Exchange anisotropic

As long as we are concerned with binuclear = 1/2 complexes, no single-ion anisotropy has to be added and this Hamiltonian describes the lowest energy levels in the absence of an external magnetic field. 

Symmetric anisotropy The basis of this Hamiltonian can no longer be restricted to determinants with the same Ms value as was done for the isotropic interactions. The inclusion of magnetic anisotropy in the model causes the removal of the degeneracy of the different Ms levels and eventually mixing of the wave functions with different spin moment. Here, we have to consider four CSFs the three components of the triplet plus the singlet. To facilitate the determination of the matrix elements of the model 

Hamiltonian, it is common practice to consider the basis of uncoupled determinants 

12 Perform the matrix multiplication of the anisotropic term in the model Hamiltonian. 

The uncoupled basis is formed by the determinants la o l, a, pa and The result of acting with the isotropic part of Hamiltonian on these determinants can directly be written down with the help of Eqs. 3.79, but the anisotropic part requires a little more work. Based on the relations given in Eqs. 1.16a and 1.20a, the following is easily derived for the products of one-electron operators 

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Finally, it is noteworthy that in addition to the isotropic part of spin coupling as treated above, there may also be an anisotropic contribution due to the presence of anisotropic exchange [114] or dipole interaction [106]. In this case, the exchange coupling constant is replaced by a tensor... 

D SAxSB (asymmetric exchange or Moriya-Dzialoshinski interaction +SA K SB (anisotropic exchange or pseudo-dipolar interaction)... 

Anisotropic exchange interaction within the Lines model described above. Dipole-dipole magnetic interaction between centres is calculated fully ah initio and is added to the exchange matrix. 

Perhaps the most detailed studies of both the isotropic and anisotropic exchange interactions have been made for InP, where the anisotropic (pseudo-dipolar) part of the 31P-115In exchange coupling was shown to reduce the measured 31P second moments from theoretical values calculated from dipolar couplings alone [255, 256]. Double and triple resonance 31P MAS-NMR experiments have also provided much information as a result of the existence of the two nearly identical isotopes of... 

To model several other electronic situations (e.g., in mixed-valence compounds), the Heisenberg Hamiltonian can be augmented by specific exchange operators that accommodate, for example, double exchange, antisymmetric exchange, anisotropic exchange, or biquadratic exchange. 

In noncubic materials, A must be replaced by the 3x3 exchange-stiffness tensor A v, and the energy is X J A v dM/dxy dM/dxv dV. Here the indices ju and v denote the spatial coordinates x, y, and z of the bonds. The energy is anisotropic with respect to the nabla operator = d/d (bond anisotropy) but isotropic with respect to the magnetization M. By contrast, the relativistic anisotropic exchange Xa i Ha(i VMa VMP dV is isotropic with respect to V but anisotropic with respect to M. 

It can be seen from the following discussion that this can lead to a three-dimensional spectrum in which two-dimensional anisotropic exchange spectra for the different chemical sites are separated according to the isotropic chemical shifts associated with the different sites. Consider first a two-dimensional VACSY experiment (Fig. 26), in which a simple FID is collected under sample spinning at different spinning angles. The signal acquired has the form... 

The interactions are of the ferromagnetic-Heisenberg type with 1% XY-square anisotropy arising from both anisotropic exchange and dipole-dipole interactions99. The xOz plane is the easy one and the spins are weakly bound in the Ox direction80,". ... 

Anisotropic exchange exchange coupling between electrons that is dependent on direction that is, exchange along the x coordinate, for example, differs from exchange along the y or z coordinates... 

The anisotropic exchange Hamiltonian appears as eqnation (29). It retains the form of a prodnct of spins 5, and Sj bnt the spin components are given different weights. The x- and... 

Fig. 7 Dinuclear model complex of symmetry, used for the calculation of the anisotropic exchange coupling (adopted from [27])...

Modelling of Anisotropic Exchange Coupling in Rare-Earth-Transition-Metal Pairs Applications to Yb3+-Mn2+ and Yb3+-Cr3+ Halide Clusters and Implications to the Light Up-Conversion (M. Atanasov, C. Daul H. U. Gtidel)... 

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