MAGNETIC METHOD FOR OBTAINING VERY LOW TEMPERATURES *

P. Debye, Leipzig

Not long ago the lowest temperature attained was \(0.7^\circ\mathrm{K}\). To reach this point on the temperature scale it was necessary first to liquefy helium and then to evaporate it as rapidly as possible—helium being the gas whose condensation is achieved with the greatest difficulty. Other gases, in which the forces of mutual attraction between molecules are still smaller than in helium, are unsuitable for the indicated purpose. For technical reasons it is hardly possible to pump off more rapidly than was done by Keesom \(^{1}\), who attained the above-mentioned temperature of \(0.7^\circ\). It therefore seems expedient to turn to other—not mechanical—methods that would make it possible to approach absolute zero still more closely.

Kamerlingh-Onnes and his collaborators showed that in gadolinium sulfate, at liquid-helium temperatures, a very sharply expressed phenomenon of paramagnetic saturation is observed; subsequently it was established that the magnetization curve in this case can, to a very good approximation, be determined by means of the classical Langevin theory. Both of these circumstances prompted me to consider this phenomenon from the point of view of Nernst’s heat theorem. According to the Langevin formula, saturation with increasing field should be reached relatively slowly (the difference between the magnetization at saturation and the magnetization in a field of strength \(H\) at temperature \(T\) is inversely proportional only to the first power of the ratio \(H/T^{2}\)). It follows from this that the entropy of a magnetic body, as it approaches absolute zero, increases logarithmically to infinitely large values. But this contradicts Nernst’s theorem, and consequently one must assume that, in sufficiently strong fields or at sufficiently low temperatures, the course of the magnetization curve can no longer be expressed by the Langevin function. As is known, the reason for the inapplicability of the classical theory should be sought in the fact that, according to Langevin, all possible orientations of elementary magnets relative to the field can exist. According to quantum theory there exists

* Report at the X German Congress of Physicists. Published in Physik. Z., 35, 923, 1934. For a survey of the question see Meissner, Physik. Z., 35, 303, 1934. Translated by N. N. Malov.

only a finite number of levels of magnetic energy; hence it follows that in this case the theoretically determined magnetization curve will be closer to reality, and that at saturation there will exist only a finite difference of entropies between the state of saturation at absolute zero and at any other temperature.

For the question considered in the present report, another consequence is more important. The more closely a body, in its properties, approaches what classical theory predicts, the easier it is, by choosing the appropriate experimental method, to approach absolute zero with the aid of this body. Therefore the idea arises of trying to penetrate into the region of very low temperatures, lower than those obtained by the evaporation of helium, by making use of the process of demagnetization. Calculation shows that very favorable results may be expected along this path.

The idea of the method and the limits of its application are most easily clarified by comparing the properties of a paramagnetic body with the properties of a gas. According to the first law of thermodynamics, the change in energy \(u\) (calculated for \(1\ \mathrm{g}\)) is equal to the sum of the quantity of heat supplied and the work done on the body:

\[ du = \delta q + \delta a. \]

If it is desired that a body doing work should, if possible, diminish its energy, care must be taken that no thermal energy be communicated to it; therefore the process must be adiabatic \((\delta q = 0)\). Since the work of a mechanical process is equal to \(pdv\) (\(p\) is the pressure, \(v\) the volume), one has to use adiabatic expansion. In this case the energy will decrease, but the goal we have set will not yet be attained. The point is that the energy \(u\) is, generally speaking, a function of the volume \(v\) and the temperature \(T\); and it is not yet clear which of these variables is the more important. As is known, thermodynamics gives a relation between the “caloric” equation of state, which determines the energy as a function of volume and temperature, and the “thermal” equation of state, which makes it possible to determine the pressure as a function of the same variables. This relation is expressed by the equation:

\[ \frac{\partial u}{\partial v} = T^{2}\frac{\partial}{\partial T}\left(\frac{p}{T}\right). \]

For an ideal gas, for which \(\frac{p}{T}\) depends only on the volume, one obtains

\[ \frac{\partial u}{\partial v} = 0. \]

Thus in this case the volume does not enter at all into the expression for the energy; therefore, if the energy changes owing to the perfor—

MAGNETIC METHOD OF OBTAINING LOW TEMPERATURES

work, then this change occurs exclusively at the expense of a change in temperature.

All this has long been well known; but these well-known facts can also be applied to the case of adiabatic demagnetization that interests us. Indeed, if \(\sigma\) is the magnetic moment per unit mass arising in a field \(H\), then

\[ \delta a = H\,d\sigma, \]

and the expression for the change in energy

\[ du = \delta q - p\,dv, \]

which was valid for the case of a changing volume, may be replaced by the analogous expression for magnetization:

\[ du = \delta q + H\,d\sigma. \]

Here the magnetization \(\sigma\) plays the role of volume, while the field strength appears in place of the pressure, taken as negative. Therefore the following relation must also hold:

\[ \frac{\partial u}{\partial \sigma} = -T^2 \frac{\partial}{\partial T}\left(\frac{H}{T}\right). \]

As is known, it has repeatedly been established that down to the lowest temperatures attained, magnetization is a function only of the ratio

\[ \frac{H}{T}. \]

Langevin’s theory requires the fulfillment of this relation, and quantum theory in its simplest form leads to the same conclusion, although the functions \(u\) take on a somewhat different form. The reason for this course of magnetization is clarified by the following considerations. If the individual paramagnetic atoms are independent of one another and are not excited, then the ratio of the magnetic moment produced to the Bohr magneton is determined by only two energy values. These are, first, the values of the magnetic energy corresponding to the different levels of the magnetic energy of atoms in an external field, proportional to \(\mu H\) (\(\mu\)—the magneton, \(H\)—the field strength). The second energy, determining the process, is the thermal energy \(kT\) (\(k\)—Boltzmann’s constant, \(T\)—temperature). Since the required ratio is a dimensionless quantity, it must be only a function of the ratio \(\mu H\) to \(kT\). Deviations will arise only in the case where excitation of the atoms exists, as a result of which the process will be subject to the influence of some other energy values connected with this excitation.

Therefore I would propose to call an ideal paramagnetic body one in which the magnetization is proportional to some function of

\[ \frac{H}{T}, \]

analogously to the concept of an ideal gas. In an ideal paramagnetic body the susceptibility must follo-

obey Curie’s law, as long as the field is so small that the influence of saturation may be neglected.

If the paramagnetic body is ideal in the sense indicated above, then from the generalized thermodynamic formula we find

\[ \frac{\partial u}{\partial \sigma}=0, \]

i.e. every expenditure of work performed by the body upon demagnetization must necessarily be accompanied by a lowering of temperature.

Practically, in addition to the fundamental possibility of the process indicated, it is of course necessary to determine the order of magnitude of the quantities governing it. At ordinary temperatures the demagnetization work that can be obtained in practically attainable fields is so small that it has no significance. It is approximately 100,000 times smaller than the thermal energy of motion of the atoms, at least for paramagnetic bodies. For ferromagnetic bodies, which possess a considerably greater magnetization, the relation is more favorable, and in this case, as is well known, the thermomagnetic effect, especially sharply expressed near the Curie point, was discovered experimentally by Weiss and Piccard³. On passing to lower temperatures, first of all, according to Curie’s law, the possible demagnetization work of a paramagnetic body increases considerably. At the same time the thermal energy of its atoms decreases. Both these causes lead to the fact that in the region of liquid-helium temperatures the magnetic and thermal energies become quantities of one and the same order.

The magnetic method for obtaining very low temperatures was proposed independently of me by Giauque⁴, who likewise possessed the means for the experimental realization of this idea. Experiments in this direction were subsequently undertaken by de Haas in Leiden and by Simon in Oxford.

TABLE

A rather long time was required before favorable results were achieved, but recently the success of the experiments has proved very considerable. I should like to illustrate this with the above table, which contains the results of the experiments of de Haas and his collaborators⁵.

In this table the first three dates refer to the time when the experiments were carried out, and the last two to the time of their publication.

In addition to this table, which I demonstrated in my report, I should now like to add the communication kindly made to me by de Haas concerning the attainment of a temperature of \(0.018^\circ\). In order to assess correctly the successes achieved, it is necessary to compare not the temperature differences, but their ratios, just as Lord Kelvin does in the logarithmic temperature scale he proposed. Making this comparison, it is easy to see that the temperature of liquid hydrogen (\(20^\circ\)) lies closer to the lowest temperature attained in the experiments with helium (\(0.7^\circ\)) than this temperature does to the lowest temperature attained by de Haas (\(0.018^\circ\)).

How were these temperatures measured? This question, naturally, arises before all others. Here it is no longer possible to use the method of measuring the pressure of an ideal gas. Simon, to whom I am indebted for providing me with two as yet unpublished manuscripts, which I used in part later on, calculated, with the aid of the constants determined by Keesom for the equation of the vapor pressure of helium, that this pressure, amounting at \(0.7^\circ\) to only \(3.2 \cdot 10^{-3}\) mm Hg, at \(0.3^\circ\) falls to \(7 \cdot 10^{-10}\) mm, and at \(0.1^\circ\) amounts to only \(3 \cdot 10^{-31}\) mm Hg. Whereas pressure measurements can be applied only in that temperature region which is accessible to gas-expansion experiments, magnetic methods must, of course, prove applicable in those regions where we use the method of demagnetization. Just as in pressure measurements we proceed from the Boyle–Gay-Lussac law for an ideal gas, in the region of application of magnetic methods we must proceed from Curie’s law for an ideal paramagnetic body. Thus, for example, if, beginning an experiment at \(1^\circ\), we obtain an unknown, still lower temperature characterized by the fact that, to create the same magnetization as at \(1^\circ\), a magnetic field already 10 times smaller is required, then on the basis of Curie’s law we may conclude that this temperature is \(0.1^\circ\).

In a similar way the temperatures indicated in the table were determined (of course taking into account deviations from Curie’s law observed in the region of liquid-helium temperatures). For measuring susceptibility in the first four experiments the specimen was placed in that part of the magnetic field where there existed a gradient of magnetic-energy density. The force with which the specimen was drawn into the region of weak field remaining after the main field had been switched off is, as is known, proportional to the susceptibility. By studying the change of this force with time, de Haas was able to measure the gradual heating that occurs after adiabatic demagnetization. In the later experiments the specimen was placed in a homogeneous part of the field and was rapidly removed from it. This method has the advantage that all parts of the specimen are subjected to the same action and that it allows the possibility of obtaining equally low—

temperature in a considerably larger quantity of substance. In the graph containing the weight of the preparation, this is seen very clearly. The measurements of susceptibility needed to determine the temperatures were made here by the induction method.

A further success, discernible from the table, is the circumstance that in the most recent experiments the complete suitability of cheap, readily available salts for obtaining low temperatures by means of the magnetic method was demonstrated. Of course, the search for still more suitable substances should be continued. The directions for these searches are indicated in Gorter’s discussion contribution, printed at the end of the report.

Naturally, against the indicated method of measuring temperatures one may raise the objection that deviations from ideal magnetic properties distort the results of the measurements. This objection is entirely similar to that which was repeatedly made in ordinary temperature measurements by the pressure method; it is refuted in an entirely similar way.

In Planck’s thermodynamics it is shown in detail how, on the basis of known caloric measurements of a gas, an absolute temperature scale may be established. One may proceed in an analogous way in the case of paramagnetic bodies as well. Thus, for example, the following formula may be derived:

\[ \ln \frac{T}{T_0} = \int_{t_0}^{t} \frac{ \left(\dfrac{\partial \sigma}{\partial t}\right)_{H} }{ \left(\dfrac{\partial q}{\partial H}\right)_{t} } \, dt . \]

Therefore, if on an arbitrary temperature scale \(t\) the temperature coefficient of magnetization is measured and, in addition, it is established what thermal effect is produced by a change of the field at constant temperature, then the absolute temperature scale is determined by the above integration. In the discussion of the report, Keesom proposed another construction, remarkable for its exceptional clarity and given at the end of the report.

Taking into account the considerable successes achieved with respect to approaching absolute zero by means of the magnetic method, it is of interest to determine how far one can go in this direction. To illuminate the most important points of this reasoning we shall make use of three entropy diagrams, which do not claim quantitative accuracy (in the region of very low temperatures, the part of the entropy corresponding to the thermal motion of atoms is shown on an enlarged scale).

In the first diagram (Fig. 1) the course of the entropy of a paramagnetic body is shown under the assumption that the Langevin law of magnetization is valid down to the lowest temperatures. The diagram gives curves determining the entropy \(S\) as a function of temperature

MAGNETIC METHOD OF OBTAINING LOW TEMPERATURES

\(T\) at an unchanged magnetic field \(H\). The curve corresponding to \(H = H_2\) pertains to the greatest field strength.

For a smaller field strength \(H\) the middle curve is obtained, and in the limit, at \(H = 0\), the curve coincides with the ordinate axis from \(S = -\infty\) to \(S = 0\) and then is represented by the uppermost curve, constructed on the assumption that the entropy of the nonmagnetic body is proportional to the 3rd power of the temperature. The adiabatic process is characterized by a horizontal straight line \((S = \mathrm{const})\).

If demagnetization is carried out starting from the field \(H_2\) at a high initial temperature, then the process will be determined by the upper straight line \(AB\) and will lead, although to a lower, nevertheless still finite temperature. If, however, demagnetization is carried out at a sufficiently low initial temperature, then, as Fig. 1 shows, one may hope to approach absolute zero (the lower straight line \(AB\)).

Fig. 1. Entropy diagram of an ideal paramagnetic body in the case of applicability of Langevin’s law down to a temperature \(T = 0^\circ\) absolute

To prove that the possibility indicated above of reaching absolute zero is due not only to the fact that, in contrast to quantum theory, the Langevin magnetization curve admits an infinitely large number of orientations of elementary magnets, the diagram in Fig. 2 was constructed. Here the basis is an atom whose moment has only two possible orientations, namely: parallel to the field and antiparallel to it. In this case the entropy remains finite, and we may take it at absolute zero to be equal to zero, so that all entropy curves emerge from the origin of coordinates. The curve for \(H = 0\) still partly coincides with the ordinate axis; this proves that, at a sufficiently low initial temperature, the method of demagnetization also makes it possible to approach absolute zero (the straight line \(A'B'\)).

The magnetization functions on which the diagrams considered were based were assumed to be functions only of the ratio \(\frac{H}{T}\), i.e., they referred to an ideally paramagnetic body. If the deviations actually existing are taken into account, then the question of the attainability of absolute zero will stand somewhat differently.

Figure 3 shows what effect the Lorentz molecular magnetic field will have. In this case the curve for \(H = 0\) will no longer coincide with the ordinate axis, but at \(T = \Theta\) there exists a point

Curie, and the entropy curve for \(H=0\) at this temperature undergoes a break and continuously descends to the origin. In this case absolute zero cannot be attained by any adiabatic demagnetization process.*

In a similar manner other distortions of the magnetic energy levels will also have an effect, such as, for example, the splitting of the magnetic moment caused by the electric field of neighboring atoms, etc.; hence it follows that any splitting is very significant for the effectiveness of the demagnetization process. The smaller the differences in energy of the individual levels, the closer the process will lead to absolute zero. In Simon’s work this question is analyzed in great detail and a simple formula is derived which connects the lowest attainable temperature with the difference of the energies of the perturbed levels and with the intensity of the initial magnetic field. As is known, the presence of splitting is already noticeable in measurements of specific heats (in a field equal to zero); this was shown by Kürti \(^{7}\) for gadolinium sulfate. If the mean difference of the energies of the levels is characterized by the temperature \(\Theta\), taking it to be equal to \(k\Theta\) (\(k\) being Boltzmann’s constant), then an anomaly of the specific heat arises near \(T=\Theta\), the magnitude for magnetic atoms being of the order of the gas constant per gram-atom. At temperatures corresponding to this anomaly, the energy of motion of the atom, which determines the specific heat under ordinary conditions, plays practically no role. In this region an increase in temperature determines above all changes in the distribution of atoms among their various modifications corresponding to different energy levels. The supplied heat is expended mainly on

Fig. 2. Entropy diagram of an ideal paramagnetic body when only two possible orientations of the magnetic moment are allowed

Fig. 3. Entropy diagram of a non-ideal paramagnetic body (Lorentz magnetic molecular field)

* The last two curves are taken from Debye’s article \(^{6}\).

“activation” of the atoms, and only a negligibly small part of it goes into increasing the thermal motion. Although these anomalies do hinder the rapid approach to absolute zero in the production of adiabatic demagnetization, they also exert a favorable influence, by slowing down the subsequent warming. This may be explained by the fact that heat exchange can occur only through radiation or through direct contact of solids, since below \(0.4^\circ\) all gases prove to be frozen. But radiation is very insignificant, and heat transfer by direct contact can be reduced by using paramagnetic substances in powdered form. Simon calculated that the warming following magnetic cooling cannot exceed \(0.25 \cdot 10^{-3}\) degrees per minute.

If it is desired, by mixing other bodies with paramagnetic ones, to bring them to as low a temperature as possible—for example for investigations of superconductivity—then the removal of the thermal energy of these bodies presents no difficulty. The thermal motion of their atoms will decrease, and near absolute zero the heat energy released will be insignificant in comparison with the energy released in the paramagnetic atoms when they pass from one level of magnetic energy to another. Thus the difficulties will consist only in creating a sufficiently reliable thermal contact, which in practice, apparently, can be achieved.

In the discussion of the paper, Gorter noted that at the low temperatures that have now been attained, one must take into account the influence of the magnetic moment of the nucleus. A preliminary calculation shows that quite interesting possibilities do in fact open up here. The Bohr magneton has a magnitude of the order of \(10^{-20}\). If one considers a field of the order of \(10\,000\) oersteds, then its magnetic energy relative to the field will have a magnitude of the order of \(10^{-16}\) erg. On the other hand, the thermal energy, equal to \(kT\), has a magnitude of the order of \(10^{-16}T\). Therefore, for ordinary paramagnetic bodies, saturation should set in at temperatures close to \(1^\circ\). As is known, it is in fact observed in the temperature range from 1 to \(10^\circ\). Further, the nuclear moment, determined, for example, from the hyperfine structure of spectral lines, is approximately a thousand times smaller than the Bohr magneton. Therefore one should expect that in the temperature range from \(0.001\) to \(0.01^\circ\) the magnetism of the nucleus will exert a substantial influence. If, on the other hand, one takes into account that de Haas has already reached a temperature of \(0.018^\circ\), then it may be concluded that in the near future direct measurements of nuclear magnetism will be possible.

Remarks in the Discussion

Keesom (Leiden). I should like to make two remarks. The first is that at low temperatures, in order to create thermal conductivity between different crystals, one may add to them a little

helium. Helium will create thermal contact either by penetrating into the gap between the bodies owing to capillary phenomena, or as a surface layer adsorbed on the surface of the crystal.

The second remark is the following: to determine the temperature obtained upon adiabatic demagnetization, on the Kelvin scale, it is simplest of all to use calorimetric measurements carried out together with measurements of adiabatic demagnetization. For this purpose let us imagine that two adiabatic demagnetizations are performed: one starting from \(T, H\), and the other starting from \(T, H + dH\) (Fig. 4).

Using a suitable thermoscope, we note the deviations corresponding to the temperatures \(T'\) and \(T' + dT'\) attained as a result of demagnetization. If, at temperature \(T\), the dependence of the entropy on the temperature is known, then the difference \(dS\) can be calculated. If now one calorimetrically measures the quantity of heat \(dQ\) necessary to heat a body having temperature \(T'\) to the temperature \(T' + dT'\), then, for determining the value of the temperature on the Kelvin scale, we obtain:

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

Fig. 4. Method of establishing a temperature scale for very low temperatures

Gorter (Haarlem). The \(\Theta\)-value indicated by Simon refers to magnetic ions which, at low temperatures, have no more than two modifications of levels. Still smaller values for \(\Theta\) may be expected if ions of the type \(\mathrm{Cu}^{++}\) or \(\mathrm{Ce}^{++++}\) are investigated, for which, at low temperatures, double excitation of Kramers is still possible.

Further, I should like to point out that temperatures have already been reached experimentally which are sufficient for setting up experiments whose aim will be the attainment of still lower temperatures by making use of the magnetic moment of the nucleus.

Finally, I should like to note that, independently of the external form of the magnetic-interaction apparatus between the magnetic ions, it determines the lower limit of the temperature attainable in experiments with the given ions. The magnetic field of neighboring ions in most of the more common substances has a magnitude of the order of hundreds of oersteds. Reducing the external field to values smaller than this magnitude will hardly make it possible to obtain any further reduction of temperature.

LITERATURE

1. W. H. Keesom, Leiden Comm., 219a.
2. P. Debye, Ann. Physik, 81, 1154, 1926.
3. P. Wess u. A. Piccard, C. R. 166, 352, 1918; Journ. de Phys., 2, 161, 1921.
4. W. F. Giauque, Journ. Amer. Chem. Soc., 49, 1864, 1870, 1927.
5. W. J. de-Haas, E. C. Wiersma, H. A. Kramers, Physica, 1, 1, 1933; 1, 779, 1934.
6. P. Debye, Sächs. Acad. Ber., 86, 105, 1934.
7. N. Kürti, Z. Phys. Chem., 20, 305, 1933; see also F. Giauque and D. P. McDougall, Phys. Rev., 43, 768, 1933; 44, 235, 19, 1933.