Anisotropic Heisenberg model

Temperature dependence of magnetic susceptibility of the PF6 salt was measured from 300 to 4 K at 5 T [35], The spin susceptibility of this salt gradually decreases from 300 to 50 K. Below 50 K, the susceptibility exhibits a rapid decrease accompanied by anisotropic temperature dependence, which is an indication of the long-range antiferromagnetic ordering. A one-dimensional Heisenberg model is... 

The coefficients a and b (see Table 3.2) take into account the restrictions in spin dimensionality. For a = b= 1, the Heisenberg model with isotropic exchange interaction and isotropic susceptibility results. The combination of a = 1 and b = 0 yields the strictly anisotropic Ising model, in which the orientation of the spins is restricted to the z-axis. Consequently, the susceptibility is strongly orientation dependent and one needs to differentiate between x" in the direction of the z-axis ( easy axis ) and x perpendicular to z. The molar susceptibilities are then related as... 

Alkynyl-bridged macrocycUc metal complexes, 704 a-phase, 110 Alternate layers, 731. 751 Alternating Heisenberg model, 786 Amphiphilic molecule, 728 Analogues of TCNQ, 229. 245 Aniline linked fullerenes, 444 Anisotropic conductivity, 758 Anisotropy, 100. 113. 667 Anodic intercalation, 651 of graphite, 653... 

The Heisenberg approach remains valid as long as the magnetic centers act as spin-only centers and represents an entirely empirical model. Orbital contributions can be accommodated as perturbations. For example, ligand field effects can be effectively approximated by the following anisotropic spin Hamilton operator ... 

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 2-D and 1-D systems even in non-uniaxial materials, the Neel model can still be used since the anisotropic field in the plane where the moments can rotate is small compared with the exchange field in the perpendicular direction. After first considering the particular case of CsNiF3, we will described several 2-D and 1-D Heisenberg fluorides showing spin-flop behavior. 

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)...

The elimination of the anisotropic part in Eq. 3.22 leads to the Heisenberg Hamiltonian for isotropic magnetic interactions. The spins are considered as co-linear vectors whose principal quantization axis has no spatially preferred orientation. An even simpler model Hamiltonian can be obtained by putting Axx and Ayy to zero in Eq. 3.21. Then, the spin reduces to a classical vector whose orientation in space is not defined and the resulting model Hamiltonian describes the isotropic coupling of two (anti-)parallel spins. Replacing A z by -J, the following expression is obtained... 

Apart from the isotropic Heisenberg Hamiltonian Eq. (49), a very anisotropic form of interaction called the Ising model,... 

Models have become separate research areas in statistical and solid-state physics. In addition to Hubbard models, familiar examples are two-level models, impurity models, and networks of spins in one, two, or three dimensions with isotropic, anisotropic, or frustrated interactions. Exact results for Hubbard chains with uniform t and for 5 = 1/2 Heisenberg chains with uniform exchange J are based on the Bethe ansatz, while exact results for two-dimensional Ising lattices come from transfer matrix techniques. Exact results for extended... 

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