Heisenberg exchange parameters

The geometry of the lattice determines the structure of the Heisenberg exchange parameter matrix. In most cases only nearest-neighbor interactions are taken into account and all non-vanishing Heisenberg exchange parameters axe set equal, i.e. we set... 

By analogy to the liquid-vapor equilibrium the system is said to be condensed for 2 > 0. The critical temperature is expressed in terms of a Heisenberg exchange parameter... 

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. 

Table 5 Spin levels (Ei,i = 1 ), /T (in cm. K/mol) values at (in °K) corresponding to maximum anisotropy of the main values of the susceptibility /ii(ii = xx, yy, zz) tensor, /avr[Xav = (l/3)(Xxx + Xyy + Xzz)] and anisotropic D12 and A12 tensor parameters of a Fe -CN-Cu" model complex with and without trigonal Jahn-TeUer distortions from model calculations accounting for the difference of Je and Jb2 exchange coupling parameters (exchange anisotropy) or alternatively, assuming an isotropic Heisenberg exchange (1 = 1e = 1b2)...

J-J ) being the effective exchange parameter, we note a significant disagreement with previously reported results [50]. Actually, we do believe that the exchange Hamiltonian cannot be, in the present case, Heisenberg-like. Such an approximation may only be accounted for D >J as shown in Fig. 3,... 

Our second second objective to illustrate how MC-SCF data on chemical reactivity can be visualized and conceptualized using this VB model. A Heisenberg VB hamiltonian is obtained from an MC-SCF computation in terms of coulomb and exchange parameters that have the same interpretation as in the HL theory of the H2 molecule. This data can then be fitted and visualized using contour diagrams. 

Fig. 330. Cu40(py)4Clg. Temperature dependence of and 1/Zm. Magnetism fitted on the basis of simple Heisenberg exchange theory using J//ic=7.65 cm , g=2.250, z (an intercluster exchange term) is — 4.0cm Low-temperature (< 8 K) data were not used in obtaining the best-fit parameters [74D16],...

In general, the magnetic energy levels of a magnetic system are described by employing a Heisenberg spin Hamiltonian Hspi defined in terms of several different spin exchange parameters. 

Here, a is called the alternation parameter specifying the ratio of the exchange energies, /ex and J ex = aJex. For 0 < a < 1, both exchange energies have the same sign, a = 1 results in the equidistant chain, and a = 0 describes isolated dimers. As with uniform chains, the susceptibility of such Heisenberg systems can only be approximated. 

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