Direct Exchange Interactions in Solids
In a crystal, generalization of the Heisenberg Hamiltonian in which the sum is taken over the exchange Hamiltonians for all the (i,j) pairs of atoms of the many-electron system gives:.
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(14)
The 1/2 factor is introduced because the interaction between the same two atoms is counted twice in performing the sums. Note that the J in Eq.(14) is the exchange constant Jab above not the exchange integral Jex. The exchange integral Jex is related to yet another quantity, called the exchange stiffness constant (A) which serves as a characteristic of a ferromagnetic material. The relationship is dependent on the crystal structure. For a simple cubic lattice with lattice parameter ,
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(15)
For a body-centered cubic lattice,
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(16)
and for a face-centered cubic lattice,
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(17)
The form of Eq. (14) corresponds identically to the Ising model of ferromagnetism except that in the Ising model, the dot product of the two spin angular momenta is replaced by the scalar product SijSji. The Ising model was invented by Wilhelm Lenz in 1920 and solved for the one-dimensional case by his doctoral student Ernst Ising in 1925. The energy of the Ising model is defined to be:
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(18)
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