Electron distribution function

Instead of plotting tire electron distribution function in tire energy band diagram, it is convenient to indicate tire position of tire Fenni level. In a semiconductor of high purity, tire Fenni level is close to mid-gap. In p type (n type) semiconductors, it lies near tire VB (CB). In very heavily doped semiconductors tire Fenni level can move into eitlier tire CB or VB, depending on tire doping type. 

Instead of plotting the electron distribution function in a band energy level diagram, it is convenient to indicate the Fermi level. For instance, it is easy to see that in -type semiconductors the Fermi level Hes near the valence band. 

Consider electrons of mass m and velocity v, and atoms of mass M and velocity V we have mjM 1. The distribution function for the electrons will be denoted by /(v,<) (we assume no space dependence) that for the atoms, F( V), assumed Maxwellian as usual, in the collision integral, unprimed quantities refer to values before collision, while primed quantities are the values after collision. In general, we would have three Boltzmann equations (one each for the electrons, ions, and neutrals), each containing three collision terms (one for self-collisions, and one each for collisions with the other two species). We are interested only in the equation for the electron distribution function by the assumption of slight ionization, we neglect the electron-electron... 

We may solve for the electron distribution function by expanding it in Legendre polynomials in cos 6 (where v = (v,6,Fourier series in cot we shall use here only the first-order terms ... 

The eigenfunction 100, the electron density p = s10o, and the electron distribution function D = 4 x rJ p of the normal hydrogen atom as functions of the distance r from the nucleus. 

Since every atom extends to an unlimited distance, it is evident that no single characteristic size can be assigned to it. Instead, the apparent atomic radius will depend upon the physical property concerned, and will differ for different properties. In this paper we shall derive a set of ionic radii for use in crystals composed of ions which exert only a small deforming force on each other. The application of these radii in the interpretation of the observed crystal structures will be shown, and an at- Fig. 1.—The eigenfunction J mo, the electron den-tempt made to account for sity p = 100, and the electron distribution function the formation and stability D = for the lowest state of the hydr°sen of the various structures. 

Fig. 5. The electron distribution function for a Dirac 2s electron in atoms with the indicated atomic numbers. The vertical broken line shows the position of r for...

In an early attempt, Mozumder (1968) used a prescribed diffusion approach to obtain the e-ion geminate recombination kinetics in the pure solvent. At any time t, the electron distribution function was assumed to be a gaussian corresponding to free diffusion, weighted by another function of t only. The latter function was found by substituting the entire distribution function in the Smoluchowski equation, for which an analytical solution was possible. The result may be expressed by... 

Fig. 1. Electron-electron distribution functions for single-configuration He wavefunction (a) radial probability distribution P(ri2) (b) intracule function h(ri2)- In both graphs, the curve with largest maximum is for the closed-shell wavefunction that of intermediate maximum is for the split-shell wavefunction that of smallest maximum is for the wavefunction containing exp( —yri2).

For simplicity, we shall commonly refer to the Q-electron distribution function as the 2-density and the 2-electron reduced density matrix as the 2-ntatrix. In position-space discussions, the diagonal elements of the 2-ntatrix are commonly referred to as the 2-density. In this chapter, we will also refer to the diagonal element of orbital-space representation of the Q-vaatnx as the 2-density. 

The argument in Eq. (77) can be generalized to higher-order electron distribution functions [28]. Unfortunately, the other A -representability conditions in Section III.G do not seem amenable to this approach. 

P. W. Ayers, S. Golden, and M. Levy, Generalizations of the Hohenberg—Kohn theorem I. Legendre transform constructions of variational principles for density matrices and electron distribution functions. J. Chem. Phys. 124, 054101 (2006). 

First let us recall the more important electron distribution functions and their origin in terms of corresponding density matrices. The electron probability density ( number density in electrons/unit volume) is the best known distribution function others refer to a pair of electrons, or a cluster of n electrons, simultaneously at given points in space. 

Substituting the new Veff(r) for Eq. (4.33), and obtaining a new set of singleelectron wavefunctions, v(r) is recomputed and the process is iterated to convergence. The final n(r) then is the correct ground-state electron distribution function. 

An elementary treatment of a three-level system under pulsed excitation was given in Sec. II.C. Pollack treats the steady-state condition and the turn-off condition as well. These cases are quite interesting and deserve further discussion. Figure 51 shows the three-level system he analyzed. The quantities Ni(f), N2(f), and N3(f)are the electron-distribution functions (populations) and a9 b, and c represent the total transition probabilities per unit time. The boundary condition Nx + N2 + N = JV =constant is assumed. 

The electron distribution function for the molecule-ion is shown in Figure 1-5. It is seen that the electron remains for most of the time in the small region just between the nuclei, only rarely getting on the far side of one of them and we may feel that the presence of the electron between the two nuclei, where it can draw them together, provides some explanation of the stability of the bond. The electron dis-... 

Fig. 1-5.—The electron distribution function for the hydrogen molecule-ion. The upper curve shows the value of the function along the line through the two nuclei, and the lower figure shows contour lines, increasing from 0.1 for the outermost to 1 at the nuclei.

The electron distribution function as given by quantum mechanics for the normal hydrogen atom has been discussed briefly in Chapter 1. The corresponding electron distribution functions for other orbitals will be discussed in the following chapter. 

Since the electron distribution function for an ion extends indefi-finitely, it is evident that no single characteristic size can be assigned to it. Instead, the apparent ionic radius will depend upon the physical property under discussion and will differ for different properties. We are interested in ionic radii such that the sum of two radii (with certain corrections when necessary) is equal to the equilibrium distance between the corresponding ions in contact in a crystal. It will be shown later that the equilibrium interionic distance for two ions is determined not only by the nature of the electron distributions for the ions, as shown in Figure 13-1, but also by the structure of the crystal and the ratio of radii of cation and anion. We take as our standard crystals those with the sodium chloride arrangement, with the ratio of radii of cation and anion about 0.75 and with the amount of ionic character of the bonds about the same as in the alkali halogenides, and calculate crystal radii of ions such that the sum of two radii gives the equilibrium interionic distance in a standard crystal. 

Plasmas typical of C02 laser discharges operate over a pressure range from 1 Torr to several atmospheres with degrees of ionization, that is, nJN (the ratio of electron density to neutral density) in the range from 10-8 to 10-8. Under these conditions the electron energy distribution function is highly non-Maxwellian. As a consequence it is necessary to solve the Boltzmann transport equation based on a detailed knowledge of the electron collisional channels in order to establish the electron distribution function as a function of the ratio of the electric field to the neutral gas density, E/N, and species concentration. Development of the fundamental techniques for solution of the Boltzmann equation are presented in detail by Shkarofsky, Johnston, and Bachynski [44] and Holstein [45]. 

For typical laser conditions, only the electrons in the tail of the distribution enter into ionization [88], dissociation [51, 89], and dissociative attachment [90] reactions. Since the rates for these reactions are a strong function of EfN, the rates must be determined by using known cross sections [51,88-90] and electron distribution functions determined by techniques outlined in Section II. Heavy particle rate constants are obtained from refs. 86 and 91. 

The Dutch physicist J.D. van der Waals found that in order to explain some of the properties of gases it was necessary to assume that molecules have a well defined size, so that two molecules undergo strong repulsion when, as they approach, they reach certain distance from one another. [...] It has been found that the effective sizes of molecules packed together in liquids and crystals can be described by assigning Van der Waals radii to each atom in the molecule. The Van der Waals radius defines the region that includes the major part of the electron distribution function for unshared [electron] pairs. Cf. Fig. l.A [2],... 

The pa and components of the electron distribution function for benzene are equal ... 

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