CALORIMETRIC STUDIES AT VERY LOW TEMPERATURES*

W. Keesom, Leiden

The author gives a brief survey of recent work at the Leiden laboratory devoted to thermal measurements at the temperatures of liquid helium. He gives a description of the apparatus and of several types of resistance thermometers.

The specific heat of certain metals (for example, silver and zinc) at temperatures of 4–5° K exhibits a certain anomaly—it is larger than the value it ought to have if Debye’s law were valid. Special investigations have confirmed the reality of this anomaly. The hypothesis is advanced that at such low temperatures the heat capacity of free or quasi-free electrons begins to play a noticeable role.

The specific heat of superconducting metals undergoes a discontinuity at the point of transition of the metal into the superconducting state. Rutgers’ equation is tested, relating this jump in heat capacity to the change of the magnetic field with temperature. Since this equation is derived under the assumption of the reversibility of the process of transition into the superconducting state, we arrive at the conclusion that the transition into the superconducting state is a reversible process.

The author, in collaboration with Kok, measured the latent heat associated with the transition from the superconducting state to the normal state, for the case when this transition takes place in a magnetic field; results were obtained that confirm the hypothesis of the reversibility of the transition into the superconducting state. The discrepancies found in the experiment are explained by the fact that not the entire specimen passed into the superconducting state.

The author then gives a summary of work on the equation of state of liquid helium. The summary is given in the form of a diagram (entropy—temperature). The diagram clearly shows that the transition of liquid helium I into liquid helium II is a transition of the second kind, i.e. a transition occurring without expenditure of heat. The density passes continuously from state I into state II; whereas the heat capacity, compressibility, thermal coefficients of expansion, and pressure undergo jumps. In conclusion the author discusses the method for establishing a temperature scale below 0.9° K, consisting of a combination of adiabatic demagnetization and calorimetric observations.

1. Introduction

In 1912, studies of the thermal properties of bodies at low temperatures began in the Leiden cryogenic laboratory. The first measurements concerned the heats of evaporation of oxygen and hydrogen; these were followed by measurements of the heat capacities of a number of metals.

* Journ. de Physique, V, No. 7, 1934. Report read before the French Physical Society on April 10, 1934.

Thanks to the works of Behn¹, Dewar² and, chiefly, Nernst³ and his school, it was already known that the heat capacity of a body decreases appreciably as the temperature is lowered. Einstein⁴, proceeding from Planck’s ideas on the quantum character of energy, gave an explanation of this experimental fact; somewhat later Debye⁵ gave the theory of the heat capacity of homogeneous solids that form which even now remains the basis for any theory of thermal motion in crystal lattices.

It could be foreseen that thermal measurements at low temperatures would be able to reveal a number of phenomena and regularities important for understanding the properties of the crystal lattices of solids, and also the structure of liquids, since such liquids still exist at such low temperatures.

In the very last few years we have come to the conclusion that such investigations may clarify for us what takes place inside atoms, in particular in processes connected with transitions from one energy level to another.

I should like to set forth here, in very brief form, some results obtained in this field in the Leiden laboratory.

2. Method

According to the methods developed by Nernst and Eucken⁶, to a definite mass of the substance under investigation, thermally insulated in the best possible way, a known portion of energy is supplied, usually by means of an electric current; the rise in temperature is then measured. In order to exclude changes in resistance arising from deformation of the vessel during sharp changes of temperature (cooling), the resistance thermometers were suspended in an atmosphere of helium, and not soldered into the vessel. Experience nevertheless shows that the thermometer immediately assumes the temperature of the mass under investigation. It goes without saying that the heat capacity of the furnace was measured separately.

For thermal insulation of the vessel it was suspended in a vacuum. The vacuum was produced by a powerful pump. The vessel was cooled by a small quantity of gaseous helium; before the start of the calorimetric measurements this helium was pumped out by the same pump. In recent years we have developed a special technique which freed us from the necessity of introducing helium. This is a very important circumstance, since the introduction of helium gave rise to certain apprehensions, to which I shall return later.

In these measurements it is very difficult to ensure satisfactory thermal insulation, since near absolute zero the heat capacities of bodies are very small. Kok⁷ and the author succeeded in developing a method which makes it possible to calculate heat capacities from the readings of the thermometer even in the case when the thermal insulation was incomplete. I shall not now dwell on the details of this method.

Let us briefly discuss the resistance thermometers which we used. The platinum or gold thermometers usually employed cannot be used at liquid-helium temperatures, since their resistances become practically constant. Nor can a lead thermometer be used: at such low temperatures lead becomes superconducting. Constantan, on the other hand, is a very suitable material, although at temperatures below \(7^\circ\) K it is less sensitive than phosphor bronze. The latter, however, cannot be used for higher temperatures, since its resistance curve, beginning at \(7^\circ\) K, becomes horizontal. In Fig. 1 the resistance curve is shown for a sample of such bronze, beginning at \(0.75^\circ\) K.

Fig. 1. Resistance of phosphor bronze

A phosphor-bronze thermometer, however, has its drawbacks: the drop in the resistance of this alloy is, in all probability, caused by the presence in it of small amounts of lead. A layer or thin threads of lead, beginning at \(7.2^\circ\) K, become superconducting and reduce the resistance. This means that the resistance will depend on the density of the measuring current and, what is still more important, on the applied magnetic field. The latter circumstance presents great inconveniences in cases where residual circulating currents may arise. Indeed, the magnetic-field strength in which the thermometer is located is then unknown. It would therefore be highly desirable to have a thermometer whose readings do not depend on the magnetic field. Quite recently we found that the resistance of cerium still changes noticeably at liquid-helium temperatures and depends almost neither on the current density nor on the magnetic field. In Fig. 2 the resistance curve of cerium as a function of temperature is shown. Near \(1.15^\circ\) K there is a certain curvature, probably due to a small amount of aluminum present in the cerium. Unfortunately, cerium is very difficult to work and oxidizes easily. It is very difficult to make from it a thin wire with sufficiently high resistance. Moreover, it is difficult to attach to cerium the leads needed for measuring the resistance. However, all these difficulties are experimen-

of an elementary kind, which, it is to be hoped, can be eliminated.

Figure 3 shows the dependence of resistance on temperature for magnesium. As we see, for liquid-helium temperatures the dependence is very sharp. It should be noted that the resistance increases markedly as the temperature is lowered. Apparently, magnesium can serve as a very sensitive thermometer. We have not yet had time to check whether its resistance depends on the density of the current flowing through it and on the magnetic field.

Fig. 2. Resistance of cerium

3. Specific Heat of Metals

Generally speaking, metals follow Debye’s formula very well; for low temperatures it assumes the following form

\[ c = \beta \frac{T^3}{\Theta^3}, \]

where \(\beta\) is a constant, \(\Theta\) is a certain temperature called the Debye temperature, and \(c\) is the atomic heat capacity. We found several slight deviations from this formula, which are not worth dwelling on here. At very low temperatures (near \(4—5^\circ\mathrm{K}\)) very interesting deviations were found. Figure 4 gives the values of \(T \sqrt[3]{c}\) for various metals as a function of temperature. If Debye’s formula were correct, then straight lines should have been obtained in the diagram. The curves

Fig. 3. Resistance of magnesium

for some metals show a very noticeable bend in the region of very low temperatures. In Fig. 5 a similar curve for silver is given on a larger scale.

These data are quite unexpected—after all, it was to be expected that precisely at low temperatures Debye’s theory should agree best with experiment. It was therefore natural to ask whether the results obtained might be the consequence of some experimental errors.

Fig. 4. Value of \(\Theta = T \sqrt[3]{\frac{c}{c}}\) as a function of temperature

One can vouch that in these experiments there is no gross error. Only one circumstance could have aroused some doubt as to the validity of the data obtained—namely, the presence of gaseous helium introduced during the cooling of the sample before the beginning of the calorimetric measurements. It might have been supposed that the metal adsorbed a layer of helium, and that this absorbed helium, evaporating from the surface of the metal, absorbed some amount of heat, causing an apparent increase in heat capacity. I shall take the liberty of referring to a paper\(^8\), where arguments are given that refute this possibility. Nevertheless, we considered it necessary to investigate this question specially.

Fig. 5. Value of \(\Theta = T \sqrt[3]{\frac{c}{c}}\) for silver

First, we carried out a series of experiments with a zinc sample having a smaller surface area, i.e., a greater mass. It was found that in this case, indeed, the heat capacity per unit mass was somewhat greater. Consequently, the heat of evaporation is measurable, but it proved to be considerably smaller than that which would have been required to explain the effect.

Second, we developed a method for cooling the sample that does not require the presence of helium. The apparatus is shown in Fig. 6. The specimen under investigation is suspended in vacuum on thread \(A\). It can be lowered until it comes into contact with the silver polished cone \(B\), which forms part of the jacket. The pressure which the specimen exerts on the cone by virtue of its own weight is already sufficient to remove the heat required in cooling the specimen. To be sure, with this method 5 hours are required for the specimen to reach the temperature of liquid hydrogen. Starting from this temperature, however, owing to the low values of the heat capacity (at low temperatures), cooling proceeds more rapidly (15 min.).

Fig. 6. Apparatus for cooling by contact \(B\)

All the data obtained under such conditions for silver fully confirmed our previous experiments and compel us to regard the deviations from Debye’s law at very low temperatures as proven.

The curves for \(\Theta\) as functions of \(T\), shown in Fig. 4, tend to merge into a single point at very low temperatures. If this is indeed true, it means that the atomic heat capacities approach a certain value independent of the nature of the metal, i.e., not determined by the atomic weight and intermolecular forces, as is the case for heat capacities at higher temperatures.

Let us note that the explanation of deviations from Debye’s law given by Simon\(^{9}\) cannot be used in this case (Simon calculated quantum transitions from one energy level to another, which explains certain deviations from Debye’s law), since these deviations are considerably smaller than those for which Simon’s corrections are valid. It would have to be assumed that only a very small fraction of atoms undergoes such transitions, but this is not very plausible.

An attractive hypothesis has been proposed: that at such low temperatures the heat capacity of free electrons begins to play a noticeable role. The task of theorists now is to develop such a theory of metals that it would be possible to test this supposition quantitatively. From the experimental point of view it would be very interesting to measure the heat capacities of poor heat conductors at such low temperatures.

4. Heat Capacities of Superconducting Metals

The heat capacity of superconductors shows that electrical conductivity somehow also affects the heat capacity of a metal. For tin, for example, Fan-den-Ende \(^{10}\) and the author found that the heat capacity undergoes a very sharp jump, perhaps even a discontinuity, at the point of transition to the superconducting state, i.e., the heat capacity for temperatures just above the transition point is greater than for temperatures just below it. Together with Kok \(^{11}\), the author succeeded in confirming and refining these results by measuring heat capacities with

Fig. 7. Atomic heat capacity of tin

heating of the specimen of the order of \(0.01^\circ\). The data obtained are shown in Fig. 7. The atomic heat capacity of tin experiences a jump from 0.0078 to 0.0054 on passing through the transition point. Very recently, investigations of the heat capacity of thallium carried out by the author together with Kok \(^{12}\) showed that at \(2.36^\circ\mathrm{K}\), at the point of transition of thallium to the superconducting state, the atomic heat capacity of thallium falls from 0.0132 to 0.0118.

Rutgers \(^{13}\) derived an equation relating the fall in heat capacity to the change in the transition temperature into the superconducting state under the influence of a magnetic field. Table 1 gives theoretical and experimental data for this jump; we see that experiment confirms the theoretical data very well.

Rutgers’ equation was derived thermodynamically for the transition from the superconducting state to the normal, i.e., non-superconducting, state, and conversely, under the condition of reversibility of the process. To

TABLE 1

Rutgers equation
\[ \left(\frac{dH}{dT}\right)^2=-\frac{4\pi \Delta c}{T V} \]

until very recently, however, no one regarded such transitions as reversible. The experiments of Meissner and Ochsenfeld \(^{14}\) compelled a change in this orthodox view.

The residual currents freely existing in a superconducting metal, as is known, die out when the metal is heated. The cited experiments confirmed the validity of the assumptions of Gorter and Casimir \(^{15}\) concerning the possibility of residual circulating currents arising in a metal placed in a magnetic field when the metal is cooled below the transition point. Such an interpretation of the question reduces the difficulties associated with the reversibility of the process of transition into the superconducting state. We shall not assert that it finally removes these difficulties.

In any case, if the indicated phenomenon exists and if the transition into the superconducting state took place in a magnetic field, then a certain absorption of heat will be observed. If, however, the sign of the process is reversed, then a certain quantity of heat will be liberated. We shall denote this quantity of heat by \(r\) and shall call it the heat of transition. Let us consider the change in entropy and see whether the equality

\[ \Delta S=\frac{r}{T}. \]

holds.

If this equality is valid, we shall say that the transition was indeed reversible.

With this method of reasoning, the validity of the Rutgers equation shows that the transition process is reversible not only in the case of a magnetic field equal to zero, but also in the case of weak magnetic fields. More precisely,
\[ \frac{d^2\sigma}{dT^2}=0 \]
for \(H=0\), if by \(\sigma\) we denote the difference \(\Delta S-\frac{r}{T}\).

Kok \(^{16}\) and the author investigated the behavior of thallium in a constant magnetic field. The results of a series of such observations are given in Fig. 8. These measurements made it possible for us to calculate, for a certain temperature interval, the difference in atomic heat capacities for the case when the metal is in the superconducting state, and for the case when the presence of a magnetic field prevents the occurrence of superconductivity. From thermodynamics it is known that

the difference of the free energies in this temperature interval, taken for the two states of the metal, is equal to the energy produced by the magnetic field in the body, under the indispensable condition that the transition is reversible. This gives us the possibility of verifying the reversibility of the process for finite magnetic fields. The experiment gave an affirmative answer to this question. The measurements were made in fields reaching 60 gauss.

We have already mentioned the latent heat of the transition from the superconducting state to the normal one. This question was the subject of discussion at the Solvay Congress[^17] in 1924. Taking into account the reversibility of the transition, one can derive relations connecting the magnitude of the latent heat with the temperature slope of the critical-magnetic-field curve. From this relation it follows that the heat of transition must

Fig. 8. Apparent atomic heat capacity of thallium in a magnetic field of 33.6 gauss

be equal to zero if the transition takes place in a field equal to zero. This consequence was checked very carefully in our work with tin and was then confirmed by the results with thallium; for thallium, however, we succeeded in showing the presence of this heat for the transition in a magnetic field. We measured the magnitudes of the heat of transition in two different magnetic fields.

If these experimental data are compared with the thermodynamic ones, a certain discrepancy can be found. The possible cause of this discrepancy is as follows: applying the principles of thermodynamics to the given case, we assume that, on passing through the value of the critical magnetic field at decreasing temperature, the whole metal passes into the superconducting state as a homogeneous body. But we have grounds to suppose that in our experiments the transition occurred at once only in some part of the specimen under investigation. I shall not go into further details.

We see that the thermal investigation of the state of superconductivity, which is only just beginning, seems to promise to yield very much for understanding the processes occurring in super-

conductors. Let us note that we have carried out a number of calorimetric measurements on superconducting thallium in variable magnetic fields, which produced measurable residual currents in the metal. I hope that we shall soon be able to publish the results of these investigations, in which the disappearance of circulating residual currents is connected with thermal effects.

5. Equation of the Thermal State of Liquid Helium

In 1924 Kamerlingh Onnes and Boks[^18] discovered an interesting anomaly in the density curve of liquid helium as a function of temperature near the point \(2.19^\circ\mathrm{K}\) (Fig. 9). In 1927 Wolfke[^19] and the author

Fig. 9. Density of liquid helium under the pressure of its vapors saturating the space (K. Onnes and Boks)

investigated a number of properties of helium near this temperature and came to the conclusion that two separate states of liquid helium should be distinguished: state I—above the temperature \(2.19^\circ\), and state II—below \(T = 2.19^\circ\mathrm{K}\). New light was shed on these phenomena in 1932 by the work of the author and Clusius[^20] on the heat capacity of liquid helium. The results are shown in the curve of Fig. 10. One clearly sees the rapid increase of the heat capacity and the subsequent sharp break at the temperature \(2.19^\circ\). The curve resembles the Greek letter \(\lambda\); therefore this point, at Ehrenfest’s suggestion, is called the lambda point.

Figure 11 gives the data of the latest work by the author and Mrs. Keesom,[^21] in which the heat inputs were of the order of only \(0.01^\circ\), so that it was possible to approach the \(\lambda\)-point more closely. The principal conclusions from these works are as follows.

The transition of liquid helium from state II to state I at the \(\lambda\)-point occurs without any release of latent heat, but is accompanied by a jump in heat capacity from a value of 3.0 to a value of 1.1,

Fig. 10. Heat capacity of liquid helium under the pressure of its own vapors

occurring, judging from the accuracy of the experiment (of the order of thousandths of a degree), instantaneously.

A series of measurements was carried out under increased pressure, of the order of 19 atm*. The same jump in heat capacity was found, but at another temperature, which is in full agreement with the data obtained by Clusius²² and by the author, who studied the displacement of the \(\lambda\)-point under the influence of pressure. In connection with these results, Mlle Keesom²³ and the author undertook a measurement of the density of helium as a function of temperature and pressure. In Fig. 12 the results of these measurements are given in the form of a \(p,T\)-diagram. One may note the curve of pressure for the saturated vapors, the solidification curve, and

* This work was not printed, since an experimental error was found—the vessel with liquid helium had not been hermetically sealed.

Fig. 11. Jump in the heat capacity of liquid helium

the $\lambda$-curve—the locus of $\lambda$-points separating the region of helium II from helium I. The remaining lines are isochores, i.e., curves of equal density. I shall not dwell on other interesting data that can be obtained from this diagram. Let us note only that the isochores intersect the $\lambda$-curve at a certain angle. This means that the thermal pressure coefficient has a discontinuity at this point. In Fig. 13 isobars are shown in the coordinates $\rho, T$ (density—temperature). We see that the expansion coefficient also undergoes a jump. Consequently, the compressibility coefficient must have the same kind of jump.

Fig. 12. Isochores of liquid helium on the $p,T$ diagram

The $\lambda$-curve has the form of a transition curve from one phase to another on the phase diagram. However, there are also essential differences. For an ordinary transition, the entire course of the curve is associated with latent heat. As we have already noted, in the case of the $\lambda$-curve this is not so—here the latent heat of transition is equal to zero. On the other hand, this transition is associated with jumps in the heat capacity, the coefficients of expansion, pressure, and compressibility. Therefore we have every reason to regard this transition as a transition from one phase to another. In order to distinguish these transitions from ordinary ones, Ehrenfest^24 proposed the following terminology: transitions from one phase

into another, in which the first derivatives of the thermodynamic potential, i.e. the entropy and the volume, undergo jumps, are called transitions of the 1st kind; transitions in which the first derivatives of the potential remain continuous, whereas the second derivatives are discontinuous, are called transitions of the 2nd kind. According to this terminology, the transition of liquid helium into the gaseous or solid phases is a transition of the 1st kind, while the transition of liquid helium II into liquid helium I is a transition of the 2nd kind.

Transitions of the 2nd kind are already known in several cases; thus, for example, Clusius and Perlick^25 studied in great detail the transition of the 2nd kind in methane.

For transitions of the 1st kind, thermodynamics gives a relation between the slope of the transition curve in the coordinates \(p, T\), on the one hand, and the jumps in entropy and volume, on the other; this is the Clapeyron–Clausius equation. In an analogous way, for transitions of the 2nd kind one can obtain two equations that relate the slope of the curve to the corresponding jumps (Fig. 14).

Fig. 13. Isobars of liquid helium in the \(p,T\)-diagram

Let us give one of them:

\[ \left(\frac{\partial S}{\partial p}\right)_{\mathrm{II}} dp+ \left(\frac{\partial S}{\partial T}\right)_{\mathrm{II}} dT = \left(\frac{\partial S}{\partial p}\right)_{\mathrm{I}} dp+ \left(\frac{\partial S}{\partial T}\right)_{\mathrm{I}} dT, \]

whence

\[ \left(\frac{dp}{dT}\right)_{\lambda} = \frac{\Delta\left(\frac{\partial S}{\partial T}\right)_p} {\Delta\left(\frac{\partial V}{\partial T}\right)_p} = \frac{\Delta C_p}{T V \Delta \alpha}. \]

This equation was tested for helium under the pressure of its saturated vapor; from the slope of the \(\lambda\)-curve and the jump in heat capacity one can calculate the jump in the coefficient of expansion; Fig. 15 shows that the calculated value agrees with the data obtained from the measurements of Kamerlingh Onnes and Boks.

Finally, the curve in Fig. 16 summarizes, on an entropy diagram, all the experimental data for liquid helium. This diagram differs from the usual entropy diagrams in that it is, as it were, folded along the \(\lambda\)-curve, i.e. in some region the two halves of the diagram are, as it were, superimposed one on the other. I shall not derive from

diagrams all those peculiarities of liquid helium that were discussed above.

The question naturally arises—can one somehow explain what occurs at a transition of the second kind? For methane and certain other substances Pauling^27 assumes that the free rotation of molecules or parts of molecules is replaced, as the temperature is lowered, by rotational oscillations. For helium this hypothesis is unsuitable. I shall allow myself to propose the following explanation^28: liquid helium,

Fig. 14. Relation of the slope of the λ-curve to jumps in the heat capacity and in the coefficient of thermal expansion

Fig. 15. Verification of the relations between the slope of the λ-curve and jumps in the heat capacity and in the coefficient of expansion, from measurements by K. Onnes and Boks of the density of helium

Fig. 16. Entropy diagram of liquid helium

crossing the λ-curve at decreasing temperature, passes into a quasi-crystalline state; the helium atoms are arranged more or less regularly in a lattice, but the lattice does not remain solid,

as in crystals, but rather is fluid. The lattice will remain regular only in individual volume elements, where the number of atoms is small and constantly changes.

I must immediately admit that I cannot offer any proof of the validity of this hypothesis, nor can I derive from it any predictions that could subsequently be tested experimentally. Here is yet another problem that we leave to the theorists.

6. Temperature scale for temperatures below 0.9°K

In conclusion I shall make one more observation. Down to a temperature of 0.9°K the temperature scale was established with the aid of a helium thermometer. At lower temperatures helium can practically no longer be used, since it is adsorbed on the surface of the thermometer vessel. This adsorption was specially studied by Schmidt^29 and by the author in order to determine how it might be eliminated—it was supposed, for example, that the thermometer could be surrounded by a layer of noble gas. However, attempts of this kind did not yield anything good.

Fig. 17. Establishment of the temperature scale for ultralow temperatures

The question naturally arises—how, then, can one establish a temperature scale for the region of very low temperatures, where the helium thermometer cannot be used?

At the present time we are studying a thermometer based on the laws of thermomolecular phenomena. In addition, the temperature scale can also be established by another route.

De Haas, Wiersma, and Kramers^30, and also Giauque^31, showed that adiabatic demagnetization of certain paramagnetic salts makes it possible to obtain ultralow temperatures; this, however, had already been predicted in 1926 by Debye and, independently of him, by Giauque in 1927.

Suppose that we carry out adiabatic demagnetizations once under the initial conditions \(T\) and \(H\), and a second time at \(T\) and \(H + \Delta H\) (Fig. 17). With the aid of a suitable thermoscope we record the readings obtained, corresponding to the temperatures \(T'\) and \(T' + \Delta T'\). If we know how the magnetization of the given paramagnetic salt depends on the magnetic field for the temperature \(T\), then the corresponding \(dS\) can be calculated. Let us now measure calorimetrically what amount \(dQ\) of heat is necessary to heat the cooled body from \(T'\) to \(T\). Then the temperature on the Kelvin scale will be expressed as:

\[ T' = \frac{dQ}{dS}. \]

Of course, such an establishment of the thermometric scale is very difficult to carry out experimentally. But all the difficulties that arise here do not seem to me insurmountable, at least for part of the temperature interval under consideration.

REFERENCES

1. U. Behn, Ann. Phys. Chem., 66, 237, 1898; Ann. Phys., 1, 257, 1900.
2. J. Dewar, Proc. Roy. Soc., March 25, 1904; A 36, 325, 1905.
3. W. Nernst, Koreff, Lindemann, Sitz. Ber., Berlin, 247, 262, 306, 1910; Ann. d. Phys., 36, 395, 1911.
4. A. Einstein, Ann. d. Phys., 22, 180, 800, 1907.
5. P. Debye, Ann. d. Phys., 39, 789, 1912.
6. A. Eucken, Phys., Z., 10, 586, 1910.
7. W. Keesom u. J. Kok, Com. Leiden, No 219e, 1932.
8. W. Keesom Z. Ges. Kälteindustrie, 40, 49, 1933.
9. F. Simon u. R. Bergmann, Z. Physik. Chem., 8, 255, 1930.
10. W. Keesom, Com. Leiden, No 219b, 1932.
11. W. Keesom u. Kok, Com. Leiden, No 221e, 1932.
12. W. Keesom u. Kok, Com. Leiden, No 230c. 1934.
13. A. Rutgers see Ehrenfest — Com. Leiden Suppl., No 75b, 1933.
14. Meissner u. Ochsenfeld, Naturwiss., 21, 787, 1933.
15. Gorter et Casimir, Physica, 1, 305, 1934.
16. Keesom u. Kok, Com. Leiden, No 230c. 1934.
17. Rapp, Cons. de Physique Sol., p. 288, 1924.
    17a. See Com. Leiden, No 232a, 1934.
18. Kamerlingh-Onnes et Boks, Com. Leiden, No 170b, 1924.
19. Wolfke u. Keesom, Com. Leiden, No 190a, 1927, No 192a, 1906 u. 1928.
20. Keesom u. Clusius, Com. Leiden, No 219e, 1932.
21. Keesom u. Mlle Keesom, Com. Leiden, No 221d, 1932.
22. Keesom u. Clusius, Com. Leiden, No 216b, 1931.
23. Keesom u. Mlle Keesom, Com. Leiden, No 224d, g, 1933; Suppl., No 76b, 1933.
24. Ehrenfest, Com. Leiden Suppl., No 75b, 1933.
25. Clusius u. Perlich, Z. physik. Chem., 24, 13, 1933.
26. Keesom, Com. Leiden Suppl., No 75a, 1933.
27. Pauling, Phys. Rev., (2) 36, 430, 1930.
28. Keesom, Com. Leiden Suppl., No 71e, 1932.
29. Keesom u. Schmidt. Com. Leiden, No 226a, 1932.
30. W. de Haas, Wiersma, Kramers, Physica, 1, 175, 1933; Naturwiss., 24, 467, 1933.
31. Glauque a. Mac Dougall, Phys. Rev., (2) 43, 768, 1933.