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1) the quantum statistical mechanics reduces to classical statistical mechanics and we can use a classical, Maxwell-Boltzmann distribution function (instead of Fermi– Dirac or Bose–Einstein). For magnetic materials, the situation is somewhat different. The Hamiltonian of a magnetic system is a function of spin operators which can usually not be directly approximated by classical vectors. Quantum models in d dimensions can be mapped onto classical models in d = (d + 1) dimensions. The classical limit of the Heisenberg model, however, can be constructed for large eigenvalues of the spin operator by replacing the spin operators by three-dimensional classical vectors.
There is an important difference between the lattice gas and the Ising models of magnetism. The number of atoms is constant (provided no atoms evaporate from or condense on the lattice), we say that the order parameter in the lattice gas models is conserved whereas it is not conserved in magnetic systems. This has important consequences for dynamics. 2 Binary Alloys A binary alloy is composed of two metals, say A and B, arranged on a lattice. We introduce the occupations: pi = 1 if i is occupied by an A atom 0 if i is occupied by a B atom qi = 1 if i is occupied by a B atom 0 if i is occupied by an A atom pi + qi = 1.
In this model, the spin operator S is replaced by a number, which represents the z− component of the spin and is usually S = ±1 (”up” or ”down”). The order parameter m = S is thus one-dimensional, it is a two-component scalar. 15) H=− 2 i,j i Jij is the interaction energy between the spins i and j. Usually (not always) the sum is only over nearest neighbouring (NN) pairs, then Jij = J. H is the external magnetic field (in energy units). The case J > 0 corresponds to ferromagnetism and J < 0 to antiferromagnetism.