Dielectric function tensor

Let us consider for simplicity a sphere composed of a uniaxial material (see Section 9.3). We denote by k ( and kx the wave numbers corresponding to the two principal values of the dielectric function tensor. It is reasonable to assert on physical grounds that anisotropy is only a perturbation if... 

Up to this point we have restricted consideration to materials for which the dielectric function is a scalar. However, except for amorphous materials and crystals with cubic symmetry, the dielectric function is a tensor therefore, the constitutive relation connecting D and E is... 

In the preceding sections the optical response of matter has been described by a scalar dielectric function e, which relates the electric field E to the displacement D. More generally, D and E are connected by the tensor constitutive relation (5.46), which we write compactly as D = e0e E. The dielectric tensor is often symmetric, so that a coordinate system can be found in which it is diagonal ... 

Thus the spectral function L(z) of an isotropic medium is represented as a linear combination of two spectral functions determined for an anisotropic medium pertinent to longitudinal ( ) ) and transverse (K ) orientations of the symmetry axis with respect to the a.c. field vector E. It is shown in GT, Section V, that these spectral functions are proportional to the main components of the dielectric-susceptibility tensor. 

X. Gonze and C. Lee, "Dynamical matrices, Bom effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory," Phys. Rev. B 55 (1997), 10355-10368. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

The dielectric tensor in a cubic crystal is reduced, as is well known, to the scalar dielectric function e(u>) when spatial dispersion is neglected. In the region of the band of two-particle states, this function can be presented in the form... 

For the analysis of the reflection spectra, we consider as usual the frequency-dependent complex dielectric tensor s (co). We restrict ourselves here to the case that the direction of polarisation is parallel to the stacking axis, and denote the real and imaginary parts of the complex dielectric function as usual s (jo) = i(co) -i- is2(co). It is related to the complex index of refraction, n = ni -i-in2, via n = The real part of... 

Anisotropy in the optical properties of a layer complicates the analytical expressions for reflectance since the complex dielectric function and the refractive index of the layer become tensors (1.1.2°). Determination of a film s anisotropy from its spectrum provides a wealth of information about the structure and molecular orientation in ultrathin films and therefore is of great importance in various areas of science and technology (Section 3.11). The theoretical approaches of Schopper [112] and Kuzmin et al. [113] (see the review in Ref. [114]) are... 

A wide variety of molecular properties can be accurately obtained with ADF. The time-dependent DFT implementation " yields UV/Vis spectra (singlet and triplet excitation energies, as well as oscillator strengths), frequency-dependent (hyper)polarizabilities (nonlinear optics), Raman intensities, and van der Waals dispersion coefficients. Rotatory strengths and optical rotatory dispersion (optical properties of chiral molecules ), as well as frequency-dependent dielectric functions for periodic structures, have been implemented as well. NMR chemical shifts and spin-spin couplingsESR (EPR) f-tensors, magnetic and electric hyperfme tensors are available, as well as more standard properties like IR frequencies and intensities, and multipole moments. Relativistic effects (ZORA and spin-orbit coupling) can be included for most properties. 

In noncubic sohds, the phonon mode frequencies of the polar lattice vibrations depend, in general, on the phonon mode propagation direction. Likewise, directionally dependent free-chargescattering rates and the anisotropic inverse effective freecarrier mass tensor will produce nonscalar free-charge-carrier contributions. The infrared dielectric function is then represented by a complex-valued second-rank tensor s, which can be expressed in Cartesian coordinates (x,y,z) as ... 

Light scattering is due to fluctuations in the local dielectric tensor e of the medium. In fluids these fluctuations are dynamic and the scattered intensity will be a function of time and the frequency spectrum of the scattered light will differ from that of the incident light. The time dependence of the total scattered intensity is analyzed by measuring the intensity autocorrelation function... 

The dielectric tensor e in a viscoelastic medium is a function of the frequency at which it is measured. It can be represented in terms of a real and imaginary part e (co) = e (co) -ie"(a>). If the frequency dependence of e is determined by a single relaxation time, then the relationship between e and r is... 

The 13 C powder NMR line shapes of the carboxyl group chemical shift tensor are shown Fig. 115 as a function of temperature. At 27 °C, a nearly regular powder spectrum is found, with o = 268 ppm, <722 = 150 ppm and (733 = 112 ppm. As temperature rises, increasingly pronounced line-shape changes are observed, which are indicative of large motions with rates exceeding 10 kHz. The motional rates estimated at the various temperatures fit quite well on the relaxation map of PMMA obtained from mechanical and dielectric measurements (Fig. 111). Thus, the motions of the carboxyl groups observed by NMR are directly related to the f3 transition. 

Fluctuations in the optical properties of a material can induce spatial and temporal inhomogeneities that scatter light. In general, the dielectric tensor is taken as the following function of space and time,... 

Fourier transform of the dielectric tensor autocorrelation function, (4.89). 

Puschnig, P. and Ambrosch-Draxel, C. (1999) Density-functional study of the oligomers of poly-para-phenylene Band structures and dielectric tensors. Physical Review. E, Condensed Matter, 60, 7891-8. 

X. Gonze, D.C. Allan and M.P. Teter, "Dielectric tensor, effective charges and phonons in a-quartz by variational density-functional perturbation theory," Phys. Rev. Lett. 68 (1992), 3603-3606. 

For further progress it is necessary to specify how E varies with D, or how P depends on Ea. For this purpose, we introduce the constitutive relations D - e(T,V)E or P - ot0(T,V)F0, where e is the dielectric constant and a0 is a modified polarizability. (Conventionally, the polarizability is defined through the relation P - oE, but no confusion is likely to arise through the introduction of this variant.) Note several restrictions inherent in the use of these constitutive relations. First, the material under study is assumed to be isotropic. If this is not the case, e and c 0 become tensors. Second, the material medium must not contain any permanent dipole moments in the preceding constitutive relations P or E vanishes when E0 or D does. Third, we restrict our consideration to so-called linear materials wherein e or a0 do not depend on the electric field phenomena such as ferroelectric or hysteresis effects are thus excluded from further consideration. These three simplifications obviously are not fundamental restrictions but render subsequent manipulations more tractable. Finally, in accord with experimental information available on a wide variety of materials, e and aQ are considered to be functions of temperature and density assuming constant composition, these quantities vary with T and V. 

If, as is mostly the case in experiment, the analysing field E and polarizing field Ep are applied in the same direction, the electric permittivity variation tensor (12) possesses but one non-zero component, in the field direction, usually denoted by Ae,( p). This particular case is referred to as the effect of electric saturation in an dectric field. In molecular liquids Ae,( ) is in general a quadratic function of the field strength Ep, as proved r )eat y by Piekara and his co-workers. Lately, Davies has published a review on aspects of recent tfielectric studies, particularly dielectric saturation in liquids and molecular solutions, as well as in solutions of macromolecules where complete dielectric saturation has been observed. 

Gajdos et alP have put forward a method of calculating dielectric tensors using density functional perturbation theory which they extended for the PAW method. 

In this equation, po is the permanent dipole moment of the molecule, a is the linear polarizability, 3 is the first hyperpolarizability, and 7 is the second hyperpolarizability. a, and 7 are tensors of rank 2, 3, and 4 respectively. Symmetry requires that all terms of even order in the electric field of the Equation 10.1 vanish when the molecule possesses an inversion center. This means that only noncentrosymmetric molecules will have second-order NLO properties. In a dielectric medium consisting of polarizable molecules, the local electric field at a given molecule differs from the externally applied field due to the sum of the dipole fields of the other molecules. Different models have been developed to express the local field as a function of the externally applied field but they will not be presented here. In disordered media,... 

Along with the functions g and G a number of other Green s functions must be known in order to calculate the dielectric tensor of a crystal in the overtone frequency region, as well as the RSL cross-section and the cross-section of nonlinear optical processes. Among these others we required the two-particle Green s function G 4k/ (t), which is determined by the relation... 

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