Planar spin model

If b = 0, a = 1 the Ising model, if b = 1, a = 1 the isotropic Heisenberg model, and if b = 1, a = 0, the planar Heisenberg model or X-Y model is valid. In the following, the values of the exchange constants were calculated, or recalculated assuming a spin Hamiltonian of this form, instead of = - J 2 SjSi, occasionally used by certain au-... 

Figure 7.9. Magnetization energy of planar spin spirals at a volume of 6.78 a.u., calculated directly with the BGFM (full circles) or using the Heisenberg model with either FM (dashed lines) or DLM (full lines) reference states. The open squares and triangles represent force theorem calculations from the ferromagnetic and anti-ferromagnetic states respectively.

In Fig. (7.8) are shown direct calculations of the planar spin spirals as a function of moment and of volume, the moment is seen to increase monotonically with the volume. One can see that the spin spiral qxw is stable for low volumes and moments, while the spin spiral qrx is stable at higher volumes and moments. Now, we calculate the spin spirals for fixed moments, both with the BGFM and with with the Heisenberg model with both FM and DLM interactions, just as described above for bcc Fe, fee Ni, and fee Co. We also calculate the spin spirals directly with the force theorem without the Heisenberg expansion by doing one calculation with the potentials from either the ferromagnetic or anti-ferromagnetic solution and the new spin spiral structure. The results can be seen in Fig.(7.9). 

Generalizations [71] of the bare anisotropic-planar-rotor model (2.5) include other multipolar interactions such as dipolar and octopolar terms with and without in-plane crystal-field modulations. Several such combinations were analyzed in the mean-field approximation, Landau theory, and spin-" wave expansion [71]. The quadrapole-quadrupole model written in the form... 

Currem field characteristics measured wiih conjugated polymers sandwiched between an indium-tin oxide (ITO) anode and an aluminum cathode are usually hole dominated and are, consequently, appropriate for testing injection/lransport models for the case of unipolar current How. Data shown in Figure 12-1 refer to injection-limited currents recorded on typically 100 nm thick spin-coated films of derivatives of poly(y d/"fi-phenylenevinylene) (PPV) and a planarized poly(/ /" -pheny-leue) employing a Keilhley source measure unit. The polymers were ... 

Any computational treatment of TM systems must account for the LFSE. QM methods achieve this implicitly but d-electron effects must be explicitly added to MM (4). Some effects can be modeled within conventional MM. For example, low-spin d8 complexes are planar by virtue of the LFSE (21,22), but a planar structure can also be enforced using a normal out-of-plane term (22). However, the simplest general model for describing d-orbital energies is ligand field theory (LFT) (23) which was itself derived from the earlier electrostatic crystal field theory (CFT) (24) approach. 

Fig. 24 Magnetic susceptibility for BABI at 10,000 Oe external field. A-D represent fits to the experimental data (A) square planar AFM system with J/k = — 1.6 K (B) square planar bilayer AFM system with J2olk = —1.4 K and interlayer Jik = —1.3 K (C) AFM spin pairing with Jjk = —3.8 K C is same model as C, with Jik = — 2.4 K D is same model as B, with J2T>lk = — 1.2K and interlayer Jik = — 1.9K (from calorimetric analysis).

Fig. 27 Magnetic heat capacity for PhBABI for 7 < 100 K showing variation with external magnetic field (left) zero-field magnetic heat capacity showing fits (right) to ID AFM chain, 2D AFM square planar, 2D AFM square planar bilayer, singlet-triplet spin pairing (ST), and spin ladder models.

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