Physics of Magnetism

34 CHAPTER 4. THE MAGNETICALLY ORDERED STATE

parallel arrangement of the two sublattice moments, as in the high-field part of curve (a)

in Fig. 4.3.3, is not possible here. Therefore, the total magnetization remains low up to the

highest field applied. In the case of curve (b), the total magnetization remains low for low

fields. However, at a certain critical value of the applied field, the total magnetization jumps

directly to the forced parallel configuration. We will compare now the free energy of the

antiparallel sublattice-moment arrangement in the applied field with the parallel sublattice­

moment arrangement in the applied field. Using Eqs. (4.3.1) and (4.3.2) for calculating

for both situations and noting that K = 0 for all situations

on curve (b), one easily derives the critical field as

This formula expresses the fact that the sudden change from antiparallel to parallel

sublattice-moment arrangement occurs when the applied field is able to overcome the anti­

ferromagnetic coupling between the two sublattice moments. This phenomenon is called

metamagnetic transition.

4.4. FERRIMAGNETISM

In ferrimagnetic substances, in contrast with the antiferromagnets described in the

previous section, the magnetic moments of the A and B sublattices are not equal. The mag­

netic atoms (A and B) in a crystalline ferrimagnet occupy two kinds of lattice sites that have

different crystallographic environments. Each of the sublattices is occupied by one of the

magnetic species, with ferromagnetic (parallel) alignment between the moments residing on

the same sublattice. There is antiferromagnetic (antiparallel) alignment, however, between

the moments of A and B. Since the number of A and B atoms per unit cell are generally

different, and/or since the values of the A and B moments are different, there is nonzero

spontaneous magnetization below At zero Kelvin, it reaches the value

As in Eq. (4.1.2), we can represent the exchange interaction between the various spins

and in the lattice by means of the Hamiltonian

where is the exchange constant describing the magnetic coupling of two moments
residing on the same magnetic sublattice A (or B) or on different sublattices A and B.
Indicating the exchange constant between two nearest-neighbor spins on the same sublattice
by (or and between two nearest-neighbor spins on different sublattices by
we can represent the three types of cooperative magnetism leading to ordered magnetic
moments as follows:

Ferromagnetism



Antiferromagnetism

 and

Ferrimagnetism and