Energy Heisenberg exchange

Fig. 8. Orientational dependence of the average (spin-independent) exchange interaction energy AE = Via) ,

The phenomenon is closely related to the resonance phenomenon treated by Heisenberg. While in resonance however, electrons with a different motion in one and the same eigenfunction series exchange their energy here electrons of the same excitation stages (the same energy) but in different eigenfunction systems (V and... 

And the combination of Heisenberg superexchange plus Zener double exchange results in zero-field energy levels in terms of the system spin S... 

Various predictive methods based on molecular graphs of Jt-systems as described in Section 3 have been critically compared by Klein (Klein et al., 1989) and can be extended to more quantitative treatments. In principle, the effective exchange integrals /ab in the Heisenberg Hamiltonian (4) for the interaction of localized electron spins at sites a and b are calculated as the difference in energies of the high-spin and low-spin states. It was Hoffmann who first tried to calculate the dependence of the M—L—M bond... 

Figure 9. Energy level diagram for reduced FcsSa cluster according to Equation (12) plotted for antiferromagnetic (J > 0) Heisenberg exchange. For B/J > 2 the system ground state have S = 2 and Sab = 9/2. The extra electron was allowed to delocalize between sites A and B.

Exchange coupling, Heisenberg, 43 265-270 Exchange energy, 38 438-439 Exchange reactions, see also specific elements binuclear complexes, formation of, 10 183-188, see also Binuclear complexes one-equivalent, 10 154-178, 188-212 of actinide ions, 10 177, 178 of cerium, 10 176, 177, 206, 207, 211 of chromium, 10 163-168, 174, 175, 188-198, 206, 208... 

Due to the dependence of / on active nuclear distortions, the Heisenberg exchange interaction, equation (38), summed up over all nearest neighbour centres in a chain, should be added from the start to the energy functional in equation (32). Equation (41) shows, however, that this addition merely leads to the renormalization of the effective force constant Kt in equation (32) by... 

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. 

This relationship, known as Bloch s rule, applies in the localized-electron limit where the interatomic spin-spin interaction is described by the superexchange perturbation theory of eq. (23) with 7n J J is the Heisenberg exchange energy. Calculations by Shrivastava and... 

Fig. 23. Schematic energy diagram of the Heisenberg exchange energy J vs. bandwidth W at the crossover from localized-electron to band magnetism, after Zhou and Goodenough (2002a).

On a continuum level, the Heisenberg exchange energy of a cubic material is... 

In the approach of micromagnetism the quantum mechanical spin operators are substituted by classical vectors, S. Thus, the exchange energy in the Heisenberg model [34]... 

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