Molecular field model

For systems comprising magnetic centers that are both exchange coupled and affected by ligand field effects (i.e., non-Curie paramagnetic centers), the susceptibility (above an ordering temperature) can be approximated by molecular field models. 

When increasing the value of H, the moments gradually rotate towards the applied field direction and when H = Hc, they are parallel to the field. In the molecular field model, the spin-flop is a first-order transition and the spin flop 5 paramagnetic transition a second-order process. Between HSF and Hc magnetization increases linearly with H M/Ms = H (2HE - Ha) where Ms is the saturation magnetization of a sublattice (Ms = A NgMBS). At T = 0 K, the critical field can be written as HSF = [2HE HA/(1 - a)]1/2... 

Those curves that do not approach T = 0°K with zero slope are not realized in nature. The N6el model is a molecular field model, and is subject to the same criticisms as the Weiss field model for ferromagnets. Kaplan (325) has applied spin wave theory to ferri-magnets and worked out a Bloch Tz/2 law, similar to equation 98, for low temperatures. In this approximation M /M% remains constant,... 

Fig. 22. Possible magnetization curves for X = fi — 0.5, according to N6el (471) collinear, molecular field model. (Along the line SD, Ms = 0 for all temperatures.)...

Fig. 28. Paramagnetic susceptibilities of (a) yttrium and (b) gadolinium ferrogarnets. Solid lines are theoretical curves based on molecular field model and empirically determined molecular field constants. Dashed lines are experimental curves indicating effects of short-range order in the neighborhood of the Curie temperature. (After A16onard (5).)...

Tetrahedral (Mootz and Staben, 1992) and octahedral (Rustad etal., 2003) coordination environments are known for HO in crystalline hydrates. Evidently the numbers and arrangements of water molecules coordinating an HO ion are flexible enough to be decided by a crystal environment. Therefore development of the self-consistent molecular field models suggested by Section 7.8 would be valuable. Proximity of a specific cation is an issue, in general, for crystals. But it is interesting that, in crystalline NaOH hydrates beyond the monohydrate, the counter-ion is excluded from the inner hydration shell of both Na and HO (Rustad et al., 2003). The latter work used the PBE electron density functional, and found overall excellent results for crystalline NaOH hydrates. So that electron density functional model is able to properly characterize higher coordination structmes where they are known to exist. 

The molecular-field model is commonly used to describe the variation of the Curie temperature in the series R2Fe14B. In this model Tc can be written as... 

It was mentioned already that the analysis of the Curie temperature in terms of the molecular-field model does not lead to a wholly satisfactory description when compounds of light rare earths and heavy rare earths are considered. In the former compounds a higher value of /RFe seems to be required than in the latter. The same deficiency in the molecular-field description was reported by Hirosawa et al. (1986),... 

For TmV04 and TmAs04 the ground state in each case is a non-Kramers doublet that can be split by the true Jahn-Teller effect. For the vanadate the cooperative transition occurs at To = 2.156 K in this region the lattice heat capacity is negligible, and the co-operative anomaly dominates, as shown by the measurements of Cooke et al. (1972) in fig. 11. The distinctive triangular shape is a typical result for a molecular field model in which the splitting J of the doublet varies with temperature as... 

Within the molecular field model, the relation between the magnetic entropy and the magnetic equation of state is simply defined. Let us consider that the magnetic equation of state is a generalized / fimction, and so M = /[(H -I- A(M, T)M)/T]. We can en integrate the magnetic entropy relation ... 

At low temperatures, B Jiy) becomes exponentially small and pu>ng T) may be neglected, reflecting the insignificance of longitudinal spin fluctuations as the moment approaches saturation at T = 0. The transverse spin fluctuations, however, remain but are more accurately described within the spin wave approximation, because of the inadequacy of the molecular field model in describing the collective excitations. Calculations (Kasuya 1959, Yamada and Takada 1972) of the electron - magnon scattering cross-section yield a resistivity contribution ... 

Furrer and Kaldis (1976) have measured the excitations in a single crystal of HoP at a variety of temperatures. Even though this material orders in a flopside structure (see section 3.5), very little -dependence of the modes was observed. Some temperature renormalization for some of the modes was obser- ved. The spectra were fitted to a single ion molecular field model including... 

Fig. 187. KCuQj. Temperature dependence of j[ . The broken curve fitted according to Bleaney-Bowers (for a binuctear cluster) with J/k= —25.6 K, g=2.10. The dotted curve fitted for the same model with g=2.07. The solid curve calculated for the molecular field model — 25.6 K, =0.05,2=2, g=2.09). For details see original...

Fig. 27. Comparison of experimental (solid triangles) and calculated magnetic transition temperatures for the RCu Si, family. Dashed smooth line represents the de Gennes rule. Broken lines are trends obtained on the basis of the molecular field model (Noakes and Sbenoy 1982) including CEF effects. Broken solid line (open circles) indicates T s predicted by the mode with A as for GdCu Si, (tjtka et al. 1979). Broken dashed line (open squares) represents calculations with full CEF Hamiltonian (5AJ parameters are, according to Stewart and Zukrowski (1982) - and according to Kozlowski (1986) -O).

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