Microscopic tensor properties

J. A. Weil, T. Buch and J. E. Clapp, Crystal point group symmetry and microscopic tensor properties in magnetic resonance spectroscopy. Adv. Magn. Reson., 1973, 6,183-257. 

The problem becomes more complex when studying solid phases because the microscopic NLO responses do not provide the full information about their macroscopic coimterparts, the second- and third-order nonlinear susceptibilities, and To make the transition between the microscopic and macroscopic, it is necessary to know the structure of the condensed phases as well as the nature and the effects of the intermolecular interactions in the bulk of the material. In both the Physics and Chemistry arena, several schemes have been proposed to characterize the NLO responses of solid phases. One of the authors has recently contributed to review these approaches [3] of which one of the extremes is occupied by the oriented gas approximation that consists in performing a tensor sum of the microscopic NLO properties to obtain the macroscopic responses of the crystal. The other extreme consists in performing a complete treatment of the solid by using the supermolecule method or by taking advantage of the spatial periodicity in crystal orbital calculations. In between these techniques, one finds the interaction schemes and the semi-empirical approaches. 

The magnetization was only taken as an example. Many other properties (dielectric susceptibility, electric and thermal conductivity, molecular diffusion, etc.) are also described by second rank tensors of the same (quadrupolar) type Microscopically, such properties can be described by single-particle distribution functions, when intermolecular interaction is neglected. There are also properties described by tensors of rank 3 with 3 = 27 components (e.g., molecular hyperpolarizability Yijk) and even of rank 4 (e.g., elasticity in nematics, ATiju) with 3 = 81 components. Microscopically, such elastic properties must be described by many-particle distribution functions. 

One property of these tensors is quite evident now. The trace of a tensor is simply the sum of the diagonal components. For the microscopic tensor, the trace of T is just... 

Effective collision cross sections are related to the reduced matrix elements of the linearized collision operator It and incorporate all of the information about the binary molecular interactions, and therefore, about the intermolecular potential. Effective collision cross sections represent the collisional coupling between microscopic tensor polarizations which depend in general upon the reduced peculiar velocity C and the rotational angular momentum j. The meaning of the indices p, p q, q s, s and t, t is the same as already introduced for the basis tensors In the two-flux approach only cross sections of equal rank in velocity (p = p ) and zero rank in angular momentum (q = q = 0) enter die description of the traditional transport properties. Such cross sections are defined by... 

To understand and optimize the electro-optic properties of polymers by the use of molecular engineering, it is of primary importance to be able to relate their macroscopic properties to the individual molecular properties. Such a task is the subject of intensive research. However, simple descriptions based on the oriented gas model exist [ 20,21 ] and have proven to be in many cases a good approximation for the description of poled electro-optic polymers [22]. The oriented gas model provides a simple way to relate the macroscopic nonlinear optical properties such as the second-order susceptibility tensor elements expressed in the orthogonal laboratory frame X,Y,Z, and the microscopic hyperpolarizability tensor elements that are given in the orthogonal molecular frame x,y,z (see Fig. 9). 

The oriented gas model was first employed by Chemla et al. [4] to extract molecular second-order nonlinear optical (NLO) properties from crystal data and was based on earlier work by Bloembergen [5]. In this model, molecular hyperpolarizabilities are assumed to be additive and the macroscopic crystal susceptibilities are obtained by performing a tensor sum of the microscopic hyperpolarizabilities of the molecules that constitute the unit cell. The effects of the surroundings are approximated by using simple local field factors. The second-order nonlinear response, for example, is given by... 

The existence of this relation should be no surprise since, as we have demonstrated, the tensor ejj(u ,k) determines the frequencies of all normal electromagnetic waves in a condensed medium. But this relation, as we show below, permits a simplified consideration of some properties of Coulomb excitons including the dependence of Coulomb exciton energies on s for k —> 0, which by using microscopic theories is quite tedious. 

It is well-known that the simplest approach to the study of the optical properties of condensed media is the macroscopic electrodynamics approach, making use of the concept of the dielectric constant tensor f..tJ (oj, k) where w and k are the frequency and the wavevector of the light wave. Calculation of this tensor for a specific medium is, however, a problem of microscopic theory. For instance, the procedures for calculating the tensor k) for the excitation region of the... 

The original Placzek theory of Raman scattering [30] was in terms of the linear, or first order microscopic polarizability, a (a second rank tensor), not the third order h3q)erpolarizability, y (a fourth rank tensor). The Dirac and Kramers-Heisenberg quantum theory for linear dispersion did account for Raman scattering. It turns out that this link of properties at third order to those at first order works well for the electronically nonresonant Raman processes, but it cannot hold rigorously for the fully (triply) resonant Raman spectroscopies. However, provided one discards the important line shaping phenomenon called pure dephasing , one can show how the third order susceptibility does reduce to the treatment based on the (linear) polarizability tensor [6, 27]. 

In order to obtain a useful material possessing a large second order nonlinear susceptibility tensor % 2) one needs to use molecules with a large microscopic second order nonlinear hyperpolarizability tensor B organised in such a way that the resulting system has no centre of symmetry and an optimized constructive additivity of the molecular hyperpolarizabilities. In addition, the ordered structure thus obtained must not loose its nonlinear optical properties with time. The nonlinear optical (NLO) active moieties which have been synthesized so far are derived from the donor-rc system-acceptor molecular concept (Figure 1). 

Microscopic expression for die stress tensor Let us now study the viscoelastic properties using molecular models. As was discussed in Chapter 3, the macroscofnc stress of the polymer solutions is written as (see eqn (3.133))... 

One application of the stress theorem is the study of elastic properties of solids, which becomes straightforward when a suitable finite macroscopic strain is applied to the solid. When the wavefunctions of the distorted solid are known, the stress tensor is evaluated with the stress theorem. In the harmonic approximation elastic constants are defined as the ratio of stress to strain, and it is furthermore possible to go to large strains to obtain all nonlinear elastic properties. In general it is necessary to be concerned with internal strains that may appear microscopically owing to the lower symmetry of the strained solid. In section 6 we show in detail how this problem is solved by combining the stress and force theorems. 

While an explicit treatment of the microscopic, atomistic degrees of freedom is necessary when a locally realistic approach is required, on a macroscopic level the continuum framework is perfectly adequate when the relevant characteristics are scalar, vector, or tensor fields (e.g., displacement fields for mechanical properties). A combination of the two levels of description would be useful [29,30]. Here we focus on the deformation of solids The elastic mechanics of homogeneous materials is well understood on the atomistic scale and the continuum theory correctly describes it. Inelastic deformation and inhomogeneous materials, however, require techniques that bridge length and time scales. 

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