FERROMAGNETIC TRANSFORMATIONS1

THE PROBLEM OF THE CURIE TEMPERATURE

Walter Gerlach, Munich

Ferromagnetic metals and alloys above a certain definite temperature have normal metallic properties, although quantitatively these properties may differ from the same properties of other metals. Thus, for example, the heat capacities of ferromagnets are above their theoretical value; the paramagnetism of ferromagnets is greater and depends more strongly on temperature than in most other metals.

We imagine that ferromagnetic bodies, like all other bodies, consist of carriers of elementary magnetic moments which, in the case of ferromagnets, are linked with one another by a molecular (inneres) field. In each crystallite2, free of external stresses, the resultant magnetic moment lies along some one definite crystallographic direction. For example, in iron it lies along axes of the type \([100]\), and in nickel along \([111]\). The molecular field according to Weiss is proportional to the magnitude of the resultant moment per unit volume. Each domain at low temperatures is magnetized completely to saturation. This magnetization is usually called internal or spontaneous.

Thanks to the chaotic distribution of the domains (with respect to the directions of their magnetization), and in a polycrystal, in addition, also thanks to the chaotic distribution of the crystallographic axes, any ferromagnetic specimen, despite its internal saturation, proves as a whole to be unmagnetized. Under the action of an external magnetic field the moments are oriented parallel, so that poles appear in the specimen. If all the elementary magnets throughout the specimen are oriented identically, one speaks of the state of saturation. The magnetic moment of a body in the state of saturation makes it possible to measure the magnitude of the spontaneous magnetization.

The anomalous behavior of almost all the physical properties of a ferromagnet is connected with spontaneous magnetization. These “ferromagnetic anomalies” do not depend on the external magnetization1, which does not affect them at all, or, if it does affect them, does so only very weakly.

The magnitude of the spontaneous magnetization depends strongly on temperature, decreasing as the latter is raised—at first slowly, and then very rapidly.

The temperature interval in which this transition from the ferromagnetic to the normal metallic state takes place is called the Curie temperature. In this relatively narrow temperature interval there occurs a very sharp change in all the properties of the ferromagnet. Therefore, approximately, one may consider that all specifically ferromagnetic properties occur below a certain temperature. Since up to the present only crystalline ferromagnets are known, it is quite possible that the crystalline structure is also essential for ferromagnetism. The only representative of liquid ferromagnets is amalgam; however, the question of the presence in amalgam of undissolved remnants of crystal has not yet been clarified.

For a long time it was thought that the disappearance of spontaneous magnetization occurs suddenly at some quite definite temperature. This seemed to be indicated by the sharp fall in the magnitude of magnetic saturation. It turns out, however, that the external magnetic field required for measuring saturation at all not too low temperatures makes it possible not only to observe spontaneous magnetization, but also changes the very magnitude of the latter. The parallel orientation of the resultant spontaneous magnetic moments of the domains throughout the specimen is now usually called “technical magnetization”; its limit is called “technical saturation”; while the change in the magnitude of the spontaneous magnetization under the influence of an external field is termed “true (wahre) magnetization”; in the limit it reaches “absolute saturation.”

Consequently, observations of spontaneous magnetization should in principle be carried out in the absence of an external magnetic field.

A change in spontaneous magnetization, i.e., in the magnitude of the true magnetization, is especially large precisely near the Curie temperature, i.e., in that temperature region where the spontaneous magnetization itself disappears. Moreover, in this same temperature interval the true magnetization is an unknown and apparently very complicated function of the magnetic-field strength \(H\), which rules out the possibility of an exact extrapolation of magnetic measurements to \(H \to 0\). Therefore, for observing the disappearance of spontaneous magnetization, methods that make use of a magnetic field are unsuitable.

But since spontaneous magnetization is responsible for almost all anomalies in the properties of a ferromagnet, both in respect to their magnitude and in respect to the temperature dependence of these properties, it is possible to determine the disappearance of spontaneous magnetization from the transition of these properties from the anomalous state to the normal one.

However, a second way is also possible. The point is that the form of the technical magnetization curve, i.e., the dependence of the external magnetization on the magnetizing field at constant temperature, depends rather strongly on external actions on the ferromagnet (in particular, such actions may be elastic stresses); the true magnetization, however, in all such experiments involving external actions, does not change, since here we are dealing with the influence of an external force on the initial distribution and orientation of the spontaneous regions, and not on the magnitude of the spontaneous magnetization itself. Consequently, one may assume that spontaneous magnetization disappears at that temperature at which the influence, for example, of elastic stresses on the form of the magnetization curve ceases.

Yet another possibility for determining the disappearance of spontaneous magnetization consists in investigating the change in the electrical resistance of ferromagnets in an external magnetic field. Although here too we use a magnetic field, it is nevertheless possible to distinguish technical and true magnetization. The point is that in the first case the increase in external magnetization occurs through a change in the orientation of the spontaneous resultant moments of the domains, which leads to an increase in electrical resistance in the direction of the magnetic field; in the second case (true magnetization) there is an increase in spontaneous magnetization, which, on the contrary, lowers the magnitude of the resistance.

Similar experiments are in principle possible for all anomalies of a ferromagnet. However, so far they have been carried out only for resistance and for the ferromagnetic thermomagnetic effect. Why precisely such experiments are of special interest is clear from the following considerations: ferromagnetic anomalies are caused by spontaneous magnetization and are in an experimentally observed relation with the latter. In sufficiently strong fields, especially near or above the temperature at which spontaneous magnetization disappears, the latter grows in the external field and can be measured by magnetic methods. At the same time, the properties of the body possessing ferromagnetic anomalies also change. Consequently, it is possible

approaches linearity; at room temperatures the increase in magnetization is only \(2\%_0\) at 20,000 oersteds.

Thus, in the magnetization curves with field there is a continuous transition from the ferromagnetic to the paramagnetic state and back. It follows from this that purely magnetic measurements give no grounds for concluding that the changes at \(TK\) are discontinuous in character.

In studying the dependence of ferromagnetic saturation on temperature, the question arises of extrapolating the curve near its sharpest fall (for Ni at \(T > \sim 300^\circ\)). Here the matter does not reduce to the question of whether or not the greater part of the spontaneous magnetization disappears in a narrow temperature region; rather, one asks whether the curve of spontaneous magnetization as a function of temperature approaches the temperature axis at a finite angle (Fig. 2, curve \(a\)) or asymptotically (Fig. 2, curve \(b\)). But, as already indicated above, the spontaneous magnetization (or ferromagnetic, technical saturation) in the region of the greatest fall of the temperature curve, because of the inevitable superposition of the true magnetization, cannot be determined by purely magnetic methods.

Fig. 2

As will be seen repeatedly from the further exposition, to this one may also add that the true magnetization is connected with a change in volume, which depends on temperature, so that, owing to the change in volume during magnetization, the boundary of the ferromagnetic region may shift if the measurements, as is usually done, are carried out at constant pressure.

In §§ 4–10 nonmagnetic methods of investigation and analysis of the magnetic transformation will be considered. They are based on the fact that ferromagnets possess anomalies in most of their properties, and these anomalies reach a very sharp maximum precisely at the temperature at which the magnetic properties also change just as sharply.

§ 4. HEAT CAPACITY DURING THE MAGNETIC TRANSFORMATION

Since the spontaneous magnetization \(\sigma\) decreases with increasing temperature, this means that, when a ferromagnet is heated, part of the heat goes to compensate the vanishing energy of the molecular field. Consequently, the heat capacity of ferromagnetic bodies must increase the more strongly, the greater the magnitude of the negative temperature coefficient of the spontaneous magnetization \(\left[\text{or, better, of the energy of the molecular field } \frac{1}{2}N\frac{d\sigma^2}{dT}, \text{ where } N \text{ is the Weiss molecular-field constant (see § 13)}\right]\).

Experiment shows that the heat capacity of Ni, \(C_p\), with increasing temperature at first grows slowly, then more rapidly up to the sha-

Particular caution is required with the question of the purity of specimens. The absolute value of \(TK\) and, evidently, also the extent of the transition region depend very strongly on the purity of the material. The exact course of the magnetic anomalies near \(TK\) likewise depends on purity (see § 4). On the other hand, the purity of the specimen does not affect the coincidence of the maxima of different anomalies at one and the same temperature.

After we later (see § 19) make use of the concept of the disappearance of “long-range order” at \(TK\) and of “short-range order” in the transition region, we shall understand why the latter changes with the dissolution of “foreign” atoms (or with impurities).

It must also be clarified whether the magnetic transformation, with which the anomaly of the coefficient of expansion is associated, affects the relief of stresses from cold working and deformation of the crystal. It turns out that the relief of cold-worked nickel proceeds especially rapidly precisely in the transition region. However, it must be admitted that up to now we do not know the nature of the connection between \(TK\) and the relief of the material.

The absence of the influence of the effects considered in §§ 1 and 2 gives a certain simplification for the development of our problem.

§ 3. DETERMINATION OF THE SPONTANEOUS MAGNETIZATION FROM MAGNETIZATION CURVES

If one measures the dependence of magnetization on the magnetizing field at high temperatures (\(\gg TK\)), a direct proportionality between them is found: the metal is paramagnetic (Fig. 1). As the temperature is lowered, the magnetization in a given external field is higher than is required by the Curie–Weiss law1. For Ni, at approximately below \(410^\circ\), the Curie–Weiss law loses its force. At this temperature the magnetization still increases proportionally to the field, but below \(400^\circ\) the proportionality disappears and the magnetization rises sharply even in weak fields, and the more sharply the lower the temperature. At these temperatures (\(\sim 400^\circ\)) a linear dependence no longer holds even at the highest attainable fields (according to Weiss and Forrer, \(\sim 22\,000\) oersteds). Between 360 and \(350^\circ\) the greater part of the magnetization is already attained in weak fields; as the field is increased the increase in magnetization decreases, and the dependence between magnetization and field

Fig. 1

Introducing these reservations, we minimally alter the usual conceptions, but in return obtain the precision that had previously been lacking in them.

§ 1. TEMPERATURE HYSTERESIS OF THE CURIE TEMPERATURE

Let us first clarify whether a ferromagnetic transformation has temperature hysteresis. At present this can be denied with certainty. In all cases in which this effect is observed, it is caused by structural transformations of the ferromagnet¹). Such changes in structure may occur in a not entirely pure ferromagnet, or in an alloy, if precipitation or dissolution of impurities takes place near the magnetic transformation. Upon precipitation of impurities the Curie temperature rises, and upon dissolution it falls. Only upon dissolution of a small amount of iron in nickel is an increase in \(TK\) obtained²).

§ 2. THE EFFECT OF MECHANICAL TREATMENT
ON THE CURIE TEMPERATURE

It has not yet been established whether cold working affects the magnetic transformation under elastic and under plastic deformation. If such an effect is observed, it is due to secondary causes—changes in composition. For example, cold working may be the cause of the precipitation of some constituent part of an alloy.

On the other hand, under hydrostatic pressure or tension, a shift of \(TK\) can probably occur. However, as yet there are no detailed measurements of magnetic properties at high pressures (see § 16).

It is known that under severe cold working the density of a metal decreases, but it is still not clear whether this occurs on account of macroscopic cracks or on account of molecular distortions of the crystal lattice. The effect on \(TK\) can hardly be appreciable, since at relatively high \(TK\) relaxation of the material already occurs during the measurement itself. Alloys with low \(TK\) are unsuitable for resolving this question, because the danger of changing the alloy composition during cold working is too great.

The fact that, under strong tensions, the Curie temperature does not change during measurement is proved by measurements of the anomaly of electrical resistance. The magnitude of the true magnetization near \(TK\) is also completely independent of tension.

It has not yet been clarified whether mechanical treatment affects the magnitude of the transition region, i.e., the character of the disappearance of spontaneous magnetization.

¹) A striking example of this is provided by the recent experiments of Sykesmith with ternary Fe—Ni—Al alloys (see Proc. Roy. Soc., 171, 525, 1939). Translator’s note.

²) Another example is the dissolution of vanadium (from 0 to 5%) in iron, likewise raising the Curie point of pure iron (see the work of Fallot in Ann. d. Phys., 1, 305, 1936).

again compare both effects and verify whether they obey one and the same law, and also whether the relationship between them depends on temperature.

These experiments, moreover, make it possible to determine with complete certainty whether the true magnetization is also connected with orientation processes. If the latter were the case, then the relations mentioned above would have to depend on the direction of magnetization (and, in the case of true magnetization, for example, at least in the case of a change in electrical resistance). However, the latter, under true magnetization, is entirely independent of direction.

Our task is to discuss all these experiments and to obtain answers to the following questions.

1. Does the spontaneous magnetization disappear suddenly, by a jump, at some definite temperature, or is there a certain temperature region in which the spontaneous magnetization (after a very steep fall, but not down to zero) disappears gradually?

2. What properties change during the magnetic transformation?

3. Is it possible, on the basis of the experimental facts known up to the present time, to understand those internal processes which determine the magnetic transformation?

The majority of the experiments described below were carried out with nickel, a smaller part (mostly not very precise) with iron, and only a few with alloys.

In order to obtain the greatest possible clarity and to avoid any ambiguity in all that follows, we shall introduce the following definition.

Let us call the “Curie temperature” \((T_k)\) that temperature at which all the physical properties of a ferromagnet that are conditioned by the natural spontaneous magnetization have a maximum anomaly.

We shall support this definition by the fact that: 1) this maximum of anomalies can be determined very precisely, and 2) all anomalies of one and the same material that have so far been investigated have their maximum at one and the same temperature.

The fact that at this temperature the magnetic properties as well (both the ferromagnetic properties and the true magnetization) undergo sharply expressed changes is a further confirmation of the definition given.

Although, as the totality of experimental data shows, the spontaneous magnetization above the Curie temperature does not disappear completely at once, but decreases gradually over a large temperature interval, it nevertheless makes no sense to introduce specially yet another temperature1 (for it would depend on the accuracy of measurement), associated with how strongly the asymptotic disappearance of the spontaneous magnetization affects the physical properties of the material. Therefore we shall call this temperature region the transition region.

...of a certain maximum, after which \(C_p\) at first falls very rapidly, and then more slowly, and at a temperature of \(\sim 405^\circ\) (50° above \(TK\)) reaches a certain value which, within the accuracy of measurement, no longer changes up to \(450^\circ\) (Fig. 3).

It has been established experimentally that both the shape of the curve \((C_p, T)\) and the magnitude of the anomaly itself depend very strongly on the purity of the material under investigation. Most striking is the fact that purer Ni has a larger anomaly than less pure Ni. In all probability this is due to the fact that in less pure Ni, owing to local fluctuations in the degree of purity of the material, there are many \(TK\)’s lying very close to one another. Such a “splitting” of \(TK\), with a sharp drop in the heat capacity, must lead to a noticeable decrease1 of the maximum.

The calculated (theoretical) values of the maximum of the magnetic anomaly of \(C_p\) are not reached in any of the numerous investigations. The best of these investigations—those of Lapp and Arens—agree well with one another; the discrepancies that occur accidentally are explained exclusively by Lapp’s extrapolation corrections. The results obtained by Lapp for Fe are in full agreement with the results for Ni. The measured value of the anomaly must always be regarded as a lower limit, since the measurement method itself presupposes averaging over an interval from 0.3 to 0.5°, which naturally, in the case of a sharp maximum, leads to a lowering of the measured heat capacity at \(TK\) in comparison with its true value. But the theoretical calculations also have a defect, since in them it is assumed that the molecular-field constant \(N\) does not depend on temperature (see § 13). However, it seems quite impossible that the entire transition region from \(TK\) to \(405^\circ\) should likewise be explained by some accidental properties of the substance, and that at \(TK\) there is a discontinuous change from its maximum value to the normal value. As we shall see from the subsequent exposition, such a transition region exists in all materials and for all properties. This is very weighty evidence that the energy of spontaneous magnetization at \(TK\) does not disappear at once, but that after a sharp drop in a narrow temperature interval, amounting to several degrees, it begins to tend asymptotically to zero and practically disappears at a temperature about 50° above \(TK\).

Fig. 3

In addition to the magnetic anomaly, which becomes unobservable (according to Arens’s measurements, less than 2%) at a temperature 50° above its maximum (at \(TK\)), nickel has yet a second anomaly of the heat capacity. This anomaly consists in the fact that the constant value of the heat capacity, after the disappearance of its magnetic anomaly, is 1.5 cal/mole higher than the theoretical value. In our opinion,

this anomaly is not directly connected with the ferromagnetism of Ni, but is due to the structure of the Ni atom itself. According to the modern theory of metals, this anomaly is connected with the degeneracy temperature of the electron gas and with the systematics of energy levels in the solid state (the “\(d\)-band” in the case of Ni). These questions go beyond the scope of our review; therefore we simply reckon with this anomaly as an experimental fact. A similarly abnormally high heat capacity is also observed in the nonferromagnetic metal palladium, which is very similar to nickel in its other properties (see § 18); thus it should also be so on the basis of theoretical considerations.

§ 5. ELECTRICAL RESISTANCE IN THE TRANSITION REGION1

The temperature coefficient of the electrical resistance of ferromagnets undergoes an especially sharp change. Below \(TK\) it tends toward a sharp maximum, while above \(TK\) it does not at once attain its low “normal” value; at first it falls rapidly, then more slowly, and reaches the region with normal temperature behavior only above \(400^\circ\), up to \(450^\circ\) (Fig. 4).

Fig. 4

An old controversial question is whether the resistance curve has a “kink” at the transformation temperature, which would indicate an actually definite Curie point. Then the curve of the temperature coefficient would have to break off sharply at the maximum and then begin from lower values. But then it is further required that this lower value of the temperature coefficient remain practically constant as the temperature increases.

Neither the one nor the other has been observed by us. The values measured by us, as by all other investigators, lie very close to one another, and therefore the “averaging” of the temperature coefficient is carried out only over intervals from \(0.1\) to \(0^\circ.5\), and, consequently, in practice we are dealing with true values (in comparison with the width of the maximum). A sharp jump in the temperature coefficient, which would correspond to a “kink” on the resistance–temperature curve, was not observed.

As for the other question, whether the entire anomaly of the resistance disappears immediately above \(TK\), and consequently attai—

...whether the temperature coefficient of its normal value decreases—here there is no disagreement of opinion. All investigators observe a transition region with a width of at least \(50^\circ\).

The anomaly of the electrical resistance is a consequence of the existence of the energy of spontaneous magnetization. This can be substantiated both by the direct connection between the square of the spontaneous magnetization and the resistance anomaly, and by the detection of a change in the resistance when the spontaneous magnetization is increased (and likewise with true magnetization) below \(TK\), as well as by creating true magnetization above \(TK\). It still remains unclear what role is played by the molecular-field constant above and below \(TK\). Here one must await exact measurements. But there is no doubt that above \(TK\) there exists a broad region with an anomalous temperature coefficient, and this once again confirms that spontaneous magnetization disappears only gradually.

By means of electrical resistance one can detect the presence of spontaneous magnetization above \(TK\) in yet another way. The point is that at sufficiently high temperatures the resistance decreases owing to true magnetization (independent of the direction of the magnetizing field), this decrease being proportional to the square of the magnetization. But above \(TK\), by about \(7^\circ\), the resistance at first increases and only then decreases (Fig. 5), but in such a way that its decrease is proportional not to \(\sigma^2\), but to \((\sigma^2-\sigma_0^2)\). It is natural to consider that \(\sigma_0\) is nothing other than the spontaneous magnetization remaining above \(TK\).

Fig. 5

Thus, experiment says that the normal resistance and the normal dependence of the resistance on true magnetization are attained only considerably above the temperature at which the anomaly has its maximum. And this is exactly the same result as follows from the course of the heat capacity of a ferromagnet.

§ 6. THE INFLUENCE OF THE MAGNETIC TRANSFORMATION ON THE RADIATION OF A FERROMAGNET

The Hagen–Rubens relation between resistance and radiation is well fulfilled for Ni in the long-wave region and at low temperatures. Later measurements of the intensity of radiation as a function of temperature also show a considerable anomaly at the magnetic transformation; namely, the temperature coefficient of radiation has a maximum at the same place as the temperature coefficient of electrical resistance. However, in this case the measurements are too complicated and do not make it possible to attain the same accuracy as in resistance measurements. One can only

to say that there is complete qualitative agreement between the results of both experiments.

The intensity of the radiation, like the resistance, reaches its normal value above \(TK\) only gradually; the transition region ends approximately around \(400^\circ\). The fact that the anomalously small radiation intensity is a consequence of spontaneous magnetization can also be proved by the fact that in strong fields above \(TK\), hence with large true magnetization, there occurs a decrease in the radiation intensity proportional to the square of the magnetization.

Near \(TK\) the optical properties of iron were also studied. Ornstein measured the reflecting power \((\lambda \sim 6500\,\text{\AA})\) as a function of temperature and found an increase of the latter by approximately \(4\%\) at temperatures between \(750\) and \(850^\circ\). Above this interval, up to \(930^\circ\), it remains independent of temperature. The increase in reflection means a simultaneous decrease in emissivity \((\varepsilon = 1-\rho)\).

It has been observed, however, that the emissive power at short wavelengths decreases, i.e. behaves oppositely to what takes place at long wavelengths. Final proof that in Ornstein’s experiment we are indeed dealing with spontaneous magnetization will be obtained only after a change in the reflecting power under true magnetization has been detected. For Ni this proof was obtained by us through direct determination of the intensity of long-wave radiation. It should also be noted that Ornstein showed that the change in the radiation intensity of iron extends far beyond \(TK\), as determined by the magnetic method.

§ 7. THERMOELECTROMOTIVE FORCE AND THE THOMSON COEFFICIENT AT THE MAGNETIC TRANSFORMATION

The course of the thermo-emf \(\dfrac{dE}{dT}\) in Ni, in comparison with that for Cu, shows a distinct anomaly at \(TK\) (Fig. 6, curve \(a\)). This is especially clear on the curve for the Thomson coefficient

\[ T\frac{d^2E}{dT^2}=f(T) \]

(Fig. 6, curve \(b\)). The form of curve \(b\) is in all details similar to the curve of the temperature coefficient of electrical resistance. Namely, in both cases there is a sharp maximum at exactly the same temperature; at higher temperatures there is at first a very rapid drop, and then a slower decrease, and only at approximately \(410^\circ\) is the normal course with temperature reached.

Fig. 6

The same measurements were also carried out on Ni—Cu alloys, which have \(TK\) of the order of \(250^\circ\). According to Foster, complete ...

qualitative similarity to pure Ni, only with a much lower and less sharp maximum of the anomaly. All this is in complete agreement with measurements of the electrical resistance on the same alloy.

§ 8. FERROMAGNETIC ANOMALY OF THERMAL CONDUCTIVITY

The thermal resistance of Ni at low temperatures increases normally with temperature; starting from 250°, a small anomalous increase first appears, which at \(TK\) reaches a sharp maximum. Above \(TK\) the resistance falls. But, in view of the fact that it is very difficult to measure the true thermal resistance at temperature gradients within the range from 0.5 to 0.7° per 1 cm, these measurements make it possible only to establish qualitative agreement with the results of measuring other anomalies.

§ 9. CHANGE IN ELASTIC PROPERTIES DURING A FERROMAGNETIC TRANSFORMATION

In the ferromagnetic state, the resultant magnetic moments of the domains (the directions of spontaneous magnetization), when elastic deformations are applied, leave their natural positions and thereby cause magnetostriction, which is added to the purely mechanical elastic deformation (Kersten). Therefore one must expect that, during a magnetic transformation, the elastic constants of a ferromagnet should change. In particular, when a ferromagnet is cooled near \(TK\), where the magnetic transformation begins, i.e. spontaneous magnetization appears, an anomaly of the elastic coefficients should be observed, the magnitude of which, as a function of temperature, depends on the change of spontaneous magnetization with temperature.

Fig. 7

Such an anomaly was indeed found; however, the measurements do not make it possible to say anything about the width of the transition region, i.e. about the temperature at which spontaneous magnetization first appears.

However, another aspect of this anomaly is important. If a ferromagnet is subjected to the action of a strong external field so that complete orientation of all domains along the field occurs and the orienting elastic forces do not act against the magnetic forces, then the anomaly should disappear. Experiment fully confirms this.

There is, however, an undoubted and notable exception to this rule, which occurs in the alloy \(58\%\ \mathrm{Fe} — 42\%\ \mathrm{Ni}\). Namely, the modulus of elasticity of this alloy decreases strongly under the action of

of the magnetic field after passing \(TK\) (Fig. 7). Dehring, considering this case, indicated that in this alloy the appearance of spontaneous magnetization must have a special influence on the elastic coefficients. The reason for this lies in the extremely large volume magnetostriction of the alloy in question under true magnetization, which indicates an especially strong change of the spontaneous magnetization for small changes of the lattice constant, and consequently also under elastic stresses. The anomaly consists in the fact that the load in measuring the modulus \(E\) changes not only the direction, but also the magnitude of the magnetization, which leads to an additional magnetostrictive expansion.

If this exception is disregarded, the modulus \(E\) does not change at \(TK\). If we take into account the observation cited above, then ferromagnets that possess volume magnetostriction at \(TK\), and correspondingly a change of magnetization under pressure (smaller than in the case of the alloy mentioned), should have at least a small anomaly of the modulus \(E\). But up to now no change of the true magnetization above \(TK\) under hydrostatic compression has been found.

§ 10. THERMAL EXPANSION IN THE TRANSITION REGION

Unfortunately, up to now very few investigations have been carried out on the thermal expansion of ferromagnets in the region of the magnetic transition, and, moreover, they do not agree with one another. In all probability both Ni and Fe have an anomaly in this case as well, but it is better not yet to draw any conclusions about the character of this anomaly.

Fig. 8
\(a\)—thermal measurements, \(b\)—calculations from magnetostriction

As Williams showed, in the case of Ni an increase was found in the coefficient of thermal expansion

\[ \frac{d \dfrac{l}{l_0}}{dT} = a \]

in the region of the magnetic transition (with a maximum at \(TK\)) and above the transition region (Fig. 8). Since for Ni (at lower temperatures) an increase of volume under true magnetization has been found, the coefficient of expansion with rising temperature should become smaller because of the decrease of the spontaneous magnetization, i.e. should have a course opposite to that observed by Williams. On the contrary, Dehring found a decrease of the length under true magnetization. The difficulty here is that one must exclude the rise of temperature caused by the magnetocaloric effect. Dehring’s results lead to a value of the thermal expansion which, in order of magnitude, agrees with Williams’s data, but they do not give an anomaly at \(TK\) (see § 17).

This question is closely connected with the problem of the variation of \(TK\) with pressure and therefore requires complete clarification (in § 16 we shall return to it once more).

§ 11. X-RAY STUDIES BELOW AND ABOVE THE TRANSITION REGION

All studies of changes in the lattice structure lead to negative results. An especially careful investigation was made of the fine structure of the \(K\)-absorption edge. Even in this case no changes were found on passing through the region of the magnetic transformation. This has been established especially rigorously for Ni (Koster).

The corresponding experiments with iron seem at first sight to indicate that the fine structure changes. But it has now been established that the change found in the fine structure between 600 and \(700^\circ\) has no relation whatever to the magnetic transformation.

We now turn to studies of the magnetic transformation in which typical magnetic methods are used. Despite the impossibility of determining the spontaneous magnetization by these methods, especially near the transformation, nevertheless from them one can obtain far-reaching conclusions, which in no way contradict the results obtained by nonmagnetic methods. Moreover, with the aid of these measurements one can obtain information on the nature of the carriers of the elementary magnetic moment and on its magnitude above and below the magnetic transformation.

§ 12. DETERMINATION OF THE MAGNETIC TRANSFORMATION BY THE MAGNETOCALORIC EFFECT

We have seen that the problem which must be solved in connection with the magnetic transformation can be solved if we succeed in separating the magnetization in the transition region into technical and true magnetization; the transformation ends when the first of these disappears. It is obvious that they cannot be separated by purely magnetic measurements (see above). In § 15 it will be shown that the presence of technical magnetization can be detected by applying an external action that acts only on the spontaneous magnetization and does not affect the true magnetization; elastic stresses, for example, may be chosen as such an action.

But one may choose another path as well, namely, to use effects that are caused only by the internal magnetic energy or by its changes. This condition is satisfied first of all by the magnetocaloric effect: the heating of a ferromagnetic body upon true magnetization.

If the external field produces only true magnetization, then the rise in temperature is \(\Delta T \sim \sigma^2\). If, however, there is also spontaneous magnetization, then the external field produces technical saturation equal to \(\sigma_0\), and, in addition, true magnetization \(\sigma\). The increase in temperature in this case is equal to \(\Delta T \sim (\sigma^2 - \sigma_0^2)\). Measuring simul-

temporarily $\Delta T$ and $\sigma^2$ and plot them on a graph (Fig. 9), then in the first case we obtain a straight line passing through the origin, while in the second case the straight line is obtained only at large values of $\sigma^2$; at smaller values of $\sigma^2$ the straight line bends and intersects the $\sigma^2$ axis not at the origin. Continuing the rectilinear part to the point of intersection with the $\sigma^2$ axis, we find the magnitude of the spontaneous magnetization corresponding to the given temperature.

It has been experimentally proved that considerably above $TK$ the proportionality between $\Delta T$ and $\sigma^2$, or $H^2$, holds for Ni, Fe, and Heusler alloys. Over the whole range of fields it ends approximately $15$–$20^\circ$ above $TK$. After this there begins a region of curves $(\Delta T,\sigma^2)$ of another form, in which the linear course of $\Delta T$ with $\sigma^2$ occurs only at large $\sigma^2$, while at smaller $\sigma^2$ the curve asymptotically approaches the $\sigma^2$ axis. Since there is a continuous transition between both types of these curves, it is quite understandable that no exact boundary can be established between them. Very important is Potter’s assertion that a single crystal of iron behaves (in this respect) exactly the same as pure electrolytic iron.

Fig. 9

Thus, it may be stated that, at least $15^\circ$ above $TK$, the magnetocaloric effect reveals a magnetization which does not manifest itself in the thermal effect and therefore is not produced by the external magnetic field, but exists spontaneously. This, then, is proof of the existence of spontaneous magnetization above $TK$.

§ 13. CHANGE OF THE CONSTANT OF THE “MOLECULAR FIELD” AT THE MAGNETIC TRANSFORMATION

According to Weiss, inside a ferromagnet—or, more precisely, inside each domain—there acts an internal molecular field $H_i$, proportional in magnitude to the spontaneous magnetization itself $\sigma_T$. Weiss further assumed that the coefficient of proportionality $N$ does not depend on magnetization and temperature: $H_i=N\sigma_T$. The magnitude $N$ is very considerable and has its own value in each ferromagnet; for example, in Ni $N\sim 5000$, in Fe $N\sim 1500$ (calculated per $1\ \mathrm{cm}^3$).

The disappearance of ferromagnetism depends on the disappearance of the molecular field, but not because $N$ tends to zero, but because the spontaneous magnetization disappears. Consequently, if above the magnetic transformation, where spontaneous magnetization disappears, an external field $H$ produces a magnetization $\sigma$, then inside the substance there will act a field $(H+N\sigma)$. Owing to the large value of $N$, at least near the transformation, we obtain $H\ll N\sigma$. The internal magnetic energy is then equal to $N\sigma^2$.

The determination of \(N\) can be obtained directly from the magnetocaloric effect (see § 12), for in the coefficient of proportionality between \(\Delta T\) and \(\sigma^2\), or \(\sigma^2-\sigma_0^2\) (when there is also spontaneous magnetization \(\sigma_0\)), apart from known quantities only \(N\) enters.

If Weiss’s assumption concerning the independence of \(N\) from temperature is correct, then the curves \((\Delta T,\ \sigma^2)\) at all temperatures should be straight lines parallel to one another. However, this is not so; moreover, in the transition region the straight lines turn noticeably, becoming steeper, and only far above it do they become parallel to one another. For Ni at \(420^\circ\) (i.e., \(60^\circ\) above \(TK\)), and for Fe even in single crystals (Potter’s measurements) at \(830^\circ\), there is still no parallelism between the straight lines. The largest value of \(N\) is approximately three times greater than in the ferromagnetic state.

Consequently, in the transition region \(N\) is a function of temperature (Fig. 10), but it does not depend on \(\sigma\), since otherwise the dependence \((\Delta T,\ \sigma^2)\) would not be rectilinear (Fig. 9).

Another determination of \(N\) according to Weiss’s theory follows from the behavior of the susceptibility \(\chi\) in the paramagnetic region, where \(\chi(T-\theta)=C\), and \(\theta \sim N\). In the factor of proportionality between \(N\) and \(\theta\), besides universal constants, there enters the square of the absolute saturation or (in other units) the square of the elementary magnetic moment. If the value of \(N\) is determined from the paramagnetic Curie constant \(C\), it does not coincide exactly with the value of \(N\) obtained from the magnetocaloric effect, but the difference is not very large. With the ferromagnetic value of the elementary moment, a substantially larger value of \(N\) is obtained.

Fig. 10

The change of \(N\) (and of the magnetic moment, see § 14) during a magnetic transformation has not yet been explained. Kornetzky, from his theory of the shift of \(TK\) upon a change of volume during spontaneous magnetization, derived consequences both for the calculation of the magnetic moment and for the magnetocaloric effect. In doing so Kornetzky showed that the relation between \(\Delta T\) and \(\sigma^2\) contains an additional term, which increases \(N\).

If near \(TK\) the thermal expansion has an anomaly (see § 10), then in the transition region the magnitude of this factor, which depends on the volume, changes, and together with it \(N\) also changes. It is still not clear whether we are on the right path here. Moreover, the calculations still contain very many extrapolations, for the measurements both of the magnetocaloric effect and of the magnitude \(N\) are still not very accurate.

Here it is also necessary to say a few words about Heisenberg’s theory, which first introduced into consideration the so-called “exchange forces” and made it possible to reduce ferromagnetism to general physical

physical conceptions. As far as I know, this theory requires that the coefficient of the molecular field \(N\) depend both on temperature and on magnetization.

The excessively large temperature dependence obtained in measuring the magnetocaloric effect can also be expected from the formulas of Heisenberg’s theory. But we cannot particularly rely on this, since it is not yet clear how the “volume effect” of magnetostriction, caused by true magnetization, affects the value of \(N\). On the contrary, the dependence of \(N\) on the magnitude of the magnetization is incompatible with all the measurements considered.\(^1\)

§ 14. THE MAGNETIC MOMENT OF NICKEL BELOW AND ABOVE THE CURIE TEMPERATURE\(^2\)

A fundamental question in the theory of ferromagnetism is the question of the origin of the magnetic moment of ferromagnets. Our ideas about the structure of the atom make it possible to connect the origin of this moment either with the electron spin or with the orbital moment. If both possibilities occur, then we have

\(^1\) In connection with what has been set forth in this paragraph, we shall allow ourselves to make the following remark of a general character. It must always be remembered that the Heisenberg–Weiss theory of ferromagnetism makes it possible to understand all the basic ferromagnetic properties, strictly speaking, only from the qualitative side. Therefore the quantitative formulation of the theory cannot in any way, and from no point of view, be regarded as correct. Those crude approximations which underlie this theory make it possible to claim quantitative agreement with experiment only for \(T \to \infty\). Therefore, rather, obtaining quantitative agreement (and not disagreement!) of the theory with experiment should require an additional explanation—since agreement has been obtained with so crude a theory, this conclusion may be obtained from some assumptions of a much more general (for example, thermodynamic) order.

Unfortunately, this circumstance is not always taken into account, especially by experimentalists. The simplicity and visual clarity of the Heisenberg–Weiss theory often lead to a whole series of experimental formulations of questions which, from the theoretical point of view, have no meaning.

The content of § 13 is a concrete example of such “temptation by theoretical speculations” of so major and demanding a specialist in the “purity” of experiment in the field of ferromagnetism as V. Gerlach. The point is that the concepts of “molecular field,” and, consequently, of the constant \(N\), referring to it, have no real physical meaning. It is clear that in exact calculations they would not enter the theory at all. Therefore, if it follows from experiment that \(N\) depends on temperature, this means only that in the given temperature range the quantitative formulas of the theory are roughly incorrect. Any attempts to squeeze the experimental data into the framework of these “incorrect” formulas, introducing artificial hypotheses about the temperature dependence of the constants entering these formulas, can hardly be considered fruitful, since the whole theoretical discussion in this case is reduced not to explaining, for example, why \(N\) changes with temperature, but to explaining why it was possible to “fit” the experimental results to the formulas of the Weiss theory with variable \(N\). Translator’s note.

\(^2\) This question, like the question of the different \(N\) above and below \(TK\), cannot be solved within the framework of the crude Heisenberg–Weiss model. Translator’s note.

FERROMAGNETIC TRANSFORMATIONS

a body with a resultant moment equal to the geometric sum of the components.

The nature of the carriers of the magnetic moment can be determined by means of the so-called gyromagnetic effect, by which we mean the magnetization of a specimen as a result of rotation about some axis of its own. If each carrier of an elementary moment is imagined as a rotating system, then gyroscopic forces must arise when the body rotates. Owing to the coupling that exists between the mechanical and magnetic moments, their ratio can be determined. For all ferromagnets, except pyrrhotite, this ratio is equal to 2 (instead of 1, as was expected at first). The explanation of this “gyromagnetic anomaly” is given by the quantum theory of the Zeeman effect. The factor 2 (the so-called \(g\)-factor) in the ratio mentioned is due to its electron origin. Thus, the carrier of the magnetic moment of ferromagnets is the electron with its own, so-called spin, magnetic moment.

There also arises the question of the magnitude of the magnetic moment falling on one atom of the metal (the “elementary” moment). Its magnitude is obtained by dividing the absolute saturation (i.e. its greatest value, attained at very low temperatures), per mole, by Loschmidt’s number \(L\): in the case of Ni, for example, we have

\[ \frac{\sigma_{\infty}\ \text{per } 1\text{ atom}}{L}=5.58\cdot 10^{-21}\ \mathrm{CGS} \]

per atom, or \(3\,385\) CGS per mole.

If the so-called Bohr magneton \(\mu_B\) is taken as the unit, then this quantity is almost exactly equal to \(0.6\,\mu_B\). The Weiss magneton \(\mu_W\) (the magnetic elementary moment per mole in \(1\,125.6\) absolute units) is exactly three times smaller than the measured value of the elementary moment of Ni, and therefore the latter is equal to \(3.0\,\mu_W\).

Measurements of the gyromagnetic effect have so far not revealed an orbital moment in ferromagnets. Therefore, in all likelihood, the magnetic moment consists exclusively of spins, but in the case of Ni not all atoms possess an electron participating in the composition of this moment, but only 60% of them. More precisely, one should say that in the lattice, for every 100 atoms there are 60 electron spins participating in ferromagnetism. This conclusion can be reached from the fact that, when atoms of other metals that give up outer valence electrons are added, ferromagnetism decreases. For the complete destruction of the ferromagnetism of Ni, it is necessary to add, of monovalent elements (Cu, Au), 60 atoms per 100; of divalent elements (Be, Zn), 30 per 100 (i.e. 30 Be—70 Ni); of trivalent Al, 20 per 100; and, finally, of tetravalent Sn, 15 per 100.

What, then, happens to the elementary moment during magnetic transformations? Above \(TK\), until very recently, only the magnitudes of the magnetic moments were measured, and only recently Sucksmith investigated the gyromagnetic effect in Ni.

Nickel at temperatures sufficiently exceeding \(TK\), over a very wide temperature range, possesses normal paramagnetism. Its susceptibility \(\chi\) then depends on temperature (though not very strictly) according to the Curie–Weiss law:

\[ \chi(T-\theta)=C=0.0055_5. \]

The temperature \(\theta\) is not equal to the measured \(TK\), which is \(358^\circ\), but lies considerably higher. From extrapolation of the experimental curve \(\left(\frac{1}{\chi}, T\right)\) one obtains \(\theta=372^\circ\).

From these measurements one can also calculate the number of magnetons from the paramagnetic measurements; one obtains

\[ \text{according to Weiss}\quad p_w=14.06\sqrt{C_M}=8\mu_W, \]

\[ \text{according to Bohr}\quad p_B=2.84\sqrt{C_M}=1.6\mu_B. \]

Consequently, in the case of Ni the elementary magnetic moment above \(TK\) is 2.67 times greater than in the ferromagnetic region for absolute saturation at low temperatures.

This difference does not yet mean that the elementary moment of Ni changes during the transformation. The point is that, according to quantum theory, there is the following fundamental distinction between magnetization in very strong and in very weak fields. In the case of saturation all magnetic moments are oriented identically. The magnitude of an individual moment is determined (see above) as the quotient of the saturation divided by the number of atoms. In the case of magnetization in weak fields, however, which is what occurs in measurements in the paramagnetic state considerably above \(TK\), spatial quantization must be taken into account. But then essentially different values are obtained for \(p\) than those required by classical theory. If only the electron spins are taken into account, then for the number of Bohr magnetons the result is the well-known Hund formula:

\[ p_B=g\sqrt{j(j+1)}, \]

which, because \(g=2\), and \(j=s=\frac{1}{2}\), passes into the old formula of Sommerfeld’s spatial quantization:

\[ p_B=\sqrt{4s(s+1)}. \]

The value \(g=2\) follows from the gyromagnetic effect of all ferromagnets. Thus, the ratio of the “effective moment” \(\mu_{\mathrm{eff}}\) (calculated in Bohr magnetons) above \(TK\) to the moment \(\mu\) obtained from saturation is equal to

\[ \frac{\mu_{\mathrm{eff}}}{\mu}=\frac{p_B\cdot\mu_B}{s\cdot g\cdot\mu_B}=\sqrt{3}, \]

i.e. it is greater than that obtained in the classical calculation.

Consequently, if the number of Bohr magnetons is calculated from the value of the Curie constant measured above \(TK\), then we obtain

0.9 instead of 0.6 from the saturation measurement. Thus, the discrepancy is reduced, but does not disappear. The same calculations for iron lead to an even smaller discrepancy.

The most acceptable assumption (for the possibility of another explanation see § 16) would be that above $TK$ orbital moments come into play. But it is precisely at this point that the full importance of Secksmith’s measurements is revealed; they showed that above $TK$ we are dealing with the same carrier of the magnetic moment—the electron spin $(g = 2)$—as at temperatures below $TK$.

In connection with Secksmith’s measurements, however, it must nevertheless be noted that they were not carried out on pure Ni. The high $TK$ of pure Ni makes it impossible to perform this very difficult experiment on it. Therefore the author carried out his experiments on three different mixed Ni—Cu crystals, whose $TK$ lies below zero; the measurements were conducted as though Cu had absolutely no influence on the magnetic properties of Ni in the paramagnetic state of these alloys. It was assumed only that, owing to the copper, there is a decrease in the density of Ni; because of this $TK$ is lowered, but the magnetic moment per Ni atom should remain unchanged. On the basis of this premise, Secksmith explained the results of his experiments.

It is necessary, however, to draw attention to the following. The normal paramagnetic behavior of Ni begins only at temperatures considerably higher than the temperature at which the spontaneous magnetization disappears, which earlier was taken as the beginning of normal paramagnetism. As the measurements show, the formula

\[ \chi (T - \theta) = C \qquad (\theta = 372^\circ), \]

holds only above $412^\circ$. And we have already seen above that the “normal” electrical resistance is reached only above $400^\circ$, whereas the maximum of the temperature coefficient lies at $360^\circ$.

If we take into account, moreover, that the ferromagnetic properties disappear at temperatures far above the maximum of the anomalies, then, strictly speaking, experiments to determine the nature of the carriers of paramagnetic moments should be carried out only in this temperature region. From Secksmith’s experiments, however, it is not clear whether they belong to the temperature region of “normal” paramagnetism. The author merely states that the experiments were carried out at room temperature and that $TK$, determined by the usual method (from the sharp drop of the saturation magnetization), lies between $-14$ and $-2^\circ$. But the measured values of the factor $g$ for all the alloys turned out to be identical.

Thus, as a result of all that has been set forth above, one may say that it is highly probable that both below and above the magnetic transformation the carrier of the magnetic moment of a ferromagnet is the electron (spin magnetism). The question remains unresolved why the elementary magnetic moment has different values in the ferromagnetic and paramagnetic regions. One possible explanation (according to Kornetskii) will be given in § 16.

§ 15. DETECTION OF FERROMAGNETIC PROPERTIES
IN THE TRANSITION REGION

In the introduction it was already indicated that it is possible to detect spontaneous magnetization in the transition region by using the characteristic properties of technical magnetization. One of these is the change in the shape of the curve of the technical magnetization of Ni under elastic deformations. According to Becker, under uniaxial tension or compression the spontaneous magnetization of the domains leaves its natural direction and is oriented along directions determined by the external stresses.

Such an influence is observed even at temperatures much higher than that at which the maxima of the anomalies lie, i.e., in the transition region, where, as was formerly believed, all the observed magnetization is exclusively true magnetization. Measurements of magnetization under longitudinal tensions of various magnitudes showed that in weak fields (from 1 to 10 oersteds) the magnetization curve is the flatter, the greater the elastic tension. In higher fields the differences are smoothed out more and more and disappear completely in fields from 300 to 900 oersteds.

The magnetization measured in higher fields is indisputably true and, consequently, does not depend on elastic stresses, which in other respects (apart from that mentioned above) do not affect the magnetic transformation. As for the change of magnetization under tension in weak fields, it undoubtedly indicates the presence of remnants of ferromagnetic spontaneous magnetization; for if rotation of the moments occurs under the influence of tension, then spontaneous magnetization must exist. The same fact—that, despite the difference in the values of the magnetization obtained in any fields under longitudinal tension, in higher fields one and the same magnetization is always obtained—indicates the independence of the saturation magnetization, and at the same time of the magnitude of the spontaneous magnetization itself, from elastic stresses.

§ 16. EFFECT OF PRESSURE ON THE CURIE TEMPERATURE

In true magnetization (obtained in sufficiently large fields above technical saturation and at temperatures below \(T_K\)) there occurs an increase in the volume of iron proportional to the field \(H\) (volume magnetostriction). This change in volume upon magnetization is thermodynamically connected with the change in true magnetization under pressure. But this, obviously, also means that when the lattice constant changes (as a result of pressure), the spontaneous magnetization also changes1.

At the same time the question arises of the dependence of \(T_K\) on pressure. Experimentally this question has been investigated many times, but so far no reliable results have been obtained.

In the Weiss theory, \(TK\) is given by the well-known formula

\[ \theta=\frac{N\rho s_0^2}{3MR} \]

(\(N\) is the molecular-field constant, \(\rho\) is the density, \(s_0\) is the saturation, \(M\) is the molecular weight, \(R\) is the gas constant). If one assumes that \(s_0\) is determined exclusively by electron spins (see § 14), then, upon a change in volume (with a constant lattice), only the constant \(N\) can change.

Kornecký considered a consequence of this assumption, according to which the molecular-field constant may also change upon heating, owing to thermal expansion, and therefore the Curie point, measured at constant volume, will be a function of temperature. This should be understood as follows: if, when heating a ferromagnet from some definite low temperature, the volume is kept constant, then \(TK\) will change because of the pressure arising in this process. The value of \(TK\), obviously, will be a function of the initial temperature, since the constant \(N\) no longer changes at constant volume.

Kornecký derived a formula connecting this shift of \(TK\), when the volume is changed, with the change in volume under true magnetization. From such measurements on iron at low temperatures, it can be extrapolated that at constant volume \(TK\) is displaced from its “normal” value (i.e., that measured at \(4\ \mathrm{atm}\)) of \(770^\circ\) to \(395^\circ\); the pressure that must be applied in order to maintain the volume constant must then reach \(6000\ \mathrm{atm}\).

Experimentally it was found that \(TK\) in Ni—Fe alloys decreases, while in Ni—Cu alloys it increases with pressure. In both cases \(TK\) lies near room temperature. In pure Ni and Fe and other alloys, measurements of the effect of pressure on \(TK\) gave a negative result.

Since measurements of the volume effect require that the change in spontaneous magnetization under pressure be eliminated, the question is only whether extrapolation to \(TK\) is legitimate. It is not legitimate if the volume effect changes sign, or if it decreases with increasing temperature and vanishes at the magnetic transformation. The latter, according to Döring’s measurements, does not occur in the case of pure Ni; here it has been shown that under true magnetization there is at least a shortening of length, and probably also of volume.

In connection with the problem considered here, the new experiments of Ebert and Kussmann are of great importance, although they too still do not give an unambiguous answer. These authors measured the change of magnetization curves as a function of pressure (of several thousand atmospheres) for various alloys and at temperatures up to the magnetic transformation.

Since these measurements at high temperatures are very difficult, binary and ternary alloys whose \(TK\) lies between room temperature and \(200^\circ\) were chosen as the objects of investigation.

For all alloys a decrease in the saturation magnetization was found and, consequently, also in the spontaneous magnetization under pressure. The authors discovered an increase of the spontaneous magnetization at higher fields (and, consequently, of the true magnetization), as if independent of pressure; this result is surprising. Above \(TK\) there is purely true magnetization, likewise not dependent on pressure.

The influence of a given pressure on the saturation magnetization undoubtedly decreases with increasing temperature, i.e. \(TK\) does not change under pressure. Extrapolating the measured curves, the authors find that at very low temperatures the influence of pressure on the magnetization also ceases.

How great the disagreement in the experimental results still is can best be seen from the fact that Steinberger obtained, for an alloy of \(30\%\ \mathrm{Ni} — 70\%\ \mathrm{Fe}\), a strong decrease of \(TK\) at room temperature (normal \(TK \simeq 125^\circ\), and at \(10\,000\ \mathit{atm}\) less than \(20^\circ\)), whereas Ebert and Kussmann found for an alloy of the same composition a very large decrease of the magnetization under pressure (while \(TK\) remained unchanged). It is very probable that the metallographic state of the alloy plays an essential role; therefore measurements should certainly be made on pure metals.

Although in this question there are still very many obscure points, it is nevertheless very important that, in principle, it can provide a new point of view. Thus, for example, it seems very suitable for eliminating a difficulty that until now has been associated with the explanation of magnetic transformation. This difficulty was already set out in § 14 and reduces to the existence of a difference in the magnitude of the magnetic moment (per atom) below and above \(TK\), despite one and the same carrier of this moment. Owing to the above-mentioned change of \(TK\) at constant volume with temperature, the formula for determining the number of magnetons acquires a correction term which considerably lowers the classically calculated values. However, because of the clearly extrapolational character of the calculations, it is of course impossible to make any precise quantitative statements.

Similar considerations arise in the calculation of the magnetocaloric effect. Potter (see §§ 12, 13) found here that near \(TK\) the Weiss molecular-field constant \(N\) increases strongly. Since near \(TK\) there is an anomaly of thermal expansion, a correction must likewise be made because of the anomaly of the volume effect, which accounts for the change in the constant \(N\). But all these arguments will remain academic until the fundamental question of any theory of spontaneous magnetization is resolved by means of measurements: do there correspond to the firmly established fact (from magnetic measurements and the volume effect) of a change of spontaneous magnetization under pressure in the region of low temperatures changes of this magnetization at \(TK\), and also a displacement of \(TK\) under pressure?

§ 17. MAGNETOSTRICTION DURING MAGNETIC TRANSFORMATION

With technical magnetization, Ni possesses negative magnetostriction. In addition, magnetostriction also occurs with true magnetization, which was studied in detail by Döring. Measurements of this latter effect are very difficult because, simultaneously with it, there is also a magnetocaloric effect, which produces heating and, in connection with this, leads to an increase in volume. However, Döring succeeded in separating these two effects to such an extent that it became possible to draw qualitative conclusions about magnetostriction under true magnetization.

Ordinary magnetostriction, caused by the orientation of magnetic moments, decreases with increasing temperature, but at \(TK\) it does not disappear, instead approaching zero asymptotically. It can be observed up to 395°. This agrees with the discovered influence of elastic stresses on magnetization in the transition region (see § 15) and is new proof of the existence of this very transition region above \(TK\), where spontaneous magnetization disappears asymptotically, since negative magnetostriction is due to the presence of spontaneous magnetization.

Döring showed that, contrary to earlier measurements in which the magnetocaloric effect was not taken into account, the change in length under true magnetization is negative. However, here too not everything is yet clear. The question concerning the displacement of \(TK\) under pressure (§ 16) remains unclear, but qualitative proof of the presence of spontaneous magnetization in the transition region by means of magnetostriction is useful for the future solution of this question.

§ 18. CHEMICAL ACTIVITY AND THE CURIE TEMPERATURE

Hedvall investigated the catalytic activity of nickel and nickel alloys for a large number of reactions as a function of temperature: for example, for \(N_2O \to N_2 + O\), \(N_2O + H_2 \to N_2 + H_2O\); \(C_2H_4 + H_2 \to C_2H_6\), etc. In the ferromagnetic region, a clearly expressed change in the reaction rate was always found, irrespective of how high \(TK\) was (the latter was lowered, for example, by adding copper to nickel). Exactly the same influence of the magnetic transformation on the reaction rate was also found in the case of iron, iron alloys, and Heusler alloys (“magnetochemical effect”).

§ 19. TEMPERATURE OF MAXIMAL ANOMALIES AND THE TRANSITION REGION OF A FERROMAGNET IN THE NORMAL STATE

The experimental facts considered in §§ 4–8 lead with complete obviousness to the conclusion that the magnetic transformation is divided into two stages. The first of them occupies a very narrow but finite temperature interval, while the second occupies a rather broad temperature region, at least 50° wide.

This separation is also obtained from purely magnetic measurements, namely even from the fact that the normal paramagnetic behavior of nickel begins only at \(410^\circ\), so that between the normal ferromagnetic and paramagnetic regions of existence of this moment there is a broad transition region. Such a separation, of course, is meaningful only if, after passing through this transition region, the ferromagnetic metal actually acquires normal properties. This we shall now show for nickel on the basis of reliable measurements.

Above \(450^\circ\) the curve of the dependence of resistance on temperature is only weakly curved. The resistance of palladium has exactly the same property. If one considers not the specific resistance itself, but its so-called \(r\)-value (i.e., the ratio of the resistance at temperature \(T\) to the resistance at \(0^\circ\),

\[ r_T=\frac{R_T}{R_0}, \]

then even quantitative agreement is obtained between the values of \(r\) for Ni and Pd.

The same is also true for the paramagnetic susceptibility of Ni and Pd at \(T \gg \theta\). Palladium approximately follows the Curie–Weiss law, but only with a “negative Curie temperature”:

\[ \chi(T+\theta)=\text{const.} \]

If one plots \(1/\chi\) as a function of temperature, then parallel curves are obtained for Ni and Pd. In other words, if not \(T\) but \(T-\theta\) is chosen for the abscissa, then the curves for Ni and Pd simply coincide. The magnetic moments of Pd and Ni for \(T \gg \theta\) are likewise identical. Finally, there is still another analogy in the behavior of the heat capacities of Ni and Pd at temperatures above the transition region. In both cases the heat capacities increase with temperature much more than is the case for other metals. Above \(TK\) nickel has an excessively large heat capacity (approximately \(1.5\ \mathrm{cal}/\mathrm{mol}\) greater than that of other metals).

Thus we see that between the properties of paramagnetic Ni and Pd at temperatures above \(450^\circ\) there is a great analogy and even quantitative coincidence. Deviations from the properties of palladium (“Palladium character”) at temperatures from \(400^\circ\) and below, down to the beginning of the clearly ferromagnetic region, must be connected with the magnetic transformation. Consequently, one may assert that the complete disappearance of spontaneous or internal magnetization occurs gradually.

The temperature of the maxima of the anomalies also has a fully real physical meaning. On the basis of numerous investigations carried out at the Munich Institute, it may be regarded as firmly established that, for pure Ni, the maxima of all anomalies coincide at the same temperature at which the sharpest change in ferromagnetic properties also occurs. In various experiments with the purest nickel that we could find (verification of the actual purity was carried out by comparing the measured \(r\)-values with the limiting values of the residual resistance according to Meissner’s measurements), a number of its properties were measured, which had a maximum anomaly at one and the same

and at the same temperature. All measurements joined in the table by a brace were made on one and the same wire specimen.

Measured quantities	Temperature, °C, of the anomaly maximum
Heat capacity	353
Temperature coefficient of resistance $\left(\dfrac{dR}{dt}\right)_{\max}$	$353 \pm 1$
Saturation magnetization $\left(\dfrac{dI}{dt}\right)_{\max}$	354.5
True magnetization $\left(\dfrac{dI}{dH}\right)_{\max}$	354
Temperature coefficient of resistance $\left(\dfrac{dR}{dt}\right)_{\max}$	353.8
Temperature coefficient of resistance	354.9
Thermo-emf and the Thomson coefficient $\left(\dfrac{d^2E}{dt^2}\right)_{\max}$	354

Recently the initial permeability $\mu_0$ has also been measured; it has a steep drop at $354^\circ$.

This raises the question why, in earlier observations, there was such a large difference between the temperature corresponding to $\left(\dfrac{dR}{dt}\right)_{\max}$ and the disappearance of the initial permeability. Apparently this is explained very simply. There is no doubt that all earlier measurements were made with materials that were not entirely pure. The specimen may, moreover, be not entirely homogeneous; its separate, more or less pure parts have different Curie points. In such a specimen, the disappearance of the “mean initial permeability” will under all circumstances occur at a higher temperature than the temperature corresponding to the “mean maximum” of the temperature coefficient of resistance. This follows simply from the different form of the temperature dependence of the two quantities.

It is also somewhat dangerous to draw any conclusions about the disappearance of spontaneous magnetization on the basis of measurements obtained on specimens of insufficiently pure metal. The same applies to alloys, even in the case where they constitute a stable mixed crystal, for small fluctuations in composition from crystallite to crystallite have the same effect as impurities in a pure metal.

Never, even in the case of the purest nickel, has a discontinuous change in its properties been observed, i.e. neither kinks nor jumps have been found. Of course, we are always still entitled to say that an ideally pure single crystal might perhaps possess these properties.

For a clear understanding of the results obtained, one may point to the following. A sharp change in the magnetic properties in a narrow temperature interval near $TK$ corresponds to the disappearance

magnetic order at large distances, but at the same time there still remains an interaction between the nearest regions, which disappears only gradually.

Perhaps, in connection with this view, the following observation should be made. In the case of alloys, broader transition regions and more diffuse maxima of magnetic anomalies are always observed than, for example, in pure Ni. Such mixed crystals in large volumes may be regarded as homogeneous, i.e., for a sufficiently large number of Ni atoms in different parts of the specimen there is the same number of impurity atoms. But if smaller volumes are considered, the fluctuations of concentration become greater in passing from one region of the specimen to another; therefore the interaction between neighboring regions in different places will differ strongly. But this, according to the statement made above, must lead to an increase of the transition region (see the addition to the translation).

LITERATURE

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Electrical resistance and true magnetization

1. W. Gerlach and K. Schneiderhan, Ann. Physik, 6, 772, 1930.
2. W. Gerlach, Ann. Physik, 8, 649, 1931; 12, 849, 1932.
3. K. Schneiderhan, Ann. Physik, 11, 385, 1931.
4. E. Englert, Ann. Physik, 14, 589, 1932; Z. Physik, 74, 748, 1932.

Influence of magnetic transformation on emissive power

1. E. Hagen and H. Rubens, Berl. Ber., 778, 1909; 467, 1910.
2. W. Gerlach, Ann. Physik, 25, 209, 1936.
3. E. Löwe, Ann. Physik, 25, 213, 1936.
4. V. A. Suydam, Phys. Rev., 5, 437, 1915.
5. M. Kahanovicz, Lincei Rend., 30, 2, 132, 1921.
6. L. S. Ornstein and I. H. v. d. Veen, Physica, 3, 289, 1936.

Thermo-emf and Thomson coefficient in a magnetic transformation

1. A. W. Foster, Phil. Mag., 18, 470, 1934.
2. A. Hammer, Ann. Physik, 30, 728, 1937.
3. I. Dorfmann and R. Janus, Z. Physik, 54, 277, 1929.
4. K. E. Grew, Phys. Rev., 41, 356, 1932.

Thermo-emf and resistance of an alloy

1. H. Bittel and W. Gerlach, Ann. Physik (in press).

Ferromagnetic anomalies of thermal conductivity

1. A. Hammer (see 40).
2. K. Honda and T. Simidu, Sc. Rep. Tôhoku Univ., 6, 219, 1917.
3. F. H. Schofield, Proc. Roy. Soc., London, 107, 206, 1925.
4. M. van Dusen and S. M. Shelton, Bureau of Standard, 12, 429, 1914.

Change of elasticity upon transformation

1. E. Giebe and E. Blechschmidt, Ann. Physik, 11, 905, 1931.
2. M. Kersten, Z. Physik, 85, 708, 1933.
3. O. von Auwers, Ann. Physik, 17, 83, 1933.
4. F. Förster and W. Köster, Z. Metallk., 29, 116, 1937.
5. O. Engler, Ann. Physik, 31, 145, 1938.
6. W. Döring, Ann. Physik, 32, 465, 1938.

Thermal expansion of the transition region

1. E. P. Harrison, Phil. Mag., 7, 626, 1904.
2. C. L. Williams, Phys. Rev., 46, 1011, 1934.
3. P. Hidnert, Bureau of Standard, 5, 1305, 1930.
4. H. Esser and G. Müller, Arch. Eisenhüttenwesen, 7, 215, 1933.
5. M. Kornetzki, Z. Physik, 97, 662, 1935.
6. W. Döring, Z. Physik, 163, 560, 1936.

X-ray studies above and below the transition region

1. D. Coster (personal communication).
2. I. D. Hanawalt, Z. Physik, 70, 293, 1931.

Magnetocaloric effect

1. P. Weiss et R. Forrer, C. R., 178, 1347, 1446, 1670, 1924.
2. P. Weiss, Ann. d. Phys., 17, 97, 1932.
3. H. H. Potter, Proc. Roy. Soc., London, 146, 362, 1934.

Change of the “internal-field” constant during the magnetic transformation

1. H. H. Potter (see 64).
2. W. Gerlach, H. Bittel u. S. Velayos (see 26).
3. M. Kornetzki, Z. Physik, 98, 289, 1935.
4. W. Heisenberg, Z. Physik, 49, 619, 1928.

Gyromagnetic effect

1. W. de Haas, Proc. Amsterd. Acad., 18, 1281, 1916.
2. E. Beck, Ann. Physik, 60, 109, 1919.
3. W. Sucksmith and Bates, Proc. Roy. Soc., London, 104, 499, 1923.
4. W. Sucksmith, Nature, 134, 936, 1934; Helv. phys. Acta, 8, 205, 1935.

Magnetization of paramagnetic nickel

1. W. Sucksmith and R. P. Pearce, Nature, 140, 970, 1937; Proc. Roy. Soc., London, 167, 189, 1938.

Detection of ferromagnetic properties in the transition region

1. R. Becker, Z. Physik, 62, 253, 1930.
2. R. Becker u. M. Kersten, Z. Physik, 64, 660, 1930.
3. G. Scharff, Z. Physik, 97, 73, 1935; Ann. Physik, 25, 223, 1936.

Change of TK with pressure

1. R. L. Steinberger, Physics, 4, 153, 1933.
2. Chi-Sun Yeh, Proc. Am. Acad., 60, 503, 1925.
3. H. Ebert u. A. Kussmann, Physik. Z., 38, 437, 1937; 39, 598, 1938.
4. L. Hadam s u. J. W. Greene, Phil. Mag., 12, 361, 1931.
5. D. R. Ray-Chhudhuri, Z. Physik, 71, 473, 1931.
6. E. Englert, Z. Physik, 97, 94, 1935.
7. M. N. Michejew, Sow. Phys., 3, 393, 1933.
8. A. Michels, A. Jaspers, J. de Boer u. J. Strijland, Physica, 4, 1007, 1937.
9. M. Kornetzki, Z. Physik, 87, 560, 1934; 97, 662; 98, 289, 1935.
10. W. Döring, Z. Physik, 103, 560, 1936.

Magnetostriction during the magnetic transformation

1. W. Döring, Z. Physik, 103, 560, 1936.

Temperature of the maximum anomalies and the transition region under normal conditions

1. W. Gerlach, H. Bittel u. S. Velayos, Münchner Ber., 81, 1936.
2. H. Bittel u. W. Gerlach, Ann. Physik (in press).

Analogy in properties

1. W. Gerlach, H. Bittel u. S. Velayos, Münchner Ber., 81, 1936.
2. F. W. Jaeger u. W. A. Veenstra, Acad. Wet. Amsterd., 37, 280, 1934.

Difference in the position of anomalies in insufficiently pure specimens

1. A. Kussmann u. A. Schulze, Physik. Z., 38, 42, 1937.
2. H. Bittel u. W. Gerlach, Ann. Physik (in press).

TRANSLATOR’S ADDENDUM

Soon after the publication of the report by W. Gerlach translated above, a very interesting article by L. S. Stilbans appeared: “Short-Range and Long-Range Order in Ferromagnetic Bodies” (ZhETF, 9, 432, 1939).

In this article the author quite rightly notes that “usually, in describing ferromagnetism one starts from long-range order and, in this way, obtains results that are only a rough approximation to reality.” Indeed, as Dirac showed [see, for example, P. A. M. Dirac, Principles of Quantum Mechanics, ONTI, 1937, § 61 (38), p. 244], the energy operator of the exchange forces can be written in the following form:

\[ W=\sum_{f<f'} I_{ff'}(1-\boldsymbol{\sigma}_f\boldsymbol{\sigma}_{f'}), \tag{1} \]

where \(\boldsymbol{\sigma}_f\) and \(\boldsymbol{\sigma}_{f'}\) are the operators of the electron spins \(f\) and \(f'\). This expression can also be interpreted quasi-classically, understanding by \(\boldsymbol{\sigma}_f\) and \(\boldsymbol{\sigma}_{f'}\) ordinary numbers corresponding to unit vectors along the directions of the magnetic moments of the spins. Then it follows at once from (1) that “the energy, and consequently also the additional heat capacity, of a ferromagnetic body is wholly determined by the short-range order; the long-range order manifests itself here only insofar as it influences the short-range order.”

If the state of complete order is taken as the zero of energy and the interaction only between nearest neighbors is taken into account, then the quasi-classical form of (1) will be:

\[ W=2In_{ab}, \tag{2} \]

where \(I\) is the exchange integral between nearest neighbors, and \(n_{ab}\) is the number of antiparallel neighboring spins. Let us further denote the total number of right spins by \(N_a\), of left spins by \(N_b\), the number of neighbors of right ones by \(n_{aa}\), and of left ones by \(n_{bb}\). Then the following obvious relations hold:

\[ 2n_{aa}=zN_a-n_{ab} \quad\text{and}\quad 2n_{bb}=zN_b-n_{ab}, \tag{3} \]

where \(z\) is the number of nearest neighbors. To determine \(n_{ab}\), it is necessary to add one more equation to (3). Stilbans uses for this the so-called quasichemical equilibrium, namely, he applies the “law of mass action” to the “reaction”

\[ \begin{aligned} (aa)+(bb) &\to 2(ab)\\ (\downarrow\downarrow)+(\uparrow\uparrow) &\to 2(\uparrow\downarrow). \end{aligned} \]

Then

\[ \frac{n_{aa}n_{bb}}{n_{ab}^{2}} = \frac{z_{aa}z_{bb}}{z_{ab}^{2}} = \frac{1}{4}e^{\frac{4I}{kT}} = c \tag{4} \]

(where \(z_{ik}\) is the phase sum). From (3) and (4) we find that

\[ n_{ab}=\frac12 Nz\left[\frac{\sqrt{1+4\frac{N_aN_b}{N^2}(4c-1)}-1}{4c-1}\right]. \tag{5} \]

To calculate the free energy \(F=W-TS\), one must also know the entropy \(S\). Stilsbans assumes that the latter is determined exclusively by the long-range order

\[ S=\left(k\ln\frac{N!}{N_a!N_b!}\right)_{\max}. \]

From the conditions \(\delta F=0\) and with the additional constraint \(N_a+N_b=N=\mathrm{const}\), we find

\[ \frac{N_a}{N_b} = \exp\left\{ \frac{2zI}{kT}\, \frac{N_a-N_b} {N\sqrt{1+4\frac{N_aN_b}{N^2}(4c-1)}} \right\}, \]

or the degree of long-range order (Fig. 1)

\[ \eta=\frac{N_a-N_b}{N} = \operatorname{tgh}\left[ \frac{zI}{kT} \frac{\eta}{\sqrt{1+(4c-1)(1-\eta^2)}} \right]. \]

Thus, for the energy \(W\), the author obtains the following limiting values:

\[ W=\frac12 NzI(1-\eta^2) \qquad \text{for } T<T_{kp}, \]

\[ W=\frac12 NzI-\frac{2}{1+e^{\frac{2I}{kT}}} \qquad \text{for } T>T_{kp} \]

(Fig. 2). Comparing these formulas with formula (7) in the presentation by R. Becker (printed below), we see that in Stilsbans’s theory an essentially different result is obtained above \(T_{kp}\). The heat capacity for \(T<T_{kp}\)

\[ C_m=\frac{dW}{dT} = -\frac{NzI}{2}\frac{d\eta^2}{dT}, \]

and for \(T>T_{kp}\)

\[ C_m=\frac{NzI}{2}\, \frac{4} {kT^2\left(1+e^{\frac{2I}{kT}}\right)^2}. \]

Fig. 1  Fig. 2  Fig. 3  Fig. 4

From Fig. 3 it is clear that the theoretical graph of the temperature dependence is in complete qualitative agreement with the experimental curve.

The temperature dependence of short-range order

\[ \sigma=\frac{n_{aa}+n_{bb}-n_{ab}}{\frac{1}{2}Nz} \]

according to (5) and (3) has the form (Fig. 4):

\[ \sigma \sim \eta^{2} \qquad (T<T_{\mathrm{cr}}), \]

\[ \sigma \sim \tanh \frac{I}{kT} \qquad (T>T_{\mathrm{cr}}). \]

From Stilbans’ work, namely from the fact that for \(T>T_{\mathrm{cr}}\), \(\sigma \ne 0\), it follows that above the Curie point there are still “residues” of spontaneous magnetization.

1. This circumstance is taken into account, for example, in Becker’s thermodynamic theory of volume magnetostriction; Z. Physik, 87, 547, 1934. Translator’s note. ↩↩↩↩↩↩↩

2. Here the author adheres to the original formulation of Weiss’s hypothesis that each individual crystallite of a polycrystalline ferromagnet is uniformly magnetized to saturation. However, now, after detailed studies of ferromagnetic single crystals, it has been established beyond doubt that both single crystals and individual crystallites of a polycrystal, in the absence of an external magnetic field (and of residual magnetization), break up into separate—usually smaller than the crystallite, and even for very fine materials—regions, the so-called regions of spontaneous magnetization, or domains. The magnitude, form, and distribution of the domains in a specimen are determined by the competition of various kinds of interaction energies in a ferromagnet; therefore only in very special, exceptionally small-sized cases may one assume that domains coincide with crystallites. In connection with this, we take the liberty of translating kristallit everywhere as “domain,” or “spontaneous region.” Translator’s note. ↩