Cullity B.D. Introduction to Magnetic Materials. Second Edition (2008)

hence the susceptibility:

(MA þ MB)T ¼ 2rC0H rC0g (MA þ MB), MT ¼ 2rC0H rC0gM,

This relation is equivalent to Equation 5.2, found experimentally, with

Note that, when a field is applied above TN, each sublattice becomes magnetized in the same direction as the field, but each sublattice then sets up a molecular field, in the opposite direction to the applied field, tending to reduce both MA and MB . The result is that the susceptibility x is smaller, and 1/x larger, than that of an ideal paramagnetic in which the molecular field is zero. The two are compared graphically in Fig. 5.3, which also shows how x varies with T in a substance with a large positive molecular field, such as a ferromagnetic above its Curie point.

Fig. 5.3 Dependence of the susceptibility x on the molecular field Hm for a fixed value of the Curie constant C.

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5.2.2Below TN

In the antiferromagnetic region, each sublattice is spontaneously magnetized, in zero applied field, by the molecular field created by the other sublattice. When H is zero,

M ¼ MA þ MB ¼ 0,

and

at any temperature below TN. At a temperature infinitesimally below TN we may assume that M is still proportional to the total field, because saturation effects are unimportant near TN. Then Equations 5.7 and 5.8 are still valid. At T ¼ TN and H ¼ 0, Equation 5.7 becomes

The Ne´el temperature, at which the maximum in the x, T curve occurs, should therefore equal the u value found from the high-temperature susceptibility measurements.

Below TN, each sublattice is spontaneously magnetized to saturation just as a ferromagnetic is, and we can compute its magnetization in the same way. As in Section 4.2, we prefer to consider the specific magnetization s (¼M/r) rather than M, because a range of temperature is involved. The fractional specific magnetization of the A sublattice will then be given, according to Equation 4.13, for any temperature and field by

where B is the Brillouin function. The field H which appears in Equation 5.14 is the total field acting on the A sublattice. Since we are computing the spontaneous magnetization, the applied field is zero, and we include only the molecular field due to the B sublattice:

Therefore, the fractional spontaneous magnetization of the A sublattice is given by

with a similar expression for the B sublattice. This equation can be solved graphically by the method of Fig. 4.4. A plot of the two sublattice magnetizations is given in Fig. 5.4.

Although the net spontaneous magnetization is zero below TN, an applied field can produce a small magnetization. The resulting susceptibility is found to depend on the angle which the applied field makes with the axis of antiparallelism marked D in

Fig. 5.4 Spontaneous magnetizations of the A and B sublattices at temperatures below TN (schematic).

Fig. 5.2, an axis which usually coincides with an important crystallographic direction in the crystal. For brevity we will call this the spin axis. (In most antiferromagnetics, the orbital contribution is almost entirely quenched, so that the net moment per magnetic ion is due essentially to spin.) We will consider two limiting cases.

Field at Right Angles to Spin Axis. The effect of the applied field H is to turn each sublattice magnetization away from the spin axis by a small angle a, as shown in Fig. 5.5a, where the vectors representing the magnetizations of the two sublattices are drawn from one

Fig. 5.5 Magnetization changes in an antiferromagnet when the field H is applied (a) at right angles and (b) parallel to the spin axis D.

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point. This rotation immediately creates a magnetization s in the direction of the field and sets up an unbalanced molecular field Hm in the opposite direction. The spins will rotate until Hm equals H, or

We have assumed here that the sublattice magnetizations ssA and ssB change only their directions and not their magnitude when a field is applied; this is a good approximation because a is very small. We note from Equation 5.18 that the susceptibility at right angles to the spin axis is inversely proportional to the molecular field constant, as might be expected, and it is independent of the temperature. The high-temperature susceptibility Equation 5.9 should thus give the same result as Equation 5.18 at TN. This can be shown by combining Equations 5.9, 5.10, and 5.13.

Field Parallel to Spin Axis. Suppose the field is applied in the direction of the A sublattice magnetization. Then the effect of the field is to increase the zero-field value of the A sublattice magnetization ssA by an amount DsA and decrease the corresponding value ssB of the B sublattice by an amount DsB, as shown in Fig. 5.5b. The balance between the two sublattices is upset, and a net magnetization in the direction of the field is produced:

Now the magnetization of either sublattice is governed by the Brillouin function B(J, a0) of Equation 4.13, as shown in Fig. 5.6. Here P represents the spontaneous magnetization ss, in the absence of an applied field, of either sublattice; it is determined by the particular value a00 of the variable a0. The effect of an applied field is to move the point P up on the curve for the A sublattice and down on the curve for the B sublattice. To simplify the calculation, we replace the Brillouin curve by its tangent at P, which amounts to assuming that the applied field produces equal changes in both sublattices:

The value of DsA will be given by the product of Da0 and the slope of the magnetization curve:

Fig. 5.6 Magnetization changes when the field is applied parallel to the spin axis.

where B0(J, a00) is the derivative of the Brillouin function with respect to its argument a0, evaluated at a00 . To find Da0, we must remember that the field H in the variable a0(¼mHH/kt) can include both an applied field, which we will now write explicitly as Ha, and a molecular field. In the present problem, the increase in a0 is caused by the application of the field Ha less the amount the molecular field due to the B sublattice has decreased due to Ha, or

where we have put s0A, which represents absolute saturation of the A sublattice, equal to (ng/2)mH, where ng is the number of magnetic ions per gram. After solving Equation 5.23 for DsA, we arrive finally at an expression for the susceptibility parallel to the spin axis:

This equation may be put in another form by making the substitution

which follows from Equation 3.48 and where C is the Curie constant and J the quantum number.

The magnetization in the direction of the field is then

s¼ sk cos u þ s? sin u

¼ xkH cos2u þ x?H sin2u

or

This susceptibility of one crystal must then be averaged over all possible values of u to give the susceptibility of the powder:

A plot of the temperature variation of the powder susceptibility is included in Fig. 5.7.

5.2.3Comparison with Experiment

The molecular field theory outlined above leads to three predictions, easily compared with experiment:

1.u/TN should equal 1, according to Equation 5.13.

2.The values of xk and x? for a single crystal should vary with temperature as in Fig. 5.7.

3.The ratio of the susceptibility of a powder at 0K to its value at TN should equal 23, according to Equation 5.28, since xk ¼ 0 and x? ¼ xTN at 0K.

Table 5.1 shows that observed values of u/TN range from 1 up to 5 or 6. This departure from unity does not mean that the molecular field theory has failed but that our initial assumption was too restrictive. We assumed that the only molecular field acting on the A ions was due to the B sublattice. Actually, there is no reason to exclude the possibility that AA and BB exchange forces are also acting. Equations 5.3 and 5.4 would then be replaced by

where there are now two molecular field constants; gAB defines the strength of the AB interaction, and gAA, usually assumed equal to gBB, defines the strength of the AA interaction. The constant gAA (¼gBB) can be positive, negative, or zero. When gAA is not zero, the ratio u/TN can take on larger values than unity; the ratio gAA/gAB can then be computed from the observed value of u/TN. Negative values of u, which mean that x ¼ C/(T 2 u) above TN, observed for some chlorides like FeCl2, imply that gAA is negative.

The generally good agreement between theory and experiment is all the more remarkable when we realize that one of our initial assumptions is often not satisfied. This is the assumption that any A ion has only B ions as nearest neighbors. This requirement can be satisfied by a body-centered cubic arrangement of metal ions, in which the cube-center ions form the A sublattice and the cube-corner ions the B sublattice, and by a body-centered tetragonal arrangement, provided the axial ratio c/a of the unit cell lies between certain limits. It is not satisfied for the face-centered cubic arrangement, as will become clear in the next section.

The crystal anisotropy of antiferromagnetics and the attendant phenomenon of metamagnetism are discussed in Section 7.9.

5.3NEUTRON DIFFRACTION

The first substance to be clearly recognized as antiferromagnetic was MnO, in 1938, when the results shown in Fig. 5.8 were published. Between 1938 and 1949 the evidence which had accumulated for the assumed spin arrangement in antiferromagnets below TN was good but rather indirect: it consisted solely in the agreement of the susceptibility data with what could be predicted from the model. In 1949 the first direct evidence was provided, when C. G. Shull and J. S. Smart [Phys. Rev., 76 (1949) p. 1256] succeeded in showing by neutron diffraction that the spins on the manganese ions in MnO are divided into two groups, one antiparallel to the other. Neutron diffraction has wide applicability as a research tool, but here we are interested only in its application to magnetic studies. These are not confined to antiferromagnetic substances, because neutron diffraction can furnish important information about ferroand ferrimagnetics as well.

A stream of particles has many attributes of wave motion, in particular a wavelength l, given by l ¼ h/p, where h is Planck’s constant and p is the momentum of the particles. A stream, or beam, of neutrons can therefore be diffracted by a crystal just like a beam of X rays, provided that the neutron wavelength is of the same order of magnitude as the interplanar spacings of the crystal. The neutrons in the core of a nuclear reactor have just the right wavelength, about 0.1nm, and a beam of them can be obtained simply by cutting a narrow hole through the shielding of the reactor.

The diffraction of neutrons is governed by the same Bragg law that governs the diffraction of X rays

where n is an integer (0, 1, 2, . . .) called the order of reflection, d is the spacing of atomic planes in the crystal, and u is the angle between the incident beam and the atomic planes involved (Fig. 5.10). When neutrons, or X rays, encounter an atom, they are scattered in all directions, and what we call a diffracted beam is simply a set of scattered beams which are in phase, so that they reinforce one another. The Bragg law states the condition that rays scattered in the direction shown in the sketch, making an angle u with the atomic planes equal to the angle of incidence, will be in phase with one another. In all other directions of space the phase relations between the scattered beams are such that they cancel one another. In experimental work, the angle 2u, rather than u, is usually measured; it is the angle between the diffracted beam and the incident or the transmitted beam.

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Fig. 5.10 Diffraction geometry.

Although both X rays and neutrons obey the Bragg law, they are scattered by atoms in markedly different ways. X rays are scattered by the electrons of the atom, because X rays are electromagnetic radiation which can interact with the electronic charge. Neutrons are uncharged, easily penetrate the electron screen, and are scattered only by the nucleus. There is one important exception to this statement: if the scattering atom or ion has a net magnetic moment, that moment will interact with the neutron beam, because the neutron has a small magnetic moment of its own, equal to about 1023 Bohr magneton. Neutron scattering from a magnetic ion therefore has two parts, one nuclear, the other magnetic; the magnetic part is due to the electrons of the ion, because it is the ion’s electrons that are responsible for its magnetic moment. Neutrons can thus “see” elementary magnetic moments, whereas X rays cannot. Furthermore, both the magnitude and direction of the magnetic moment of an atom or ion can be determined from measurements of the intensity of the magnetic scattering. It is this feature of neutron diffraction which makes it so valuable in the study of magnetic materials.

Electron diffraction is a fairly common means of investigating crystals, and the reader may wonder why it also is not effective in revealing magnetic structure. After all, each electron in the beam of electrons incident on the solid, in an electron diffraction experiment, has a magnetic moment of one Bohr magneton and therefore of the same order of magnitude as the net magnetic moment of each atom of the solid, rather than a moment of less than 1023 times an atomic moment, as is typical of neutrons. The answer lies in the fact that electrons are charged. The incident electrons are scattered by atomic electrons because of the very large electrostatic (Coulomb) repulsion between them. This electrostatic interaction is so much stronger than the magnetic interaction that the latter is normally unobservable. Neutrons, on the other hand, are uncharged, and their magnetic interaction with the scattering atoms, although much weaker than that of electrons, can easily be observed because it is not overwhelmed by an electrostatic interaction.

When an antiferromagnet is cooled below TN, what was previously a random arrangement of spins becomes an ordered arrangement, with one set of spins antiparallel to the other. This change is very similar, especially from a diffraction point of view, to the chemical ordering which takes place in certain solid solutions when cooled below a critical temperature. Consider X-ray diffraction from the (100) planes of such a solid solution, consisting of elements C and D in equal atomic proportions and assumed to have a body-centered cubic structure. These planes are marked X and Z in Fig. 5.11a, which applies to the disordered state. If the incoming X rays make an angle u such that the path difference abc between scattered rays 1 and 3 equals one whole wavelength, then rays 1 and 3 will be in phase and reinforce each other. But if abc ¼ l, the path difference def between rays 1 and 2 is l/2, so that these rays are exactly out of phase. Moreover, the amplitudes of 1