Bethe equation

Besides, when y° = 0, the Davydov-Serikov equation reduces the Bethe equation [93], that is,... 

In this expression /3 is the velocity of the electrons divided by the velocity of light, and I is the average excitation potential of the material. Equation 2 differs from the complete Bethe equation in that a combination of constant factors multiplying the logarithmic term, which would cancel in Equation 1, has been omitted. The I of a compound can be calculated from values of its atomic constituents by the approximate formula ... 

The final nuclear detector makes possible a separation of isobars based upon the principle that the range and rate of energy loss for particles of a given energy is atomic number dependent. Ions such as 14C and 14N have ranges in solids or gases that differ by over 20 percent at energies of about 14 MeV. The basis for this separation is the Bethe-Bloch equation [26,27], which can be simplified to read ... 

The linear response i plays a fundamental role. It can be evaluated using the Bethe-Salpter equation (Equation 24.82) where the screened response x is evaluated from Kohn-Sham equations (Equation 24.80). It is remarkable that any nonlinear response can be computed using the linear one and the hardness kernels [26,32]. For instance, y3(r, r1 r2, r3) (see diagram 52a in Ref. [26]) is... 

On the other hand, functional derivatives of the Bethe-Salpeter equation allows to evaluate the nonlinear responses using the interaction kernels h only (which depend on the Hartree and exchange-correlation energies). The relations between the screened nonlinear responses and the bare ones are derived by using nonlinear PRF [32],... 

The account of two-particle correlations in nuclear matter can be performed considering the two-particle Green function in ladder approximation. The solution of the corresponding Bethe-Salpeter equation taking into account mean-field and Pauli blocking terms is equivalent to the solution of the wave equation... 

In the framework of the Brueckner theory a rigorous treatment of TBF would require the solution of the Bethe-Faddeev equation, describing the dynamics of three bodies embedded in the nuclear matter. In practice a much simpler approach is employed, namely the TBF is reduced to an effective, density dependent, two-body force by averaging over the third nucleon in the medium,... 

Being based on lowest-order perturbation theory, Bethe theory approximates equation (3) by T—2 Zie lp) lmv so that... 

Figure 1 shows a comparison between experimental stopping data and equations (7) and (11). It is seen that over a broad range of beam energies, Bohr s classical formula is superior to Bethe s when the same /-value is inserted. 

As it stands, ALbiocB. equation (16), when added to the Bethe formula, extends its range of validity into the classical regime. Alternatively one may introduee an inverse-Bloch correetion [10,11]... 

Shell corrections can also be evaluated without recourse to an expansion in powers of v, but existing calculations such as Refs. [13,14] are based on specific models for the target atom and, unlike equation (19), do not end up in expressions that would allow to identify the physical origin of various contributions. It is clear, however, that orbital motion cannot be the sole cause of shell corrections The fact that the Bethe logarithm turns negative at 2mv /I< 1 cannot be due to the neglect of orbital motion but must be of a purely mathematical nature. Unfortunately, the uncertainty principle makes it impossible to eliminate orbital motion in an atom from the beginning. 

The calculation of n k) is an important and very nontrivial many-body problem and requires the approximate solution of a Bethe-Salpeter equation. Our present objective is to generate gradient corrections to the LDA so (A.7) can be simplified by assuming that Sn(r) is so slowly varying in space (in addition to being of small magnitude) that 8n k) is essentially zero except for very small k. 

Van der Horst JW, Bobbert PA, Michels MAJ, Bassler H (2001) Calculation of excitonic properties of conjugated polymers using the Bethe-Salpeter equation. J Chem Phys 114 6950... 

L(0) = Z In 7, where I is the mean excitation potential appearing in Bethe s stopping power equation [Eq. (4)]. 7,(2) is proportional to the logarithm of average excitation energy, which is also involved in the Lamb shift [26]. 7,(—1) has been shown to be an optical... 

According to the optical approximation, which was shown by Platzman [4] to be based on the Born-Bethe theory, a radiation chemical yield of channel G, may be estimated from optical data as shown in the following equation ... 

The details of the stopping power equations have been very well summarized elsewhere [22,34-38]. Most applications involve ions of low charge and high velocity in which the Bethe formalism is valid. The nonrelativistic stopping power equation of Bethe for heavy ions is given approximately by. 

Bethe-Salpeter Equation and the Effective Dirac Equation 5... 

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