Spin-only model

It will be recalled that in Section 9.1 it was shown that the spin-only model for molecular paramagnetism gave rise to the equation... 

The value of the Ising model lies therein that it is the only model of disorder to have produced valid theoretical predictions of spontaneous phase changes. To understand the role of symmetry it is noted that spontaneous magnetization, starting from a random distribution of spins, amounts to a process of ordering that destroys an existing isotropic arrangement. 

All three models may be fit to the thermodynamic data for aqueous solutions of NiClp, LiCl, and their mixtures by adjusting their Gurney parameters. Thus, it is difficult to tell from the thermodynamic data whether one model is more realistic than the others. However only model B is in satisfactory agreement with the spin relaxation data. (Ll)... 

Transitions from a localized to an itinerant state of an unfilled shell are not a special property of actinides they can, for instance, be induced by pressure as they rue in Ce and in other lanthanides or heavy actinides under pressure (see Chap. C). The uniqueness for the actinide metals series lies in the fact that the transition occurs naturally almost as a pure consequence of the increase of the magnetic moment due to unpaired spins, which is maximum at the half-filled shell. The concept has resulted in re-writing the Periodic Chart in such a way as to make the onset of an atomic magnetic moment the ordering rule (see Fig. 1 of Chap. E). Whether the spin-polarisation model is the only way to explain the transition remains an open question. In a very recent article by Harrison an Ander-... 

Room-temperature Mossbauer data revealed that fast electron transfer (ti, < 10 s) is retained throughout the compositional range 0 < x < 0.8. Samples with x = 0.4 and 0.8 exhibited room-temperature neutron-diffraction data consistent with collinear FeA -ion spins below T, but with a significantly reduced moment (ca. 1 Hb vis a vis a saturated spin-only value) on the B sites. However, at 4.2 K, the spontaneous magnetization could be interpreted with a Neel ferrimagnetic model ... 

Several simple models exist5 that approximately describe the temperature dependence of x for transition metal cations that do not represent spin-only centers. As one example that is applicable to coordination complexes at low temperatures, the Kotani theory6 incorporates the effects of spin-orbit coupling into the Van Vleck equation and describes y(T) as a function of the spin-orbit coupling energy C,. 

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 ... 

In the model of localized magnetic moments for the spin-only state (orbital moment is quenched) the interrelation between the effective magnetic moment and the moment in the magnetically ordered state is given by... 

The spin-only value, 2.80 + 0.17 BM, for the magnetic moment of the V1 cation of V(mesitylene)+ cannot be reconciled with a ligand-field model, even allowing for very substantial distortions from Cmv symmetry.10... 

The vector model cannot be interpreted in such a simple way in the case of a spin system with more than one nucleus. For weakly coupled spin systems, the single spin vector model may be applied for each nucleus, one after the other. Thus the coupling with the other nuclei can be incorporated into its precession frequency, since the definition of the weak coupling (J -C vM v,/1) means that the transitions of a nucleus only depend on the spin states of the other nuclei in the first order. The detected signal is the sum of the sine curves provided by the individual environment of the nuclei. 

The single spin vector model demonstrates well the relationship between spin set and spin system. In the spin system, all possible environments of the nucleus (i.e. all frequencies) are incorporated. Meanwhile, the spin set contains only one nucleus the different frequencies of which are corresponding to the different conformers. So the spin set is rather taken as the extension of the definition of a nucleus than the simplification of a spin system.102... 

Most of the simple fitness functions, including the spin-glass models and the AW-modcl, are isotropic (Schuster and Stadler, 1994). However, it is expected that real sequence spaces are highly anisotropic with most regions devoid of fitness, and functional regions that are rich with landscape features. A real protein landscape can appear isotropic in two ways. The first is if a subset of the space is defined—for instance, only the functional sequences are considered. Further, the landscape can... 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

The impact of this is tremendous. No long-range order (LRO) can exist at finite temperature in one dimension no crystals, no magnets, no superconductors. Only special transitions are possible in two dimensions. The Ising model (n = 1 component) is an example [7]. The Kosterlitz-Thouless transition [8], without LRO, is another case for d = 2 and n = 2, discussed in Section V.C. The thermal fluctuations are very destructive in lower dimensions. Quantum fluctuations (i.e., those associated with the dynamics of a system) also tend to suppress LRO and can sometimes destroy it even at 0 K when the Mermin-Wagner theorem does not apply. Such is the case of the quantum spin- antiferromagnetic models [9] in one dimension. 

---