居里定律

title: 用量子力学推导.
content: A simple model of a paramagnet concentrates on the particles which compose it which do not interact with each other. Each particle has a magnetic moment given by formula_6. The energy of a magnetic moment in a magnetic field is given by
title: Two-state (spin-1⁄2) particles.
content: To simplify the calculation, we are going to work with a 2-state particle: it may either align its magnetic moment with the magnetic field, or against it. So the only possible values of magnetic moment are then formula_8 and formula_9. If so, then such a particle has only two possible energies  and  When one seeks the magnetization of a paramagnet, one is interested in the likelihood of a particle to align itself with the field. In other words, one seeks the expectation value of the magnetization formula_8: where the probability of a configuration is given by its Boltzmann factor, and the partition function formula_14 provides the necessary normalization for probabilities (so that the sum of all of them is unity.) The partition function of one particle is:  This is magnetization of one particle, the total magnetization of the solid is given by formula_17 where "n" is the number density of magnetic moments. The formula above is known as the Langevin paramagnetic equation. Pierre Curie found an approximation to this law which applies to the relatively high temperatures and low magnetic fields used in his experiments. Let's see what happens to the magnetization as we specialize it to large formula_4 and small formula_19. As temperature increases and magnetic field decreases, the argument of hyperbolic tangent decreases. Another way to say this is  with a Curie constant given by formula_24.  In the regime of low temperatures or high fields, formula_25 tends to a maximum value of formula_26, corresponding to all the particles being completely aligned with the field. Since this calculation doesn't describe the electrons embedded deep within the Fermi surface, forbidden by the Pauli Exclusion principle to flip their spins, it does not exemplify the quantum statistics of the problem at low temperatures. Using the Fermi-Dirac distribution, one will find that at low temperatures formula_25 is linear dependent on the magnetic field, so that the magnetic susceptibility saturates to a constant.
title: General case.
content: When the particles have an arbitrary spin (any number of spin states), the formula is a bit more complicated. At low magnetic fields or high temperature, the spin follows Curie's law, with where formula_29 is the total angular momentum quantum number and formula_30 is the spin's g-factor (such that formula_31 is the magnetic moment).  For this more general formula and its derivation (including high field, low temperature) see the article: Brillouin function. As the spin approaches infinity, the formula for the magnetization approaches the classical value derived in the following section.
title: 用经典统计力学推导.
content: An alternative treatment applies when the paramagnetons are imagined to be classical, freely-rotating magnetic moments. In this case, their position will be determined by their angles in spherical coordinates, and the energy for one of them will be:  where formula_33 is the angle between the magnetic moment and the magnetic field (which we take to be pointing in the formula_34 coordinate.) The corresponding partition function is  This function would appear to be singular for small formula_47, but it is not, since the two singular terms cancel each other. In fact, its behavior for small arguments is formula_48, so the Curie limit also applies, but with a Curie constant three times smaller in this case. Similarly, the function saturates at formula_49 for large values of its argument, and the opposite limit is likewise recovered.

Summary: 居里定律是指在顺磁性材料中,材料的磁化强度大致与施加的磁场强度成正比。然而,若加热材料,则比值减小。对于固定场强的磁场,磁化率大致与温度成反比。

Curie's law: In a paramagnetic material, the magnetization of the material is (approximately) directly proportional to an applied magnetic field. However, if the material is heated, this proportionality is reduced: for a fixed value of the field, the magnetization is (approximately) inversely proportional to temperature. This fact is encapsulated by Curie's law, after Pierre Curie: