Neutron Diffraction Study of the Magnetic Structure of Antiferromagnets

R. P. Ozerov

The presence of a magnetic moment in the neutron makes the neutron diffraction study of the magnetic structure of substances whose atoms possess a permanent magnetic moment extremely fruitful. At present this method is the only direct method for studying magnetic microstructure, one that has confirmed and refined certain assumptions and hypotheses advanced on the basis of a large amount of indirect data.

1. Scattering of Neutrons by Para-, Ferro-, and Antiferromagnets

Usually, in presenting the theory of X-ray scattering in gases and liquids (see, for example, ¹), one first considers scattering by an atom and by a monatomic gas at low pressures, representing \(N\) independent scattering centers; then one passes to diatomic gases and intramolecular interference; next, with increasing order, one passes from gases to liquids and, finally, with the establishment of long-range order, to crystalline bodies. Approximately the same plan may also be followed in presenting the conclusions of the theory of magnetic scattering of neutrons in substances whose atoms possess a permanent magnetic moment, i.e., in para-, ferro-, and antiferromagnets.

As is known, in an ideal paramagnet the magnetic moments of atoms or ions are oriented completely at random. In the scattering of neutrons by such a substance, the resulting pattern is a superposition of \(N\) independent scattering events, as in the case of nuclear scattering of neutrons by a rarefied gas. If in the latter case the independence is due to the chaotic distribution of atoms in space, then in the former it is due to the chaotic orientation of the magnetic moments.

In contrast to the spherically symmetric nuclear scattering of thermal neutrons (the linear dimensions of the nucleus are 5

orders of magnitude smaller than the neutron wavelength), magnetic scattering is characterized by an angular dependence, since the electrons responsible for the magnetic moment of the atom are distributed in a volume whose linear dimension is comparable with the neutron wavelength.

This angular dependence of magnetic scattering can be characterized (by analogy with X-ray scattering) by the amplitude \(F_{\mathrm{M}}\), or by the factor \(f^2\) of magnetic scattering (the form factor). Theoretically, the amplitude of the magnetic scattering factor can be calculated from the formula known for X-rays

\[ f(\mathbf{k}\cdot \mathbf{s})=4\pi \int_{0}^{\infty}\rho(r)\,\frac{\sin ksr}{ksr}\,r^2\,dr, \tag{1} \]

where \(\rho(r)\) is the electron density at a distance \(r\) from the nucleus, and

\[ \mathbf{k}\cdot \mathbf{s}=4\pi \frac{\sin\theta}{\lambda}. \]

Conversely, from the experimentally found form factor one can determine the radial distribution of electrons in the atom. However, in doing this one must remember the following: when formula (1) is used for X-rays, the function \(\rho(r)\) describes the distribution of all the electrons of the atom, whereas in the case of neutrons \(\rho(r)\) must take into account the density of the electron cloud only in the \(3d\)-shell, since only \(3d\)-electrons are responsible for the presence of a magnetic moment in atoms of transition metals such as Mn, Fe, Ni, Co, Cr, and others. Consequently, having determined the form factor experimentally, one can calculate the distribution of electrons separately in the \(3d\)-shell.

It is obvious that, in magnetic scattering of neutrons in ideal paramagnets (analogously to the scattering of X-rays in gases at low pressures), the dependence of the intensity of the scattered radiation on the angle will correspond to the form factor. Starting from the simple dipole–dipole interaction of the neutron and atomic magnetic moments, a formula was derived for the differential cross section of magnetic scattering \(d\sigma_{\mathrm{M}}\) by an atom²:

\[ d\sigma_{\mathrm{M}}=\frac{2}{3}S(S+1)\left(\frac{e^2\gamma}{mc^2}\right)^2 f^2 d\Omega, \tag{2} \]

where \(S\) is the total spin of the atom, \(\gamma\) is the neutron magnetic moment in nuclear magnetons, \(f^2\) is the magnetic form factor; the remaining symbols have their usual meanings. It is clear from the formula that the entire angular dependence of \(d\sigma_{\mathrm{M}}\) is contained in the form factor.

It is well known that ferro- and antiferromagnets above their Curie points behave as typical paramagnets. However, a theoretical consideration of neutron diffraction in ferro- and antiferromagnets shows that these substances must give interference patterns that differ from scattering in ideal paramagnets. In the first approximation, the intensity of scat-

STUDY OF THE MAGNETIC STRUCTURE OF ANTIFERROMAGNETS

…the scattering of monochromatic neutrons \(k\) as a function of angle can be represented as the sum of two terms\(^3\):

\[ I(\theta)=I_0(\theta)+\frac{1}{T}I_1(\theta). \tag{3} \]

The first term depends only on \(\dfrac{\sin\theta}{\lambda}\), and characterizes scattering in

Fig. 1. Dependence of the intensity of neutron scattering on \(\dfrac{\sin\theta}{\lambda}\) for ferro- and antiferromagnets at high temperatures.

an ideal paramagnet and is therefore proportional to the form factor. The term \(I_1(\theta)\) is the expression

\[ I_1(\theta)=\frac{2}{3}S(S+1)\cdot I_0(\theta) \left\{ AZ_n\,\frac{\sin ksl_n}{ksl_n} + A'Z_{nn}\,\frac{\sin ksl_{nn}}{ksl_{nn}} \right\}, \tag{4} \]

where \(A\) and \(A'\) are quantities proportional to the exchange integral, \(k\) is the neutron wave number, \(s=2\sin\theta\), \(l_n\) and \(l_{nn}\) are the radii of the first and second coordination spheres, and \(Z_n\) and \(Z_{nn}\) are the numbers of atoms on these spheres.

It is seen that \(I_1(\theta)\) is a periodic function which is added to \(I_0(\theta)\) for ferromagnets or subtracted from \(I_0(\theta)\) for antiferromagnets (Fig. 1). Of particular interest is the second case, when the subtraction of \(I_1(\theta)\) from \(I_0(\theta)\) gives a broadened maximum at values

\[ ks=\frac{\pi}{l}. \]

The physical meaning of this phenomenon is that, at temperatures even considerably above the Curie points, there are regions in ferro- and antiferromagnets in which short-range order is preserved, while the substance as a whole exhibits paramagnetic properties.

An obvious analogy suggests itself between the picture described and the scattering of X-rays in liquids and in compressed gases. Indeed, all formulas describing the diffraction of X-rays and neutrons in gases and liquids—see formulas (1.48), (1.48′), (1.49), and (1.49′) of article\(^4\)—contain a term

\[ \frac{\sin ksl}{ksl}, \]

which causes a similar behavior of the curves \(I(\theta)\) in nuclear scattering of neutrons in gases and liquids and in magnetic scattering in ferro- and antiferromagnets at high temperatures. In the first case this term arises as a result of atomic short-range order, and in the second—as a result of magnetic short-range order.

As the temperature is lowered in ferro- and antiferromagnets, ever greater ordering occurs, with the establishment of long-range order below the Curie point. In this case the magnetic scattering of neutrons becomes almost entirely coherent, just as does nuclear scattering upon the transition of substances from the liquid to the crystalline state. Consequently, the neutronogram of a ferro- or antiferromagnetic substance at low (below the Curie point) temperatures, obtained with unpolarized neutrons, must contain components of magnetic and nuclear coherent scattering. The separation of these components, the need for which arises quite often, is carried out comparatively simply for antiferromagnets and is considerably more difficult for ferromagnets. The validity of this will become clear below.

For polarized neutrons such a conclusion is incorrect. In this case it is necessary to take into account the interference between nuclear and magnetic scattering. It was shown that the differential cross section of scattering by a magnetic ion \(d\sigma\) is equal to:

\[ d\sigma = F_{\mathrm{я}}^{2} + 2F_{\mathrm{я}}\cdot F_{\mathrm{м}}\cdot \lambda q + q^{2}F_{\mathrm{м}}^{2}, \tag{5} \]

where \(F_{\mathrm{я}}\) is the amplitude of nuclear scattering, and \(F_{\mathrm{м}}\) is the amplitude of magnetic scattering, \(\lambda\) is the unit vector describing the polarization of the neutron wave,

\[ \mathbf{q}=\mathbf{e}\cdot(\mathbf{e}\cdot\mathbf{x})-\mathbf{x} \tag{6} \]

and

\[ q^{2}=1-(\mathbf{e}\cdot\mathbf{x})^{2}; \tag{7} \]

STUDY OF THE MAGNETIC STRUCTURE OF ANTIFERROMAGNETS

e — the unit scattering vector, equal to

\[ \mathbf{e}=\frac{(\mathbf{k}-\mathbf{k}')}{|\mathbf{k}-\mathbf{k}'|}, \tag{8} \]

\(\mathbf{k}\) and \(\mathbf{k}'\) are the wave vectors of the incident and reflected neutron waves, and \(\boldsymbol{\chi}\) is the unit vector of the direction of orientation of the magnetic moments of the atoms. For unpolarized neutrons the averaged product \(\boldsymbol{\lambda}\cdot\mathbf{q}\) is equal to zero and the interference term vanishes; magnetic and nuclear scattering become additive. Equation (5) then takes the form:

\[ d\sigma=F_{\text{n}}^{2}+q^{2}F_{\text{m}}^{2}. \tag{9} \]

It should be noted that the amplitude of nuclear scattering \(F_{\text{n}}\) at present cannot be calculated theoretically because of insufficient knowledge of the nature of nuclear forces; however, it has been determined experimentally for many nuclei (see Table VII\(^4\)). The magnitude of the magnetic-scattering amplitude \(F_{\text{m}}\), as already noted above, can be determined theoretically from the formula

\[ F_{\text{m}}=\frac{e^{2}\gamma}{mc^{2}}\,S\cdot f. \tag{10} \]

From (7) it is seen that the quantity \(q^{2}\), which depends on the mutual orientation of the scattering vector and the vector characterizing the direction of the magnetic moments in the lattice, can take various values from 0 to 1. For a random arrangement of elementary moments \(q_{\text{random}}^{2}=\frac{2}{3}\). The described dependence of \(q^{2}\) provides a means for direct experimental determination of the magnetic structure of crystalline substances.

2. DETERMINATION OF THE FORM FACTOR

In order to be able to estimate the intensity of coherent magnetic scattering of neutrons, it is first necessary to determine the form factor. For this purpose, according to the foregoing, it is necessary to measure the magnetic scattering of neutrons by an ideal paramagnet. Measurements were carried out with salts of the divalent manganese ion\(^{5,6}\): MnO and MnF\(_2\).

The diffuse magnetic scattering by these substances after subtraction of coherent nuclear scattering, spin incoherence, thermal diffuse and multiple scattering is presented in Fig. 2 as the dependence of the differential cross section of magnetic scattering on angle. The magnetic scattering of MnF\(_2\) decreases regularly with increasing angle, as required by the form factor, whereas for MnO the presence of a maximum is clearly seen.

Thus, in accordance with what was set forth above, in antiferromagnetic MnO at room temperature (i.e., 180° above the Curie point) there exists short-range order in the orientation of the magnetic moments of the Mn ions—a fact of exceptional importance for the interpretation of magnetic transformations.

Unlike MnO, the scattering in \(\mathrm{MnF}_2\) shows no coherence. To verify this, a \(\mathrm{MnF}_2\) specimen was heated to a temperature of \(400^\circ\mathrm{C}\) (which should have affected the second term of equation (3)), but no changes in the magnetic scattering were detected. This indicates the absence

Fig. 2. Dependence of the differential cross section of magnetic scattering on the scattering angle for \(\mathrm{MnF}_2\) and MnO.

of magnetic ordering in \(\mathrm{MnF}_2\) and shows that the dependence of the scattering on the angle in this case represents the pure magnetic form factor of the \(\mathrm{Mn}^{++}\) ion. Measurement of the differential cross section in the forward direction gives a value 10% smaller than that calculated from formula (2) with \(S = 5/2\). Such a discrepancy may be explained by the difficulty of measurements at small scattering angles.

Figure 3 presents the amplitude of the magnetic scattering factor of the divalent manganese ion, \(\mathrm{Mn}^{++}\). This curve was used to estimate the intensity of the magnetic scattering of MnO, and it was also used in the study of compounds of other transition-metal ions: NiO, CoO, FeO, \(\mathrm{Fe}_2\mathrm{O}_3\). The latter is not entirely justified, since, unlike \(\mathrm{Mn}^{++}\), which has \(L = 0\) and is in an \(S\)-state, the named ions have \(L > 0\) and are in \(F\)-(\(\mathrm{Co}^{++}\) and \(\mathrm{Ni}^{++}\)) and \(D\)-(\(\mathrm{Fe}^{++}\)) states. The latter circumstance entails an asymmetry of the ion, and the form factor for them will be not only a function of \(\frac{\sin \theta}{\lambda}\), but will also depend on the orientation of the ion’s asymmetry with respect to the scattering atomic plane. However, the error caused by this is the smaller, the smaller the difference in \(Z\), and, moreover,

is extremely small for small \(\dfrac{\sin\theta}{\lambda}\). Experimental data have shown that the use of the form factor of \(\mathrm{Mn}^{++}\) for the ions \(\mathrm{Ni}^{++}\), \(\mathrm{Co}^{++}\), \(\mathrm{Fe}^{++}\), and \(\mathrm{Fe}^{+++}\) is legitimate, at least in first approximation.

Starting from the experimentally determined form factor, according to equation (1) (preliminarily transformed with respect to \(\rho(r)\) and \(f(k\cdot S)\)), the electron distribution in the \(3d\)-shell of the \(\mathrm{Mn}^{++}\) ion was found. The result differs somewhat from the theoretical calculation, evidently because the experiment gives the electron distribution in an ion of the crystal lattice, whereas the calculation was made for a free ion.

Fig. 3. Amplitude of the magnetic scattering factor of the \(\mathrm{Mn}^{++}\) ion. The dashed curve is the amplitude of the x-ray atomic factor of the Mn atom.

3. MAGNETIC STRUCTURE OF MnO, NiO, CoO, and FeO

The possibility of determining the magnetic structure of antiferromagnets is based on their already known crystal structure. Otherwise, in the present state of neutronography, such a problem would be insoluble.

The determination of the magnetic structure of antiferromagnets is divided mainly into two stages: finding, first, the mutual orientation of the magnetic moments relative to one another and, second, the orientation of the magnetic ordering relative to the crystal lattice. From this point of view the indicated oxides are also considered.

All of them are antiferromagnets. This is indicated by the anomalous behavior of certain magnetic and thermal properties

of these compounds. The magnetic susceptibility of MnO, for example, increases with decreasing temperature according to the Curie–Weiss law down to 120°K; below this temperature the magnetic susceptibility changes its course and begins to decrease^7. L. D. Landau, proceeding from the antiferromagnetic interaction, showed^8 that for these substances precisely such behavior of the magnetic susceptibility should be expected.

Table I

Curie temperatures of certain substances

Table I gives data on the temperatures of the magnetic transformations of all the oxides mentioned above and, in addition, of Cr₂O₃. The figures in the second column were obtained on the basis of thermal or magnetic measurements. The data in the third column will be explained below.

In view of the fact that for unpolarized neutrons magnetic and nuclear scattering are additive (equation (9)), neutron diagrams of antiferromagnets obtained at temperatures below the Curie point must contain maxima of coherent magnetic and nuclear scattering. Figures 4, 5, 6, and 7 show neutron diagrams of all the indicated oxides, obtained at temperatures

Fig. 4. Neutron diagrams of MnO.

above and below the Curie point. The structure of these oxides is isomorphous with the rock-salt structure; therefore, on neutron diffraction patterns obtained at temperatures above the Curie point there must be present

Fig. 5. Neutron diffraction patterns of FeO.

Fig. 6. Neutron diffraction patterns of CoO.

maxima of coherent nuclear scattering with indices of the same parity. At low sample temperature \((80^\circ\mathrm{K})\), in the neutron diffraction patterns, in addition to the nuclear maxima there arise maxima of coherent magnetic scattering, which

can be indexed only by taking the period of the magnetic lattice to be twice that of the chemical one. Because of the presence of the form factor, their intensity decreases strongly with increasing reflection angle.

Because Mn and O have nuclear scattering amplitudes of opposite signs, the (200) maximum on the MnO neutronogram is weaker than (111). In addition, the magnetic maximum (311) is superposed on the nuclear (111), which does not make it possible to measure the intensity of the former with sufficient accuracy.

On the FeO neutronograms the magnetic maximum (111) is almost completely absent; this will subsequently permit a definite conclusion to be drawn concerning the magnetic structure of this compound.

Fig. 7. Neutronogram of NiO.

The presence of the magnetic maximum (111) on the CoO neutronogram obtained at room temperature indicates considerable magnetic ordering in this compound, although this temperature is above the Curie point.

The Curie point of NiO lies far above room temperature, and therefore in Fig. 7 only one neutronogram is given, corresponding to the antiferromagnetic state of nickel oxide. Individual nickel isotopes were also investigated[^9]: because \( \mathrm{Ni}^{58} \), \( \mathrm{Ni}^{60} \), and \( \mathrm{Ni}^{62} \) have different nuclear-scattering amplitudes, the intensity of the nuclear maxima changed sharply, whereas the magnetic maxima remained unchanged for all isotopes.

The fact that for all the oxides now under discussion the parameter of the magnetic cell is twice the parameter of the atomic cell shows that in the lattice neighboring magnetic moments are oriented antiparallel to one another. On the other hand, it is known that a cubic face-centered lattice may be regarded as four simple cubic lattices (as

this is done, for example, when considering atomic ordering \(^{10}\). Therefore such an antiferromagnetic lattice may be regarded as four independent simple lattices inserted into one another and shifted relative to one another by half the face diagonal (Fig. 8, a). If these lattices are identically oriented with respect to the crystal lattice, then the picture shown in Fig. 8, b is obtained. Let us note at once that the choice between these two models cannot be made by neutronography. However, model a cannot explain the experimentally observed distortions of the cubic symmetry of the structure of antiferromagnets, which arise when their temperature changes with passage through the Curie point. (This will be discussed in more detail below.) This makes it possible to choose between the two possibilities in favor of model b.

Fig. 8. Two variants of the magnetic structure of MnO.

The determination of the orientation of magnetic ordering with respect to the crystal lattice can be made as a result of the fact that the quantity \(q^2\) (equation (7)) depends on the mutual orientation of the scattering vector \(\mathbf{e}\), parallel to the normal to the reflecting plane, and the vector characterizing the direction of the magnetic moments \(\boldsymbol{\chi}\).

For the model shown in Fig. 8, b, there are three most probable cases of orientation of the magnetic ordering with respect to the axes of the lattice:

A. The magnetic moments are oriented along the cubic axis.

B. The magnetic moments are oriented perpendicular to the plane \((111)\). For this case \(q^2\) for the maximum \((111)\) is equal to 0 because of the collinearity of the vectors \(\mathbf{e}\) and \(\boldsymbol{\chi}\).

C. The magnetic moments are situated in the plane \((111)\). For this case, on the contrary, \(q^2_{(111)} = 1\), since the vectors \(\mathbf{e}\) and \(\boldsymbol{\chi}\) are perpendicular to one another.

R. P. OZEROV

In Table II the experimental data on the intensity of the maxima of the magnetic scattering of MnO are compared with the values calculated for these three cases.

Table II

Comparison of the experimental intensity of the magnetic maxima of the neutron diffraction pattern of MnO with that calculated for different orientations of the magnetic moments

This comparison shows that the magnetic moments in the lattice of manganese oxide are directed along the cubic axis \([100]\) and that, consequently, the magnetic structure of MnO corresponds to the structure shown in Fig. 8,b. Comparison of the neutron diffraction patterns of the other oxides with the neutron diffraction pattern of MnO shows that CoO and NiO are magnetically isomorphous with MnO. A neutron diffraction study of MnS and MnSe showed that these compounds also possess the same magnetic structure. In the neutron diffraction patterns of FeO the magnetic maximum \((111)\) is absent. This corresponds to case B, since only for this orientation of the moments \(q_{(111)}^{2}=0\). Consequently, the magnetic moments of the ions in FeO are oriented perpendicular to the \((111)\) plane.

In Fig. 8,b it is seen that in the structure shown there are planes in which all magnetic moments are oriented parallel, and planes with antiparallel orientation of the moments, i.e., ferromagnetic and antiferromagnetic planes. This leads to the idea that all metamagnetics should also be assigned to the class of antiferromagnetics, since there is no need to distinguish these two classes of magnetic substances.

For comparison of the magnetic properties of the ions Mn++, Fe++, Co++ and Ni++, the neutron diffraction data were recalculated into the differential cross section of magnetic scattering in order to exclude all other factors (multiplicity factor, sample density, etc.). The results are plotted in Fig. 9.

From certain effects (for example, from measurement of the gyromagnetic ratio) it followed that the magnetic moment of the atoms of some ferromagnetics is not entirely spin in origin, but consists of a spin moment with some admixture of orbital moment (see, for example, 11)). By neutron diffraction it proved possible to check this assumption.

In Fig. 9, together with the experimental values, are presented the calculated values of the differential cross sections of magnetic scattering as a function of angle, the solid cur-

…correspond only to the spin moment, while the dashed curves correspond to the sum of the spin and orbital moments. For the \( \mathrm{Mn}^{++} \) ion, which is in the \(S\)-state, it has been established spectroscopically that its magnetic moment is due only to the spins of the five \(3d\)-electrons. Neutronographic data confirm this well. For the more

Fig. 9. Dependence of the transverse magnetic-scattering cross section of various ions on \(\dfrac{\sin\theta}{\lambda}\). Also shown is the result of a theoretical calculation for the purely spin magnetic moment (solid curve) and the total—spin and orbital—moment (dashed curve).

complex cases of \( \mathrm{Fe}^{++} \), \( \mathrm{Co}^{++} \), and \( \mathrm{Ni}^{++} \) ions, neutronography gives: a completely spin moment for nickel ions, a certain admixture of orbital moment for iron ions*) and a significant component of orbital moment for cobalt ions.

4. TEMPERATURE DEPENDENCE OF ORDERING

As is known, any ordering, whether atomic or magnetic, is characterized by a quantity \(Q\), which is defined as follows:

\[ Q=\frac{r-w}{r+w}, \tag{11} \]

where the quantity \(r\) is the probability of finding correctly positioned ions at a given site, and \(w\) is that of incorrectly positioned ions. At absolute zero, the state with the lowest energy, i.e. the equilibrium state, will be complete ordering (the ideal antiferromagnetic state), for which \(r=1\), and \(w=0\), and, consequently, \(Q=1\). With increasing temperature \(Q\) at first changes little; upon approaching the phase-transition point \(T_k\), \(Q\) sharply

*) The loss of the point corresponding to the maximum (111) of FeO is explained by the orientation effect and should not be taken into account here.

decreases, becoming equal to zero at the Curie point and approaching the abscissa axis at a right angle. If, however, during the transition some ordering—short-range order—is retained, then the curve \(Q(T)\) does not intersect the abscissa axis at the Curie point, but changes slope and approaches the axis at a small angle at a higher temperature (Fig. 10).

The question of the behavior of the function \(Q(T)\) near the temperature axis long remained open, in particular also for magnetic ordering. Later, on the basis of a large number of thermal and electrical data (magnetic measurements for ferromagnets could not be used, since spontaneous ordering is being investigated), it was concluded\(^{12}\) that magnetic short-range order is preserved at temperatures approximately \(50^\circ\) above the Curie point.

Fig. 10. Dependence of the ordering coefficient \(Q\) on temperature.

Neutron diffraction provides a simple and at the same time direct and convincing method for verifying this conclusion. An example of maxima of coherent magnetic scattering, whose intensity is proportional to \(Q\), permits the direct construction of a graph of this function. Even a simple inspection of neutron diffraction patterns of CoO and especially MnO indicates that magnetic short-range order is preserved at temperatures more than \(100^\circ\) above the Curie point.

Figure 10 gives the result of such measurements, and the bend of \(Q(T)\) at the transition temperature is quite evident. The Curie point itself must be defined as the place of the steepest decline of this function.

From the width of the maximum (111) of the neutron diffraction patterns of MnO at the transition temperature, one can determine the size of the regions in which short-range order is preserved. The calculation gives a value of approximately \(50\ \text{\AA}\).

5. DEFORMATION OF THE STRUCTURE OF ANTIFERROMAGNETS DURING THEIR TRANSITION TO THE PARAMAGNETIC STATE

B. Rügemann established\(^{13}\) that when the temperature is lowered below the Curie point, certain changes occur in the structure of MnO, expressed in the broadening of definite lines of the X-ray diffraction patterns. Later works\(^{14,15,16,17}\) showed that cu-

...the face-centered cubic lattice, upon transition to the antiferromagnetic state, is somewhat deformed, and for different substances these changes proved to be different. In other words, the symmetry of the lattices of the oxides of manganese, nickel, iron, and cobalt changes in different ways at the magnetic transition.

Upon cooling NiO to the temperature of liquid nitrogen, it was found\(^{16}\) that its lattice changes from cubic to slightly rhombohedral. This means that the lengths of the different body diagonals become somewhat different. The model of the magnetic structure established above (Fig. 8, b) corresponds precisely to such a change in lattice symmetry, since it is quite natural to expect that the bonding forces behave differently in the ferromagnetic and antiferromagnetic planes. With the described deformation of the NiO lattice, the angle \(\alpha\) of the rhombohedral lattice becomes somewhat larger than \(60^\circ\). (The equivalent angle of the undeformed lattice is \(60^\circ\).) MnO behaves in exactly the same way.

The lattice of iron oxide, upon cooling below \(203^\circ\) K, undergoes approximately the same change, with the sole exception that the angle \(\alpha\) becomes smaller than \(60^\circ\).

In contrast to the oxides mentioned above, the CoO lattice, upon cooling below the Curie point, transforms from cubic to tetragonal\(^{15}\). Precision measurements of the lattice constants of CoO at a temperature of \(90^\circ\) K gave the following values: \(a = 4.2552 \pm 0.0005\), \(c = 4.2058 \pm 0.0005\) kX, and \(c/a = 0.988\). As the temperature is raised to \(203^\circ\) K, the ratio \(c/a\) increases to 0.995, and at room temperature all deviations from cubic symmetry disappear.

Thus, according to the change in the symmetry of the crystal lattice, the four antiferromagnets described fall into three groups, characterized by: 1) deformation of the cubic lattice into a rhombohedral one with an angle \(\alpha\) somewhat greater than \(60^\circ\) (MnO and NiO; MnS\(^{18}\) also belongs to this same group), 2) the same deformation, but with an angle less than \(60^\circ\) (FeO), and 3) transformation of the cubic lattice into a tetragonal one (CoO). It is noteworthy that the magnetic properties of these lattices stand in the same relation: the analogous magnetic structure of MnO and NiO with purely spin magnetic moments, the significant contribution of the orbital moment to the magnetic moment of the \(\mathrm{Co}^{++}\) ion, and the anomalous magnetic structure of FeO. However, a theoretical explanation and a connection between the effects described have still not been obtained.

For the same purpose, the antiferromagnetic \(\mathrm{Cr_2O_3}^{18}\) and \(\alpha\)-\(\mathrm{Fe_2O_3}^{19}\) were investigated by X-ray methods. Both these substances possess the same corundum-type lattice. The investigations showed that in \(\mathrm{Cr_2O_3}\) there occurs a compression of its lattice along the \([111]\) axis, which becomes noticeable at a temperature of \(318^\circ\) K and appears more sharply at \(307^\circ\) K. On the other hand, for the isomorphous

$\alpha$-Fe$_2$O$_3$, in which a magnetic transformation was found at $-20^\circ$C, showed no changes whatever by X-ray diffraction. We shall dwell in more detail on these two structures in the following section.

On the basis of all the facts set forth, it can now be said with confidence that structural changes of the type described, when the temperature is lowered below the Curie point, are an inalienable property of antiferromagnetic substances. The data given in Table I (third column) concerning the Curie points for various antiferromagnetic substances, obtained on the basis of X-ray investigations, do not in any case contradict thermal and magnetic measurements.

It should be noted that deformation of the crystal lattice of alloys upon their atomic ordering had also been discovered earlier[^20]. This only once more emphasizes the analogy between atomic and magnetic superstructures.

6. THE MAGNETIC STRUCTURE OF HEMATITE ($\alpha$-Fe$_2$O$_3$)

In the main, the following was known about the magnetic properties of hematite. Hematite is antiferromagnetic below 250°K; above this

Fig. 11. Neutronogram of $\alpha$-Fe$_2$O$_3$. For comparison, an X-ray diffraction pattern is also shown.

point and up to 950°K various investigators attributed to hematite various magnetic properties (weakly ferromagnetic or paramagnetic); these properties changed at a temperature of 950°K,

which was also regarded as its Curie point. However, this temperature is extremely close to the Curie point of magnetite, and therefore some investigators doubted its reality.

The neutron diffraction patterns of hematite obtained over a wide temperature interval (80–1000° K) indicate the existence of an antiferromagnetic lattice throughout the entire temperature range, but with some difference in the details of the structure. In Fig. 11 a neutron diffraction pattern of \(\alpha\)-Fe\(_2\)O\(_3\) is presented and compared with an X-ray diffraction pattern. All maxima can be indexed on the basis of the atomic unit cell \((a = 5.42\ \text{Å},\ \alpha = 55^\circ 17')\), with two molecules in the cell. Consequently, the dimensions of the magnetic and atomic unit cells are the same.

Fig. 12. Neutron diffraction patterns of \(\alpha\)-Fe\(_2\)O\(_3\) at temperatures 80° and 293° K.

However, if it is assumed that all Fe atoms are identical, the (111) and (100) maxima should not be present on the neutron diffraction pattern (compare with the X-ray diffraction pattern). On the other hand, they can differ only in the orientation of their magnetic moments, which also indicates the presence of antiferromagnetic properties in this substance. Therefore, in order to determine the magnetic transformations in hematite it is necessary to investigate the temperature dependence precisely of these magnetic maxima.

Investigation at high temperatures (up to 1000° K) showed that the magnetic maxima (111) and (100), decreasing considerably in magnitude, continue to remain at all these temperatures. This indicates the absence of a Curie point at 950° K, the existence of which had earlier been regarded as doubtful.

Studies at low temperatures, on the contrary, showed that at 250° K a certain change in the magnetic structure occurs, although antiferromagnetism is preserved both above and below this point. In Fig. 12 neutron diffraction patterns obtained at

300 and 80°K. Against the background of the constant nuclear maximum (110), the disappearance of one of the magnetic maxima—(111)—and an increase of the other—(100)—are clearly visible. Such a change in the intensity of the magnetic maxima indicates a reorientation of the magnetic moments, with antiferromagnetism preserved throughout the entire temperature range.

To clarify the details of the magnetic structure, it is necessary, as in the case of MnO, to compare various assumptions with the experimental intensity of the maxima.

○ iron ions
(oxygen ions are located at the lattice nodes)

Fig. 13. Unit cell of α-Fe₂O₃.

As it turned out, since the dimensions of the magnetic and atomic cells coincide, there are four inequivalent positions of iron atoms in the cell, denoted in Fig. 13 by the letters A, Б, В, and Г. On the other hand, for the mutual antiferromagnetic orientation of the moments, three models are possible: model (a) \(+--+\), model (б) \(++--\), and model (в) \(+-+-\). Here model (a) indicates, for example, that the moments of atoms A and Г are directed to one side, while the moments of atoms В and Б are directed to the other. Let us note at once that the structural factor of model (в) for the maxima (111) and (100) that disappear is small and, consequently, this model is excluded from consideration.

As in the case of MnO, the second question in determining the magnetic structure is the determination of the direction of orientation of the magnetic moments relative to the crystal lattice. In this respect there are, essentially, three most reasonable possibilities:

A. The magnetic moments are parallel to the edges of the unit cell.

Б. The magnetic moments are oriented along the body diagonal of the unit cell and, consequently, are perpendicular to the planes (111).

В. The magnetic moments are located in the planes (111), that is, perpendicular to the body diagonal and directed toward one of the three nearest neighbors in the plane.

For all these six cases, the intensities of the two magnetic maxima (111) and (100) were theoretically calculated in the form of the differential cross section of magnetic scattering per Fe₂O₃ molecule. Table III gives a comparison of the calculated values with the experimental ones. It is seen that at room tempe-

temperature the data for model (a) with orientation B give the best agreement with experiment, whereas at low temperatures the same model corresponds, but with orientation Б. Thus, the low-temperature magnetic transformation consists in a reorientation of the magnetic moments from the (111) planes to a direction perpendicular to these planes as the temperature is lowered.

Table III

Comparison of the experimental intensities of magnetic maxima of the neutronogram of \(\alpha\)-Fe\(_2\)O\(_3\) with those calculated for various models
\((d\sigma\) in \(10^{-24}\ \mathrm{cm}^2\) per molecule)

The magnetic lattice of \(\alpha\)-Fe\(_2\)O\(_3\) possesses certain properties inherent also in the MnO lattice. If one imagines the cells adjacent to that shown in Fig. 13, it is seen that the structure consists of a series of (111) planes, within which all moments are oriented parallel to one another (ferromagnetic planes), but with antiparallel orientation to the moments in the neighboring plane. It is interesting that between planes containing only iron atoms there lie planes containing only oxygen atoms. Such an alternation of ferromagnetic planes with nonmagnetic ones was also observed for cubic oxides.

If it is assumed that the magnetic structure of Cr\(_2\)O\(_3\) is identical to the structure of \(\alpha\)-Fe\(_2\)O\(_3\) (the magnetic moments are oriented perpendicular to the body diagonal), the deformation of the Cr\(_2\)O\(_3\) lattice described above becomes readily understandable. On the other hand, no changes in the \(\alpha\)-Fe\(_2\)O\(_3\) lattice were detected by X-ray diffraction when its temperature was changed in the region of \(-20^\circ\)C, although everything suggests that they should be expected. Possibly more precise experiments are required in order to detect these distortions.

7. MAGNETIC STRUCTURE OF MAGNETITE

Magnetite—Fe\(_3\)O\(_4\)—is a ferrimagnet. Its structure belongs to the spinel type. In the Fe\(_3\)O\(_4\) molecule there are one divalent and two trivalent iron ions; the molecular formula of magnetite may be represented in the form:

\[ \mathrm{Fe}_3\mathrm{O}_4=\mathrm{Fe}^{++}\mathrm{Fe}_2^{+++}\mathrm{O}_4. \]

The atomic structure of magnetite (Fig. 14) is a doubled face-centered cubic lattice composed of oxygen atoms; it contains, per 32 oxygen atoms, 24 iron atoms located in 16 octahedral and 8 tetrahedral interstices (see, for example, \(^{21}\)).

Figure labels: tetrahedral ions; octahedral ions; oxygen ions.

Fig. 14. Two octants of the unit cell of \(\mathrm{Fe}_3\mathrm{O}_4\).

To explain the ferromagnetic properties of magnetite, one hypothesis assumed \(^{22,23}\) that part of the trivalent iron ions occupy the tetrahedral interstices of the lattice formed by the oxygen atoms, while the remaining part of the trivalent ions, together with the divalent ions, occupy the octahedral interstices in a disordered manner (inverse spinel structure). All magnetic moments of the ions occupying tetrahedral interstices are oriented ferromagnetically. The octahedral ions are oriented in the same way. But relative to one another these two lattices are oriented antiferromagnetically. The observed ferromagnetic properties are due to the fact that there are twice as many octahedral ions in the magnetite lattice as tetrahedral ones.

Alongside this, several other hypotheses were also put forward concerning the magnetic structure of magnetite. Neutronographic investigation made it possible to decide which of them corresponds to reality.

Figure labels: X-rays; \(\mathrm{Fe}_3\mathrm{O}_4\); neutrons; intensity; \((111)\), \((220)\), \((311)\), \((222)\), \((400)\), \((331)\); \(\theta\).

Fig. 15. Neutronogram of \(\mathrm{Fe}_3\mathrm{O}_4\). For comparison, the X-ray diagram is also shown.

Figure 15 shows a neutronogram of magnetite obtained at room temperature \(^{24}\). Since nuclear and magnetic scat-

scattering are incoherent with one another, the latter can be determined by subtracting the nuclear-scattering neutronogram. Since the amplitudes of coherent nuclear scattering by iron and oxygen are known, and since the structure of magnetite is known from X-ray data, the nuclear component of the neutronogram can be readily calculated. The remaining part can be attributed to magnetic scattering and compared with the calculation for the magnetic structure described.

Such a comparison between the experimental and theoretical intensities of the maxima for the described model is given in Table IV; the same table also gives the structure factors for various maxima.

Table IV

Comparison of the experimental intensity of the maxima of the Fe\(_3\)O\(_4\) neutronogram with that calculated for the model described in the text (neutr./min.)

\(f_T\), \(f_O\), and \(f_K\) are the scattering amplitudes of the tetrahedral, octahedral, and oxygen ions.

various maxima. In the calculations the values of the nuclear-scattering amplitudes of iron and oxygen, \(0.956\) and \(0.575 \cdot 10^{-12}\) cm, respectively, and the magnetic form factor of the divalent manganese ion were used; the amplitude of magnetic scattering was determined from formula (10) with \(S = \frac{5}{2}\) and \(2\) for the ions Fe\(+++\) and Fe\(++\), respectively. The quite satisfactory agreement between calculation and experiment confirms the model of the magnetic structure of magnetite described above. On the other hand, the considerable

the discrepancy between the results of calculations carried out for other models (for example, for the normal spinel structure) and experiment makes it possible to regard the remaining hypotheses as not corresponding to reality.

It is also known that magnetite exhibits certain interesting changes in its magnetic and electrical properties when the temperature is lowered below 120–150° K. These phenomena have been explained by the ordering of Fe+++ and Fe++ ions in octahedral positions^25. Neutron-diffraction analysis did not reveal changes in the structure of Fe₃O₄ upon cooling. However, this result is consistent with the assumption of ordering, which is very difficult to detect neutronographically in Fe₃O₄, since the scattering power of divalent and trivalent iron ions is almost identical (just as it is difficult to detect, by X-ray methods, ordering in alloys consisting of elements with close atomic numbers).

8. EXPERIMENTS WITH MAGNETIZED SPECIMENS

The magnetic component of neutron scattering, according to equation (9), depends on the mutual orientation of the scattering vector \((\mathbf{e})\) (8) and the vector of the ionic magnetic moment \((\boldsymbol{\chi})\). In ferromagnetic specimens the direction of the latter can be controlled by applying a sufficiently large external field. In this case the magnetic scattering will depend essentially on the magnitude and direction of the external magnetic field. By comparing the intensity of neutron scattering in fields differing in magnitude and direction with theoretical calculations, it is possible to determine the magnetic component of the scattering most accurately. This is extremely important, if only for determining the amplitudes of nuclear scattering by magnetic atoms.

For the experimental investigation of neutron scattering in magnetized specimens on an ordinary neutron spectrograph, an electromagnet was mounted that was capable of producing, in the gap, a field with an intensity of 8000 oersteds^24. For observing neutron magnetic diffraction the following two orientations of the magnetic field are of greatest interest: 1) the field perpendicular to the scattering vector \((\mathbf{e}\perp\boldsymbol{\chi})\); for such an orientation \(q_\perp^2=1\), and 2) the field parallel to the scattering vector \((\mathbf{e}\parallel\boldsymbol{\chi})\), \(q_\parallel^2=0\). To these measurements should be added the results obtained in the absence of a field \(\left(q_0^2=\frac{2}{3}\right)\). On the other hand, it follows from equation (9) that, for the study of magnetic effects, one should choose, as far as possible, those maxima which have the largest magnetic and the smallest nuclear components. From Tables IV it is seen that such is the (111) maximum of magnetite, which was chosen for the experiment.

In Fig. 16 this maximum is shown, obtained for various directions of the magnetic field and without it. It is seen that the intensity of the maximum changes sharply in this case.

Fig. 16. The (111) maximum of the neutronogram of Fe$_3$O$_4$, obtained for different orientations of the magnetic field and without it.

In Fig. 17 the data obtained by varying the field strength for two different orientations are summed. The magnitude of the total differential cross section in the absence of a field is equal to the sum $F_{\mathrm{n}}^2 + \frac{2}{3}F_{\mathrm{m}}^2$. As the strength of the field applied parallel to the scattering vector is increased, the quantity $q_{\parallel}^{2}$ tends to 0, and the cross section becomes purely nuclear. If, however, the field is applied perpendicular to the scattering vector, $q_{\perp}^{2}$ tends to 1, and the cross section becomes the sum of the nuclear and magnetic cross sections. The magnitude of nuclear scattering ($F_{\mathrm{n}}$) is much smaller than that of magnetic scattering ($F_{\mathrm{m}}$) for the (111) maximum of Fe$_3$O$_4$, which is in good agreement with the calculation of the structure factor for this reflection. In Fig. 17 the saturation effect is clearly seen, characterizing the limiting orientation of the moments in strong fields.

By means of the experiments described, it proved possible to verify the correctness of one of the two theories that operate with the vector $\mathbf q$.

Fig. 17. Dependence of neutron scattering on the magnitude and direction of the magnetic field (maximum (111) $\mathrm{Fe_3O_4}$).

Fig. 18. Dependence of the quantity $q^2$ on $\alpha$.

According to one of them[^26], the numerical value of the square of this vector is equal to:

$$ q^2 = 1 - \cos^2 \alpha = \sin^2 \alpha, \tag{12} $$

according to the other[^27]

$$ q^2 = \cos^2 \alpha. \tag{13} $$

For the check, the same maximum was used. After all the introduced corrections, the value of the differential transverse cross section was recalculated in terms of \(q^2\) for four angles \(\alpha\): \(0^\circ\), \(23^\circ\), \(54^\circ\), and \(90^\circ\).

These values of \(q^2\), normalized by means of the value \(q_{\text{nonmag}}^2 = \dfrac{2}{3}\) for a nonmagnetic specimen, are shown in Fig. 18 together with two theoretical curves corresponding to equations (12) and (13). It is evident that the experiment confirms quite satisfactorily the theory based on the quantity \(q^2 = \sin^2 \alpha\).

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