居里-外斯定律

居禮-外斯定律為居禮定律的修正公式，用來補足該公式的不足。 當一物質的溫度大於居禮溫度($T_{c}$)時，其磁化率$\chi$與溫度$T$的關係式為：

$\chi ={\frac {C}{T-T_{c}}}$

其中：

$C$是該物質的居禮常數

$T$是絕對溫度，單位為$K$

$T_{c}$是該物質的居禮溫度，單位為$K$

根據該定律，當$T=T_{c}$時，磁化率為無窮大。而當低於此溫度時，鐵磁性物質將會自發磁化。

基本概念

一個磁性物質的磁化向量(或稱磁化強度)代表單位體積該物質的磁矩強度。而磁矩可能由原子內的電子運動或者是自旋所產生。而淨磁化向量可能由外加磁場誘導產生，甚至可能不需要施加外加磁場也能產生磁化向量(例如足夠低溫狀態下的鐵)，其中後者被稱之為自發磁化。

而其他與鐵擁有同樣性質的物質，如鎳與磁鐵礦，被稱之為鐵磁性物質。鐵磁性物質在足夠低的溫度下將會不須施加外部磁場也會產生磁性，而該臨界溫度稱之為居禮溫度。

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

$\chi \sim {\frac {1}{(T-T_{c})^{\gamma }}}$

with the critical exponent γ. However, at temperatures T ≫ T_c the expression of the Curie–Weiss law still holds true, but with T_c replaced by a temperature Θ 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 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. However we list some classical approaches to it as they are easy to understand and relate to even though they are incorrect.

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 external magnetic field. When present, such 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 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 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 half. In the presence of an external magnetic field the ground state will split into two states having 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 $\langle A\rangle =Tr(A\rho )$. 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 $Z=Tr\exp(-H/T)$.

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

$\langle J_{x}\rangle =\langle J_{y}\rangle =0,\langle J_{z}\rangle =-{\frac {\hbar }{2}}\tanh(\gamma \hbar B/(2T)).$