Momentum density electron number densities

This expression has been superseded by the expression derived by Bethe and Bloch based on momentum transfer in a quantum mechanically correct formalism. Their expression with the expanded form of the electron number density is... 

Free-space light wave vector Plasmon momentum Electron mass Electron number density Ambient refractive index... 

Another manifestation of the reciprocity of densities in r- and p-space is provided by Fig. 19.2. It shows the radial electron number density D r) = Aiir pir) and radial momentum density /(p) = Aitp nip) for the ground state of the beryllium atom calculated within the Hartree-Fock model in which the Be ground state has a ls 2s configuration. Both densities show a peak arising from the Is core electrons and another from the 2s valence electrons. However, the origin of the peaks is reversed. The sharp,... 

In the present calculation the SIC potential is introduced for each angular momentum in a way similar to the SIC one for atoms [9]. The effects of the SIC are examined on the CPs of three materials, diamond, Si and Cu compared with high resolution CP experiments except diamond [10, 11]. In order to examine the quasi-particle nature of the electron system, the occupation number densities of Li and Na are evaluated from the GWA calculation and the CPs are computed by using them [12, 13]. 

Figure 7. The occupation number densities as functions of wave vector for Na. The thick curves labeled (100), (110) and (111) represent the three principal directions within the first Brillouin zone, obtained by the FLAPW-GWA. The thin solid curve is obtained from an interacting electron-gas model [27]. The dash-dotted line represents the Fermi momentum. 

Three-dimensional reconstruction of electron momentum densities and occupation number densities of Cu and CuAl alloys... 

The reconstruction of the electron momentum densities and the occupation number functions of Cu and Cuq.953A10 047 could not produce results on an equal profound base as those based on the results of Li and LiMg reconstructions. This would need approximately 100 times the number of counts per spectrum which was not achieved. 

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

This article provides an introduction to the momentum perspective of the electronic structure of atoms and molecules. After an explanation of the genesis of momentum-space wave functions, relationships among one-electron position and momentum densities, density matrices, and form factors are traced. General properties of the momentum density are highlighted and contrasted with properties of the number (or charge) density. An outline is given of the experimental measurement of momentum densities and their computation. Several illustrative computations of momentum-space properties are summarized. 

It is clear from Eqs. (18) and (19) that the number and momentum densities are not related by Fourier transformation. This is most readily understood for a one-electron system where the r-space density is just the squared magnitude of the orbital and the p-space density is the squared magnitude of the Fourier transform of the orbital. The densities are not Fourier transforms of one another because the operations of Fourier transformation and taking the absolute value squared do not commute. Moreover, there is no known direct and practical route from one density to the other even though the Hohenberg-Kohn theorem [32] guarantees that it must be possible to obtain the ground state n(p) from p(r). 

There are two main methods for the reconstruction of 7T(p) from the directional Compton profile. In the Fourier-Hankel method [33,51], a spherical harmonic expansion of the directional Compton profile is inverted term-by-term to obtain the corresponding expansion of /T(p). In the Fourier reconstraction method [33,34], the reciprocal form factor B0) is constructed a ray at a time by Fourier transformation of the measured J(q) along that same direction. Then the electron momentum density is obtained from B( ) by using the inverse of Eq. (22). A vast number of directional Compton profiles have been measured for ionic and metallic solids, but none for free molecules. Nevertheless, several calculations of directional Compton profiles for molecules have been performed as another means of analyzing the momentum density. 

The calculations include, as said previously, overlaps, conditional probability distributions of the electron probability densities, and these observables oscillator strengths, quadrupole moments (for states with total angular momentum quantum numbers of 1 or more) and expectation values (pi p2)/( pi i>2 )- (Distributions of this last quantity have also been computed, in preparation for two-electron ionization experiments by electron impact, but are not reported here.) We can proceed to summarize these indicators and then examine them and ask how well each model performs. 

P Electron momentum component of the particle number density... 

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