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The Brillouin and Langevin functions are a pair of special functions that appear when studying an idealized paramagnetic material in statistical mechanics. ==Brillouin function == The Brillouin function〔C. Kittel, ''Introduction to Solid State Physics'' (8th ed.), pages 303-4 ISBN 978-0-471-41526-8〕 is a special function defined by the following equation:
The function is usually applied (see below) in the context where ''x'' is a real variable and ''J'' is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as and -1 as . The function is best known for arising in the calculation of the magnetization of an ideal paramagnet. In particular, it describes the dependency of the magnetization on the applied magnetic field and the total angular momentum quantum number J of the microscopic magnetic moments of the material. The magnetization is given by:〔 : where * is the number of atoms per unit volume, * the g-factor, * the Bohr magneton, * is the ratio of the Zeeman energy of the magnetic moment in the external field to the thermal energy : :: * is the Boltzmann constant and the temperature. Note that in the SI system of units given in Tesla stands for the magnetic field, , where is the auxiliary magnetic field given in A/m and is the permeability of vacuum. :/Z where ''Z'' (the partition function) is a normalization constant such that the probabilities sum to unity. Calculating ''Z'', the result is: :. All told, the expectation value of the azimuthal quantum number ''m'' is :. The denominator is a geometric series and the numerator is a type of (arithmetic-geometric series ), so the series can be explicitly summed. After some algebra, the result turns out to be : With ''N'' magnetic moments per unit volume, the magnetization density is :. |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Brillouin and Langevin functions」の詳細全文を読む スポンサード リンク
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