Hamiltonian exchange

With these data, we calculated numerically the (mi, m2 He[t m u m 2) matrix elements of the effective exchange Hamiltonian, which are presented in Table 1. Each group of matrix elements can be associated with a definite equivalent spin operator, which has the same matrix elements in the basis set as those... 

Table 1. Non -zero (ni],m2 Haflm3,m4) matrix elements of the effective exchange Hamiltonian and the equivalent spin operators (all parameters are in cm-1)...

Exact solutions do not exist for the exchange Hamiltonian of two-dimensional layers and three-dimensional networks. These situations can, however, be described using approximation methods such as molecular field approximation and high-temperature series expansion. 

The perfectly aligned N6el state is not an eigenstate of the antiferromagnetic exchange Hamiltonian, except in the case where its anisotropy is infinite. Consequently, even at T = OK, the spins will be subject to deviations. 

A. WEISS MOLECULAR FIELD AND HEISENBERG EXCHANGE HAMILTONIAN... 

Generalization of equation 89 to the many-electron system of a crystal is given by the Heisenberg exchange Hamiltonian... 

It was pointed out in Chapter II that the Heisenberg exchange Hamiltonian of equation 90, which can be directly related to the Weiss field parameters at T = 0°K by equation 94, is an excellent formal expression for the interactions between atomic spins (or moments) of neighboring atoms. There remains the problem of establishing the various spin-dependent mechanisms that contribute to the Jij. In general, there are two types of interaction cation- -cation and cation-anion-cation (or even cation-anion-anion-cation) interactions. 

The application of second-order perturbation theory to the Hiickel-Hubbard Hamiltonian for z close to one (Section 4.2) yields the freeon, antiferromagnetic, Heisenberg exchange Hamiltonian,... 

The equivalent Heisenberg, antiferromagnetic, spin-exchange Hamiltonian... 

In the previous Sections we have made considerable use of the freeon exchange Hamiltonian. In this Section we give a little history of this concept. 

In this Section the ferromagnetic and the antiferromagnetic Heisenberg exchange Hamiltonians are derived from the Hiickel-Hubbard Hamiltonian in which the value of the coupling constant, z takes on values from - 1 to + 1. The M = N = 2, linear, extended Hiickel-Hubbard spectrum is plotted in Fig. 4.1. (Compare with Fig. 3.1)... 

By analogy to the ordinary-spin Heisenberg exchange Hamiltonian the isospin exchange Hamiltonian is... 

---