SATURATION MAGNETIZATION AND CRYSTAL CHEMISTRY OF FERRIMAGNETIC OXIDES*

E. W. Gorter

CONTENTS

VI. Saturation magnetic moment and crystal chemistry of ferrimagnetic spinels containing titanium . . . . . . . . . . . . . . . . . . . . . . 435
VII. Ferrimagnetic oxides containing chromium: the system
$Li_{0.5}Fe^{III}_{2.5-a}Cr^{III}_aO_4\,(Li_{0.5}Fe_{2.5}O_4 — Li_{0.5}Fe_{0.5}Cr_2O_4)$ . . . . . . . . . . . . . . . . . . . 453
VIII. Ferrimagnetic spinels containing aluminum . . . . . . . . . . . . . . . . . . 463
IX. Other ferrimagnetic oxides containing chromium: the system
$Mn^{II}Fe^{III}_{2-a}Cr_aO_4\,(MnFe_2O_4 — MnCr_2O_4)$ . . . . . . . . . . . . . . . . 475
Cited literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

VI. SATURATION MAGNETIC MOMENT AND CRYSTAL CHEMISTRY OF FERRIMAGNETIC SPINELS CONTAINING TITANIUM

VI.1. Introduction to Sections VI—VIII

The investigations that will be set forth in Sections VI, VII, and VIII were begun with the aim of obtaining a series of mixed crystals in which the difference of the partial saturation magnetizations of the sublattices at $0^\circ K$, $m_b - m_a$, changes sign. Our discussion of Néel’s theory (Section II.2.1) showed that, for compositions close to those for which $m_b - m_a = 0$, one may expect the appearance of anomalous curves $\sigma = \sigma(T)$.

By replacing, in an inverse ferrite $Fe^{3+}[Me^{2+}Fe^{3+}]O_4$, the iron ion in sublattice $B$ wholly or partly by a nonmagnetic ion or by a magnetic ion with a small magnetic moment, the saturation moment is lowered, and by this method it may in principle be possible to bring about a change in the sign of the difference $m_b - m_a$.

* For the beginning, see UFN, Vol. LVII, issue 2.

If it is assumed that the distribution of the other ions remains unchanged and that all magnetic moments in sublattice \(B\) are antiparallel to all magnetic moments in sublattice \(A\), then replacement of an iron ion in sublattice \(B\) gives a change in the magnitude of the saturation magnetic moment that can be simply calculated. An iron ion in sublattice \(B\) can be replaced in the following ways:

1. Replacement of \(1\mathrm{Fe}^{3+}\) by \(1\mathrm{Al}^{3+}\) could, under the above assumption, give
   \[ m_b - m_a = m_{\mathrm{Me}^{2+}} - 5 \]
   and thus lead to a change in the sign of the moment if the magnetic moment of the divalent ion \(m_{\mathrm{Me}^{2+}} < 5\). Aluminum, however, as is known, does not have a greater tendency toward sixfold coordination than toward fourfold coordination (see silicates and \(\mathrm{Al}[\mathrm{Li}_{0.5}\mathrm{Al}_{1.5}]\mathrm{O}_4\)*). Moreover, it was known to us that in the system \(\mathrm{MgFe}_2\mathrm{O}_4\)—\(\mathrm{MgAl}_2\mathrm{O}_4\) a certain miscibility interval is observed (Section I.2, reference \(^{7}\)). In view of the large difference in the lattice constants of ferrites and the corresponding aluminates, this may prove to be a general phenomenon. On this basis, the indicated substitution was not the only one used in the investigation.

The system \(\mathrm{NiFe}_2\mathrm{O}_4\)—\(\mathrm{NiAl}_2\mathrm{O}_4\), as well as the systems \(\mathrm{MgFe}_2\mathrm{O}_4\)—\(\mathrm{MgAl}_2\mathrm{O}_4\) and \(\mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}_2\mathrm{O}_4\)—\(\mathrm{Fe}^{\mathrm{II}}\mathrm{Al}_2\mathrm{O}_4\), has been investigated by other authors and will be discussed in Section VIII.

1. Replacement of \(1\mathrm{Fe}^{3+}\) by \(1\mathrm{Cr}^{3+}\), which has a strong tendency toward sixfold coordination, gives
   \[ m_b - m_a \simeq m_{\mathrm{Me}^{2+}} - 2 \]
   (if \(m_{\mathrm{Cr}^{3+}} = 3\) is assumed) and should therefore lead to a change in the sign of \(m_b - m_a\) if \(m_{\mathrm{Me}^{2+}} < 2\). Therefore, by replacing \(\mathrm{Fe}^{3+}\) with \(\mathrm{Cr}^{3+}\) in Mg-ferrite or in Cu-ferrite, such a change can be achieved; however, these ferrites are already partly normal. \(\mathrm{Fe}[\mathrm{Li}_{0.5}\mathrm{Fe}_{1.5}]\mathrm{O}_4\) has a completely inverse ion arrangement; nevertheless, with the above assumptions, replacement in the indicated manner of \(1.25\mathrm{Fe}^{3+}\) by \(1.25\mathrm{Cr}^{3+}\) gives
   \[ m_b - m_a = 0, \]
   and replacement of \(> 1.25\mathrm{Fe}^{3+}\) by \(> 1.25\mathrm{Cr}^{3+}\) changes the sign of \(m_b - m_a\). This system will be described in Section VII.

2. Replacement of \(\mathrm{Fe}^{3+}_{B}\) by \(0.5\mathrm{Me}^{2+} + 0.5\mathrm{Ti}^{4+}\) should, under the above-mentioned assumptions, give
   \[ m_b - m_a = 1.5m_{\mathrm{Me}^{2+}} - 5, \]
   and thus change the sign of \(m_b - m_a\) when \(m_{\mathrm{Me}^{2+}} < 3.33\), i.e., if \(\mathrm{Me}^{2+} = \mathrm{Ni}^{2+}\) (or \(\mathrm{Cu}^{2+}\), or \(\mathrm{Mg}^{2+}\)). The system with \(\mathrm{Me}^{2+} = \mathrm{Ni}^{2+}\) is one of the systems described in Section VI.

However, it will be seen that neither \(\mathrm{Al}^{3+}\) nor \(\mathrm{Ti}^{4+}\) occupy only octahedral sites, and that replacement of \(\mathrm{Fe}^{3+}\) by \(\mathrm{Cr}^{3+}\) changes the distribution of the other cations.

In the first system, which will be described in the present section, we replaced \(1\mathrm{Fe}^{3+}\) in \(\mathrm{Fe}[\mathrm{NiFe}]\mathrm{O}_4\) by \((0.5\mathrm{Ni}^{2+} + 0.5\mathrm{Ti}^{4+})\).

* Scandium has such a tendency, but it is very expensive to use.

Nickel, as is known, has a strong tendency toward sixfold coordination: nickel titanate $\mathrm{Ni_2TiO_4}$ does not exist $^{81,82}$*). In another series, namely $\mathrm{Zn}_{2-a}\mathrm{Ni}_a\mathrm{TiO}_4$, studied by Birnbaum and Schott $^{81}$, the spinel phase exists only up to $a = 1.0$, which suggests that $\mathrm{Ni}^{2+}$ and $\mathrm{Ti}^{4+}$ will occupy only octahedral sites, corresponding to the formula $\mathrm{Zn[NiTi]O_4}$ at $a = 1$. For this reason we extended the series $\mathrm{Ni}_{1+a}\mathrm{Fe}_{2-2a}\mathrm{Ti}_a\mathrm{O}_4$ (nickel ferrite—nickel titanate) only to $a = 0.5$, i.e. to the composition $\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4$.

Materials with $a = 0.60,\ 0.70,\ 0.75$ and $0.80$ were prepared. X-ray diffraction patterns of these preparations showed reflections from the ilmenite phase ($\mathrm{NiTiO_3}$), with intensities increasing in the higher orders, near the spinel lines, and, probably, reflections from $\mathrm{NiO}$, which, however, coincide with strong reflections of the spinel. This behavior is analogous to that found $^{81}$ in the series $\mathrm{Zn}_{2-a}\mathrm{Ni}_a\mathrm{TiO}_4$ at $a > 1$ and indicates the instability of $\mathrm{Ni_2TiO_4}$ even in mixed crystals.

Two further large series of mixed crystals were studied in which, according to the assumptions given above, $m_b - m_a$ should have changed sign. These systems, $\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4$—$\mathrm{NiZn}_{0.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4$**) and $\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4$—$\mathrm{Mn}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4$, are described in Sections VI.3 and VI.4, respectively.

VI.2. System $\mathrm{Ni}^{\mathrm{II}}_{1+a}\mathrm{Fe}^{\mathrm{III}}_{2-2a}\mathrm{Ti}^{\mathrm{IV}}_a\mathrm{O}_4$ ($\mathrm{NiFe_2O_4}$—$\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4$)

VI.2.1. Experimental data

Materials with the above general formula were prepared with $a = 0,\ 0.1,\ 0.2,\ 0.3,\ 0.4$ and $0.5$.

These materials were prepared from
Fe (C 0.03%), Ni (Si 0.02%, Co 0.01%, Pb 0.01%),
$\mathrm{TiO_2}$ (Si 0.13%, Fe 0.02%, Al 0.01%, Mg 0.01%) by method (B); preliminary calcination was carried out at $600^\circ\mathrm{C}$ until complete decomposition of the nitrates, then grinding was performed, preliminary calcination for 4 hours at $1000^\circ\mathrm{C}$, grinding again, and final sintering for 4 hours at $1200^\circ\mathrm{C}$ in $\mathrm{O_2}$.

Two series were prepared:

1) cooled slowly from $1200^\circ\mathrm{C}$ in $\mathrm{O_2}$ and annealed in $\mathrm{O_2}$ at low temperatures for 3 hours at $800^\circ\mathrm{C}$, 7 hours at $700^\circ\mathrm{C}$, and 30 hours at $600^\circ\mathrm{C}$;

2) quenched from $1200^\circ\mathrm{C}$ by rapid immersion in a solution of sodium chloride and then washed with boiling distilled water.

*) Unpublished observations of the author at various sintering temperatures.

**) Here $m_b - m_a$ should change sign if an additional assumption is made, namely that $\mathrm{Zn}^{2+}$ atoms occupy only tetrahedral sites.

Chemical analysis did not establish the presence of Fe\(^{2+}\) ions in both series.

X-ray patterns obtained on a Norelco diffractometer showed that the materials are pure spinels; however, because of the small difference in the scattering intensities of the ions present, no data were obtained on the distribution

Fig. 16. a) Saturation magnetic moments of the mixed-crystal series \(Ni_{1+a}Fe_{2-2a}Ti_aO_4\) for \(a=0\)—0.5. Curve \(I\)—annealed samples; curve \(II\)—samples quenched from \(1200^\circ C\); line \(III\)—saturation magnetic moments calculated for the case in which all Ti\(^{4+}\) and Ni\(^{2+}\) ions are in the octahedral sublattice. b) The number of Ti\(^{4+}\) ions in the tetrahedral sublattice per molecular unit \((x)\), calculated from the saturation magnetic moments shown in Fig. 16, a, under the assumption that Ni\(^{2+}\) ions occupy octahedral sites throughout the series. Curve \(I\)—annealed samples; curve \(II\)—samples quenched from \(1200^\circ C\); line \(III\)—the line \(x=0\), for which line \(III\) in Fig. 16, a was drawn.

of the ions. The temperature dependences of magnetic saturation were measured for all materials*). The saturation magnetic moments in Bohr magnetons are given in Fig. 16, a and in Table X (p. 441). The Curie temperatures \((\Theta)\) for all samples and para-

*) In magnetic fields up to 5900 oersteds and at \(20^\circ K\).

meter for the annealed materials are also indicated in the table.

The curves of the dependence of the relative saturation magnetization on the reduced temperature (that is, \(\sigma_T/\sigma_{T=0}=f(T/\Theta)\)) for annealed materials are shown in Fig. 17. Beljers carried out measure-

Fig. 17. Curves of the dependence of the relative magnetization on the reduced temperature \((\sigma/\sigma_0\) versus \(T/\Theta)\) for a series of mixed crystals
\(\mathrm{Ni}_{1+a}\mathrm{Fe}_{2-2a}\mathrm{Ti}_a\mathrm{O}_4\).

ments of the effective \(g\)-factor \((g_{\mathrm{eff}})\) of annealed \(\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4\) by the method of ferromagnetic resonance at wavelengths of \(3.18\ \mathrm{cm}\) and \(1.24\ \mathrm{cm}\) on spherical specimens about \(0.5\ \mathrm{mm}\) in diameter. The following results were obtained:

\[ \mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4 \]

\[ \text{(annealed)} \]

Wavelength in cm	Temperature in °K	\(g_{\mathrm{eff}}\)
1.24	300	2.70—2.71
3.18	293	2.73
3.18	83	2.85
(extrapolation to 0° K)	(extrapolation to 0° K)	2.90

VI.2.2. Discussion of the results for annealed specimens and neutron-diffraction data

The saturation magnetic moments, calculated under the assumption that all \( \mathrm{Ni}^{2+} \) and \( \mathrm{Ti}^{4+} \) ions occupy octahedral sites and that the magnetic moments of the ions in the sublattices \(A\) and \(B\) are oriented completely antiparallel to one another, are indicated in Fig. 16, \(a\), by the dashed curve III. It is seen from the figure that the experimental curves deviate appreciably from this line*). In addition, the curves \( \sigma = \sigma(T) \) for the compositions studied do not reveal anomalous behavior. It is obvious that one of the assumptions made in calculating curve III is incorrect.

K. F. Nissen \(^{83}\), using experimental data for another series of mixed crystals \((\mathrm{NiFe}_2\mathrm{O}_4 — \mathrm{ZnFe}_2\mathrm{O}_4)\) and relying on Néel’s molecular-field theory, extended to the case of the presence of magnetic ions of two different kinds, showed that the case of nonparallelism of the magnetic moments of the ions in the \(B\) (or \(A\)) sublattice is improbable for the series \((\mathrm{NiFe}_2\mathrm{O}_4 — \mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4)\). Therefore we shall adhere to our assumption that, in the series under study, the magnetic moments of the ions within each of the sublattices \(A\) and \(B\) are parallel.

Consequently, the deviations of the experimental values of the magnetic moments from curve III must be explained by the transfer of some \( \mathrm{Ni}^{2+} \) or \( \mathrm{Ti}^{4+} \) ions, or of both together, into the tetrahedral sublattice. The general formula will then have the form:

\[ \mathrm{Ni}_y \mathrm{Ti}_x \mathrm{Fe}_{1-x-y} [\mathrm{Ni}_{1+a-y}\mathrm{Fe}_{1-2a+x+y}\mathrm{Ti}_{a-x}] \mathrm{O}_4, \tag{6.1} \]

where, for antiparallelism of the magnetic moments of the sublattices \(A\) and \(B\),

\[ n_B = |m_b - m_a|, \]

with

\[ m_b - m_a = 2.3 - 7.7a + 10x + (7.7 - p)y. \tag{6.2} \]

In this equation \(p = (g_{\mathrm{Ni}^{2+}})_A\) is the \(g\)-factor for \( \mathrm{Ni}^{2+} \) in the \(A\) sublattice**).

*) With the exception of \(a = 0.5\), where, moreover, the following cation distribution seems a priori probable:
\(\mathrm{Fe}[\mathrm{Ni}_{1.5}\mathrm{Ti}_{0.5}]\mathrm{O}_4\), owing to the possibility of long-range order of the type
\(\mathrm{Fe}[\mathrm{Li}_{0.5}\mathrm{Fe}_{1.5}]\mathrm{O}_4\).

**) In view of the difference between the crystalline electric fields in the sublattices \(A\) and \(B\), \((g_{\mathrm{Ni}^{2+}})_A\) will differ from (and, apparently, be larger than) \((g_{\mathrm{Ni}^{2+}})_B\). D. S. Smart (private communication, to be published in Phys. Rev.) uses the value \((g_{\mathrm{Ni}^{2+}})_A = 3.5\), obtained from measurements of the susceptibility of \(\mathrm{NiAl}_2\mathrm{O}_4\).

Using the experimental values \(n_B\) from Table X, on the basis of this formula, for the values \(a = 0.1,\ 0.2,\ 0.3,\ 0.4\) and \(0.5\), we obtain
\[ 10x + (7.7 - p)y = 0.35,\ 0.73,\ 1.26,\ 2.02,\ \text{and } 3.00 \]
or \(0.1\), respectively. (For \(a = 0.5\) there are two solutions: the second value gives \(m_b - m_a < 0\).)

Table X

Magnetic saturation moments, Curie temperatures, and lattice constants of the series
\[ \mathrm{Ni}_{1+a}^{\mathrm{II}}\mathrm{Fe}_{2-2a}^{\mathrm{III}}\mathrm{Ti}_{a}^{\mathrm{IV}}\mathrm{O}_4 \]

Ti content \((a)\)	Formula	\(n_B\) annealed	\(n_B\) quenched	\(\Theta\) annealed (°C)	\(\Theta\) quenched (°C)	Lattice constant, annealed (Å)
0	\(\mathrm{NiFe_2O_4}\)	2.29	2.29	585	585	8.337
0.1	\(\mathrm{Ni_{1.1}Fe_{1.8}Ti_{0.1}O_4}\)	1.88	1.85	550	550	8.337₅
0.2	\(\mathrm{Ni_{1.2}Fe_{1.6}Ti_{0.2}O_4}\)	1.49	1.48	500	500	8.338
0.3	\(\mathrm{Ni_{1.3}Fe_{1.4}Ti_{0.3}O_4}\)	1.25	1.23	440	440	8.338₅
0.4	\(\mathrm{Ni_{1.4}Fe_{1.2}Ti_{0.4}O_4}\)	1.24	1.10	375	356	8.339
0.5	\(\mathrm{Ni_{1.5}Fe_{1.0}Ti_{0.5}O_4}\)	1.45	1.12	293	265	—

It is sufficiently obvious that it is mainly precisely the \(\mathrm{Ti}^{4+}\) ions, and not the \(\mathrm{Ni}^{2+}\) ions, that pass into the tetrahedral sublattice.

1) Nissen \(^{34d}\) calculated, by the method indicated above, that the Curie temperatures given in Table X can be explained only by an increase in the transfer of \(\mathrm{Ti}^{4+}\) ions into the \(A\) sites as \(a\) increases*).

2) The quantity \(g_{\mathrm{eff}}\), extrapolated to \(T = 0^\circ\mathrm{K}\), can be used, together with \(n_B\), to determine the cation distribution; in this case formula (4.6) is used for \(g_{\mathrm{eff}}\). Starting from the values of the quantities for \(\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4\) \((a = 0.5)\), we have:
\[ n_B = \left| -1.55 + 10x + (7.7 - p)y \right| = 1.45 \]
and
\[ g_{\mathrm{eff}} = \left| \frac{n_B}{-1.5x + 3y} \right| = 2.90. \]

These equations give only one physically possible solution:
\[ x = 0.30,\qquad y = 0. \]

*) This result was obtained before the experimental evidence set forth in points 2) and 3) was obtained.

This means that only (or, in view of the uncertainty in the extrapolation of \(g_{\mathrm{eff}}\) to \(0^\circ\mathrm{K}\), practically only) \(\mathrm{Ti}^{4+}\) ions pass into the tetrahedral sublattice, and that \(m_b-m_a>0\).

We wish to draw attention to the fact that, in general, for ferrimagnetics containing two different kinds of magnetic ions, the distribution of cations can be calculated from \(g_{\mathrm{eff}}\) and \(n_B\), if only the \(g\)-factors of the ions are known and sufficiently different, and if the moments of the ions within each of the sublattices are parallel\(^{84*}\).

Although the theory still cannot explain certain details, such as, for example, the frequency dependence of the effective \(g\)-factor, we consider it justified to use \(g_{\mathrm{eff}}\), by the method described above, to determine the approximate distribution of cations.

3) Neutron diffraction patterns, kindly made for us by Shull, for annealed materials with \(a=0\), 0.3 and 0.5 showed a regular decrease in the intensity of the 220 reflection (structural factor \(=8f_{\mathrm{tetr}}\)), which indicates an increase in the number of \(\mathrm{Ti}^{4+}\) ions at the \(A\) sites in this arrangement, in view of the very small neutron scattering cross section of Ti nuclei.

We shall give elsewhere a comparison of the experimental intensities of the neutron diffraction patterns of \(\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4\) with the intensities calculated for the distribution

\[ \mathrm{Fe}_{0.7}\mathrm{Ti}_{0.3}\,[\mathrm{Ni}_{1.5}\mathrm{Fe}_{0.3}\mathrm{Ti}_{0.2}]\,\mathrm{O}_4 \tag{6.3} \]

with allowance for magnetic contributions.

On the basis of the evidence presented above, we consider it justified to assume \(y \simeq 0\) and to calculate \(x\) from the saturation magnetic moments for the entire series of mixed crystals. The values obtained are shown in Fig. 16, \(b\). The distribution in the annealed specimens is very close to that distribution in which the \(\mathrm{Ni}^{2+}\) ions occupy only the sites of the octahedral sublattice, while \(\mathrm{Fe}^{3+}\) and \(\mathrm{Ti}^{4+}\) are distributed at random over the remaining sites, i.e.,

\[ \mathrm{Fe}_{0.667}\mathrm{Ti}_{0.333}\,[\mathrm{Ni}_{1.5}\mathrm{Fe}_{0.333}\mathrm{Ti}_{0.167}]\,\mathrm{O}_4. \]

The presence of \(\mathrm{Ti}^{4+}\) ions in tetrahedral coordination is somewhat unexpected. Up to now, to our knowledge, the only indication of the presence of \(\mathrm{Ti}^{4+}\) ions in tetrahedral coordination has been the inclusion of \(\mathrm{Ti}^{4+}\) ions as an activator in phosphors \(\mathrm{SiO}_2\) and \(\mathrm{Zn}_2\mathrm{SiO}_4^{85}\); in the structures of the latter compounds the cations occur only in tetrahedral coordination.

\(*\) D. C. Smart independently used this method for finding the distribution of cations in the \(\mathrm{NiFe}_2\mathrm{O}_4\)–\(\mathrm{NiAl}_2\mathrm{O}_4\) series (see Section VIII) (a personal communication from Smart will be published in Phys. Rev.). The case of nonparallel ionic moments within one sublattice is considered in Section VI.3.2.

Smart*) showed for another system of ternary spinels \((\mathrm{NiFe}_2\mathrm{O}_4 — \mathrm{NiAl}_2\mathrm{O}_4)\) that in two Boltzmann-type formulas of the definition of the type of equation (1.1) of Section 1.2.6, the values of the energies \(E\) of exchange of two cations between two crystallographic sublattices can be determined so that the curves of the theoretical distribution of cations coincide with the experimental curves. These curves are of the same type as the curves found for annealed and quenched specimens of the system studied by us (Fig. 16, б).

This suggests that our distribution curves can be qualitatively explained without taking into account the existence of short-range order. We shall see, however, that the decrease of \(n_B\) upon quenching cannot be explained without taking into account the influence of short-range order.

VI.2.3. Discussion of the results for quenched specimens

The saturation magnetic moments and Curie temperatures for quenched specimens proved to be smaller than the corresponding quantities for annealed specimens near \(a = 0.5\).

Quenching, in general, leads to the preservation of a considerably more disordered distribution of cations over sites \(A\) and \(B\). Equation (6.2) shows that an increase in \(y\) gives an increase in \(n_B\), and even for an arrangement in which the \(\mathrm{Fe}^{3+}\) and \(\mathrm{Ti}^{4+}\) ions tend to be distributed almost randomly over the sites not occupied by \(\mathrm{Ni}^{2+}\) ions, i.e. for the case \(x \simeq (1 - y)a/(2 - a)\), the calculated values of \(n_B\) are higher than the experimental values for annealed specimens. Therefore an investigation of only one saturation magnetic moment leads to the conclusion that quenching entails, above all, a decrease in \(x\) and thus leads to stronger deviations from the statistical distribution of the \(\mathrm{Ti}^{4+}\) and \(\mathrm{Fe}^{3+}\) ions among the available lattice sites. This paradox is removed only by the assumption that the \(\mathrm{Ti}^{4+}\) ions have a certain preferential tendency to occupy \(B\) sites as compared with the \(\mathrm{Fe}^{3+}\) ions, which is found in the quenched specimen \((x = 0.27)\). During slow cooling, the change in entropy should have caused the transfer of a larger number of \(\mathrm{Ti}^{4+}\) ions into \(B\) sites; however, in reality a larger number of \(\mathrm{Ti}^{4+}\) ions move into \(A\) sites \((x = 0.30)\), because, apparently, the increase in short-range-order energy of the \(\mathrm{Ti}^{4+}\) and \(\mathrm{Fe}^{3+}\) ions in \(A\) sites exceeds the combined effect of the entropy and certain losses of short-range-order energy in \(B\) sites.

Our information on the magnitude of the various energetic terms is insufficient to explain this effect.

*) See the footnote on the preceding page.

Fig. 18. a) Saturation magnetic moments of a series of mixed crystals
\(\mathrm{Ni}_{1.5-x}\mathrm{Zn}_{\alpha}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}\) for \(\alpha = 0\)–\(0.5\). Curve \(I\)—annealed specimens; curve \(II\)—specimens quenched from \(1200^\circ\mathrm{C}\); line \(III\)—saturation magnetic moments calculated for the case when all \(\mathrm{Ti}^{4+}\) and \(\mathrm{Ni}^{2+}\) ions are in the octahedral sublattice, and all \(\mathrm{Zn}^{2+}\) ions are in the tetrahedral sublattice. b) Number of \(\mathrm{Ti}^{4+}\) ions in the tetrahedral sublattice per molecular unit \((x)\), calculated from the saturation magnetic moments given in Fig. 18, a, under the assumption that \(\mathrm{Zn}^{2+}\) ions occupy tetrahedral sites and \(\mathrm{Ni}^{2+}\) ions occupy octahedral sites throughout the entire series. Curve \(I\)—annealed specimens; curve \(II\)—specimens quenched from \(1200^\circ\mathrm{C}\); line \(III\)—\(x=0\), along which line \(III\) in Fig. 18, a, is drawn.

VI.3. System

\[ \mathrm{Ni}^{II}_{1.5-a}\mathrm{Zn}^{II}_{a}\mathrm{Fe}^{III}\mathrm{Ti}^{IV}_{0.5}\mathrm{O}_{4} \quad (\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_{4} -\mathrm{NiZn}_{0.5}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}) \]

VI.3.1. Experimental data

Materials with the general formula given above were prepared with \(a = 0^{*}\), 0.1, 0.2, 0.3, 0.4, and 0.5.

The preparation was carried out by the same method as the preparation of the materials described in the preceding system (VI.2). ZnO was used as an additional starting material. Two series of specimens were prepared: annealed at low temperatures and quenched from \(1200^\circ\)C, analogous to what was described in Section VI.2. Chemical analyses did not reveal any appreciable traces of \(\mathrm{Fe}^{2+}\) in either series.

X-ray diffraction patterns obtained on the Norelco diffractometer showed that the materials are pure spinels; no data concerning the distribution of ions were obtained. The values of the saturation magnetic moments\(^{**}\) in Bohr magnetons are given in Fig. 18, a and in Table XI; the Curie temperatures for all specimens

Table XI

Magnetic saturation moments, Curie temperatures, and lattice constants of the series
\(\mathrm{Ni}^{II}_{1.5-a}\mathrm{Zn}_{a}\mathrm{Fe}^{III}\mathrm{Ti}^{IV}_{0.5}\mathrm{O}_{4}\)

Zn content (\(a\))	Formula	\(n_B\), annealed	\(n_B\), quenched	\(\Theta\), annealed (°C)	\(\Theta\), quenched (°C)	Lattice constant, annealed (Å)
0	\(\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}\)	1,44	1,12	293	265	—
0,1	\(\mathrm{Ni}_{1.4}\mathrm{Zn}_{0.1}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}\)	1,02	1,04	267	250	8,348
0,2	\(\mathrm{Ni}_{1.3}\mathrm{Zn}_{0.2}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}\)	1,07	1,34	265	235	8,355
0,3	\(\mathrm{Ni}_{1.2}\mathrm{Zn}_{0.3}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}\)	1,32	1,52	264	220	8,365
0,4	\(\mathrm{Ni}_{1.1}\mathrm{Zn}_{0.4}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}\)	1,73	1,82	240	195	8,372
0,5	\(\mathrm{NiZn}_{0.5}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}\)	2,1	2,1	225	172	8,382

and the lattice parameters only for the annealed materials are also given in Table XI.

Curves of the dependence of the relative saturation on the reduced temperature \((\sigma_T/\sigma_{T=0} = f(T/\Theta))\) for the annealed materials

* This material is from the preceding series with \(a = 0.5\).
** Fields up to 5900 oersteds at 20°K were used.

shown in Fig. 19. For the annealed sample \(\mathrm{NiZn}_{0.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4\), Beljers measured the effective \(g\)-factor.

Fig. 19. Curves of the dependence of the relative saturation magnetization on the reduced temperature \((\sigma/\sigma_0\) on \(T/\Theta)\) for a series of mixed crystals \(\mathrm{Ni}_{1.5-a}\mathrm{Zn}_a\mathrm{FeTi}_{0.5}\mathrm{O}_4\).

The results of these measurements are as follows:

\[ \mathrm{NiZn}_{0.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4 \]
\[ \text{(annealed)} \]

Wavelength in cm	Temperature in °K	\(g_{\mathrm{eff}}\)
3.18	301	2.33
3.24	90	2.44
1.24	298	2.19
1.24	143	2.19
	(extrapolation to \(0^\circ\) K)	\(\approx 2.19\)

VI.3.2. Discussion of the Results

The results obtained for the annealed samples of this system are very similar to the results obtained in the preceding section. The saturation magnetic moments do not pass through zero along the dotted curve III, which was calculated for the case in which the \(\mathrm{Zn}^{2+}\) ions are located in tetrahedral sites,

and the ions Ni\(^{2+}\) and Ti\(^{4+}\) in octahedral sites. None of the materials studied has an anomalous curve \(\sigma=\sigma(T)\) (see Fig. 19).

This result is understandable from the fact that, for Ni\(_{1.5}\)FeTi\(_{0.5}\)O\(_4\), as we have seen, \(m_b>m_a\). We shall now try to draw conclusions from the experimental data about the distribution of cations. In view of the fact that the number of Fe\(^{3+}\) ions in the \(A\) sites at \(a=0.5\) is approximately only 0.5, we must allow for the possibility of nonparallelism of the magnetic moments of the ions within sublattice \(B\).

The general formula NiZn\(_{0.5}\)FeTi\(_{0.5}\)O\(_4\) (\(a=0.5\)) is represented in the form

\[ (\mathrm{Zn}+\mathrm{Ti})_{x}\mathrm{Ni}_{y}\mathrm{Fe}_{1-x-y} [\mathrm{Ni}_{1-y}\mathrm{Fe}_{x'+y}(\mathrm{Zn}+\mathrm{Ti})_{1-x'}]\mathrm{O}_{4}. \]

The usual equations for \(n_B\) and \(g_{\mathrm{eff}}\) with parallel magnetic moments of the ions in sublattice \(B\) give one physically possible solution for the annealed specimen, namely: \(x'=0.45,\ y=0.07\) (for \((g_{\mathrm{Ni}^{2+}})_A=3.5\)). Since the simultaneous location of a noticeable quantity of Ni\(^{2+}\) ions in \(A\) sites and Zn\(^{2+}\) ions in \(B\) sites in an annealed specimen can be excluded, because this is improbable in every respect from the crystallochemical point of view, another arrangement, with incompletely parallel ionic magnetic moments in sublattice \(B\), namely \(x'\simeq 0.5\) and \(y=0\), appears more probable*). It may be noted that, in contrast to the saturation magnetic moments, the Curie temperatures for annealed and quenched specimens are different, so that the cation distributions in the two specimens are probably not quite identical.

The number of Ti\(^{4+}\) ions in sublattice \(A\), which we denote by \(x\), is equal to \(x=0.30\) when the Zn content \(a=0\) and, probably, \(x\simeq 0\) when \(a=0.5\); both cases are for \(y=0\). It is assumed that for intermediate compositions also \(y\simeq 0\). The values of \(x\), calculated from the values of \(n_B\) for \(y=0\), are shown in Fig. 18, b.

*) For the case when angles \(180^\circ-2\psi\) occur between the directions of the magnetic moments of ions within sublattice \(B\), we saw from equation (4.5) that \((M_{\mathrm{full}})_B\) and \((M_{\mathrm{spin}})_B\) are the resultant moments of sublattice \(B\), i.e. we use the expressions

\[ n_B=\left| \sum_i (x_i g_i S_i)_A-\sum_i (x_i g_i S_i)_B \sin\psi \right|, \tag{6.3} \]

\[ g_{\mathrm{eff}}= \left| \frac{n_B}{ \sum_i (x_i S_i)_A-\sum_i (x_i S_i)_B \sin\psi } \right|. \tag{6.4'} \]

The values of \(u_B\) for quenched specimens with \(a = 0.2\text{–}0.4\) can be explained mainly by the statistical distribution of cations, whereas in the case of the material with \(a = 0.5\) the influence of short-range order may again become appreciable.

The system under consideration is too complicated for one to attempt to understand its crystal chemistry more deeply.

VI.4. System

\[ \mathrm{Ni}^{II}_{1.5-a}\mathrm{Mn}^{II}_{a}\mathrm{Fe}^{III}\mathrm{Ti}^{IV}_{0.5}\mathrm{O}_{4} \quad (\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_{4} - \mathrm{Mn}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}) \]

VI.4.1. Experimental data

Materials with the above general formula were prepared with \(a = 0^*)\), 0.2, 0.4, 0.575, 0.675, 0.775, 0.95, 1.1, and 1.5.

The indicated materials were prepared by the very same method as the materials described in the preceding sections (VI.2 and VI.3); \(\mathrm{MnCO}_3\) was used as an added raw material (\(\mathrm{Na} <\) approximately 0.1%; Mg 0.01%).

The specimens were sintered at \(1200^\circ\mathrm{C}\) in an atmosphere containing \(\mathrm{N}_2\), \(\mathrm{H}_2\), and \(\mathrm{CO}_2\) in various proportions; X-ray diffraction patterns and chemical analysis showed that materials with different \(a\), i.e. with different \(\mathrm{Mn}^{2+}/\mathrm{Ni}^{2+}\) ratios, require different compositions of the sintering atmosphere for their preparation.

Adjustment of the oxygen content was carried out by heating, for three hours at \(1100^\circ\mathrm{C}\), fired rods with a known oxygen deficiency in sealed evacuated quartz tubes, also containing a certain amount of \(\mathrm{BaO}_2\), calculated so as to give exactly as much \(\mathrm{O}_2\) as is necessary to oxidize the rods to the correct composition.

The materials on which the measurements reported below were carried out contained the following amounts of excess\(^{**)}\) oxygen:

\(a=\)	0.2	0.4	0.575	0.675	0.775	0.95	1.1	1.3	1.5
Oxygen (wt. %)	0.04	0.11	0.04	−0.04	0.06	0.05	−0.015	0.0	−0.1

X-ray diffraction patterns obtained on a Norelco diffractometer with Fe \(K_{\alpha}\) radiation indicate that the materials are pure spinels. From these diffraction patterns it is impossible to obtain any data on the distribution of the cations.

\[ \begin{aligned} &*)\ \text{This is the material of section VI.2 with } a = 0.5.\\ &**)\ \text{An oxygen deficiency is denoted by a minus sign.} \end{aligned} \]

The saturation magnetization measurements were carried out using fields up to 6000 oersteds up to a temperature of \(77^\circ\text{K}\); at 77 and \(20^\circ\text{K}\) the measurements were carried out by Jongebreur in fields up to 23,000 oersteds by a method that will be described by him elsewhere.

The saturation magnetic moments, Curie temperatures, and lattice constants are indicated in Figs. 20, \(a\), \(b\) and in Table XII.

Table XII

Saturation magnetic moments, Curie temperatures, and lattice constants of the series \(\mathrm{Ni}_{1.5-a}\mathrm{Mn}_a\mathrm{FeTi}_{0.5}\mathrm{O}_4\)

Mn content \((a)\)	Formula	\(n_B\)	\(\Theta\) (°C)	Lattice constant (Å)
0	\(\mathrm{Ni}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	1.45	293	—
0.2	\(\mathrm{Ni}_{1.3}\mathrm{Mn}_{0.2}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	0.32	277	\(8.3752 \pm 10\)
0.4	\(\mathrm{Ni}_{1.1}\mathrm{Mn}_{0.4}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	0.30	265	\(8.4033 \pm 4\)
0.575	\(\mathrm{Ni}_{0.925}\mathrm{Mn}_{0.575}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	0.10	249	\(8.4456 \pm 10\)
0.675	\(\mathrm{Ni}_{0.825}\mathrm{Mn}_{0.675}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	0.445	192	\(8.4667 \pm 10\)
0.775	\(\mathrm{Ni}_{0.725}\mathrm{Mn}_{0.775}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	0.60	181	\(8.4834 \pm 4\)
0.95	\(\mathrm{Ni}_{0.55}\mathrm{Mn}_{0.95}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	1.11	152	\(8.5096 \pm 7\)
1.1	\(\mathrm{Ni}_{0.4}\mathrm{Mn}_{1.1}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	1.28	126	\(8.5412 \pm 7\)
1.3	\(\mathrm{Ni}_{0.2}\mathrm{Mn}_{1.3}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	1.42	109	\(8.5691 \pm 10\)
1.5	\(\mathrm{Mn}_{1.5}\mathrm{FeTi}_{0.5}\mathrm{O}_4\)	1.68	89	\(8.6025 \pm 10\)

Measurements of the temperature dependence of the saturation magnetization are given in the form of curves \(\sigma=\sigma(T/\Theta)\) in Fig. 21.

For the material with \(a=0.775\), Beljers measured the effective \(g\)-factor on a specimen in the form of a sphere with a diameter of about \(0.5\) mm. The results of the measurements are as follows:

\[ \mathrm{Ni}_{0.725}\mathrm{Mn}_{0.775}\mathrm{FeTi}_{0.5}\mathrm{O}_4 \]

Wavelength in cm	Temperature in °K	\(g_{\mathrm{eff}}\)
\(1.25_5\)	293	2.58
\(1.25_5\)	209	2.68
\(1.25_5\)	143	2.75
	(extrapolation to \(0^\circ\text{K}\))	\((2.80\text{--}2.88)\)

Fig. 20. Properties of a series of mixed crystals \( \mathrm{Ni}_{1.5-a}\mathrm{Mn}_{a}\mathrm{FeTi}_{0.5}\mathrm{O}_{4} \).

a) Curie temperatures \(\Theta\) (\({}^{\circ}\mathrm{K}\), scale on the left) and lattice constants (Å, scale on the right).
b) Saturation magnetic moments \(m\); the sign of \(m\) is not known with certainty. Curve \(I\) is the most probable curve; line \(II\) gives the saturation magnetic moments calculated for the case in which all \(\mathrm{Ti}^{4+}\) and \(\mathrm{Ni}^{2+}\) ions are in octahedral sites and there are no angles between the ionic magnetic moments within one of the sublattices; curves \(III\), \(IV\), \(V\) are other possible curves.
c) Number of \(\mathrm{Ti}^{4+}\) ions in the tetrahedral sublattice per molecular unit \((x)\) for cases \(I\), \(III\), \(IV\) (scale on the left). \(\sin\psi\) for angles \(180^\circ - 2\psi\) occurring between the directions of the ionic magnetic moments in sublattice \(B\) for cases \(I\) and \(V\) (scale on the right).

Fig. 21. a) and b) Curves of the dependence of saturation magnetization on reduced temperature for the system \(\mathrm{Ni}_{1.5-a}\mathrm{Mn}_{x}\mathrm{FeTi}_{0.5}\mathrm{O}_{4}\).

VI.4.2. Discussion of the Results

The general formula of the indicated series of mixed crystals is as follows:

\[ \mathrm{Ti}_x\mathrm{Ni}_y(\mathrm{Fe}^{\mathrm{III}}+\mathrm{Mn}^{\mathrm{II}})_{1-x-y} [\mathrm{Ni}_{1.5-a-y}(\mathrm{Fe}^{\mathrm{III}}+\mathrm{Mn}^{\mathrm{II}})_{a+x+y}\mathrm{Ti}_{0.5-x}] \mathrm{O}_4^{*}). \]

As long as the magnetic moments of the ions in the sublattices \(A\) and \(B\) are mutually antiparallel, the saturation magnetic moment will be equal to

\[ n_B=|m_b-m_a|; \]

in the case under consideration

\[ m_b-m_a=-1.55+2.7a+10x+(7.7-p)y. \tag{6.5} \]

For all values of \(a\), with the exception of \(a=0\), the sign of \(m_b-m_a\) is unknown.

The values of \(m_b-m_a\) for antiparallel resultant magnetic moments of the ions in the sublattices \(A\) and \(B\) for the case in which all \(\mathrm{Ni}^{2+}\) and \(\mathrm{Ti}^{4+}\) ions occupy the \(B\) sites are represented by the dashed line \(II\) (Fig. 20, b).

For \(a=0.77\), the equation given above and the equation for the effective \(g\)-factor (the value \(g_{\mathrm{eff}}=2.88\) is used) give a single solution \((x=-0.02;\ y=-0.02)\), close to the physically possible arrangement \((x\approx 0;\ y\approx 0)^{**}\); in this case \(m_b-m_a\) is positive and lies on line \(II\).

If it is assumed that there are angles between the magnetic moments of the ions within the \(B\) sublattice, then, using equations (6.3) and (6.4′), we find that there exist solutions for which \(m_b-m_a\) is negative (line \(V\)), but with higher values of \(y\), for example, \(y\approx 0.2\) at \(x\approx 0\). We shall reject these solutions as improbable in comparison with the preceding solution \((x\approx 0,\ y\approx 0,\) without angles in the \(B\) sublattice), on the basis of the fact that \(y=0\) at \(a=0\) and that \(\mathrm{Ni}^{2+}\) ions will certainly not be displaced from octahedral sites by \(\mathrm{Mn}^{2+}\) ions, owing to their great individual tendency toward sixfold coordination.

From the fact that, in materials with \(a=0.4,\ 0.57_5\), and \(0.67_5\), curves of type \(c\) are observed (see Fig. 8), while in materials with \(a=0.2\) and \(0.77_5\) curves of type \(b\) are observed (see the same figure), we conclude that, in all these materials, the difference \(m_b-m_a\) has the same

*) Since \(m_{\mathrm{Mn}^{2+}}=m_{\mathrm{Fe}^{3+}}=5\), interchange of \(\mathrm{Mn}^{2+}\) and \(\mathrm{Fe}^{3+}\) ions from sublattice \(A\) to sublattice \(B\), and conversely, does not affect the magnitude of \(n_B\).

**) The fact that no physically possible solution was found is not unexpected if one takes into account the sensitivity of \(n_B\) to small variations, for example, of \(x\); the measurements of \(n_B\) and \(g_{\mathrm{eff}}\) were carried out on different parts of one specimen, so that very small inhomogeneities, i.e., deviations in the mean value of \(x\) of the order of \(\pm 0.01\), may cause such a small shift of values.

sign, i.e., is positive. Consequently, \(m\) is positive for the entire system, and curves III and IV may likewise be discarded.

The values of \(n_B\) up to \(a = 0.95\) can be explained by a continuous decrease in the number of \(\mathrm{Ti}^{4+}\) ions in the tetrahedral sublattice \((x)\) (see Fig. 20, c). The curve of the dependence of \(x\) on the \(\mathrm{Ni}^{2+}\) content has the same form as in the preceding Sections VI.2 and VI.3.

The fact that the experimental values of \(n_B\) for \(a = 1.1\)–\(1.5\) lie below line II must, in our opinion, be explained by incomplete antiparallelism of the moments of sublattices \(A\) and \(B\).

The Curie temperatures show a decline near \(a = 0.57\), but from the values of the saturation magnetic moment we found that this material has fewer \(\mathrm{Ti}^{4+}\) ions in the tetrahedral sublattice than the amount indicated on the smooth line I drawn through the other points; this could explain the higher Curie temperature for this material than the value lying on the smooth dashed line I passing through the other points.

The lattice constants vary practically linearly from \(a = 0\) to \(a = 1.5\), although the amount of \(\mathrm{Ti}^{4+}\) in the \(A\) sites varies nonlinearly as a function of \(a\). The same behavior could be observed in the two preceding mixed-crystal systems (Sections VI.2 and VI.3); apparently, mutual transfers of \(\mathrm{Ti}^{4+}\) and \(\mathrm{Fe}^{3+}\) from sublattice \(A\) to sublattice \(B\) and conversely do not have any appreciable effect on the value of the lattice constant.

The distribution of \(\mathrm{Mn}^{2+}\)—\(\mathrm{Fe}^{3+}\) between the two sublattices cannot be determined from the available data.

Despite the fact that in the system under consideration one type of anomalous curves \(\sigma = \sigma(T)\) is encountered, no change in the sign of the difference \(m_b - m_a\) was found.

VII. FERRIMAGNETIC OXIDES CONTAINING CHROMIUM: SYSTEM \(\mathrm{Li}_{0.5}\mathrm{Fe}^{III}_{2.5-a}\mathrm{Cr}^{III}_{a}\mathrm{O}_4\), \((\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5}\mathrm{O}_4—\mathrm{Li}_{0.5}\mathrm{Fe}_{0.5}\mathrm{Cr}_2\mathrm{O}_4)\)

VII. 1. Experimental data

A series of materials with the general composition

\[ \mathrm{Li}_2\mathrm{O}\cdot(5 - 2a)\mathrm{Fe}_2\mathrm{O}_3\cdot 2a\mathrm{Cr}_2\mathrm{O}_3 \]

was prepared with \(a = 0,\ 0.50,\ 0.75,\ 1.0,\ 1.25,\ 1.50,\ 1.60,\ 1.70,\ 2.00,\) and \(2.50\).

The materials were prepared from
\(\mathrm{Fe}\) (\(\mathrm{Ni}\ 0.01_{6}\%\)),
\(\mathrm{CrO}_3\) (\(\mathrm{Ca} <\) approximately \(0.2\%\), \(\mathrm{Cu}\ 0.01\%\), \(\mathrm{Si}\ 0.01\%\)),
\(\mathrm{Li}_2\mathrm{CO}_3\) (\(\mathrm{Mg}\) and \(\mathrm{Ca}\ 0.02\%\), \(\mathrm{Fe}\), \(\mathrm{Al}\), and \(\mathrm{Si}\ 0.01\%\)).

The \(\mathrm{Cr}^{III}\) nitrate solution was prepared by dissolving \(\mathrm{CrO}_3\) in water and reducing it in ethyl alcohol at \(40^\circ\mathrm{C}\); after being left overnight, the gelatinous mass was dissolved in concentrated \(\mathrm{HNO}_3\). To the mixed solution of \(\mathrm{Fe}^{III}\) and \(\mathrm{Cr}^{III}\) nitrates, ammonium hydroxide was added in excess; after boiling, the precipitate was...

filtered, thoroughly washed until the removal of \(NH_4NO_3\), dried in vacuum, and slowly heated to \(500^\circ C\). Mixed oxide crystals having the formula \(Fe_{2-0.8a}Cr_{0.8a}O_3\) were ground together with \(Li_2CO_3\) in the presence of benzene in a Bloch-Rosetti agate mill. The mixture was dried and heated for 2 hours at \(700^\circ C\) in an atmosphere containing equal volumes of \(CO_2\) and \(O_2\). After secondary grinding, rods and disks were pressed from the resulting powder; these were sintered for 2 hours at \(1150^\circ C\) in \(O_2\) and then slowly cooled. By means of this method of preparation it was possible to avoid excessive losses of \(Li_2O\) during evaporation.

\(Li_{0.5}Fe_2Cr_{0.5}O_4\) (\(a=0.5\)) was also prepared by another method: \(1Li_2CO_3\) and \(1Cr_2O_3\) were ground in an agate ball mill in the presence of benzene; then the mixture was dried and heated for 2 hours at \(1100^\circ C\) in \(O_2\). X-ray diffraction patterns showed that the resulting powder contained no \(Li_2O\), \(Li_2CO_3\), \(Cr_2O_3\), or spinel phase and that it probably consisted of a mixture of \(LiCrO_2\). This powder was mixed with \(Fe_2O_3\) in the required proportion and then again ground, dried, and fired at \(1150^\circ C\) in \(O_2\).

Study of the X-ray diffraction patterns obtained on a Norelco diffractometer showed that the spinel \(Li_{0.5}Cr_{2.5}O_4\) is not formed; specimens with \(a=2.5\) practically exhibited only the lines of \(Cr_2O_3\)*. Some of the additional weak reflections observed probably came from \(LiCrO_2\).

Other materials prepared as described above were pure spinels, with the exception of the material with \(a=2.00\), which showed weak reflections from \(Cr_2O_3\).

A pressed disk of unsintered material with \(a=2.0\) was then placed on a plate pressed from the same material (in order to maintain the necessary vapor pressure of \(Li_2O\)) and was sintered by the same method as indicated above. Thanks to this, the \(Cr_2O_3\) lines became considerably weaker (for details see \(^{14}\)).

The distribution of \(Li^+\) ions in the series of mixed crystals

\[ Li_{0.5}Fe_{2.5}O_4 - Li_{0.5}Fe_{0.5}Cr_2O_4 \]

was determined by Braun \(^{14}\). His results are reproduced in Fig. 22, б.

Prolonged annealing of the materials at low temperatures (24 hours at \(500^\circ C\), 96 hours at \(450^\circ C\)) did not change the diffraction pattern at all. Measurements of the magnetic properties were carried out on specimens that had only been slowly cooled.

The material \(Li_{0.5}Fe_2Cr_{0.5}O_4\) (\(a=0.5\)) exhibited the same superstructure lines as the material \(Fe[Li_{0.5}Fe_{1.5}]O_4\); the material \(Li_{0.5}Fe_{0.5}Cr_2O_4\) (\(a=2\)) revealed a new superstructure

*) The statement that “\(Li_{0.5}Cr_{2.5}O_4\) is a noncubic compound” \(^{86}\) should be read as: it does not have the cubic structure of \(Li_{0.5}Cr_{2.5}O_4\).

Fig. 22. a) Saturation magnetic moments of the system \(\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5-a}\mathrm{Cr}_{a}\mathrm{O}_{4}\). Curve \(I\)—experimental values; curve \(II\)—values calculated from the cation distribution determined by means of X-ray diffraction (Fig. 22, b, curve \(II\)); line \(III\)—values calculated for the case when all \(\mathrm{Li}^{+}\) and \(\mathrm{Cr}^{3+}\) ions are in octahedral sites. Line \(IV\)—values calculated for the case when all \(\mathrm{Fe}^{3+}\) ions are in tetrahedral sites. b) Number of \(\mathrm{Li}^{+}\) ions in tetrahedral sites per molecular unit \((x)\) for the system

\[ \mathrm{Li}_{0.5}\mathrm{Fe}_{2.5-a}\mathrm{Cr}_{a}\mathrm{O}_{4} \left(\mathrm{Fe}_{1-x}\mathrm{Li}_{x}\left[\mathrm{Li}_{0.5-x}\mathrm{Fe}_{1.5+x-a}\mathrm{Cr}_{a}\right]\mathrm{O}_{4}\right). \]

Curve \(II\)—experimental values obtained by means of X-ray diffraction; curve \(I\)—values calculated from the saturation magnetic moments (Fig. 22, a, curve \(I\)); lines \(III\) and \(IV\)—values for which lines \(III\) and \(IV\) in Fig. 22, a, were obtained, for the case when all \(\mathrm{Cr}^{3+}\) ions are in octahedral sites. Curve \(V\)—theoretical distribution curve (see text).

\((1\mathrm{Li}^{+}:1\mathrm{Fe}^{3+}\) in the tetrahedral sublattice), already described in Section 1.2.3[^14]. The occurrence of such long-range order was found in the material with \(a = 1.6\).

The relative distribution of \(\mathrm{Fe}^{3+}\) and \(\mathrm{Cr}^{3+}\) ions cannot be determined by X-ray diffraction. The fact that, despite the transition of all \(\mathrm{Li}^{+}\) ions into the tetrahedral sublattice when \(a\) increases to 2, the spinel \(\mathrm{Li}_{0.5}\mathrm{Cr}_{2.5}\mathrm{O}_{4}\) is not formed clearly indicates the presence of \(\mathrm{Cr}^{3+}\) ions only in octahedral sites.

Curves of the dependence of the saturation magnetization on temperature were measured for all materials of the mixed-crystal series.

Fig. 23. Curves of the dependence of the relative magnetization on the reduced temperature \((\sigma/\sigma_{0}\) versus \(T/\Theta)\) for the system \(\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5-a}\mathrm{Cr}_{a}\mathrm{O}_{4}\). For clarity, the values of \(\sigma/\sigma_{0}\) below and above the compensation temperature \((\sigma = 0)\) are plotted with different signs, although the measurements determine only absolute values. Most of the measured points are given in the cited work[^88].

Fig. 23 shows the dependence of the relative saturation magnetization \((\sigma/\sigma_{0})\) on the reduced temperature \((T/\Theta)\). It is seen that, for a number of materials, anomalous curves \(\sigma = \sigma_{s}(T)\) of the Néel type \((N)\) are observed (see Fig. 8, curve \(e\)), possessing a compensation temperature \(T_{\mathrm{comp}}\), at which \(\sigma = 0\)[^87].

In Fig. 23 the saturation magnetization has different signs both below and above the compensation temperature, although measurements of the saturation magnetization give, of course, only the absolute values of the spontaneous magnetization. To show that the spontaneous magnetization indeed changes sign at \(T_{\mathrm{comp}}\), we measured the residual induction \((B_r)\) of \(\mathrm{Li}_{0.5}\mathrm{Fe}_{1.25}\mathrm{Cr}_{1.25}\mathrm{O}_4\) in the absence of a magnetic field, perpendicular to the Earth’s magnetic field. Fig. 25 shows that the residual magnetization indeed changes sign at \(T_{\mathrm{comp}}\).

For \(\mathrm{Li}_{0.5}\mathrm{Fe}_{0.5}\mathrm{Cr}_2\mathrm{O}_4\) \((a=2.0)\) it was found that \(\sigma\) decreases almost linearly from the value \(\sigma = 1\,CGSM \cdot \mathrm{cm}^{3/2}\), approximately at \(100^\circ\mathrm{K}\), to \(\sigma \approx 0\) at the Curie temperature. Nevertheless, measurements of the residual magnetization clearly revealed the compensation temperature. The reason for this apparent paradox is discussed in Section VII.2.4 for the mixed crystals \(\mathrm{NiFe}_2\mathrm{O}_4 — \mathrm{NiAl}_2\mathrm{O}_4\).

The saturation magnetic moments (in Bohr magnetons) are given in Fig. 22, \(a\) (curve 1) and in Table XIII.

Table XIII

Cation distributions, lattice constants, Curie temperatures, compensation temperatures, and saturation magnetic moments for the system \(\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5-a}\mathrm{Cr}_a\mathrm{O}_4\)

Cr content \((a)\)	Ion distribution	Lattice constant (Å)	\(\Theta\) (°C)	\(T_{\mathrm{comp}}\) (°C)	\(n_B\)
0	\(\mathrm{Fe}_{1,00}[\mathrm{Li}_{0,50}\mathrm{Fe}_{1,50}]\mathrm{O}_4\)	8,331	680	—	2,47; 2,60*)
0,50	\(\mathrm{Fe}_{1,00}[\mathrm{Li}_{0,50}\mathrm{Fe}_{1,00}\mathrm{Cr}_{0,50}]\mathrm{O}_4\)	8,306	500	—	1,62; 1,50**)
0,75	\(\mathrm{Fe}_{0,98}\mathrm{Li}_{0,02}[\mathrm{Li}_{0,48}\mathrm{Fe}_{0,77}\mathrm{Cr}_{0,75}]\mathrm{O}_4\)	8,296	410	—	1,35
1,00	\(\mathrm{Fe}_{0,94}\mathrm{Li}_{0,06}[\mathrm{Li}_{0,44}\mathrm{Fe}_{0,56}\mathrm{Cr}_{1,00}]\mathrm{O}_4\)	8,292	315	205	0,84
1,25	\(\mathrm{Fe}_{0,91}\mathrm{Li}_{0,09}[\mathrm{Li}_{0,41}\mathrm{Fe}_{0,34}\mathrm{Cr}_{1,25}]\mathrm{O}_4\)	8,290	214	\(+38\)	0,61
1,50	\(\mathrm{Fe}_{0,80}\mathrm{Li}_{0,20}[\mathrm{Li}_{0,30}\mathrm{Fe}_{0,20}\mathrm{Cr}_{1,50}]\mathrm{O}_4\)	8,287	119	\(-16\)	0,55
1,60	\(\mathrm{Fe}_{0,64}\mathrm{Li}_{0,36}[\mathrm{Li}_{0,14}\mathrm{Fe}_{0,26}\mathrm{Cr}_{1,60}]\mathrm{O}_4\)	8,288	167	\(+11\)	0,42
1,70	\(\mathrm{Fe}_{0,54}\mathrm{Li}_{0,46}[\mathrm{Li}_{0,04}\mathrm{Fe}_{0,26}\mathrm{Cr}_{1,70}]\mathrm{O}_4\)	8,290	155	\(+20\)	0,22
2,00	\(\mathrm{Fe}_{0,50}\mathrm{Li}_{0,50}[\mathrm{Cr}_{2,00}]\mathrm{O}_4\)	8,288	\(80 \pm 16\)	\(+37 \pm 15\)***)	0,10

) For the material described in Section IV.
) Obtained for material prepared from \(\mathrm{LiCrO}_2\) and \(\mathrm{Fe}_2\mathrm{O}_3\).
**) Obtained from measurements of the residual magnetization.

Curie temperatures \((\Theta)\) and compensation temperatures \((T_{\mathrm{comp}})\) are shown in Fig. 24 and in Table XIII. The lattice constants determined by Braun are also given in Table XIII.

The effective \(g\)-factor of a material possessing a compensation temperature exhibits anomalous behavior as a function of temperature. Figure 25 shows the results of measurements by Van Uitert[^59,^87] on \(\mathrm{Li}_{0.5}\mathrm{Fe}_{1.25}\mathrm{Cr}_{1.25}\mathrm{O}_4\).

VII.2. Discussion of the Results of Magnetic Measurements

VII.2.1. Saturation Magnetic Moments

Knowing the distribution of \(\mathrm{Li}^+\) ions and assuming that the \(\mathrm{Cr}^{+3}\) ions occupy only octahedral sites, one can calculate the saturation magnetic moments for the case of complete antiparallelism between the magnetic moments of the ions of sublattices \(A\) and \(B\). The moments calculated in this way (with \(g = 2\) and \(S = 3/2\) for the \(\mathrm{Cr}^{3+}\) ion) are represented by the dashed curve II in Fig. 22,a.

For \(0 \leq a \leq 1.25\), the experimental and calculated values of the magnetic moments are practically identical. We have seen that there is indirect X-ray evidence that the \(\mathrm{Cr}^{3+}\) ions occupy only octahedral sites. The magnitude of the magnetic moment of \(\mathrm{Li}_{0.5}\mathrm{Fe}_{0.5}\mathrm{Cr}_{0.5}\mathrm{O}_4\) \((a = 0.5)\), where the presence of angles between ionic magnetic moments within sublattice \(B\) is very improbable in view of the high Curie temperature \((500^\circ \mathrm{C})\), which indicates a very large and therefore predominant \(AB\) interaction, shows that the \(\mathrm{Cr}^{3+}\) ion indeed contributes 3 Bohr magnetons to ferrimagnetism. The good agreement between the observed and calculated saturation magnetic moments for \(0 \leq a \leq 1.25\) shows that throughout this entire region the magnetic moments of the ions in sites \(A\) and \(B\) are practically all antiparallel to one another. The discrepancy between the observed and calculated values for materials with \(a > 1.25\) must be explained by the assumption that here the ionic magnetic moments within one crystallographic sublattice are no longer parallel to one another: the fact that \(m\) is lower than the calculated value shows that at \(0^\circ\mathrm{K}\) angles occur in sublattice \(B\) (see equations (2.18) and (2.19)).

If this arrangement were preserved also at high temperatures, then, according to the simplest theory (see Section II.2.2), no anomalous curves \(\sigma = \sigma(T)\) should be obtained. The fact that for \(a = 1.6\text{--}2.0\) anomalous curves do in fact occur may be due either to the disappearance of the angles at high temperatures, which is possible according to the theory of Yafet and Kittel, or to the temperature dependence of \(\gamma_2\), owing to which \(\gamma_2 = -1\) at \(T_{\mathrm{comp}}\). The temperature dependence of the ratio of the interactions has so far been introduced into the theory only by Smart[^90].

VII.2.2. Curie Temperatures

The Curie-temperature curve (Fig. 24) shows an almost linear decrease between \(a=0\) and \(a=1.50\); then a rise is observed, and after passing through a maximum at \(a=1.6\) the curve again begins to fall. The lattice constants decrease between \(a=0\) and \(a=1.50\) and again increase slightly near \(a=2.00\) (Table XIII). The anomaly in the change of the Curie temperature with changing \(a\) cannot be ascribed to an anomaly in the change of the lattice constant, since a decrease in the lattice constant should be expected to increase the Curie temperature, and conversely.

If the \(AB\) interaction is large in comparison with the \(BB\) interaction, i.e. so long as no angles occur between the magnetic moments of ions at the \(B\) sites, the Curie temperature is determined chiefly by the strength of the interactions between iron ions at the \(A\) sites and iron or chromium ions at the \(B\) sites, and by the number of the corresponding neighbors. The numbers of neighbors are proportional to the products of the numbers of ions in each of the sublattices per molecular unit. These products, taken from column 2 of Table XIII, are given in Table XIV.

Table XIV

Cr content \((a)\)	Product	Product	Product
	\(\mathrm{Fe}_A\cdot \mathrm{Fe}_B\) (1)	\(\mathrm{Fe}_A\cdot \mathrm{Cr}_B\) (2)	\(\mathrm{Fe}_A\cdot(\mathrm{Fe}_B+\mathrm{Cr}_B)\) (3)
0	1.50	0.00	1.50
0.50	1.00	0.50	1.50
0.75	0.75₅	0.73₅	1.49
1.00	0.52₅	0.94	1.46₅
1.25	0.31	1.13₅	1.44₅
1.50	0.16	1.20	1.36
1.60	0.16₅	1.02₅	1.19
1.70	0.14	0.92	1.06
2.00	0.00	1.00	1.00

The material with \(a=0.5\) has a noticeably lower Curie temperature than the material with \(a=0\), despite the somewhat smaller value of the lattice constant. From the fact that in both cases the products \(\mathrm{Fe}_A\cdot(\mathrm{Fe}_B+\mathrm{Cr}_B)\) are equal, it follows that the \(\mathrm{Fe}_A-\mathrm{Fe}_B\) interaction is considerably stronger than the \(\mathrm{Fe}_A-\mathrm{Cr}_B\) interaction.

This can only partially explain the sudden rise of the Curie temperature at \(a>1.5\) (Fig. 24), since the figures in Table XIV

Fig. 24. Curie temperatures ($\Theta$) and compensation temperatures ($T_{\mathrm{comp}}$) for the system
$\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5-a}\mathrm{Cr}_{a}\mathrm{O}_{4}$.

Fig. 25. Residual induction ($B_r = 4\pi I_r$, scale at right) of $\mathrm{Li}_{0.5}\mathrm{Fe}_{1.25}\mathrm{Cr}_{1.25}\mathrm{O}_{4}$ as a function of temperature: the change of sign is shown. Effective $g$-factor ($g_{\mathrm{eff}}$, scale at left) of $\mathrm{Li}_{0.5}\mathrm{Fe}_{1.25}\mathrm{Cr}_{1.25}\mathrm{O}_{4}$, measured by Van Wieringen$^{87}$. The hyperbolic curve corresponding to equation (4.5) is shown.

(column 1) reveal an irregularity, but not an increase at \(a > 1.5\). Moreover, the Curie temperature at \(a = 2.00\) is somewhat greater than one third of the Curie temperature at \(a = 0\), whereas neglecting all interactions except the interaction \(\mathrm{Fe}_A—\mathrm{Fe}_B\) should have given \(\Theta = 0^\circ\mathrm{K}\).

The figures in column 3 of Table XIV reveal a sharp drop at values \(a > 1.5\); this indicates that including all \(AB\) interactions in the calculation explains the increase of the Curie temperature still less. Consequently, here an important role will be played by the \(BB\) interactions; from the values of \(n'_B\) for \(a > 1.25\) it was already clear that these \(BB\) interactions cannot be discarded. Only a molecular-field theory generalized to the case of the presence of magnetic ions of two kinds, similarly to the way in which Néel developed it for other cases, will be able to explain all these anomalies quantitatively more correctly.

The compensation temperatures also exhibit a minimum near the value \(a = 1.5\). A discussion of the behavior of \(T_{\mathrm{comp}}/\Theta\) as a function of \(a\) will be given in Section VII.2.3.

VII.2.3. Anomalous temperature dependence of the saturation magnetization and the effective \(g\)-factor

We saw in Section II.2 that, for a ferrimagnetic spinel containing magnetic ions of only one kind, a Néel-type curve \(\sigma = \sigma(T)\) of type \(N\) (Fig. 8, type \(e\)) occurs for values of \(x_a/x_b\) lying between 1 and \((1+\beta)/(1+\alpha)\). Since in this case \(|\alpha|\) and \(|\beta|\) are small compared with 1, owing to the geometry of the spinel lattice the occurrence of this type of curve, like that of other anomalous curves, is limited to a narrow range of compositions. This means that \(T_{\mathrm{comp}}\) will vary from \(0^\circ\mathrm{K}\) to the Curie temperature with a small change in composition.

In the present system \(T_{\mathrm{comp}}\) does not depend strongly on composition above \(a \simeq 1.25\), where it passes through a minimum. This is connected, first, with the anomaly on the Curie-temperature curve, which was discussed in Section VII.2.2, and, second, with the minimum in the value of the ratio \(T_{\mathrm{comp}}/\Theta\).

Fig. 8 indicated that, in general, a Néel-type curve \(N\) (Fig. 8, curve \(e\)) will occur in a series of mixed crystals formed from two materials characterized respectively by \(\sigma = \sigma(T)\) curves of types \(d\) and \(f\), i.e. the ratio \(T_{\mathrm{comp}}/\Theta\) in this case will vary within the limits from 0 to 1. Fig. 23 shows that, in the present system, the ratio \(T_{\mathrm{comp}}/\Theta = 1\) at a value of \(a\) (Cr content) slightly greater than 0.75, decreases up to the composition with \(a = 1.25\), and then increases again.

This characteristic behavior is due above all to the sudden transition of \(\mathrm{Li}^+\) into the tetrahedral sublattice, which prevents the passage of the saturation magnetic moment through zero during the motion

along the dashed line III in Fig. 22,a, calculated under the assumption that both the Li+ and Cr3+ ions would have to be located in octahedral sites. The theory of the compensation temperature, generalized to the case of a material containing magnetic ions of two kinds, was given by K. F. Niessen 34c.

A magnetized rod made of a material having a compensation temperature, suspended in a weak horizontal field (less than the coercive force), for example above a permanent magnet made of “ferrox-dure” *), will rotate when passing through the compensation temperature (upon heating). This very simple experiment is the first visual demonstration of the existence of uncompensated antiferromagnetism, or ferrimagnetism.

The practical consequence of the fact that \(T_{\mathrm{comp}}\) in the system does not change strongly with composition is that it becomes possible to make the value of \(T_{\mathrm{comp}}\) less sensitive to the exact composition and to the homogeneity of the material than it would have been in the case where the phenomenon occurred only in a narrow range of compositions. Consequently, it is easy to prepare a material for the experiment indicated above with a convenient \(T_{\mathrm{comp}}\), i.e. with \(T_{\mathrm{comp}}\) slightly above room temperature (for example, \(\mathrm{Li}_{0.5}\mathrm{F}_{1.25}\mathrm{Cr}_{1.25}\mathrm{O}_{4}\)).

The effective \(g\)-factor for the type of material studied, measured as a function of temperature, shows hyperbolic behavior; this is in agreement with the formula

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

This formula gives \(g_{\mathrm{eff}}=0\) at \(T_{\mathrm{comp}}\), but the denominator becomes zero at a slightly different temperature, leading to values \(g_{\mathrm{eff}}=\pm\infty\).

VII.3. Discussion of the cation distribution

Three series of mixed crystals of Section VI gave curves \(x=x(a)\) (Figs. 16,b, 18,b and 20,b) of the type that should be regarded as normal for ternary or complex spinels. This means that in simple Boltzmann expressions, such as those used by Néel and Smart, the exchange energies of two cations between the two sublattices of the crystal are usually not very large, and therefore we must keep in mind that the ion distributions found in practice are fixed in a definite way depending on the cooling rate.

*) A material for permanent magnets with high coercive force, having the following approximate composition: \(\mathrm{BaFe}_{12}\mathrm{O}_{19}\) 78.

In the present system, the presence of all the \(\mathrm{Li}^{+}\) ions in the tetrahedral sublattice in \(\mathrm{Li}_{0.5}\mathrm{Fe}_{0.5}[\mathrm{Cr}_{1}]\mathrm{O}_{4}\) is easy to understand, proceeding from the very strong tendency of \(\mathrm{Cr}^{3+}\) ions toward sixfold coordination.

If we assume that for \(\mathrm{Li}^{+}_{A}—\mathrm{Cr}^{3+}_{B}\) the exchange energy is very large, and for \(\mathrm{Li}^{+}_{B}—\mathrm{Fe}^{3+}_{A}\) small, then the stability of the inverse arrangement in \(\mathrm{Fe}[\mathrm{Li}_{0.5}\mathrm{Fe}_{1.5}]\mathrm{O}_{4}\) is determined by the order \(1:3\) in the octahedral sublattice. One cannot expect that the replacement of \(\mathrm{Fe}^{3+}\) ions by \(\mathrm{Cr}^{3+}\) ions in this material will change to any significant degree the factors determining the most stable \(\mathrm{Me}^{+}—\mathrm{Me}^{3+}\) distribution (\(a\), \(u\), the ordering energy, etc.), so that the most stable distribution at \(a=1.5\) is still \(\mathrm{Fe}[\mathrm{Li}_{0.5}\mathrm{Cr}_{1.5}]\mathrm{O}_{4}\).

The fact that in this material \(0.20\,\mathrm{Li}^{+}\) was found in tetrahedral sites is due to the strong tendency of \(\mathrm{Cr}^{3+}\) ions toward \(B\) sites, which changes the \(\mathrm{Li}^{+}—\mathrm{Fe}^{3+}\) distribution frozen in the remaining sites.

If the exchange energy \(\mathrm{Li}^{+}_{A}—\mathrm{Cr}^{3+}_{B}\) is taken equal to \(E=\infty\), and the other exchange energy \(\mathrm{Li}^{+}_{A}—\mathrm{Fe}^{3+}_{B}\) is determined so as to make the theoretical distribution curve coincide with the Brown experimental curve at \(a=1\div 1.5\) (Fig. 22), then a curve similar to curve \(V^{*}\) is obtained. The fact that at \(a=0\div 0.75\) and \(a=1.6\div 1.7\) the experimental values of \(x\) are closer to 0 and 0.5, respectively, than the theoretical values clearly indicates the influence of short-range order in the \(A\) and \(B\) sublattices, respectively. It is clear that in the region \(a=1\div 1.5\) short-range order cannot be neglected; here, however, the ordering energies attained in the two sublattices probably compensate one another\(^{14}\).

VIII. FERRIMAGNETIC SPINELS CONTAINING ALUMINUM

VIII.1. Discussion of published data on the systems \(\mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}^{\mathrm{III}}_{2-a}\mathrm{Al}_{a}\mathrm{O}_{4}\) and \(\mathrm{Mg}\mathrm{Fe}^{\mathrm{III}}_{2-a}\mathrm{Al}_{a}\mathrm{O}_{4}\)

A small part of the iron ferrite—iron aluminate system \(\left(\mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}^{\mathrm{III}}_{2-a}\mathrm{Al}_{a}\mathrm{O}_{4}\right)\) was studied by Giyó and Michel\(^{55,91}\) in cases where \(a=0\div 0.2\).

The saturation magnetic moments determined for four materials decreased linearly with increasing \(a\) and were very close to the values \(n_{B}=4-3a\).

* The author thanks D. S. Smart for discussion of this question and for carrying out the indicated calculations.

Giyō^55 assumes that one of the following two ionic distributions takes place: 1) \( \mathrm{Al}^{3+} \) gradually replaces \( \mathrm{Fe}^{3+} \) in the octahedral sublattice; 2) \( \mathrm{Al}^{3+} \) gradually replaces \( \mathrm{Fe}^{2+} \) in the octahedral sublattice, while an equal number of \( \mathrm{Fe}^{3+} \) ions in the tetrahedral sublattice is converted into \( \mathrm{Fe}^{2+} \) ions.

He states that the experimental results agree with Néel’s theory, i.e. can be explained on the basis of these ionic distributions, if the additional assumption is made that the ionic magnetic moments of sublattices \(A\) and \(B\) are completely antiparallel; the latter assumption in the present case is undoubtedly justified.

The general formula of the system under consideration is as follows:

\[ \mathrm{Fe}^{\mathrm{II}}_{y}\mathrm{Al}_{x}\mathrm{Fe}^{\mathrm{III}}_{1-x-y} \left[\mathrm{Fe}^{\mathrm{II}}_{1-y}\mathrm{Fe}^{\mathrm{III}}_{1-a+x+y}\mathrm{Al}_{a-x}\right]\mathrm{O}_{4}. \tag{8.1} \]

Hence, taking \(g_{\mathrm{Fe}^{2+}}=2\), we find the saturation magnetic moment:

\[ n_B=\left|m_b-m_a\right|,\quad \text{and}\quad m_b-m_a=4-5a+10x+2y. \tag{8.2} \]

Giyō’s first assumption, i.e. \(x=0,\ y=0\), would give \(n_B=4-5a\); consequently, it is incorrect.

This does not mean, however, that his second assumption (\(x=0;\ y=a\)) is the only correct assumption, because there is one more cation distribution that can explain the experimental data, namely: \(x=0.2a;\ y=0\). The latter distribution seems more probable for the following reasons.

1) If all \( \mathrm{Al}^{3+} \) ions occupy \(B\) sites and the exchange energy \( \mathrm{Fe}^{3+}_{A}—\mathrm{Fe}^{2+}_{B} \) is not exceptionally small, one may expect that exchange of this kind, which is merely an electronic transition, is so easy that at low temperatures the practically most stable distribution is the distribution with \( \mathrm{Fe}^{2+} \) ions in \(B\) sites, analogous to what occurs in \( \mathrm{Fe}_{3}\mathrm{O}_{4} \).

2) In \( \mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}^{\mathrm{III}}\mathrm{AlO}_{4} \) (\(a=1\)), the large decrease in conductivity upon annealing^22, as well as the appearance of the X-ray photographs^9, indicate that this compound has the formula

\[ \mathrm{Fe}^{\mathrm{III}}_{1-x}\mathrm{Al}_{x} \left[\mathrm{Fe}^{\mathrm{II}}\mathrm{Fe}^{\mathrm{III}}_{x}\mathrm{Al}_{1-x}\right]\mathrm{O}_{4} \]

with \(x\leq 0.1\).

Therefore we expect that also for \(0.04<a<0.2\) in the tetrahedral sites we shall find rather a small fraction of \( \mathrm{Al}^{3+} \) ions than \( \mathrm{Fe}^{2+} \) ions.

The system \( \mathrm{MgFe}_{2-a}\mathrm{Al}_{a}\mathrm{O}_{4} \) was investigated by Jones and Robertson^57; in contrast to Jonker^92, they did not find here

interval of miscibility. We can explain this discrepancy only by assuming that their materials*) were cooled after sintering considerably more rapidly. The saturation magnetic moments (values at \(20^\circ\mathrm{K}\)) are shown in Fig. 26, a.

Fig. 26. a) Saturation magnetic moments of the system \(\mathrm{MgFe}_{2-a}\mathrm{Al}_a\mathrm{O}_4\) (Jones and Roberts). The experimental points lie around line \(I\) and between lines \(II\) and \(III\). Curve \(V\) is a hypothetical curve passing through the experimental point for which \(m\) changes sign as a result of the presence of angles in sublattice \(B\). The lines denoted by indices from \(\gamma_2=-0.2\) to \(\gamma_2=-1.4\) were calculated from (2.19) for the cation distribution of line \(I\) in Fig. 26, b.

b) Values of \(x+y\), and in some cases only \(x\), in the formula

\[ \mathrm{Mg}_x\mathrm{Al}_y\mathrm{Fe}_{1-x-y} [\mathrm{Mg}_{1-x}\mathrm{Al}_{a-y}\mathrm{Fe}_{1-a+x+y}]\mathrm{O}_4, \]

calculated from the values of the saturation magnetic moments. Lines \(I\), \(II\), and \(III\) correspond to the same lines in Fig. 26, a. Curve \(IV\) gives the values for completely antiparallel magnetic moments of sublattices \(A\) and \(B\). \(\square\) denotes values determined by neutron diffraction (Bacon and Roberts).

*) The method of preparation has not yet been published. (See \(^{57}\).)

The general formula of these materials is as follows:

\[ \mathrm{Mg}_x\mathrm{Al}_y\mathrm{Fe}_{1-x-y} \left[\mathrm{Mg}_{1-x}\mathrm{Al}_{a-y}\mathrm{Fe}_{1-a+x+y}\right]\mathrm{O}_4 . \tag{8.3} \]

Assuming, together with the authors named above, the existence of complete antiparallelism of the ionic magnetic moments in the \(A\) and \(B\) sublattices, we obtain:

\[ \begin{aligned} n_B&=\lvert m_b-m_a\rvert,\\ m_b-m_a&=10(x+y)-5a. \end{aligned} \tag{8.4} \]

The quantities \(x+y\), calculated in this way from the experimental values of \(n_B\), are given in Fig. 26, б. The points obtained are scattered about the straight line \(I\), passing through \((x+y=1;\ a=2)\) and varying according to the law \(x+y=0.11+0.44_5a\), and lie between the lines \(II\) and \(III\), for which \(x+y=0.11+(0.44_5\pm 0.04_5)a\).

The quantities \(x\) for \(a=0\) and \(a=2\) were determined by Bacon and Robertson neutronographically\(^{58,93}\), in the same way as the quantity \(x+y\) for \(a=1\)*; the authors assumed that in the latter case \(y=0\) and that the \(\mathrm{Al}^{3+}\) ions remain in the octahedral sublattice throughout the whole series, i.e. that the transition of \(\mathrm{Mg}^{2+}\) ions into the tetrahedral sublattice occurs almost linearly as a function of \(a\), i.e. of the Al content (Fig. 26, б, line \(I\)).

The scatter of the points about line \(I\) (Fig. 26, а and б) occurs, according to the supposition of the authors\(^{57}\), because of the different heat treatment of the individual materials of the series.

Let us briefly discuss the probability that, for some materials, the quantity \(m\) is negative. Such a possibility could occur in two cases.

1) With completely antiparallel magnetic moments of the \(A\) and \(B\) sublattices and for \(a=0.4-1.0\), equation (8.4) gives a second solution for \(x+y\), for which the quantity \(m_b-m_a\) is negative; these quantities are represented by line \(IV\) in Fig. 26, б. In view of the single-valuedness of the quantities \(x+y\) for \(a=0,\ 0.2\), and \(2.0\), however, this solution can be excluded.

2) For the values of \(x+y\) plotted on line \(I\) (Fig. 26, б), \(m\) could decrease along line \(V\) (Fig. 26, а), passing through zero near \(a=0.7\), as a result of the appearance of angles between the magnetic moments in the \(B\) sublattice. When such angles occur,

\[ m=-m_a\left(1+\frac{1}{\gamma_3}\right). \tag{2.19} \]

* The latter data have recently been confirmed by X-ray diffraction\(^{94}\).

and this expression changes sign when $\gamma_2=-1$. In the case under consideration, when there is only one type of magnetic ion in the materials, we may expect that $|\gamma_2|$ remains considerably less than unity (for example, $<0.3$) because of the geometry of the spinel lattice, so that $m$ will in no way change sign.

Indeed, according to equation (2.19), the angles between the ionic magnetic moments in sublattice $B$ will not occur at all throughout the system $\mathrm{MgFe_2O_4—MgAl_2O_4}$, as long as $|\gamma_2|<0.8$.

VIII.2. System $\mathrm{Ni^{II}Fe^{III}_{2-a}Al_aO_4}$ ($\mathrm{NiFe_2O_4—NiFeAlO_4}$)

VIII.2.1. Experimental data

A series of mixed crystals $\mathrm{NiFe_{2-a}Al_aO_4}$ was prepared with $a=0,\ 0.25,\ 0.45,\ 0.50,\ 0.62_5,\ 0.66,\ 0.68,\ 0.68_6,\ 0.68_8,\ 0.70,\ 0.75$ and $1.00$.

These materials were prepared from

$\mathrm{Ni}$ (Si 0.02%, Co 0.01$_8$%, Pb 0.01%),

$\mathrm{Fe}$ (C 0.03%),

$\mathrm{Al}$ (Si 0.01$_3$%, Cr 0.01$_5$%, Pb $\sim$0.01%, Ti 0.01$_3$%) by method (B). The preliminary firing was carried out for 2 hours at $700^\circ\mathrm{C}$ in $\mathrm{O_2}$, and sintering for 2 hours at $1350^\circ\mathrm{C}$ in $\mathrm{O_2}$.

Three series of specimens were prepared:

1) slowly cooled from $1350^\circ\mathrm{C}$ (only with $a=0.66—0.75$);

2) annealed for 16 hours at $1250^\circ\mathrm{C}$, 16 hours at $1150^\circ\mathrm{C}$, 16 hours at $1000^\circ\mathrm{C}$, and 45 hours at $600^\circ\mathrm{C}$, with a cooling rate between these temperatures and below them of about $\frac{1}{4}^\circ\mathrm{C}$ per minute;

3) quenched from $1350^\circ\mathrm{C}$ (only for $a=0—0.62_5$ and $0.75—1.00$). According to analysis, the content of $\mathrm{Fe}^{2+}<0.1\%$.

X-ray diffraction patterns obtained on a Norelco diffractometer showed that all materials are pure spinels. No attempts were made to obtain information on the cation distribution from the X-ray diffraction patterns.

The saturation magnetization was measured in fields up to 9000 oersted down to $77^\circ\mathrm{K}$, and in the temperature range from 77 to $20^\circ\mathrm{K}$ the measurements were carried out by Djonnebourg in fields up to 23,000 oersted. The saturation magnetic moments for

annealed and quenched specimens are shown in Fig. 27 and in Table XV.

Table XV

Saturation magnetic moments, Curie temperatures, and lattice constants in the system $\mathrm{NiFe}_{2-a}\mathrm{Al}_a\mathrm{O}_4$
$(\mathrm{NiFe}_2\mathrm{O}_4—\mathrm{NiFeAlO}_4)$

Al content $(a)$	Formula	Lattice constant, annealed (Å)	$\Theta$ (°C)	$n_B$ annealed	$n_B$ quenched from 1350°C
0	$\mathrm{NiFe_2O_4}$ . . . .	$8{,}3370 \pm 4$	580	2,29	2,29
0,25	$\mathrm{NiFe_{1,75}Al_{0,25}O_4}$	$8{,}3062 \pm 7$	506	1,30	1,59
0,45	$\mathrm{NiFe_{1,55}Al_{0,45}O_4}$	$8{,}2769 \pm 4$	465	0,61	1,19
0,50	$\mathrm{NiFe_{1,50}Al_{0,50}O_4}$	$8{,}2705 \pm 6$	430	0,44	0,99
0,62$_5$	$\mathrm{NiFe_{1,37_5}Al_{0,62_5}O_4}$	$8{,}2521 \pm 7$	360	0—0,045	—
0,66	$\mathrm{NiFe_{1,34}Al_{0,66}O_4}$	$8{,}2420 \pm 20$	368	*)	—
0,68	$\mathrm{NiFe_{1,32}Al_{0,68}O_4}$	$8{,}2485 \pm 1$	356	*)	—
0,68$_6$	$\mathrm{NiFe_{1,31_4}Al_{0,68_6}O_4}$	$8{,}2388 \pm 4$	$\approx 340$	*)	—
0,68$_8$	$\mathrm{NiFe_{1,31_2}Al_{0,68_8}O_4}$	$8{,}2387 \pm 2$	$\approx 340$	*)	—
0,70	$\mathrm{NiFe_{1,30}Al_{0,70}O_4}$	$8{,}2385 \pm 7$	$\approx 340$	*)	—
0,75	$\mathrm{NiFe_{1,25}Al_{0,75}O_4}$	$8{,}2329 \pm 7$	294	0,38	0,58
1,0	$\mathrm{NiFeAlO_4}$ . . .	$8{,}1951 \pm 7$	198	0,64	0,42

*) The interpolation is either too unreliable, or measurements at liquid-hydrogen temperature were not carried out at all.

The curves $\sigma/\sigma_{T=0}=f(T/\Theta)$ for annealed specimens at $a=0—0{,}62_5$ and $0{,}75—1{,}00$ are given in Fig. 28). For $a=0{,}66—0{,}75$ the curves $\sigma=\sigma(T)$ are shown in Figs. 29 $a—d$*). The Curie temperatures and lattice constants of the annealed specimens are given in Table XV.

The material with $a=1$, $\mathrm{NiFeAlO_4}$, was quenched from various temperatures; its saturation magnetic moments are given in Table XVI**).

) These experiments were carried out by F. Hershleb.
*) These experiments were carried out by J. Shulkes.

Fig. 27. Saturation magnetic moments \((m)\) of the series of mixed crystals \(\mathrm{NiFe}_{2-a}\mathrm{Al}_a\mathrm{O}_4\) within the range from \(a=0\) to \(a=1\). Curve \(I\)—experimental values for annealed specimens; \(\times\)—our measurements; \(+\)—measurements of Maxwell and Pickart. Curve \(II\)—experimental values for quenched specimens; \(\bigcirc\)—our measurements on specimens quenched from \(1350^\circ\mathrm{C}\); \(\bullet\)—measurements of Maxwell and Pickart on specimens quenched from \(1420^\circ\mathrm{C}\). Line \(III\)—values calculated for the case in which all \(\mathrm{Ni}^{2+}\) and \(\mathrm{Al}^{3+}\) ions are in the octahedral sublattice.

Fig. 28. a) Curves of the dependence of the relative magnetization on the reduced temperature \((\sigma/\sigma_0\) as a function of \(T/\Theta)\) for the system \(\mathrm{NiFe}_{2-a}\mathrm{Al}_a\mathrm{O}_4\) at \(a=0\text{–}0.625\) and \(a=0.75\text{–}1.00\). b) Curve of the dependence of the saturation magnetization on the reduced temperature \((\sigma\) as a function of \(T/\Theta)\) for \(\mathrm{NiFe}_{1.375}\mathrm{Al}_{0.625}\mathrm{O}_4\).

Fig. 29. Curves of the dependence of saturation magnetization on temperature \((\sigma\) on \(T)\) for the system \(\mathrm{NiFe}_{2-a}\mathrm{Al}_a\mathrm{O}_4\) at \(a = 0.66\text{--}0.75\). \(\square\)—before annealing (Section VIII.2.1, item (1)); \(\circ\)—after annealing (Section VIII.2.1, item (2)).

Maxwell, Pickart, and Hall^95,96 investigated the very same system up to \(a = 2.0\). Their most recent data for

Table XVI

Magnetic moments of saturation
of \(\mathrm{NiFeAlO_4}\) as a function
of quenching temperature

Quenching temperature (°C)	Magnetic moment of saturation
1350	0.42
1120	0.08
1020	0.05
1000	0.02
700	0.52
annealed	0.64

\(a = 0,\ 0.25,\ 0.50,\ 0.63,\ 0.75,\ 0.85\) and \(1.00\) are in very good agreement with ours (see Fig. 27). Mac-Guaig^97 measured the effective \(g\)-factor of their annealed samples and gave the following values*):

\[ \begin{array}{c|ccccc} a &= 0 & 0.25 & 0.50 & 0.63 & 0.75\\ g_{\mathrm{eff}} &= 2.3 & 2.7 & 6.9 & 3.8 & 1.5 \end{array} \]

VIII.2.2. Discussion of the magnetic moments of saturation

Our results for \(a = 0.25,\ 0.45,\ 0.50,\ 0.62_5,\ 0.75\) and \(1.00\) were obtained for the first time. Table XV shows that the values of the magnetic moments of saturation of the annealed samples have a minimum near \(a = 0.62_5\). If one assumes that at \(a\) equal to 0.75 and 1.00 the value of \(m\) is negative, then the curve \(m = m(a)\) will be smooth (Fig. 27, curve I). Among the curves of the dependence of saturation magnetization on temperature shown in Fig. 28, there is one curve having an anomalous form of the type of curve \(c\) in Fig. 8. It may be noted that the shape of the curve changes discontinuously between \(a = 0.62_5\) and 0.75, which indicates that \(m\) changes sign between these compositions. Further discussion of the curves \(\sigma = \sigma(T)\) is given in Section VIII.2.4.

The magnetic moments of saturation of quenched samples do not show a minimum and lie on the smooth curve II, from which we conclude that in the quenched samples \(m\) remains positive throughout the entire series of mixed crystals. The curves \(\sigma = \sigma(T)\) of these materials—

*) For details see work^93.

of the samples not reproduced here, all of the Neél type \((Q)\) and bent with respect to the \(T\) axis.

To prove that in annealed specimens with \(a > 0.75\,m\) is negative, material with \(a = 1.0\) was quenched from various intermediate temperatures. Table XVI shows that the minimum in the values of the saturation magnetic moments occurs at a quenching temperature close to approximately \(1000^\circ\) C. Since a discontinuous change in the cation distribution as a function of temperature may be ruled out as highly improbable, this confirms that in the annealed specimen, and, consequently, also in the material with \(a = 0.75\), \(m\) is negative. Further evidence that \(m\) changes sign near \(a = 0.62_5\) follows from measurements of \(g_{\mathrm{eff}}\) carried out by McGuire and showing a hyperbolic dependence of \(g_{\mathrm{eff}}\) on composition, similar to that obtained in the measurements of \(g_{\mathrm{eff}}\) as a function of temperature described above\({}^{87}\).

VIII.2.3. Discussion of the Cation Distribution

The general formula of the indicated mixed crystals has the form

\[ \mathrm{Ni}_y\mathrm{Al}_x\mathrm{Fe}_{1-x-y} [\mathrm{Ni}_{1-y}\mathrm{Fe}_{1-a+x+y}\mathrm{Al}_{a-x}]\mathrm{O}_4, \tag{8.5} \]

whence, for antiparallel ionic magnetic moments of the sublattices \(A\) and \(B\), assuming \((g_{\mathrm{Ni}^{2+}})_B = 2.3;\ (g_{\mathrm{Ni}^{2+}})_A = p^{*})\), we have:

\[ n_B = |m_b - m_a| \quad \text{and} \quad m_b - m_a = 2.3 - 5a + 10x + (7.7 - p)y. \tag{8.6} \]

Therefore, at \(x = 0,\ y = 0\), the saturation magnetic moments should have followed the dotted line III in Fig. 27. Experiment shows that in annealed specimens (curve I), and to an even greater extent in quenched specimens (curve II), some quantity of \(\mathrm{Ni}^{2+}\) or \(\mathrm{Al}^{3+}\) ions, or of both together, must be located in the tetrahedral sublattice.

Since the transfer of \(\mathrm{Ni}^{2+}\) ions to the \(A\) sites contributes much less to the increase of \(m\) than the transfer of \(\mathrm{Al}^{3+}\) ions, and in view of the recently found X-ray distribution of ions in \(\mathrm{NiAl}_2\mathrm{O}_4\) (namely \(\mathrm{Al}_{1.76}\mathrm{Ni}_{0.24}\mathrm{Ni}_{0.76}\mathrm{Al}_{1.24}\mathrm{O}_4\))\({}^{b}\), it seems beyond doubt that in the tetrahedral sites it is rather \(\mathrm{Al}^{3+}\) ions that are partially located than \(\mathrm{Ni}^{2+}\) ions.

The cation distributions can be calculated from the values of \(n_B\) and the effective \(g\)-factors by the method described in Section VI.2.2. This was done independently of us by Smart\({}^{99}\) for the \(\mathrm{NiFe}_2\mathrm{O}_4\)—\(\mathrm{NiAl}_2\mathrm{O}_4\) system, using data obtained at the Naval

\[ \text{*) See the footnote in Section VI.2.2.} \]

artillery laboratory of the USA \(^{96,97,98}\); the results show that, indeed, the \(\mathrm{Al}^{3+}\) ions, rather than the \(\mathrm{Ni}^{2+}\) ions, are located in tetrahedral sites.

Maxwell and Pickart have recently carried out a study of the system \(\mathrm{NiFe}_{2-a}\mathrm{Ga}_a\mathrm{O}_4(\mathrm{NiFe}_2\mathrm{O}_4-\mathrm{NiGa}_2\mathrm{O}_4)\).

The general formula of the compounds of this system is equivalent to formula (8.5) with the replacement of the \(\mathrm{Al}^{3+}\) ion by the \(\mathrm{Ga}^{3+}\) ion, and so long as no angles arise between the ionic magnetic moments in sublattice \(B\), the saturation magnetic moments \(n_B\) are determined by formula (8.6). It is known that gallium has a strong tendency toward tetrahedral sites (i.e. \(a \gg x \gg 0.5a\)), so that it is assumed that \(n_B\) increases together with the growth of \(a\), starting from \(\mathrm{NiFe}_2\mathrm{O}_4\).

Such behavior was in fact observed: at \(a = 0.25\), from (8.6) one can derive, assuming that all \(\mathrm{Ni}^{2+}\) ions occupy \(B\) sites, that the \(B\) sites are also occupied by \(32\%\) of the \(\mathrm{Ga}^{3+}\) ions in annealed specimens and by \(38\%\) in quenched specimens, provided that the above assumptions are correct.

VIII.2.4. Discussion of the Shape of the Curves \(\sigma = \sigma(T)\)

In order to find materials for which anomalous curves \(\sigma = \sigma(T)\) of types \(b—f\) occur (see Fig. 8), materials with \(a = 0.66,\ 0.68,\ 0.68_6,\ 0.68_8,\ 0.70\) were prepared; the curves \(\sigma = \sigma(T)\) were measured both before annealing, i.e. after the slow-cooling process, and after annealing specially indicated in Section VIII.2.1. Results were also obtained for \(a = 0.75\). Figs. 29, \(a—д\) show that at \(a = 0.66\), both before and after annealing, a curve of type \(b\) is obtained. At \(a = 0.68\), before annealing a curve of type \(c\) (Néel type \((P)\)) appears; annealing transforms this curve into a curve of anomalous type, which is entirely unlike any curve predicted by Néel. The latter type of curve is also found in materials with \(a = 0.68_6\) and with \(a = 0.68_8\) before annealing; after annealing, curves similar to curves of type \(f\) are obtained. A curve of the latter type appears again at \(a = 0.70\) before annealing and changes upon annealing, becoming a curve of Néel type \((Q)\), which, however, is still convex in the direction of the \(T\) axis. We obtained a curve of this last type also at \(a = 0.75\) before annealing. Annealing transforms this curve into a linear curve, also of type \((Q)\). If we include in the general consideration the curves obtained at other values of \(a\) (Fig. 28), it will be seen that with increasing \(a\) there occurs a change of curves of types \(a, b\), and \(c\) into curves of types \(f\) and \(g^*)\).

* The fact that, for an annealed specimen with \(a = 0.62_5\), a curve of type \(c\) was obtained, whereas with \(a = 0.66\) a curve of type \(b\) was obtained, is probably due to the fact that the cooling rates during annealing treatment of materials with \(a = 0.66—0.75\) were somewhat different from the cooling rates used in the treatment of the other materials.

Curves of type \(d\) (Néel type \((L)\)) and \(e\) (Néel type \((N)\)) were not found, but typical curves of this type are shown in Fig. 28, \(b\) and \(v\).

We shall show that this phenomenon can be understood by taking into account the inhomogeneity of the materials.

From equation (8.2) it is easily inferred that a change in the quantity \(x\) by 0.01 leads to a change in \(n_B\) by 0.1, i.e., to a change in \(\sigma_{T=0}\) by 2.6. From the fact that our annealing treatment leads to a change in \(\sigma_{T=0}\) of the order of 3, we may conclude that this is caused by a change—namely a decrease—of \(x\) by slightly more than 0.01. Fig. 29, \(a\)—\(d\), shows that this is sufficient to produce a noticeable difference in the type of the curve \(\sigma=\sigma(T)\). A change in the form of the curve from type \(d\) (Néel type \((L)\)) to type \(f\), between which there should occur curves of type \(e\) (Néel type \((N)\)) with compensation temperatures ranging from \(0^\circ\mathrm{K}\) to the Curie temperature, will be caused by a change in \(x\) of only a few hundredths.

The inhomogeneity of the material, caused by local changes in \(x\) of less than 0.01, will consequently already lead to a superposition of curves of type \(d\) with compensation temperatures varying within narrow limits. Since measurements of magnetic saturation give only the absolute value of the spontaneous magnetization, such a superposition results precisely in the appearance of curves of the type of “hybrid” curves, found at \(a=0.68\) (annealed material) and at \(a=0.68_6\) and \(0.68_8\) (slowly cooled material).

In order to show that this explanation is correct, we measured the residual magnetization of the materials under consideration at several temperatures. The residual magnetization does indeed change sign with temperature.

Samples of \(\mathrm{FeNiAlO}_4\), quenched from \(1020^\circ\mathrm{C}\) and \(1000^\circ\mathrm{C}\), also exhibit “hybrid” curves \(\sigma=\sigma(T)\) of the type indicated above; here too the residual magnetization changes sign with temperature.

The small increase of \(\sigma\) on approaching \(T=0^\circ\mathrm{K}\) in the annealed sample with \(a=0.62\) (Fig. 28, \(b\)) indicates that this curve \(\sigma=\sigma(T)\) is also “hybrid,” with an average value of \(m_b-m_a\) very close to zero.

The above-mentioned small local changes in \(x\) will, of course, exist throughout the entire series of mixed crystals and, indeed, in all ferrimagnetic spinels the existing distribution of cations changes with temperature. The curve \(\sigma=\sigma(T)\), consequently, is in all these cases an averaged curve; however, only in materials for which \(m_b-m_a\) changes sign with temperature can the inhomogeneity be detected from the curves of the dependence of the saturation magnetization on temperature.

IX. OTHER FERRIMAGNETIC OXIDES CONTAINING CHROMIUM: THE SYSTEM \( \mathrm{Mn^{II}Fe^{III}_{2-a}Cr_aO_4} \) \((\mathrm{MnFe_2O_4}—\mathrm{MnCr_2O_4})\)

IX.1. Introduction

Figure 30 gives the properties of a series of mixed crystals \( \mathrm{MnFe_{2-a}Cr_aO_4} \).
a) Curie temperatures (\(\Theta\) in °K, scale on the left) and lattice constants (in Å, scale on the right).
b) Saturation magnetic moments. Curve \(I\) — saturation magnetic moments (most probable curve); curves \(I—III\) — saturation magnetic moments (other possible curves); line \(II\) — values calculated for the case in which all \( \mathrm{Cr^{3+}} \) ions are in the octahedral sublattice and there are no angles between the ionic moments within either sublattice; line \(IV\) — saturation magnetic moments calculated under the same assumptions as line \(II\), but with the moments of the \( \mathrm{Cr^{3+}} \) ions (in sublattice \(B\)) parallel to the moments of the ions in sublattice \(A\).
c) \(\sin \psi\) for the angles \(180^\circ - 2\psi\) occurring between the directions of the ionic magnetic moments in sublattice \(B\) for cases \(I\) and \(III\) of Fig. 30,b. On curve \(III\), above \(a = 1.8\),

\[ \sin \psi = 1,\qquad \text{but}\qquad \sin \varphi < 1. \]

The general formula of the series of mixed crystals \( \mathrm{MnFe_2O_4}—\mathrm{MnCr_2O_4} \) has the form

\[ \mathrm{Cr}_x\left(\mathrm{Fe^{III}}+\mathrm{Mn^{II}}\right)_{1-x} \left[\left(\mathrm{Fe^{III}}+\mathrm{Mn^{II}}\right)_{2-a+x}\mathrm{Cr}_{a-x}\right]\mathrm{O}_4, \tag{9.1} \]

whence, for completely antiparallel ionic magnetic moments of the sublattices \(A\) and \(B\), we obtain:

\[ n_B = |m| = |m_b - m_a| \]

and

\[ m = m_b - m_a = 5 - 2a + 4x. \]

We have seen that \( \mathrm{Cr^{3+}} \) ions have a strong tendency toward sixfold coordination; at \(x = 0\), \(m_b - m_a = 5 - 2a\) (Fig. 30, line \(I'\)).

For \( \mathrm{MnCr_2O_4} \) (\(a = 2\)), the value known from the literature is\({}^{100}\) \(n_B = 1.5\), measured at the temperature of liquid helium. We suppose that the discrepancy with the value \(5 - 2a = 1.0\) should be explained not by assuming \(x = 0.12_5\), but by the existence of angles between the ionic magnetic moments within sublattice \(A\) or \(B\).

If there are angles in sublattice \(A\), equation (2.18) gives \(\alpha_2 = -1.33\), but such a value is very unlikely on geometrical grounds. It was assumed that the magnitude of the \( \mathrm{Mn—O—Mn} \) interaction usually does not greatly exceed

Fig. 30. Properties of the series of mixed crystals \( \mathrm{MnFe}_{2-a}\mathrm{Cr}_a\mathrm{O}_4 \).

interaction Mn—O—Cr, which is equivalent to the indicated geometric argument.

If angles between magnetic moments occur in sublattice \(B\), then \(m\) will be less than 1, i.e., \(-1.5\). In this case, in the system \(\mathrm{MnFe_2O_4}\)—\(\mathrm{MnCr_2O_4}\), as a result of the existence of these angles, \(m\) will pass through zero.

IX.2. Experimental data

A series of mixed crystals of composition \(\mathrm{MnFe}_{2-a}\mathrm{Cr}_a\mathrm{O}_4\) was prepared with \(a = 0,\ 0.12;\ 0.25;\ 0.50;\ 1.00;\ 1.08_5;\ 1.25\) and \(1.50\).

The following materials were used:
\(\mathrm{MnCO_3}\) (\(\mathrm{Na} <\) approximately \(0.1\%\), \(\mathrm{Mg}\ 0.01\%\)),
\(\mathrm{Fe}\) (\(\mathrm{Ni}\ 0.01\%\)),
\(\mathrm{CrO_3}\) (\(\mathrm{Ca} <\) approximately \(0.2\%\), \(\mathrm{Cu}\ 0.01\%\), \(\mathrm{Si}\ 0.01\%\)).

The solution of chromic nitrate was prepared as described in Section VII.1. The required weight amount of this solution was added to a solution of specified weight amounts of Fe and \(\mathrm{MnCO_3}\) in dilute \(\mathrm{HNO_3}\). The subsequent preparation of the material was carried out by method (C); preliminary calcination was performed for 2 hours at \(700^\circ\mathrm{C}\), and sintering for 2 hours at \(1250^\circ\mathrm{C}\).

Sintering was carried out in an atmosphere containing \(\mathrm{N_2}\), \(\mathrm{H_2}\), and \(\mathrm{CO_2}\) in various proportions. X-ray patterns were obtained for all samples; the oxygen content was determined for those samples whose X-ray patterns showed only spinel lines. The final determination of the oxygen content was performed as described in Section VI.4.

Magnetic investigations were carried out on samples that had the following excess oxygen content:

\(a\)	0	0.12	0.25	0.50	1.00	\(1.08_5\)	1.25	1.50
wt. % oxygen	\(+0.007\)	\(-0.075\)	\(-0.025\)	\(+0.05\)	\(-0.015\)	\(+0.05\)	\(+0.03\)	\(+0.01\)

(The minus sign indicates an oxygen deficiency.)

The X-ray patterns, obtained on a Norelco diffractometer using Mo \(K_\alpha\) radiation, showed, for all compositions, only reflections from the spinel structure. No data on the distribution of cations were obtained from these X-ray patterns.

The saturation magnetizations were measured as a function of temperature. The curves \(\sigma = \sigma(T;\vartheta)\) are shown in Fig. 31, \(a\) and \(b\).

Fig. 31. a) and b) Curves of the dependence of the saturation magnetization on the reduced temperature for the system \(\mathrm{MnFe}_2-a\mathrm{Cr}_a\mathrm{O}_4\). Values up to \(77^\circ\mathrm{K}\) were measured in fields up to 6000 oersteds, and values from \(77^\circ\mathrm{K}\) to \(20^\circ\mathrm{K}\) in fields up to 2300 oersteds. For \(a=1\text{–}1.5\), the solid curves represent measurements at 6000 oersteds, and the dashed lines represent measurements at 23000 oersteds.

The saturation magnetic moments are given in Fig. 30, б and in Table XVII; Curie temperatures and lattice constants are indicated in Fig. 30, а.

Table XVII

Curie temperatures, lattice constants, and saturation magnetic moments
in the system \(\mathrm{MnFe}_{2-a}\mathrm{Cr}_a\mathrm{O}_4\)

Content \(\mathrm{Cr}_a\)	Formula	Curie temperature \(\Theta\) (°C)	Lattice constant (Å)	Saturation magnetic moment \(n_B\)
0	\(\mathrm{MnFe}_2\mathrm{O}_4\)	330	\(8{,}5074 \pm 2\)	\(4{,}85\)
\(0{,}12_5\)	\(\mathrm{MnFe}_{1{,}87_5}\mathrm{Cr}_{0{,}12_5}\mathrm{O}_4\)	272	\(8{,}5185 \pm 1\)	\(4{,}04\)
0,25	\(\mathrm{MnFe}_{1{,}75}\mathrm{Cr}_{0{,}25}\mathrm{O}_4\)	247	\(8{,}5107 \pm 3\)	\(3{,}28\)
0,50	\(\mathrm{MnFe}_{1{,}50}\mathrm{Cr}_{0{,}50}\mathrm{O}_4\)	210	\(8{,}4977 \pm 2\)	\(1{,}73\)
1,00	\(\mathrm{MnFeCrO}_4\)	97	\(8{,}4809 \pm 2\)	\(0{,}25\)
\(1{,}08_5\)	\(\mathrm{MnFe}_{0{,}91_5}\mathrm{Cr}_{1{,}08_5}\mathrm{O}_4\)	88	\(8{,}4869 \pm 2\)	\(0{,}37\)
1,25	\(\mathrm{MnFe}_{0{,}75}\mathrm{Cr}_{1{,}25}\mathrm{O}_4\)	15	\(8{,}4680 \pm 2\)	\((>)\,0{,}45\)
1,50	\(\mathrm{MnFe}_{0{,}50}\mathrm{Cr}_{1{,}50}\mathrm{O}_4\)	\(-49\)	\(8{,}4567 \pm 1\)	\(0{,}77\)
2,00	\(\mathrm{MnCr}_2\mathrm{O}_4\)	\(-233^{*)}\)	\(8{,}426^{**)}\)	\(1{,}5^{*)}\)

*) See \(^{106}\).

**) See Section I.2.5.1.

IX.3. Discussion of the results

Table XVII shows that the saturation magnetic moments \(m\) in the system \(\mathrm{MnFe}_2\mathrm{O}_4\)—\(\mathrm{MnCr}_2\mathrm{O}_4\) pass through a minimum and, probably, through zero, similarly to what is shown in Fig. 30, б (curve I): in any case all the values, except the value for \(a = 2\), lie below line II, so that the assumption \(x \ne 0\) is not essential for explaining them.

The maximum in the values of the lattice constants at \(a = 0{,}12_5\) is probably caused by the transfer of a rather large number of \(\mathrm{Mn}^{2+}\) ions into the tetrahedral sublattice as a result of the introduction of a small quantity of \(\mathrm{Cr}^{3+}\) ions into the \(B\) sublattice. The linear decrease of the lattice constants at \(a > 0{,}12_5\) shows that the transfer of \(\mathrm{Mn}^{2+}\) into the \(A\) sites occurs more smoothly in this region.

It is possible that the steep fall of \(m\) for \(a<0.5\) should be attributed to the appearance of angles between the magnetic moments at the \(B\) sites. Nevertheless, it seems rather unexpected that even the introduction of only \(0.12_{5}\) Cr lowers the saturation magnetic moment by \(0.8 n_B\). This phenomenon can be explained, similarly to the maximum in the lattice-constant values, by the transfer of a certain number of \(\mathrm{Mn}^{2+}\) ions to the \(A\) sites and, thus, by a further decrease in the number of \(\mathrm{Fe}_A-\mathrm{Fe}_B\) interactions, which make the largest contribution to the total \(AB\) interaction.

If the saturation magnetic moment \(m\) changes sign (line I), then from (2.19) it follows that \(\gamma_2=-1\) for \(m=0\) and \(\gamma_2=-1.43\) for \(m=-1.5\) (\(a=2.0\)). This is quite possible if the \(\mathrm{Cr}^{2+}-\mathrm{Cr}^{3+}\) interaction is much stronger than the \(\mathrm{Mn}^{2+}-\mathrm{Cr}^{3+}\) interaction, so that the geometrical arguments in favor of such small values of \(\gamma_2\) lose their force.

If \(m\) did not change sign (lines I—III), this would mean that \(\gamma_2\) first decreases almost to \(-1\), i.e. \(\sin\psi\) decreases to 0.6, then \(\sin\psi\) increases to 1 at \(a=1.8\), after which angles \(180^\circ-2\varphi\) should arise between the magnetic moments of ions in the \(A\) sublattice and \(\sin\varphi\) should decrease to 0.9, i.e. \(\alpha_2\) would tend to \(-1.33\); such behavior is extremely unlikely.

In the end, all this suggests that the possibility of the existence of a positive \(\mathrm{Fe}_A-\mathrm{Cr}_B\) interaction, which would be positive, cannot be excluded.

In Section II.3.3 we saw that the superexchange interaction between an ion having fewer than five \(3d\)-electrons and an ion having five \(3d\)-electrons may be either positive or negative: in the system \(\mathrm{Li}_{0.5}\mathrm{Fe}_{2.5}\mathrm{O}_4-\mathrm{Li}_{0.5}\mathrm{Fe}_{0.5}\mathrm{Cr}_2\mathrm{O}_4\) (the lattice constants are approximately \(8.3\) Å) the \(\mathrm{Fe}_A-\mathrm{Cr}_B\) interactions are negative; however, it is possible that in the system considered by us, where for \(a<0.5\) the lattice constants are of the order of \(8.5\) Å, this interaction is positive.

Line IV represents the saturation magnetic moments for the case in which the \(\mathrm{Cr}^{3+}_B\) ions would have magnetic moments parallel to the magnetic moments of the ions in the \(A\) sublattice. As we have seen, it is highly probable that the \(\mathrm{Mn}^{2+}\) ions pass into the \(A\) sublattice when \(a\) increases, so that in this case the number of positive \(\mathrm{Fe}_A-\mathrm{Cr}_B\) interactions may rapidly decrease with increasing \(a\), i.e. the saturation magnetic moments will, as may be supposed, deviate from line IV as \(a\) increases.

Among the curves of the dependence of saturation magnetization on temperature there is none of the anomalous curves predicted by Néel: for \(a=1.25\), \(\sigma\) decreases with decreasing temperature, but this is probably only the result of the fact that a saturating field of magnitude 23,000 oersteds is already insufficient.

A change in the sign of \(m\) without the appearance of anomalous curves of the types shown in Fig. 8 was to be expected for a series of mixed crystals in which there are angles between the magnetic moments of ions in one of the sublattices (for example, \(B\)) at all temperatures (Section II.2.2).

It is possible that the present series is an example of such a case.

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