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Plane-wave first approximation

The longitudinal current density in vacuo is investigated, first in the plane-wave first approximation, by taking the real part of the potential... [Pg.34]

In the plane-wave first approximation, the current density is therefore... [Pg.34]

The vacuum charge density is also structured in general, but in the plane wave, first approximation is given by... [Pg.35]

In SI units, the transverse vacuum current densities are given in the plane-wave first approximation by... [Pg.35]

Fig. 11.5. The 1500 eV noncoplanar-symmetric momentum profiles for the argon ground-state transition (15.76 eV), first excited state (29.3 eV) and the total 3s manifold (McCarthy et ai, 1989). Hartree—Fock curves are indicated DWIA, distorted-wave impulse approximation PWIA, plane-wave impulse approximation. Experimental data are normalised to the 3p distorted-wave curve with a spectroscopic factor Si5.76(3p) = 0.95. The experimental angular resolution has been folded into the calculations. Fig. 11.5. The 1500 eV noncoplanar-symmetric momentum profiles for the argon ground-state transition (15.76 eV), first excited state (29.3 eV) and the total 3s manifold (McCarthy et ai, 1989). Hartree—Fock curves are indicated DWIA, distorted-wave impulse approximation PWIA, plane-wave impulse approximation. Experimental data are normalised to the 3p distorted-wave curve with a spectroscopic factor Si5.76(3p) = 0.95. The experimental angular resolution has been folded into the calculations.
Different methods exist to compute wave functions 01(7 ) that are the solutions of equations of the type Eq.(2.5.) If the potential K(r) is small compared with the kinetic energy, it is useful to consider 0t(7) approximately as a sum of plane waves, an approximation often used in solid physics. If 0j(r) is approximated by a single plane wave the method is called the free-electron method. Such solutions are discussed later. We will first consider the wave function 0t( r ) to be a linear combination of atomic orbitals centered on the lattice atoms ... [Pg.28]

A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. [Pg.392]

Match the main features of the rocking curve first, using plane wave calculations and a single (sigma) polarisation. It shonld be possible to fit all the main peaks accnrately in spacing and approximately in intensity. Then begin refinement to match the intensities and widths. [Pg.123]

To calculate the vibrational frequency of CO using DFT, we first have to find the bond length that minimizes the molecule s energy. The only other piece of information we need to calculate is a = (d2E/db2)h hlj. Unfortunately, plane-wave DFT calculations do not routinely evaluate an analytical expression for the second derivatives of the energy with respect to atomic positions. However, we can obtain a good estimate of the second derivative using a finite-difference approximation ... [Pg.115]

The first (and still the foremost) quantum theory of stopping, attributed to Bethe [19,20], considers the observables energy and momentum transfers as fundamental in the interaction of fast charged particles with atomic electrons. Taking the simplest case of a heavy, fast, yet nonrelativistic incident projectile, the excitation cross-section is developed in the first Born approximation that is, the incident particle is represented as a plane wave and the scattered particle as a slightly perturbed wave. Representing the Coulombic interaction as a Fourier integral over momentum transfer, Bethe derives the differential Born cross-section for excitation to the nth quantum state of the atom as follows. [Pg.13]

In general, the vacuum current density has a definite structure in the vacuum that is much richer than in the first plane-wave approximation a structure that has to be computed because analytical solutions to Eq. (183) are not available. [Pg.34]

In this section, we illustrate the self-consistent calculation of these charge current densities in the plane-wave approximation, using plane waves in the X, Y, and Z directions. In general, the solution of the field equation (459) must be found numerically, and it is emphasized that the plane-wave approximation is a first approximation only. In the internal space, there is the real vector ... [Pg.74]

Further, the first term in the above approximation, which is an exact propagation constant for a plane wave propagating along the z-axis, is reexpressed as a sum of its two lowest-order Taylor expansion terms plus the rest ... [Pg.267]

In first Born approximation, the wave function is the product of a plane wave and a part which contains only the spin structure (ip = we1,

momentum conserving (5-functions, which are explicitly removed in the definiton of the T-matrix. The main point is now the connection between the 16 x 16 Tif = u 1u 2Tu U2 and its 8x8 form M. Defining in analogy to the one-particle case... [Pg.742]

The positions of hydrogen in hydrides are sometimes difficult to be determined experimentally. So, in this study, the crystal structures of hydrides are optimized by the total energy minimization using the plane-wave pseudopotentital method. For this purpose, the first-principle calculations based on the DFT are performed with a generalized gradient approximation (GGA) by Perdew et al. [5]. The implementation of DFT employed here combines a plane-wave basis set with the total energy pseudopotential method, as is embodied in the CASTEP code [6]. [Pg.146]

We shall see below that the transport properties above are consistent with the picture of Luttinger chains in (a - b) planes. The exponent a = 0.7 derived from the constant volume transverse data leads to K = 0.22. This value of allows in turn a prediction for the constant volume T-dependence for p . The only scattering process through which electron-electron collisions can contribute to resistivity in this 1-D electron gas occurs when the total momentum transfer is commensurate with a Brillouin zone wave vector. For the situation of a 1/4-filled 1-D band which is likely to apply to (TMTSFljPF as the dimerization can be forgotten in first approximation (A ,[Pg.254]

In first Born approximation, the wave fnnction is the prodnct of a plane wave and a part which contains only the spin strnctnre (t/> = we, 4> = =... [Pg.742]


See other pages where Plane-wave first approximation is mentioned: [Pg.323]    [Pg.11]    [Pg.716]    [Pg.1362]    [Pg.495]    [Pg.40]    [Pg.52]    [Pg.95]    [Pg.603]    [Pg.194]    [Pg.210]    [Pg.81]    [Pg.582]    [Pg.231]    [Pg.266]    [Pg.159]    [Pg.172]    [Pg.184]    [Pg.241]    [Pg.26]    [Pg.247]    [Pg.80]    [Pg.200]    [Pg.357]    [Pg.548]    [Pg.7]    [Pg.193]    [Pg.115]    [Pg.450]    [Pg.331]    [Pg.110]    [Pg.436]   


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