Ferromagnetism

(7.244) |

where is the Bohr magneton, and the -factor, , is a dimensionless number of order unity. (See Section 7.9.) In the presence of an externally-applied magnetic field, , directed parallel to the -axis, the Hamiltonian, , representing the interaction of the atoms with this field is written

(7.245) |

Each atom is also assumed to interact with neighboring atoms. The interaction in question is not simply the
magnetic dipole-dipole interaction due to the magnetic field produced by one atom at the position of another.
Such an interaction is, in general, much too small to produce ferromagnetism. Instead, the predominant interaction is known as
the *exchange interaction*. This interaction is a quantum-mechanical consequence of the Pauli exclusion
principle. (See Section 8.2.) Because electrons cannot occupy the same state, two electrons on neighboring atoms
that have parallel spins (here, to simplify the discussion of the origin of the exchange interaction, we are temporarily assuming that the electrons all have zero orbital angular momenta, so that their
total angular momenta are solely a consequence of their spin angular momenta) cannot come too close to one another in space (else they would simultaneously occupy identical spin
and orbital states, which is forbidden). On the other hand, if the electrons have antiparallel spins then they are
in different spin states, so they are allowed to occupy the same orbital state. In other words, there is no exclusion-related
restriction on how close they can come together. Because different spatial separations of the electrons
give rise to different electrostatic interaction between them, the electrostatic interaction
between neighboring atoms (which is much stronger than any magnetic interaction) depends on the relative
orientations of their spins. This, then, is the origin of the exchange interaction, which for two atoms labelled
and
can be written in the form

Here, the so-called

To simplify the interaction problem, we shall replace the previous expression by the simpler functional form

(7.247) |

This approximate expression, which is known as the

The Hamiltonian, , representing the interaction energy between atoms is written in the form

where is the exchange energy for neighboring atoms, and the index refers to the nearest neighbor shell surrounding the atom . The factor is introduced because the interaction between the same two atoms is counted twice in performing the sums.

The total Hamiltonian of the atoms is

(7.249) |

The task in hand is to calculate the mean magnetic moment of the system parallel to the applied magnetic field, , as a function of its absolute temperature, , and the applied field, . The presence of the interatomic interactions makes this task difficult, despite the extreme simplicity of expression (7.248). Although the problem has been solved exactly for a linear one-dimensional array of atoms, and for two-dimensional array of atoms when , the three-dimensional problem is so complex that no exact solution has ever been found. Thus, in order to make further progress, we must introduce an additional approximation.

Let us focus attention on a particular atom, , which we shall refer to as the ``central atom.'' The interactions of this atom are described by the Hamiltonian

The final term represents the interaction of the central atom with its nearest neighbors. As a simplifying approximation, we shall replace the sum over these neighbors by its mean value. In other words, we shall write

where is a parameter with the dimensions of magnetic field-strength, which is called the

Thus, the influence of neighboring atoms has been replaced by an effective magnetic field, . The problem presented by expression (7.252) reduces to the elementary one of a single atom, of normalized angular momentum, , and -factor, , placed in an external -directed magnetic field, . Recall, however, that we already solved this problem in Section 7.9.

According to the analysis of Section 7.9,

where

and . Here, is the Brillouin function, for normalized atomic angular momentum , that was introduced in Equation (7.113).

Expression (7.253) involves the unknown parameter . To determine the value of this parameter in a self-consistent manner, we note that there is nothing that distinguishes the central th atom from any of its neighboring atoms. Hence, any one of these atoms might equally well have been considered the central atom. Thus, the corresponding mean value is also given by Equation (7.253). It follows from Equations (7.251) and (7.253) that

(7.255) |

The previous two equations can be combined to give

which determines (and, hence, ). Once has been found, the mean magnetic moment of the system follows from Equation (7.253):

The solution of Equation (7.256) can be found by drawing, on the same graph, both the Brillouin function and the straight-line

(7.258) |

and then finding the point of intersection of the two curves. See Figure 7.9.

Consider the case where the external field, , is zero. Equation (7.256) simplifies to give

Given that , it follows that is a solution of the previous equation. This solution is characterized by zero magnetization: that is, . However, there is also the possibility of another solution with , and, hence, . (See Figure 7.9.) The presence of such spontaneous magnetization in the absence of an external magnetic field is, of course, the distinguishing feature of ferromagnetic materials. In order to have a solution with , it is necessary that the curves shown in Figure 7.9 intersect at a point when both curves pass through the origin. The condition for the existence of an intersection with is that the slope of the curve at should exceed that of the straight-line . In other words,

However, when , the Brillouin function takes the simple form specified in Equation (7.119):

Hence, Equation (7.260) becomes

(7.262) |

or

(7.263) |

where

Thus, we deduce that spontaneous magnetization of a ferromagnetic material is only possible below a certain critical temperature, , known as the

As the temperature, , is decreased below the Curie temperature, , the slope of the dash-dotted straight-line in Figure 7.9 decreases so that it intersects the curve at increasingly large values of . As , the intersection occurs at . Because, [see Equation (7.117)], it follows from Equation (7.257) that as . This corresponds to a state in which all atomic magnetic moments are aligned completely parallel to one another. For temperatures, , less than the Curie temperature, , we can use Equations (7.257) and (7.259) to compute in the absence of an external magnetic field. We then obtain a magnetization curve of the general form shown in Figure 7.10.

Let us, finally, investigate the magnetic susceptibility of a ferromagnetic substance, in the presence of a small external magnetic field, at temperatures above the Curie temperature. In this case, the crossing point of the two curves in Figure 7.9 lies at small . Hence, we can use the approximation (7.261) to write Equation (7.256) in the form

(7.265) |

Solving the previous equation for gives

(7.266) |

where use has been made of Equation (7.264). Now, making use of the approximation (7.261), Equation (7.257) reduces to

(7.267) |

The previous two equations can be combined to give

(7.268) |

where is the dimensionless magnetic susceptibility of the substance, and is its volume. This result is known as the

Experimentally, the Curie-Weiss law is well obeyed at temperatures significantly above the Curie temperature. See Figure 7.11. It is, however, not true that the temperature, , that occurs in this law is exactly the same as the Curie temperature at which the substance becomes ferromagnetic.

The so-called *Weiss molecular-field model* (which was first put forward by Pierre Weiss in 1907) that we have just described is
remarkably successful at explaining the major features of ferromagnetism. Nevertheless, there are some serious discrepancies
between the predictions of this simple model and experimental data. In particular, the magnetic contribution to the
specific heat is observed to have a very sharp discontinuity at the Curie temperature (in the absence of an external field),
whereas the molecular field model predicts a much less abrupt change. Needless to say, more refined models have been
devised that considerably improve the agreement with experimental data.