The l-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy

In a recent paper¹ Fröhlich has tried to interpret the l-phenomenon of liquid helium as an order-disorder transition between n holes and n helium atoms in a body-centred cubic lattice of 2n places. He remarks that a body-centred cubic lattice may be considered as consisting of two shifted diamond lattices, and he assumes that below the l-point the helium atoms prefer the places of one of the two diamond lattices. The transition is treated on the lines of the Bragg-Williams-Bethe theory as a phase transition of second order in close analogy to the transition observed with b-brass. Jones and Allen in a recent communication to Nature² also referred to this idea. In both these papers, use is made of the fact, established by the present author, that with the absorbed abnormally great molecular volume of liquid helium (caused by the zero motion³) the diamond-configuration has the lowest potential energy among all regular lattice structures⁴.

In this note, I should like first to show that the mechanism proposed by Fröhlich cannot be maintained and then to direct attention to an entirely different interpretation of this strange phenomenon.

(1) According to Fröhlich, a diamond lattice of He atoms, when partly formed, should offer, to any other He atom, a preference for being attached at those points which belong to the same diamond lattice, that is, the binding energy at a diamond point should be greater than anywhere else. It is, however, easy to see that just those points, which according to Fröhlich should become less favoured for low temperatures, have an appreciably greater binding energy. It is true these holes have four nearest neighbours at exactly the same distance (3.08 A.), as the lattice points of the diamond lattice have, but in addition they possess six second neighbours at the distance of 3.57 A. which the diamond lattice points do not possess, and these second neighbours contribute considerably to the binding energy just at the hole-places (about 50 per cent to the potential energy). Therefore, actually no such co-operative phenomenon will appear as supposed by Fröhlich. The atoms would rather frequent the holes as much as the proper diamond lattice points, and this would signify that we have a body-centred lattice of 2n places for n atoms, every place being occupied with the probability ½ only—even at the absolute zero. In this configuration every atom has on the average four nearest neighbours at a distance of 3.08 A., as in the diamond configuration, but in addition there are here on the average three second neighbours at the distance of 3.57 A. In the diamond lattice there are twelve second neighbours but at a distance of 5.04 A., where there is almost no Van der Waals field. It might be mentioned, by the way, that a face-centred lattice of 2n places for n atoms (on the average 6 first neighbours at a distance of 3.17 A.) has been found to have a still little lower energy than the configuration just discussed of the co-ordination number 4.

Complete numerical details cannot be given here; in any event it can be shown by such energetic discussions that a static spatial model of liquid He II of whatever regular configuration is certainly not possible. This has been previously suggested in consideration of the great zero point amplitude calculated for He⁴. The determination of the most favourable co-ordination numbers of the first and second neighbours, however, maintains a good physical meaning: it may be considered as a rough Hartree calculus which yields the self-consistent field and the corresponding probability distribution of the atoms belonging to the minimum of energy.

(2) It seems, therefore, reasonable to imagine a model in which each He atom moves in a self-consistent periodic field formed by the other atoms. The different states of the atoms may be described by eigen functions of a similar type to the electronic eigen functions which appear in Bloch's theory of metals; and, as in Bloch's theory, the energy of the lowest states will roughly be represented by a quadratic function of the wave number K,

the effective mass m* being of the order of magnitude of the mass of the atoms. But in the present case we are obliged to apply Bose-Einstein statistics instead of Fermi statistics.

(3) In his well-known papers, Einstein has already discussed a peculiar condensation phenomenon of the 'Bose-Einstein' gas; but in the course of time the degeneracy of the Bose-Einstein gas has rather got the reputation of having only a purely imaginary existence. Thus it is perhaps not generally known that this condensation phenomenon actually represents a discontinuity of the derivative of the specific heat (phase transition of third order). In the accompanying figure the specific heat (C_v) of an ideal Bose-Einstein gas is represented as a function of T/T₀ where

With m* = the mass of a He atom and with the mol. volume N_l/n = 27.6 cm.³ one obtains T₀ = 3.09°. For T T₀ the specific heat is given by

and for T T₀ by

The entropy at the transition point T₀ amounts to 1.28 R independently of T₀.

Figure 1 Specific heat of an ideal Bose-Einstein gas. | high-resolution version |

(4) Though actually the l-point of helium resembles rather a phase transition of second order, it seems difficult not to imagine a connexion with the condensation phenomenon of the Bose-Einstein statistics. The experimental values of the temperature of the l-point (2.19°) and of its entropy (~0.8 R) seem to be in favour of this conception. On the other hand, it is obvious that a model which is so far away from reality that it simplifies liquid helium to an ideal gas, cannot, for high temperatures, yield but the value C_v = 3/2 R, and also for low temperatures the ideal Bose-Einstein gas must, of course, give too great a specific heat, since it does not account for the gradual 'freezing in' of the Debye frequencies.

According to our conception the quantum states of liquid helium would have to correspond, so to speak, to both the states of the electrons and to the Debye vibrational states of the lattice in the theory of metals. It would, of course, be necessary to incorporate this feature into the theory before it can be expected to furnish quantitative insight into the properties of liquid helium.

The conception here proposed might also throw a light on the peculiar transport phenomena observed with He II (enormous conductivity of heat⁵, extremely small viscosity⁶ and also the strange fountain phenomenon recently discovered by Allen and Jones²).

A detailed discussion of these questions will be published in the Journal de Physique.

F. LONDON.

Institut Henri Poincaré, Paris.
March 5.

1. Fröhlich, H., Physica, 4, 639 (1937).
2. Allen, J. F., and Jones, H., Nature, 141, 243 (1938).
3. Simon, F., Nature, 133, 529 (1934).
4. London, F., Proc. Roy. Soc., A, 153, 576 (1936).
5. Rollin, Physica, 2, 557 (1935); Keesom, W. H., and Keesom, H. P., Physica, 3, 359 (1936); Allen, J. F., Peierls, R., and Zaki Uddin, M., Nature, 140, 62 (1937).
6. Burton, E. F., Nature, 135, 265 (1935); Kapitza, P., Nature, 141, 74 (1938); Allen, J. F. and Misener, A. D., Nature, 141, 75 (1938).

Transport Phenomena in Helium II

F. London¹ has recently proposed a new conception of helium II, according to which this liquid can be regarded as a degenerate Bose-Einstein gas, that is, as a system in which one fraction of the substance—say, n atoms per cm.³—is distributed over the excited states in a way determined by the temperature, while the rest—n₀-n atoms per cm.³—is 'condensed' in the lowest energy level. If T₀ denotes the temperature of degeneracy, the ratio n/n₀ is given by

For an ideal Bose-Einstein gas, according to London, s = 3/2, but for the real fluid one should rather insert s = 5 in order to fit Keesom's specific heat measurements.

The calculation of viscosity of a degenerate Bose-Einstein gas is in progress. A preliminary estimation shows that the atoms belonging to the lowest energy state do not take part in the dissipation of momentum. Thus, the viscosity of the system is entirely due to the atoms in excited states. This simple description allows us to account in a qualitative way for the well-known transport phenomena in helium II, though in the more exact treatment certain modifications have to be introduced, which are to be discussed in a detailed publication.

(1) Measurements of the damping of an oscillating cylinder will show the effect of atoms in excited states only. To a first approximation the viscosity, measured in this way, should be the same as the viscosity of helium gas at the same temperature. The value of the viscosity coefficient of helium gas at T ~ T₀ is m ~ 10^-5 C.G.S. units⁶, while measurements in helium II just below the l-point² give m = 3 × 10^-5. Parallel measurements of the viscosity of helium II and helium gas in the corresponding range of temperature would provide us with an indication as to what extent the simple picture adopted here has to be refined.

(2) It is, however, not the viscosity coefficient m which determines the flow velocity of helium II moving under the influence of a pressure gradient through a capillary. In this case, the fraction of substance consisting of atoms in the lowest energy state will perform—like a 'superfluid' liquid of viscosity m ~ 0—some sort of turbulent motion the flow velocity of which does not depend on the bore of the capillary. In addition, the atoms in excited states, behaving like a gas of pressure ~ nkT, will diffuse in a similar way to, say, molecules in a solution under the influence of a gradient of osmotic pressure. Thus, the total flow represents a rather complex combination of both these effects. It will largely depend on the ratio (n₀-n)/n and, therefore, on T, in agreement with experiment³. According to this interpretation, a temperature gradient should arise during the flow of helium II through a thin capillary.

(3) The so-called fountain phenomenon⁴ of helium II is an inverse process to (2). If one maintains a temperature difference between the ends of a capillary, a gradient of density of excited atoms, n, and, thus, of pressure is produced. In consequence, (a) the excited atoms will diffuse towards the colder end, and (b) the super-fluid fraction of the liquid moves in the opposite direction. In the case of a wide tube, these currents must be equal and no resulting flow will be observed. If, however, the capillary is sufficiently narrow, the rate of the process (a) becomes reduced and the temperature gradient causes a surplus convection current opposite to heat flow. This picture can account for the great values of the heating current required to maintain a temperature difference at the ends of the capillary⁵. Simultaneous measurements of the heating current and the total convection of substance could provide us with information about the relative magnitude of the processes (a) and (b).

A detailed discussion of the problem will be given in the Journal de Physique. I am greatly indebted to Dr. F. London for the opportunity of seeing his paper before publication.

L. Tisza.

Laboratoire de Physique Expérimentale, Collège de France, Paris.
April 16.

1. London, F., Nature, 141, 643 (1938).
2. Burton, E. F., Nature, 135, 265 (1935). Wilhelm, Misener and Clark, Proc. Roy. Soc., A, 151, 342 (1935).
3. Kapitza, P., Nature, 141, 74 (1938). Allen, J. F., and Misener, A. D., Nature, 141, 75 (1938).
4. Allen, J. F., and Jones, H., Nature, 141, 243 (1938).
5. Rollin, Physica, 2, 557 (1935). Keesom, and Miss Keesom, Physica, 3, 359 (1936). Allen, J. F., Peierls, R., and Uddin, M. Z., Nature, 140, 62 (1937). Keesom, W. H., Keesom, Miss A. P., and Saris, B. F., Physica, 5, 281 (1938).
6. Itterbeck, A. van, and Keesom, W. H., Physica, 5, 257 (1938).

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