PROPERTIES OF THE LIGHT HELIUM ISOTOPE He$^3$ AT LOW TEMPERATURES. I

R. A. Chentsov

INTRODUCTION

The present review is devoted to the properties of the light helium isotope He$^3$ at low temperatures. This field of low-temperature physics has undergone considerable development in recent years: in the last five years about 150 experimental and theoretical papers have been published devoted to the investigation of the low-temperature properties of liquid and solid He$^3$ and of mixtures of He$^3$ with ordinary helium He$^4$. Within the scope of a short article it is impossible even briefly to set forth all the work devoted to this subject. We have therefore decided, in our account, to dwell only on the most interesting and essential results. We have tried to make up for this shortcoming of the exposition by adding, insofar as possible, a complete bibliography.

At the beginning of the review we recall the properties at low temperatures of the ordinary helium isotope. The following sections discuss the abundance of He$^3$, methods of enriching helium with the light isotope and of obtaining He$^3$ in pure form. Next, the phase diagram of He$^3$, the properties of liquid He$^3$, and liquid solutions He$^3$—He$^4$ are described. In the second part of the review, to be published in the next issue, the principal theoretical works devoted to He$^3$ and to mixtures He$^3$—He$^4$ are set forth. The bibliography is also given there.

1. PROPERTIES OF He$^4$ AT LOW TEMPERATURES

We shall assume that the reader is familiar with the basic properties of the ordinary helium isotope, for example from the review by E. L. Andronikashvili published earlier in this journal$^1$. We shall therefore confine ourselves only to listing, as a reminder, some of these properties, comparison with which will be essential when considering the properties of He$^3$.

He\(^4\), before the discovery of He\(^3\), was the substance with the lowest boiling temperature (4.2° K) and critical point (5.25° K). Helium remains liquid when cooled under the pressure of its own vapor down to arbitrarily low temperatures; to obtain solid helium it is necessary to subject liquid helium to a pressure exceeding 25 atmospheres. The most interesting feature of the phase diagram of He\(^4\) is the existence of regions corresponding to two different liquid phases: He I and He II. The latter exists only at temperatures below 2.19° K and is distinguished by a number of unusual properties. The most essential of the properties of He II is the “superfluidity” discovered by P. L. Kapitsa\(^2\)—the ability to flow without friction through the finest slits and capillaries. In this case the flowing superfluid helium carries no heat with it, and the process is accompanied by cooling of the vessel into which it flows (“mechanocaloric effect”). A peculiarity of He II is also the “substantial” character of heat-flow propagation in it: P. L. Kapitsa\(^3\) discovered the phenomenon of an invisible “heat jet” propagating over considerable distances in He II when heat is released in a small vessel immersed in liquid helium and ending in a capillary nozzle. The heat jet emerging from the capillary exerts pressure on obstacles placed opposite it and has a reactive action. Of interest is the surface film of liquid helium, which covers to a considerable height the walls of a vessel containing He II. The film has a thickness of \(\sim 10^2\) atomic layers and possesses considerable mobility: the levels of liquid helium in vessels communicating through the wall film rapidly equalize.

According to the modern views developed by L. D. Landau\(^4\), liquid helium at lower temperatures is a weakly excited quantum system, the thermal energy in which is associated with individual “quanta of excitation,” whose number increases with increasing temperature. At low temperatures (below 1° K) the predominant role is played by excitations called phonons—sound vibrations of very high frequency. At higher temperatures (above 1.5°) the main role belongs to excitations of another type—rotons, which have a different (quadratic, not linear) dependence of energy on momentum. To excite rotons a certain minimum energy is required (about \(8.9k\), where \(k\) is Boltzmann’s constant), and therefore the expressions for the number of rotons, as well as for the thermodynamic quantities determined by them (for example, heat capacity, entropy), contain factors that depend exponentially on temperature. Elementary thermal excitations possess an effective mass; the latter quantity for a roton is of the order of the mass of a He\(^4\) atom. The total mass \(\rho_n\) of all excitations present in 1 cm\(^3\) of He II changes from zero at 0° K to \(\rho_n\)—

densities of liquid helium at the He II–He I transition temperature. Experimentally the temperature dependence \(\rho_n/\rho\) was first determined by E. L. Andronikashvili\(^5\) in experiments with torsional oscillations of a stack of parallel disks placed in He II. The totality of all elementary excitations may be regarded as the “normal component” of He II, in contrast to the remaining part of it—the “superfluid component.” The superfluid component has no viscosity and easily penetrates through slits practically impermeable even to gases. The normal component, on the contrary, has a viscosity of \(\sim 10^{-5}\) poise, which can be measured from the damping of a disk oscillating in He II\(^6\), or by the viscometer method with stationary (suspended) and rotating cylinders\(^7\). The temperature dependence of the viscosity finds a satisfactory theoretical explanation\(^8\). The so-called “thermomechanical effect” is also easily explained—the appearance of a mechanical pressure when two vessels filled with He II and maintained at different temperatures are connected through a very narrow slit or film. Here we are dealing with a kind of osmotic pressure of thermal excitations, the slit (film) playing the role of a semipermeable partition.

The theory of He II made it possible to predict an interesting new phenomenon of “second sound”—thermal waves propagating in He II with considerable velocity and without noticeable attenuation. Experimentally, second sound was first observed by V. P. Peshkov\(^9\). Investigations of the velocity of second sound at very low temperatures, carried out by Peshkov and other researchers\(^10\), brilliantly confirmed its temperature behavior predicted on the basis of Landau’s theory.

2. THE LIGHT ISOTOPE OF HELIUM—He\(^3\)

In 1933–1934 Rutherford, Oliphant, and others\(^11\) reported nuclear reactions, one of whose products was a new, lighter isotope of helium: \(\mathrm{Li}^6(\mathrm{p}, \alpha)\mathrm{He}^3\); \(\mathrm{H}^2(\mathrm{d}, \mathrm{n})\mathrm{He}^3\). However, at first nothing was known about the stability of the \(\mathrm{He}^3\) nucleus. Only in 1939 Alvarez and Cornog\(^12\), working on a 60-inch cyclotron with a variable magnetic field, found between the peaks corresponding to protons and \(\alpha\)-particles a small but distinct peak which, as they showed, corresponded to the ion \((\mathrm{He}^3)^{++}\). The beam intensity at first amounted to only about a thousand particles per second. Having increased it, the authors carried out the first nuclear reaction with \(\mathrm{He}^3\) as the working particle: \(\mathrm{Si}^{28}(\mathrm{He}^3,\mathrm{p})\mathrm{P}^{30}\). These and subsequent investigations showed that \(\mathrm{He}^3\) is a stable but extremely rare isotope of helium.

According to modern data\(^13\), the content of \(\mathrm{He}^3\) in atmospheric helium is \((1.2\text{–}1.3)\cdot 10^{-6}\) (by number of \(\mathrm{He}^3\) atoms

and $\mathrm{He}^4$). Let us recall that the content of helium in atmospheric air is about $5 \cdot 10^{-6}$; thus, the content of $\mathrm{He}^3$ in air is less than $10^{-11}$. The relative content of $\mathrm{He}^3$ in helium from natural gas sources is still smaller—from $5 \cdot 10^{-8}$ to $5 \cdot 10^{-7}$. Helium obtained from radioactive minerals contains only about one atom of $\mathrm{He}^3$ per $10^3$ atoms of $\mathrm{He}^4$. The $\mathrm{He}^3$ content is somewhat higher in helium extracted from nonradioactive minerals: from $5 \cdot 10^{-6}$ to $1.2 \cdot 10^{-5}$ (in the mineral spodumene).

The origin of $\mathrm{He}^3$ on Earth is most often associated with the action of cosmic rays on various elements. Thus, under the action of the neutron component of cosmic radiation in the lithosphere, the reaction $\mathrm{Li}^6 (n,\alpha) \mathrm{H}^3$ may occur; “superheavy hydrogen”—tritium $\mathrm{H}^3$—is $\beta$-active and, decaying with a half-life of 12.5 years, is transformed into $\mathrm{He}^3$. Some authors believe that the slow neutrons necessary for this reaction may be not of cosmic origin, but may appear as a result of the emission by uranium, thorium, and other naturally radioactive elements of $\alpha$-particles, which, in interaction with the nuclei of light elements (Si, Al, etc.), cause the formation of fast neutrons that are then slowed down in granite massifs. In the atmosphere, reactions on fast neutrons of the type $\mathrm{N}^{14}(n,\mathrm{C}^{12})\mathrm{H}^3$ or $\mathrm{N}^{14}(n,3\alpha)\mathrm{H}^3$ may occur, which should lead to the formation of $\mathrm{He}^3$ in the atmosphere. Direct formation of $\mathrm{He}^3$ in “stars” arising under the action of cosmic rays is also possible. In this connection it is interesting to note$^{14}$ that in helium extracted from meteorites the content of $\mathrm{He}^3$ may be considerably higher than in helium from terrestrial sources, and may exceed 30%. In all, apparently, on the order of 1 atom of $\mathrm{He}^3$ is formed in the atmosphere per $1 \ \mathrm{cm}^2$ of the Earth’s surface per second. The atoms of $\mathrm{He}^3$ gradually leave the atmosphere—on the average over a time measured in several million years.

In view of the extremely small content of $\mathrm{He}^3$ in ordinary helium, the properties of the latter at practically all temperatures are determined by the properties of the isotope $\mathrm{He}^4$. In order for the presence of $\mathrm{He}^3$ to affect the properties of helium, it is necessary to carry out a substantial enrichment of He with the light isotope. The first of the methods of enrichment of $\mathrm{He}^4$—$\mathrm{He}^3$ mixtures to be applied was the thermodiffusion method, which makes it possible to achieve substantial initial enrichment. However, to obtain significant concentrations by this method requires a great deal of time and large energy expenditures. For example, one of the described installations of this type$^{15}$ consisted of three successive thermodiffusion columns consuming a power of $16.5 \ \mathrm{kW}$; the yield of helium with a $\mathrm{He}^3$ content of about 0.2% was only $14 \ \mathrm{cm}^3$ of gas per day.

A very effective method of enriching helium with the light isotope is the low-temperature method, based on the fact that $\mathrm{He}^3$ atoms, being in solution in liquid He II, do not take part

tions in superfluid motion and thus enter into the normal component (this question is discussed in more detail in Sec. 4). If He II is forced to flow, by means of superfluidity, through a narrow slit or a wall film, then He\(^3\) will be retained, and its concentration in the vessel from which the outflow occurs will increase. B. N. Esel’son and B. G. Lazarev\(^{16}\), using a very simple apparatus constructed by them and based on this phenomenon, as well as on the thermomechanical effect, achieved an enrichment by a factor of 2000. This enabled them, in a few days, to obtain several liters of helium with a He\(^3\) content of 0.01%.

The low-temperature method is also applicable for further enrichment of already richer initial mixtures; thus, an apparatus has been described\(^{17}\) in which the concentration of He\(^3\) in He II was raised from 2 to 30–40%. However, this method is characterized by one fundamental shortcoming: the concentration of He\(^3\) in the liquid cannot exceed a certain limiting value, dependent on the temperature of the separating apparatus, a value determined by the curve of \(\lambda\)-transitions in liquid solutions of He\(^4\)—He\(^3\) (see below).

Fig. 1. Rectification column for separating helium isotopes.

For further separation of already considerably enriched mixtures of He\(^4\) with He\(^3\), it is convenient to use the method of rectification in a rectification column. The application of this method to the isotopic mixture He\(^4\)—He\(^3\) was first described in the work of B. N. Esel’son and B. G. Lazarev\(^{16}\). The simple and effective separation column constructed by them is shown in Fig. 1. The initial mixture is condensed at the bottom of a small Dewar vessel \(D\). By passing current through the heater \(H\), the liquid mixture is heated intensely. The vapor thus formed condenses in the upper part of the column (maintained at the temperature of a helium bath) and flows downward in the form of a film along the screw \(B\). At the same time, molecular exchange occurs between the liquid and the counterflowing gaseous phases, leading to depletion in He\(^4\) and enrichment in He\(^3\) of the part of the mixture located at the top of the column. The enriched mixture is withdrawn through tube \(T\).

The column permitted enrichment by \(\sim 150\) times. With the aid of this apparatus the authors obtained an amount of helium with a He\(^3\) content of 1.5%, sufficient for carrying out a number of essential experiments. With the aid of secondary rectification one can obtain a further considerable enrichment of the mixture in the light isotope (B. N. Esel’son, B. G. Lazarev\(^{18}\)). Fairbank and others\(^{19}\) reported the successful application of this method for obtaining a helium mixture containing more than 99% He\(^3\).

It should be noted that, although obtaining very pure He\(^3\) from natural helium by successive enrichment using various methods is practically quite possible, this route is rather cumbersome. This is evident, for example, from the fact that to obtain \(\sim 10\ \mathrm{cm}^3\) of gaseous He\(^3\) under normal conditions (the minimum quantity with which experiments with condensed He\(^3\) can be carried out), it is necessary to process \(\sim 100\) cubic meters of helium obtained from natural sources.

The development of nuclear physics led to an entirely new possibility for the artificial production of pure He\(^3\), bypassing the need to enrich natural helium, which is poor in the light isotope \({}^{20}\). The method consists in using the already mentioned reaction for producing tritium from lithium irradiated with neutrons:

\[ \mathrm{Li}^6 + \mathrm{n} \to \mathrm{He}^4 + \mathrm{H}^3 . \]

Irradiation is carried out in a nuclear reactor. The tritium obtained is separated from the He\(^4\) formed by diffusion through a heated palladium cap, which is readily permeable to hydrogen but retains helium. The collected pure tritium is allowed to decay spontaneously with formation of He\(^3\):

\[ \mathrm{H}^3 \to \mathrm{He}^3 + e^- . \]

When a sufficient quantity of He\(^3\) has accumulated in the resulting mixture of tritium and the light helium isotope, the two elements are again separated with the aid of the same palladium, or by freezing out the tritium in a coil immersed in liquid helium \({}^{21}\). In this way, practically pure samples of He\(^3\) were obtained, on which the principal thermodynamic and other properties of the light isotope of helium at low temperatures were determined.

3. PHASE DIAGRAM OF PURE He\(^3\) AND ITS PROPERTIES. ABSENCE OF SUPERFLUIDITY

Helium was the last of the so-called “permanent” gases; it was first liquefied only in 1908. The difficulty of condensing helium is connected, on the one hand, with the weakness of the molecular attraction forces that characterize the interaction of helium atoms, which have filled electron shells; this circumstance alone already ensured a very low liquefaction temperature for He\(^4\). On the other hand, because of the small mass of the He\(^4\) atom (4.0039 units on the physical scale of atomic weights) and the low liquefaction temperature, the role of zero-point oscillations increases significantly, manifesting itself, in particular, in the appearance of additional repulsive forces when atoms approach one another, i.e., in a further weakening of the effective intermolecular attraction. As a result, the critical temperature of He\(^4\) is exceptionally low: about \(-268^\circ\mathrm{C}\). The He\(^3\) atom has the same electronic structure as He\(^4\), but an even smaller mass (3.0170 units on the physical scale). In view of this, the energy of the zero-point oscillations that arise upon liquefaction of He\(^3\) must be still more

more than in the case of He\(^4\). On the basis of similar considerations, the suggestion was even made\(^{22}\) that, apparently, liquefying He\(^3\) is in principle impossible: the gain in energy of intermolecular interaction when atoms approach one another might prove smaller than the accompanying increase in zero-point energy. However, these considerations were mainly qualitative.

A different theoretical approach to the problem of liquefying light gases was developed in the so-called “quantum theory of corresponding states” of de Boer\(^{23}\). The author of the theory proceeded from the assumption that the interaction of the molecules of the gases under consideration can be described by the Lennard-Jones potential:

\[ U(r) = 4\varepsilon \left[\left(\frac{r}{\sigma}\right)^{-12} - \left(\frac{r}{\sigma}\right)^{-6}\right]. \]

Here \(r\) is the distance between the centers of mass of the molecules; \(\varepsilon\) and \(\sigma\) are constants characterizing the magnitude and “range” of the molecular interaction: \(\varepsilon\) is the absolute value of the interaction energy at the minimum point, \(\sigma\) is the intermolecular distance at which the interaction energy becomes zero. \(\varepsilon\) and \(\sigma\) may be determined, for example, by studying gas isotherms. De Boer expresses all macroscopic thermodynamic characteristics of a substance—pressure, volume, temperature, etc.—through the molecular parameters \(\varepsilon\), \(\sigma\), and their combinations. For example, instead of the absolute temperature \(T\), the dimensionless reduced temperature \(T^* = T/(\varepsilon/k)\) is introduced (\(k\) is Boltzmann’s constant); instead of the molar volume \(V\), the reduced volume \(V^* = V/(\sigma^3 N)\) is introduced (\(N\) is Avogadro’s number); instead of the pressure \(p\), the reduced pressure \(p^* = p/(\varepsilon/\sigma^3)\), and so on. The participation of forces of quantum origin is taken into account by introducing an additional dimensionless quantity

\[ \lambda^* = \lambda/\sigma = (h\sqrt{\varepsilon m})/\sigma \]

(\(m\) is the molecular mass, \(\lambda\) is the de Broglie wavelength of the molecules at the temperature at which the mean energy of thermal motion of the gas molecules becomes of the order of \(\varepsilon\)).

According to de Boer, the equation of state of a substance described by this theory is a universal function:

\[ f(T^*, p^*, V^*, \lambda^*) = 0. \]

Therefore the reduced critical temperature \(T^*_{\mathrm{cr}}\), pressure and volume, the molar volume at absolute zero, the liquefaction temperature, vapor elasticity, and other thermodynamic quantities, expressed in the corresponding reduced units, must, for different substances of this class, be described by a single universal function of the parameter \(\lambda^*\). Figure 2 shows an example of such a dependence—the curve \(T^*_{\mathrm{cr}}(\lambda^*)\). Its consideration shows that for a considerable number of monatomic and even diatomic gases a universal dependence is indeed observed. At the same time, for heavier molecules \(T^*_{\mathrm{cr}}\) is approximately constant, while for lighter ones it begins to decrease because of the influence of the quantum-mechanical energy of zero-point vibrations and the corresponding increase in the value of the quantum

of the parameter \(\lambda^*\). Extrapolating the plotted curve to the value \(\lambda^*=3.05\), corresponding to \(\mathrm{He}^3\), we at once arrive at the value \(T_{\mathrm{cr}}\) for this element.

In a similar way de Boer predicted that the critical constants for \(\mathrm{He}^3\) should have the values: \(T_{\mathrm{cr}}=3.3^\circ\pm0.2^\circ\ \mathrm{K}\),

Fig. 2. Dependence of the reduced critical temperature on the quantum parameter \(\lambda^*\).

\(p_{\mathrm{cr}}=1.1\pm0.2\ \mathrm{atm}\), and so on. The vapor elasticity curve was even predicted and its equation was given. For the expected temperature (under atmospheric pressure) the value \(T_{\mathrm{boil}}\sim3.1^\circ\ \mathrm{K}\) was predicted; consequently, He not only was expected to be condensable, but this could even be accomplished rather easily. The subsequent experiments proved to be an excellent confirmation of the predictions made by de Boer. For example, for the critical temperature and pressure the values \(T_{\mathrm{cr}}=3.35^\circ\ \mathrm{K}\), \(p_{\mathrm{cr}}=890\ \mathrm{mm\ Hg}\) were obtained; the boiling temperature at atmospheric pressure was \(T_{\mathrm{boil}}=3.195^\circ\ \mathrm{K}\); the saturated-vapor pressures at different temperatures, within the limits of the experimental errors, fell on the theoretical curve. This demonstrates the success of de Boer’s semiempirical approach to the problem of taking into account quantum-mechanical properties manifested in the behavior of a substance at very low temperatures.

\(\mathrm{He}^3\) was first liquefied by Sydoriak, Grilly, and Hammel\(^{24}\) in 1949. They had at their disposal \(20\ \mathrm{cm}^3\) of practically pure \(\mathrm{He}^3\). From such an amount of gas only a few tens of cubic millimeters of liquid could be obtained. Therefore the liquefaction was carried out in a capillary immersed in a bath of ordinary liquid helium, whose temperature was regulated in the usual way (by vapor pressure) and kept constant. The pressure of \(\mathrm{He}^3\) in the capillary was gradually increased by means of

movement of the mercury piston in the external vessel to which the capillary was connected; at the same time the pressure in the He\(^3\) system was measured continuously. The onset of condensation was detected by the cessation of the pressure increase upon further movement of the piston. In a second experiment, using a glass capillary instead of a steel one, the authors established that the condensed phase is indeed liquid. Liquid He\(^3\) is a colorless, transparent liquid, similar in appearance to He I—liquid He\(^4\) at a temperature above the \(\lambda\)-point. The authors determined the vapor-pressure curve, the boiling point, and the critical point of He\(^3\).

Fig. 3. Vapor pressure over liquid He\(^3\).

The dependence of the vapor pressure of He\(^3\) on temperature is shown graphically in Fig. 3; the corresponding curve for He\(^4\) is also given there for comparison. It is seen that, as the temperature is lowered, the ratio of the vapor pressures of He\(^3\) and He\(^4\) increases rapidly; at \(1^\circ\) K it already exceeds 70. The high vapor pressure of He\(^3\) makes it possible to use it as the working substance for constructing a sensitive condensation thermometer, quite suitable for measuring temperatures of \(\sim 1^\circ\) K and below.

The liquid–vapor equilibrium curve was investigated in greater detail in the work of Abraham, Osborne, and Weinstock\(^{21}\). Measurements of the vapor pressure of liquid He\(^3\) at various temperatures were carried out in a helium cryostat that possessed interesting features. To increase the pumping speed, the Dewar connection (whose internal diameter was about 6 cm) to the system was made by means of wide flanges. The setting and regulation (with an accuracy up to \(0.001^\circ\)) of the temperature was carried out by a diaphragm installed in a tube about 10 cm in diameter, ве-

flowing to the pump, according to the readings of a dibutyl-phthalate differential manometer with an inclination of \(1:10\). The manometric tube entered the Dewar by \(43\ \text{cm}\), so that the lower end was always in the region of laminar gas flow.

\(\mathrm{He}^3\) with a \(\mathrm{He}^4\) content of \(0.03 \pm 0.03\%\) was condensed in a volume of \(50\ \text{mm}^3\) bored out in a copper block. The vessel was connected to the external system by a monel tube \(0.5\ \text{mm}\) in diameter. The authors determined the vapor-pressure curve above liquid \(\mathrm{He}^3\) in the temperature interval from \(1.025^\circ\ \mathrm{K}\) to the critical point. The pressure values averaged by the method of least squares are expressed by the formula: \(\lg p = 0.97796/T + 2.5 \lg T + 0.000302T^3 + 1.91594\). Here the pressure \(p\) is expressed in millimeters of mercury, and the absolute temperature \(T\) is on the so-called “agreed temperature scale”\({}^{25}\), determined from the vapor pressure of \(\mathrm{He}^4\), with the corrections of Kistemaker\({}^{26}\). Exact values of the vapor pressure of \(\mathrm{He}^3\) at various temperatures, according to work\({}^{24}\), are given in Table 1.

Table 1

Temperature dependence of the vapor pressure of \(\mathrm{He}^3\)

The density of liquid \(\mathrm{He}^3\) was determined in the above-mentioned work\({}^{24}\) by the following indirect method. A portion of the liquid \(\mathrm{He}^3\) condensed in a glass capillary was made to evaporate, increasing the external volume; the magnitude of the latter was measured. Simultaneously, the amount of evaporated volume of liquid was measured. From the change in both volumes, the temperature of the external volume, and the vapor pressure of \(\mathrm{He}^3\), it was possible to determine the difference between the densities of \(\mathrm{He}^3\) in the liquid state and of the saturated vapor of \(\mathrm{He}^3\), \(\rho_{\mathrm{l}} - \rho_{\mathrm{v}}\). To exclude the vapor density, the authors used the Mathias rule, according to which \((\rho_{\mathrm{l}} + \rho_{\mathrm{v}})/2\) is a linear function of temperature, attaining at the critical point the value of the critical density \(\rho_{\mathrm{cr}}\). Taking \(\rho_{\mathrm{cr}} = 0.0420\) (a value obtained from data on the critical temperature and pressure) and extrapolating the Mathias rule to low temperatures, where \(\rho_{\mathrm{v}} \ll \rho_{\mathrm{l}}\), the authors obtained values of the density of liquid \(\mathrm{He}^3\) situated

in equilibrium with the vapor, are shown in Fig. 4. The accuracy of the determination of \(\rho_{\text{l}}\), according to the authors’ estimate, is \(\sim 3\%\)*).

Using their results for measurements of the elasticity of He\(^3\) vapor, Abraham, Osborne, and Weinstock \(^{21}\) determined, from the Clapeyron–Clausius equation, the heat of vaporization of He\(^3\) and its temperature dependence. The specific volume of the liquid was determined from the above values of the density; the specific volume of the vapor was determined from the equation of state of gaseous He\(^3\), with allowance for the second virial coefficient. According to these calculations, in the interval between 1 and \(2.2^\circ\) K the heat of vaporization of He\(^3\) is about \(10\ \mathrm{cal/mol}\)—two times smaller than for He\(^4\). At \(T \simeq 2.1^\circ\) K there is a maximum of \(11.4\ \mathrm{cal/mol}\); extrapolation to \(T = 0\) gives the value \(4.47\ \mathrm{cal/mol}\). According to other calculations (E. M. Lifshitz \(^{27}\)), the heat of vaporization at low temperatures is equal to \(5.5\ \mathrm{cal/mol}\). Direct measurements of this quantity have not yet been made.

Fig. 4. Density of liquid He\(^3\).

The phase equilibrium solid He\(^3\)—liquid He\(^3\) was investigated in the works of Osborne, Abraham, and Weinstock \(^{28}\). It turned out that, in order to bring He\(^3\) into the solid state, as in the case of He\(^4\), rather high pressures must be applied. The solidification pressure of He\(^3\) at different temperatures was determined by the so-called method of the blocked capillary. A U-shaped capillary, in which He\(^3\) had been condensed, was immersed in a He\(^4\) bath. The pressure on both sides of the capillary was measured. The pressure was changed by moving a mercury piston in a vessel connected with one of the capillaries. As long as the pressure did not exceed the value corresponding to solidification of He\(^3\), the readings of both manometers corresponded to one another. However, as soon as this value was reached, He\(^3\) solidified and blocked the capillary; therefore further movement of the piston caused an increase of pressure only in the limb connected with it. The temperature dependence found for the solidification pressure of He\(^3\) between 0.5 and \(1.5^\circ\) K is described by the formula \(P_{\text{m}} = 26.8 + 13.1T^2\ \mathrm{atm}\).

*) In work \(^{57}\) it is indicated that Kerr carried out measurements of the density \(\rho_{\text{l}}\) with an accuracy of 0.1%; the results were reported at the Third Conference on Low Temperature Physics in Houston (USA).

Below \(0.5^\circ\) K the solidification pressure ceases to depend on temperature and remains equal to \(29.3\) atm. The authors initially supposed that this might be connected with a disturbance of the thermal contact between the capillary and the paramagnetic salt used to obtain ultralow temperatures. However, it later became clear that the capillary-blocking method is unsuitable if the curve of the dependence of melting pressure on temperature has a minimum (the existence of such a minimum had been predicted by I. Ya. Pomeranchuk\(^ {29}\)). At temperatures below the temperature of the minimum, blocking of the capillary will still occur at the minimum pressure; in this case the capillary will become plugged not in the coldest part, but somewhat higher: where it has the temperature corresponding to the minimum.

An indirect determination of the melting pressure at temperatures above \(1.5^\circ\) was carried out by B. N. Eselson and B. G. Lazarev\(^ {18}\). Extrapolating data on the solidification of mixtures \( \mathrm{He}^3\)—\( \mathrm{He}^4 \) (see Sec. 4), they obtained a smooth continuation of the curve of the preceding authors up to a temperature of \(\sim 2.3^\circ\) K. At the latter temperature the melting pressure is somewhat less than \(100\) atm. The curve of the temperature dependence of the melting pressure according to the data of works\(^ {28,18}\) is shown in Fig. 5.

Fig. 5. Melting pressure of solid \(\mathrm{He}^3\).

Despite the uncertainty in the course of the curve below \(0.5^\circ\) K, in any case one may note the absence of any tendency for the melting pressure to decrease to zero as the absolute zero of temperature is approached. Consequently, \( \mathrm{He}^3 \), like \( \mathrm{He}^4 \), apparently cannot be transformed into the solid state by any cooling if it is under the pressure of its own vapor. This is one of the most vivid manifestations of zero-point oscillations caused by the action of forces of quantum nature.

Comparatively recently, de Vries and Daunt\(^{30}\) measured, over a wide temperature range \((0.5—2.3^\circ\mathrm{K})\), the heat capacity of liquid \(\mathrm{He}^3\) containing a small admixture \((4\%)\) of \(\mathrm{He}^4\). Despite the small amount of \(\mathrm{He}^3\) at the authors’ disposal (the working volume of their adiabatic calorimeter was \(\sim 14—16\ \mathrm{mm}^3\)) and the small absolute magnitude of the heat capacity of the sample \((\sim 10^{-4}\ \mathrm{cal/deg})\), they succeeded in measuring the heat capacity \(c_{\mathrm{l}}\) of liquid \(\mathrm{He}^3\) in the indicated temperature interval with an accuracy of \(\sim 10—20\%\). It turned out that \(c_{\mathrm{l}}\), as the temperature is lowered, decreases monotonically from \(3.6\ \mathrm{cal/deg\cdot mol}\) at \(2.3^\circ\mathrm{K}\) to \(0.9—1.1\ \mathrm{cal/deg\cdot mol}\) at temperatures below \(1^\circ\mathrm{K}\). It is interesting that at low temperatures the dependence of the heat capacity on temperature becomes very weak: \(c_{\mathrm{l}}\) seems to tend to a constant, nonzero limit. Such behavior, however, would contradict the general proposition of statistical physics according to which the heat capacity of all bodies must vanish as \(T \to 0\). This indicates the presence of anomalies on the curve of the heat capacity of \(\mathrm{He}^3\) at very low temperatures—anomalies still subject to experimental determination.

Later data on the heat capacity of liquid \(\mathrm{He}^3\) at ultralow temperatures were also reported by Osborne, Abraham, and Weinstock\(^{31}\), and also by Roberts and Sydoriak\(^{32}\). In work\(^{31}\), \(c_{\mathrm{l}}\) was determined in the temperature interval \(0.42—1.06^\circ\mathrm{K}\) with an accuracy (according to the authors’ estimate) of \(\sim 5\%\). The measurements were carried out by the usual method of an adiabatic calorimeter; to obtain He at ultralow temperatures, demagnetization was carried out of a paramagnetic salt surrounding a copper vessel of volume about \(2\ \mathrm{cm}^3\), containing \(0.2—0.7\ \mathrm{cm}^3\) of liquid \(\mathrm{He}^3\). In the indicated temperature interval, the authors found the heat capacity of liquid \(\mathrm{He}^3\) to be given by the following interpolation formula: \(c_{\mathrm{l}} = 0.53 + 0.48\cdot T\ \mathrm{cal/deg\cdot mol}\).

Roberts and Sydoriak\(^{32}\) used an original calorimetric method. Their calorimeter was a copper sphere of volume \(3/4\ \mathrm{cm}^3\), filled with liquid \(\mathrm{He}^3\) and cooled relative to the surrounding \(\mathrm{He}^4\) bath by pumping off \(\mathrm{He}^3\) vapor. Simultaneous measurements were made of the heating rate \(\dot T\) and the heat input \(\dot Q\) at various values of \(N\)—the total number of moles of \(\mathrm{He}^3\) in the system. Plotting the quantity \(\dot Q/\dot T\) as a function of \(N\) (after introducing corrections for the release of latent heat and for the heat capacity of the vapor) gave a straight line, whose slope, as is easy to see, is precisely equal to the molar heat capacity of liquid \(\mathrm{He}^3\). The advantage of the method is the elimination of the influence of the heat capacity of gaseous helium outside the sphere and of the heat capacity of the calorimeter itself, since these quantities do not depend on \(N\). By this method values of the heat capacity \(c_{\mathrm{l}}\) were found in the temperature interval \(0.54—1.695^\circ\mathrm{K}\). The preliminary determinations carried out by the authors-

measurements of the heat capacity \(C_{\text{ж}}\) at \(T \sim 0.4^\circ\text{K}\) showed that at this temperature the entropy of liquid \(\mathrm{He}^3\), in accordance with the prediction of I. Ya. Pomeranchuk\(^{29}\), falls to values smaller than the value \(R \ln 2\), corresponding to a chaotic spatial distribution of the spins of \(\mathrm{He}^3\) nuclei at low temperatures (for more detail on this, see Section 5, devoted to theoretical work).

The heat-capacity curve of liquid \(\mathrm{He}^3\) (in equilibrium with its own vapor), constructed taking into account the data of various authors, is given in Fig. 6. For comparison, let us recall that the heat capacity of liquid \(\mathrm{He}^4\) has a sharp peak at the \(\lambda\)-point, and below the \(\lambda\)-point varies according to a law close to \(T^{5.5}\), except at the very lowest temperatures, where it becomes proportional to \(T^3\). The difference between the properties of the two helium isotopes appears here with great clarity.

Fig. 6. Heat capacity of liquid \(\mathrm{He}^3\), in equilibrium with vapor.

Various authors have also studied the magnetic properties of liquid \(\mathrm{He}^3\) (Hammel and others\(^{33}\); Fairbank and others\(^{19}\)). The nucleus of the \(\mathrm{He}^3\) atom, unlike the nucleus of the \(\mathrm{He}^4\) atom (the \(\alpha\)-particle), has spin equal to \(1/2\) and a magnetic moment equal to \(1.07 \cdot 10^{-23}\). Therefore one might expect, at low temperatures, magnetic effects in liquid \(\mathrm{He}^3\) associated with the mutual orientation of the magnetic moments of \(\mathrm{He}^3\) nuclei. The first group of authors, carrying out measurements of magnetic susceptibility by the inductance-bridge method, showed that in liquid \(\mathrm{He}^3\) no ferromagnetic phenomena are observed, at least at temperatures above \(0.9^\circ\text{K}\).

The character of the temperature dependence of the susceptibility of liquid \(\mathrm{He}^3\) was established in the work of Fairbank and co-workers\(^{19}\). \(\mathrm{He}^3\) was condensed inside a coil that formed part of a high-frequency resonant circuit and was placed in a magnetic field of strength 10,000 oersteds. The system was held for a long time so that equilibrium would be established between thermal motion and the orientation of the magnetic moments. The measurements were made at a frequency of 30 Mc. The amplitude of the observed signal (which was kept sufficiently weak so as not to cause a noticeable disturbance of equilibrium) was, under these conditions, proportional to the volume magnetic susceptibility associated with the orientation of the nuclei. For conversion to the molar susceptibility

the above data on the density of liquid He³ were used. The authors estimate the accuracy of the measurements at 10%. The results of these measurements (in arbitrary units) are shown in Fig. 7. It is seen that, in the temperature interval studied, between 1.2 and 2.8° K, He³ possesses a paramagnetic susceptibility associated with the nuclei, definitely following Curie’s law, \(\chi T=\mathrm{const}\). The point at 4.2° K was taken for gaseous helium under a pressure of 900 mm Hg. As we see, this point also lies fairly well on the straight line \(\chi(1/T)\). Thus, in liquid He³ above 1.2° K no deviations from Curie’s law are observed that would have had to accompany correlation phenomena; the spatial distribution of the orientations of the magnetic moments of He³ nuclei in the absence of a field is chaotic at these temperatures*).

Fig. 7. Magnetic susceptibility of liquid and gaseous He³.

In the same work \({}^{19}\), the relaxation time was determined, characterizing the rate at which equilibrium is established between the nuclear spins and the thermal motion of liquid He³. It proved to be equal to \(\sim 2\)–\(3\) minutes.

In considering the phase diagram of He³, which very much resembles the phase diagram of ordinary He⁴, the question naturally arises: are there any phase transitions in He³ in the liquid state? It is interesting that applying to the triple points of xenon, krypton, deuterium, hydrogen, and helium a procedure analogous to that described above for the critical point, i.e., plotting the curve \(T^{*}_{\mathrm{tr.p.}}(\lambda^{*})\), leads \({}^{13}\), for He⁴, directly to the \(\lambda\)-point, while for He³ it predicts a transition at \(T \sim 1^\circ\) K. Nevertheless, there are as yet no convincing experimental indications of the existence of such a transition. An attempt to detect the presence of a transition at \(T>0.8^\circ\) K by observing the rate of heating of a reservoir with liquid He⁴ at constant—

*) In a recently published work \({}^{57}\), Fairbank, Ard, and Walters established that at \(T<1^\circ\) K, \(\chi T\) becomes a function of temperature, tending to zero as \(T\to 0^\circ\) K. This indicates the establishment of antiparallel orientation of the nuclear spins of atoms in liquid He³ at the lowest temperatures.

supplied power gave a negative result: heating between 0.84 and 3.21°K proceeded quite monotonically; no kinks or bends were observed on the heating curve.

On the other hand, there are definite theoretical indications that liquid He³ at temperatures below 1°K must undergo some substantial rearrangement. The point is that, according to the third law of thermodynamics, the entropy of liquid He³ must go to zero at absolute zero. The limiting value of the entropy can be estimated from the value of the chemical potential of He³, which in turn can be calculated from the vapor-pressure data. If, for such a calculation, one uses the vapor-pressure data obtained in work²¹, extrapolating the authors’ curve to 0°K, it turns out that the entropy of liquid He³ does not go to zero, but tends to a certain positive value of the order of several tenths of a cal/degree·mole. Consequently, the indicated extrapolation of the vapor-pressure data is illegitimate; this points to the probable occurrence in liquid He³, upon further cooling, of certain processes that lead to a fall of the entropy to zero and at the same time somehow affect the liquid–vapor phase equilibrium. Whether the data mentioned on the fall of the entropy, obtained on the basis of preliminary heat-capacity measurements (see above), may be regarded as the first experimental confirmation of the existence of rearrangement processes in liquid He³ at very low temperatures will, one must suppose, be shown in the very near future*).

Last (in the order of our exposition, but by no means in importance) are the hydrodynamic properties of He³ at low temperatures. First let us consider the viscosity of gaseous He³, although it was measured later³⁴ than the viscous properties of liquid He³. The viscosity was measured from the increase in the damping decrement of a quartz rod set into oscillation when it was placed in the gas. It turned out that the viscosity of gaseous He³ is of the same order of magnitude (\(\sim 10^{-5}\) poise) as the viscosity of gaseous He⁴, but exceeds the latter by a factor of two or three. In addition, in contrast to the viscosity of He⁴, the viscosity of gaseous He³ depends only weakly on temperature. According to the classical kinetic theory, the viscosity of a gas is \(\eta = \mathrm{const}\cdot nmvl = \mathrm{const}\cdot \dfrac{mv}{\sigma}\); here \(m\) is the mass, \(v\) the molecular velocity, \(n\) the number of molecules per unit volume, \(\sigma\) the effective cross section of molecular collisions, and \(l = 1/n\sigma\) the mean free path. The classical values of \(\sigma\) for the cases of gaseous He⁴ and He³ coincide, and there-

*) See the footnote on p. 63.

PROPERTIES OF THE HELIUM ISOTOPE He³ AT LOW TEMPERATURES

since \(v=\operatorname{const}/\sqrt{m}\), the viscosities of the two gases should have been in the ratio \(\sqrt{m_3/m_4}\simeq 0.87\); thus the viscosity of gaseous He³ should have been somewhat smaller. The observed excess of the viscosity of He³ over the viscosity of gaseous He⁴ is a quantum effect. It is connected with the fact that He³ atoms, which possess nuclear spin equal to \(1/2\), obey the Fermi–Dirac quantum statistics, whereas He⁴ atoms (whose nuclear spin is zero) obey Bose–Einstein statistics. The fundamental role of quantum statistics in the behavior of helium isotopes at low temperatures will be discussed in greater detail later (sec. 5).

Qualitative determinations of the viscosity of liquid He³ were carried out by Abraham, Osborne, and Weinstock³⁵. They compared the rate of flow through a narrow slit (“superleak”) of liquid He⁴ and liquid He³ at various temperatures. The “superleak” was formed by a gap, arising on cooling, between a platinum wire and a capillary made of Pyrex glass seated on it in the hot state. By measuring the rate of flow of gaseous He⁴ through the “superleak” it was established that the size of the gap was less than \(1\,\mu\). The “superleak” was sealed into the lower end of one of the limbs of a U-shaped capillary. He³ (or He⁴) condensed in the part of the capillary situated above the “superleak” and began to flow through the latter under its own vapor pressure into the second limb of the U-shaped capillary, which was connected with a large volume located outside the cryostat and initially evacuated. The rate of increase of the pressure in this vessel served as a measure of the amount of liquid flowing through the “superleak,” a quantity related to the viscosity. The results of such measurements at various temperatures are shown in Fig. 8. As the temperature is lowered, in both He³ and He⁴ above the \(\lambda\)-point the rate of flow decreases monotonically with decreasing temperature, corresponding to a decrease of the pressure difference across the “superleak.” However, in the case of He⁴ the character of the curve changes sharply at the \(\lambda\)-point: with further lowering of the temperature a sharp increase in the rate of flow begins. This corresponds to the appearance of superfluidity in the transition from He I to He II. At the same time, in the case of liquid He³ there are not the slightest signs of cessation of the decrease in the rate of flow with decreasing temperature down to \(1.05^\circ\) K. According to the estimate made by the authors (not a very reliable one), the viscosity of liquid He³ with decreasing temperature even increases from \(2.2\cdot 10^{-5}\) poise at \(2.8^\circ\) K to \(3.0\cdot 10^{-5}\) poise at \(1.0^\circ\) K.

Thus, direct measurements show that at temperatures above \(1^\circ\) K He³ is not superfluid. Experiments by Daunt and Heer³⁶ with liquid mixtures He³—He⁴, described in detail in the next section of the review, show that superfluidity does not set in in He³ even at considerably lower temperatures.

temperatures, at least down to \(0.3\)—\(0.25^\circ\mathrm{K}\). Thus, the light isotope \(\mathrm{He}^3\) turns out to differ sharply in its properties from the ordinary isotope \(\mathrm{He}^4\): it does not possess the property of superfluidity.

\[ \frac{dm}{dt}\;(10^{-8}\,\mathrm{g/sec}) \]

\(T(^{\circ}\mathrm{K})\)

Fig. 8. Fluidity of liquid \(\mathrm{He}^3\) and \(\mathrm{He}^4\).

The enormous theoretical significance of this fact for the problem of superfluidity is quite evident. This question will be considered in greater detail in the concluding part of the review.

4. PROPERTIES OF LIQUID MIXTURES \(\mathrm{He}^3\)—\(\mathrm{He}^4\)

The properties of dilute solutions of \(\mathrm{He}^3\) in liquid \(\mathrm{He}^4\) began to be studied earlier than the properties of pure \(\mathrm{He}^3\), since achieving enrichment of natural helium to a relative content of the light isotope of \(\sim 10^{-4}\)—\(10^{-2}\) presented no special difficulties. Many investigations were devoted to the study of the distribution of \(\mathrm{He}^3\) in dilute solutions between the liquid phase and the vapor. It is known that for ideal classical solutions Raoult’s law must hold: \(c_{\mathrm{p}}/c_{\mathrm{ж}}=p_3/p_4=\mathrm{const}\) at constant temperature \((c_{\mathrm{p}}\) and \(c_{\mathrm{ж}}\) are the concentrations of \(\mathrm{He}^3\) in the vapor and liquid, \(p_3\) and \(p_4\) are the vapor pressures of the pure isotopes at the given temperature). Investigations with solutions of concentration \(c_{\mathrm{ж}}\sim 10^{-6}\)—\(10^{-2}\) in the \(\mathrm{He\ I}\) region\({}^{37}\) showed that Raoult’s law is in general obeyed here, though only very approximately. With the transition to the \(\mathrm{He\ II}\) region the situation became considerably less clear. The data of different authors, especially in early works, differ from one another by several times.

orders of magnitude: \(c_{\text{п}}/c_{\text{ж}}\) is either much larger than according to Raoult’s law, or proves to be practically equal to zero. It was found that the cause of such a large discrepancy is effects associated with evaporation and condensation of the wall film of He II.

Let us consider the apparatus shown in Fig. 9. The solution under investigation, \(P\) He\(^3\) in He\(^4\), is in a small vessel connected with the upper part of the apparatus by a tube \(T\), through which a sample of the gaseous phase is taken in order to determine the concentration of He\(^3\) in it. The small vessel is surrounded by a bath of liquid helium and has the temperature of the bath. The wall film, having risen to a level exceeding the level of the liquid helium in the bath (and consequently to the more heated regions of the tube), evaporates. Under the action of the increased pressure corresponding to the higher temperature, the evaporated helium rushes downward and condenses in the small vessel. As will be described below, He\(^3\) atoms do not take part in the superfluid motion of He\(^4\). Therefore the resulting convection—small vessel—film—vapor—small vessel—leads to filling the space above the liquid solution with pure He\(^4\). As a result, in such an arrangement the vapor proves to be strongly depleted with respect to He\(^3\). In other devices the opposite distortion occurs.

Fig. 9. Circulation of He\(^4\) associated with the mobility of the wall film.

When the causes of the erroneousness of a number of results had finally been clarified, it was established that Raoult’s law for He\(^3\) solutions in the region below the \(\lambda\)-transition temperature is certainly not obeyed: as the temperature is lowered below the \(\lambda\)-point, deviations from Raoult’s law in the direction of an increase begin to be observed, so that \(c_{\text{п}}/c_{\text{ж}}\) at low temperatures becomes approximately twice as large as for an ideal solution.

A study of the vapor pressure over liquid He\(^3\)—He\(^4\) solutions in the region below the \(\lambda\)-point was the subject of the work of Sommers\(^{38}\), who worked with mixtures containing up to 13% He\(^3\). One of the difficulties of such experiments is that even when the concentration of the initial mixture is known, the exact concentration of He\(^3\) in the liquid and in the vapor is unknown, since He\(^3\) is distributed between the two phases in a not sufficiently well known ratio. In part of the experiments\(^{38}\), measurements of the vapor pressure over the solution were made under conditions in which the total mass of helium in the vapor was very small compared with the mass of the liquid. In this case, evidently, the concentration of the liquid is simply equal to the concentration of the initial-

of the mixture. Another series of measurements consisted in determining the dew point: the mixture was admitted in small identical portions into a vessel immersed in a liquid-helium bath, and the corresponding rise in pressure was measured. The appearance of the first drops of liquid was noted from the break in the curve of the dependence of pressure on the amount of mixture admitted into the vessel. In this experiment, evidently, it could be assumed that the concentration of the initial mixture was equal to the concentration of the vapor above the liquid that had appeared. Measurements of the second type were convenient for studying the region of small concentrations, while measurements of the first type were convenient for the region of higher concentrations of He³. Both types of measurements gave mutually consistent results. In Fig. 10 are shown curves of the dependence of vapor pressure on the concentration in the liquid \(c_{\mathrm{l}}\) and in the vapor \(c_{\mathrm{v}}\) for two temperatures, according to the data of work 38. The considerable “divergence” of the branches is striking (especially at the lower temperature), a circumstance very favorable for separating the components of the mixture by rectification. The results of such measurements can be used to construct detailed state diagrams of mixtures, which are necessary for calculating and monitoring the operation of rectification columns. However, Sommer’s results, as was indicated, refer only to temperatures below the \(\lambda\)-point of He II.

Fig. 10. Dependence of the vapor pressure over a liquid mixture of He⁴—He³ on the concentration in the liquid and in the vapor.

Very careful measurements of the vapor pressure of mixtures of helium isotopes have recently been carried out by B. N. Esel’son \(^{39}\). Noting that the data of previous measurements are for the most part erroneous because of insufficient allowance for the influence of the film in measurements below the \(\lambda\)-point and because of the absence of established equilibrium above the \(\lambda\)-point, the author reports a number of precautions he took in order to ensure the reliability of the results. Esel’son’s apparatus is shown in Fig. 11. After thorough evacuation of the apparatus, all stopcocks were closed, the Dewar was filled with liquid helium, and the temperature was lowered to \(\sim 1.35^\circ\)K. Then condensation of the gaseous mixture was carried out in one of the bulbs \(c, d\).

Before condensation the gas passed through a system of coils cooled by liquid hydrogen and liquid helium; this led to the removal of traces of foreign gases which otherwise would have been adsorbed on the walls of tubes \(a\), \(b\), causing enhanced creep of the wall film of He II. In the second bulb pure \(\mathrm{He}^4\) was condensed. An important precaution was also the use of considerable quantities of gas (\(\sim 0.4\ \mathrm{l}\)). It was shown that condensation of an insufficient quantity of gas leads to large distortions. The volume of the gas phase was very small (not more than \(5\ \mathrm{cm}^3\)). Finally, in measurements above the \(\lambda\)-point, stirring by means of the magnetic stirrer \(N\) was used. This precaution is

Fig. 11. Apparatus for precise determination of the vapor pressure over solutions of \(\mathrm{He}^3\) in \(\mathrm{He}^4\).

Fig. 12. Deviations of the behavior of a solution of \(\mathrm{He}^3\) in \(\mathrm{He}^4\) from ideality.

very essential, since without the use of a stirrer, establishment of equilibrium requires a time measured in hours.

The measurements were carried out in the temperature interval \(\sim 1.3\text{–}3.2^\circ\mathrm{K}\) and at concentrations \(c \sim 0.5\text{–}8\%\ \mathrm{He}^3\). Experimentally, the curves \(\Delta p(T)\) were determined for specimens of different concentrations (\(\Delta p\) is the difference of the vapor pressures of the mixture and of pure \(\mathrm{He}^4\)); an example of such a dependence for \(c = 8.08\%\ \mathrm{He}^3\) is shown in Fig. 12. From these data were constructed

isotherms \(p(c)\). Consideration of the curves of both types clearly shows that \(\mathrm{He}^3—\mathrm{He}^4\) solutions are not ideal either below or above the \(\lambda\)-point: in all regions of \(T\) and \(c\) appreciable deviations from Raoult’s law are observed, and these deviations at low temperatures and concentrations occur in one direction (the vapor pressure of the mixture is higher than according to Raoult’s law), while at higher ones—in the other. The author indicates that the effect must have a quantum nature.

B. N. Esel’son and B. G. Lazarev \(^{18}\) also carried out important investigations of solidification curves in \(\mathrm{He}^3—\mathrm{He}^4\) mixtures. The measurements were made by the same blocked-capillary method as the measurements of the solidification pressure of pure \(\mathrm{He}^3\) described above. Since a very thin capillary was used (diameter \(0.1\) mm), it had to become blocked practically upon the appearance of the first crystals of solid helium. Therefore the curves obtained give the pressures at the onset of solidification. The measurements were made in the temperature interval \(1.5—4.2^\circ\mathrm{K}\) and at \(\mathrm{He}^3\) concentrations \(0—64\%\). The solidification curves \(p(c)\) constructed on the basis of these measurements show that the phase diagram for the liquid and solid phases of the \(\mathrm{He}^3—\mathrm{He}^4\) system is a cigar-shaped form with a small region of stratification.

One of the interesting effects observed in \(\mathrm{He}^3—\mathrm{He}^4\) solutions is the effect of isotopic osmosis: an admixture of only \(\sim 10^{-5}\) atoms of \(\mathrm{He}^3\) in \(\mathrm{He}^4\), contained in one part of an apparatus connected by a narrow slit with another part in which pure \(\mathrm{He}^4\) is located, leads to a difference in the liquid levels in the two parts of height \(\sim 1\) cm. A detailed investigation of this phenomenon at concentrations \(c_{\mathrm{ж}}\sim 10^{-3}\) showed \(^{40}\) that the well-known classical van ’t Hoff law

\[ p=n_3 kT=n_4 c_{\mathrm{ж}} kT=\frac{\rho RT}{\mu_4}\cdot c_{\mathrm{ж}} \]

is also fulfilled in the present case. This shows that, in some respects, the behavior of a liquid mixture of helium isotopes is nevertheless similar to the behavior of ideal solutions.

The study of phenomena occurring during the artificial mixing of liquid \(\mathrm{He}^3\) with liquid \(\mathrm{He}^4\) leads to an analogous conclusion. These experiments were published recently by Sommers, Keller, and Dash \(^{41}\). Their apparatus consisted of two volumes, one (\(\simeq 1\ \mathrm{cm}^3\)) for liquid \(\mathrm{He}^4\) and the other (\(\simeq 0.1\ \mathrm{cm}^3\)) for liquid \(\mathrm{He}^3\). The second vessel was inside the first and was made thin-walled. For thermal insulation, both vessels were surrounded by a vacuum jacket. To condense liquid helium into both vessels, thin capillaries were led in from above. In a straight vertical capillary leading into the larger vessel there was placed a tungsten rod sharpened at the bottom, which could move along the capillary. The entire device was surrounded by a bath of liquid helium. After the vessels were filled, the temperature of the bath was lowered to \(1.02^\circ\mathrm{K}\), and after that

PROPERTIES OF THE HELIUM ISOTOPE He\(^3\) AT LOW TEMPERATURES

as the temperature of the vessels took on the same value, mixing was carried out. For this purpose the tungsten rod was rapidly lowered and broke the inner vessel containing liquid He\(^3\). The process of mixing, as experiments showed, was completed within a two-minute interval of time, after which the concentration of He\(^3\) assumed one and the same value throughout the entire volume of the mixture (8.6%). It turned out that with such mixing the temperature falls from the initial value \(1.02^\circ\mathrm{K}\) to \(0.78^\circ\mathrm{K}\). Thus, mixing liquid He\(^3\) and He\(^4\) is accompanied by an effect of considerable cooling.

In another experiment the authors increased the amount of He\(^3\) in the gas phase. Since, upon mixing, the vapor pressure above the resulting mixture becomes smaller than above the pure light isotope, gaseous He\(^3\) begins to condense. The authors succeeded in selecting the amount of He\(^3\) in the gas phase so that the heat released upon condensation approximately compensated the cooling effect during mixing, and the total temperature change was small. In this way they determined the heat of mixing. For the aforementioned final concentration of 8.6% He\(^3\) it proved to be equal to \(0.17\ \mathrm{cal}/\mathrm{mole}\) of the solution formed. A calculation of the increase in entropy occurring upon mixing showed that the change in entropy, within a small 10% discrepancy, coincides with the classical value of the change in entropy upon mixing for the case of ideal solutions.

Thus, a peculiar situation has arisen: some properties of isotopic liquid mixtures He\(^3\)—He\(^4\) turn out to be rather close to the properties of classical ideal solutions, whereas in other cases the quantum nature of these mixtures is very sharply manifested and their behavior deviates noticeably from “ideal” behavior. Quantitative study of these deviations represents one of the promising directions in the investigation of the properties of He\(^3\) and He\(^4\) at low temperatures.

Let us turn to the consideration of the superfluidity of liquid mixtures He\(^3\)—He\(^4\).

The principal result obtained by Daunt and co-workers as early as 1947\(^ {42}\) is that He\(^3\), present in ordinary He II (liquid He\(^4\) below the \(\lambda\)-point) in the form of a small impurity, does not take part in the superfluid motion. One of the experiments (Fig. 13) was as follows. Into a bath \(B\) of liquid helium, maintained at a temperature below the \(\lambda\)-point, there was immersed an apparatus consisting of an outer volume \(A\) and a small Dewar vessel \(D\),

Fig. 13. Experiment proving the nonparticipation of He\(^3\) in the superfluidity of He\(^4\).

of a closed hollow ground stopper \(P\), tightly lapped to \(D\) (in order to make transfer of substance through the gas phase impossible). The leads of an electric heater, placed in \(D\), passed through the stopper. It turned out that, when liquid helium flowed from \(A\) into the initially empty volume \(D\) through the slit \(K\), which occurred by superfluid motion and was caused by the release of heat in the heater (the thermomechanical effect), the liquid helium in \(A\) became enriched in \(\mathrm{He}^3\). On the contrary, the liquid helium \(\Gamma\), collected in \(D\), proved, within the accuracy of the mass-spectroscopic determination, to be pure \(\mathrm{He}^4\), containing no \(\mathrm{He}^3\). It followed from this that the flowing wall film of He II does not carry along \(\mathrm{He}^3\) atoms and transports only \(\mathrm{He}^4\). An analogous experiment showed that also under conditions in which the slit is completely immersed in liquid He II, superfluid mass transfer is connected exclusively with the transfer of \(\mathrm{He}^4\). The phenomenon of nonparticipation of \(\mathrm{He}^3\) in superfluid motion, as was indicated in Sec. 2, was widely used for the development of low-temperature methods of enriching helium in the light isotope.

It is appropriate here to describe a very interesting experiment by which it was established that the heavy isotope of helium—\(\mathrm{He}^6\)—also does not participate in the superfluid motion of \(\mathrm{He}^4\) (as is known, \(\mathrm{He}^5\) and other isotopes of helium, apart from the mentioned \(\mathrm{He}^4\), \(\mathrm{He}^3\), and \(\mathrm{He}^6\), do not exist). This experiment was carried out by Gutman and Arnold\({}^{43}\).

Fig. 14. Experiment proving the nonparticipation of \(\mathrm{He}^6\) in superfluidity.

The principal difficulty lay in the fact that \(\mathrm{He}^6\) is a short-lived \(\beta\)-active isotope with a half-life of only \(0.82\) sec. The experiment was arranged in the following way (Fig. 14). Gaseous helium of natural composition (i.e., practically pure \(\mathrm{He}^4\)) in a continuous stream passed successively through the “\(\mathrm{He}^6\) generator,” \(\Gamma\), and then, in the form of a mixture \(\mathrm{He}^4\)—\(\mathrm{He}^6\), passed through the ring Geiger–Müller counter \(C_1\) into the helium Dewar \(D\). Here it was cooled and seeped through narrow (\(\sim 1\)–\(2\mu\)) slits between stacks of copper rings rubbed against one another. The evaporated helium, having passed through the Dewar, entered the second counter \(C_2\) of large volume (\(0.8\) l) and then went to the pump and to the flowmeter. Passing descr...

a path, the heavy isotope admixed with He⁴ gradually decayed. Most of the He⁶ nuclei that had not had time to decay decayed in counter \(C_2\). Knowing the flow velocity and the geometry of the apparatus, it was possible to calculate the expected counting rate in \(C_2\), corresponding to the initial concentration of He⁶, regulated according to the readings of counter \(C_1\).

The He⁶ generator was an apparatus consisting of a neutron source (a beryllium sphere coated on the inside with a layer of polonium), immersed in a suspension of powdered beryllium in a nonvolatile oil, through which helium was bubbled. The advantage of the radioactive source over a cyclotron is its great simplicity and stability. The apparatus operated successfully at bath temperatures of \(\sim 1.9\text{--}2.0^\circ\) K (at higher temperatures the gas density became too high, which led to an impermissible increase in the time for helium to pass through the system; at lower temperatures the vapor elasticity became less than the pressure required for sufficiently rapid forcing of helium through the slits). The authors obtained the following result: in four experiments out of five, each lasting from 20 to 100 minutes, within the limits of measurement accuracy the effect that should have been produced in counter \(C_2\) by the He⁶ reaching it was completely absent. The ratio of the observed counting rate in \(C_2\) to the expected one was \(4\% \pm 7\%\). Despite the considerable errors (associated with a large background in \(C_2\), 10–20 times exceeding the expected effect), the experiment described apparently proves convincingly that He⁶, like He³, does not participate in the superfluid motion of He II, in accordance with the prediction of L. D. Landau and I. Ya. Pomeranchuk\(^{44}\).

The experiments described and others showed that although impurity atoms do not take part in the superfluidity of He⁴, the property of superfluidity itself is also preserved in solutions. However, the presence of an He³ impurity leads to a shift of the \(\lambda\)-point toward lower temperatures. From the theoretical point of view, the magnitude of this shift \(dT_\lambda/dc\) in the region of small concentrations \(c\) is of great interest. One determination of this quantity belongs to B. N. Eselson, B. G. Lazarev, and I. M. Lifshitz\(^{45}\). The mixtures studied were condensed into a glass inverted U-shaped tube with sealed ends. The arms of the tube were made of different lengths, and therefore the levels of the condensed mixture in the two arms proved to be different. If the temperature of the helium bath into which the tube was immersed was below the \(\lambda\)-point of the solution, equalization of the levels immediately began owing to the superfluid flow of He⁴ from one arm to the other along the wall film. At temperatures above the \(\lambda\)-point no flow occurred. The transition temperature of a mixture containing \(1.5 \cdot 10^{-2}\) He³ proved to be \(0.03^\circ\) lower than the \(\lambda\)-point of ordinary helium; the corresponding value

the displacement \(dT_{\lambda}/dc\) is about \(-2.0^\circ\). These same authors, studying the character of the flow of the film under the indicated conditions, found and explained a number of curious features. The displacement of the \(\lambda\)-point upon adding \(\mathrm{He}^3\) in the region of small concentrations of the light isotope has also recently been investigated by King and Fairbank \(^{46}\), who studied second sound in solutions with \(\mathrm{He}^3\) concentrations up to \(\sim 4\%\). The temperature of the \(\lambda\)-transition was determined by extrapolating the velocity of second sound to zero. It was found that the temperature \(T_{\mathrm{D}}\) depends linearly on the concentration (in the indicated concentration range), with

\[ dT_{\lambda}/dc=-1.5^\circ . \]

The dependence of the temperature \(T_{\mathrm{D}}\) on the \(\mathrm{He}^3\) concentration in the region of large concentrations was investigated by Abraham, Weinstock, and Osborne \(^{47}\), and by Daunt and Heer \(^{36}\). The first authors used the same “superleak” technique that was applied in the above-described work \(^{35}\) of these authors on the study of the viscous properties of pure \(\mathrm{He}^3\); the \(\lambda\)-point corresponded to the temperature below which, upon cooling, a sharp increase began in the rate of flow of helium through the “superleak.” A concentration interval up to \(28\%\,\mathrm{He}^3\) was investigated. It turned out that the temperature of the \(\lambda\)-transition at a concentration of \(28.2\%\,\mathrm{He}^3\) is already \(1.56^\circ\mathrm{K}\). Daunt and Heer \(^{36}\) investigated the \(\lambda\)-transition at concentrations of \(42\text{—}89\%\,\mathrm{He}^3\). It turned out that these concentrations correspond to \(\lambda\)-transition temperatures between \(1.15\) and \(0.38^\circ\mathrm{K}\). The mixtures studied were condensed in a small evacuated volume, thermally insulated from a helium bath, except for heat input through a capillary filling tube. As the temperature was lowered below the \(\lambda\)-point, this latter began to increase sharply, owing to the occurrence of helium circulation through the wall film and the gas phase, with a corresponding release of considerable heat of condensation on the surface of the liquid mixture. The results of various authors who determined the form of the curve \(T_{\lambda}(c)\) are given in Fig. 15. From examination of the graph it is evident that even the most “unfavorable” extrapolation to \(c=1\) leads to the value \(T_{\lambda}=0.25^\circ\mathrm{K}\), so that pure \(\mathrm{He}^3\) in any case proves not to be superfluid upon cooling to this temperature. It is more than probable that \(\mathrm{He}^3\) does not possess superfluidity even at absolute zero, so that the curve must fall at the point \(T_{\lambda}(1)=0\).

In considering the phenomenon of the \(\lambda\)-transition in \(\mathrm{He}^3\)—\(\mathrm{He}^4\) mixtures, the natural question arises whether this transformation is a phase transition of the first kind (as was assumed, for example, by de Boer and Gorter \(^{48}\)) or else a transition of the second kind. This question was resolved by the experiment of Weinstock, Abraham, and Osborne \(^{49}\), who found that at all temperatures below the \(\lambda\)-point the vapor pressure above a solution with \(\mathrm{He}^3\) concentration of about \(25\%\) is greater than above a solution with concentration of about \(20\%\,\mathrm{He}^3\). In the case where the transition were a transition of the first kind, below the \(\lambda\)-point there would have to be

have phase separation. Application of the Gibbs phase rule (derived, as is well known, from the most general considerations) to this case, where in a two-component system three phases would be present simultaneously (two liquid and one gas-like), would directly lead to the existence of only one degree of freedom, i.e. to equal pressures at equal temperatures. The theoretical question of the nature of $\lambda$-transitions in He$^3$—He$^4$ solutions has been considered in a number of works. In particular,

Fig. 15. Curve of $\lambda$-transitions in liquid mixtures He$^4$—He$^3$.

in the cited work of B. N. Esel’son, B. G. Lazarev, and I. M. Lifshitz$^{45}$ it was shown that, at small concentrations of He$^3$, this transition in any case remains a transition of the second kind. In the same work a number of predictions were made concerning changes in the properties of solutions upon transition through the $\lambda$-point, and, in particular, a jump in the heat of solution was predicted ($\Delta Q_{\text{theor}} = 40$ cal/mole). Experimentally this phenomenon has not yet been observed.

The diffusion of He$^3$ atoms in He II was investigated in the work of Beenakker, Taconis, Lynton, Dokoupil, and van-Sust$^{50}$. The authors discovered and studied the phenomenon of an apparent decrease in the thermal conductivity of He II upon addition of an He$^3$ impurity to it. The apparatus consisted of a cylindrical volume 8 mm in diameter and 1 mm high, which was placed in vacuum and whose upper bottom was in thermal contact with a helium bath. On the lower bottom there was an electric heater, and a capillary also led there, through which the volume was filled with the mixture under investigation. When the heater was switched on, an additional pressure difference arose between the capillary and the bath, which could be recalculated as a change in tempera-

ature. It turned out that the “thermal resistance” of the layer of solution located in the vessel, at all temperatures, is strictly proportional to the concentration of He\(^3\) in the initial mixture. The results of the experiments were used by the authors to determine the diffusion coefficient of He\(^3\) in He\(^4\), which proved to be \(\sim 10^{-4}\) at \(T \simeq 1.9^\circ\mathrm{K}\), \(\sim 10^{-3}\) at \(T \simeq 1.5^\circ\mathrm{K}\), and \(\sim 10^{-2}\) at \(T \simeq 1.3^\circ\mathrm{K}\). The experimental data of this work are discussed in detail in the paper by V. N. Zharkov and I. M. Khalatnikov,\(^{15}\) who pointed out the incorrect processing of the results by the authors of paper \(^{50}\).

The subject of a number of investigations was the propagation of thermal waves (second sound) in dilute He\(^3\)—He\(^4\) solutions below the \(\lambda\)-point. Linton and Fairbank\(^{52}\) measured the velocity of second sound at He\(^3\) concentrations up to 0.8%. A small cylindrical reservoir was immersed in a helium bath and filled with the mixture. On both bottoms there were carbon resistors, one serving as the emitter and the other as the receiver of the thermal waves. The experiments were carried out by a pulse method: the emitter emitted a short heat pulse, the receiver detected it, and the velocity of second sound was determined from the time taken by the pulse to travel from the heater to the receiver. It was found that the presence even of such comparatively small amounts of He\(^3\) in He II appreciably changes the velocity of second sound: for example, at \(1.26^\circ\mathrm{K}\) the velocity of second sound in a solution containing 0.8% He\(^3\) was \(26.5\ \mathrm{m/sec}\), whereas for pure He\(^4\) at this temperature it is \(19.4\ \mathrm{m/sec}\). The authors note that their results confirm the theoretical predictions of I. Ya. Pomeranchuk.\(^{53}\) On the basis of the data of Linton and Fairbank,\(^{52}\) I. M. Khalatnikov\(^{54}\) found that the effective mass of a He\(^3\) atom present as an impurity in He II is approximately three times greater than its actual mass, and also drew certain conclusions about the nature of the energy spectrum of dissolved He\(^3\) atoms. Later, King and Fairbank,\(^{55}\) using the same pulse method, measured the velocity of second sound in mixtures containing 0.017–4.3% He\(^3\) at temperatures from the \(\lambda\)-point down to \(\sim 0.25^\circ\mathrm{K}\). Below \(1^\circ\mathrm{K}\) the mixture was cooled by demagnetization of a paramagnetic salt (chromium potassium sulfate). It is interesting to note that replacing pure He\(^4\) with the mixture caused a noticeable (by more than a factor of two) increase in the warm-up period from superlow to ordinary helium temperatures. The results for two concentrations are shown in Fig. 16. The ordinate axis gives the velocity of second sound, and the abscissa axis the absolute temperature. For comparison, the same figure gives part of the curve for pure He\(^4\); in this case, as is known, the velocity of second sound as \(T \to 0\) tends to the constant value \(\sim 150\ \mathrm{m/sec}\). A characteristic feature of the curves for the mixtures is that the region of the second (low-temperature) rise

PROPERTIES OF THE HELIUM ISOTOPE He$^3$ AT LOW TEMPERATURES

the second-sound velocity ends in a maximum, and at lower temperatures the velocity apparently tends to a constant, comparatively small value. Such behavior of solutions of He$^3$ in He$^4$ was predicted by I. Ya. Pomeranchuk$^{53}$ and is in contradiction with Dingel’s predictions$^{54}$. A characteristic feature of measurements of the second-sound velocity at ultralow temperatures in He$^3$—He$^4$ solutions, as noted by the authors of the work$^{46}$, is the complete absence of dispersion and of the distortions of the heat-pulse shape caused by it, which greatly hinder the determination of the second-sound velocity at very low temperatures in ordinary He II.

Fig. 16. Second-sound velocity in solutions of He$^3$ in He$^4$.

In conclusion we shall mention interesting experiments with a “Rayleigh heat disk” in a He$^3$—He$^4$ solution, carried out by Wainston and Pellam$^{56}$. The method is analogous to the well-known absolute method for measuring sound intensity proposed by Rayleigh. In the present case, in a small cylindrical cavity of volume $3/4\ \mathrm{cm}^3$, filled with the liquid mixture under investigation (or with pure He$^4$), a mirror disk of diameter $2.8\ \mathrm{mm}$ was suspended on a fine elastic thread. By passing an alternating current through a carbon resistance in the cavity, standing waves of second sound were excited. At certain frequencies a resonance was observed, revealed by a considerable deflection of the disk from its equilibrium position. The cause of the deflection was the force acting on the disk as it was flowed around by the He II components, which in the heat wave at resonance possessed considerable velocities. To increase the sensitivity of the apparatus, modulation of the power was employed—

...of an alternating current with a frequency equal to the frequency of the disk’s natural torsional oscillations. In this way the authors achieved a sensitivity of \(10^{-8}\) dyn·cm; the maximum twisting moment acting on the disk from the liquid helium was of the order of \(10^{-6}\) dyn·cm. The second-sound velocity was easily determined from the product of the cavity length and the resonance frequency corresponding to the occurrence of standing thermal waves. The data obtained are in agreement with the more accurate determinations of the second-sound velocity by King and Fairbank cited above \(^{46}\).

(Conclusion in the next issue.)

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