Weiss model

P. Lansky, M. Weiss. Modeling heterogeneity of particles and random effects in drug dissolution. Pharm. 

R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss. Model for photon migration in turbid biological media. Journal of the Optical Society of America A, 4(3) 423-432, 1987. 

Source Equation numbers in [ ] as in J. E. Kiefer, V. A. Parsegian, and G. H. Weiss, "Model for van der Waals attraction between spherical particles with nonuniform adsorbed polymer," J. Colloid Interface Sci., 51, 543-545 (1975). 

Rose and Waite [5] have developed a detailed kinetic model to describe the oxidation of Fe(II) in the presence of the well-characterised Suwannee River fulvic acid by extension of the Haber-Weiss model [91]. In this model, O2 reduces to OH by oxidation of Fe(II) (either inorganic or complexed with an organic ligand L) in a four-step process ... 

It has been pointed out that any relationship between the exchange integral and the Weiss field is only valid at 0 K, since the former considers magnetic coupling in a pair-wise manner and the latter results from a mean-field theory (Goodenough, 1966). Finally, it is also essential to understand that Eq. 8.43 is strictly valid only for localized moments (in the context of the Heitler-London model). One might wonder then whether the Weiss model is applicable to the ferromagnetic metals, in which the electrons are in delocalized Bloch states, for example, Fe, Co, and Ni. This will be taken up later. 

The discussion of the preceding two sections relied on the presumption that localized (atomic-like) moments were present. However, valence s and p electrons are always best described by Bloch fimctions, while 4/electrons are localized and 5/are intermediate. Valence d electrons, depending on the intemuclear distance, are also intermediate -neither free nor atomic-Uke. In such cases, the dilemma is that the Heisenberg exchange interaction of Eq. 8.43, which is the physical basis for the Weiss field, is not strictly applicable in the case of delocalized electrons in metallic systems, in spite of the success of the Weiss model. 

Treating a two-level system in an individual way is, of course, a delicate matter and can be quite misleading, as will be seen later in our discussion of the Curie-Weiss model. This point cannot be stressed often enough. It is only for reasons of simplicity that two-level systems are used here for purposes of illustration. 

In summary, statistical quantum mechanics permits us to derive strictly classical observables (such as the classical specific magnetization operator) by appropriate limit considerations (such as a limit of infinitely many spins in case of the Curie-Weiss model). However, statistical quantum mechanics cannot cope with fuzzy classical observables (for finitely many degrees of freedom) since different decompositions of a thermal state Dp are considered to be equivalent. The introduction of a canonical decomposition of Dp into pure states will give rise to an individual formalism of quantum mechanics in which fuzzy classical observables can be treated in a natural way. 

It is, unfortunately, not simple to compute the maximum entropy decomposition in molecular situations. We shall therefore consider again the simpler example of the (quantum-mechanical) Curie-Weiss model with the Hamiltonian ... 

FIGURE 11 An entropy function in the sense of fluctuation (i.e., large-deviation) theory, describing how fast the mean magnetization of a spin system gets classical with an increasing number of spins. The figure is based on an approximate calculation for the Curie-Weiss model. The temperature is fixed and has been taken here as one third of the critical (Curie) temperature. Above the Curie temperature the respective entropy Sn ,an would only have one minimum, nameiy, at m = 0. 

Bearing in mind the large-deviation considerations for the Curie-Weiss model, one could try to characterize molecules by some large-deviation entropy that describes how fast a nuclear molecular structure appears with increasing molecular nuclear masses. Such a large-deviation entropy would describe the decrease in fuzziness of the molecular nuclear structure when the nuclear masses increase. In the limit of infinite nuclear masses one expects a strictly classical nuclear framework, this not being fuzzy anymore at all. Such a large-deviation entropy would also nicely describe the quantum fluctuations round the strictly classical nuclear structure. 

Anderson-Weiss model of relaxation [Andl] the decay of the Hahn-echo amplitudes of the transverse H magnetization as a function of the echo time tE has been derived [Siml] ... 

The eonsiderations to be presented here seem to be readily extendible to any model that can be exactly soluble by molecular field methods (e.g., BCS). The physical drawbacks of these models are well-known. We retain here the Weiss model for antiferromagnetism for its didactie value, and consider simply the interactions of the form... 

Order-disorder transitions are examples of a second-order transformation. An order-parameter can be assigned that goes from one for a perfectly ordered state to zero for a completely random state, i.e., a solid solution. Using a technique similar to the Curie-Weiss model for ferromagnetism (see Chapter 25), it can be shown that the order parameter goes from 1 at low temperatures to 0 at the transition temperature and the system then becomes a solid solution again. 

Polarization versus temperature predicted using the Curie-Weiss model for spontaneous polarization. 

See Ref. 35 for (a) further consideration of the Pauling-Coryell, McClure and Weiss models of Fe-02 bonding in oxyhemoglobin, and (b) Refs. 19, 24 and 62 therein for reviews of computational studies of heme-02 and heme-NO bonding. 

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