MAGNETIC PROPERTIES OF ALUMS AT TEMPERATURES BELOW 0.1°K

In 1949 several papers appeared\(^{1—4}\) devoted to the study of the magnetic properties of alums and Tutton salts at temperatures below \(0.1^\circ\)K.

To obtain temperatures below \(0.5^\circ\)K, the method of adiabatic demagnetization of paramagnetic salts is used. However, it is impossible to measure the temperature directly (with a thermodynamic method) during demagnetization. Therefore one usually proceeds as follows.

First, some magnetic parameter \(x\) is chosen. As such a parameter, as the authors indicate, one may choose either \(\chi'\) (the real part of the complex magnetic susceptibility), or \(\chi''\) (the imaginary part of the complex magnetic susceptibility), or the residual magnetization \(\Sigma\) (which is nonzero below the Curie point; see below).

To find the entropy as a function of the magnetic parameter, the authors carry out adiabatic demagnetization from temperatures greater than \(1^\circ\)K (at such temperatures the entropy can be calculated). In the final state they leave a small field (of the order of one oersted), and in this field they measure the magnetic parameter \(x\). In this way the function \(S(x)\) is obtained (since the entropy does not change during demagnetization).

Then a definite amount of heat \(\Delta Q\) is imparted to the paramagnetic salt and the corresponding change \(x\) is measured. Using these data and the second law of thermodynamics, one can determine the absolute thermodynamic temperature of the final state, obtained as a result of adiabatic demagnetization.

There are several ways of imparting heat to the paramagnetic salt; at temperatures above the Curie point the authors placed the specimen in an alternating magnetic field; in the case of temperatures below the Curie point, heat was supplied by passing through a certain number of hysteresis loops (see below).

As a result of the experiments the authors determine the magnetic susceptibility \(\chi\), the remanent magnetization \(\Sigma\), and the entropy \(S\) as functions of the absolute temperature \(T\).

The following substances were studied: potassium-chromium alum \((\mathrm{KCr}(\mathrm{SO}_4)_2 \cdot 12\mathrm{H}_2\mathrm{O})\), iron-ammonium alum \(\mathrm{NH}_4\mathrm{Fe}(\mathrm{SO}_4)_2 \cdot 12\mathrm{H}_2\mathrm{O}\), and two Tutton salts \((\mathrm{K}_2\mathrm{Cu}(\mathrm{SO}_4)_2 \cdot 6\mathrm{H}_2\mathrm{O}\) and \((\mathrm{NH}_4)_2\mathrm{Mn}(\mathrm{SO}_4)_2 \cdot 6\mathrm{H}_2\mathrm{O})\). As the authors indicate, in the case of Tutton salts the accuracy of the results obtained is low, and therefore we shall confine ourselves to presenting the results obtained by the authors for the alums.

The experiments showed that the magnetic susceptibility of the alums increases as the temperature is lowered, reaching a maximum at a certain temperature.

Further, it turned out that the alums possess a Curie point, below which remanent magnetization appears.

The table gives the values of the temperature corresponding to the maximum magnetic susceptibility \((T_m)\), and the Curie temperature \((T_k)\), for two of the alums studied.

The authors point out that the difference between \(T_m\) and \(T_k\), possibly, is not real, but is due to the conditions of the experiment (different methods of supplying heat).

The maximum values of the magnetic susceptibility are*) 1.9 for potassium-chromium alum and 9 for iron-ammonium alum. The corresponding values of the magnetic permeability are approximately 25 and 115.

Thus, the experiments showed that chromium and iron alums at very low temperatures become ferromagnetic, the ferromagnetism of the iron alums being much stronger than that of the chromium alums (a higher Curie point and larger \(\chi\)).

It should be noted that the ferromagnetism of the alums is caused not by exchange interaction of spins, which in alums plays no role owing to the large distances between the paramagnetic ions, but by the magnetic interaction of the paramagnetic ions \((\mathrm{Cr}^{+++}, \mathrm{Fe}^{+++})\) with one another. Therefore one should expect a strong magnetic anisotropy of these alums below the corresponding Curie points.

A simple estimate gives that the Curie point caused by the magnetic interaction of paramagnetic spins should, in the case of alums, be of the order of \(0.05^\circ\ \mathrm{K}\). Thus, iron alums behave normally; as for chromium alums, their magnetic properties are anomal—

*) The authors give not the values of the susceptibility, but the values of the so-called magnetic temperature, which is readily converted into susceptibility if the Curie constant is known.

... At lower temperatures the appearance of ferromagnetism in them is, for some reason, delayed.

As experiments have shown, below the corresponding Curie points both iron and chromium alums possess a residual magnetization of the order of several

\[ \frac{\text{gauss}\cdot \text{cm}^3}{\text{mol}} . \]

In the case of chromium alums the authors also observed hysteresis loops.

The figure shows the entropy curve of potassium chromium alums as a function of temperature, obtained by Gorter and coworkers. In order to understand the form of this curve, one must consider the splitting of the energy levels of the paramagnetic ion caused by the electric field of neighboring molecules (the so-called crystalline Stark effect).

According to X-ray studies\(^5\), the chromium or iron ions in the alums are each surrounded by six water molecules, which form an almost regular octahedron, slightly deformed in the direction of a space diagonal of the elementary cube. Therefore the electric field in which these ions are situated is cubic with a small trigonal addition (in the elementary cell there are four paramagnetic ions, and they have different directions of the trigonal axes, corresponding to the four space diagonals of the elementary cube).

The state of the free ion \(\mathrm{Cr}^{+++}\) is \(3d^3\,{}^4F_{3/2}\). However, the crystalline field completely quenches the orbital moment \(L=3\), and the free spin alone remains (at temperatures considerably exceeding \(0.25^\circ\mathrm{K}\), see below), \(S=\frac{3}{2}\). The state of the free ion \(\mathrm{Fe}^{+++}\) is \(3d^5\,{}^6S_{5/2}\), corresponding to the free spin \(S=\frac{5}{2}\).

In the case of the \(\mathrm{Fe}\) ion the cubic field splits the sixfold-degenerate spin level into two- and fourfold-degenerate levels (the lower one being the twofold-degenerate level). In the case of the \(\mathrm{Cr}\) ion, however, the cubic field does not cause a splitting of the fourfold-degenerate spin level; this level is split only by the trigonal part of the field into two doubly degenerate levels.

The splitting \(\delta\) of the spin level of the paramagnetic ion by the electric field of the lattice is determined by the authors from the requirement that the entropy observed by them experimentally at temperatures above \(0.2^\circ\mathrm{K}\) coincide with the theoretically calculated entropy\(^ {6,7}\). This gives:

\[ \delta = 0.25^\circ\mathrm{K}\quad \text{for KCr alums,} \]

\[ \delta = 0.20^\circ\mathrm{K}\quad \text{for } \mathrm{NH}_4\mathrm{Fe}\text{-alums.} \]

At temperatures considerably exceeding \(\delta\), the spin is free and has \(2S+1\) possible orientations. Accordingly, the entropy (per mole, divided by the gas constant) must be equal to \(\ln 6\) for iron alums and to \(\ln 4\) for chromium alums, which agrees with experiment (at temperatures below \(2^\circ\) K one may neglect the lattice entropy obtained by the authors by comparison with the spin entropy).

According to Nernst’s theorem, as the temperature is lowered the entropy must tend to zero. In fact, there are two reasons for this: first, at temperatures below \(\delta\) the entropy must become equal to \(\ln 2\), since only two possible orientations remain for the spin. Second, owing to the magnetic interaction of the spins, the entropy falls to zero.

A drop in the entropy is indeed present on the curve; however, the following circumstance is puzzling: the curve has a rather long plateau (the temperature scale is logarithmic). If a plateau is present, one would expect the value of

\[ \frac{S}{R} \]

on the plateau to be equal to \(\ln 2\); in reality, however,

\[ \frac{S}{R} \]

on the plateau is less than \(\ln 2\) by about 0.25.

The authors suggest that this may possibly be explained by the freezing of the spin of the chromium nucleus as a result of its magnetic interaction with the spin of the chromium ion (9.5% of natural chromium consists of the isotope \(\mathrm{Cr}^{53}\), whose nucleus has spin).

We note in conclusion that the lowest temperature obtained by Gorter’s group (by demagnetizing potassium chromium alums) is \(0.003^\circ\) K, which is a record-low temperature.

G. R. Khutsishvili

References Cited

1. D. De Klerk, M. J. Steenland and C. G. Gorter, Physica 15, 649 (1949).
2. M. J. Steenland, D. De Klerk and C. G. Gorter, ibid. 15, 711 (1949).
3. M. J. Steenland, L. C. Van-Der-Marel, D. De Klerk and C. G. Gorter, ibid. 15, 906 (1949).
4. A. H. Cooke, Proc. Phys. Soc. 62, A, 269 (1949).
5. H. Lipson and C. A. Beevers, Proc. Roy. Soc. 148, 664 (1935).
6. J. H. Van-Vleck, Journ. Chem. Phys. 5, 320 (1937).
7. M. H. Hebb and E. M. Purcell, Journ. Chem. Phys. 5, 338 (1937).