Physics of Magnetism

20 CHAPTER 4. THE MAGNETICALLY ORDERED STATE

In most cases, it is sufficient to consider only the exchange interaction between spins on

nearest-neighbor atoms. If there are Z magnetic nearest-neighbor atoms surrounding a given

magnetic atom, one has

with the average spin of the nearest-neighbor atoms. Relation (4.1.3) can be rewritten
by using which follows from the relations and
(Fig. 2.1.2):

Since the atomic moment is related to the angular momentum by (Eq. 2.2.4),

we may also write

where

can be regarded as an effective field, the so-called molecular field, produced by the average

moment of the Z nearest-neighbor atoms.

Since it follows furthermore that is proportional to the magnetization

The constant is called the molecular-field constant or the Weiss-field constant. In fact,
Pierre Weiss postulated the presence of a molecular field in his phenomenological theory
of ferromagnetism already in 1907, long before its quantum-mechanical origin was known.
The exchange interaction between two neighboring spin moments introduced in
Eq. (4.1.2) has the same origin as the exchange interaction between two electrons on
the same atom, where it can lead to parallel and antiparallel spin states. The exchange
interaction between two neighboring spin moments arises as a consequence of the overlap
between the magnetic orbitals of two adjacent atoms. This so-called direct exchange inter­
action is strong in particular for 3d metals, because of the comparatively large extent of the
3d-electron charge cloud. Already in 1930, Slater found that a correlation exists between
the nature of the exchange interaction (sign of exchange constant in Eq. 4.1.2) and the ratio
where represents the interatomic distance and the radius of the incompletely
filled d shell. Large values of this ratio corresponded to a positive exchange constant, while
for small values it was negative.
Quantum-mechanical calculations based on the Heitler–London approach were made
by Sommerfeld and Bethe (1933). These calculations largely confirmed the result of Slater
and have led to the Bethe–Slater curve shown in Fig. 4.1.1. According to this curve, the
exchange interaction between the moments of two similar 3d atoms changes when these
are brought closer together. It is comparatively small for large interatomic distances, passes
through a maximum, and eventually becomes negative for rather small interatomic dis­
tances. As indicated in the figure, this curve has been most successful in separating the