Molecular field theories

Molecular field theory (MFT) is the simplest theory of ordered magnetic moments in solids. Proposed in 1907 by P. Weiss, its main assumption is 

A simple estimate shows that magnetic order is not the result of a real magnetic field. The order of magnitude of the magnetic interaction energy for one spin, PBHy, can be compared with the thermal energy at the Curie transition, lc7  

The molecular field can be introduced in Eq. (4.18) to investigate thermal effects on spontaneous magnetisation  

Antiferromagnetic materials have two identical magnetic sublattices. Their resultant susceptibility is small and shows a maximum at the Neel temperature. Fig. 4.14. MFT is applied to antiferromagnets by considering 

The minus sign accounts for the antiparallel order between sublattices X is taken as positive). Fig. 4.15. Susceptibility can be expressed by  

---

Luckhurst G R, Zannoni C, Nordic P L and Segre U 1975 A molecular field theory for uniaxial nematic... 

Luckhurst G R 1985 Molecular field theories of nematics systems composed of uniaxial, biaxial or flexible molecules Nuclear Magnetic Resonance of Liquid Crystals ed J W Emsiey (Dordrecht Reidel)... 

Pratt, L. R. Ashbaugh, H. S., Self-consistent molecular field theory for packing in classical liquids, Phys. Rev. E 2003, 68, 021505... 

Most of the experimental results on CJTE can be explained on the basis of molecular field theory. This is because the interaction between the electron strain and elastic strain is fairly long-range. Employing simple molecular field theory, expressions have been derived for the order parameter, transverse susceptibility, vibronic states, specific heat, and elastic constants. A detailed discussion of the theory and its applications may be found in the excellent review by Gehring Gehring (1975). In Fig. 4.23 various possible situations of different degrees of complexity that can arise in JT systems are presented. 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

It is known that the classical molecular field theory discussed above is not suited for describing a close vicinity of the critical point. Experimentally obtained values of the parameter (3 (called the critical exponent) are essentially less than (3q = 1/2 predicted by the mean-field theory. On the other hand, the experimental values of (3 = 0.33-0.34 turn out to be universal for many different systems (except for quantum liquid-helium where (3... 

There are further models which do not introduce new ideas but use the above frame for more complicated types of interactions. Bari and SivardiSre215) discussed a two sublattice model in analogy to the molecular field theory of antiferromagnetism. In this case there are two different interaction constants, viz. the intrasublattice and the intersublattice interaction. They also expand the one-sublattice model by an Heisenberg type magnetic interaction term between the HS states. Such an interaction may only become important for degenerate spin states. 

Pyshnograi GV (1996) An initial approximation in the theory of viscoelasticity of linear polymers and non-linear effects. J Appl Mech Techn Phys 37(1) 123—128 Pyshnograi GV (1997) The structure approach in the theory of flow of solutions and melts of linear polymers. J Appl Mech Techn Phys 38(3) 122—130 Pyshnograi GV, Pokrovskii VN (1988) Stress dependence of stationary shear viscosity of linear polymers in the molecular field theory. Polym Sci USSR 30 2624—2629 Pyshnograi GV, Pokrovskii VN, Yanovsky YuG, Karnet YuN, Obraztsov IF (1994) Constitutive equation on non-linear viscoelastic (polymer) media in zeroth approximation by parameter of molecular theory and conclusions for shear and extension. Phys — Doklady 39(12) 889-892... 

Neutron diffraction experiments have confirmed that KCrF3 exhibits antiferromagnetism of type A with the spins lying in the pseudo-tetragonal (001) plane41) (TN = 40 K 6P = + 5 K42)). The exchange constants calculated on the basis of the molecular field theory have the same order of magnitude within and between two layers J3/k = 1.4 K and J2/k = - 2.2 K42). 

The field dependence of magnetization and of the susceptibility of a weakly anisotropic antiferromagnet - according to the molecular field theory - is described in Fig. 33 and the H = f(T) diagram in Fig. 34. 

I believe, however, that the molecular-field theory is supported by a sufficient number of facts that one can be certain that it contains an important part of the truth and that the difficulty of interpretation should be considered less an objection than a stimulus for research on new hypotheses of atomic structure . 

The quantities K R) describe occupancy transformations fully involving the solution neighborhood of the observation volume. These coefficients are known only approximately. Building on the preceding discussion, however, we can go further to develop a self-consistent molecular field theory for packing problems in classical liquids. We discuss here specifically the one component hard-sphere fluid this discussion follows Pratt and Ashbaugh (2003). 

Figure 7.12 Excess chemical potential of the hard-sphere fluid as a function of density. The open and filled circles correspond to the predictions of the primitive quasi-chemical theory and the self-consistent molecular field theory, respectively. The solid and dashed lines are the scaled-particle (Percus-Yevick compressibility) theory and the Carnahan-Starling equation of state, respectively (Pratt and Ashbaugh, 2003).

Figure 7.14 Comparison of Iny, with y the Lagrange multiplier of Eq. (7.30), p. 160, against computed excess chemical potential, /3yu = -ln/ (0), demonstrating the thermodynamic consistency of these quasi-chemical theories. The open circles are the primitive quasi-chemical theory (Eq. (7.29), p. 160), and the filled circles are the present self-consistent molecular field theory (Pratt and Ashbaugh, 2003).

Chen, Y.-G., Kaur, C., and Weeks, J. D., Connecting systems with short and long ranged interactions local molecular field theory for ionic fluids. J. Phys. Chem. B 108, 19 874-19 884 (2005). 

Applying superposition approximations to the Ising model, one finds an evidence for the phase transition existence but the critical parameter to is systematically underestimated (To is overestimated respectively). Errors in calculation of to are greater for low dimensions d. Therefore, the superposition approximation is effective, first of all, for the qualitative description of the phase transition in a spin system. In the vicinity of phase transition a number of critical exponents a, /3,7,..., could be introduced, which characterize the critical point, like oc f-for . M oc (i-io), or xt oc i—io for the magnetic permeability. Superposition approximations give only classical values of the critical exponents a = ao, 0 = f o, j — jo, ., obtained earlier in the classical molecular field theory [13, 14], say fio = 1/2, 7o = 1, whereas exact magnitudes of the critical exponents depend on the space dimension d. To describe the intermediate order in a spin system in terms of the superposition approximation, an additional correlation length is introduced, 0 = which does not coincide with the true In the phase... 

---