Curie–Weiss law
The Curie–Weiss law describes the magnetic susceptibility of a ferromagnet in the paramagnetic region above the Curie point: where is a material-specific Curie constant, is the absolute temperature, and is the Curie temperature, both measured in kelvin. The law predicts a singularity in the susceptibility at . Below this temperature, the ferromagnet has a spontaneous magnetization. The name is given after Marie Curie and Pierre-Ernest Weiss. Brief summary of related concepts. The magnetic moment of a magnet is a quantity that determines the torque it will experience in an external magnetic field. A loop of  electric current, a bar magnet, an electron, a molecule, and a planet all have magnetic moments. The magnetization or magnetic polarization of a magnetic material is the vector field that expresses the density of permanent or induced magnetic moments. The magnetic moments can originate from microscopic electric currents caused by the motion of electrons in individual atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic moment that may be present even in the absence of the external magnetic field, for  example, in sufficiently cold iron. The latter is called spontaneous magnetization. Other materials that share this property with iron, like Nickel and magnetite, are called ferromagnets. The threshold temperature below which a material is ferromagnetic is called the Curie temperature and varies between materials. 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 with the critical exponent . However, at temperatures the expression of the Curie–Weiss law still holds true , but with replaced by a temperature that is somewhat higher than the actual Curie temperature. Some authors call 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. 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 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, formula_3, 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, formula_4, over the ensemble is formula_5. In terms of a complete set of states, formula_6, one can write Von Neumann's equation tells us how the density matrix evolves with time. In equilibrium, one has formula_9, and the allowed density matrices are formula_10. The canonical ensemble has formula_11 where formula_12. For the 2-state system, we can write formula_13. Here formula_14  is the gyromagnetic ratio. Hence formula_15, and From which Explanation of para and diamagnetism using perturbation theory. In the presence of a uniform external magnetic field formula_18 along the z-direction, the Hamiltonian of the atom changes by where formula_20 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. formula_21 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 formula_22 is quite small w.r.t. atomic excitation energies. Degeneracy of the original Hamiltonian is handled by choosing a basis which diagonalizes formula_22 in the degenerate subspaces. Let formula_24 be such a basis for the state of the atom (rather the electrons in the atom). Let formula_25 be the change in energy in formula_26. So we get In our case we can ignore formula_28 and higher order terms. We get In case of diamagnetic material, the first two terms are absent as they don't have any angular momentum in their ground state. In case of paramagnetic  material all the three terms contribute. Adding spin-spin interaction in the Hamiltonian: Ising model. So far, we have assumed that the atoms do not interact with each other. Even though this is a reasonable assumption in the case of diamagnetic and paramagnetic substances, this assumption fails in the 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 the atom. Ising model is one of the simplest approximations of such pairwise interaction. Here the two atoms of a pair are at formula_31. Their interaction formula_32 is determined by their distance vector formula_33. In order to simplify the calculation, it is often assumed that interaction happens between neighboring atoms only and formula_32 is a constant. The effect of such interaction is often approximated as a mean field and, in our case, the Weiss field. Modification of Curie's law due to Weiss field. The Curie-Weiss law is an  adapted version of Curie's law, which for a paramagnetic material may be written in SI units as follows, assuming formula_35: formula_36 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: formula_37 where is Boltzmann's constant, the number of magnetic atoms (or molecules) per unit volume, the Landé g-factor, the Bohr magneton, the angular momentum quantum number. For the Curie-Weiss Law the total magnetic field is where is the Weiss molecular field constant and then formula_38 which can be  rearranged to get formula_39 which is the Curie-Weiss Law formula_40 where the Curie temperature is formula_41