Lindhard dielectric function

The analysis of the properties of the linear theory to be considered in the following will make use of this formulation, using for e(k, ru) the dielectric function obtained by Lindhard [13],... 

First, we show calculations for bare ions, using the following methods (i) the linear response formalism described in Section 3.1, based on the exact Lindhard s dielectric function e(k, ), and (ii) following the non-linear approach described in Section 3.2. 

In the examples presented here, the extension to the Lindhard RPA [23] suggested by Mermin [24] is used for the bulk dielectric function. This allows one to use non-zero values of the electron gas damping, keeping the number of electrons in the system constant. We want to emphasize that this description incorporates both single-particle excitations (creation of electron-hole pairs) and collective excitations (bulk and surface plasmons). 

Dielectric constant, relation to susceptibility, 110. See also Dielectric susceptibility Dielectric function Fermi-Thomas, 378 Lindhard, 378... 

Several different models for (k,CO) were explored by them, including the (local) Drade model, a hydrodynamic model, and the Lindhard-Mermin dielectric function. At low frequencies, much below the plasma frequency, they found that the imaginary part of the polarizability was actually enhanced, but that at higher frequencies this enhancement was not as pronounced. The enhancement was attributed to the excitation of particle-hole pairs in the metal. 

When the molecule is close to the metal, the spatially inhomogeneous field can carry the needed momentum. To take into account these electron-hole pair excitations and, more generally, to improve the description of the q-dependence of the dielectric constant, one can use a modified Lindhard-Mermin electric permittivity [84, 85], The Lindhard-Mermin dielectric function is the Lindhard function... 

In Eq. (5.63), kp is the Fermi wavevector. This dielectric constant can be used in the framework of the specular scattering or semiclassical infinite-barrier (SCIB) model for planar metal surface [88]. Within the framework of such approach, the different descriptions of the metal response discussed so far (local dielectrics, hydrod3mamic dielectric function and Lindhard-Mermin dielectric function) can be compared. This has been done in Ref. [89] for the metal-induced non radiative rate of a molecule close to an Ag surface. The results are summarized in Fig. 5.9. 

Kliewer, K. L, and Fuchs, R. (1969] Lindhard dielectric functions with a finite electron lifetime, Phys. Rev., 181, 552-558. 

Fermi level that is, we must have k — k/2 < fcp and k + k/2 > kp (or vice versa), since otherwise the occupation numbers are either both 0 or both 1. These considerations indicate that the contributions to the integral will come from electron-hole excitations with total momentum k. At T = 0, the integral over k in the Lindhard dielectric function can be evaluated to yield... 

In this equation, n is the conduction electron density, Ep the Fermi energy and kp the radius of the Fermi sphere. e(Q) given by (4.159) is also known as the Lindhard dielectric constant [4.69]. For Q 0, the quantity in the square brackets is equal to 1 and e(Q) then reduces to the Thomas-Fermi dielectric function [4.69],... 

Figure 5.9 Non radiative decay rate Ynr of biacetyl in an ammonia matrix as a function of the metal-molecule distance from a silver surface described through a local, a modified hydrodynamic [Eq. (5.61)] and a modified Lindhard-Mermin [Eq. (5.62)] dielectric constant. The molecular plane is parallel to the metal surface. Reprinted with permission from Ref [89]. Cop5n lght [2003], American Institute of Physics.

This is called the Lindhard dielectric response function. Notice that, at T = 0, in order to have a non-vanishing integrand, one of the occupation numbers must correspond to a state below the Fermi level, and the other to a state above the... 

We wish to derive the Lindhard dielectric response function for the free-electron gas, using perturbation theory. The charge density is defined in terms of the single-particle wavefunctions as... 

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