E. F. HAMMEL University of California, Los Alamos Scientific Laboratory, Los Alamos,
N. M.
Helium-3 and the Liquid Helium Problem Is a break-through into the liquid helium problem imminent? I N 1908 when Kamerlingh Onnes first liquefied helium (70), the Leyden workers must have been exhilarated also by the achievement of having a t last produced what was expected to be the simplest liquid imaginable-one in which the molecules were spherically symmetric, chemically unreactive, and bound together by forces an order of magnitude smaller than in any other known liquid. Since liquids generally had proved intractable to theory, it appeared that here at last was a model liquid which theoreticians and experimentalists alike had been eagerly awaiting. The first suggestion that these expectations might not be realized came 3 years later in 1911 when Kamerlingh Onnes found a maximum in the liquid density a t 2.2' K. (70). But the effect was small and, unfortunately, little attention was paid to it. Oddly enough, some 20 years passed from the date of its first liquefaction before it was recognized that this most normal liquid, this potential paragon of liquids, was in fact most peculiar and anything but normal. Within the next few years, liquid helium was found to exhibit so many remarkable properties that present-day authors of monographs on liquids have found it most expedient barely to admit its existence. Actually, other liquefied gases of low molecular weight have also been found to deviate from normal liquid behavior. These deviations have recently been accounted for by a quantum mechanical treatment of corresponding states developed by De Boer (5), but even in this scheme, properties of liquid helium below 2.2' K. find no place. Liquid helium has, therefore, become a special topic in physics and chemistry. And, despite a n almost exponential increase in the amount of research since 1945 on its properties, the theoretical basis for its behavior is still imperfectly understood. Some Properties of liquid Helium
For the experimentalist, striking demonstration of one of the peculiar properties of liquid helium is obtained simply by reducing the pressure on the liquid in order to lower its temperature. As the pressure drops from 1 atm., the liquid boils in quite a normal fashion; but
when a vapor pressure of 38 mm. of mercury or 2.19' K. is reached, this bubbling dramatically disappears. The uninitiated observer will quickly check to see if the pump is still going or if a valve in the pumping line has been closed. Actually, the pump still works, the vapor pressure and temperature continue to drop a t about the same rate as before, but there is no boiling. We now know what has happened. In a few hundredths of a degree, the effective thermal conductivity of the liquid has increased by a factor of a million or so. Consequently, vaporization takes plact. entirely a t the surface ; a temperature difference of only a few ten thousandths of a degree is enough to provide sufficient heat flow to the surface where the rate of evaporation is the same as formerly occurred into the bubbles throughout the body of the liquid. The density of liquid helium shown in Figure 1 as a function of temperature (70, 77) is about one seventh that of water or about twice that of liquid hydrogen. O n this scale, the density maximum a t 2.19' K. is apparent. Related to the liquid density is, of course, the molar volume. For most liquids, a fairly good estimate of this quantity may be obtained by computing the volume of Avogadro's number of close packed spheres with diameters equal to the gas kinetic diameter. If this is done for liquid helium, about 8 cc. is obtained for the molar volume, whereas the actual value is about 27 cc. or more than three times this calculated value. Therefore, compared with other liquids, liquid helium is "blown up." Each of its molecules already occupies a space about three times larger than its actual volume. Since the process of evaporation consists physically of pulling apart molecules stuck together to form a liquid, in liquid helium the job of vaporization is already partially accomplished. I n 1932 the specific heat of liquid helium shown in Figure 2 was measured at Leyden by Keesom and Clausius (70). This was the first quantitative evidence that the transition a t 2.19' K. was not a trivial matter. As may be seen, the shape of the specific heat curve bears some resemblance to a reversed lambda, whence the designations
such as A-point, A-transition, and A-line, to identify various aspects of this phenomenon. In recent years, A-type transitions have been discovered and investigated in a number of solid substances such as ammonium halides and methane (25). Although statistical mechanics have not been able to provide a quantitative explanation of any A-transition, a considerable amount of effort has been expended both on experimental and theoretical aspects of phase changes of this type. I n general, as the temperature increases, they seem to be associated with the gradual disappearance of some sort of order leading to complete disorder a t the A-temperature. I n helium, however, it is surprising to find this anomaly occurring in a liquid, for it appears that liquids intrinsically constitute a disordered state of matter. Now it is known that, according to the third law of thermodynamics, systems in thermal equilibrium must have lost all their entropy (disorder) a t absolute zero. Ordinary liquids satisfy this requirement a t their triple point by forming an orderly array of molecules in a solid crystal. But liquid helium possesses no such triple point and is almost certainly still liquid a t absolute zero. Therefore, the questions should be asked: Does liquid helium violate the third law and if not, what sort of an ordering process causes it to lose its entropy as absolute zero is approached? In answer to the first question there is probably no doubt about its satisfying the third law. Measurements of the melting curve (70, 27) (Figure 3 ) show that the liquid-solid transition line in a P-T diagram intersects the P axis a t T = 0 with a horizontal slope. According to the Clausius Clapeyron equation, we have--dP . dfliq. sol. T
AS 0 = -.
AV
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Sihce A V # 0, the difference
in entropy between liquid and solid helium a t absolute zero must be equal to zero. Assuming no frozen-in disorder in the solid (which is probably reasonable), its entropy and hence that of the liquid must a t absolute zero equal zero. Even a t 1' K., the slope of the melting line is almost zero; it would therefore appear that the liquid has lost almost all of its residual entropy or VOL. 48, NO. 1 1
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if one reservoir of this same system having the levels initially equalized and at uniform temperature is heated, flow will occur from the cold to the hot reservoir until a certain level difference obtains which is a function of both temperature difference and absolute temperature. There are, of course, many other properties of liquid helium which could be discussed ; but the selection made will suffice to illustrate the sort of phenomenon for which an explanation must be found. Theories of Liquid Helium
T, "K. Figure 1.
Density of liquid helium as a functicn of temperature
disorder between 1' K. and the A-point. As to how the entropy is removed from this substance, a satisfactory answer is to be found only in a satisfactory theory; as shown subsequently, the theories thus far advanced are incomplete and sometimes inconsistent. Although these state and thermal properties exhibited by liquid helium are anomalous, the changes occurring in so-called transport properties such as viscosity and thermal conductivity are even more remarkable. First of all, below the A-point they can no longer even be defined. Obviously, if a liquid flows through a pipe a t a rate independent of length and pressure gradient, the concept of liquid viscosity becomes meaningless. Similarly, thermal conductivity is usually defined as the heat flow per unit cross-sectional area per unit temperature gradient. But in liquid helium below the A-point when the heat flow is more or less proportional to the temperature gradient, it is not proportional to the cross-sectional area and vice versa. The transport properties of liquid helium(I1) (liquid helium below its A-point) are too involved even to outline the observed phenomena in a brief review. Some insight into this problem may be derived from the fact that results from heluim (11) experiments involving transport properties, depend upon both the type of experiment and the dimensions of the apparatus. For example, in ordinary liquids the viscosity can be obtained either by observing the rate a t which the
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liquid flows through a slit or a pipe and then calculating the viscosity from Poiseuille's law, or by observing the drag or an oscillating disk or cylinder immersed in the liquid and again employing the appropriate hydrodynamical equations. In liquid helium (11), vastly different results are obtained depending upon the method used. This article must, therefore, be confined to a brief discussion of one aspect of the problem-e.g., the flow of helium(I1) through narrow capillaries. Consider two reservoirs connected by a narrow capillary with a diameter of 1 micron or less. If one reservoir is filled with helium(I1) and the system is maintained a t constant temperature, the flow from one reservoir to the other will proceed until the levels are equalized. However, the flow rate through the capillary will be independent of the level difference-i.e., it will be just as rapid when the levels are almost equalized as it was at the beginning of the experiment. If this experiment is repeated with each reservoir and the capillary thermally insulated and with the temperature of the system initially uniform, the flow process will generate a temperature difference across the capillary. The temperature of the reservoir from which the liquid flows will rise, while that of the other reservoir will fall. Although the flow rate will still be independent of level difference, the levels will not equalize. Instead, a permanent Ah will persist so long as the temperature difference is maintained. Alternatively,
INDUSTRIAL AND ENGINEERING CHEMISTRY
London-Tisza Two-Fluid Theory. Several theories, none of which is entirely satisfactory, have been developed which endeavor to account for this strange behavior. Actually, it is asking too much of theoretical physics or chemistry to produce at the present time a satisfactory theoretical treatment of liquid helium, since not even a general theory of relatively normal liquids has been advanced. Sufficient progress has been made, however, on the helium problem to permit provisional descriptions of the microscopic character of the liquid at least in different temperature ranges. Also, we shall see later how recently determined properties of the rare isotope, helium-3, have also contributed to our understanding of liquid helium. Liquids obviously occupy an intermediate position between gases and solids each of which can be treated theoretically in a fairly effective fashion. Consequently, most theories of the liquid state represent extrapolations of the well grounded theory available at one or the other of these two extremes. Thus, either gaslike or solidlike theories of liquids are discussed. In 1938 the late Fritz London, being influenced probably by the gaslike (blown up) character of liquid helium mentioned earlier, introduced a convincing model of the liquid based on the behavior of an ideal gas (75). I t is well known that the equation of state for an ideal gas is PV = RT and that real gases approach this ideal gas behavior as the temperature is raised or as the gas density is reduced. More accurately, however, real gases actually approach in their behavior either socalled ideal Bose-Einstein gases or ideal Fermi-Dirac gases. Molecules consisting of an even number of elementary particles form Bose gases; those consisting of an odd number of elementary particles are Fermi gases. Thus, helium4 (2 protons, 2 neutrons, and 2 electrons), is a Bose gas and helium-3 (2 protons, 1 neutron, and 2 electrons) is a Fermi gas. Although the behavior of these quantum ideal gases is indistinguishable from that of the familiar classical ideal gas over most of the usually accessible temperature range, deviations do occur for sufficiently light molecules at suffi-
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Figure 3. helium
Specific heat of liquid helium under its own vapor pressure
ciently low temperatures. Without going into theoretical detail, the results can be shown simply. Consider first a Bose-Einstein ideal gas. The first consequence of the theory is that a peculiar type of condensation occurs at a certain well defined temperature, To, which depends upon both the molecular weight and the density of the gas. In many respects this is similar to the condensation of a real gas below its critical temperature to form a liquid; here, however, it is a condensation not in ordinary coordinate space, but in momentum space. Above TO, the gas molecules are distributed among the various energy states-some with high, most with average, and a few with low energies. As the temperature is lowered below To, this classical distribution is seriously distorted by progressive crowding of the molecules into the lowest energy state. This situation may be crudely compared to a boxful of flies. If the effect of gravity were somehow removed, it would make little difference to the flies; they would merely have to do no work to change their altitude. If this represents the situation above To and if this box is cooled, a temperature, identified as To will eventually be reached where some of the less hardy flies will perish. They will just stop moving. But they will not reduce the density of flies in the box; there will be just as many flies per cubic centimeter as before. The dead flies have simply “condensed” to their lowest energy state. As the temperature is lowered further, more flies will condense until finally, when all have condensed, a state exists comparable to the ideal Bose gas state a t absolute zero. This fanciful description of the condensation phenomena exhibited by an
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Equilibrium diagram for solid-liquid
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Figure 5 . p , J p as a function of temperature. Circles are experimental values of Andronikashvilli ( I )
ideal Bose-Einstein gas when translated back into mathematical language is capable of accounting qualitatively for a number of the properties of liquid helium. The quantitative agreement will not be good, but we have entirely neglected in this ideal gas approximation the interaction forces which actually bind the helium molecules together to form a liquid. Consider first the thermal properties of the liquid. In Figure 4, the experimental specific heat is compared with that of an ideal Bose-Einstein gas having the mass, and density of liquid helium-4. The value of T Oturns out to be 3.13' K. which is surprisingly close to the observed A-temperature of 2.19' K. Although the actual discontinuity in specific heat is reproduced here only as a discontinuity in slope, there is some reason to believe that this discrepancy can be removed by the introduction of interaction forces and a modification of the energy spectrum. There is fair agreement also between the calculated and the experimental value of the entropy; a t T Othe entropy of the ideal Bose gas is 2.5 cal. deg.-l mole-' compared with the experimental value of 1.5 cal. deg.-' moleM1for the entropy of liquid helium a t its A-point. The London theory was quickly extended by Tisza (75, 27) to yield what is known as the two-fluid theory. Tisza pointed out that many of the unusual properties of helium (11) could be explained by assuming that liquid helium below its A-point is composed of two interpenetrating fluids-a so-called normal and a superfluid. At absolute zero, all the helium atoms are superfluid; a t the A-point and above, all the atoms are normal. I n terms of the behavior of an ideal Bose-Einstein gas, the superfluid is identified with the condensed atoms,
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while the normal a toms are obviously those uncondensed. Since the condensed atoms are in their lowest energy state, it follows that all of the thermal excitation of the liquid is carried by the normal molecules. The entire entropy of the liquid is therefore resident in the normal fluid. The superfluid is assumed to possess zero entropy or disorder. Finally it is assumed that the viscosity of the superfluid atoms is equal to zero. T o summarize, the London-Tisza theory utilizes the condensation phenomena of an ideal Bose-Einstein gas to account roughly for the thermal behavior of the liquid and superimposes upon this framework a consistent two-fluid theory in order to explain the transport properties as follows : 1. The high thermal conductivity of liquid helium (11) is assumed to be not heat conduction in the usual sense of the word but a kind of inner convection. Heat is transported by actual mass transport by superfluid atoms moving rapidly to the heat source, accepting energy, and thus transforming themselves into excited (normal) atoms which then, carrying heat with them, move toward the colder regions of the liquid, With this model, comparable to a conveyor belt system for transporting heat, a marked dependence of the effective thermal conductivity on the physical dimensions of the liquid can be anticipated. If the thermal conductivity of liquid helium (11) is measured as a function of the cross-sectional area-e.g., the heat flowing through liquid helium columns of decreasing diameter-a marked drop in the conductivity will be observed after reaching diameters of about 10 microns. The reason for this is clear in terms of the two-fluid theory. Since the heat is actually transported by countercurrent flow of superfluid and normal atoms, any process which impedes the flow of one or the other will alter the heat transport. Inasmuch as the superfluid is assumed to possess zero viscosity, reducing the channel size \vi11 have no effect whatsoever upon its ability to flow. The normal fluid does, however, exhibit a viscosity and, for sufficiently small channels, its drag on the walls will be sufficient to immobilize it. Hence, the effective value of the thermal conductivity should drop sharply for sufficiently small diameters and this is what is observed (4, 75), 2. The viscosity measurements in liquid helium (11) also find a ready explanation in terms of the two-fluid theory. I n flow measurements through sufficiently small slits or channels, it is assumed again that only the superficial flows. If Poiseuille's law for fluid flow through narrow channels is used to calculate the viscosity, an extremely small value will be obtained. But this viscosity will not be unique because of the previously mentioned independence of the flow velocity on channel length and pressure drop. A disk or cylinder oscillating in the liquid, however, will still interact only with normal molecules. From the experimental data, the
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product of the viscosity and the density is obtained. Since independent methods of determining the actual normal molecule concentration in the liquid exist, it is easy to obtain values of the normal fluid viscosity and, although they show a minimum at about 1.8' K., their over-all behavior is relatively normal (2, 4, 9, 75). 3. Although the pressure-independent isothermal flow of liquid helium through narrow channels is only partially described by the London-Tisza two-fluid theory, this hypothesis at least provides a reasonably consistent outline within which the details are gradually being developed. As in the previous cases, it is probably only the superfluid that is actually moving through the channel. But for an ideal fluid-i.e., one with zero viscosity-the velocity of flow should by Torricelli's theorem be derivable from equating the loss in potential energy of head to the kinetic energy of the flowing fluid; this will lead to a velocity dependent on head. The observed flow velocities are independent of head and much less than those predicted by Torricelli's theorem. This fact had led to the suggestion that a critical velocity exists for superfluid flow. Any forces tending to increase the flow beyond this critical value are apparently opposed by some kind of frictional force, the nature of which is not understood, but which seems to increase rapidly after the critical velocity is reached. Thus, no matter how great the driving pressure, the velocity cannot be increased much beyond this critical value. As the pressure gradient along the channel decreases, the velocity approaches a limiting value but the pressure range in which this drop occurs is small and has not been systematically investigated. 4. The so-called thermomechanical effects mentioned previously are also readily explicable in terms of the twofluid theory. Since narrow channels permit only flow of superfluid helium atoms, they act as filters capable of separating normal from superfluid. If the ratio of normal to superffuid atoms is determined by the temperature, it is equally correct for rhe temperature to be determined by the ratio of normal to superfluid atoms. Consequently, when flow of liquid helium (11) occurs from one vessel to another through a fine capillary, the vessel from which the liquid flows will become richer in normal atoms; hence its temperature will tend to rise. Similarly, the other vessel will become richer in superfluid atoms and hence colder. In order to maintain isothermal conditions during the flow process, heat will have to be added to one vessel and removed from the other. If these vessels are insulated, the temperature of each vessel will necessarily change as the flow proceeds. In each of these cases, since the flow involves only superfluid which carries no entropy, A s for the process equals zero. Reversible thermodynamics may, therefore, be applied to obtain a relationship between the difference in pressure associated with a difference in temperature. This expression, first proposed
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by London (76), has since been completely substantiated by experiment. 5 . Perhaps the most significant contribution from the two-fluid theory was the recognition that classical hydrodynamics could no longer be used to describe the motion of liquid helium (11). A new set of equations, for which not all terms have been determined even yet, one for the normal fluid and one for the superfluid were set u p and from these, Tisza (75, 26), in 1938, was able to predict that liquid helium (11) should exhibit a new type of phenomenon calied second sound, which consists of nondispersive and nonattenuative propagation of temperature waves in the liquid. From Tisza’s theory, it was possible to obtain the velocity of these waves as a function of the absolute temperature. I n 1944 the first observation of these temperature waves reported by Peshkov (79) showed a temperature dependence of the second sound velocity in essential agreement with that computed by Tisza. From the velocity of these second sound waves, it was possible to obtain experimentally in terms of the two-fluid theory, the variation of pn (density of the normal fluid) and p B (density of the superfluid) with temperature. Finally, in a beautifully conceived experiment, the existence of quantities which could be called p . and p n was demonstrated most convincingly in 1946 by Andronikashvilli (7). His apparatus consisted of a pile of closely spaced disks immersed in the liquid and hanging from a torsion fiber. The disks are so closely spaced that if the disk assembly is set into torsional oscillation in liquid helium (I) (above the Apoint) the liquid between the disks will be dragged along with them by ordinary viscous forces. The moment of inertia of the disk assembly which depends on the mass of the oscillating system will, therefore, involve the weight of the disks plus the weight of all of the liquid helium enclosed between the disks. If now the temperature of the system is reduced below the A-point, some of the helium enclosed between the disks will become superfluid with zero viscosity and hence will no longer interact either with the disks or with the remaining normal moleclues. Returning to the analogy of flies, the material bodies of the condensed flies are replaced by ghosts, but nevertheless, they must be counted with the live ones. The live ones can, of course, pass or be dragged right through the departed spirits without interaction. The normal atoms confined between disks in the oscillating bob simply continue to stick to the disks and likewise pass right through any superfluid atoms standing in their way. As the temperature is reduced, the moment of inertia of the assembly is observed to decrease, indicating that the mass of the rotor has decreased or that less and less liquid is being dragged between the plates. By the time the temperature has been reduced to 1 K., the moment of inertia is about the same as in vacuum, indicating that at this temperature practically all the liquid is superfluid. Figure 5 shows the results of Andronikashvilli’s experiment where
the fraction of normal atoms is givcn as a function of temperature. The temperature dependence of p , / p in this temperature range is given fairly well by T5.6. This relationship is entirely empirical, however, and the exponent changes when the temperature falls below 1 O K. The results of Peshkov and Andronikashvilli, therefore, have introduced seemingly convincing evidence for the two-fluid hypotheses and indirectly for the Bose-Einstein condensation theory upon which it is based. Despite its successes, however, the two-fluid theory is almost sure to be literally incorrect and, if the description is carried too far, many difficulties will be encountered. It can be said only that liquid helium behaves as if it were composed of two interpenetrating fluids.
landau Theory
In 1941 Landau (73) introduced another theory of liquid helium, basically quite different from the London-Tisza work. From a consideration of hydrodynamics applicable to a quantum liquid, Landau derived an energy spectrum for the whole assembly of particles from which the following results may be obtained : 1. At absolute zero, liquid helium is an assembly of atoms in a single lowest energy state devoid of any thermal energy. 2. As the temperature is raised, thermal exciwtions begin to appear in the liquid as quantized phonons or sound waves and as quantized rotational motions called rotons. I n other words, the liquid can accept energy only as compressional waves of sound and by the creation of vortex motions throughout the liquid. 3. If these thermal excitations are assumed to play ~e part of the normal fluid and the unexcited atoms the part of the superfluid, Landau’s description leads also to a two-fluid theory. When liquid helium flows past a wall slowly enough, its interaction is insufficient to generate thermal excitations; hence, it behaves as a superfluid with zero viscosity (no wall interaction). The Landau theory thus provides a better basis for superfluidity than does the LondonTisza theory in which the assumption of zero viscosity for the superfluid was introduced rather arbitrarily.
I n the Landau theory, the concept of two fluids is different from that of the London-Tisza theory. In the former case, a picture of a background fluid consisting of all the atoms through which excitations move must be imagined. Identifying the background fluid as the superfluid, the excitations constitute the normal fluid. Thus in a sense the superfluid changes into normal fluid and back again as the excitation moves through the liquid. If flies are used again to represent the helium atoms, Landau proposes that a t absolute zero the entire
collection of flies be identified as the background superfluid in its lowest energy state. Whatever movement the flies exhibit is assumed to be caused by their zero point energy. Then, as the temperature is raised, energy (as quanta or in fixed amounts) must be introduced into this assembly. The lowest units of energy are Landau’s phonons, or compressional waves of sound which travel through the flies like little coherent clouds bouncing about within the box. The higher the temperature the more clouds. Eventually the temperature is high enough to give some rotary motion to these clouds and quantized wave packets of somewhat higher energy called rotons are introduced. As a phonon cloud passes through the flies, it is assumed that those enveloped by it are momentarily attracted toward each other-Le., their local density is momentarily increased-as in a sound wave. They are, therefore, temporarily removed from the superfluid state. If a roton cloud passes through them, they are likewise temporarily excited by some sart of vortex motion. Thus, the greater the number of excitations the smaller the concentration of undisturbed superfluid. The excitations (flies or atoms) enveloped by them play the role of the normal fluid. I t turned out that Landau’s theory was successful, especially a t very low temperatures. I t predicted the existence of second sound, and in addition it correctly predicted the course of the second sound velocity below 1’ K., whereas the London-Tisza theory indicated that the second sound velocity went to zero at T = 0. But the Landau theory is unable to account satisfactorily for the A-point transition in the specific heat, although it does give the right temperature dependence of specific heat within a few tenths of a degree from the A-point. According to Landau, the filtering effect of narrow slits and the associated thermal effects are easily explained with the aid of a few additional assumptions. I n the analogy, those flies not enveloped in a cloud (superfluid ones) are allowed to pass out through a small hole in the box while the clouds themselves and the flies within them are unable to follow. Another triumph of the Landau theory was the prediction with Khalatnikov (74)of the temperature dependence of the normal fluid viscosity below about 1.8’ K. As the A-point is approached, however, this theory is inapplicable because of the increased concentration of excitations. I n summary, so long as the number of excitations (clouds) is small, they may be effectively treated as an ideal gas of energy packets. As the temperature increases, however, the concentration excitations also increase and the ideal gas approximation no longer holds any more than it does for real gases at high densities. VOL. 48, NO. 11
NOVEMBER 1956
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Figure 7. Entropy of liquid helium-3. The solid line i s calculated for ideal Fermi-Dirac fluid. Experimental points are from literature (6, 78, 20)
Figure 6. Rate o f flow through a superleak for liquid helium-3 and helium-4 ( 1 7 )
Perhaps the most significant difference between the Landau and the London theories has not been mentioned. The London theory depends entirely upon the statistics obeyed by the individual particles making up the liquid, whereas the role of statistics is completely neglected in the Landau theory. Since helium-4 atoms contain an even number of elementary particles, in the London formulation, the Bose-Einstein formulas must be used. I n fact this is the whole basis of the theory.
viewpoint, a two-fluid theory was impossible for helium-3 ; consequently, the most striking property of helium-4 namely, its superfluidity-was ruled out for helium-3. Landau's theory on the other hand, being entirely independent of the statistics, would predict similar properties for helium-3 such as superfluidity, a X-transition, and thermomechanical effects. The only thing preventing an immediate test of the theories was that, although appreciable quantities of helium-3 existed, it was mixed with ordinary helium in the ratio of 10,000,000 to 1 and the task of separating isotopes in such small initial concentrations was formidable. Fortunately, at about this time the Atomic Energy Commission began producing tritium, the radioactive isotope of hydrogen which decays with a half life of about 12 years to give helium-3 and a fiparticle. This instability of the nucleus, while something of a nuisance to the AEC, was a blessing to low-temperature physicists and chemists for, as the first stocks of tritium accumulated, it became possible to milk them for their helium-3 content from time to time and so
Helium-3 and the Theories of liquid Helium
It was recognized early that the properties of the rare isotope, helium-3, might provide essential information for determining which of these two theories held the key to the problem of liquid helium. According to the London theory, helium3 would obey Fermi-Dirac statistics and an ideal Fermi-Dirac gas should exhibit no discontinuities in any of its properties as the temperature approaches absolute zero. Thus, according to London's
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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procure appreciable quantities of pure helium-3. Twenty standard cubic centimeters of this material were liquefied at Los Alamos in the fall of 1948, and some of its properties investigated (22). Some crude specific heat measurements indicated no A-transition (20, 23). Then at the Argonne National Laboratory in 1949, Osborne, Weinstock, and Abraham (78) showed that pure helium-3 showed no superfluid properties down to 1.05' K. Their demonstration was spectacular. They simply condensed helium-3 on one side of a tiny leak and measured the amount passing through as the temperature was reduced. Their results are shown on Figure 6 where the flow rates through the leak for helium-3 and for helium-4 are contrasted. Superfluidity was subsequently shown by Daunt and Heer (3) to be nonexistent in helium-3 down to 0.25' K. Next came more precise specific heat measurements (6, 77, 20)) and with them, theoretical calculations for the specific heat and the entropy for an ideal Fermi-Dirac gas whose particles were assumed to possess a mass equal to that of a helium-3 atom and a density equal to that of the liquid. The calculated entropy curve is shown in Figure 7 as a solid line together with the points calculated from experimental specific heat data. Actually every property of helium-3 reported until the fall of 1353 seemed to substantiate the LondonTisza statistical model of liquid helium. I n fact London's idea that statistics played the key role in determining the liquid properties, yielded better quantitative agreement between theory and experiment for helium-3 than for helium-4. In October 1953, however, these views were dramatically challenged. I t has been known for some time that electrons circulating around nuclei give rise in certain atomic systems to the elementary magnets which in bulk matter
combine to yield the well known phenomena of para- and ferromagnetism. The spinning of the electrons themselves contributes a second and much smaller elementary magnet to the atomic system, and in certain cases the nuclei spin also, contributing still a third magnetic moment to the system. For helium-4, all the electronic moments cancel; the nucleus has a spin of zero, hence thisatom possesses no magnetic moment. Helium3, on the other hand, has a nuclear spin of ‘/2 and hence, although the electronic moments still cancel, a magnetic moment is associated with each atom. For an assembly of helium-3 atoms, the third law of thermodynamics requires that a t absolute zero the spins all be aligned-Le., perfectly ordered either parallel or antiparallel. I t turns out that the antiparallel arrangement is the one possessing the lowest energy. Now recognition of the existence of these spins and the way in which they must become progressively ordered as the temperature falls is a n essential part of the ideal Fermi-Dirac gas theory. Furthermore, to summarize briefly the thesis discussed thus far: If the Bose-Einstein ideal gas theory provides the foundation for the behavior of helium-4, then the behavior of liquid helium-3 must be equally well described by the FermiDirac ideal gas theory. It was necessary in the actual quantitative application of the Bose-Einstein theory to helium-4, to insert into the appropriate formulas the mass of the helium-4 atom and the density of the liquid. From the introduction of these parameters it was possible to obtain such data as To and the specific heat curve. For the FermiDirac theory, it is likewise necessary to introduce the mass of the helium-3 atom and the density of liquid helium-3. This was done in order to obtain the agreement mentioned earlier between the calculated and experimental values of the thermal properties of the liquid. I n October 1953, Fairbank and his associates a t Duke University (7) described measurements of the nuclear magnetic susceptibility of liquid helium3. Their results, subsequently extended a t 0 . 2 3 O K.. are shown in Figure 8 where the observed product of susceptibility and temperature is plotted against temperature. The horizontal line, 1, represents the expected behavior if helium-3 were an ideal Boltzman gas. For such an assembly of particles, the susceptibility itself should be inversely proportional to the absolute temperature according io Curie’s law. Curve 3 is the predicted behavior for an ideal Fermi-Dirac gas, the same theory which gave such good agreement in the entropy curve. The agreement between this curve and the experimental points is not spectacular. I n order to obtain agreement with experiment in terms of the ideal Fermi gas model, it was necessary to assume that
the molecules or, more accurately, the particles actually composing this fluid weighed about 11 times as much as a single helium-3 atom. On this assumption, curve 2 is obtained. This idea considered by itself, might have been acceptable because the elementayy liquid particles can be pictured as somehow consisting of groups of about 11 helium-3 atoms. But if this is done, calculations of the thermal data requiring the more reasonable assumption of single helium-3 atomic liquid particles were hopelessly out of line. Thus, it is impossible to describe even approximately both the thermal and the magnetic properties of liquid helium-3, both of which are fundamental, in terms of a consistent ideal Fermi-Dirac gas theory. There is certainly no basis a t present for accepting the implausible idea that the elementary liquid particle is 1 atom for one property and about 11 for the other.
Conclusion Whatever the role statistics play in the problem of liquid helium, it is clearly not so simple as was originally believed. The interpretation of any property of liquid helium-3 or helium-4, based on statistics alone, is probably invalid or a t least highly suspect. O n the other hand, it seems equally improbable that the statistics can be ignored unless one is prepared to revise a considerable fraction of the theoretical foundations of modern chemistry and physics. Thus, it appears that the ultimate solution of the liquid helium problem will be found in a synthesis resulting from the ideas of London and Tisza on the one hand and Landau on the other. Recently, several steps in this direction have been taken by a number of theoretical physicists such as Temperley (24), Kronig and Thellung (72), Ziman (28), and finally with a considerable degree of success for liquid helium-4, by Feynman (8). The path these theoreticians are following is rather rugged and tortuous in terms of mathematical complexity and many unsolved problems remain, but, since interest and effort directed toward solution of this problem are so prodigious and in such competent hands, it seems certain that a break-through may not be far away. Such a solution of the liquid helium problem may lead the way toward a better understanding of the liquid state and eventually realize the early hopes of the Leyden workers for a model liquid,
literature Cited (1) Andronikashvilli, E., J.Phys. ( U . S. S. R.) 10, 201 (1946); J. Exptl. Theoret. Phys. ( U . S. S. R . ) 18,424 (1948). (2) Dash, J. G., Taylor, R. D., Phys. Rev., to be published.
Daunt, J. G., Heer, C. V., Phys. Rev. 79, 46 (1950). Daunt, J. G., Smith, R. S., Revs. Mod. Phys. 26, 172 (1954). De Boer, J., Physica 14,139 (1948); Prigogine, I., Philippot, J., Zbid., 18, 729 (1952); Prigogine, I., Bingen, R., Bellemans, A,, Zbid., 20, 633 (1954); Chester, G. V., Phys. Rev. 100,446 (1955). De Vries, G., Daunt, J. G., Zbid, 92, 1572 (1953); 93, 631 (1954). Fairbank. W. M.. Ard. W. B.. Dehmelt, H. G., Gordy, ‘W., Williams, S. R., Zbid., 92, 208 (1953); Fairbank, W. M., Ard, W. B., Walters, G. K., Zbid., 95, 566 (1954). Feynman, R. P., Zbid., 91, 1291,1301 (1953); 94, 262 (1954); Feynman, R. P., Cohen, M., Zbid., 102, 1189 (1956). (9) Hdllis Hallett, A. C., “Progress in Low Temperature Physics” (C. J. Gorter, editor), Chap. IV, North Holland Publishing Go., Amsterdam, 1955. (IO) Keesom, W. H., “Helium,” Elsevier Press. Amsterdam. 1942. Reference’for all work on liquid helium prior to 1942. Kerr, E. C., J. Chem. Phys., to be published; Atkins, K. R., Edwards, M. H., Phys. Rev. 97, 1429 (1955). Kronig, R., Thellung, A., Phyrica 18, 749 (1952). (13) Landau, L.,‘J.Phys. (U.S.S.R.) 5, 71 11941). Landau, L., Khalatnikov, I., J . Exptl. Theoret. Phys. (U.S.S.R.) 19, 709 (1949). London. F.. “Suaerfluids.” vol. 11. Wiley, Ndw Yirk, 1954.‘ London, H., Nature 142, 612 (1938); Proc. Roy. Sac. (London), A 171, 484 (1939). Osborne, D. W., Abraham, B., Weir?stock, B., Phys. Rev. 94, 202 (1 954). Osborne, D. W., Weinstock, B., Abraham, B., Zbid., 85, 988 (1949). Peshkov, V., J. Phys. (U.S.S.R.) 8, 131 (1944); Zbid., 10, 389 (1946); J. Exptl. Theoret. Phys. (U.S.S.R.) 18, 857, 951 (1948); 19, 270 (1949). (20) Roberts; T. R., Sydoriak, S. G., Phys. .Rev. 93, 1418 (1954); 98, 1672 (1955). (21) Swenson, C. A., Zbid., 79, 626 (1950). (22) Sydoriak, S. G., Grilly, E. R., Hammel. E. F.. Zbid.. 75. 303 (1949); Grilly, E. R.,‘ Hammel, E. F., Sydoriak, S. G., Zbid., 1103 (1949). (23) Sydoriak, S. G.,Hammel, E. F., Proc. Intern. Conf. on Physics of Very Low Temp., p. 42, MIT, 1949. Temperle , H. N. V., Proc. Phys. Sot. (London! A 65, 490, 619 (1952); 66,995 (1953). Tisza, L., “Phase Transformations in Solids” (R, Smoluchowski, ed. ) Chap. I, Wiley, New York, 1951. Tisza. L.. Combt. rend. 207. 1035 (1$38);‘Landau, L., J. Phys. ( U S . S.R.) 5 , 77 (1941). Tisza, L., Phys. Rev. 72, 838 (1947). Ziman, J. M., Proc. Roy. Soc. (London) A 219, 257 (1953). \ - -
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RECEIVED for review January 13, 1956 ACCEPTED September 6,1956 Work performed under the auspices of the Atomic Energy Commission. Discussion based on an ACS California Section Award address, Berkeley, Calif., October 10,1955. VOL. 48, NO. 1 1
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