Microscopic theories

4 Fermi resonance with polaritons 6.4.1 Microscopic theory 

Let us consider the effects that arise when the branch of C phonons corresponds to dipole-active vibrations. In the region of small values of k = 2-k/X, where 

A is the wavelength of light with frequency oj = t o /h, C phonons of this kind strongly interact with the transverse photons. As a result, in the region of long wavelengths, new elementary excitations - phonon-polaritons (see Ch. 4) - are formed instead of C phonons and transverse photons. 

The spectrum of polaritons can be found by means of Maxwell s macroscopic equations (see Ch. 4), provided that the dielectric tensor of the medium (44) is assumed to be known. Without going into details, we emphasize here that always a gap appears in the polariton spectrum (here we ignore spatial dispersion) in the region of the fundamental dipole-active vibration (C-phonon, exciton, etc.). At present, there is a sufficiently detailed theory for RSL by phonon-polaritons, taking many phonon bands into consideration. With this theory the RSL cross-section can be calculated for various scattering angles provided that the dielectric tensor of the crystal is known, as well as the dependence of the polarizability of the crystal on the displacement of the lattice sites and the electric field generated by this displacement (45). 

It is an essential fact that the above-mentioned gaps in the polariton spectrum, if they arise, as well as the corresponding interaction between the photon and phonon, are nonzero within the framework of linear theory and, in general, do not require that anharmonicity be taken into account. Therefore, it makes sense to denote as a polariton Fermi resonance only such situations where vibrations of overtone or combination tone frequencies resonate with the polariton. We now turn our attention to an analysis of such rather complex situations, requiring that multiparticle excited states of the crystal be taken into consideration. Shown schematically in Fig. 6.6 is a typical polariton spectrum, as well as a band of two-particle states of B phonons. If, under the effect of anharmonicity, biphonons with energy E = E are formed, these states also resonate with the polariton, influencing its spectrum. 

For a solid to interact with a magnetic field intensity, it must possess a net magnetic moment which, as discussed momentarily, is related to the angular momentum of the electrons, as a result of either their revolution around the nucleus and/or their revolution around themselves. The former gives rise to an orbital angular moment whereas the latter is the spin angular moment /ij. The sum of these two contributions is the total angular moment of an atom or ion, /ijon- 

From elementary magnetism a current i going around in a loop of area A will produce an orbital magnetic moment /Zorb given by 

Assuming the electron moves in a circle of radius r, then combining Eqs. (15.13) and (15.14), one obtains 

But since MgUjQr is nothing but the orbital angular momentum Ho of the electron, it follows that 

The ratio eh/Anm occurs quite frequently in magnetism and has a numerical value of 9.27 x lO A m. This value is known as the Bohr magneton /x, and, as discussed below, it is the value of the orbital angular momentum of a single electron spinning around the Bohr atom. In terms of Eq. (15.17) can be succinctly recast as  

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Microscopic theory yields an exact relation between the integral of the radial distribution function g(r) and the compressibility... 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

Petukhov A V 1995 Sum-frequency generation on isotropic surfaces general phenomenology and microscopic theory for ]ellium surfaces Phys. Rev. B 52 16 901 -11... 

Mendoza B S, Gaggiotti A and Del Sole R 1998 Microscopic theory of second harmonic generation at Si(IOO) surfaces Phys. Rev. Lett. 81 3781-4... 

R. Evans. Microscopic theories of simple fluids and their interfaces. In J. Charvolin, J. F. Joanny, J. Zinn-Justin, eds. Liquids at Interfaces. Amsterdam North-Holland, 1990, pp. 4-98. 

Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a sohd can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables Hi = 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. (The connection with magnetic systems is made by a transformation to spin variables cr, = 2n, — 1.) In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model Hamiltonian which, for the simplest system of a one-component adsorbate with one adsorption site per unit cell, is... 

Microscopic theory of nucleation in metastable alloy states... 

Martensitic traasfonnation Master ec[uations Mean field crossover to Ising Mechanical properties Metallic alloys Metallic glasses Metastable alloys Microhardness test Microscopic theory of nucleation... 

A spin-gas microscopic theory has been pursued by Berker et al. [32] to explain the multiplicity of smectic ordering and the re-entrance phenomenon in strongly polar mesogens. They have used a model Hamiltonian of the form... 

The concentrations of the reactants and reaction prodncts are determined in general by the solution of the transport diffusion-migration equations. If the ionic distribution is not disturbed by the electrochemical reaction, the problem simplifies and the concentrations can be found through equilibrium statistical mechanics. The main task of the microscopic theory of electrochemical reactions is the description of the mechanism of the elementary reaction act and calculation of the corresponding transition probabilities. 

Statistical mechanical and microscopic theories of the double layer have often been intensively reviewed. Our focus is only on features related to the problem of negative C. 

Further progress in understanding membrane instability and nonlocality requires development of microscopic theory and modeling. Analysis of membrane thickness fluctuations derived from molecular dynamics simulations can serve such a purpose. A possible difficulty with such analysis must be mentioned. In a natural environment isolated membranes assume a stressless state. However, MD modeling requires imposition of special boundary conditions corresponding to a stressed state of the membrane (see Refs. 84,87,112). This stress can interfere with the fluctuations of membrane shape and thickness, an effect that must be accounted for in analyzing data extracted from computer experiments. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

Simonson, T. Perahia, D. Briinger, A.T., Microscopic theory of the dielectric properties... 

D. The Microscopic Theory of Long-Range Coulomb Forces. 195... 

No attempt will be made here to extend our results beyond the simple lowest-order limiting laws the often ad hoc modifications of these laws to higher concentrations are discussed in many excellent books,8 11 14 but we shall not try to justify them here. As a matter of fact, for equilibrium as well as for nonequilibrium properties, the rigorous extension of the microscopic calculation beyond the first term seems outside the present power of statistical mechanics, because of the rather formidable mathematical difficulties which arise. The main interests of a microscopic theory lie both in the justification qf the assumptions which are involved in the phenomenological approach and in the possibility of extending the mathematical techniques to other problems where a microscopic approach seems necessary in the particular case of the limiting laws, obvious extensions are in the direction of other transport coefficients of electrolytes (viscosity, thermal conductivity, questions involving polyelectrolytes) and of plasma physics, as well as of quantum phenomena where similar effects may be expected (conductivity of metals and semi-... 

In Section III, we discuss the equilibrium properties of dilute strong electrolytes we first give a brief critical summary of the macroscopic approach and we consider next the microscopic theory, following the work of Balescu,1 and we try to make as clear as possible the approximations involved. 

In this chapter, we shall first make a brief review of the phenomenological aspect of Brownian motion and we shall then show how the general transport equation derived in Section II allows an exact microscopic theory to be developed. 

In this section, we shall first give a brief review of the phenomenological theory of these effects.5 -6 26 We shall then show how the methods we have discussed in the previous sections may be extended to derive a microscopic theory of the relaxation effect the microscopic theory of electrophoresis will be considered in the next section. 

VI. MICROSCOPIC THEORY OF ELECTROPHORESIS AN EXAMPLE OF HYDRODYNAMICAL LONG-RANGE... 

From our present point of view, it will suffice to stress that Eqs. (443) and (444) have been obtained here through a microscopic analysis of the iV-body problem involving the fluid and the heavy B-particle. We are led to the conclusion that microscopic theory indeed allows us to show that ... 

Microscopic Theory of Amontons s Laws for Static Friction. 

MSN.94. I. Prigogine, The microscopic theory of irreversible processes, in Proceedings, 11th Symposium Parked Gas Dynamics, Vol. 1, CEA, Paris, 1979, pp. 1-27. 

MSN. 103.1. Prigogine and C. George, The second law as a selection principle The microscopic theory of dissipative processes in quantum systems, Proc. Natl. Acad. Sci. USA 80,4590-4594 (1983). 

In a mode coupling approach, a microscopic theory describing the polymer motion in entangled melts has recently been developed. While these theories describe well the different time regimes for segmental motion, unfortunately as a consequence of the necessary approximations a dynamic structure factor has not yet been derived [67,68]. 

Thereby, the features of the a-relaxation observed by different techniques are different projections of the actual structural a-relaxation. Since the glass transition occurs when this relaxation freezes, the investigation of the dynamics of this process is of crucial interest in order to understand the intriguing phenomenon of the glass transition. The only microscopic theory available to date dealing with this transition is the so-called mode coupling theory (MCT) (see, e.g. [95,96,106] and references therein) recently, landscape models (see, e.g. [107-110]) have also been proposed to account for some of its features. 

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