On the theory of liquid helium, Adv. in phys., 3, No. 10, 131—148.

10. I. Prigogine, R. Bingen, A. Bellemans, Isotopic effects and thermodynamic properties of the condensed phase, part III, Physica, 20, No. 9, 633—654 (French).

11. I. Prigogine, R. Bingen, J. Jeener, Isotopic effects and thermodynamic properties of the condensed phase, part I, Physica, 20, No. 7, 383—394 (French).

12. I. Prigogine, J. Jeener, Isotopic effects and thermodynamic properties of the condensed phase, part II, Physica, 20, No. 8, 516—520 (French).

13. O. K. Rice, Nature of phase transitions of high order, with application to liquid helium, Phys. Rev., 93, 1161—1168.

14. H. N. V. Temperley, A possible model of liquid He³, Proc. Phys. Soc. A 67, No. 6, 414, 495—504.

5. Miscellaneous Works Not Included in the Second Part of the Review

1. R. Kronig, A. Thellung, Physica, 18, 749 (1952); A. Thellung, Physica, 18, 217 (1952).
2. F. M. Zinman, Proc. Roy. Soc. (London), A219, 257 (1953).
3. R. P. Feynman, Phys. Rev., 91, 1291, 1301 (1953); 94, 262 (1954).
4. F. London, Nature, 141, 643 (1938).
5. I. A. Yakovlev, J. of Phys. USSR, 7, 307 (1943).
6. A. D. Galanin, ZhETF, 16, 12 (1946).
7. I. Ia. Pomeranchuk, Phys. J. USSR, 11, 77 (1947); J. of Phys. USSR, 11, 25 (1947); ZhETF, 18, 657 (1948).

Proof note. Recently a number of experimental studies of surface tension in the system He$^4$—He$^3$ have been published. B. N. Eselson and N. G. Berezniak94–95 established that an admixture of He$^3$ in liquid He$^4$ leads to a noticeable decrease in the value of the surface tension. This indicates positive adsorption of He$^3$ atoms at the phase boundary. The same authors95 and K. N. Zinov’eva (ZhETF, 28, No. 1, 100–101, 1955) measured the surface-tension coefficient $\alpha$ of pure He$^3$. In comparison with He$^4$, the curve $\alpha(T)$ in general retains the same character, but is shifted by $\simeq 2^\circ$ (the difference of the critical temperatures) toward absolute zero and has a smaller slope. At a temperature of $1^\circ$K, the surface tension of He$^3$ is two and a half times smaller than that of He$^4$.