Curie–Weiss law

Background

A magnetic moment which is present even in the absence of the external magnetic field is called spontaneous magnetization. Materials with this property are known as ferromagnets, such as iron, nickel, and magnetite. However, when these materials are heated up, at a certain temperature they lose their spontaneous magnetization, and become paramagnetic. This threshold temperature below which a material is ferromagnetic is called the Curie temperature and is different for each material.

The Curie–Weiss law describes the changes in a material's magnetic susceptibility, \chi, near its Curie temperature. The magnetic susceptibility is the ratio between the material's magnetization and the applied magnetic field.

Limitations

In many materials, the Curie–Weiss law fails to describe the susceptibility in the immediate vicinity of the Curie point, since it is based on a mean-field approximation. Instead, there is a critical behavior of the form : \chi \propto \frac{1}{(T - T_{\rm C})^\gamma} with the critical exponent γ. However, at temperatures T ≫ TC the expression of the Curie–Weiss law still holds true, but with TC replaced by a temperatureΘ that is somewhat higher than the actual Curie temperature. Some authors call Θ the Weiss constant to distinguish it from the temperature of the actual Curie point.

Classical approaches to magnetic susceptibility and Bohr–van Leeuwen theorem

According to the Bohr–van Leeuwen theorem, when statistical mechanics and classical mechanics are applied consistently, the thermal average of the magnetization is always zero. Magnetism cannot be explained without quantum mechanics. That means that it can not be explained without taking into account that matter consists of atoms. Next are listed some semi-classical approaches to it, using a simple atom model, as they are easy to understand and relate to even though they are not perfectly correct.

The magnetic moment of a free atom is due to the orbital angular momentum and spin of its electrons and nucleus. When the atoms are such that their shells are completely filled, they do not have any net magnetic dipole moment in the absence of an external magnetic field. When present, such a field distorts the trajectories (classical concept) of the electrons so that the applied field could be opposed as predicted by the Lenz's law. In other words, the net magnetic dipole induced by the external field is in the opposite direction, and such materials are repelled by it. These are called diamagnetic materials.

Sometimes an atom has a net magnetic dipole moment even in the absence of an external magnetic field. The contributions of the individual electrons and nucleus to the total angular momentum do not cancel each other. This happens when the shells of the atoms are not fully filled up (Hund's Rule). A collection of such atoms however, may not have any net magnetic moment as these dipoles are not aligned. An external magnetic field may serve to align them to some extent and develop a net magnetic moment per volume. Such alignment is temperature dependent as thermal agitation acts to disorient the dipoles. Such materials are called paramagnetic.

In some materials, the atoms (with net magnetic dipole moments) can interact with each other to align themselves even in the absence of any external magnetic field when the thermal agitation is low enough. Alignment could be parallel (ferromagnetism) or anti-parallel. In the case of anti-parallel, the dipole moments may or may not cancel each other (antiferromagnetism, ferrimagnetism).

Density matrix approach to magnetic susceptibility

We take a very simple situation in which each atom can be approximated as a two state system. The thermal energy is so low that the atom is in the ground state. In this ground state, the atom is assumed to have no net orbital angular momentum but only one unpaired electron to give it a spin of the half. In the presence of an external magnetic field, the ground state will split into two states having an energy difference proportional to the applied field. The spin of the unpaired electron is parallel to the field in the higher energy state and anti-parallel in the lower one.

A density matrix, \rho , is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states (here several similar 2-state atoms). This should be contrasted with a single state vector that describes a quantum system in a pure state. The expectation value of a measurement, A , over the ensemble is \langle A \rangle = \operatorname{Tr} (A \rho) . In terms of a complete set of states, |i\rangle , one can write : \rho = \sum_{ij} \rho_{ij} |i\rangle \langle j| .

Von Neumann's equation tells us how the density matrix evolves with time. : i \hbar \frac d {dt} \rho (t) = [H, \rho(t)]

In equilibrium, one has [H, \rho] = 0 , and the allowed density matrices are f(H) . The canonical ensemble has \rho = \exp(-H/T)/Z , where Z =\operatorname{Tr} \exp(-H/T) .

For the 2-state system, we can write H = -\gamma \hbar B \sigma_3 . Here \gamma is the gyromagnetic ratio. Hence Z = 2 \cosh(\gamma \hbar B/(2T)) , and : \rho(B,T) = \frac 1 {2 \cosh(\gamma \hbar B/(2T))} \begin{pmatrix} \exp (-\gamma \hbar B/(2T)) & 0 \ 0 & \exp (\gamma \hbar B/(2T)) \end{pmatrix}. From which : \langle J_x \rangle = \langle J_y \rangle = 0, \langle J_z \rangle = - \frac \hbar 2 \tanh (\gamma \hbar B/(2T)).

Explanation of para and diamagnetism using perturbation theory

In the presence of a uniform external magnetic field B along the z-direction, the Hamiltonian of the atom changes by : \Delta H = \alpha J_z B + \beta B^2 \sum_i (x_i^2 + y_i^2 ), where \alpha, \beta are positive real numbers which are independent of which atom we are looking at but depend on the mass and the charge of the electron. i corresponds to individual electrons of the atom.

We apply second order perturbation theory to this situation. This is justified by the fact that even for highest presently attainable field strengths, the shifts in the energy level due to \Delta H is quite small w.r.t. atomic excitation energies. Degeneracy of the original Hamiltonian is handled by choosing a basis which diagonalizes \Delta H in the degenerate subspaces. Let |n\rangle be such a basis for the state of the atom (rather the electrons in the atom). Let \Delta E_n be the change in energy in |n \rangle . So we get : \Delta E_n = \langle n | \Delta H | n \rangle + \sum_{m, E_m \neq E_n} \frac

References

1. {{harvnb. Hall. 1994
2. {{harvnb. Levy. 1968