Specific Heat of Ferric Ammonium Alum Below \(1^\circ\mathrm{K}\) and a Preliminary Determination of the Thermodynamic Temperature Scale

N. Kurti and F. Simon

Up to the present time, data on the behavior of paramagnetic salts in the range of temperatures attainable by the method of magnetic cooling have been obtained exclusively by means of magnetic measurements. These measurements consisted of demagnetization carried out in various initial fields and at various initial temperatures, and of the subsequent determination of the final temperature on an arbitrary temperature scale \((T^*)\) with the aid of extrapolation of Curie’s law. Combining these data with the easily established expression for the entropy in the initial state as a function of the field, one may obtain the entropy of the salt at a field equal to zero as a function of \(T^*\).

The introduction of calorimetric measurements would make it possible for us to establish the thermodynamic temperature scale, since, according to the second law of thermodynamics, the simultaneous knowledge of the change in the quantity of heat contained in a body and of the entropy in the transition from one state to another makes it possible to determine the absolute temperature. In addition, the heat capacity is a very sensitive indicator of changes occurring in the body as a result of the interaction of magnetic dipoles with the lattice or with one another, since they

always accompanied by an anomaly in the behavior of the specific heat[^20].

These anomalies occur near the temperature

\[ \theta_m=\frac{U}{R} \]

(where \(U\) denotes the interaction energy).

For the reasons set forth above, we undertook an investigation of the specific heat of iron ammonium alums, the results of which are reported here briefly. Regarding the technical side of the experiments, we note only that heat was supplied to the body under investigation by irradiation with \(\gamma\)-rays[^21]. This method of heating is very convenient also because the substance under study is heated quite uniformly*.

In Fig. 2 are shown the results obtained with a cylinder 20 mm long and 8 mm in diameter. The abscissas indicate the values \(T\), measured by means of a field of approximately 1 gauss directed parallel to the axis of the cylinder; along the ordinate axis is plotted

\[ C^*=\frac{\partial U}{\partial T^*}, \]

calculated for 1 grammion.

Combining this curve with the entropy curve mentioned above, we can determine the absolute temperature \(T\), which is the quotient obtained by dividing \(C^*\) by \(\dfrac{\partial S}{\partial T^*}\).

In this way we find that, within the limits of the experimental errors, which in our case amounted to about \(2\%\), for temperatures above \(0.10^\circ K\), \(T\) coincides with \(T^*\). At lower temperatures the experimental errors increase, but we can nevertheless assert that, under the conditions of our experiment, beginning with this temperature the absolute temperature \(T\) is higher than the temperature \(T^*\) obtained from Curie’s law; thus, for example, at \(T=0.06^\circ\) this excess is approximately \(10\%\). It may be asserted that upon further lowering down to \(0.03\), the tendency toward an excess increases*.

Fig. 2.

Although we are not yet able, from the diagram \(C^*, T^*\), to construct the diagram \(C, T\) with a high degree of accuracy, it may nevertheless be said that the character of the curve should not thereby undergo any substantial change.

We may therefore turn to consideration of the principal characteristic features of this curve, in particular the anomalies \(A\) and \(B\) exhibited by it.

According to Kramers, the ground state of a magnetic ion with an odd number of electrons can split for two reasons. First, such splitting can occur in each ion taken separately, owing to the Stark effect in the electric field of the lattice.

The remaining twofold degeneracy is no longer affected by the electric field and can be destroyed only by the direct interaction of magnetic dipoles with one another. Therefore one may expect²³ that the anomaly in the course of the specific heat due to the first effect should be expressed by a curve of flat form (a Schottky-type curve). The anomaly caused by the second effect, however, should be of the “interaction type,” i.e., should be expressed by a curve in which, for example, as at the Curie point. Consequently, we must attribute anomaly A to the interaction of the field of the crystal with the ion, and anomaly B, for which only the rising part of the curve has been observed, to the interaction of the magnetic dipoles with one another.

Such an interpretation is in agreement with experiments on diluted solutions of paramagnetic salts. Working with these salts in the temperature region lying somewhat below the maximum of anomaly A, we do not obtain appreciably lower temperatures²⁴, because the action of the crystal field on each ion does not change upon dilution. However, by using stronger magnetic fields, i.e., by working in the region of anomaly B, we diminish the effectiveness of dilution. The interaction energy between the dipoles becomes smaller, the sharp drop of the entropy curve shifts toward lower temperatures, and as a result, by adiabatic demagnetization we can obtain temperatures lower than when working with undiluted salts²⁵.

In conclusion we should like to point out that, in experiments of this kind, attention must be paid to the shape of the body under investigation, since at high susceptibility and at low temperatures it is of great importance. Thus, for example, in demagnetizing a cylinder made of ferric ammonium alum and starting from a temperature of \(1.2^\circ K\) and a field of 16 thousand gauss, we attain a final temperature \(T^*\) equal to \(0.025^\circ\) when the length of the cylinder is 2.5 times its diameter, and \(0.043^\circ\) when the ratio of length to diameter is 1.25.

At present we are repeating these experiments with a new apparatus which permits greater accuracy in magnetic measurements, and we are extending the investigation to other salts. A more detailed communication on all the work will be made after its completion.

1. It should be emphasized that there is no need at all to determine the absolute amount of energy transferred to the body by absorption measurements. It is quite sufficient to measure relative values and then calculate one absolute value in the region in which, over a considerable interval, \(T\) is equal to \(T^*\). ↩

2. This is explained by the fact that with our electromagnet, at strong fields it is impossible to attain sufficient constancy of the field. Because of this, an inaccuracy appears in the determination of \(\dfrac{\partial S}{\partial T^*}\). ↩

3. After this report had been read, Giock and MacDougall[^22] published a short communication on similar experiments carried out with certain gadolinium salts. The authors report similar results. ↩