The Molecular Field Theory

In the halls of chemistry departments, we often hear how difficult it is to understand quantum chemistry. It is hard to know if this situation reflects a problem on the side of the teacher or the student. Somehow it should be easier to study quantum chemistry than classical mechanics, since so many areas of study in the 20th century have been shaped by quantum theory. In other words, the essence of quantum theory has invaded our conventional ideas. For example, we say If there is no conflict, it has to be right. The logic of this statement is obviously related to the formulation of quantum theory. One would think that it should be easier to understand a field based on contemporary logic than fields rooted in classical logic. Pedagogues often believe one should study history systematically from the past to the latest events. This does not mean that the concept of a field is easy to understand, but it reflects the fact that history is a human drama. If many people took this attitude, the study of quantum theory could become quite enjoyable. 

In the limited space of this book we cannot explain in detail the large number of equations presented in this section. We will assume that the reader has learned quantum chemistry in his 

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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 . 

In order to understand the concept of Molecular Photonics, it is crucial for the reader to undertake a study of fundamental principles. Chapter 1 Fundamentals of Molecular Photonics includes four sections dedicated to optics, the molecular field theory, the radiation field theory, and the interactions between the molecular field and the radiation field. Fundamental principles are often treated in an introductory chapter, leading the reader to think that they are of little importance and that they can be understood with ease. This trend of relegating the fundamentals to a brief introduction is getting increasingly common in natural... 

The theories of photo- and opto-related areas can be classified into three categories the fundamentals of optics, the molecular field theory, and the radiation field theory. As we defined molecular photonics by Equation (0.1) which relates the interaction of the radiation field with the molecular field, it may seem sufficient to restrict our discussion to the molecular field theory and the radiation field theory. However we believe that the fundamentals of optics are also very important to understand and appreciate all the photo and opto concepts described in this book. To support this view, consider the following. 

The usual antiferromagnet can be viewed as a system consisting of two interpenetrating sublattices. Each sublattice is uniformly magnetised with spins aligned parallel, but the magnetisations of the sublattices are cancelled exactly. The molecular field theory assumes that the field acting on sublattice A is... 

The molecular field theory assumes that the field acting on sublattice A is... 

In the molecular field theory, the exchange energy per ion is given from eq. (7.3), by... 

The Bogoliubov approximation described above obtains the same result without the (Pq-Pi) factor. At low temperatures, Po= 1, Pi = 0, and for the situation where (q) is a maximum at q = 0, we see that the collective mode will go soft at q = 0 when A (defined in eq. (7.44)) reaches 1, which is also the threshold value for induced moment ordering in the molecular field theory. The temperature renormalization of w, in this theory is contained in the factor (Po—Pi) which may be evaluated in the molecular field approximation,... 

The molecular field theory has been quite successful in describing the magnetic properties of many simple ferro-ferri- and antiferromagnetic spin arrangements in solids. According to this, the expression for the magnetic ordering temperature of various types of order and crystal structures is... 

When applied to a ferromagnetic material the molecular field theory gives for the Curie temperature... 

If the spins in a lattice are aligned spontaneously in the same direction thus producing a finite magnetization, the ordered state is ferromagnetic. In the molecular field theory of ferromagnetism, the existence of an internal field ifi , is assumed which is proportional to the magnetization... 

The molecular field theory of this particular system has been worked out by Blume (1966) in order to explain the magnetism of UOa. Only for relatively small level separation Fi-Fs is the phase transition of second order. Larger level separation leads either to a first order phase transition or no transition depending on the coupling strength. 

The magnetic properties of the rare earth metals systems with localized magnetic moments, can be described by means of the molecular field theory. In the frame of this model the Curie temperature Tq and the paramagnetic transition temperature Tp may be expressed as (Smart 1966)... 

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