Nuclear Cooling*

N. Kurti, F. N. H. Robinson, F. Simon, and D. A. Spohr

Temperatures lower than absolute zero by an amount of the order of \(0.001^\circ\) have been obtained in a number of laboratories by the method of magnetic cooling, which consists in the adiabatic (isentropic) demagnetization of substances possessing electronic paramagnetism. The lowest temperature that can practically be reached in this way is determined by the presence of interactions between the electron spins (dipole–dipole, exchange, etc.). Soon after the first successful experiments on magnetic cooling, the suggestion was made\(^{1,2}\) that, by demagnetizing a system of nuclear spins, it would be possible to attain considerably lower temperatures, namely those at which ordering of the nuclear spins occurs.

The question of the feasibility of experiments on nuclear demagnetization was considered in considerable detail by Simon\(^{3}\) in 1939. He pointed out that, in order to produce an appreciable decrease in the entropy of a system of nuclei, magnetic fields of the order of 50,000 oersteds and temperatures of the order of \(0.01^\circ\) K would be required.

The first part of an experiment on nuclear cooling must consist in magnetization, i.e., polarization of the nuclear spins. Halban\(^{4}\) in 1937 drew attention to the possibility of using systems of oriented nuclei in studies of nuclear reactions, and Spiers\(^{5}\) calculated the expected angular distribution of the radiation intensity of oriented radioactive nuclei. Since experiments of this kind are not connected with the necessity of carrying out demagnetization, atomic and molecular fields can in this case be used for orienting the nuclei. A whole series of such methods was proposed and experimentally realized; their description may be found in the review article by Ambler and Hudson, published in 1955\(^{6}\). Later, Dabbs, Roberts, and Bernstein\(^{7}\) (1955), using an external magnetic field, polarized sodium nuclei. From various experiments of this type, information was obtained on the spins and parities of the excited states of approximately ten nuclei.

In view of the successful application of these methods, work on nuclear demagnetization in the Clarendon Laboratory, after the first experiment of this kind carried out by Hatton and Rollin\(^{8}\) (1949), was naturally pushed into the background. These authors demagnetized a crystal of calcium fluoride from an initial field of 4000 oersteds at \(1.2^\circ\) K to 500 oersteds. The temperature of the system of fluorine nuclear spins was determined from the magnitude of the nuclear-resonance signal. It was found that this temperature falls to \(0.17^\circ\) K and then rises again to \(1.2^\circ\) K with a relaxation time, due to the interaction of the spins with the lattice, equal to 60 sec. However, in these experiments the decrease in the entropy of the system of nuclear spins was extremely small (\(10^{-4}\%\)) compared with the entropy possessed by the lattice, and therefore no appreciable cooling of the system as a whole could occur. These observations

* Nature 178, 450 (1956). Translated by R. A. Chentsova.

also belong to the series of ingenious experiments described in 1951 by Pound, Purcell, and Ramsey[^9].

At a later time interest in this problem revived, and we succeeded in making some progress toward the realization of the ultimate goal. This goal is to achieve a noticeable decrease in the entropy of a system of nuclear spins under the action of an external magnetic field and, by measuring the temperature that will be attained upon subsequent demagnetization, thereby to obtain information about nuclear interactions in solids.

Nuclear cooling includes the following stages. A substance possessing nuclear paramagnetism is magnetized in a strong magnetic field, and the heat of magnetization released is absorbed by a “heat absorber” at a temperature of about \(0.01^\circ\text{K}\). The field is then reduced to zero and, if this process takes place adiabatically, the system of nuclear spins is cooled to a temperature determined by the initial temperature, the applied field, and the nuclear interactions.

The role of a heat absorber at a temperature of \(0.01^\circ\text{K}\) can be played only by a material possessing electronic paramagnetism and previously subjected to demagnetization (from several tens of thousands of oersteds at a temperature of about \(1^\circ\text{K}\)) down to a practically zero field. Since the nuclear specimen must be subjected to magnetization, the heat absorber has to be placed somewhere to the side of this specimen. Daniels[^10] constructed a water-cooled solenoid capable of producing the fields required for experiments on nuclear demagnetization and arranged in such a way that at a distance of \(23\ \text{cm}\) from the center of the coil the field falls to a value less than \(0.1\%\) of the field at the center, in a region large enough to accommodate the heat absorber. Such a magnet was used in the experiments described below.

In order that the heat of magnetization of the nuclei could be transferred to the heat absorber in a reasonable time (say, no more than half an hour), it is necessary first of all that the relaxation time of the nuclear spin–lattice system be sufficiently short. This is easy to achieve if a metal specimen is used, since it is known[^8,^11] that in metals this time is of the order of seconds at \(1^\circ\text{K}\) and, as may be expected, will be of the order of minutes at \(0.01^\circ\text{K}\).

However, if a metal is used, precautionary measures against heating of the specimen by Foucault currents, arising from accidental fluctuations in the magnitude of the magnetic field and from changes in the magnetic flux during demagnetization, become substantially necessary. Heating caused by the first source can be eliminated to some extent if a very stable direct-current source is used to supply the magnet and the cryostat is surrounded by a thick copper shield. Heating associated with the action of the second cause can be avoided only by dividing the specimen into a large number of insulated parts. Let us illustrate the importance of these precautions by a numerical example: the total heat of nuclear magnetization of a copper cylinder \(1\ \text{cm}\) in diameter and \(5\ \text{cm}\) long in a magnetic field of intensity \(30{,}000\) oersteds at \(0.01^\circ\text{K}\) is of the order of \(10^3\) ergs, whereas the heat released in the cylinder in the process of reducing this field to zero over \(2\ \text{min}\) exceeds this value by approximately a factor of 100.

Kurti[^12] considered various types of heat conductors and came to the conclusion that a copper heat conductor of reasonable dimensions should be suitable for transferring the heat of magnetization. Copper is not only a material readily available in the form of thin insulated wire, but also possesses desirable nuclear properties (see, for example, work[^3]). It is therefore convenient to use copper both as the heat conductor and as the working substance itself in the nuclear stage of cooling. In this case the nuclear specimen can be realized simply as a continuation of the heat conductor.

Heat transfer from the heat conductor to the paramagnetic salt, which is the heat absorber, and the parasitic inflow of heat to the specimen were the two main stumbling blocks in this investigation.

It is known\(^{13,14}\) that heat exchange between strips of copper foil and compressed paramagnetic salt, when they are in contact with one another over a macroscopic area \(A\), can be expressed by the formula

\[ Q=-\alpha A\left(T_1^3-T_2^3\right), \tag{1} \]

in which \(T_1\) and \(T_2\) are the temperatures of the contacting surfaces, and \(\alpha\) is of order \(10^3\ \text{erg}\ \text{cm}^{-2}\ \text{deg}^{-3}\ \text{sec}^{-1}\). Since we wished to remove amounts of heat of the order of 1000 ergs over several minutes at \(0.01^\circ\text{K}\), the contact area obviously had to amount to several hundred square centimeters. Attempts undertaken at an early stage of the investigations by Robinson to achieve such heat transfer, using strips of copper foil or copper wire and pressing them into powdered chrome-potassium alum, ended unsuccessfully. It was relatively simple to obtain contact surfaces of the order of \(10\)–\(50\ \text{cm}^2\), suitable for work at temperatures above \(0.1^\circ\text{K}\), but it proved practically impossible to increase this area substantially. However, Robinson\(^{15}\) found that one can achieve a thermal-contact efficiency corresponding to the same formula (1), but without applying pressure, if the alum is mixed with glycerin and water, thereby converting it into a claylike mass. In such a case it is possible to make a construction with the paramagnetic salt into which a comparatively complex system of foil strips or wires is immersed and thus to obtain a large contact area. Using this method, Kurti and Spore constructed a specimen and carried out with it an experiment, to the description of which we now turn. (This experiment forms part of a dissertation for the degree of Doctor of Philosophy which will be submitted by Spore to Oxford University.)

Fig. 1.

In Fig. 1 a photograph is reproduced of the specimen and the holder in which it rests. A schematic drawing is also given here, showing the essential features of the construction. The heat conductor connecting the upper and lower parts of the apparatus consists of 1540 enamelled copper wires (diameter \(0.121\ \text{mm}\)), the upper ends of which are uniformly distributed over the cross section of a mixture of glycerin with chrome-potassium alum (in an amount of about \(16\ \text{g}\)), enclosed in a Perspex container. It was calculated that the contact area between the metal and the mixture is \(400\ \text{cm}^2\). The lower ends of the wires are folded four times over a length of about \(7\ \text{cm}\), forming a “nuclear specimen”; thus the latter contains approximately \(0.75\) gram-atom of copper.

In this experiment, which was preliminary in character, there was no thermal switch between the two stages. The presence of thermal resistance between the heat conductor and the paramagnetic salt, as well as the assumed magni-

values of the spin-lattice relaxation time made it possible to measure the temperature of the system of nuclei after demagnetization and before this system had again warmed up to the temperature of the heat absorber (a process that took 2–3 min).

The system shown in the photograph was mounted inside a freely suspended “capsule,” consisting of a brass tube with double walls, the upper part of which contained 25 g of manganese ammonium sulfate; the latter was cooled by demagnetization simultaneously with potassium chromium alum. The lower part of the capsule, surrounding the nuclear stage, was kept in good thermal contact with the manganese ammonium salt by means of liquid helium, as well as by thin copper wires arranged longitudinally between the walls of the brass tube. The purpose of the capsule was to reduce the parasitic heat input to the system associated with vibrations, gas recondensation, and heat supply from above along the glass suspensions. Preliminary experiments using the capsule and a sample in which the coiled part of the wires was replaced by a cylinder of cerium magnesium nitrate (used as a magnetic thermometer) showed that the heat input to the lower part of the heat conductor was about 1 erg per minute. In addition, these experiments showed that, 10 min after demagnetization, the lower end of the heat conductor reaches a temperature of \(0.011^\circ\mathrm{K}\). From the slope of the cooling curves and the known heat capacity of cerium magnesium salt it was possible to establish that the heat transfer between both salts and the heat conductor approximately obeyed formula (1).

The temperature of the system of nuclear spins was determined from the results of measurements of the magnetic susceptibility of this system, carried out by the ballistic method. It was assumed that the susceptibility obeys Curie’s law. The calibration was obtained by calculation, starting from the theoretical value of the nuclear susceptibility of copper \(\left(\chi = \dfrac{3.74\cdot 10^{-7}}{T}\ \text{per gram-atom}\right)\) and from the sensitivity of the measuring circuit. The latter was determined by measurements, performed at hydrogen temperature, with a sample of manganese ammonium sulfate of the same size as the nuclear sample. For a primary-coil field of 7.8 oersted the calibration can be expressed by the formula

\[ \delta_N = 3.4\cdot 10^{-5}/T\ \text{cm}. \tag{2} \]

Unfortunately, it proved impossible to calibrate the measuring circuit in the usual way, by observing ballistic deflections of the galvanometer at various known temperatures above \(0.01^\circ\mathrm{K}\). This is explained partly by the extremely small magnitude of the nuclear susceptibility in this calibration region, and partly by the fact that at these temperatures the nuclear susceptibility is masked by the influence of a susceptibility apparently associated with paramagnetic impurities in the copper wire (compare \(^{16}\)) and in the insulation. The interpretation of the results of the ballistic measurements is also complicated by the proximity of considerable masses of paramagnetic material in the heat absorber and in the capsule. The influence exerted on the measured value of the susceptibility by these masses and by the paramagnetic impurities was determined in a control experiment, in which the nuclear sample was placed outside the region influenced by the field of the primary coil.

In the nuclear-cooling experiment, the electronic stage (heat absorber) was first cooled by adiabatic demagnetization from approximately \(1^\circ\mathrm{K}\) and 20,000 oersted to \(\sim 0.01^\circ\mathrm{K}\). The nuclear sample was then slowly magnetized in fields of various magnitudes—up to 28,000 oersted. The resulting decrease in entropy for this field was about 1% of the total nuclear entropy, equal to \(2.8\ \mathrm{cal/deg}\) per gram-atom. The field was maintained at its maximum value for some

time, varying within the limits from 5 to 20 min, and then decreased to zero at a rate varying within the limits from 500 to 1000 oersteds per second. The readings of the ballistic deflection began approximately 13 sec after the field reached zero, and were then made every 10 sec.

In Fig. 2 the ballistic deflections \(\delta_N\), obtained upon reversing the direction of a field of 7.8 oersteds, are shown as a function of the time elapsed from the moment of complete demagnetization. The figure also shows the corresponding values of \(T\), calculated from formula (2). The different curves correspond to different values of the initial field, indicated in the table (see below). By making a reverse extrapolation of these curves to zero time, one can determine the magnetic temperature attained immediately after demagnetization. As is seen from consideration of the indicated curves, in the case of the largest of the fields used this temperature was about \(20 \cdot 10^{-6}\ ^\circ\mathrm{K}\).

We have already mentioned that paramagnetic impurities may give a susceptibility of the same order as the susceptibility due to the system of nuclei. However, it appears unlikely that the large relative changes in susceptibility observed after demagnetization could have been caused by this reason. In order for such large ballistic deflections to be observed, the electronic paramagnet would have had to be close to ideal behavior; but even in this case one could not expect a noticeable change in the entropy (and, consequently, a further increase in the deviations) for values of \(H/T\) greater than approximately 50,000 oersteds/degree (the difference in the heat of magnetization of a gram-atom of an ideal electronic paramagnet for \(H/T = 300,000\) and \(H/T = 2,300,000\) oersteds/degree is \(10^{-9}\) erg at \(T = 0.012^\circ\mathrm{K}\)). However, examination of Fig. 2 shows that for increasing values of \(H/T\), up to 2,300,000 oersteds/degree, ever lower temperatures were obtained.

Fig. 2.

If it may be assumed that Curie’s law is satisfied even at the very lowest temperatures reached in the course of this experiment, then one can calculate the value of the temperature \(\theta_N\) characterizing the nuclear interactions\(^3\). If \(T_i\) is the initial temperature, \(H_i\) is the value of the field before demagnetization, and \(T_f\) is the temperature attained immediately after demagnetization, then \(\theta_N\) is determined by the relation

\[ \theta_N = T_f \frac{\mu H_i}{kT_i}, \tag{3} \]

in which \(\mu\) is the magnetic moment of the nucleus. Taking \(T_i = 0.012^\circ\mathrm{K}\) and using the dashed lines in Fig. 2, obtained by extrapolation, to determine \(T_f\),

we obtain the values of \(\theta_N\) given in the table for the different curves shown in Fig. 2:

From consideration of this table it is evident that the results obtained in the various series of measurements are in approximate agreement with one another. The mean value of \(\theta_N\), namely \(2.3 \cdot 10^{-6}\ ^\circ\mathrm{K}\), may be compared with the value \(4 \cdot 10^{-6}\ ^\circ\mathrm{K}\) calculated for copper by Fröhlich and Nabarro’s theory\({}^{17}\) for the temperature corresponding to the anomaly associated with the interaction of the nuclei.

On the basis of the experiment described it is impossible to decide whether the conduction electrons and the lattice also take part in the cooling. The rate of warming is compatible both with the estimated value of the relaxation time of the system nuclear spins—lattice and with the coefficient of heat transfer between the nuclear stage and the heat absorber. Two arguments may be adduced in favor of the hypothesis that the electrons take part in the cooling and that, consequently, thermal equilibrium is established throughout the entire specimen (the lattice, in any case, has an insignificantly small heat capacity). One argument is as follows: if the cooling and the subsequent warming were due only to the nuclear spins, then \(\delta_N\) (i.e. \(1/T\)) would have to depend linearly on time, since the nuclear heat capacity depends on temperature as \(1/T^2\). As may be seen from consideration of Fig. 2, this is not the case (at least at higher temperatures), and the curves indicate an increase of the heat capacity with temperature. Further, in the mechanism which gives rise to the anomaly considered by Fröhlich and Nabarro, the electron spins also take part. If this were not so, the expected anomaly would have to be observed at a considerably lower temperature (\(\sim 10^{-7}\ ^\circ\mathrm{K}\)), which contradicts the value calculated on the basis of the results obtained.

The experiments described have shown that, by using nuclear demagnetization, it is possible to obtain temperatures lower than those which can be reached by the demagnetization of electronic paramagnets. In addition, they have made it possible to obtain direct data on the interaction between nuclei in metals. It was therefore considered expedient to describe these first exploratory experiments, which we hope in due course to improve (for example, by using a heat switch), and also to extend, in addition to copper, to other materials. It should be emphasized that the significance of experiments of the type described lies not so much in the smallness of the temperature obtained as in the fact that one may hope in this way to obtain information on the behavior of nuclear spins in solids and on their interaction with the surrounding medium.

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