Magnetic Machine for Achieving Ultra-Low Temperatures

R. A. Chentsov

Although the principle of magnetic cooling has been known since 1925, when it was proposed by Debye1, and although this method has found broad practical application2, only very recently has it proved possible to create a magnetic refrigerator of continuous (cyclic) operation. This machine, which makes it possible to obtain and use for experimental purposes temperatures below \(1^\circ\mathrm{K}\) (down to \(0.25^\circ\mathrm{K}\)), is based on the use of the large thermal effects that accompany the magnetization and demagnetization of certain paramagnetic salts at low temperatures. To understand how the machine works, it is best to turn to the \(T\)—\(S\) diagram of such a paramagnetic salt (Fig. 1; here \(S\) is the entropy of the salt, \(T\) the absolute temperature). The two dependences \(S(T)\) shown in the figure correspond to two cases: when the salt is placed in a constant magnetic field \(H = H_1\) (lower curve) and when the field is equal to zero (upper curve). At \(T = 0\) both curves converge, and the entropy becomes zero in accordance with the third law of thermodynamics.

First let us consider what happens during a single magnetization and demagnetization of the salt. Such a process is carried out in the apparatus shown very schematically in Fig. 2. A sample of salt \(A\) (for example, iron-ammonium or chrome-potassium alum, taken in the form of a compressed powder or a crystal) is placed inside a vacuum jacket \(E\). Owing to the thermal conductivity of the materials located between \(A\) and the bath of liquid helium \(B\) (this thermal contact is conventionally denoted by \(K\)), the salt is maintained at the temperature of this bath, \(T_1 \sim 1^\circ\mathrm{K}\). A current is passed through the winding of the electromagnet \(N—S\), producing a magnetic field in the space where the salt is placed. On the diagram (Fig. 1) this corresponds to a transition from the upper curve \(S(T)\) to the lower one. If the thermal contact between the salt \(A\) and the bath

is sufficiently good, the transition occurs isothermally, along the straight line \(1 \to 2\). The entropy of the salt thereby decreases by the amount \(\Delta S_1 = S(1) - S(2)\), which, according to the thermodynamic equality \(\Delta Q_1 = T \Delta S\), means that the salt gives up to the bath a certain quantity of heat \(\Delta Q_1 = T_1 \Delta S_1\); in other words, the magnetization of the salt proceeds with the liberation of heat.

The physical mechanism of the decrease in entropy upon magnetization of the salt (and, consequently, of the corresponding caloric effect) is easy to understand if one recalls that the entropy of any system is a measure of the molecular “disorder” existing in the system. The application of a sufficiently strong magnetic field (\(\sim 10\) kilooersteds) causes the paramagnetic ions responsible for the paramagnetism of the salt to pass from various states, which in the absence of an external field are equally probable, into a single state corresponding to the orientation of the magnetic moments of all ions along the field. Obviously, this change leads to an “increase in order” in the salt, i.e., to a decrease of its entropy. In an actual apparatus, owing to the finite thermal conductivity of the thermal contact between the salt and the bath, the temperature of the salt rises somewhat when the field is applied, and upon going over to the curve \(S(T)|_{H=H_1}\), the configurational point in Fig. 1 does not go strictly along the line \(1 \to 2\), but deviates somewhat to the right. In order for the salt to cool to the initial temperature \(T_1\) (point 2), it is necessary, after magnetization, to wait for some time (\(\sim 5\)–\(10\) min.) until the heat of magnetization has been removed into the helium bath.

Before carrying out adiabatic demagnetization, it is necessary to create adiabatic conditions for the salt (complete thermal insulation). This is achieved in such installations by using a “thermal switch” \(K_1\), whose thermal resistance must, at the experimenter’s discretion, be variable over wide limits. The thermal switch is “opened,” and then the magnetic field is slowly reduced to zero. Since the demagnetization takes place under adiabatic conditions, in this case \(\Delta Q = T \Delta S = 0\); the entropy remains constant during the transition process, and the transition itself is represented in the diagram of Fig. 1 by the straight line \(2 \to 3\). In reality, here too the conditions are not fully ideal: between the salt and the bath (through the thermal contact, which in practice is not broken up to infinite resistance), and also within the salt itself, irreversible processes of heat exchange and heat conduction occur, leading to a certain increase in the entropy of the salt, so that the final point is not 3, but another point on the curve \(S(T)|_{H=0}\), lying somewhat higher.

From examination of Fig. 1 it is evident that the process of adiabatic demagnetization is accompanied by a considerable lowering of temperature—practically to tenths or hundredths of a degree on the absolute scale (a record lowering of temperature—to \(0.0012^\circ\) K—was also achieved in this way). The mechanism of this effect is already clear from what has been set forth above: when the magnetic field is removed, its orienting action on the magnetic moments of the ions disappears, and the latter again acquire freedom to be distributed among several states. This “increase in disorder” is expressed in a noticeable increase of the “magnetic” part of the entropy of the salt. Since, however, the entropy of the entire salt does not change during adiabatic demagnetization, this increase must be equal in magnitude to the decrease of another part of the entropy, namely the entropy associated with the thermal vibrations of the crystal lattice of the atoms forming the salt. And this means that the intensity of the thermal vibrations of the lattice decreases, and its temperature, consequently, is lowered.

In order that the temperature effect of adiabatic demagnetization be considerable and lead to the attainment of “ultralow” temperatures, the diagram shown in Fig. 1 must evidently be characterized by a substantial “divergence” of the upper and lower curves \(S(T)\) in the temperature region.

tures of \(\sim 1^\circ\mathrm{K}\) and below. It is precisely for this reason that, for magnetic cooling, one has to use “magnetically diluted” salts, in which the ions possessing magnetic moments are at a considerable distance from one another. For example, in iron-ammonium alum \(\mathrm{Fe(NH_4)(SO_4)_2\cdot 12H_2O}\), one paramagnetic ion \(\mathrm{Fe}^{3+}\) is surrounded by about fifty “nonmagnetic” atoms and atomic groups. This leads to the fact that, when the salt is cooled even to very low temperatures, when the thermal motion of the atoms has already largely ceased (the horizontal portion of the curve \(\left. S(T)\right|_{H=0}\)), the energy of this motion still proves sufficient to overcome the weak forces of magnetic interaction of the iron ions and to distribute the paramagnetic ions uniformly among the various possible states (the entropy of the salt is different from zero). Only when the temperature is lowered to \(T \sim 0.1^\circ\mathrm{K}\) and below does the thermal energy become less than the interaction energy of the ions, and the number of states occupied by the ions decreases (the descending portion of the curve \(\left. S(T)\right|_{H=0}\)^*).

The thermal effect of adiabatic demagnetization proves to be very considerable. To give an idea of this effect, let us note that, using only \(1\ \mathrm{g}\) of such a paramagnetic salt, it would be possible to cool from \(T \sim 1^\circ\mathrm{K}\) to ultralow temperatures several kilograms of diamagnetic substance. However, in the described variant of a single demagnetization, the method has an essential fundamental drawback: immediately after demagnetization, owing to practically unavoidable effects of parasitic heat input, as well as the release of power associated with the measurements being carried out, the salt begins to warm up. As a result, all measurements have to be made “on the fly,” while the temperature is changing, i.e., under deliberately nonequilibrium conditions. Although in practice, owing to the large heat capacity \(C\) of the demagnetized paramagnetic salt (caused by the great steepness of the curve of the temperature dependence of the entropy, related to \(C\) by the direct proportionality relation \(C = T \dfrac{dS}{dT}\)), the entire warming-up period takes tens of minutes or even hours, the idea arose rather long ago^3 of creating, at ultralow temperatures, conditions of thermostating by constructing an apparatus for adiabatic demagnetization operating continuously (cyclically).

The principle of operation of such a machine can be clarified by considering Fig. 1 and Fig. 3. The former device is supplemented by a “cold reservoir” \(R\), which is used as a thermostating medium for carrying out various measurements with substances brought into good thermal contact with \(R\). The jacket \(E\) is surrounded by a bath of liquid helium; the helium Dewar, in turn, is immersed in a Dewar with liquid \(\mathrm{H_2}\) or liquid air (not shown in the figure). \(A\) and \(R\) are connected to one another by a second “thermal switch” \(K_2\).

Fig. 3.

At \(T = T_1 \sim 1^\circ\mathrm{K}\), the salt \(A\) is magnetized isothermally—with the “switch” \(K_1\) open—(line \(1 \to 2\)), then switch \(K_1\) is opened, and adiabatic demagnetization is carried out. In doing this, however, the field is not brought to zero; at some intermediate value of the field corresponding to the temperature \(T_0\) (point 4 in Fig. 1), the switch \(K_2\), which up to this time had been open, is closed, and then the field is reduced to zero at \(T = T_0\) (segment \(4 \to 5\)). In this process the salt \(A\) takes from the reservoir \(R\) (which initially was also at \(T = T_1\))

^*) The picture of the change of entropy with decreasing temperature is presented here in simplified form. In reality, complex effects of the interaction of magnetic ions with the electric field of the crystal lattice play an important role; however, discussion of this question would take us too far.

heat \(Q_0 = T_0 \Delta S_0\), where \(\Delta S_0 = S(5) - S(4)\). After this the thermal contact \(K_2\) is opened, the salt is magnetized adiabatically to point 6 and then—with the switch \(K_1\) open—to the maximum value \(H = H_1\) \((6 \to 2)\). Thus the salt performs the thermodynamically most advantageous cycle—the Carnot cycle \(2 \to 4 \to 5 \to 6 \to 2 \ldots\), each time taking from the reservoir the amount of heat \(Q_0\). The reservoir is gradually cooled to the temperature \(T_0\) and subsequently maintains this temperature (with small “sawtooth” deviations from cycle to cycle). The “cold” \(Q_0\) is thereby used to compensate the heat received by the reservoir \(R\) (per cycle) through imperfect thermal insulation, and also the heat released during measurements.

In the practical development of such a scheme, the center of gravity lay in the problem of variable thermal contacts—the “switches” \(K_1, K_2\). It is quite evident that these contacts must meet very stringent requirements. Thus, the switch \(K_1\) must, on the sections \(2 \to 4\) and \(5 \to 6\), provide complete thermal insulation of the salt \(A\) from the bath, despite the fact that large temperature differences exist between them. However, on the section \(6 \to 2\) the same switch \(K_1\) must have high thermal conductivity, ensuring the removal into the bath of the considerable heat of magnetization of the salt \(Q_1 = T_1 [S(6) - S(2)]\) at a small temperature difference between the salt and the bath. The same requirements are imposed on the switch \(K_2\). In single demagnetization, “heat-exchange gas” is usually used as the thermal switch \(K\): a small amount of gaseous helium is first admitted into the vacuum jacket \(E\), and is then pumped out. Despite the fact that powerful diffusion pumps are usually used to evacuate \(E\) (pumping is carried out through the special tube \(T\) in Fig. 3), this operation takes 10–15 minutes. The cumbersome nature of this method makes it of little use for a continuously operating machine. A second method sometimes used in single demagnetization is mechanical thermal contact: to close the switch, metal parts connected, for example, to the salt \(A\) and to the liquid-helium bath are pressed against one another by corresponding surfaces or springs, and when the switch is opened they are separated. However, such a switch has a serious drawback: at each closing and (what is more important) at each opening, a fairly large amount of parasitic heat is released at the contact owing to friction.

Practically the most convenient for creating a cyclic machine proved to be the third method\(^4\), based on the strong dependence of the thermal conductivity of a superconductor (for example, lead) on the magnitude of the magnetic field in which it is placed. A sufficiently strong magnetic field can, as is known, bring a superconductor out of the superconducting state (in which its electrical resistance is equal to zero) and transfer it into the so-called “normal” state, characterized by finite electrical conductivity. It turned out that, in such a transition, the behavior of the thermal conductivity is in a certain sense opposite to the behavior of the electrical conductivity: the thermal conductivity of pure lead in the superconducting state is tens and hundreds of times smaller than its thermal conductivity in the normal state (at ultralow temperatures) and reaches very small values, \(\sim 10^{-2} — 10^{-3}\ \mathrm{W}/\mathrm{cm}\cdot\mathrm{deg}\). By placing a lead strip or wire in a sufficiently strong magnetic field produced by an electromagnet, one can “destroy” the superconductivity of lead and “close” the thermal switch. When the field is switched off, such a thermal switch opens.

In 1953–1954, reports were published on the first magnetic refrigerator of cyclic operation to be realized, built by Daunt and co-workers\(^5\). The apparatus had the following parameters. The working substance (paramagnetic salt) was 15 g of iron-ammonium alum in the form of compressed powder (for mechanical binding, water retention, and improvement of the thermal conductivity of the salt, a small amount of silicone grease was mixed in).

pieces and finely chopped copper wire). The magnetic field that performs the work of magnetizing the salt is produced by an electromagnet powered by a current of up to 75 A and giving a field of up to 7000 oersteds. The heat switches \(K_1\), \(K_2\) are made of lead and are controlled by auxiliary electromagnets (not shown in Fig. 3). All the magnets are provided with iron shields that reduce the stray field. The liquid-helium bath is maintained at \(1^\circ\) K. The “cold reserve” \(R\) consisted of 15 g of chrome-potassium alum; to increase the heat capacity it was usually kept in a field of intensity 3000 oersteds. The cycle period is 2 minutes. Switching was performed automatically by a clock mechanism.

The operating characteristics of the installation are as follows. The time for cooling reservoir \(R\) to \(0.3^\circ\) K is 40 minutes (if the field is removed, less than 10 minutes). The refrigerating capacity (in ergs per cycle) is \(6.2\cdot 10^{-4}\) at \(0.65^\circ\) K, \(4.2\cdot 10^{-4}\) at \(0.55^\circ\) K, \(3.6\cdot 10^{-4}\) at \(0.45^\circ\) K, and \(2.0\cdot 10^{-4}\) at \(0.35^\circ\) K. At still lower temperatures the refrigerating capacity begins to fall more rapidly and at \(0.26^\circ\) amounts to \(8.5\cdot 10^{-5}\) ergs per cycle. Thus this experimental refrigerating machine made it possible to carry out investigations under approximately isothermal conditions at lower temperatures than had previously been possible (it is known that by pumping the vapors of ordinary liquid helium—\(\mathrm{He}^4\)—it was not possible to lower the temperature below \(0.7^\circ\) K). At the same time, the experimenter can dissipate, in the part of the apparatus at the ultra-low temperature, powers quite sufficient for carrying out a large number of electrical, magnetic, and other measurements.

Finally, a brief report has recently been published\(^6\) on the beginning of series production of a similar magnetic refrigerating machine by Arthur D. Little, Inc. (Cambridge, USA). The report states that the machine is a technical version of the installation of Daunt and co-workers\(^5\). It may be assumed that its operating characteristics are close to those given above. The vacuum jacket \(E\) (Fig. 3) is extended downward, so that an “experimental space” has been formed (diameter \(\sim 2.5\) cm, length \(\sim 25\) cm), where the materials under investigation, in thermal contact with the “cold reserve,” can be placed. All its elements (magnets, pumps, power supply, cryostat, control, etc.) are conveniently combined in a single mobile unit.

The creation of continuously operating machines for obtaining ultra-low temperatures is a major achievement of cryogenic technology. There is no doubt that these machines will develop and improve and will help in the solution of many questions of superconductivity, superfluidity, and other problems in the physics of very low temperatures.

References

1. P. Debye, Ann. Phys. 81, 1154 (1926).
2. Garrett, Magnetic Cooling. New York, 1954.
3. J. G. Daunt, C. V. Heer, Phys. Rev. 76, 985 (1949).
4. C. J. Gorter, Les phénomènes cryomagnetiques. Paris, 1948, p. 76.
5. C. V. Heer, C. B. Barnes, J. G. Daunt, Phys. Rev. 91, 412 (1953); 93, 362 (1954); Rev. Sci. Instr. 25, 1088 (1954).
6. New ADL Magnetic Refrigerator, Rev. Sci. Instr. 26, 793 (1955).