Dielectric function wave-vector-dependence

P. A. Bopp, A. A. Kornyshev and G. Sutmann, Frequency and wave-vector dependent dielectric function of water collective modes and relaxation spectra, J. Chem. Phys., 109 (1998) 1939-58. 

Another point worth mentioning is the use of the metal long-wavelength dielectric function. No dispersion was allowed in the model. The wave-vector dependence of the dielectric constant is important at close proximity to the surface as was already remarked in the context of the image calculations. This approximation may be reasonable away from the surface, distances for which the LFE is most appropriate. 

In order to improve the accuracy of the calculated acceptor levels in silicon and germanium, particularly for the even-parity ones, Lipari et al. [38] have used a screened point-charge impurity potential based on the wave-vector-dependent dielectric function calculated for Si, Ge, GaAs and ZnSe [65]. They make use of a phenomenological parameter a, adjusted to fit the calculated q-dependent dielectric function e(q), in this potential. The resulting potential in real space is ... 

Contents Introduction. - Volume Plasmons. - The Dielectric Function and the Loss Function of Bound Electrons. -Excitation of Volume Plasmons. - The Energy Loss Spectrum of Electrons and the Loss Function. - Experimental Results. - The Loss Width. - The Wave Vector Dependency of the Energy of the Volume Plasmon. - Core Excitations. -Application to Microanalysis. - Energy Losses by Excitation of Cerenkov Radiation and Guided Light Modes. - Surface Excitations. - Different Electron Energy Loss Spectrometers. - Notes Added in Proof - References. - Subject Index. 

The wave vector, k , and the screening length, 1/ , depend only on the density of the free-electron gas through the poles of the approximated inverse dielectric response function, whereas the amplitude, A , and the phase shift, a , depend also on the nature of the ion-core pseudopotential through eqs (6.96) and (6.97). For the particular case of the Ashcroft empty-core pseudopotential, where tfj fa) = cos qRc, the modulus and phase are given explicitly by... 

Inelastic Scattering Cross Section. The inelastic scattering cross section that defines the probability that an electron loses energy T per unit energy loss and per unit path length traveled is one of the key parameters in quantitative peak shape analysis. In the dielectric response formalism of solid electron interaction, the cross section may be evaluated from the wave vector and frequency dependent complex dielectric function ( , [Pg.42]

In 1.1.3°-1.1.7°, we have assumed the medium to be nonabsorbing and, thus, the parameters e and n to be constant and o =0. However, if the medium absorbs electromagnetic radiation, these quantities become dependent on the frequency of incident radiation, the function e((o) termed the dielectric function. Below we will neglect the so-called spatial dispersion effects [18] connected with the dependence of the dielectric function on the wave vector (k). This is permissible for the IR range (the limit k 0). 

It should be emphasized that these derivations can be fully extended for time dependent perturbations. For that purpose it is sufficient to introduce a frequency w, apart from the wave vectors, in the densities and potentials, in the polarizability matrices and dielectric matrices, and in the exchange and correlation function which then becomes G[q,u). This latter point will be discussed in more detail in the next chapter. Note also that, strictly speaking, causality and the related dispersion relations only hold for the inverse dielectric matrices, since they describe the response. For the dielectric matrix itself, to the best of our knowledge no proof of the dispersion relations has been given. 

The parameters of the Hamiltonian (3), i.e. the frequencies and the number of oscillators (more specifically, the strength of oscillators) are determined by the imaginary part of a complex dielectric function a(k, co) which characterizes the dielectric losses for polarization fluctuations in a medium. This model is, strictly speaking, applicable to homogeneous isotropic media in which the spatial correlations of polarization fluctuations SP r)dP r ), which determine the dependence of s k, co) on the wave vector k, depend on the difference of coordinates r—r only. 

The solid particle can have an arbitrary shape, but attention for now will focus on the case of a sphere. The radius of the sphere is denoted by a and its complex dielectric function by s((o). The dielectric function will be assumed to be local in this discussion so there is no dependence on the wave-vector of the photon. The dielectric function for the solvent is denoted by the local function Ss( ) particle size is assumed to be sufficiently small compared to the wave length of the relevant photons that the electrostatic approximation to electrodynamics is warranted. Thus retardation effects will be neglected here. 

Close to a surface, the description of screening effects is more difficult than in the bulk, because the periodicity of the system is broken in one direction. Even for an electron gas, assumed to be homogeneous in a half-space, the dielectric function is non-local. It is characterized by two wave vectors q and q with the same projection q in the surface plane e( ll,qz,q, (u). In the classical macroscopic limit, image effects and the value of the surface plasmon energy will be analyzed first. Then, the relationship between the surface electronic structure and the dielectric function will be discussed. Finally the spatial dependence of screening efiTects in the vicinity of a surface will be exemplified. 

Another important feature of the intermolecular contributions to the relaxation energy is the dependence of their average value on the dielectric response of the medium. If e ( , to ) is the non-local dielectric function of the medium associated with wave vector and frequency v = d)/2 x, then the longitudinal polarization fluctuations of the medium are defined by... 

---