SATURATION MAGNETIZATION AND CRYSTAL CHEMISTRY OF FERRIMAGNETIC OXIDES

E. W. Gorter *)

CONTENTS

Introduction ........................................................................ 279
I. Spinel structure ............................................................. 282
II. Theory of ferrimagnetism .................................................... 300
III. Experimental methods ...................................................... 315
IV. Experimental proof of the correctness of Néel’s hypothesis: ferrites .... 322
V. Experiments on the angular dependence of superexchange interaction ........ 335
VI. Saturation magnetic moment and crystal chemistry of ferrimagnetic spinels containing titanium. VII. Ferrimagnetic oxides containing chromium: the system $\mathrm{Li}_{0.5}^{\mathrm{I}}\mathrm{Fe}_{2.5-a}^{\mathrm{III}}\mathrm{Cr}_{a}^{\mathrm{III}}\mathrm{O}_4(\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5}\mathrm{O}_4 — \mathrm{Li}_{0.5}\mathrm{Fe}_{0.5}\mathrm{Cr}_2\mathrm{O}_4)$. VIII. Ferrimagnetic spinels containing aluminum. IX. Other ferrimagnetic oxides containing chromium: the system $\mathrm{Mn}^{\mathrm{II}}\mathrm{Fe}_{2-a}^{\mathrm{III}}\mathrm{Cr}_a\mathrm{O}_4(\mathrm{MnFe}_2\mathrm{O}_4 — \mathrm{MnCr}_2\mathrm{O}_4)$.

INTRODUCTION

To a considerable extent thanks to Snoek’s work, “ferrites,” i.e., oxides with the formula $\mathrm{MeO}\cdot\mathrm{Fe}_2\mathrm{O}_3$ or $\mathrm{Me}^{\mathrm{II}}\mathrm{Fe}_2^{\mathrm{III}}\mathrm{O}_4$, are becoming increasingly important as materials for high-frequency cores, because they combine useful ferromagnetic properties with high electrical resistivity. In the formula given above, $\mathrm{Me}^{\mathrm{II}}$ denotes either a divalent ion Mn, Fe, Co, Ni, Cu, Zn, Cd, Mg $(0.5\mathrm{Li} + 0.5\mathrm{Fe}^{\mathrm{III}})$, or two or more ions of these metals in the case of mixed crystals. In recent years the principal ferromagnetic properties of ferrites have been investigated in detail. In what follows we shall deal with the saturation magnetization of these ferrites and related materials.

) E. W. Gorter, Philips Research Reports 9*, No. 4, 295—320; No. 5, 321—365; No. 6, 403—443 (1954). Translated from the English by A. S. Pakhomov and N. A. Smol’kov.

Practically all these materials have the same crystal structure as the mineral spinel, whose structure (Bragg, 1915) may be regarded as a cubic, almost close packing of oxygen ions \((r = 1.32\ \text{Å})\) with metallic ions, having radii \(0.4\text{--}1.0\ \text{Å}\), distributed over interstices of two kinds \((A\) and \(B)\), which are surrounded by 4 and 6 oxygen ions, respectively. (In what follows we shall call the indicated interstices the sites of the sublattices \(A\) and \(B\).) The spinel structure is described in all details in Section I.1.

X-ray and theoretical investigations, chiefly by Verwey and his collaborators, have shown that the distribution of different metallic ions between the \(A\) and \(B\) sites is determined mainly by the Coulomb energy, including also the ordering energy within each sublattice \(A\) and \(B\), and by the individual tendency of certain ions toward fourfold or sixfold coordination. Often identical ions occur in both types of lattice sites. These investigations are summarized in Section I.2.

The ferromagnetic moment per molecular unit, obtained from measurements of the saturation magnetization \(\sigma\) at low temperatures, is not simply the sum of the magnetic moments of the individual ions, but is considerably smaller than it; \(\mathrm{ZnFe_2O_4}\) and \(\mathrm{CdFe_2O_4}\) are not even ferromagnets. Néel in 1948 gave a theory of these materials, using the basic assumption that in them the negative \(AB\) interaction between the magnetic moments of the sublattices \(A\) and \(B\) predominates, causing an antiparallel orientation of these moments, so that the resultant magnetic moment \(m\) is equal to the difference between the magnetic moments of the sublattices \(A\) and \(B\). Néel calls this uncompensated antiferromagnetism ferrimagnetism*). With the aid of this theory he was able to explain the data on susceptibility and saturation that were at his disposal at that time, and to predict the discovery of several types of anomalous curves of the temperature dependence of magnetization, which he expected to encounter in the case when the partial magnetic moments of the sublattices \(A\) and \(B\) differ little from one another. Such anomalous behavior can occur only in the case of ferrimagnetism.

Equally negative interactions \(AA\) and \(BB\) may be either insignificant in comparison with the interaction \(AB\), so that the magnetic moment of each sublattice is the sum of the moments of the ions entering into this sublattice, or they may be

*) In translation we retain Gorter’s term “ferrimagnetism,” but consider it necessary to note that this term has not received general recognition in the scientific literature (see on this question the article by Ya. G. Dorfman in Izv. AN SSSR, ser. fizich., 16, 4, 412 (1952)). (Translator’s note.)

comparable with the interaction \(AB\), and in this case the magnetic moment of one of the sublattices is smaller than the sum of the magnetic moments of the ions composing it. Yafet and Kittel (1952) showed theoretically that in this case the indicated sublattice is split into 2 or 4 different parts: in each of these parts the ionic magnetic moments are parallel to one another, but the moments of different parts form angles with one another.

The nature of the exchange coupling consists in an indirect exchange interaction involving diamagnetic oxygen ions; the theory of this exchange was set forth by Kramers in 1934. Anderson’s theory (1950) indicates that the strength of this superexchange interaction depends on the angle (metal ion)—(oxygen ion)—(metal ion). Both theories are considered in Section II.

The purpose of the present investigation was above all to obtain experimental evidence for Néel’s theory. The results of our measurements of the saturation magnetization of a number of series of mixed crystals with the spinel structure can almost all be explained by Néel’s theory. For the group of materials which for the most part have relatively high Curie temperatures, the resulting magnetic moments are in agreement with the theory developed for the case of a predominant \(AB\) interaction.

In a number of cases the picture of the ion distribution obtained from our measurements proves to differ from that which might have been expected on the basis of information on the structure of binary spinels. The distribution of cations in most cases can be understood from statistical considerations, assuming a definite order in the tendency of the cations to occupy \(A\) positions within a series of compounds. In some cases the influence of the degree of short-range order is noticeable.

In one series this could be confirmed by means of X-ray diffraction; in another case, in order to find the ion distribution, in addition to the saturation data, data on the effective \(g\)-factor were used. Certain anomalous curves \(\sigma=\sigma(T)\), predicted by Néel, were found in a series in which \(m\) changes sign with change in composition. It was found that the absence in this series of those curves \(\sigma=\sigma(T)\) for which \(\sigma\) is equal to zero at some temperature between \(0^\circ\) K and the Curie temperature is due to the nonuniformity of the cation distribution. In another system we found materials in which such curves \(\sigma=\sigma(T)\) occur. The results of measurements for another group of materials, having relatively low Curie temperatures, are in agreement with the theory developed for the case of comparable magnitudes of the interactions \(AB\) and \(BB\). In one system the resulting moment \(m\) changes sign, which should not have occurred for any distribution of ions if parallelism of the ionic magnetic moments in each sublattice is assumed.

Experimental data are discussed on the dependence of the interaction force on the angle (metal ion)—(oxygen ion)—(metal ion), and it is shown that the presence of such a dependence makes it possible to explain the values of the resultant moments in other crystal structures.

I. SPINEL STRUCTURE

I.1. Geometry of the spinel lattice

Since we shall examine in detail the magnetic properties of oxide spinels as a function of the crystal structure, we shall give a fuller description of this structure than is available in the literature, drawing attention to those features that are of interest in connection with the magnetic properties indicated.

Since only oxides will be discussed, the expression “oxygen ion” will everywhere below be substituted for the term “anion.”

The crystal structure of the mineral magnetite (\(\mathrm{Fe_3O_4}\)) and of spinel (\(\mathrm{MgAl_2O_4}\)) was determined in 1915 by W. H. Bragg\(^1\) and, simultaneously with him, Nishikawa\(^2\).

Their space group is \(O_h^7 — F3dm\) (cubic)\(^3\). The positions of the atoms in the spinel structure are as follows\(^4\):

8-fold position: 8 metal ions in \((a)^*\)
\(0, 0, 0;\ \frac14, \frac14, \frac14;\)

16-fold position: 16 metal ions in \((d)^*\)
\(\frac58, \frac58, \frac58;\ \frac58, \frac78, \frac78;\ \frac78, \frac58, \frac78;\ \frac78, \frac78, \frac58;\)

32-fold position: 32 oxygen ions in \((e)\)

\[ u, u, u;\ u, \bar u, \bar u;\ \frac14-u, \frac14-u, \frac14-u;\ \frac14-u, \frac14+u, \frac14+u; \]

\[ \bar u, u, \bar u;\ \bar u, \bar u, u;\ \frac14+u, \frac14-u, \frac14+u;\ \frac14+u, \frac14+u, \frac14-u; \]

with translations

\[ +(0,0,0;\ 0,\tfrac12,\tfrac12;\ \tfrac12,0,\tfrac12;\ \tfrac12,\tfrac12,0). \]

A center of symmetry exists at each point of the 16-fold position.

The unit cell obviously contains eight molecular units, or “molecules,” \(\mathrm{MgAl_2O_4}\), or, in more general form, \(\mathrm{Me_3O_4}\), if \(\mathrm{Me}\) represents several metal ions in a definite proportion.

From the presence of the translations

\[ +(0,0,0;\ 0,\tfrac12,\tfrac12;\ \tfrac12,0,\tfrac12;\ \tfrac12,\tfrac12,0) \]

it is seen that the unit cell (a cube with edge equal to \(a\)) consists of two different groups, each of four cubes with edges \(\tfrac12 a\) (octants) and identical ionic positions.

\[ \text{*} \]
In several publications other positions of the unit cell have been used, in which there are 8 metal ions in \((8f)\) and 16 metal ions in \((16c)\).

Ionic positions are different in two octants separated by a face, and identical in octants separated only by an edge. Consequently, the simplest picture is obtained if only the ion positions in two adjacent octants are singled out (Fig. 1).

Fig. 1 is chosen from an “idealized” structure with oxygen parameter \(u = 3/8\). In reality \(u\) is usually somewhat larger:

Fig. 1. Unit cell of the spinel structure. The positions of ions are shown only in two octants. The dotted circles belong to other octants. Solid lines indicate the fourfold and sixfold coordination of the corresponding positions of the metal ions. Here and below, the large circles are oxygen ions; the small shaded circles are metal ions in octahedral sites; the small unshaded circles are metal ions in tetrahedral sites. The drawing is made for the case \(u = \dfrac{3}{8}\).

when \(u > 3/8\), the oxygen ions are displaced from their ideal positions in the \([111]\) direction away from the nearest tetrahedral ion. The radii of the ions in Fig. 1—5 are quite arbitrary and, perhaps, too small in scale.

It is seen from Fig. 1 that each octant contains on its body diagonals four oxygen ions (large spheres), which lie at the corners of a tetrahedron and thus form a face-centered cubic lattice for \(u = 3/8\), and four interpenetrating face-centered lattices for \(u \ne 3/8\). The octant located on the left contains, at its center, a metal ion (small sphere, unshaded) in position \((8a)\), lying at the center of a tetrahedron of oxygen ions. We shall call this ion an ion located in a tetrahedral site, or a tetrahedral ion. In the octant on the right one sees four metal ions (small shaded spheres) in positions \((16d)\), each of which is surrounded by an octahedron formed by six oxygen ions. We

we shall call each such ion an ion situated at an octahedral site, or an octahedral ion.

Each octant, moreover, contains metal ions in position \((8a)\) (i.e., tetrahedral ions) at all the remaining corners, in such a way that the nearest neighboring tetrahedral ions (i.e., the ions at the corners and center of the left octant) are not separated from one another by an oxygen ion.

Figs. 2 and 3 show the positions of the metal ions only at the tetrahedral sites and only at the octahedral sites, respectively.

Fig. 2. Positions of metal ions only at tetrahedral sites. Each ion is surrounded by a regular tetrahedron of tetrahedral ions. The asterisks refer to the superstructure discussed in Section I.2.3.

Fig. 3. Positions of metal ions only at octahedral sites. Each ion forms the vertex of two regular tetrahedra of octahedral ions having only this ion in common. The asterisks refer to the superstructure discussed in Section I.2.3.

Figs. 4 and 5 show one octahedral ion surrounded by tetrahedral ions, and one tetrahedral ion surrounded by octahedral ions, respectively; in Fig. 5 the ion under consideration is located at the point \(^{1}/_{2},\,^{1}/_{2},\,^{1}/_{2}\).

We shall see that, in the superexchange interaction to be discussed in Section II.3.2, the Me—Me distances play no role. What is important here are the Me—O distances and the Me—O—Me angles. Therefore in Fig. 6 (thin lines) a certain number of triangular configurations Me—O—Me have been outlined for \(u\), slightly greater than \(^{3}/_{8}\), which in all probability occur in all ferrimagnetic spinels. The shortest Me—O distances fall into different groups: the group of nearest neighbors, denoted by \(p\) and \(q\), and the group of more distant distances, denoted by \(r\), \(s\), and \(t\); all the remaining distances are appreciably larger. In Fig. 6 we have confined ourselves to all those triangles for which

Fig. 4. Positions of metal ions only in tetrahedral sites and of one octahedral ion; its six nearest neighbors—tetrahedral ions—are shown by solid circles. The six neighbors form two equilateral triangles, each of which lies in a plane intersected by the spatial diagonal perpendicular to it (indicated by --------) at the shortest distances

\[ \left(\pm \frac{1}{24}a\sqrt{3}\right) \]

from the octahedral ion.

Fig. 5. The octahedral sublattice and one tetrahedral ion; the ion under consideration is displaced to the point

\[ \frac{1}{2},\quad \frac{1}{2},\quad \frac{1}{2}, \]

in order to show the tetrahedral ion with its twelve nearest neighbors—octahedral ions; these ions are marked by solid circles.

one Me—O distance is equal to \(p\) or \(q\), and the second Me—O distance is equal to \(p, q, r, s\), or \(t\). The ten tri-

Fig. 6. Configurations Me—O—Me occurring in the spinel lattice with one, at least, shortest Me—O distance (\(p\) or \(q\)) and with the other distances no longer than \(r, s\), and \(t\).

angles indicated in the figure also have five distinct Me—Me distances: \(b, c, d, e\), and \(f\). The ten interionic distances introduced above are expressed in Table 1 in terms of \(a\) and \(u\), or, more precisely, for convenience in terms of \(a\) and

\[ \delta = u - \frac{3}{8}. \]

Table 1

Me—O distances	Me—Me distances
\(\displaystyle p = a\sqrt{\frac{1}{16} - \frac{1}{2}\delta + 3\delta^2}\)	\(\displaystyle b = \frac{1}{4}a\sqrt{2}\)
\(\displaystyle q = a\left(\frac{1}{8} + \delta\right)\sqrt{3}\)	\(\displaystyle c = \frac{1}{8}a\sqrt{11}\)
\(\displaystyle r = a\sqrt{\frac{11}{64} + \frac{1}{4}\delta + 3\delta^2}\)	\(\displaystyle d = \frac{1}{4}a\sqrt{3}\)
\(\displaystyle s = a\sqrt{\frac{3}{16} + \frac{1}{2}\delta + 3\delta^2}\)	\(\displaystyle e = \frac{3}{8}a\sqrt{3}\)
\(\displaystyle t = a\left(\frac{1}{4} - \delta\right)\sqrt{3}\)	\(\displaystyle f = \frac{1}{4}a\sqrt{6}\)

In Fig. 6 the shaded circles again represent octahedral ions, the unshaded circles—tetrahedral-

…ions, and the large circles are oxygen ions; the ratios of the radii and distances are correct in a rough approximation.

The number above each triangle, or the total number of oxygen ions drawn around each triangle, indicates how many times the given type of triangle occurs between two separately chosen ions. In addition, the numbers in parentheses near each ion indicate the number of neighboring cations with which it forms a triangle of a definite type (cf. pre and tqe) at a given Me—Me distance. Consequently, the number indicating how many times a definite triangle occurs for a selected ion is the product of the number indicated above each triangle and the number indicated near the ion.

Finally, the plane in which, or near which, the triangle lies is indicated below the triangle.

I. 2. Crystal chemistry of oxide spinels

I.2.1. Metallic ions occurring in oxide spinels

Below is given a list of metallic ions which, as has been found, occur in oxide spinels (their Goldschmidt radii are indicated in parentheses):

\[ \mathrm{H}^{+}(-);\quad \mathrm{Li}^{+}(0.78);\quad \mathrm{Cu}^{+}(1.01?)?;\quad \mathrm{Ag}^{+}(1.13). \]

\[ \mathrm{Cd}^{2+}(1.03);\quad \mathrm{Mg}^{2+}(0.78);\quad \mathrm{Ca}^{2+}(1.06)^{*});\quad \mathrm{Mn}^{2+}(0.91); \]

\[ \mathrm{Fe}^{2+}(0.83);\quad \mathrm{Co}^{2+}(0.82);\quad \mathrm{Ni}^{2+}(0.78);\quad \mathrm{Cu}^{2+}(0.85?);\quad \mathrm{Zn}^{2+}(0.82). \]

\[ \mathrm{Al}^{3+}(0.57);\quad \mathrm{Ti}^{3+}(0.69)?;\quad \mathrm{V}^{3+}(0.65);\quad \mathrm{Cr}^{3+}(0.64); \]

\[ \mathrm{Mn}^{3+}(0.70)?;\quad \mathrm{Fe}^{3+}(0.67);\quad \mathrm{Ga}^{3+}(0.62);\quad \mathrm{Rh}^{3+}(0.68),\quad \mathrm{In}^{3+}(0.93). \]

\[ \mathrm{Ti}^{4+}(0.69);\quad \mathrm{V}^{4+}(0.65);\quad \mathrm{Mn}^{4+}(0.52);\quad \mathrm{Ge}^{4+}(0.44);\quad \mathrm{Sn}^{4+}(0.74). \]

\[ \mathrm{Mo}^{6+}(0.62?);\quad \mathrm{W}^{6+}(0.63?). \]

A question mark in parentheses denotes uncertainty regarding the value of the ionic radius; a question mark outside parentheses denotes doubt as to the presence of this ion in oxide spinels.

It is seen that the radii of all these ions lie between 0.44 and approximately 1 Å, with the exception of the ion \(\mathrm{Ag}^{+}\). Some ions having radii lying within the indicated limits have not been found in oxide spinels; among them are all pentavalent ions. It is quite probable that some of them are capable of occurring in spinels, whereas for others the spinel would be unstable with respect to the oxides entering into it or, in the general case,

*) Only up to 0.35 per molecular unit \(\mathrm{Me}_{3}\mathrm{O}_{4}\).

of one or more compounds with different structures. Failure in obtaining binary spinels has been described for the case of Zr\(^{4+}\).

Binary spinels of the following types are known:

\[ \mathrm{Me}^{\mathrm{I}}_{1/2}\mathrm{Me}^{\mathrm{III}}_{5/2}\mathrm{O}_4,\quad \mathrm{Me}^{\mathrm{I}}_{4/3}\mathrm{Me}^{\mathrm{IV}}_{5/3}\mathrm{O}_4\,^{*}),\quad \mathrm{Me}^{\mathrm{I}}_2\mathrm{Me}^{\mathrm{VI}}\mathrm{O}_4; \]

\[ \mathrm{Me}^{\mathrm{II}}\mathrm{Me}^{\mathrm{III}}_2\mathrm{O}_4,\quad \mathrm{Me}^{\mathrm{II}}_2\mathrm{Me}^{\mathrm{IV}}\mathrm{O}_4. \]

In addition to these types, two spinels are known which, as was reported, contain only trivalent ions and vacant sites in the lattice \((\gamma = \mathrm{Fe}_2\mathrm{O}_3,\ \gamma = \mathrm{Al}_2\mathrm{O}_3)\), i.e. have the formula \(\mathrm{Me}^{\mathrm{III}}_{8/3}\square_{1/3}\mathrm{O}_4\) (\(\square\) is a vacant site in the lattice). Mixed crystalline formations of all the indicated types of spinels, containing mono-, di-, tri-, and tetravalent ions, have been described in many cases. A small number of series of mixed crystals with a definite miscibility interval are known, for example\(^5\) the system \(\mathrm{CoFe}_2\mathrm{O}_4-\mathrm{Co}_3\mathrm{O}_4\), various systems described by Romeijn\(^6\), and several systems investigated by Jonker\(^7\), for example \(\mathrm{MgFe}_2\mathrm{O}_4-\mathrm{MgAl}_2\mathrm{O}_4\).

If the metallic ions present in a spinel can be in states of different valence, then chemical analysis makes it possible to determine only the average value of the valence state. Consequently, the valence states of individual ions, for example in \(\mathrm{Fe}_3\mathrm{O}_4\), \(\mathrm{Co}_3\mathrm{O}_4\), \(\mathrm{Mn}_3\mathrm{O}_4\), \(\mathrm{MnFe}_2\mathrm{O}_4\), \(\mathrm{Mn}_2\mathrm{TiO}_4\), and \(\mathrm{Fe}_2\mathrm{TiO}_4\), cannot be found by such an analysis.

Here one should discuss the ionization potentials of the corresponding ions in addition to the other factors determining the lattice energy of a spinel and discussed in Section 1.2.5.

The difference in ionization potentials for the various possible cases is usually small, whereas the probable error in the literature data on the potentials of 3- and 4-fold ionization is very large. Nevertheless, in most spinels there is no great basis for doubts concerning the valence states of the ions, both because of known physical properties and on the basis of analogy with similar compounds.

1.2.2. Distribution of Cations in Binary Oxide Spinels

The distribution of metal ions in binary spinels \(\mathrm{MeMe}'_2\mathrm{O}_4\) may be\(^9\):

(1) “normal,” with 1 Me in the tetrahedral sublattice and 2 Me′ in the octahedral sublattice. Every time we wish to make

\(^*\) In contrast to Bertaut and Durif (C. R. Acad. Sci. Paris 236, 212–214 (1953)), who described the spinel \(\mathrm{LiTi}_{7/4}\square_{1/4}\mathrm{O}_4\), Jonker obtained firm indications that the spinel \(\mathrm{Li}_{4/3}\mathrm{Ti}_{5/3}\mathrm{O}_4\) exists (private communication from Jonker).

in the formula the indication concerning the distribution of ions, we shall henceforth write the ions located in octahedral sites in brackets, and, consequently, in the case under consideration we have \(\mathrm{Me}[\mathrm{Me}'_2]\mathrm{O}_4\).

(2) “inverse,” with \(\mathrm{Me}\) in the octahedral sublattice and one \(\mathrm{Me}'\) in the octahedral and tetrahedral sublattices. Such an arrangement will be described by the formula \(\mathrm{Me}'[\mathrm{MeMe}']\mathrm{O}_4\).

(3) “intermediate,” for example \(\mathrm{Me}'_{1-x}\mathrm{Me}_x[\mathrm{Me}_{1-x}\mathrm{Me}'_{1+x}]\mathrm{O}_4\).

In individual cases, for example in spinels containing several ions of metals of the first transition group, the distribution of the ions cannot be determined with any considerable accuracy by means of X-ray diffraction, owing to the small difference in the scattering intensities from different ions. This is true at least in the case when, in addition, ions similar to \(\mathrm{Mg}^{2+}\), \(\mathrm{Al}^{3+}\) are present, because the scattering intensities depend on \(x\), just as on the oxygen parameter \(u\).

In most cases the method of comparing the intensities of two reflections with a small difference in \(\sum h^2\) is used. These intensities vary strongly and in different ways as a function of \(x\), and are appreciably less sensitive to differences in the value of \(u\). This method was used to determine the cation distribution of a series of spinels. The factor limiting the accuracy of this method is the error in the scattering factor used\(^{10}\).

For ferrites an indirect method has also been used. According to X-ray studies, the chromites \(\mathrm{MeCr}_2\mathrm{O}_4\) have the “normal” distribution. The ferrites \(\mathrm{ZnFe}_2\mathrm{O}_4\) and \(\mathrm{CdFe}_2\mathrm{O}_4\) have the structure of a normal spinel, while \(\mathrm{MgFe}_2\mathrm{O}_4\) and \(\mathrm{CuFe}_2\mathrm{O}_4\)—the structure of an inverse spinel*). The difference in the lattice constant between chromites and the corresponding ferrites is \(0.12\ \text{Å}\) for \(\mathrm{Zn}^{2+}\) and \(\mathrm{Cd}^{2+}\), and about \(0.05\ \text{Å}\) for all the other ions, among them \(\mathrm{Mg}^{2+}\) and \(\mathrm{Cu}^{2+}\). From these data it was concluded that the ferrites \(\mathrm{MeFe}_2\mathrm{O}_4\), where \(\mathrm{Me}\) denotes \(\mathrm{Mn}^{2+}\), \(\mathrm{Fe}^{2+}\), \(\mathrm{Co}^{2+}\), \(\mathrm{Ni}^{2+}\), \(\mathrm{Cu}^{2+}\), or \(\mathrm{Mg}^{2+}\), are inverse spinels\(^{9}\).

The aluminates are all normal spinels\(^{6,8}\), with the exception of \(\mathrm{NiAl}_2\mathrm{O}_4\), for which an approximate distribution \(\mathrm{Al}_{0.75}\mathrm{Ni}_{0.25}[\mathrm{Ni}_{0.75}\mathrm{Al}_{1.25}]\mathrm{O}_4\) has recently been found\(^{6}\).

\(\mathrm{MgV}_2\mathrm{O}_4\), \(\mathrm{ZnV}_2\mathrm{O}_4\), \(\mathrm{MgRh}_2\mathrm{O}_4\), and \(\mathrm{ZnRh}_2\mathrm{O}_4\) are normal spinels\(^{11}\); \(\mathrm{MgIn}_2\mathrm{O}_4\) and, probably, \(\mathrm{MgGa}_2\mathrm{O}_4\) are inverse spinels\(^{8}\).

\(\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5}\mathrm{O}_4\) is an inverse spinel\(^{12}\) \(\mathrm{Fe}[\mathrm{Li}_{0.5}\mathrm{Fe}_{1.5}]\mathrm{O}_4\).

*) Magnetic investigation showed that these latter ferrites have only approximately an “inverse” structure; see Section IV.

The series of titanates^9 \(\mathrm{Me}^{\mathrm{II}}_2\mathrm{Ti}^{\mathrm{IV}}\mathrm{O}_4\), as well as \(\mathrm{Zn}_2\mathrm{Sn}^{\mathrm{IV}}\mathrm{O}_4\) and \(\mathrm{Mg}_2\mathrm{V}^{\mathrm{VI}}\mathrm{O}_4\), have been identified as inverse spinels^9,11; it seems almost obvious that all similar titanates and stannates are inverse spinels. However, \(\mathrm{Ni}_2\mathrm{GeO}_4\) and \(\mathrm{Co}_2\mathrm{GeO}_4\) are normal spinels^6.

I.2.3. Long-range order in oxide spinels

It was initially assumed that the distribution of different cations within a single sublattice is disordered^8. But, as was later discovered, in some spinels long-range order occurs in one of the sublattices. So far three types of such order have been found:

(1) Order \(1:1\) in the octahedral sublattice.
The positions of the ions in the cell are approximately*) as follows:

\[ \begin{aligned} \mathrm{Me}\ \text{at}\quad& \tfrac{1}{8},\tfrac{5}{8},\tfrac{1}{8};\ \tfrac{5}{8},\tfrac{1}{8},\tfrac{1}{8};\ \tfrac{3}{8},\tfrac{7}{8},\tfrac{1}{8};\ \tfrac{7}{8},\tfrac{3}{8},\tfrac{1}{8};\\ & \tfrac{1}{8},\tfrac{1}{8},\tfrac{5}{8};\ \tfrac{3}{8},\tfrac{3}{8},\tfrac{5}{8};\ \tfrac{5}{8},\tfrac{5}{8},\tfrac{5}{8};\ \tfrac{7}{8},\tfrac{7}{8},\tfrac{5}{8}. \end{aligned} \]

\[ \begin{aligned} \mathrm{Me}'\ \text{at}\quad& \tfrac{3}{8},\tfrac{5}{8},\tfrac{5}{8};\ \tfrac{5}{8},\tfrac{3}{8},\tfrac{3}{8};\ \tfrac{1}{8},\tfrac{7}{8},\tfrac{3}{8};\ \tfrac{7}{8},\tfrac{1}{8},\tfrac{3}{8};\\ & \tfrac{1}{8},\tfrac{3}{8},\tfrac{7}{8};\ \tfrac{3}{8},\tfrac{1}{8},\tfrac{7}{8};\ \tfrac{5}{8},\tfrac{7}{8},\tfrac{7}{8};\ \tfrac{7}{8},\tfrac{5}{8},\tfrac{7}{8}. \end{aligned} \]

Successive \((0,0,1)\) layers of octahedral sites are occupied by alternating \(\mathrm{Me}\) and \(\mathrm{Me}'\) ions.

Each \(\mathrm{Me}\) ion has \(4\mathrm{Me}'\) and \(2\mathrm{Me}\) ions as octahedral neighbors or, conversely, each \(\mathrm{Me}'\) ion has \(4\mathrm{Me}\) and \(2\mathrm{Me}'\) as neighbors.

Verwey and Haayman^13, on the basis of the presence of a resistance jump at \(120^\circ\mathrm{K}\), determined the structure of \(\mathrm{Fe}[\mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}]\mathrm{O}_4\) below \(120^\circ\mathrm{K}\). This structure, in their opinion, is essentially orthorhombic. It has recently been shown by various methods that below \(120^\circ\mathrm{K}\) \(\mathrm{Fe}_3\mathrm{O}_4\) does indeed have orthorhombic symmetry (see, however, the literature data^6).

(2) Order \(1:3\) in the octahedral sublattice.
The positions of the ions in the cell are approximately as follows:

\[ \begin{aligned} \mathrm{Me}\ \text{at}\quad& \tfrac{5}{8},\tfrac{5}{8},\tfrac{5}{8};\ \tfrac{1}{8},\tfrac{7}{8},\tfrac{3}{8};\ \tfrac{7}{8},\tfrac{3}{8},\tfrac{1}{8};\ \tfrac{3}{8},\tfrac{1}{8},\tfrac{7}{8}. \end{aligned} \]

\[ \begin{aligned} \mathrm{Me}'\ \text{at}\quad& \tfrac{1}{8},\tfrac{5}{8},\tfrac{1}{8};\ \tfrac{5}{8},\tfrac{1}{8},\tfrac{1}{8};\ \tfrac{3}{8},\tfrac{7}{8},\tfrac{1}{8};\ \tfrac{7}{8},\tfrac{1}{8},\tfrac{3}{8};\\ & \tfrac{3}{8},\tfrac{5}{8},\tfrac{3}{8};\ \tfrac{5}{8},\tfrac{3}{8},\tfrac{3}{8};\ \tfrac{1}{8},\tfrac{1}{8},\tfrac{5}{8};\ \tfrac{3}{8},\tfrac{3}{8},\tfrac{5}{8};\\ & \tfrac{7}{8},\tfrac{7}{8},\tfrac{5}{8};\ \tfrac{1}{8},\tfrac{3}{8},\tfrac{7}{8};\ \tfrac{5}{8},\tfrac{7}{8},\tfrac{7}{8};\ \tfrac{7}{8},\tfrac{5}{8},\tfrac{7}{8}. \end{aligned} \]

*) It is clear that long-range order is accompanied by a change of space group, so that the ion positions differ somewhat from the positions encountered in the “ideal” structure (see, for example, ^12).

The distribution of ions is shown in Fig. 3; see the asterisks. Each row of octahedral ions in the \([110]\) directions contains a Me ion in every fourth position. Each Me ion is surrounded by six \(\mathrm{Me}'\) ions; each \(\mathrm{Me}'\) ion has as neighbors 2Me and 4\(\mathrm{Me}'\) ions.

This type of order, found by Braun, occurs in
\(\mathrm{Fe}[\mathrm{Li}_{0.5}\mathrm{Fe}_{1.5}]\mathrm{O}_4\), for which Braun determined the structure.

This structure is essentially cubic (its space group is very probably \(P4_3 3\) (or \(P4_1 3\))\({}^{12}\)). Similar superstructure lines have also been found in \(\mathrm{Al}[\mathrm{Li}_{0.5}\mathrm{Al}_{1.5}]\mathrm{O}_4\) and in \(\gamma\)-\(\mathrm{Fe}_2\mathrm{O}_3\) (apparently representing\({}^{12}\) a mixed crystal \(\mathrm{Fe}[\mathrm{H}_{1/2}\mathrm{Fe}_{3/2}]\mathrm{O}_4\)—\(\mathrm{Fe}[\square_{1/3}\mathrm{Fe}_{5/3}]\mathrm{O}_4\)) and it is probable that these compounds, as well as \(\gamma\)-\(\mathrm{Al}_2\mathrm{O}_3\), have a similar superstructure.

In \(\mathrm{Fe}[\mathrm{Li}_{0.5}\mathrm{Fe}_{1.5}]\mathrm{O}_4\) the transition temperature to the disordered spinel structure, which still contains all \(\mathrm{Li}^+\) in the octahedral sublattice, lies between 1008 and \(1028^\circ\mathrm{K}\).

(3) Order \(1:1\) in the tetrahedral sublattice.
The positions of the ions in the cell are as follows:

\[ \begin{aligned} \mathrm{Me}\ \text{at}\quad &0,\ 0,\ 0;\ 0,\ \tfrac12,\ \tfrac12;\ \tfrac12,\ 0,\ \tfrac12;\ \tfrac12,\ \tfrac12,\ 0.\\ \mathrm{Me}'\ \text{at}\quad &\tfrac14,\ \tfrac14,\ \tfrac14;\ \tfrac14,\ \tfrac34,\ \tfrac34;\ \tfrac34,\ \tfrac14,\ \tfrac34;\ \tfrac34,\ \tfrac34,\ \tfrac14. \end{aligned} \]

Each Me ion is surrounded by four \(\mathrm{Me}'\) ions and, conversely, each \(\mathrm{Me}'\) ion is surrounded by four Me ions. This superstructure\({}^{14}\), found recently, occurs in our material \(\mathrm{Li}_{0.5}\mathrm{Fe}^{\mathrm{III}}_{0.5}[\mathrm{Cr}_2]\mathrm{O}_4\) and will be considered in Section VII. The distribution of ions is shown in Fig. 2 (see the asterisks).

For all these three superstructures it has been shown that long-range order gradually disappears when the ratios \(1:1\), \(1:3\), or \(1:1\), respectively, are departed from.

1) Verwey and Heilmann showed that when the ratio \(\mathrm{Fe}^{3+}/\mathrm{Fe}^{2+}\) in magnetite increases above 2.0, i.e. the ratio \(\mathrm{Fe}^{3+}/\mathrm{Fe}^{2+}\) in the octahedral sublattice exceeds 1.0 (with, as a rule, some vacant cation sites then appearing), the jump in electrical resistance becomes smaller and shifts toward lower temperatures. It disappears when the ratio \(\mathrm{Fe}^{3+}/\mathrm{Fe}^{2+}\) in the octahedral sublattice is equal to 1.1.

2) The superstructure lines of \(\mathrm{Fe}[\mathrm{Li}_{0.5}\mathrm{Fe}_{1.5}]\mathrm{O}_4\) become weaker as soon as the ratio \(\mathrm{Fe}/\mathrm{Li}\) in the octahedral sublattice rises above 3.0, as, for example, in the formation of a mixed crystal with \(\mathrm{ZnFe}_2\mathrm{O}_4\). In \(\mathrm{Zn}_{0.15}\mathrm{Fe}_{0.85}[\mathrm{Li}_{0.42}\mathrm{Fe}_{1.57}]\mathrm{O}_4\) the superstructure lines disappear.

3) Order \(1:1\) in the tetrahedral sublattice, manifested in the reflection from the (200) plane, is undoubtedly observed\({}^{14}\) in
\(\mathrm{Fe}_{0.64}\mathrm{Li}_{0.36}[\mathrm{Li}_{0.14}\mathrm{Fe}_{0.26}\mathrm{Cr}_{1.60}]\mathrm{O}_4\).

I.2.4. Structure of hausmannite

Some compounds of the type \(\mathrm{Me}_3\mathrm{O}_4\), namely \(\mathrm{CuFe}_2\mathrm{O}_4\), \(\mathrm{CuCr}_2\mathrm{O}_4\), \(\mathrm{Mn}_3\mathrm{O}_4\), and \(\mathrm{ZnMn}_2\mathrm{O}_4\), have a tetragonal crystal structure \(^{15,16,17,19}\), i.e., the hausmannite structure, so named after the mineral hausmannite \(\mathrm{Mn}_3\mathrm{O}_4\). This structure is described much more conveniently as the structure of a spinel stretched in the \([001]\) direction; such a description differs from that given in \(^{15}\). \(\mathrm{CuFe}_2\mathrm{O}_4\) has an axial ratio of 1.06 when it is annealed, for example, at \(300^\circ\mathrm{C}\); at higher temperatures the axial ratio decreases, reaching unity at \(760^\circ\mathrm{C}\) \(^{16}\). Like \(\mathrm{Mn}_3\mathrm{O}_4\), \(\mathrm{CuFe}_2\mathrm{O}_4\) forms mixed crystals with cubic spinels: in the systems \(\mathrm{Cu}_{1-a}\mathrm{Me}^{\mathrm{II}}_{a}\mathrm{Fe}_2\mathrm{O}_4\), where \(\mathrm{Me}^{\mathrm{II}}\) is Zn, Ni, or Co, the axial ratio decreases as the content of \(\mathrm{Me}^{\mathrm{II}}_{a}\) decreases \(^{18}\). In the system

\[ \left. \begin{array}{c} \mathrm{Mn}_3\mathrm{O}_4\\ \mathrm{ZnMn}_2\mathrm{O}_4 \end{array} \right\} - \left\{ \begin{array}{c} \mathrm{MnFe}_2\mathrm{O}_4\\ \mathrm{ZnFe}_2\mathrm{O}_4 \end{array} \right. \]

where the same decrease of the axial ratio in the hausmannite phase occurs, a miscibility interval between the hausmannite and spinel phases was found \(^{19}\); it is not known whether this is a general phenomenon. The cause of the appearance of the stretched spinel structure is unknown (see also \(^{6}\)); X-ray diffraction on a \(\mathrm{CuFe}_2\mathrm{O}_4\) specimen does not reveal any superstructure lines \(^{16}\).

I.2.5. Factors affecting ion distribution and the stability of oxide spinels

To a first approximation, the chemical bond in oxide spinels may be regarded as purely ionic, so that the principal part of the lattice energy consists of Coulomb energy and Born repulsion energy. Other effects that contribute additional terms to the expression for the lattice energy are polarization, the individual tendency of certain ions toward fourfold or sixfold coordination owing to their electronic configuration, as well as other factors, among which magnetic interaction will be considered.

All terms in the energy expression depend on \(a\), \(u\), and the ion distribution. The equilibrium distribution of cations could in principle be calculated by minimizing the total energy with respect to the indicated variables, but, since the quantitative relations between the various components of the energy and these three variables are unknown, this cannot be done at present.

Below, the Coulomb energy is discussed quantitatively as a function of the lattice constant \(a\), the oxygen parameter \(u\), and the distribution of charges over the cation sites in the lattice. Additions to the lattice energy from other terms are probably smaller in magnitude.

1.2.5.1 Coulomb energy and repulsion energy.

The Coulomb potential energy per “molecule”

\[ V_C=-\frac{Me^2}{a} \]

depends on the charge distribution, the oxygen parameter \(u\), and, of course, the lattice constant \(a\). The Madelung constant \((M)\) was calculated by Verwey, de Boer, and van Santen \(^{20}\) as a function of \(u\) for an average ionic charge of 4, 3, or 2 in the tetrahedral sublattice, and 2, 2.5, and 3, respectively, in the octahedral sublattice.

Since we are also interested in mixed crystals, often containing a fractional number of charges per molecular unit in both sublattices, the results are given here in generalized form.

The values of \(M\) for \(u=0.375\) (\(\delta=0\)) are expressed by the formula \(^{21}\)

\[ M=139.8-10.84q_a+2.61q_a^2, \]

where \(q_a\) is the average ion charge per molecular unit in the tetrahedral sublattice.

The curves expressing the dependence of \(M\) on \(u\) for various values of \(q_a\) are very close to straight lines \(^{20}\). We assumed that they are straight lines and, further, that the slope of these lines \((\Delta M/\Delta u)\) varies linearly as a function of \(q_a\), which for \(q_a=4, 3,\) and \(2\) is almost exactly valid.

The Madelung potentials, expressed in units of \(e^2/a\) as functions of \(q_a\), for various values of \(u\), are shown as a family of parabolas in Fig. 7. From this figure it is clear that a high Madelung potential is obtained:

1) from a small average charge in the tetrahedral position at large values of \(u\);

2) from a large average charge in the tetrahedral position at small values of \(u\).

Fig. 7. Dependence of the Madelung constant \((M)\) on the charge in position \(A\) \((q_a)\) for various values of the oxygen parameter \((u)\).

Consequently, insofar as this is connected with the Madelung constant, an ion that has a small charge and is large in ...

in comparison with other cations, will tend to occupy a tetrahedral site if, for other cation distributions, the Madelung constant calculated from Fig. 7 is appreciably smaller.

Possibly this effect could explain the fact that in CdFe$_2$O$_4$ the large Cd$^{2+}$ ions are in fourfold coordination, whereas in CdO with the NaCl structure they have sixfold coordination (see, however, section 1.2.5.4). From the same point of view, an ion with a high charge will tend to enter tetrahedral sites only when it is small in comparison with the other cations present and when, for other cation distributions, the Madelung constant calculated from Fig. 7 is appreciably smaller. Indeed, among Me$^{4+}$ ions only a single ion, Ge$^{4+}$, has been found that occupies exclusively tetrahedral sites.

It should be borne in mind that the lattice constant $a$ also depends on the cation distribution, so that the distribution with a high value of $M$ need not have the greatest value of the Coulomb energy.

The repulsive potential energy, calculated per molecule, may be represented in the form

$$ V_R=\frac{B}{a^n}. $$

At equilibrium between the Coulomb and repulsive forces, this expression gives the repulsion energy, i.e. a constant fraction $1/n$ of the Coulomb energy ($n$ of order 10).

The Born–Mayer exponential expression might give a better approximation, but both expressions contain constants ($n$ and $\rho$, respectively) calculated from compressibility data. For transition-metal oxides, compressibility measurements are not known, and the values of $n$ and $\rho$ may differ appreciably from their values for alkali-halide crystals, which have been used up to now$^{22}$. Moreover, the compressibilities of binary oxides having crystal structures without parameters unknown for trivalent-metal ions would have to be measured.

For those few cases in which not only $a$ but also $u$ are known accurately from experiment, the cation distribution can be obtained from Fig. 7 (for example, for GeCo$_2$O$_4$)$^{6}$, if the cations are known exactly to be in the interstices whose positions are calculated from the given $a$ and $u$. If the difference between the Madelung constants for different distributions is small, then, in order to find the true cation distribution, one can use purely geometrical considerations; this method was applied$^{23}$ in the study of Fe$_3$O$_4$. The parameter $u$ can, in principle, also be calculated from the condition of equilibrium of the Coulomb forces and the repulsive forces.

On the other hand, it is instructive to compare the experimental values of \(a\) and \(u\) with the values calculated for a model consisting of a close packing of hard spheres. The cation radii from Section 1.2.1 are used and 4% is subtracted from the distance \(r_{\mathrm{Me}_A}+r_{\mathrm{O}^{2-}}\) in order to obtain the distance \(\mathrm{Me}_A—\mathrm{O}\). In many spinels the cation radii are such that, in this model, contact exists only between cations and anions; in this case the lattice constant \(a\), calculated according to the indicated model, does not differ greatly from the experimental value.

For inverse ferrites the calculated value of \(a\) is 0.4–1% larger than the experimental one*); for (inverse) titanates and stannates, \(a_{\mathrm{calc}}\) is larger by 0.9–2.8% and by 0.7–1.9%, respectively. In all these cases the greatest discrepancies are found for the largest divalent ions.

If the cations in tetrahedral sites are large in comparison with the cations in octahedral sites, then anion–anion contact occurs. The discrepancy between the calculated and experimental values of \(a\) is then already considerably greater. Namely, for \(\mathrm{Cd}[\mathrm{Fe}_2]\mathrm{O}_4\) it is 4.5%, for \(\mathrm{Zn}[\mathrm{Fe}_2]\mathrm{O}_4\) 1.3%, for \(\mathrm{Cd}[\mathrm{Cr}_2]\mathrm{O}_4\) 4.5%, for (normal) Mn-), Fe-, Zn-, Co-, Ni- and Mg-chromites 3, 2.2, 1.6, 1.5, 1.1 and 1%, respectively; for (normal) Mn-), Fe-, Zn-, Co- and Mg-aluminates 5.8, 5, 5, 5 and 4%, respectively.

From the illustrations given above it is evident that Goldschmidt’s radii of the oxygen ion (1.32 Å) and of the largest divalent ions are too large to satisfactorily explain the values of the lattice constants of oxide spinels; i.e., these ions may turn out, in such a structure, to be as if “compressed” in comparison with the crystal structures having sixfold coordination from which the values of their radii were obtained.

It is clear that such a compression effect, whatever its cause, tends to increase with increasing ionic diameter and to decrease with decreasing ion charge, and especially in fourfold coordination. This effect is not sufficiently taken into account by subtracting 4% from the sum of the radii \(r_{\mathrm{Me}_A}+r_{\mathrm{O}^{2-}}\) while keeping \(r_{\mathrm{Me}_B}+r_{\mathrm{O}^{2-}}\) unchanged.

It follows from what has been said above that the pattern of cation distribution is determined primarily by the magnitude of the space required for the smallest ions and the ions having the greatest

*) For the cases \(\mathrm{MnFe}_2\mathrm{O}_4\), \(\mathrm{MnCr}_2\mathrm{O}_4\) and \(\mathrm{MnAl}_2\mathrm{O}_4\), the lattice constants were determined again on preparations with exactly known valence states, prepared by the author, and for them the values 8.499 Å, 8.425 Å, 8.202 Å, respectively, were found.

charge (cf. with work ^6). The influence of the distribution of cations on the value of the lattice constant, as noted by Verwey and Heilmann, was found from the hard-sphere model for \(\mathrm{MnFe_2O_4}\): \(a_{\text{normal}}\) is 1.5% greater than \(a_{\text{inverted}}\). However, if one uses the model of \(\mathrm{MgFe_2O_4}\), where there is no anion–anion contact, the two calculated values of the lattice constant are equal. According to Verwey and Heilmann, the experimental difference \(a_{\text{normal}}-a_{\text{inverted}}\) is constant and is about \(0.07\ \text{\AA}\), or about 0.8%. Since the repulsive potential increases, of course, less steeply than in the hard-sphere model, this may be an indication that the experimental difference \(a_{\text{normal}}-a_{\text{inverted}}\) is due to anion–anion repulsion.

The upper and lower limits of \(u\), calculated from the hard-sphere model with anion–anion contact, deviate more strongly from the ideal value \(u=0.375\) than those found experimentally.

\[ \text{for } \mathrm{Zn[Fe_2]O_4}\quad u_{\text{calc}}=0.390,\quad u_{\text{exp}}=0.385\pm0.002, \]

\[ \text{for } \mathrm{Ge[Co_2]O_4}\quad u_{\text{calc}}=0.369,\quad u_{\text{exp}}=0.375\pm0.003. \]

This can also be explained by the “compression” of large ions with smaller charge, i.e., respectively, of the ions \(\mathrm{Zn}^{+2}\) and \(\mathrm{Co}^{2+}\).

I.2.5.2. Short-range order. The formation of the superstructure described in section I.2.3 is accompanied by an increase in the Coulomb energy compared with the energy of a disordered distribution within the octahedral and tetrahedral sublattices, respectively. These “ordering energies” for the first two superstructures mentioned are given in work ^24. The position of the ions \(\mathrm{Me}\) and \(\mathrm{Me'}\) in the presence of 1:1 order in the tetrahedral sublattice is analogous to the position of Zn and S in the zinc blende structure, for which the Madelung potential is known from the literature. These ordering energies, like \(M\), expressed in \(e^2/a\) (\(a\) is the lattice parameter), are given in Table II.

Very rough calculations show that polarization tends to increase these energies by an amount reaching

Table II

Ordering energy in units of \(e^2/a\)

Charge difference \((q_{Me'}-q_{Me})\)	1:1 order in the octahedral sublattice	1:3 order in the octahedral sublattice	1:1 order in the tetrahedral sublattice
1	1.001	0.712	0.946
2	4.004	2.848	3.783

nearly one quarter.^24 The indicated ordering energies are significantly greater than the energy of thermal motion at the transition temperatures, so that it may be concluded that above these temperatures a very sharply expressed short-range order must remain.^24

Therefore the quantities given in Fig. 7 must be corrected to take account of strong short-range order, whose energy is approximately equal to the above-mentioned value of the long-range-order energy.

1.2.5.3. Energy of magnetic interaction. We shall see that, for ferrimagnetic spinels, the magnetic (exchange) interaction energy depends strongly on the distribution of magnetic ions among the crystallographic positions. Therefore the question may be raised whether, in turn, this energy affects the distribution of ions. The Curie temperatures \(\Theta\), i.e., the temperatures at which the long-range order of magnetic moments is destroyed, reach \(950^\circ\text{K}\), from which we may conclude that this energy is of the order

\[ k\Theta \simeq 2\ \text{kcal/mole}. \]

However, it must be borne in mind that the short-range order of moments may also persist above the Curie temperature, i.e., up to temperatures at which diffusion of ions takes place. An indication of this effect is the fact that very weak reflections (arising from the ordering of magnetic spins) in neutron diffraction by the antiferromagnet MnO persist stubbornly up to \(3\Theta\), where \(\Theta\) is the Néel temperature, at which the long-range order of moments is destroyed. However, as far as the energy is concerned, it seems probable that the magnetic interaction energy above the Curie temperature is very small, as is the case, for example, in nickel, for which the excess specific heat becomes insignificant above the Curie temperature.

1.2.5.4. Individual tendencies of cations toward 4- or 6-fold coordination. The influence of the diameter and charge of a cation on its tendency toward tetrahedral or octahedral sites in an oxide spinel is not quite correctly called an individual property, since the diameters and charges of the other ions present must also be taken into account. But, apart from the influence of diameter and charge on the Coulomb energy and on other energy terms, there exists an individual difference between cations, manifested in their tendency toward fourfold or sixfold coordination. This is confirmed by the fact that cations with equal diameters and equal charges, such as, for example, the cations \(\mathrm{Zn}^{2+}\) and \(\mathrm{Co}^{2+}\), or \(\mathrm{Ni}^{2+}\) and \(\mathrm{Mg}^{2+}\), exhibit completely different behavior with respect to the distribution of ions in spinels which are otherwise completely identical in composition.

One of the reasons for such individual behavior is the difference in the electronic configurations of the various ions. We shall

distinguish five groups of ions with different electronic configurations. Of course, the resulting distribution of ions cannot be discussed without taking into account the diameter and charge of the ions.

1. The ions Zn²⁺ and Cd²⁺ with a filled 3d shell are known for their tendency to form covalent bonds with Sp³ orbitals. Therefore they readily occupy tetrahedral sites in spinels. Zn²⁺ in ZnO has fourfold coordination, whereas Cd²⁺ in CdO with the NaCl structure has sixfold coordination. In the latter case the size effect, according to Pauling’s rule, counteracts the individual tendency toward fourfold coordination.

Ions with filled d shells, namely

\[ \begin{aligned} 3d:&\quad \mathrm{Cu}^{+},\ \mathrm{Zn}^{2+},\ \mathrm{Ga}^{3+},\ \mathrm{Ge}^{4+},\\ 4d:&\quad \mathrm{Ag}^{+},\ \mathrm{Cd}^{2+},\ \mathrm{In}^{3+},\ \mathrm{Sn}^{4+}, \end{aligned} \]

probably all have an individual tendency toward fourfold coordination in oxides, but this may prove to be incorrect in spinels because of the dependence of the lattice energy on \(q_a\), \(u\), and \(a\). We have seen that a large tetravalent ion in the tetrahedral sublattice gives a low value of the Madelung constant; as a consequence, the stannates \(\mathrm{Me}^{II}_{2}\mathrm{SnO}_{4}\) are inverse spinels; \(\mathrm{Ag}_{2}\mathrm{MoO}_{4}\) is probably a normal spinel, because the inverse arrangement in this case should have led to a very large value of \(a\).

1. Ions with noble-gas shells have no individual tendency toward any particular coordination. Their distribution will be determined mainly by the dependence of the lattice energy on \(q_a\), \(a\), and \(u\). Thus, among ions of this kind that occur in spinels, namely Li⁺, Mg²⁺, Al³⁺, Ti⁴⁺, the tetravalent ions are too large to occupy exclusively only tetrahedral sites.

2. Ions with half-filled 3d shells have spherical symmetry. Therefore one cannot expect that the ions Mn²⁺, Fe³⁺, and Co⁴⁺ will have an individual tendency toward any position (Co⁴⁺, if it occurs at all in spinels, which at present seems unlikely, may choose a tetrahedral position for electrostatic reasons, which were discussed above). The tendency of ions of other transition metals toward a definite coordination is determined by the influence of the crystalline electric field, arising from neighboring ions, on the average energy level and on the spatial distribution of the 3d electrons; see ⁶. As a result, these ions may be divided into two groups:

3. Ions with 3d³ and 3d⁸ shells, having a strong tendency toward sixfold coordination: Cr³⁺, Ni²⁺, and, possibly, Mn⁴⁺ and

1. ions of all other transition metals that do not have a strong tendency toward any particular site.

The scheme presented above is an extension of the rule on the distribution of cations in spinels given by Verwey and Heilmann⁹.

I.2.6. Spinels Containing Three or More Different Cations

Verwey and Heilmann found by X-ray methods that in mixed crystals formed by normal \( \mathrm{Zn[Fe_2]O_4} \) and inverse \( \mathrm{Fe[CuFe]O_4} \), the \( \mathrm{Zn}^{2+} \) ions still occupy the sites of the tetrahedral sublattice, while the \( \mathrm{Cu}^{2+} \) ions occupy the sites of the octahedral sublattice, giving, for example, the formula \( \mathrm{Zn_{0.5}Fe_{0.5}[Cu_{0.5}Fe_{1.5}]O_4} \). Since in 1947 there were practically no data on the distribution of ions in ternary spinels, it was assumed that ions always retain their individual preferences for one of the crystallographic sites as against other ions in any compounds. This means that a known order of ionic preference, for example for tetrahedral sites, may be specified; according to this order the authors placed \( \mathrm{Fe}^{3+} \) immediately after \( \mathrm{Zn}^{2+} \), \( \mathrm{Cd}^{2+} \), \( \mathrm{Ga}^{3+} \), and \( \mathrm{In}^{3+} \), and before other divalent ions. Recently, Romeijn⁶ showed that in a series of ternary spinels the behavior of the ions is quite different. Our magnetic measurements, which will be presented in Sections VI–IX, will give further examples of cases in which such simple behavior was not found.

The difference in total energies between normal and inverse arrangements can be found experimentally for those spinels in which the distribution changes appreciably with temperature. For example, for a spinel of composition \( \mathrm{Me'_{1-x}Me_x[Me_{1-x}Me'_{1+x}]O_4} \), the equilibrium distribution of ions is determined by the Boltzmann expression derived by Néel²⁵:

\[ kT \ln \frac{(1-x)^2}{x(x+1)} = E, \tag{1.1} \]

in which \( E \) is the exchange energy of a \( \mathrm{Me} \) ion in the octahedral sublattice with a \( \mathrm{Me'} \) ion in the tetrahedral sublattice. We have seen that an exact determination of \( x \) from X-ray data is usually impossible. However, Potenza and Boshroll²⁶ determined \( x \), and also \( E \), from data on the magnetic saturation of \( \mathrm{MgFe_2O_4} \) and \( \mathrm{CuFe_2O_4} \) (see Section IV); it was found that \( E \) is a constant quantity.

Knowledge of the exchange constants \( E \) of two binary spinels containing the same cation would, in general, make it possible to calculate the distribution of cations in a mixed crystal of these spinels, if it were possible to use a single constant \( E \) for a given pair of cations. There is no doubt that there does not exist

such a case, when \(E\) would not change as a function of the charge distribution \((q_i)\), the lattice constant \(a\), and the parameter \(u\); these quantities, like the correction for the nearest-neighbor effect, will be different for different compositions.

II. THEORY OF FERRIMAGNETISM

II. 1. Introduction

In contrast to paramagnets, ferromagnetic substances can be magnetized to saturation by a comparatively small magnetic field (usually \(10^2\)—\(10^4\) oersteds). This occurs because the magnetic moments of atoms with partially filled \(3d\)- or \(4f\)-shells are parallel to the magnetic moments of neighboring atoms already in the unmagnetized state, which is the result of the exchange interaction of electrons, considerably greater than the forces of interaction between magnetic dipoles. The nature of this exchange mechanism is not considered by us here. In the unmagnetized (initial) state the total magnetization is equal to zero, since in each crystal the parallelism of atomic moments takes place only inside domains (regions of spontaneous magnetization); the direction of magnetization of each domain is determined by the structure of the crystal, as well as by anisotropies of shape and stresses. The saturation magnetic field only rotates the moments of all domains, overcoming the anisotropy forces.

Complete parallelism of the atomic magnetic moments within domains exists only at \(0^\circ\) K; the saturation magnetization decreases with temperature. At the Curie temperature the long-range order of the atomic moments is destroyed, and at high temperatures only paramagnetism remains. The saturation magnetization at \(0^\circ\) K, calculated per magnetic atom or “molecule” and called by us the saturation magnetic moment, can be expressed through the spin moment of the electron—the Bohr magneton \(\mu_{\mathrm{B}}=eh/4\pi mc\), where \(e\), \(h\), \(m\), and \(c\) have their generally accepted meanings.

In ferromagnetic metals with a partially filled \(3d\)-shell there is no simple dependence between the number of unpaired \(3d\)-electrons and the total number of available electrons, so that the saturation magnetic moments cannot be simply predicted. On the other hand, in compounds which may essentially be regarded as ionic (for example, oxides, with which we shall be concerned), the situation must be considerably simpler. Here each magnetic ion contains a definite number of unpaired electrons, which do not take part in the formation of chemical bonds. For substances containing elements of the first transition group, the ferromagnetic and paramagnetic

behavior is almost completely determined by the spins of the unpaired \(3d\)-electrons. The ferromagnetic moment of each ion will thus be equal to \(2S\) (\(S\) is the total spin quantum number of the ion), if the orbital contribution is negligible, or to \(gS\), if this contribution is taken into account. \(g\) is the so-called \(g\)-factor, equal to \((2mc/e)\times\left(\dfrac{\text{magnetic moment}}{\text{angular moment}}\right)\).

Table III gives the number of \(3d\)-electrons and the number of unpaired \(3d\)-electrons \((2S)\) for these ions. These ions in compounds,

Table III

Ions	Ions	Ions	Ions	Ions	Ions	Ions	Ions	Ions	Ions	Number of \(3d\)-electrons	Number of unpaired \(3d\)-electrons
\(\mathrm{Sc}^{3+}\)	\(\mathrm{Ti}^{4+}\)	\(\mathrm{V}^{5+}\)	\(\mathrm{Cr}^{6+}\)	\(\mathrm{Mn}^{7+}\)						0	0
	\(\mathrm{Ti}^{3+}\)	\(\mathrm{V}^{4+}\)	\(\mathrm{Cr}^{5+}\)	\(\mathrm{Mn}^{6+}\)						1	1
	\(\mathrm{Ti}^{2+}\)	\(\mathrm{V}^{3+}\)	\(\mathrm{Cr}^{4+}\)	\(\mathrm{Mn}^{5+}\)	\(\mathrm{Fe}^{6+}\)					2	2
		\(\mathrm{V}^{2+}\)	\(\mathrm{Cr}^{3+}\)	\(\mathrm{Mn}^{4+}\)						3	3
		\(\mathrm{V}^{+}\)	\(\mathrm{Cr}^{2+}\)	\(\mathrm{Mn}^{3+}\)	\(\mathrm{Fe}^{4+}\)					4	4
				\(\mathrm{Mn}^{2+}\)	\(\mathrm{Fe}^{3+}\)	\(\mathrm{Co}^{4+}\)				5	5
					\(\mathrm{Fe}^{2+}\)	\(\mathrm{Co}^{3+}\)	\(\mathrm{Ni}^{4+}\)			6	4
						\(\mathrm{Co}^{2+}\)	\(\mathrm{Ni}^{3+}\)			7	3
							\(\mathrm{Ni}^{2+}\)			8	2
								\(\mathrm{Cu}^{2+}\)		9	1
								\(\mathrm{Cu}^{+}\)	\(\mathrm{Zn}^{2+}\)	10	0

as was stated above, constructed and fixed in a close-packed arrangement.

An exact calculation of the changes in the saturation magnetization with temperature in a crystal is still impossible, and all such calculations, even with simplifying assumptions, are extremely difficult\(^{27}\). Therefore one often uses the simple approximation given by P. Weiss\(^{28}\), in which each ion is regarded as being in a fictitious magnetic field, the so-called molecular field, representing the resultant action of the exchange forces of all the surrounding atoms. The complete alignment of the magnetic moments in one direction, which exists at \(0^\circ\mathrm{K}\), tends to be destroyed at higher temperatures, and the saturation magnetization decreases with temperature in accordance with the Brillouin function, in which \(H+h\) is the field acting at any given instant; the external field \(H\), necessary for attaining saturation, i.e. for the orientation of domains, at low temperatures can usually be neglected in comparison with the very large molecular field \(h\).

At the Curie temperature the long-range order of the magnetic moments is destroyed; however, the fact that ferromagnets above the Curie temperature obey the Curie-Weiss law \(\chi=C/(T-\vartheta)\), with \(\vartheta>0^{*}\), shows that here exchange interactions and, consequently, molecular fields continue to play a role. Here \(h\) is small in comparison with the external fields usually used, and \(H+h\) enters into the Brillouin function.

Exchange interactions can in certain cases lead to a minimum of the free energy in another state, i.e. with antiparallel orientation of the spins of neighboring atoms or ions, as found in antiferromagnets such as MnO, \(\alpha\)-Fe\(_2\)O\(_3\), CrSb. These materials may be regarded as consisting of two equivalent sublattices with equal, mutually antiparallel spontaneous magnetizations, decreasing with temperature according to one and the same Brillouin function, which contains only the molecular field \(h\). The long-range order of the moments is destroyed at the so-called Néel temperature, and above this temperature these materials usually obey the Curie-Weiss law \(\chi=C/(T-\vartheta)\) with \(\vartheta<0\).

In paramagnetic substances containing magnetic ions with partially filled \(3d\)-shells, in which the magnetic ions are at a large distance from one another, so that exchange interaction is impossible, the magnetic field tends to orient the moments of the ions in opposition to the thermal motion.

*) We shall see that small values of \(\vartheta\) can occur even without exchange interaction.

Thus, the magnetization per gram-ion \((I)\) is

\[ I = NgS\mu_B B_S \left( \frac{gS\mu_B H}{kT} \right), \]

where \(N\) is Avogadro’s number, \(k\) is Boltzmann’s constant, and the Brillouin function \(B_S\) is defined by the expression

\[ B_S \left( \frac{gS\mu_B H}{kT} \right) = \frac{S+\frac{1}{2}}{S} \operatorname{cth} \frac{\left(S+\frac{1}{2}\right) g\mu_B H}{kT} - \frac{\frac{1}{2}}{S} \operatorname{cth} \frac{\frac{1}{2} g\mu_B H}{kT}. \]

For small values of \(gS\mu_B H/kT\) one may write

\[ B_S \left( \frac{gS\mu_B H}{kT} \right) = \frac{S+1}{3} \frac{g\mu_B H}{kT}, \]

and, consequently,

\[ I = \frac{Ng^2\mu_B^2 S(S+1)H}{3kT} \equiv \frac{CH}{T}. \]

Some dilute paramagnetic substances, for example substances containing \(\mathrm{Fe}^{3+}\) ions (for example, \(\mathrm{Fe}^{\mathrm{III}}\mathrm{NH}_4(\mathrm{SO}_4)_2 \cdot 12\mathrm{H}_2\mathrm{O}\)) do in fact obey Curie’s law: \(I/H=\chi=C/T\) (\(\chi\) and \(C\) are referred to a gram-ion) over a wide temperature range, and the Curie constant \(C\) agrees exactly with the theoretical value. Thus it is proved that each magnetic ion contributes an increment equal to its theoretical moment \(gS\mu_B\). Other dilute paramagnetic substances, over a limited temperature range, obey the Curie–Weiss law: \(\chi=C/(T-\vartheta)\), with a small positive or negative value of \(\vartheta\), due to the interaction of the orbital moment with the crystalline electric field of the crystal.

II.2. Outline of the Theory of Ferrimagnetism

II.2.1. Néel’s Theory

In a ferromagnetic, essentially ionic, compound the magnetic moment per molecular unit should be expected to be equal to the sum of the ionic magnetic moments. Consequently, for magnetite \(\mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}^{\mathrm{III}}_2\mathrm{O}_4\), taking \(g\) equal to 2 for both ions, it is equal to: \(4+2\times5=14\mu_B\). Weiss and Forrer\(^{29}\) in 1929 found that the magnetic moment of a molecular unit of magnetite is \(4.08\mu_B\). From earlier work\(^{30}\) it is known that the (normal) ferrites \(\mathrm{Zn}[\mathrm{Fe}_2]\mathrm{O}_4\) and \(\mathrm{Cd}[\mathrm{Fe}_2]\mathrm{O}_4\) are paramagnetic, whereas the (inverse) ferrites \(\mathrm{Mn}\), \(\mathrm{Co}\), \(\mathrm{Ni}\), \(\mathrm{Cu}\), and \(\mathrm{Mg}\) are ferromagnetic, like \(\mathrm{Fe}[\mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}]\mathrm{O}_4\).

It is also known that above the Curie temperature, for a number of ferrites the curve \(1/\chi = 1/\chi(T)\) has a concavity directed toward the \(T\)-axis.

Using these data, Néel, who had already made a major contribution to the theory of antiferromagnetism, in 1948 formulated the basic hypothesis^31, which consists in the fact that a strong negative interaction, i.e., a tendency toward antiparallel orientation, exists, on the one hand, between the magnetic moments of ions located at tetrahedral sites and, on the other hand, between the magnetic moments of ions located at octahedral sites. Consequently, the atomic magnetic moment of \(\mathrm{Fe}^{\mathrm{III}}[\mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}^{\mathrm{III}}]\mathrm{O}_4\) must be \((5+4)-5=4\), which agrees well with experiment. This behavior may be interpreted as uncompensated antiferromagnetism. We shall use the term ferrimagnetism, introduced by Néel.

At the same time^31 Néel extended Weiss’s molecular-field theory to a lattice with two different groups of sites in the lattice, in which different numbers of magnetic ions (atoms) are found. The theory was developed only for the case of magnetic ions of one kind (for example, \(\mathrm{Fe}^{3+}\)). We shall set forth the essence of the theory, applying it only to spinels and introducing, consequently, from the very beginning the basic assumption of negative interaction between ions located at tetrahedral and octahedral sites.

Following Néel, we shall use the index \(a\) for tetrahedral sublattices, and the index \(b\) for octahedral sublattices. Using notation that differs somewhat from Néel’s, we obtain for the spinel the formula

\[ \mathrm{Fe}^{\mathrm{III}}_{x_a}\mathrm{Me}_{1-x_a} [\mathrm{Fe}^{\mathrm{III}}_{x_b}\mathrm{Me}_{2-x_b}]\mathrm{O}_4, \tag{2.1} \]

in which \(\mathrm{Me}\) represents nonmagnetic ions, and \(x_a\) and \(x_b\) are the numbers of iron ions in the sublattices \(A\) and \(B\), respectively.

II.2.1.1. Paramagnetic behavior. If one considers the paramagnetic behavior above the Curie temperature, then the total magnetization of the mole is equal to

\[ \mathbf{I}=x_a\mathbf{I}_a+x_b\mathbf{I}_b, \tag{2.2} \]

where \(\mathbf{I}_a\) is the magnetization per gram-ion in \(A\), and \(\mathbf{I}_b\) is the magnetization per gram-ion in \(B\). The molecular field \(h_a\) for an ion in position \(A\) arises from the resultant magnetization of its neighbors in sublattice \(B\) and also from neighbors in sublattice \(A\), and is taken to be proportional to \(\mathbf{I}_b\) and \(\mathbf{I}_a\), respectively. Consequently,

\[ \mathbf{h}_a=n(-x_b\mathbf{I}_b+\alpha x_a\mathbf{I}_a), \tag{2.3} \]

where \(n\) is a positive constant connected with the sum of exchange integrals of type \(AB\), and \(-\alpha\) is the ratio of the interactions \(AA\) and \(AB\). Similarly,

\[ \mathbf{h}_b=n(-x_a\mathbf{I}_a+\beta x_b\mathbf{I}_b), \tag{2.4} \]

where \(-\beta\) is the ratio of the \(BB\) and \(AB\) interactions. The coefficients \(n\alpha\), \(-n\), and \(n\beta\) are linearly related to the sums of the possible exchange integrals \(AA\), \(AB\), and \(BB\), respectively. The partial magnetizations of the sublattices \(\mathbf I_a\) and \(\mathbf I_b\) may be written as

\[ \mathbf I_a=\frac{C}{T}(\mathbf H+\mathbf h_a), \tag{2.5} \]

\[ \mathbf I_b=\frac{C}{T}(\mathbf H+\mathbf h_b). \tag{2.6} \]

Eliminating \(\mathbf I_a\), \(\mathbf I_b\), \(\mathbf h_a\), and \(\mathbf h_b\) from (2.3–2.6), we obtain:

\[ \frac{H}{I_{\mathrm{mol}}} = \frac{T}{C_{\mathrm{mol}}} + \frac{n\left(2x_ax_b-\alpha x_a^2-\beta x_b^2\right)} {(x_a+x_b)^2} - \frac{ n^2 C_{\mathrm{mol}} x_a x_b \left[x_a(1+\alpha)-x_b(1+\beta)\right]^2/(x_a+x_b)^4 }{ T-nC_{\mathrm{mol}}x_ax_b(2+\alpha+\beta)/(x_a+x_b)^2 } \tag{2.7} \]

or

\[ \frac{1}{\chi_{\mathrm{mol}}} = \frac{T}{C_{\mathrm{mol}}} + \frac{1}{\chi_0} - \frac{s}{T-\vartheta}, \tag{2.8} \]

where the values of \(1/\chi_0\), \(s\), and \(\vartheta\) are determined from equation (2.7). It is evident that this expression represents a relation between \(1/\chi_{\mathrm{mol}}\) and \(T\) of hyperbolic type, as is also found experimentally.

It was later found that the slope of the experimental asymptote of the hyperbola is not equal to \(1/C_{\mathrm{mol}}\), and Néel proposed that \(n\) (for simplicity, not \(\alpha\) and \(\beta\)) depends linearly on temperature,

\[ n=n_0(1+\gamma T), \]

which gives:

\[ \frac{1}{C_{\mathrm{mol}}} = \frac{1}{C'_{\mathrm{mol}}} + \frac{\gamma}{\chi_0}, \]

where \(C'_{\mathrm{mol}}\) is the theoretical Curie constant.

From the experimental curve, \(\chi_0\), the values of \(s\) and \(\vartheta\) can be obtained.

The quantity \(x_a+x_b\) is known (2 for \(\mathrm{MgFe_2O_4}\), 2.5 for \(\mathrm{Li_{0.5}Fe_{2.5}O_4}\), etc.), and from \(\chi_0\), \(s\), \(\vartheta\), and the ferromagnetic behavior of the materials, \(x_a\), \(x_b\), \(\alpha\), \(\beta\), and \(n\) can be found for those cases in which only \(\mathrm{Fe^{3+}}\) ions are present and the molecular-field theory can be used (i.e., in the absence of a sharply expressed ordering of ions in one or another sublattice and for not too small \(x_a\) or \(x_b\)). We wish to emphasize again\(^{32}\) that if \(x_a/x_b\) changes with temperature, as was found in the case, for example, of \(\mathrm{MgFe_2O_4}\), the paramagnetic measurements do not correspond to this chemical formula, so that calculations similar to the one presented have no more than qualitative interest.

Nevertheless, Néel was able to show qualitatively in this way for a number of ferrites that \(|\alpha|\) and \(|\beta|\ll 1\), and this means that the interactions \(AA\) and \(BB\) are small in comparison with the interaction \(AB\). He also found that \(\alpha\) and \(\beta\), and consequently the interactions \(AA\) and \(BB\), are negative.

II.2.1.2. Ferromagnetic behavior. In what follows we shall be interested only in ferromagnetic behavior. In the ferromagnetic case the molecular fields are as follows:

\[ \mathbf{h}_a=n(-x_b\mathbf{I}_{bs}+\alpha x_a\mathbf{I}_{as}) \tag{2.9} \]

and

\[ \mathbf{h}_b=n(-x_a\mathbf{I}_{as}+\beta x_b\mathbf{I}_{bs}), \tag{2.10} \]

where \(\mathbf{I}_{as}\) and \(\mathbf{I}_{bs}\) are the spontaneous magnetizations per gram-ion of sublattices \(A\) and \(B\), respectively. We have:

\[ I_{as}=NgS\mu_B B_s\left(\frac{gS\mu_B h_a}{kT}\right), \tag{2.11} \]

\[ I_{bs}=NgS\mu_B B_s\left(\frac{gS\mu_B h_b}{kT}\right). \tag{2.12} \]

The resultant spontaneous magnetization per mole is equal to

\[ I_s=x_bI_{bs}-x_aI_{as}. \tag{2.13} \]

Fig. 8. Curves of the dependence of spontaneous magnetization on temperature for various ratios between the numbers of iron ions in the \(A\) and \(B\) sites (respectively \(x_a\) and \(x_b\)) and \(\alpha\) and \(\beta\). The capital letters \(L, N, P\), and \(Q\) correspond to Néel’s nomenclature for these types of curves.

We shall now consider the different behaviors of \(I_s\) if the ratio \(x_a/x_b\) changes.

From equation (2.13) it is evident that the spontaneous magnetization at \(0^\circ\mathrm{K}\) (when \(I_{as}=I_{bs}\)) changes sign at \(x_a/x_b=1\). It is easy to show from (2.11)—(2.12) that \(I_{as}\) and \(I_{bs}\) remain equal to one another at all temperatures only if \(h_a\) and \(h_b\) also remain equal, and this occurs only in the case when \(\alpha=\beta\).

Consequently, for \(\alpha\ne\beta\), \(I_{as}\ne I_{bs}\) at all temperatures, with the exception of \(T=0^\circ\mathrm{K}\), and the curve \(I_s=I_s(T)\) is of the kind shown in Fig. 8, \(d\) (Néel type \(L\)).

Near absolute zero and the Curie temperature, simplified expressions for the Brillouin functions may be used. By this method Néel calculated that near the Curie temperature,

i.e., for small \(I_{as}\) and \(I_{bs}\), the slope of the curve \(I_s=I_s(T)\), i.e. \(\dfrac{dI_s}{dT}\), changes sign at \(x_a/x_b=(1+\beta)/(1+\alpha)\). \(I_s\) at \(0^\circ\mathrm{K}\) is then not equal to zero, so that \(I_s\) near the Curie temperature changes sign with respect to the sign at \(0^\circ\mathrm{K}\). The curve \(I_s=I_s(T)\) is then as shown in Fig. 8, \(f\). For \(x_a/x_b\) within the limits from 1 to \((1+\beta)/(1+\alpha)\), the curve \(I_s=I_s(T)\) becomes as shown in Fig. 8, \(e\).

The condition under which \(\dfrac{dI_s}{dT}\) changes sign near \(0^\circ\mathrm{K}\) has the form \(x_a/x_b=(1-\beta)/(1-\alpha)\). Néel noted that under this condition the shape of the curve \(I_s=I_s(T)\) is the same as for a normal ferromagnet (see Fig. 8, \(b\)). Consequently, for \(x_a/x_b\) within the limits from 1 to \((1-\beta)/(1-\alpha)\), the curve \(I_s=I_s(T)\) has the form indicated in Fig. 8, \(c\). For values of \(x_a/x_b\) more distant from 1 than \((1+\beta)/(1+\alpha)\) and \((1-\beta)/(1-\alpha)\), curves \(I_s=I_s(T)\) are obtained with a smoother variation of \(\dfrac{dI_s}{dT}\) with temperature than in a normal ferromagnet; see Fig. 8, \(a\) and \(g\).

Fig. 8 shows that a continuous change of the ratio \(x_a/x_b\) will inevitably lead to a continuous change in the form of the curve of the dependence of the spontaneous magnetization on temperature in passing from type \(a\) to type \(g\), with the exception of the improbable case when \(\alpha=\beta\).

If the ratio \(x_a/x_b\) increases in magnitude, passing through unity, then the order of change of the curves will be from \(g\) to \(a\) for negative values of \(\alpha\) and \(\beta\) and for \(|\beta|>|\alpha|\), whereas for \(|\alpha|>|\beta|\) the transition from curve to curve should occur in the reverse order, from \(a\) to \(g\).

II. 2. 2. Case of nonparallelism of the ionic magnetic moments within each of the sublattices

Néel \(^{31}\) showed that the state with minimum energy, caused by the presence of a molecular field at \(0^\circ\mathrm{K}\), is not necessarily a state in which all ionic magnetic moments within each sublattice are parallel. Two further possibilities exist, in which in each sublattice \(A\) or \(B\) the ionic magnetic moments are nonparallel as a result of negative interaction between the ionic magnetic moments within the sublattices. Néel considered the distribution of nonparallel magnetic moments to be disordered. Yafet and Kittel \(^{33}\) showed that the energy of the system is minimized if the ionic magnetic moments with nonparallel orientation form an ordered arrangement within sublattices \(A\) or \(B\), either in the form of two face-centered sublattices \(A'\) and \(A''\), or in the form of four face-centered sublattices in sublattice \(B\). In the latter case,

simpler, although a formal description is also obtained if only two sublattices \(B'\) and \(B''\) are considered. The ionic magnetic moments within each of the sublattices \(A'\), \(A''\), \(B'\), and \(B''\) are mutually parallel, but the ionic magnetic moments in the sublattices \(A'\) and \(A''\), or in the sublattices \(B'\) and \(B''\), form an angle with one another as a result of the negative interactions \(AA\) or \(BB\), which compete with the interactions \(AB\).

In the modification of Néel’s theory given by Yafet and Kittel, curves \(\sigma=\sigma(T)\) of types \(V\), \(R\), and \(M\) (see \(^{31}\), p. 154), which have a nonzero slope at \(0^\circ\) K, cannot occur, since the sublattices \(A'\), \(A''\), \(B'\), and \(B''\) are saturated at \(0^\circ\) K\(*\).

Whenever it is found that the curves fall toward low temperatures like the curves \(V\), \(R\), and \(M\), they should be regarded as curves of types \(N\), \(Q\), and \(P\), respectively. Consequently,

Fig. 9. Four different theoretically possible orientations of the ionic magnetic moments in the sublattices \(A\) and \(B\) for different ratios among the interactions \(AA\), \(AB\), and \(BB\). For each of the cases \(a\)—\(c\), the drawing is made for positive and negative \(I_s\), and for case \(d\) for

\[ I_{a's}<\text{or}>I_{b's}. \]

a) The interaction \(AA\) is comparable with \(AB\), while \(BB\) is small in comparison with \(AB\).
b) The interactions \(AA\) and \(BB\) are both small in comparison with \(AB\).
c) The interaction \(AA\) is small in comparison with \(AB\), while \(BB\) is comparable with \(AB\).
d) The interactions \(AA\) and \(BB\) are large in comparison with \(AB\).

for example, every curve \(\sigma=\sigma(T)\) which has convexity in the direction of the \(T\) axis is considered as a curve of type \(Q\).

The possible arrangements of the magnetizations of the sublattices are shown in Figs. 9, \(a\) and 9, \(c\), respectively. The resulting magnetizations are equal to:

for Fig. 9, \(a\):

\[ I_s=x_b I_{bs}-x_a I_{as}\sin\varphi, \tag{2.14} \]

for Fig. 9, \(c\):

\[ I_s=x_b I_{bs}\sin\psi-x_a I_{as}, \tag{2.15} \]

where \(180^\circ-2\varphi\) and \(180^\circ-2\psi\) are the angles between the directions of the magnet—

\(*\) This does not contradict the third law of thermodynamics.

moments in the sublattices \(A'\) and \(A''\), and in the sublattices \(B'\) and \(B''\), respectively.

For the case shown in Fig. 9, \(a\), the molecular fields in the sublattice \(A'\), due to sublattice \(B\) and sublattice \(A''\), have a resultant equal in magnitude to the following expression:

\[ n\left(x_b I_{bs}+\alpha_2 x_a I_{a''s}\right)=n\alpha_2 x_a I_{a''s}, \]

where \(\alpha_2\) is the ratio of the interactions \(A'A''/AB\). Since the angle between \(\mathbf h_{a'}\) and \(\mathbf h_{a''}\) is equal to \(180^\circ-2\varphi\), this gives \(\sin\varphi=-x_b I_{bs}/\alpha_2 x_a I_{as}\), and substitution into (2.14) leads, for Fig. 9, \(a\), to

\[ I_s=x_b I_{bs}\left(1+\frac{1}{\alpha_2}\right). \tag{2.16} \]

Analogously, for Fig. 9, \(c\), one finds:

\[ I_s=-x_a I_{as}\left(1+\frac{1}{\gamma_2}\right), \tag{2.17} \]

where \(\gamma_2\) is the ratio of the interactions \(B'B''/AB\).

The fourth, theoretically possible arrangement, namely for predominant \(A'A''\) and \(B'B''\) interactions, is shown in Fig. 9, \(d\). This case will not be encountered in practice if there is only one type of magnetic ion, because the geometry of the spinel lattice favors the interaction \(AB\).

A series of crystals consisting of a mixture of two materials which, at all temperatures, have the arrangements shown in Fig. 9, \(a\) and 9, \(c\), respectively, will not give a series of curves of the dependence of magnetization on temperature of the kind shown in Fig. 8, \(a\)—\(g\). Such anomalous curves appear only because they are differences of the temperature-dependence curves of the magnetizations of two sublattices, each of the curves being a Brillouin curve. In the case under consideration, however, where \(I_s\) is determined only by \(I_{as}\) or only by \(I_{bs}\), the theory predicts normal Brillouin curves for all materials also by virtue of \(\alpha_2\ne\gamma_2\).

Yafet and Kittel showed, however, that transitions between the arrangements shown in Fig. 9, \(a\)—\(d\), can occur in one and the same material if the temperature changes. The presence of angles between the ionic magnetic moments within a sublattice at \(0^\circ\mathrm K\) does not therefore mean that the curves \(\sigma=\sigma(T)\) cannot become anomalous at higher temperatures.

It follows from equations (2.16) and (2.17) that for small negative values of \(\alpha_2\) and \(\gamma_2\) (\(|\alpha_2|\) or \(|\gamma_2|<1\)) the direction of the resultant magnetization is opposite to the direction

sublattice magnetization that determines the magnitude of this resultant magnetization, as shown in Fig. 9, a (right) and in Fig. 9, c (left). Only for large negative values of \(\alpha_2\) or \(\gamma_2\) (\(|\alpha_2|\) or \(|\gamma_2|>1\)) are these directions the same, as is shown in Fig. 9, a (left) and Fig. 9, c (right).

Large negative values of \(\alpha_2\) or \(\gamma_2\) will not occur in a spinel containing only one type of magnetic ion, owing to the geometry of the spinel lattice; thus in this case the last two of the indicated arrangements will not be realized. Spinels containing two types of magnetic ions, \(\mathrm{Me}_1\) and \(\mathrm{Me}_2\), may have very different interactions: \(\mathrm{Me}_1-\mathrm{Me}_1\), \(\mathrm{Me}_1-\mathrm{Me}_2\), and \(\mathrm{Me}_2-\mathrm{Me}_2\); and this may considerably increase \(\alpha_2\) or \(\gamma_2\), if the simple theory is applicable to this case (see, nevertheless, Section II.2.3). It is quite probable that in this case, especially for \(\gamma_2\), values \(|\gamma_2|>1\) will appear.

In the following sections we shall use the symbols \(m_a\) and \(m_b\) for the sums of the ionic magnetic moments in sublattices \(A\) or \(B\), respectively, at \(0^\circ\mathrm{K}\), expressed in Bohr magnetons per molecular unit, irrespective of their orientation. We take both quantities to be positive. The saturation magnetic moment is \(n_B=|m|\), where in the general case \(m=m_b\sin\psi-m_a\sin\varphi\). Either \(\sin\psi\), or \(\sin\varphi\), or both sines together are equal to unity; \(m\) may be either positive or negative. Expressions (2.14)—(2.17) then take the form

\[ m=m_b-m_a\sin\varphi=+m_b\left(1+\frac{1}{\alpha_2}\right), \tag{2.18} \]

\[ m=m_b\sin\psi-m_a=-m_a\left(1+\frac{1}{\gamma_2}\right). \tag{2.19} \]

II.2.3. Spinels containing two or more magnetic ions “per molecule”

Many materials investigated by us (among them some ferrites of industrial importance) contain, in addition to \(\mathrm{Fe}^{3+}\) ions, one or several other magnetic ions “per molecule.” For this reason it is necessary to give a generalization of Néel’s theory for this case. For the case of magnetic ions of two kinds, \(\mathrm{Me}_1\) and \(\mathrm{Me}_2\), situated at the sites of both lattices, there are 10 different types of interactions, since the interactions \(AA\), \(AB\), and \(BB\) depend on the kind of interacting ions. Obviously, it is impossible to obtain any results with such a large number of parameters, so that simplifying assumptions must be made. Néel’s assumption that the interaction constants \(\mathrm{Me}_1-\mathrm{Me}_1(n)\), \(\mathrm{Me}_1-\mathrm{Me}_2(n')\), and \(\mathrm{Me}_2-\mathrm{Me}_2(n'')\) are equal to one another is, of course, crude. Nissen \(^{34}\) attempted to refine Néel’s treatment by introducing three different con-

will be \(n\), \(n'\), and \(n''\), but he made several further simplifying assumptions; among them he assumed that \(\alpha\) and \(\beta\) are constants for the three interactions mentioned, \(nn'' = n'^2\), and \(g_1 = g_2\). One may also use assumptions concerning the ionic distribution, but only with great caution. Expressions can be derived from which arise the curves of the dependence of the saturation magnetization on temperature, shown in Fig. 8, \(a\)—\(g\).

II. 3. The nature of the exchange interaction

II. 3. 1. Indirect exchange interaction

The Heisenberg exchange integral depends on the degree of overlap of the wave functions of the \(3d\)-electrons of neighboring metal atoms. Slater and Neel calculated curves indicating the dependence of the magnitude of the exchange integral on \(D/d\) and \(D-d\), respectively, where \(D\) is the distance between atoms and \(d\) is the diameter of the \(3d\)-orbital. For small values of \(D/d\) (approximately \(< 1.5\)) the exchange interaction is negative; above 1.5 it becomes positive and, after passing through a maximum at \(D/d\) equal to approximately 1.8, becomes small, but still remains positive also at values of \(D/d\) equal to approximately 3. In ferrimagnetic spinels, for which \(a \geq 8.30 \text{ Å}\), the distances between neighboring iron ions are \(A—A \geq 3.61 \text{ Å}\), \(A—B \geq 3.44 \text{ Å}\), \(B—B \geq 2.94 \text{ Å}\), while \(d\) is of the order of the ionic diameter, i.e. \(1.34 \text{ Å}\), which gives for \(D/d\) values approximately equal to 2.7, 2.6, and 2.2, respectively, i.e. values lying in the region of both weak and strong positive interaction.

It should be noted that in spinels the strongest is the negative interaction \(AB\), for which (if one considers only those configurations where the Me—O distances are shortest) the triangle \(pqc\) in Fig. 6 indicates that the oxygen ion hinders the direct bond between the metallic ions more strongly than in the cases of the \(AA\) and \(BB\) interactions (Fig. 6, triangles \(qrd\) and \(ppb\), respectively). Neel assumed that here there is a mechanism of indirect exchange, in which the anions play the role of intermediaries, as was first described by Kramers. This mechanism is usually called superexchange.

By means of neutron diffraction it was proved \(^{35}\) that in the antiferromagnet MnO with the NaCl structure the strongest negative interaction exists between \(\mathrm{Mn}^{2+}\) ions situated at a distance \(a\), the angle \(\mathrm{Mn}^{2+}—\mathrm{O}^{2-}—\mathrm{Mn}^{2+}\) being \(180^\circ\), although there is also a shorter \(\mathrm{Mn}^{2+}—\mathrm{Mn}^{2+}\) distance,

equal to \(1/2a\sqrt{2}\), for which, however, the angle \(\mathrm{Mn}^{2+}—\mathrm{O}^{2-}—\mathrm{Mn}^{2+}\) is \(90^\circ\)*). Since here direct overlap of the \(3d\)-wave functions is evidently impossible, this indicates that the intermediate oxygen ion must play a role in the exchange mechanism.

II. 3. 2. Dependence of the superexchange interaction on the angle \(\mathrm{Me}—\mathrm{O}—\mathrm{Me}\)

The collinear configuration \(\mathrm{Mn}^{2+}—\mathrm{O}^{2-}—\mathrm{Mn}^{2+}\) should be more favorable for superexchange, since in this case the distance \(\mathrm{Mn}^{2+}—\mathrm{Mn}^{2+}\) increases in comparison with the distance between the ions in the rectangular configuration

\[ \begin{array}{c} \mathrm{Mn}^{2+} \\ | \\ \mathrm{Mn}^{2+}—\mathrm{O}^{2-}. \end{array} \]

Anderson\(^{36}\) showed theoretically that this is indeed so, i.e., that the exchange interaction has a maximum for the angle \(\mathrm{Me}—\mathrm{O}—\mathrm{Me}\) equal to \(180^\circ\), and a minimum for the angle equal to \(90^\circ\). Qualitatively this can be understood as follows\(^{37}\). The ground state of the pair of ions \(\mathrm{Mn}^{2+}—\mathrm{O}^{2-}\) is as follows:

\[ \begin{array}{ccc} \mathrm{Mn}^{2+}(3d^5) & \mathrm{O}^{2-}(2p^6) & \\[2pt] \left. \begin{array}{c} \downarrow \\ \downarrow \\ \downarrow \\ \downarrow \\ \downarrow \end{array} \right\} 3d^5 & \begin{array}{c} (\uparrow\downarrow) \\ (\uparrow\downarrow) \\ (\uparrow\downarrow) \end{array} & \begin{array}{c} 2p^2 \\ (2p^2) \\ (2p^2). \end{array} \end{array} \]

The six \(2p\)-electrons of the \(\mathrm{O}^{2-}\) ion form three pairs, located on three different dumbbell-shaped \(p\)-orbitals. There exists an excited state in which one \(2p\)-electron from a pair of electrons whose dumbbell axis is directed toward the \(\mathrm{Mn}^{2+}\) ion passes over to this \(\mathrm{Mn}^{2+}\) ion, and since the \(3d\)-orbital is half filled, its spin is evidently arranged antiparallel to the spins of the electrons present on \(\mathrm{Mn}^{2+}\), which gives con-

\[ \text{——} \]

*) Neutrons are scattered by the magnetic moments of unpaired electrons, as well as by atomic nuclei, and the magnetic contribution of the former to the scattering magnitude has opposite signs for ions with magnetic moments directed in mutually opposite directions. Consequently, the total scattering magnitude has different values, for example, for \(\mathrm{Mn}^{2+}\uparrow\) and \(\mathrm{Mn}^{2+}\downarrow\). The neutron-diffraction pattern on MnO below the Néel temperature reveals superstructure lines that determine a unit cell with twice the value of the lattice constant as that found from X-ray diffraction.

configuration

\[ \begin{array}{ccccc} & \mathrm{Mn}^{+} && \mathrm{O}^{-} & \\ 3d^{6}\left\{ \begin{array}{c} \downarrow\ \uparrow\\ \downarrow\\ \downarrow\\ \downarrow\\ \downarrow \end{array} \right. && \left. \begin{array}{c} \downarrow\\ (\uparrow\ \downarrow)\\ (\uparrow\ \downarrow) \end{array} \right\} & \begin{array}{l} 2p\\ (2p^{2})\\ (2p^{2}). \end{array} \end{array} \]

The remaining \(p\)-electron, having the same axis of the dumbbell as the electron that has passed to the ion \(\mathrm{Mn}^{2+}\), will now apparently be in a position to interact with the \(\mathrm{Mn}^{2+}\) ion situated at the end of the dumbbell opposite to the first ion. Under the condition that both of the assumptions indicated are valid, Anderson’s calculation shows that the perturbing influence of the excited states creates a negative exchange coupling of the proper order of magnitude between the opposite \(\mathrm{Mn}^{2+}\) ions in the chain
\[ \mathrm{Mn}^{2+}—\mathrm{O}^{2-}—\mathrm{Mn}^{2+}. \]

Because of the dumbbell-shaped form of the \(2p\)-orbitals, the superexchange interaction for the rectangular configuration should be very weak. A small interaction in this case may also be the result of hybridization of the \(2p\)- and \(2s\)-wave functions, or of the presence of excited states in which the electron makes a transition from a \(2p\)- to a \(3d\)-orbital, which, however, requires a considerable expenditure of energy.

II.3.3. Influence of the type of magnetic ion

If both ions have fewer than five \(3d\)-electrons, then the \(p\)-electron will be located in a \(3d\)-orbital with parallel spin, and therefore Anderson expects, as the result in all these cases, the occurrence of a positive interaction, citing examples such as CrFe (ferromagnetic) and comparing them with MnFe (antiferromagnetic). There are, however, many oxides of metallic ions with a number of \(3d\)-electrons smaller than five that are antiferromagnets (for example, \(\mathrm{Cr}_{2}\mathrm{O}_{3}\)), whereas only a few truly ferromagnetic oxides are known, i.e., substances with positive interactions. Thus, for example, \(\mathrm{CaMn}^{\mathrm{IV}}\mathrm{O}_{3}\) with the perovskite structure is an antiferromagnet, whereas mixed crystals

\[ \mathrm{La}_{1-x}\mathrm{Me}_{x}\mathrm{Mn}^{\mathrm{III}}_{1-x}\mathrm{Mn}^{\mathrm{IV}}_{x}\mathrm{O}_{4}, \]

in which \(\mathrm{Me}=\mathrm{Ca}\), \(\mathrm{Sr}\), or \(\mathrm{Ba}\), are true ferromagnets, i.e., they possess interactions of positive character \({}^{38}\). According to the theory presented, the interaction between ions with a number of \(3d\)-electrons smaller than five and ions with five \(3d\)-electrons may be either positive or negative.

It may be noted that Zener\(^ {39}\) proposed, in justification of the ferromagnetism of \(\mathrm{La}_{1-x}\mathrm{Sr}_{x}\mathrm{Mn}^{III}_{1-x}\mathrm{Mn}^{IV}_{x}\mathrm{O}_{4}\), another exchange mechanism. It consists in the simultaneous transfer of an electron from one metal ion to a neighboring oxygen ion and of another electron from the oxygen ion to another metal ion (“double exchange”). This mechanism should therefore occur in semiconductors for which de Boer and Verwey\(^ {40}\) postulated the presence of two ions of one metal with different valences in one crystallographic position. It was suggested\(^ {41}\) that Zener’s double-exchange mechanism plays a role in the \(BB\)-interaction in magnetite. In Sections IV—IX it will be seen that all saturation moments can be explained on the basis of negative exchange interaction.

Only one transition-metal ion possessing fewer than five \(3d\)-electrons is used in the materials described in the present article; this is the \(\mathrm{Cr}^{3+}\) ion. It will become clear below that in the materials described in Section VII the interaction \(\mathrm{Fe}^{3+}_{A}—\mathrm{Cr}^{3+}_{B}\) is undoubtedly negative.

II.3.4. Influence of Angles and Interionic Distances on the Superexchange Interaction in Spinels

Although there is as yet no theory concerning the influence of the Me—O distance on the strength of the superexchange interaction, it is assumed that, in the general case, exchange interactions decrease rapidly with increasing interionic distance. We saw in Section I.1 that the shortest Me—O distances in the spinel structure fall into two groups: the group of nearest neighbors, formally denoted \(p\) and \(q\), and the group of more distant distances, denoted \(r\), \(s\), and \(t\) (see Section I.1). Configurations Me—O—Me with the more distant Me—O distances, as well as configurations in which both distances belong to the groups \(r\), \(s\), and \(t\), will be regarded as having negligible interactions in comparison with the interactions obtained when one distance is \(p\) or \(q\), and the other \(p\), \(q\), \(r\), \(s\), or \(t\). From Fig. 6 we see that only in the triangle \(pqc\) are both Me—O distances and the Me—O—Me angle favorable in the sense indicated above. The arrangements \(pre\) and \(tqe\) have very favorable angles, but one large interionic distance. The interaction \(AB\) will therefore be strong. In \(qrd\) the angle is very unfavorable and one of the interionic distances is large; consequently, the interaction \(AA\) will be very weak. In \(ppb\) the angle is very unfavorable, but both distances are small; in \(ptb\), \(psb\), and \(psf\) the angles are more favorable, but one of the distances is large. Consequently, it may be expected that the \(BB\) interaction will have an intermediate magnitude between those of the \(AB\) and \(AA\) interactions.

In the literature up to now only those interactions have been considered for which the distances \(AA\), \(AB\), and \(BB\) are the shortest (i.e., \(d\), \(c\), and \(b\), respectively). The relative influence on one another of the angles and the interionic distances is still unknown; there are no grounds on which other configurations could be neglected.

The virtue of the application of the molecular field is that the latter is in no way connected with the number of configurations taken into account, since the molecular-field constants \(n\), \(\alpha\), \(\beta\), etc., include all the interactions that play a role. However, if an attempt is made to connect the molecular-field constants with exchange integrals, then, apparently, one must take into account all those configurations that make contributions to the interaction energy.

P. S. Weiss\(^{42}\) made an attempt to derive an empirical quantitative relation giving the dependence of the exchange-interaction energy \(k\theta\) on the distance \(\mathrm{Me—O+O—Me}'\) \((l)\) and on the angle \(\mathrm{Me—O—Me}'\) \((\varphi)\):

\[ k\theta = C S_1 S_2 e^{-10l} \cos^6 \varphi , \]

where \(C\) is a constant, and \(S_1\) and \(S_2\) are the spin quantum numbers of the ions Me and Me′, respectively.

For the antiferromagnets MnO, FeO, and CoO, for which \(\cos \varphi = 1\), the constant \(C\) is found from the Néel temperatures; however, it is not found for NiO.

The formula was applied to spinels in order to obtain the ratio of the exchange-interaction energies \(AA\), \(AB\), and \(BB\). For \(\alpha\) and \(\beta\) values of the order \(10^{-9}\) and \(10^{-7}\), respectively, were found. Weiss therefore took \(E_{\mathrm{exch}}(AB) = k\theta\). The Curie temperature calculated in this way from an interaction of type \(AB\) agrees with the experimental value to within 2%; in an earlier communication Weiss used the factor \(e^{-7l}\), which gave agreement with experiment to within 15%. It will be seen that most of the experiments described in the following sections cannot be understood for the small values of the constants \(\alpha\) and \(\beta\) calculated by Weiss. The theoretical basis for his formula is still insufficient.

III. EXPERIMENTAL METHODS

III.1. Methods of magnetic measurements used in Sections IV–IX

In the following sections we shall set forth and discuss measurements of the saturation magnetizations of various ferrimagnetic oxides.

The measurements were carried out by the ponderomotive method described by Pauthenay and Choek\(^{43}\) (Fig. 10). The material \(p\) is fastened to a horizontal pendulum \(P\), suspended on four wires \(W\) and moving perpendicular to the inhomogeneous field. The field varies almost exactly according to the law

\[ H = H_0 - \frac{1}{2} ax^2 \]

in the direction of oscillation of the pendulum along the \(x\)-axis; these oscillations take place near the surface of pole pieces having the form of identical spherical segments \(N\) and \(S\). The constant of the force acting on the magnetic material in this inhomogeneous field is equal to \(a\sigma m_p\), where \(\sigma\) is the saturation magnetization in CGSM·\(\mathrm{cm}^3/\mathrm{g}\), and \(m_p\) is the weight of the specimen. The magnet-

the saturation magnetization $\sigma$ is found from the formula

\[ \sigma=\frac{4\pi^{2}m_p}{a m_p}\left(\frac{1}{\tau^{2}}-\frac{1}{\tau_0^{2}}\right), \]

where $m_p$ is the weight of the pendulum, $\tau_0$ is the period of oscillation of the pendulum in the magnetic field without magnetic material, and $\tau$ is the period of oscillation of the pendulum with the attached material. The constant $4\pi^2/a$ is determined by calibration, using very pure iron and nickel as the material.

Fig. 10. Apparatus for measuring saturation magnetization (see the text for the notation).

The measurements described in Sections IV, VII, and IX were carried out in fields up to 5900 oersteds; the measurements described in Sections V, VI, and VIII, in fields up to 8000 or 9000 oersteds. In the case of measurements at high temperatures, the end of the pendulum carrying the magnetic material moves in a small tubular furnace; in the case of measurements at low temperatures, the end of the pendulum moves in channel $C$ through vessel $V$, which contains liquid nitrogen. To obtain temperatures between $80^\circ\mathrm{K}$ and room temperature, a stream of liquid nitrogen is sucked from a Dewar vessel and passed through vessel $V$. By varying the rate of suction, any constant temperature can be obtained. At liquid-nitrogen temperatures, a slow stream of dry nitrogen is passed through the channel in order to prevent condensation of oxygen on the pendulum. The temperatures were measured with a thermocouple connected to a millivoltmeter by thin conductors so that they would not interfere with the motion of the pendulum.

Measurements of the saturation magnetization at the temperature of liquid hydrogen were carried out by Folger and Jochenburger by a ballistic method, using a water-cooled solenoid to produce the field. These measurements were made on rods of dimensions $3 \times 3 \times 50\ \mathrm{mm}^3$ at field strengths of about 7000 oersteds. In this method the magnetic flux $\Phi$ in the rod is measured. The magnetization in CGSM units is equal to $I=\Phi/4\pi A$ ($A$ is the cross-sectional area of the rod in sq. cm), and $\sigma$ is obtained from the equation

\[ \sigma=\frac{I}{d}=\frac{IAl}{m_p}=\frac{\Phi l}{4\pi m_p}. \]

where \(d\) is the specific gravity, equal to the weight divided by the external volume, and \(l\) is the length of the specimen in cm.

The measurements set out in Section IV were carried out when liquid hydrogen was not yet available. Consequently, the quantities given in that section were not extrapolated to \(0^\circ\mathrm{K}\), and are smaller by one decimal place than the quantities presented in the subsequent sections. The latter quantities were extrapolated according to the \(T^2\) law. Extrapolation according to the \(T^{3/2}\) law, postulated by Bloch’s spin-wave theory, does not give substantially different results.

The magnetic moment per molecular unit \(\mathrm{Me}_3\mathrm{O}_4\) (\(n_B\)), expressed in Bohr magnetons, is obtained from \(\sigma_{T=0}\) \((\sigma_0)\) for \(H \to \infty\) by the formula

\[ n_B = \frac{\sigma_0 \times \text{mol. wt. }(M)} {(\text{Avogadro’s number}) \times (\text{Bohr magneton in } \mathrm{erg/gauss})} = \frac{\sigma_0 M}{5585}. \]

III.2. Preparation of Materials

III.2.1. Final Sintering Operation

The final operation in the preparation of all the materials investigated was sintering, carried out on pressed bars, balls, etc., placed in alumina boats inside a gas-permeable tube of an electric furnace made of molybdenum wire. This operation is performed at temperatures of \(900\)—\(1350^\circ\mathrm{C}\) and in the appropriate atmosphere, which to a large extent determines the composition of the specimen. To reduce the influence of the atmosphere and of high temperatures, two concentric gas-permeable tubes were used, with air blown through the space between them.

Practically all of the materials investigated were spinels and contained iron ions, and many also contained other transition-metal ions, which could be ions of different valence—either greater or smaller than required. In equilibrium, all materials are characterized by a definite oxygen pressure, which increases with temperature and will be different for different compositions. Owing to the specific feature of the oxide spinel of forming mixed crystals containing ions ranging from tetravalent to monovalent, as well as vacant sites in the lattice, sintering in an atmosphere with an oxygen partial pressure markedly different from the equilibrium value leads to the formation of spinels either with an inhomogeneous or with a uniform composition, as, for example, in the case

\[ \mathrm{MnFe}_2\mathrm{O}_4 + \frac{1}{2}x\,\mathrm{O}_2 \to (1 - 3x)\,\mathrm{MnFe}_2\mathrm{O}_4 \cdot x\,\mathrm{Mn}_3\mathrm{O}_4 \cdot 3x\,\mathrm{Fe}_2\mathrm{O}_3, \]

where, for \(x \ll 1\), either this phase or the other phase may be formed:

\[ \mathrm{MnFe_2O_4} - \frac{1}{2}x\,\mathrm{O_2} \rightarrow (1-x)(\mathrm{Mn},\mathrm{Fe}^{\mathrm{II}})\mathrm{Fe_2O_4} + 3x(\mathrm{Mn},\mathrm{Fe}^{\mathrm{II}})\mathrm{O}. \]

Divalent oxides are usually practically insoluble in the spinel phase.

For practically all materials, with the exception of \(\mathrm{Fe_3O_4}\), the equilibrium oxygen pressure below the melting point exceeds \(1\ \mathrm{atm}\); therefore the materials cannot be melted in a stream of oxygen at a pressure of \(1\ \mathrm{atm}\) without serious decomposition. Thus the upper temperature limit for the sintering process is determined primarily by the temperature at which the equilibrium oxygen pressure reaches \(1\ \mathrm{atm}\).

Another limiting factor is the possibility of a chemical reaction between the material and the vessel. We usually used crucibles made of recrystallized alumina (fired above \(1800^\circ\mathrm{C}\)) or placed fused-alumina powder under the material. Alumina proved to be chemically very inert. To avoid any reactions with alumina, the sintering temperatures were kept below \(1350^\circ\mathrm{C}\). When necessary, the surface of the material that had been in contact with alumina was removed.

Many of the spinels which, along with \(\mathrm{Li^+}\), \(\mathrm{Mg^{2+}}\), and \(\mathrm{Al^{3+}}\) ions (having noble-gas shells), contained only \(\mathrm{Fe^{3+}}\), \(\mathrm{Ni^{2+}}\), \(\mathrm{Cr^{3+}}\), and \(\mathrm{Cu^{2+}}\) ions, possessing valences higher than required, proved to be very unstable; this was manifested in the fact that, even in an oxygen atmosphere at the minimum temperature required to obtain complete reaction, no formation of ions with higher valences was observed. Such materials were fired in oxygen at a temperature low enough not to produce any noticeable conversion of \(\mathrm{Fe^{3+}}\) into \(\mathrm{Fe^{2+}}\). In spinels containing \(\mathrm{Cu^{2+}}\), the oxygen content at temperatures above \(900\text{--}1000^\circ\mathrm{C}\) decreases appreciably. Such materials may have pores, so that any oxygen losses occurring during firing can be compensated by oxygen during cooling.

Although the preparation of spinels containing ions whose valence states are lower than those required is difficult, it is nevertheless quite easy to obtain, for example, \(\mathrm{MnCr_2O_4}\). Consequently, \(\mathrm{MnCr_2O_4}\) can be fired over a wide range of reduced pressures of the surrounding atmosphere without formation of metallic Mn (or Cr) and with only a slight amount of excess oxygen. The problem of obtaining correct compositions is considerably more difficult for spinels containing ions such as \(\mathrm{Fe^{2+}}\), \(\mathrm{Mn^{2+}}\), and, to a lesser extent, \(\mathrm{Co^{2+}}\), for which higher valence states remain stable up to high temperatures, and also for spinels containing ions such as \(\mathrm{Fe^{3+}}\), for which

at high temperatures lower valence states are formed.

To obtain the required composition (i.e., a definite oxygen content), one can choose one of two methods:

1) while maintaining a constant composition of the atmosphere, change the temperature until chemical analysis shows that the correct oxygen content has been obtained;

2) while maintaining a constant temperature, change the oxygen content in the atmosphere until the material has the correct oxygen content.

However, it must be borne in mind that an atmosphere which is in equilibrium with the material at the sintering temperature will reoxidize it at low temperatures and, mainly, on the outer surface, so that the composition becomes nonuniform during cooling. This can be avoided to a considerable extent by quenching the material from the sintering temperature; however, as we shall see, the distribution of cations between tetrahedral and octahedral sites often depends strongly on temperature, so that with such treatment different materials are obtained quite independently of the effect of possible reoxidation.

We shall be interested in the properties of a material having the most stable ion distribution and, consequently, one that has been slowly cooled. The ideal method for obtaining a homogeneous material with the correct oxygen content and with the most stable ion distribution is the method of slowly cooling the material in a continually changing atmosphere, which at all temperatures is in equilibrium with the oxygen pressure of the material. Smiltens used a stepwise method of changing the atmosphere in preparing a single crystal of \(Fe_2O_4\), using his own data on the equilibrium pressure \(^{44}\). Determining the equilibrium oxygen pressure as a function of temperature is a complicated and laborious process, and it has been carried out only for \(Fe_3O_4\).

Sometimes the method of reducing reoxidation during cooling by sintering to a high surface density can be used successfully, so that the atmosphere cannot penetrate through the pores into the material \(^{45}\). Another method consists in firing the material in a mixture of gases, the oxygen content of which changes with temperature (for example, in \(CO_2\), \(CO_2 + CO\), \(H_2O\), \(H_2O + H_2\)). From Smiltens’ work it is known that the decrease in oxygen content with decreasing temperature in a \(CO_2 - CO\) mixture is insufficient to maintain equilibrium with \(Fe_3O_4\). This, apparently, is a very general phenomenon for known gas mixtures.

When the firing temperature is lowered, the difference between the oxygen pressures at the firing temperature and at room temperature decreases. The minimum sintering temperature is determined from the condition that the reaction proceeds completely at it. This

the temperature can be lowered by the appropriate method of preparing the powder for sintering. In practice, in most cases we left the materials to cool after the furnace was switched off in the same atmosphere. The cooling time was 7–9 hours. Often we fired a specimen at low temperatures, sufficient only for ion diffusion to occur over the required time intervals. Whenever this was done, our subsequent recipes contain special notes to that effect.

To avoid errors in comparing the distribution of ions in a series of mixed crystals, we always fired the entire series at one temperature and, consequently, often used different atmospheres for individual members of the series (for example, in the series $\mathrm{MnFe_2O_4}$—$\mathrm{MnCr_2O_4}$). A single sintering temperature cannot be used if certain members of this series do not react completely at the temperature at which melting already occurs for other members (see the series $\mathrm{CaFe_2O_4}$—$\mathrm{ZnFe_2O_4}$).

In view of the volatility of $\mathrm{ZnO}$ and $\mathrm{Li_2O}$, preliminary measures were taken so that no free $\mathrm{ZnO}$ or $\mathrm{Li_2O}$ would remain at the moment when the final sintering operation began. Since these oxides volatilize from their compounds already under the conditions usually employed, sintering was carried out at a temperature not higher than $1300^\circ\mathrm{C}$ for Zn compounds and not higher than $1150^\circ\mathrm{C}$ for Li compounds.

III.2.2. Methods of preparing powder for sintering

The preparation of powders for sintering was carried out by several methods.

A. Pure oxides or, in individual cases, carbonates were ground together in a steel ball mill with a chromium-plated surface or in a Bloch–Rosetti agate mill with ethyl alcohol. The mixture was dried under an electric (infrared) lamp, was usually prefired in air at a temperature below the final sintering temperature, and was again ground in the ball mill. Before use, zinc oxide was heated to $500^\circ\mathrm{C}$ in order to decompose a certain amount of $\mathrm{ZnCO_3}$ present in it. Other oxides, before chemical analysis and use, were heated to $200^\circ\mathrm{C}$.

B. Pure metals (or, in some cases, oxides and carbonates) were dissolved in nitric acid; the solutions were evaporated to dryness under an infrared lamp (often in a sand bath), and the nitrates were decomposed to oxides. When the powder had completely dried, it was transferred from a porcelain dish to a Glinozem crucible and heated to $800^\circ\mathrm{C}$. The powder was then ground in a Bloch–Rosetti agate mill. Compounds containing titanium were prepared by adding $\mathrm{TiO_2}$ (synthetic anatase) to the nitrate solution before evaporation and drying.

B′. A method similar to B, but using sulfuric acid instead of nitric acid. The preliminary calcination was interrupted at \(600^\circ\mathrm{C}\) for the grinding operation, then continued up to \(900^\circ\mathrm{C}\), after which the powder was ground again.

B″. A method similar to B, but ammonia was added to the nitrate solution. The hydroxides were filtered, washed, dried, and decomposed at \(800^\circ\mathrm{C}\).

C. To the solution of nitrates and sulfates prepared by methods B and B′, a soda solution was added at \(100^\circ\mathrm{C}\); the precipitate was boiled, filtered, washed thoroughly, dried, and decomposed by preliminary heating at \(800^\circ\mathrm{C}\).

In the following sections we shall describe the preparation of the various materials, indicating:

a) the starting materials with the principal impurities, if they amount in total to more than \(0.01\%\);

b) the method used;

c) the temperature and duration of the preliminary calcination and the atmosphere, if it differs from air (for example, preliminary calcination for 2 hours at \(800^\circ\mathrm{C}\) in \(\mathrm{O}_2\));

d) the temperature and duration of the final sintering and the atmosphere (for example, sintering for 2 hours at \(1200^\circ\mathrm{C}\) in \(\mathrm{O}_2\)).

III.3. X-ray Analysis

X-ray diffraction patterns for all the materials mentioned were obtained using a Norelco X-ray diffractometer. All the materials listed above had a spinel pattern without additional lines, except in cases where superstructure lines occurred, or in certain other cases that will be mentioned separately. The lattice constants were determined using Co \(K\alpha_1\), Fe \(K\alpha_1\), or Mo \(K\alpha_1\) radiation, at the following wavelengths: Co \(K\alpha_1\) — \(1.78890\ \text{Å}\), Fe \(K\alpha_1\) — \(1.93597\ \text{Å}\), Mo \(K\alpha_1\) — \(0.70926\ \text{Å}\).

III.4. Chemical Analysis

Analyses of the oxygen content were carried out for most of the materials. The method used was developed by G. V. Van Oosterhout and A. Bolom.

A weighed sample was placed in a glass tube with an internal diameter of \(7\ \text{mm}\), having a constricted section. The tube was cleaned by flushing with carbon dioxide gas or very pure nitrogen. Then Mohr’s salt solution in six-normal hydrochloric acid with a known titer was introduced. After flushing with the same gas, the tube was sealed and heated to \(120\text{–}200^\circ\mathrm{C}\), until the material decomposed (usually this took from 8 to 24 hours). Then the tube was opened and the contents were transferred into a titration vessel, which was continuously flushed with carbon dioxide gas.

Divalent iron was titrated potentiometrically using a lamp voltmeter, as described by Claassen[^46], in a 0.01 or 0.1 N solution of ceric sulfate. By this method an excess or deficiency of oxygen with respect to Fe$^{3+}$, Mn$^{2+}$, Co$^{2+}$, Ni$^{2+}$, or Cu$^{2+}$ can be determined. For titrating excess oxygen, the amount of Mohr’s salt in hydrochloric acid was adjusted so that a small excess was obtained in each case. To determine an oxygen deficiency, a small amount of Mohr’s salt was added to hydrochloric acid in order to reduce the dissolved oxygen. The titer of the solution was established at least one day after preparation. It is convenient to store the solutions in vessels; a precisely measured amount of solution was taken from these vessels through a silver reductor for the reaction in a tube. The titer of the solution was established exactly as described above, independently of the particular kind of material being titrated. To determine the total iron content, the sample solution was passed, before titration, through a silver reductor.

The titrations were carried out by T. Gerharts for the materials described in Section IV, and, for the materials described in Sections V—IX, in the analytical chemistry department of our laboratory under the supervision of A. Claassen and D. Visser. In some cases, determination of the quantitative ratio of the metallic ions present was performed. It was found that, within the limits of analytical error, there was no deviation from their quantitative ratio in the starting material. A number of spinels containing volatile oxides were weighed before and after final sintering. In the case where the material had reacted practically completely at a temperature considerably below the firing temperature of 1150°C, the loss in weight of the lithium-containing spinel was small—less than 1% of the Li$_2$O present. For spinels containing zinc but not containing divalent manganese or iron ions, no losses in zinc were observed; for spinels containing Zn$^{2+}$ and Mn$^{2+}$ ions, no loss of zinc was observed after it had been possible to avoid the presence of free ZnO during final sintering. For spinels containing Zn$^{2+}$ and Fe$^{2+}$ ions, special measurements were made, which will be described in Section IV.

The amount of impurities present in the starting material was determined, in order of magnitude, by a semiquantitative spectrochemical method by H. V. Addink.

IV. EXPERIMENTAL PROOF OF THE VALIDITY OF NEEL’S HYPOTHESIS: FERRITES

IV.1. Introduction

In order to obtain experimental proof of Neel’s theory, we prepared simple ferrites MnFe$_2^{\mathrm{III}}$O$_4$, Fe$^{\mathrm{II}}$Fe$_2^{\mathrm{III}}$O$_4$, CoFe$_2^{\mathrm{III}}$O$_4$, NiFe$_2^{\mathrm{III}}$O$_4$, Cu$^{\mathrm{II}}$Fe$_2^{\mathrm{III}}$O$_4$, MgFe$_2^{\mathrm{III}}$O$_4$, Li$_{0.5}$Fe$_{2.5}^{\mathrm{III}}$O$_4$,

\(\mathrm{ZnFe_2^{III}O_4}\) and \(\mathrm{CdFe_2^{III}O_4}\) and mixed crystals of the first seven ferrites (having an inverse distribution of cations) with zinc ferrite.

The saturation magnetic moments of the named materials, expressed in Bohr magnetons, had been found earlier\(^{48,49,50}\). More complete data on most of these materials were reported later by other authors, who gave curves of the dependence of the saturation magnetization on temperature and measurements of the temperature dependence of the susceptibility above the Curie temperature. In the present section we shall not present new magnetic-measurement data, but shall discuss the experimental data currently available. As we shall see, the magnetic properties depend on the distribution of ions, and since this distribution in turn depends on the method of preparation, in Sec. IV.2 we indicate our methods for preparing the ferrites studied.

IV.2. Preparation and Analysis of Ferrites

a) Manganese-zinc ferrites \(\mathrm{Mn_{1-a}Zn_aFe_2O_4}\) were prepared from \(\mathrm{MnCO_3}\) (Ca 0.1%, Mg 0.12%, Zn 0.05%, Na \(<0.04\%\)), \(\mathrm{ZnO}\) (Mg \(<0.01\%\)), \(\mathrm{Fe_2O_3}\) (Si 0.04%, Pb 0.03%, Mn 0.1%) by method A. The preliminary calcination was carried out for 2 hours at \(1000^\circ\mathrm{C}\) in air; sintering was carried out by heating in \(\mathrm{O_2}\) to \(1100^\circ\mathrm{C}\), then to \(1250^\circ\mathrm{C}\), and for 2 hours at \(1250^\circ\mathrm{C}\) in \(\mathrm{N_2}\) containing traces of \(\mathrm{O_2}\). \(\mathrm{MnFe_2O_4}\) contained 0.05% \(\mathrm{Fe^{2+}}\); the same \(\mathrm{MnZn}\) ferrites for which data are given contained \(<0.1\%\) \(\mathrm{Fe^{2+}}\).

b) Iron-zinc ferrites were prepared from the same \(\mathrm{ZnO}\) and \(\mathrm{Fe_2O_3}\) as in item a), by method A. The preliminary calcination was carried out for 2 hours at \(900^\circ\mathrm{C}\) in \(\mathrm{N_2}\); sintering for 2 hours at \(1250^\circ\mathrm{C}\) in an \(\mathrm{N_2}\) atmosphere containing a variable amount of oxygen. The material was placed in a shell of \(\mathrm{ZnO}\). For two materials the analysis showed that the content of \(\mathrm{Fe^{2+}}\) and \(\mathrm{Zn}\) was correct within 0.1%.

Iron ferrite was calcined for 2 hours at \(1350^\circ\mathrm{C}\) in a \(\mathrm{CO_2—H_2}\) mixture and slowly cooled in a \(\mathrm{CO_2—H_2}\) mixture, the composition of which was varied stepwise, as indicated by Smiltens. The \(\mathrm{CO_2—H_2}\) ratios were the same as the \(\mathrm{CO_2—CO}\) ratios given by Smiltens. However, our \(\mathrm{Fe_3O_4}\) contained 1.04 \(\mathrm{Fe^{2+}}\) per 2.00 \(\mathrm{Fe^{3+}}\).

c) Cobalt-zinc ferrites \(\mathrm{Co_{1-a}Zn_aFe_2O_4}\) were prepared from \(\mathrm{CoCO_3}\) (\(<0.1\%\) impurities), \(\mathrm{ZnO}\) (the same as in item a)), \(\mathrm{Fe_2O_3}\) (Ni 0.03%, Mn 0.02%, Si 0.02%) by method A; the preliminary calcination was carried out for 2 hours at \(800^\circ\mathrm{C}\) in air; sintering for 2 hours in oxygen or in air at \(1250^\circ\mathrm{C}\). \(\mathrm{CoFe_2O_4}\) contained no \(\mathrm{Fe^{2+}}\) or excess oxygen; the \(\mathrm{CoZn}\) ferrite contained \(<0.1\%\) excess oxygen.

d) Nickel-zinc ferrites \(\mathrm{Ni}_{1-a}\mathrm{Zn}_a\mathrm{Fe}_2\mathrm{O}_4\) were prepared from a solution of \(\mathrm{NiSO}_4\) (\(\mathrm{Na}<0.07\%\), \(\mathrm{Ca}\ 0.03\%\), \(\mathrm{Co}\ 0.02\%\), \(\mathrm{Si}\ 0.02\%\)), \(\mathrm{ZnO}\) (the same as in item a)), and \(\mathrm{Fe}\) (\(<0.01\%\) impurities) by method B; sintering was carried out in oxygen for 2 hours at \(1200^\circ\mathrm{C}\) or for 4 hours at \(1250^\circ\mathrm{C}\). The \(\mathrm{Fe}^{2+}\) content was negligible, and the \(\mathrm{SO}_3\) content was small, i.e. \(<0.1\%\).

e) Copper-zinc ferrites proved difficult to prepare, since they readily lose oxygen at temperatures at which complete reaction takes place. The starting materials used were \(\mathrm{CuO}\) (\(\mathrm{Pb}\ 0.03\%\), \(\mathrm{Si}\ 0.02\%\)), \(\mathrm{ZnO}\) (the same as in item a)), and \(\mathrm{Fe}_2\mathrm{O}_3\) (the same as in item c)). \(\mathrm{CuFe}_2\mathrm{O}_4\): preliminary firing was carried out for 2 hours at \(700^\circ\mathrm{C}\); sintering for 2 hours at \(900^\circ\mathrm{C}\) in \(\mathrm{O}_2\); annealing for 168 hours at \(360^\circ\mathrm{C}\) in \(\mathrm{O}_2\). The sample showed a tetragonal hausmannite structure with \(c=8.68\ \text{Å}\), \(a=8.24\ \text{Å}\). It contained \(0.1\%\ \mathrm{Fe}^{2+}\) (or \(\mathrm{Cu}^{+}\)). \(\mathrm{Cu}_{0.5}\mathrm{Zn}_{0.5}\mathrm{Fe}_2\mathrm{O}_4\): preliminary firing was carried out for 2 hours at \(800^\circ\mathrm{C}\); sintering for \(3^{1}/_{2}\) hours at \(1050^\circ\mathrm{C}\). It contained \(0.1\%\ \mathrm{Fe}^{2+}\) (or \(\mathrm{Cu}^{+}\)).

f) Magnesium-zinc ferrites were prepared from \(\mathrm{MgO}\) (\(\mathrm{Ca}\ 0.2\%\), \(\mathrm{Fe}\ 0.01\%\)), \(\mathrm{ZnO}\) (the same as in item a)), and \(\mathrm{Fe}_2\mathrm{O}_3\) (the same as in item c)), by method A. Preliminary firing was carried out for 2 hours at \(900^\circ\mathrm{C}\); sintering for 4 hours at \(1200^\circ\mathrm{C}\) in \(\mathrm{O}_2\). \(\mathrm{MgFe}_2\mathrm{O}_4\) was annealed for 24 hours at \(700^\circ\mathrm{C}\) in \(\mathrm{O}_2\) and cooled slowly. The materials contained \(<0.1\%\ \mathrm{Fe}^{2+}\).

g) Lithium-zinc ferrites \(\mathrm{Li}_{0.5-0.5a}\mathrm{Zn}_a\mathrm{Fe}_{2.5-0.5a}\mathrm{O}_4\) were prepared from \(\mathrm{Li}_2\mathrm{CO}_3\) (\(\mathrm{Mg}\ 0.02\%\), \(\mathrm{Na}\ 0.1\%\), \(\mathrm{Ca}\ 0.02\%\)), \(\mathrm{ZnO}\) (the same as in item a)), and \(\mathrm{Fe}_2\mathrm{O}_3\) (the same as in item a)). Powders \(1\mathrm{Li}_2\mathrm{CO}_3 + 1\mathrm{Fe}_2\mathrm{O}_3\) were ground in an agate ball mill with pure ethyl alcohol and, after drying, were heated to \(700^\circ\mathrm{C}\) in oxygen for 5 hours. The \(\mathrm{LiFeO}_2\) thus obtained was used in method A; preliminary firing was carried out for 2 hours at \(750^\circ\mathrm{C}\), and sintering for 4 hours at \(1150^\circ\mathrm{C}\) in \(\mathrm{O}_2\). The \(\mathrm{Fe}^{2+}\) content was \(<0.1\%\).

The following materials were quenched by being immersed in a saturated \(\mathrm{NaCl}\) solution, which was then washed off with boiling water, or by immersion in oil, which was washed off with benzine: \(\mathrm{NiFe}_2\mathrm{O}_4\) from \(1250^\circ\mathrm{C}\); \(\mathrm{CuFe}_2\mathrm{O}_4\) from \(900^\circ\mathrm{C}\); \(\mathrm{MgFe}_2\mathrm{O}_4\) from \(1250^\circ\mathrm{C}\); \(\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5}\mathrm{O}_4\) from \(1150^\circ\mathrm{C}\). All these samples contained \(<0.1\%\ \mathrm{Fe}^{2+}\).

IV.3. Saturation magnetic moments of simple ferrites \((\mathrm{Me}^{\mathrm{II}}\mathrm{Fe}^{\mathrm{III}}_{2}\mathrm{O}_4\) and \(\mathrm{Li}_{0.5}\mathrm{Fe}^{\mathrm{III}}_{2.5}\mathrm{O}_4)\)

Our results, obtained earlier \(^{49,50}\), are given in Table IV together with the results obtained later by Pauthenet \(^{25,26,51,52}\) and by Gilleo with co-workers \(^{53,54a-c,55}\). Gilleo states that his materials, except for those stoichiometric to within \(0.2\%\),

SATURATION MAGNETIZATION OF FERRIMAGNETIC OXIDES

Table IV.

Saturation magnetic moments of simple ferrites

Ferrite	1 Gorter \(^{50,49}\)	2 Guillaud \(^{53,54,55}\)	3 Pauthenet \(^{26,51,52,53}\)	4 Other authors	5 \(2S_{\mathrm{Me}^{2+}}\)
\(\mathrm{MnFe_2O_4}\)	5.0	4.60	\(4.40 \pm 0.04\)	—	5
\(\mathrm{FeFe_2O_4}\)	4.2	4.03	\(4.08\ (\pm 0.01)^*\)	\(4.08^{39*}\)	4
\(\mathrm{CoFe_2O_4}\)	3.3	3.67; 3.70	\(3.94 \pm 0.002\)	—	3
\(\mathrm{NiFe_2O_4}\)	2.3	2.40	2.224	—	2
\(\mathrm{CuFe_2O_4}\)	1.3	—	1.37	\(>1.70^{56}\)	1
\(\mathrm{MgFe_2O_4}\)	1.1	\(>1.0\)	0.86	\(1.1^{57}\)	0
\(\mathrm{Li_{0.5}Fe_{2.5}O_4}\)	2.6	—	—	—	(2.5)
\(\mathrm{ZnFe_2O_4}\)	0.0	0	—	—	0
\(\mathrm{CdFe_2O_4}\)	0.0	—	—	—	0

*) Natural magnetite.

contain less than 0.05 weight percent of metal in states of lower or higher valence. Pauthenet gives no data on the preparation or purity of the materials, except in cases where the materials were pure spinels.

In contrast to our results set forth in the present section, which were calculated from the magnetization at \(77^\circ\mathrm{K}\) and a magnetic field of 5900 oersteds, these authors also carried out measurements at \(20^\circ\mathrm{K}\) and in fields up to 20000 oersteds, and extrapolated the results to \(H=\infty\) and \(T=0^\circ\mathrm{K}\).

We have seen (Section I.2) that, according to Verwey and Heilmann, the six ferrites indicated above are inverse: \(\mathrm{Fe[Me^{II}Fe]O_4}\). Consequently, in these six ferrites the antiparallelism of the ionic magnetic moments at the \(A\) and \(B\) sites, which is the result of the predominant \(AB\) interaction, leads to a saturation moment equal to \((5+m_{\mathrm{Me}^{2+}})-5=m_{\mathrm{Me}^{2+}}\). Taking \(g_{\mathrm{Me}^{2+}}=2\), we obtain \(m_{\mathrm{Me}^{2+}}=2S_{\mathrm{Me}^{2+}}\). A comparison of columns 1 and 5 of Table IV indicates that our measurements, to a first approximation, confirmed Néel’s hypothesis. Braun\(^{12}\) found that \(\mathrm{Li_{0.5}Fe_{2.5}O_4}\) has a completely inverse ion arrangement. Consequently, for \(\mathrm{Fe[Li_{0.5}Fe_{1.5}]O_4}\) the saturation magnetic moment is calculated as follows: \(7.5-5=2.5\). Agreement with the experimental value of the magnetic moment indicates that \(m_{\mathrm{Fe}^{3+}}=5\), as expected.

There are two possible reasons for the discrepancies between our values of \(m_{\mathrm{Me}^{2+}}\) and the values \(2S\): (1) the ferrites under discussion

are not completely inverted; (2) the \(g\)-factor of \(\mathrm{Me}^{2+}\) is not equal to 2. The first of these hypotheses was first proposed by Néel \(^{31}\), in order to explain the increase in the saturation magnetic moment of \(\mathrm{CuFe}_2\mathrm{O}_4\) upon quenching \(^{36}\). If a slowly cooled ferrite is not completely inverted or normal, then, consequently, it has not reached the state with the smallest free energy. Therefore one may expect that the ionic distribution will vary with temperature. The greatest entropy would be attained with a completely random arrangement, i.e. \(\mathrm{Fe}_{2/3}\mathrm{Me}^{\mathrm{II}}_{1/3}[\mathrm{Me}^{\mathrm{II}}_{2/3}\mathrm{Fe}_{4/3}]\mathrm{O}_4\). Thus, the general formula of the ferrite \(\mathrm{Me}^{\mathrm{II}}\mathrm{Fe}_2\mathrm{O}_4\) is as follows:

\[ \mathrm{Fe}_{1-x}\mathrm{Me}^{\mathrm{II}}_{x}[\mathrm{Me}^{\mathrm{II}}_{1-x}\mathrm{Fe}_{1+x}]\mathrm{O}_4. \tag{4.1} \]

The theoretical limiting value of \(x\), i.e. \(x = 1/3\), has in practice not been reached at the highest temperatures to which it was possible to heat the ferrite without decomposition through the reduction of \(\mathrm{Fe}^{3+}\) to \(\mathrm{Fe}^{2+}\). The value of \(x\), further, will be determined by the cooling rate. If the ferrite is quenched from a high temperature, then the equilibrium distribution at the quenching temperature may be fixed to a considerable extent. If the ferrite is cooled slowly, the ionic distribution will try to maintain equilibrium until a temperature is reached at which ion diffusion becomes too slow to maintain equilibrium upon further cooling. Consequently, the distributions actually observed are to some extent frozen in, and \(x\) depends on the cooling rate.

The magnetic moment calculated from formula (4.1) is as follows:

\[ n_B = 10x + m_{\mathrm{Me}^{2+}}(1 - 2x). \tag{4.2} \]

We assumed that, if \(x \ne 0\), then it should vary with temperature. We therefore subjected several ferrites to quenching treatment and found \(^{50,49}\) that in \(\mathrm{MgFe}_2\mathrm{O}_4\) and \(\mathrm{CuFe}_2\mathrm{O}_4\) the saturation magnetic moment increases owing to quenching, but in \(\mathrm{NiFe}_2\mathrm{O}_4\) and \(\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5}\mathrm{O}_4\) it remains the same. The results are given in Table V.

Table V

	Treatment	\(n_B\)	Treatment	\(n_B\)
\(\mathrm{MgFe}_2\mathrm{O}_4\)	24 hours 700° C	1.1	Quenched from 1250° C	1.4
\(\mathrm{CuFe}_2\mathrm{O}_4\)	168 hours 360° C	1.3	Quenched from 900° C	2.3
\(\mathrm{NiFeO}_4\)	Cooled slowly	2.3	Quenched from 1250° C	2.3
\(\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5}\mathrm{O}_4\)	” ”	2.47*)	Quenched from 1150° C	2.50*)

*) Material of Section VII (new data).

For \(\mathrm{MgFe_2O_4}\) we obtain from (4.2) \(x=0.11\) and \(x=0.14\) for the annealed and quenched samples, respectively. Pauthenet and Bochirol\({}^{26}\) published, after us, measurements of the saturation magnetic moments of \(\mathrm{MgFe_2O_4}\) and \(\mathrm{CuFe_2O_4}\) quenched from various temperatures. The values of \(x\), calculated from (4.2), lie near the curve representing the Boltzmann distribution formula:

\[ \frac{x(1+x)}{(1-x)^2}=e^{-\frac{E}{kT}}, \]

where \(E\) is the energy required to transfer a \(\mathrm{Me}\) ion from a \(B\) site to an \(A\) site, and an iron ion, conversely, from \(A\) to \(B\). Bertaut\({}^{10}\) qualitatively confirmed the values of \(x\) found by Pauthenet and Bochirol by measuring the intensity of X-ray scattering, but the accuracy of the X-ray method (according to Bertaut, approximately \(\pm 0.03x\)) is very low.

Neutron-diffraction patterns for a sample of \(\mathrm{MgFe_2O_4}\) with \(n_B=1.1\)\({}^{57}\) also show better agreement with theory at \(x=0.1\) than at \(x=0.0\)\({}^{58}\). For \(\mathrm{NiFe_2O_4}\), equality of \(n_B\) for quenched and slowly cooled samples leads to the assumption \(x=0\); consequently, \(m_{\mathrm{Ni}^{2+}}=2.3\).

Thus, we are forced to assume that in \(\mathrm{NiFe_2O_4}\) the discrepancy between columns 1 and 5 of Table IV is due to \(g_{\mathrm{Me}^{2+}}\) being greater than two. For \(\mathrm{Fe^{II}Fe_2O_4}\), \(\mathrm{CoFe_2O_4}\), and \(\mathrm{CuFe_2O_4}\) we likewise assumed \(g_{\mathrm{Me}^{2+}}>2\), in view of the known paramagnetic behavior of salts containing the ions \(\mathrm{Fe^{2+}}\), \(\mathrm{Co^{2+}}\), \(\mathrm{Ni^{2+}}\), and \(\mathrm{Cu^{2+}}\). The values of \(n_B\) for \(\mathrm{CuFeO_4}\) should be explained by taking for \(g_{\mathrm{Cu}^{2+}}\) a value greater than 2, and also by assuming the presence of some number of \(\mathrm{Cu^{2+}}\) ions in \(A\) sites. It should be noted that for \(\mathrm{Mn^{II}Fe_2O_4}\) and \(\mathrm{Li_{0.5}Fe_{2.5}O_4}\) the agreement between the figures in columns 1 and 5 is very good, in accordance with the fact that \(g_{\mathrm{Mn}^{2+}}=g_{\mathrm{Fe}^{3+}}=2\), since here there is no contribution from an orbital moment.

The \(g\)-factors of divalent magnetic ions can be obtained from measurements of the effective \(g\)-factors of ferrites, if the cation distribution is known. These \(g\)-factors will be discussed in Section IV.3.1, and data from such measurements described in the literature will be given.

We shall first discuss the discrepancies between the values of \(n_B\) found by different authors.

Our own values, having been obtained at \(77^\circ\mathrm{K}\) and 5900 oersteds, will prove, as might generally be expected, too low. This is in fact so, especially for \(\mathrm{CoFe_2O_4}\), which, owing to its large crystalline anisotropy and large stress anisotropy under these conditions, is far from saturation. However, our value for \(\mathrm{Fe_3O_4}\) is higher than the values of other authors.

Our material contains an excess of \(\mathrm{Fe}^{2+}\) \({}^{50,49}\). \(\mathrm{FeO}\), initially present adjacent to the spinel, apparently decomposes on cooling below \(570^\circ\mathrm{C}\) and gives \(\mathrm{Fe} + \mathrm{Fe}_3\mathrm{O}_4\). This phenomenon may just account for the deviation of our value from the value found by Potenz \({}^{26,51,52}\) and by Weiss and Forrer \({}^{29}\). Our value, 2.3, for \(\mathrm{NiFe}_2\mathrm{O}_4\) was confirmed by measurements extended down to the temperature of liquid hydrogen at 9000 oersted (see Section VIII).

Potenz, like us, reports that the saturation moment of \(\mathrm{NiFe}_2\mathrm{O}_4\) does not change appreciably as a function of quenching, and finds analogous behavior in \(\mathrm{CoFe}_2\mathrm{O}_4\). The discrepancies between the values for \(\mathrm{NiFe}_2\mathrm{O}_4\) and \(\mathrm{CoFe}_2\mathrm{O}_4\) in columns 2 and 3 could, nevertheless, be due to very slight differences in the ion distribution, since a very small amount of divalent metal can be retained in the tetrahedral sites during the formation of the spinel.

A serious discrepancy in Table IV occurs only between the values for \(\mathrm{MnFe}_2\mathrm{O}_4\). Guillaud prepared three mixed crystals \(\mathrm{MnFe}_2\mathrm{O}_4\)—\(\mathrm{NiFe}_2\mathrm{O}_4\) and one ferrite \(\mathrm{MnNiCo}\), whose saturation magnetic moments agree well with the values of the moments calculated on the assumption that \(m_{\mathrm{Mn}^{2+}}=4.60\), \(m_{\mathrm{Ni}^{2+}}=2.4\), etc., to within 0.05. This, according to Guillaud, confirms his proposed value \(m_{\mathrm{Mn}^{2+}}=4.60\). Another value, obtained by us for \(\mathrm{MnFe}_2\mathrm{O}_4\) later, is \(n_B=4.85\) (Section IX). We believe that the discrepancies in the values of \(n_B\) are real and that they are due to different distributions of cations in different materials. The fact that Guillaud’s specimen was sintered at \(1380^\circ\mathrm{C}\) \({}^{54a}\), and ours at \(1250^\circ\mathrm{C}\), somewhat supports the latter supposition, which, nevertheless, must be checked by further experiments.

We cannot, however, explain the fact that in \(\mathrm{MnFe}_2\mathrm{O}_4\) \(n_B\) remains less than 5. One might suppose that \((m_{\mathrm{Mn}^{2+}})_A>5\), i.e. that \((g_{\mathrm{Mn}^{2+}})_A>2\) (see Section IV.3.1), but the value \(n_B=4.6\) would then lead to improbably high values of \((g_{\mathrm{Mn}^{2+}})_A\), since we assume that \((m_{\mathrm{Mn}^{2+}})_A=(m_{\mathrm{Mn}^{2+}})_B=m_{\mathrm{Fe}^{3+}}=5\). Another possible explanation is that some of the \(\mathrm{Mn}^{2+}\) and \(\mathrm{Fe}^{2+}\) ions in the \(B\) sites are converted into \(\mathrm{Mn}^{3+}\) and \(\mathrm{Fe}^{2+}\) ions \((m_{\mathrm{Mn}^{3+}}\simeq m_{\mathrm{Fe}^{2+}}\simeq 4)\), the state of which may become stable owing to the occurrence of a high degree of short-range order between divalent and trivalent ions, but this seems to us rather unlikely. Finally, it may turn out that the ionic magnetic moments in the \(B\) sites are not completely parallel. We shall discuss this possibility in Section IV.4.

Direct confirmation of Néel’s hypothesis on the antiparallelism of the magnetic moments in the \(A\) and \(B\) sites was given by experiments

according to neutron diffraction in Fe\(_3\)O\(_4\) \(^{68}\), NiFe\(_2\)O\(_4\) \(^{69}\), MgFe\(_2\)O\(_4\) \(^{58,7}\). In addition, it was found that Fe\(_3\)O\(_4\) and NiFe\(_2\)O\(_4\) are qualitatively opposite to one another; for MgFe\(_2\)O\(_4\) the ion distribution was even determined quantitatively (\(x = 0.12\)). A discussion of Zn[Fe\(_2\)]O\(_4\) will be given in Section IV.4.

IV.3.1. Effective \(g\)-factors

Effective \(g\)-factors can be determined by measurements of microwave absorption \(^{60}\). The method used in our laboratory will be described elsewhere \(^{59}\). The specimen (in the present case a sphere of diameter \(0.8\)—\(0.1\) mm) is placed in a hollow resonator situated in a strong external magnetic field \(H_z\) and in a high-frequency field, perpendicular to it, with angular frequency \(\omega\). The spins execute a precessional motion about \(H_z\); owing to the high-frequency field the magnetization \(M_z\) decreases by \(\Delta M\), and the angular momentum \(J_z\) by \(\Delta J\). When the field \(H_z\) is varied, a resonance peak is observed for a fixed angular frequency \(\omega\).

The resonance condition for a sphere is as follows: \(\omega = g(e/2mc)H_z\). For polycrystalline materials with random orientation of the crystallites, the influence of crystalline anisotropy on the resonance condition may be neglected if \(H_z\) is large and the crystalline anisotropy very small. Let \(M\) be the magnetization and \(J\) the angular momentum per unit volume; Kittel \(^{61}\) showed that

\[ g \frac{e}{2mc} = \frac{\Delta M}{\Delta J} = \frac{\Delta(M_{\mathrm{spin}}-M_{\mathrm{orb}})}{\Delta J_{\mathrm{spin}}}. \tag{4.3} \]

Since \(M_{\mathrm{spin}}/J_{\mathrm{spin}} = e/mc\) and the changes in \(M\) and \(J\) are proportional to their absolute values, one obtains

\[ g = \frac{2(M_{\mathrm{spin}}+M_{\mathrm{orb}})}{M_{\mathrm{spin}}} = \frac{2(\text{total magnetic moment})}{\text{spin moment}}. \tag{4.4} \]

For the case of a spherical ferrimagnetic spinel with antiparallel ionic magnetic moments at the \(A\) and \(B\) sites,

\[ g_{\mathrm{eff}} = 2\frac{(M_{\mathrm{tot}})_A-(M_{\mathrm{tot}})_B}{(M_{\mathrm{spin}})_A-(M_{\mathrm{spin}})_B}. \tag{4.5} \]

Such a spinel, containing various magnetic ions \(i\) (\(x\) ions per molecular unit in each sublattice), has at \(0^\circ\)K

\[ g_{\mathrm{eff}} = \frac{ \sum_i (x_i g_i S_i)_A - \sum_i (x_i g_i S_i)_B }{ \sum_i (x_i S_i)_A - \sum_i (x_i S_i)_B } = \frac{ \sum_i^{\rightarrow} x_i g_i S_i }{ \sum_i^{\rightarrow} x_i S_i }. \tag{4.6} \]

In his first measurements\(^{62}\) Beljers found for \(\mathrm{NiFe_2O_4}\) \(g = 2.36\), but later it was shown that the spheres used by him were insufficiently small. In his latest measurements there were

Table VI

Experimental values of the effective \(g\)-factors of various simple ferrites*)

Ferrites	\(g_{\mathrm{eff}}\)	Temperature \((^\circ\mathrm{C})\)	Wavelength \((\mathrm{cm})\)	Authors	Specimen
\(\mathrm{MnFe_2O_4}\)	2.05	Room temp.	1.24	a)	Polycryst.
\(\mathrm{MnFe_2O_4}\)	2.16 ↓ 2.02	Room temp. ↓ Room temp.	3.14 ↓ 0.64	b)	Polycryst.
\(\mathrm{FeFe_2O_4}\)	2.06 2.08; 2.09 2.17; 2.13	−153 −143 20	3.35 3.35; 1.25 3.35; 1.25	c)	Synth. Single cryst.
\(\mathrm{CoFe_2O_4}\)	Broad peak	Room temp.	1.24	a)	Polycryst.
\(\mathrm{CoFe_2O_4}\)	2.22 ↓ 2.91 ↓ 2.08	100 ↓ 300 ↓ 480	3.2 3.2 3.2	d)	Polycryst.
\(\mathrm{NiFe_2O_4}\)	2.21	Room temp.	1.24	a)	Polycryst.
\(\mathrm{NiFe_2O_4}\)	2.19	Room temp.	1.25	e)	Single cryst.
\(\mathrm{NiFe_2O_4}\)	2.25 (mean)	−195—588	3.33	f)	Polycryst.
\(\mathrm{NiFe_2O_4}\)	2.43 ↓ 2.12	Room temp. ↓	3.14 ↓ 0.64	b)	Polycryst.
\(\mathrm{NiFe_2O_4}\)	2.20; 2.17 ↓ ↓	Room temp.	0.64
\(\mathrm{CuFe_2O_4}\)	2.05; 2.06	−195 ↓	1.25	g)	Single cryst. Polycryst.
\(\mathrm{CuFe_2O_4}\)	2.03—2.06	450	1.25
\(\mathrm{MgFe_2O_4}\)	2.08	Room temp.	1.24	a)
\(\mathrm{Li_{0.5}Fe_{2.5}O_4}\)		Room temp.	3.18	h)

a) W. A. Yager, F. R. Merrit and C. Guillaud, Phys. Rev. 81, 477—478 (1951).
b) T. Okamura, Y. Torizuka and Y. Kojima, Phys. Rev. 88, 1425—1426 (1952).
c) L. R. Bickford, Phys. Rev. 76, 137—138 (1949).
d) T. Okamura, Y. Torizuka and Y. Kojima, Phys. Rev. 84, 372 (1951).
e) W. A. Yager, J. K. Galt, F. R. Merritt and E. A. Wood, Phys. Rev. 80, 744—748 (1950).
f) D. W. Healy, Phys. Rev. 86, 1009—1013 (1952).
g) T. Okamura and Y. Kojima, Phys. Rev. 86, 1040—1041 (1951).
h) Unpublished value of H. G. Beljers.

*) Results obtained on excessively large specimens are ignored.

spheres of diameter \(0.8\)—\(0.1\) mm, prepared by Bond’s method \(^{63}\), were used.

The values of the \(g\)-factor published in the literature for several ferrites are given in Table VI.

It is clear from this table that \(g_{\mathrm{eff}}\), which for completely reversed ferrites is equal to \(g_{\mathrm{Me}^{2+}}\), as follows from equation (4.5), can qualitatively account for the discrepancies between the experimental values of the saturation magnetic moment in columns 1—4 and the figures in column 5 of Table IV.

IV.4. Saturation magnetic moments of mixed crystals of ferromagnetic ferrites with zinc ferrite \((\mathrm{Me}_{1-a}^{\mathrm{II}}\mathrm{Zn}_a\mathrm{Fe}_2\mathrm{O}_4\ \text{and}\ \mathrm{Li}_{0.5-0.5a}\mathrm{Zn}_a\mathrm{Fe}_{2.5-2.5a}\mathrm{O}_4)\)

Verwey and Heilmann showed that in mixed crystals of \(\mathrm{ZnFe}_2\mathrm{O}_4\) and \(\mathrm{CuFe}_2\mathrm{O}_4\), divalent ions occupy the same crystallographic positions as in the simple ferrites composing them. If it is assumed that the \(\mathrm{Zn}^{2+}\) ions occupy only tetrahedral sites, then the mixed crystals \(\mathrm{Zn}[\mathrm{Fe}_2]\mathrm{O}_4\) with the completely reversed ferrite \(\mathrm{Fe}[\mathrm{Me}^{\mathrm{II}}\mathrm{Fe}]\mathrm{O}_4\) have the formula

\[ \mathrm{Zn}_a\mathrm{Fe}^{\mathrm{III}}_{1-a} [\mathrm{Me}^{\mathrm{II}}_{1-a}\mathrm{Fe}^{\mathrm{III}}_{1+a}]\mathrm{O}_4, \]

which, for complete antiparallelism of the ionic magnetic moments in the \(A\) and \(B\) sublattices, gives the following value of the saturation moment:

\[ n_B = m_b - m_a = 10a + (1-a)m_{\mathrm{Me}^{2+}}, \tag{4.7} \]

where \(m_a\) and \(m_b\) are the resultant magnetic moments per molecular unit, respectively, in the \(A\) and \(B\) sublattices. If the probability of finding Zn ions in \(B\) sites and \(\mathrm{Me}^{\mathrm{II}}\) ions in \(A\) sites is taken into account in the calculation, then the general formula is as follows:

\[ \mathrm{Zn}_{a-y}\mathrm{Fe}_{1-a+y-x}\mathrm{Me}^{\mathrm{II}}_{x} [\mathrm{Me}^{\mathrm{II}}_{1-a-x}\mathrm{Zn}_{y}\mathrm{Fe}_{1+a-y+x}]\mathrm{O}_4 \]

and it corresponds to

\[ n_B = m_b - m_a = \]

\[ = 10a + (1-a)m_{\mathrm{Me}^{2+}} + (10-2m_{\mathrm{Me}^{2+}})x - 10y. \tag{4.8} \]

Fig. 11 shows the results of our measurements of the saturation magnetic moments of a series of mixed crystals:

\[ \begin{aligned} &\mathrm{Mn}_{1-a}\mathrm{Zn}_a\mathrm{Fe}_2\mathrm{O}_4,\qquad \mathrm{Cu}_{1-a}\mathrm{Zn}_a\mathrm{Fe}_2\mathrm{O}_4\quad(\text{only for } a=0 \text{ and } a=0.5),\\ &\mathrm{Fe}_{1-a}\mathrm{Zn}_a\mathrm{Fe}_2\mathrm{O}_4,\qquad \mathrm{Mg}_{1-a}\mathrm{Zn}_a\mathrm{Fe}_2\mathrm{O}_4,\\ &\mathrm{Co}_{1-a}\mathrm{Zn}_a\mathrm{Fe}_2\mathrm{O}_4,\qquad \mathrm{Li}_{0.5-0.5a}\mathrm{Zn}_a\mathrm{Fe}_{2.5-2.5a}\mathrm{O}_4.\\ &\mathrm{Ni}_{1-a}\mathrm{Zn}_a\mathrm{Fe}_2\mathrm{O}_4, \end{aligned} \]

The dashed lines represent the values of \(n_B\), calculated from equation (4.7) or from equation (4.8), under the condition that \(x\) is a constant fraction of \((1-a)\), and \(y=0\), i.e., under the condition that the relative distribution of \(\mathrm{Me}^{2+}\) between tetrahedral and octahedral sites does not change in the formation of a mixed crystal with \(\mathrm{Zn[Fe_2]O_4}\). It is seen that the magnetic

Fig. 11. Saturation magnetic moment in Bohr magnetons for various series of mixed crystals
\(\mathrm{Me^{II}Fe_2O_4 — ZnFe_2O_4}\) (author’s measurements).

saturation moments found for small values of the \(\mathrm{Zn}_a\) content do indeed increase with increasing \(a\); the initial slopes of the curves in several cases are practically equal to the slopes of the corresponding dashed lines. The agreement of these slopes is considerably better for the curves (Fig. 12) obtained by Giyô for several of these systems; these curves, as we have seen, were recorded using higher fields and lower temperatures and were extrapolated to \(H=\infty\) and \(T=0^\circ\mathrm{K}\).

The latter results prove (at least up to \(a=0.4\)) that \(\mathrm{Zn}^{2+}\) occupy, practically exclusively, tetrahedral sites and, incidentally, confirm that \(m_{\mathrm{Fe}^{3+}}=5\). However, the distribution of Zn ions depends on temperature: Potenz[^52] measured the saturation magnetic moments also of quenched specimens of nickel-zinc ferrites up to \(a=0.6\) and found that the slope of the curves in this case differs more from the slope of the dashed lines than for annealed specimens. This proves that some \(\mathrm{Zn}^{2+}\) occupy

now octahedral sites; the difference between the values of \(n_B\) for the annealed and quenched samples is equal to 10\(\mu\) (see equation (4.8)).

The decrease of \(n_B\) with increasing zinc contents, as is shown in Figs. 11 and 12, must be explained as follows: the number of iron ions in the \(A\) sites and, consequently, \(m_a\) decreases with increasing \(a\). Where the straight line \(m=m_b-m_a\) intersects the straight line

\[ m=-m_a\left(1+\frac{1}{\gamma_2}\right), \]

according to the theory, angles begin to arise between the ionic moments in the \(B\) sublattice.

The curvature of the curve \(n_B=n_B(a)\) was explained by Néel\(^ {64}\) for NiZn ferrites by fluctuations in the ratio of the numbers of \(\mathrm{Zn}^{2+}\) and \(\mathrm{Fe}^{3+}\) ions in the \(A\) sites surrounded by different \(B\) sites, i.e., by local fluctuations of the \(AB\) interaction. In these calculations Néel

Fig. 12. Saturation magnetic moment in magnetons for various series of mixed crystals \(\mathrm{Me}^{II}\mathrm{Fe}_2\mathrm{O}_4\)—\(\mathrm{ZnFe}_2\mathrm{O}_4\) (measurements by Guillaud and co-workers).

used interaction constants calculated from his own susceptibility data\(^ {65}\). If one takes into account the circumstance that these materials contain two kinds of magnetic ions and that the theory presented should be modified along the lines of the theory of Yafet and Kittel, then the agreement obtained is satisfactory. The interaction constant \(\beta\) (or \(\gamma_2\)) need not remain constant: it will depend on the lattice constant and on the oxygen parameter \(u\), and if \(\mathrm{Me}^{2+}\) has

magnetic moment, then also on the relative intensities of the interactions Fe$^{3+}$—Fe$^{3+}$, Fe$^{3+}$—Me$^{2+}$, and Me$^{2+}$—Me$^{2+}$, and on the distribution of the cations.

If the interactions, in decreasing order of their magnitude, are arranged in the order indicated above, then $|\beta|$ or $|\gamma_2|$ will increase with an increase in the number of Fe$^{3+}$ ions in sublattice $B$.

If we were to assume that in MnFe$_2$O$_4$, investigated by Giyau, there are angles between the magnetic moments in sublattice $B$, then equation (2.19) would give $\gamma_2=-0.52$, whereas for Zn$_{0.8}$Mn$_{0.2}$Fe$_2$O$_4$ we obtain $\gamma_2=-0.17$. This in itself is an argument against such an assumption, since here the interactions in fact decrease in magnitude in the order Fe$^{3+}$—Fe$^{3+}$, Fe$^{3+}$—Mn$^{2+}$, Mn$^{2+}$—Mn$^{2+}$. A more serious objection to Giyau’s assumption about the presence of angles in sublattice $B$ in MnFe$_2$O$_4$ is the linear increase of $n_B$ as a function of the Zn content $a$, which he found for MnZn ferrite. If it is assumed that angles exist and that $\gamma_2$ is constant, then $m$ should have decreased linearly as a function of $m_a$, i.e. of $1-a$. To explain the linear increase of $m$, we must assume that $|\gamma_2|$ decreases as a function of $a$ in such a way that the angle $180^\circ-2\psi$ remains very small up to $a=0.5$, which seems quite plausible.

If we suppose that in MnFe$_2$O$_4$ $n_B<m_b-m_a$ only as a result of the above-mentioned local fluctuations of the interaction $AB$, it remains still more difficult to explain why this effect should at first decrease with increasing Zn content $a$.

The decrease of the Curie temperature with the content of ZnFe$_2$O$_4$ $^{53,54\mathrm{b},c}$ can also be explained by qualitative considerations. The Curie temperature is determined, above all, by the strongest interaction (or interactions), i.e. by the interaction $AB$, since it is predominant. Consequently, the Curie temperature will decrease with a decrease in the number of $AB$ interactions, i.e. with increasing $a$. It is clear that there will be no angles between the ionic magnetic moments in sublattice $B$ in materials with a very high Curie temperature.

For normal ZnFe$_2$O$_4$, prepared by annealing, $n_B=0.0$. Here only $BB$ interactions can take place, and as a result an “antiferromagnetic” arrangement will appear, probably with different spin directions in each of the four face-centered sublattices in sublattice $B$. The Néel temperature for such a material has still not been found.

When ZnFe$_2$O$_4$ is quenched, it becomes ferrimagnetic, $^{67,55}$ with a high Curie temperature (60° C). This is evidently due to the interaction $AB$, which appears as a result-

...the presence of some amount of Fe\(^{3+}\) in sublattice \(A\), as was proposed in both of the cited papers.

Gilleo suggested that the presence of some amount of Zn\(^{2+}\) at \(B\) sites, as indicated by the X-ray scattering intensity diagrams for his NiZn ferrites\(^{55,66}\), may, without taking all other hypotheses into account, explain the drop of the curve \(n_B = n_B(a)\) at large values of \(a\). It seems safe to assume that this effect plays only a secondary role. Indeed, we shall give examples of materials whose saturation magnetic moment cannot be explained at all without assuming the existence of angles between the magnetic moments of the ions within sublattice \(B\) (Sections VI.4 and VII).

The presence of some number of Zn ions in the octahedral sublattice must be assumed in order to explain the following results. Néel and Brochet\(^{65}\) calculated the ratios of the interactions \(AA/BB\) (\(\alpha\)) and \(BB/AB\) (\(\beta\)) from their susceptibility data for NiZn ferrites, assuming that all Zn\(^{2+}\) ions occupy \(A\) sites.

In this case they found extremely high values of \(\alpha\) at large Zn contents, which according to Anderson’s theory are very improbable on geometrical grounds. If, however, it is assumed that some small percentage of Zn\(^{2+}\) occupies \(B\) sites, \(\alpha\) remains small throughout the entire series of mixed crystals.

V. EXPERIMENTS ON THE ANGULAR DEPENDENCE OF THE SUPEREXCHANGE INTERACTION

V.1. The angle \(A — O — B\) in spinels

V.1.1. Introduction

The results obtained for most of the series of mixed crystals discussed in the preceding section showed that the \(BB\) interaction was considerably weaker than the \(AB\) interaction. A theoretical justification of such a situation, nevertheless, was not possible\(^*\). Kramer’s theory of superexchange\(^{71}\) gave no clear indications either regarding the order of magnitude of the superexchange interaction or regarding the dependence of the magnitude of this interaction on the geometrical configuration. Thus Néel was the first to make a crude assumption, which he called a “deliberate inaccuracy,” that all superexchange interactions \( \mathrm{Me} — O — \mathrm{Me}\) are equal in strength for nearest-neighbor \(\mathrm{Me} — \mathrm{Me}\), and that the \(BB\) interaction is weak

\(^*\) The magnetic measurements described in Section V.1.2.1 were carried out after the publication of the results presented in Section IV.4, but before Anderson’s theory became known to the author.

because the direct positive interaction Me—Me appears simultaneously with the negative superexchange interaction Me—O—Me.

We suggested that the angle \(A—O—B\), being larger than the angle \(B—O—B\), may play an important role. The angle \(A—O—B\) varies as a function of the parameter \(u\) and depends on the size of the ions. Consequently, the angle \(A—O—B\) can be increased by introducing a large ion into the \(B\) sublattice.

We therefore attempted to introduce \(Ca^{2+}\) ions into the spinel structure. \(CaFe_2O_4\) is a fairly well-defined compound, but it does not have the spinel structure: its crystal structure is still unknown\(^{72}\). We found, however, that at high temperatures more than one third of the \(Zn^{2+}\) ions in \(ZnFe_2O_4\) can be replaced by \(Ca^{2+}\) ions; these mixed crystals do not decompose on quenching. It would seem that \(Ca^{2+}\) ions should have a considerably greater tendency toward sixfold coordination than \(Mg^{2+}\) ions, since in most oxides \(Ca^{2+}\) occurs in sixfold or still higher coordination. We therefore assumed that in these CaZn ferrites the large \(Ca^{2+}\) ions in the \(B\) sublattice should produce a \(BB\) interaction much less able to compete with the \(AB\) interaction than in other corresponding ferrites, i.e. that CaZn ferrites should have higher values of the saturation magnetic moments.

V.1.2. CaZn ferrite

V.1.2.1. Experimental data. Materials were prepared with the composition \(a\,CaO \cdot (1-a)\,ZnO \cdot 1Fe_2O_3\), where \(a = 0.20,\ 0.30,\ 0.35,\ 0.40,\ 0.50,\ 0.90\), and \(1.00\). The materials were prepared from \(CaCO_3\), \(ZnO\) (\(Pb\,0.01\%,\ Mg < 0.01\%\)), \(Fe\) (\(C\,0.03\%\)) by method c) by means of preliminary calcination for 2 hours at \(1000^\circ C\) and grinding for 4 hours. The materials were sintered at various temperatures and either slowly cooled or quenched.

The X-ray patterns gave the results presented in Table VII. These results show:

1) At \(1280^\circ C\) and \(1300^\circ C\) equilibrium is obtained for all compositions, but down to \(1250^\circ C\) an incomplete reaction is obtained, at least for \(a=0.2\).

2) Materials with \(a=0.3\) and \(a=0.35\) are pure spinels at temperatures \(\geq 1280^\circ C\), but separate into two phases at lower temperatures; only the material with \(a=0.2\) does not separate into two phases on cooling at the rate used by us.

3) The temperature at which melting begins, at least of part of the material, decreases with increasing \(a\). Only

materials quenched from \(1280^\circ\text{C}\) sintered into pure spinels at \(a = 0.20\text{--}0.35\). Materials in which partial melting occurred were not all investigated, because the alumina support was slightly porous, so that the correct composition was not completely preserved.

Among the pure spinels obtained by sintering, the spinels with \(a = 0.35\), i.e., with the formula \(\mathrm{Ca}_{0.35}\mathrm{Zn}_{0.65}\mathrm{Fe}_2\mathrm{O}_4\), obtained

Table VII

Phases found in materials \(a\,\mathrm{CaO}\cdot(1-a)\,\mathrm{ZnO}\cdot 1\,\mathrm{Fe}_2\mathrm{O}_3\)

Temperature treatment	\(a=1.00\)	\(0.90\)	\(0.50\)	\(0.40\)	\(0.35\)	\(0.30\)	\(0.20\)
\(1200^\circ\text{C}\), cooled slowly	\(\mathrm{CaFe}_2\mathrm{O}_4\)*)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)
\(1250^\circ\text{C}\), cooled slowly	Melt \((\mathrm{CaFe}_2\mathrm{O}_4)\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\mathrm{CaFe}_2\mathrm{O}_4\) and/or \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)
\(1300^\circ\text{C}\), cooled slowly	—	—	—	—	Melt (spinel + \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\))	Spinel + \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel
\(1250^\circ\text{C}\), quenched	—	—	\(\mathrm{CaFe}_2\mathrm{O}_4\) + spinel	\(\mathrm{CaFe}_2\mathrm{O}_4\) + spinel	Spinel	Spinel + \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)	Spinel + \(\alpha\)-\(\mathrm{Fe}_2\mathrm{O}_3\)
\(1280^\circ\text{C}\), quenched	—	—	Melt	Spinel + traces of \(\mathrm{CaFe}_2\mathrm{O}_4\)	Spinel	Spinel	Spinel
\(1300^\circ\text{C}\), quenched	—	—	—	—	Melt	Spinel	Spinel

*) An identical sample is obtained after quenching from \(1200^\circ\text{C}\).

by quenching from \(1250^\circ\text{C}\) and \(1280^\circ\text{C}\), gave a considerably higher saturation magnetic moment \((5.3)^{74}\) and Curie temperature (approximately \(300^\circ\text{C}\)). After heating to the Curie temperature, the saturation magnetic moment decreased by \(3\%\), which is an indication of the instability of this composition at low temperatures. Results for the other spinels are not given here; their saturation magnetic moments were below \(1.0\).

Recently P. B. Braun made an attempt to determine the distribution of cations in \(\mathrm{Ca}_{0.35}\mathrm{Zn}_{0.65}\mathrm{Fe}_2\mathrm{O}_4\). The intensities observed on a Norelco precision X-ray diffractometer using Mo \(K\alpha\) radiation are given in Table VIII.

Table VIII

\(\Sigma h^2\)	3	8	11	12	16	19	24	27	32
Observed value	28	102	383	25	62	\(\leq 1\)	41	116	147
Calculated value	28	112	372	23	71	0	38	114	150

The number of \(\mathrm{Ca}^{2+}\) ions in the tetrahedral sublattice per molecular unit \(t\) cannot be determined from the available data, because for Mo radiation the scattering intensity of \(\mathrm{Fe}^{3+}\) is intermediate between the scattering intensities of \(\mathrm{Zn}^{2+}\) and \(\mathrm{Ca}^{2+}\). We therefore write the general formula in the following form:

\[ \mathrm{Ca}_t\mathrm{Zn}_{s+t}\mathrm{Fe}_{1-s-2t} [\mathrm{Ca}_{0.35-t}\mathrm{Zn}_{0.65-s-t}\mathrm{Fe}_{1+s+2t}]\mathrm{O}_4, \tag{5.1} \]

where \(s\) is to be determined. The best agreement with the observed intensities was found for \(s = 0.3 \pm 0.05\); the intensities calculated for \(s = 0.3\) are given in Table VIII.

The oxygen parameter \(u = 0.382 \pm 0.005\) and decreases slightly with increasing \(t\). The lattice constant is \(a = 8.49\ \text{Å}\).

V.1.2.2. Discussion of results. The saturation magnetic moment of quenched \(\mathrm{Ca}_{0.35}\mathrm{Zn}_{0.65}\mathrm{Fe}_2\mathrm{O}_4\) was determined earlier. Initially we compared its value (\(n_B = 5.3\)) with the value for \(\mathrm{Ni}_{0.35}\mathrm{Zn}_{0.65}\mathrm{Fe}_2\mathrm{O}_4\) (\(n_B = 4.8\))\(^{73,74}\) and assumed that all \(\mathrm{Ca}^{2+}\) ions occupy \(B\) sites. The latter assumption, as has now been shown, is erroneous, but the comparison with the Ni compound is also inappropriate, because replacement of nonmagnetic \(\mathrm{Ca}^{2+}\) ions by magnetic \(\mathrm{Ni}^{2+}\) ions will increase the \(BB\) interaction more than the \(AB\) interaction, i.e., will increase \(|\beta|\) or \(|\gamma_2|\) in a theory that takes into account the presence of a single type of magnetic ion, even if some ionic-size effect is not considered.

In the discussion below we shall show that the results of the X-ray measurements obtained thus far do not prove, but also do not refute, the picture proposed above.

The general formula for CaZn and MgZn ferrites is as follows:

\[ \mathrm{Me}^{\mathrm{II}}_x\mathrm{Fe}_{1-x} [\mathrm{Me}^{\mathrm{II}}_{1-x}\mathrm{Fe}_{1+x}]\mathrm{O}_4, \tag{5.2} \]

where \(\mathrm{Me}^{\mathrm{II}}\) represents some mixture of two diamagnetic ions \(\mathrm{Mg}^{2+}\), \(\mathrm{Ca}^{2+}\), or \(\mathrm{Zn}^{2+}\); \(x\) is unknown. The magnetic moment

of saturation, \(n_B = 10x\), for antiparallel magnetic moments in sublattices \(A\) and \(B\), or less than \(10x\) if the moments at sites \(B\) are not mutually parallel. It follows logically that for CaZn ferrite, with \(n_B = 5.3\), \(x \gg 0.53\), while for MgZn ferrite, with \(n_B = 4.7\), \(x \gg 0.47\). For MgZn ferrites it seems admissible to assume that \(\gamma_2\) does not depend on the rearrangement of the \(Mg^{2+}\) and \(Zn^{2+}\) ions at sites \(A\) and \(B\), owing to their identical ionic radii, so that \(\gamma_2\), and consequently \(n_B\), will depend practically only on \(x\).

The curve \(n_B = n_B(x)\) for MgZn ferrites is shown schematically in Fig. 13 (I), the value of \(\gamma_2\) derived from \(n_B = 3\) for \(Zn_{0.8}Mg_{0.2}Fe_2O_4\) by means of equation (2.19) being used,

Fig. 13. Schematic representation of the saturation magnetic moments of materials
\(Fe_{1-x}Me_x[Me_{1-x}Fe_{1+x}]O_4\) with a nonmagnetic ion \(Me\). Curve I: experimental data of Gorter for the saturation magnetic moments of MgZn ferrites, plotted schematically not as a function of the Zn content but as a function of \(x\). Curve II: schematic curve drawn through the maximum \(n_B = 5.3\) with the same radius of curvature as curve I, in order to estimate \(\gamma_2\) for \(Ca_{0.35}Zn_{0.65}Fe_2O_4\); \(\gamma_2\) in this case is larger than for MgZn ferrites. The dotted curve III: schematic curve for \(Ca_{0.35}Zn_{0.65}Fe_2O_4\) with \(\gamma_2\) smaller than for MgZn ferrites; the radius of curvature here is very small and, consequently, improbable.

and with the arbitrary assumption that in the latter material \(x = 0.82\), i.e., that, for example, all \(Zn^{2+}\) ions and 10% of the \(Mg^{2+}\) ions are located at sites \(A\).

The radius of curvature of this curve is determined by local fluctuations of the \(AB\) interaction, and since these radii are similar for different series of MeZn ferrites (Fig. 12), we have drawn schematic curve II through the maximum \(n_B = 5.3\) with a similar radius of curvature for our CaZn ferrite, in order to estimate \(\gamma_2\).

It is seen that in this way a lower value of \(|\gamma_2|\) is obtained than for MgZn ferrites. Only if the radius of curvature

significantly smaller (dotted curve III), \(|\gamma_2|\) will be larger than for MgZn ferrites.

It seems tempting to us to conclude that \(|\gamma_2|\) for CaZn ferrite is in fact smaller than for MgZn ferrites. This is just what we expected in the presence of a considerable fraction of large \(Ca^{2+}\) ions in sublattice \(B\), i.e., for \(t \leq 0.17_5\). For \(t > 0.17_5\) we expected that \(|\gamma_2|\) has a larger value than for MgZn ferrites.

The minimum value of \(t\) is obtained from x-ray and magnetic measurements: \(x = s + 2t > 0.53\), where \(s = 0.30 \pm 0.05\) gives \(t > 0.11_5 \pm 0.02_5\). The values of \(x\) for which \(t <\) or \(> 0.17_5\) are obtained from the relation \(x = s + 2t\):

\[ t < 0.17_5 \quad \text{for } x < 0.6; \]

\[ t \text{ either } <, \text{ or } > 0.17_5 \quad \text{for } 0.6 < x < 0.7; \]

\[ t > 0.17_5 \quad \text{for } x > 0.7. \]

For \(x < 0.6\) the radius of curvature of the curve \(n_B = n_B(x)\) would be very small; this is highly improbable, so that we cannot prove that \(t < 0.17_5\). For \(x > 0.7\) the radius of curvature would probably be too large, so that the value \(0.6 < x < 0.7\) is the most probable. This means that we can only say that the possibility \(t < 0.17_5\) is not excluded. The error in determining the oxygen parameter \(u = 0.382 \pm 0.005\) also does not exclude this possibility.

The fact that the saturation magnetic moments of CaZn ferrites with higher Zn contents are less than 1.0 is likewise not an objection to the assumption concerning the influence of large ions on the magnitude of the angle \(A—O—B\) and, through this, on the ratio of the interactions \(BB:AB\), because in these materials the values of \(x\) may be considerably larger.

V.2. Prediction of the Magnetic Behavior of Other Crystal Structures by Means of Anderson’s Theory

V.2.1. Introduction

Anderson’s theory gives the following useful approximate rule for predicting the saturation magnetic moments of ferrimagnetics, if their crystal structure is known: if there are magnetic ions of one type, then the magnitude of the exchange interaction increases as the angle \(Me—O—Me\) increases from a minimum at \(90^\circ\) to a maximum at \(180^\circ\); the magnitude of the interaction, moreover, rapidly decreases as the \(Me—O\) distance increases.

In crystal structures such as the spinel structure and the structures discussed in the present section, the oxygen ions form an approximately close-packed arrange-

…with cations placed in the intervals between the oxygen ions. Consequently, all the shortest Me—O distances will be of the same order of magnitude (like \(p\) and \(q\) in the spinel structure, see Table I).

The relative directions of the ionic magnetic moments in ferrimagnets, i.e. in the case where only negative interactions occur, can be found by fixing an arbitrary direction of an arbitrary ionic magnetic moment and by arranging, antiparallel to this direction, the magnetic moments of those surrounding ions for which the angle Me—O—Me is the largest and the Me—O distances are equal to the distances between nearest neighbors, say \(2.5\ \text{Å}\). We shall apply this rule to several compounds with a hexagonal crystal structure, namely to \(\mathrm{BaFe}^{\mathrm{III}}_{12}\mathrm{O}_{19}\), \(\mathrm{KFe}^{\mathrm{III}}_{11}\mathrm{O}_{17}\), and \(\mathrm{BaFe}^{\mathrm{II}}_{2}\mathrm{Fe}^{\mathrm{III}}_{16}\mathrm{O}_{27}^{78}\).

V.2.2. Application of the theory to \(\mathrm{BaFe}^{\mathrm{III}}_{12}\mathrm{O}_{19}\) and \(\mathrm{KFe}^{\mathrm{III}}_{11}\mathrm{O}_{17}\)

The hexagonal unit cells of \(\mathrm{BaFe}^{\mathrm{III}}_{12}\mathrm{O}_{19}\) (or \(\mathrm{BaO}\cdot 6\mathrm{Fe}_{2}\mathrm{O}_{3}\)) and \(\mathrm{KFe}^{\mathrm{III}}_{11}\mathrm{O}_{17}\) (or \(\mathrm{K}_{2}\mathrm{O}\cdot 11\mathrm{Fe}_{2}\mathrm{O}_{3}\)) are shown schematically in Fig. 14. The “spinel blocks” in them consist of 4 horizontal planes, each containing 4 oxygen ions, with cations between these planes. The ionic positions in these “blocks” are not shown; apart from a slight difference in parameters, they are identical with the positions of the ions in the spinel lattice with a vertical orientation of the \([111]\) axis*).

\(\mathrm{BaFe}_{12}\mathrm{O}_{19}^{*}\) is isomorphous\(^{75}\) with the mineral magnetoplumbite, which has the approximate composition \(\mathrm{Pb}(\mathrm{Fe}_{7.5}\mathrm{Mn}_{3.5}\mathrm{Al}_{0.5}\mathrm{Ti}_{0.5})\mathrm{O}_{19}^{76}\). The crystal structure of this mineral was determined by Adelsköld\(^{75}\). \(\mathrm{KFe}_{11}\mathrm{O}_{17}^{*}\) is isomorphous\(^{75}\) with “\(\beta\)-alumina,” \(\mathrm{NaAl}_{11}\mathrm{O}_{17}\), whose structure was determined by Beevers and Ross\(^{77}\).

The mean distances between the horizontal oxygen planes) and the oxygen—oxygen distances within an oxygen plane) are compared with the analogous distances in spinel with \(a = 8.35\ \text{Å}\) (the mean lattice parameter for ferrite) in Table IX.

The indicated slight differences in dimensions and slight differences in the magnitudes of the parameters, which undoubtedly exist, but have not been precisely determined and are too small to have any influence on the application of the above approximate rule.

The difference between the two structures is apparent from Fig. 14. In the unit cell of the barium compound, the planes containing

* For a comparison of the unit cells as a whole, see work\(^{78}\), part III, Fig. 2.

of the \( \mathrm{Ba}^{2+} \) ion, contain in addition three \( \mathrm{O}^{2-} \) ions, thus forming 12-fold coordination of the \( \mathrm{Ba}^{2+} \) ion with oxygen ions and one \( \mathrm{Fe}^{3+} \) ion. In the unit cell of the potassium compound

Fig. 14. Unit cells of \( \mathrm{BaFe}_{12}\mathrm{O}_{19} \) and \( \mathrm{KFe}_{11}\mathrm{O}_{17} \) (schematic). The large circles denote oxygen ions or \( \mathrm{K}^+ \) and \( \mathrm{Ba}^{2+} \) ions; the small circles represent iron ions. The positions of ions inside the “spinel blocks” are not indicated; each block contains 4 horizontal planes, each containing 4 oxygen ions; between these planes there are \(9\mathrm{Fe}^{3+}\), of which 7 have ionic magnetic moments directed in one direction, and 2 have moments directed in the opposite direction.

the corresponding planes contain, in addition to \( \mathrm{K}^+ \), only one \( \mathrm{O}^{2-} \) ion, thus forming 9-fold coordination of the \( \mathrm{K}^+ \) ion, as was also found in other compounds.

Measurements of the saturation magnetization were carried out on oriented single crystals of \( \mathrm{BaFe}_{12}\mathrm{O}_{19} \) \(^{78}\)*), and measurements of the suscep-

*) Recent measurements on polycrystalline \( \mathrm{BaFe}_{12}\mathrm{O}_{19} \) at liquid-hydrogen temperature and in a field up to 26,000 oersteds showed that \( \sigma_0 \) is very close to 100 CGSM·cm\(^3\)/g, i.e. \( n_B = 40 \) (A. L. Staits and Jongeburger, being prepared for publication).

Table IX

Comparison of the structural dimensions of BaFe$_{12}$O$_{19}$ and KFe$_{11}$O$_{17}$ with the dimensions of the spinel structure with $a = 8.35$ Å

Structure	$c$ (Å)	$a$ (Å)	Average distance between O planes (Å)	O—O distances in the O plane (Å)
BaFe$_{12}$O$_{19}$	23.21$_5$ *)	5.891 *)	$\dfrac{1}{10}c = 2.321_5$	$\dfrac{1}{2}a = 2.945_5$
KFe$_{11}$O$_{17}$	23.73 *)	5.932 *)	$\dfrac{1}{10}c = 2.374$	$\dfrac{1}{2}a = 2.966$
Spinel		(8.35)	$\dfrac{1}{6}a\sqrt{3} = 2.41_0$	$\dfrac{1}{4}a\sqrt{2} = 2.95_2$

*) Values determined by Braun.

Fig. 15. Magnetic properties of BaFe$_{12}$O$_{19}$ and KFe$_{11}$O$_{17}$ as a function of temperature. Curve I: saturation magnetization ($\sigma$) of BaFe$_{12}$O$_{19}$ (left-hand scale). Curve II: $1/\chi$ per gram for BaFe$_{12}$O$_{19}$ above the Curie temperature (right-hand scale). Curve III: $1/\chi$ per gram for KFe$_{11}$O$_{17}$ (right-hand scale).

susceptibility—on these same single crystals and on a polycrystalline specimen of KFe$_{11}$O$_{17}$. The results can be compared in Fig. 15.

KFe_{11}O_{17} was originally prepared by decomposing a solution of K and Fe^III nitrates in a stoichiometric proportion, obtained by dissolving K_2CO_3 and Fe (C 0.03%) in nitric acid.

After preliminary firing at 800^\circ C in O_2, grinding, and sintering for 2 hours at 1200^\circ C in O_2, X-ray patterns of the final materials were obtained on a Norelco precision diffractometer. The X-ray patterns showed a strong reflection from \alpha-Fe_2O_3 in addition to the reflection from KFe_{11}O_{17}. The material became ferromagnetic in moist air, possibly because of the presence of KFeO_2, which, as is known, reacts with water, giving \gamma-Fe_2O_3.

The material was then treated with a supersaturated solution of KNO_3, dried, ground, and sintered for 2 hours at 1200^\circ C in O_2. The final material showed reflections only from KFe_{11}O_{17} and was stable in moist air.

There is a remarkable difference between the magnetic properties of these two materials: BaFe_{12}O_{19} exhibits ferromagnetic behavior with an extrapolated saturation magnetic moment equal to 40 Bohr magnetons per unit cell. KFe_{11}O_{17} exhibits antiferromagnetic behavior (Fig. 15). This fact can be explained by the slight difference between the two structures described above and shown in Fig. 14. In the barium compound, pairs of adjacent iron ions on both sides of the plane in which Ba lies (denoted by ⊙) have parallel magnetic moments. This is explained by the fact that the negative interaction ⊙ — oxygen — ⊙ exceeds the interaction ◉ — oxygen — ⊙, since for the first interaction the angle is close to 180^\circ. In the potassium compound there is only a large negative interaction ⊙ — oxygen — ⊙ between these pairs, so that the magnetic moments of the ions are antiparallel.

The resultant magnetic moments of the spinel blocks are antiparallel to the magnetic moments of adjacent ions, so that in the barium compound these resultant moments are parallel to the resultant moments of neighboring blocks, whereas in the potassium compound they are antiparallel.

For complete unit cells this leads to the arrangement

\overleftarrow{16Fe^{3+}} + \overrightarrow{8Fe^{3+}}

with saturation magnetic moment n_B = 40 in BaFe_{12}O_{19}, and to the arrangement

\overleftarrow{11Fe^{3+}} + \overrightarrow{11Fe^{3+}}

with saturation magnetic moment n_B = 0 in KFe_{11}O_{17}. The agreement between the experimental and theoretical values shows that the saturation magnetic moment of BaFe_{12}O_{19} can be explained without assuming the presence of

angles between the magnetic moments of ions within one of the sublattices. The curve \(1/\chi = 1/\chi(T)\) (Fig. 15, curve \(II\)) has a hyperbolic form characteristic of a ferromagnet.

The \(1/\chi\) curve for \(\mathrm{KFe}_{11}\mathrm{O}_{17}\) (Fig. 15, curve \(III\)) shows a kink near \(530^\circ\mathrm{C}\); this is almost certainly the Néel temperature of the antiferromagnetic transition. On the basis of the discussion given above, it is not unexpected that this temperature is near, or more precisely somewhat above, the Curie temperature for \(\mathrm{BaFe}_{12}\mathrm{O}_{19}\).

The anomalous drop of the curve \(1/\chi = 1/\chi(T)\) for \(\mathrm{KFe}_{11}\mathrm{O}_{17}\) below the Néel temperature cannot at present be calculated. Measurements above the Néel temperature were not continued to sufficiently high temperatures to determine whether the curve has a hyperbolic form, as is found for ferromagnets. From the fact that the tangent to the curve at high temperatures should intersect the \(T\) axis below \(0^\circ\mathrm{K}\), since all interactions are negative, one may conclude that the curve indeed has such a form.

It should be emphasized that \(\mathrm{KFe}_{11}\mathrm{O}_{17}\) is not an ordinary antiferromagnet, because within a fairly large region (namely, the spinel layers) the antiferromagnetism is not compensated, so that its magnetic behavior may differ considerably from that of other antiferromagnets.

We cannot at present explain the practically linear decrease of the curve \(\sigma = \sigma(T)\) in \(\mathrm{BaFe}_{12}\mathrm{O}_{19}\) (Fig. 15, curve \(I\)). A theoretical treatment, analogous to Néel’s theory for a lattice containing two sublattices, in order to estimate the various interaction constants, is very difficult, since the \(\mathrm{BaFe}_{12}\mathrm{O}_{19}\) lattice contains five different crystallographic sublattices\(^{78}\). Even if only interactions involving the shortest Me—O distances are included in the calculation, there still remain six different types of interactions.

We wish to draw attention to the difference between \(\mathrm{BaFe}_{12}\mathrm{O}_{19}\) and spinel structures, which may be relevant to our problem. We have seen (Section II.3.4) that the geometry of the spinel lattice is such that the \(AB\) interaction is predominant in comparison with the \(AA\) and \(BB\) interactions. In \(\mathrm{BaFe}_{12}\mathrm{O}_{19}\), one of the interactions that does not determine the orientation of the magnetic moments of the ions is probably still rather strong. The interaction between the sublattices \(\odot\) and \(\oslash\) (\(^{78}\), part II, Fig. 2), which have mutually parallel magnetic moments of the ions as a result of the predominant (negative) interactions \(\odot\)—\(\oslash\) and \(\odot\)—\(\oslash\), must be rather strong, since the angle \(\odot\)—oxygen—\(\oslash\) is approximately \(125^\circ\).

Néel’s simple theory for the case of magnetic ions of one kind, located in two sublattices, shows (see Fig. 8) that the anomalous course of the curve $\sigma=\sigma(T)$ (including the practically linear dependence observed between types $f$ and $g$) occurs over a wide range of $x_a/x_b$, when $|\beta|$ is large and $|\alpha|$ is small (or conversely). This may suggest that the existence of one strong “competing” interaction in $\mathrm{BaFe}_{12}\mathrm{O}_{19}$ gives rise to a linear curve when all sublattices are completely occupied by magnetic ions.

V.2.3. Application of the theory to $\mathrm{BaFe}^{\mathrm{II}}_{2}\mathrm{Fe}^{\mathrm{III}}_{16}\mathrm{O}_{27}$

Another compound with an analogous crystal structure has been described,⁷⁸ namely $\mathrm{BaFe}^{\mathrm{II}}_{2}\mathrm{Fe}^{\mathrm{III}}_{16}\mathrm{O}_{27}$, or $\mathrm{BaO}\cdot2\mathrm{FeO}\cdot8\mathrm{Fe}_{2}\mathrm{O}_{3}$, or $\mathrm{BaFe}_{12}\mathrm{O}_{19}\cdot2\mathrm{Fe}_{3}\mathrm{O}_{4}$. Its crystal structure, which was determined by Braun,⁷⁹ differs from $\mathrm{BaFe}_{12}\mathrm{O}_{19}$ chiefly in that the spinel layer is thicker here: two molecular units $\mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}^{\mathrm{III}}_{2}\mathrm{O}_{4}$ are inserted into each spinel block in the unit cell (see ⁷⁸, part III). The saturation magnetization of this compound was studied on a single crystal prepared by Wijn using the recently described high-frequency melting method.⁸⁰ The saturation magnetic moment was found to be approximately $n_B=48$ per unit cell,* whereas we calculated $n_B=56$, considering the $\mathrm{Fe}^{2+}$ ions to be situated at octahedral sites inside the spinel blocks, i.e., assuming the arrangement

$$ \overrightarrow{20\mathrm{Fe}^{3+}\,4\mathrm{Fe}^{2+}}\;+\;\overleftarrow{12\mathrm{Fe}^{3+}}. $$

Other antiparallel arrangements of the ions, for which $n_B$ is closer to 48 (for example, 44), are very improbable.

* Up to now there have been no measurements at liquid-hydrogen temperature and in high fields.

(To be continued in the next issue)