居里-外斯定律

content: 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).
title: Density matrix approach to magnetic susceptibility.
content: 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.
title: Explanation of para and diamagnetism using perturbation theory.
content: In the presence of a uniform external magnetic field formula_27 along the z-direction, the Hamiltonian of the atom changes by where formula_29 are positive real numbers which are independent of which atom we are looking at but depends on the mass and the charge of the electron. formula_30 corresponds to individual electrons of the atom.
title: Adding spin-spin interaction in the Hamiltonian: Ising model.
content: So far we have assumed that the atoms do not interact with each other. Even though this is a reasonable assumption in case of diamagnetic and paramagnetic substances, this assumption fails in case of ferromagnetism where the spins of the atom try to align with each other to the extent permitted by the thermal agitation. In this case we have to consider the Hamiltonian of the ensemble of the atom. Such a Hamiltonian will contain all the terms described above for individual atoms and terms corresponding to the interaction among the pairs of atom. Ising model is one of the simplest approximation of such pairwise interaction.  Here the two atoms of a pair are at formula_40. Their interaction formula_41 is determined by their distance vector formula_42. In order to simplify calculation it is often assumed that interaction happens between neighboring atoms only and formula_41 is a constant. The effect of such interaction is often approximated as a mean field and in our case the Weiss field.
title: Modification of Curie Law due to Weiss field.
content: The Curie-Weiss Law is an adapted version of Curie's Law, which for a paramagnetic material may be written in SI units as  Here μ is the permeability of free space; "M" the magnetization (magnetic moment per unit volume), is the magnetic field, and "C" the material-specific Curie constant:  For the Curie-Weiss Law the total magnetic field is where is the Weiss molecular field constant and then

Summary: 居礼-外斯定律为居里定律的修正公式,用来补足该公式的不足。当一物质的温度大于居里温度(formula_1)时,其磁化率formula_2与温度formula_3的关系式为: 其中:

Curie–Weiss law: The Curie–Weiss law describes the magnetic susceptibility of a ferromagnet in the paramagnetic region above the Curie point: