Bethe’s formula

Bethe s formula requires that the velocity of the incident particle be much larger than that of the atomic electrons, a condition not easily fulfilled by the K-electrons except in the lightest elements. The required correction, called the shell correction, is denoted by subtracting a quantity C from the stopping number. In the penetration of high-Z material, even L-shell correction may be required. In that case, C denotes the sum total of all shell corrections. The subject of shell correction has been extensively treated by several authors, and various graphs and formulas are available for its evaluation (see, e.g., Bethe andAshkin, 1953). 

In the region 104-109 eV, where the energy loss is via electronic excitation and ionization, Bethe s formula with corrections (Eq. 2.11) describes the stopping power quite accurately. In the interval 104-106 eV, the decrease of stopping power with energy is attributable to the v-2 term. It reaches a minimum of —0.02 eV/A at -1.5 MeV then it shows a relativistic rise before the restricted part rides to the Fermi plateau at -40 MeV. 

I like to emphasize that Fig. 1 is not meant to indicate any fundamental limitation of quantum mechanics both Bohr s and Bethe s formulae invoke mathematical approximations to the underlying physical models, and Bethe s formula in particular relies on first-order perturbation theory for both distant and close collisions. 

If the energy is transferred is small portions, leading to excitation of a molecule from the ground state into the nth quantum state with transition energy hw0n (this sort of collisions are called glancing), the cross section is given by Bethe s formula... 

This formula exactly coincides with the one obtained by integrating the Bethe s cross section (4.13) over q, provided we take qmin = (o0Jttv [when deriving (4.33) we took qmin = relation between q and b, namely, small q corresponds to large b, and vice versa qmax b f a l. Fano150 has made the transformation from momentum representation to a representation in terms of the impact parameter b in the Bethe s formula (4.13) and has obtained an expression for the differential cross section coinciding with (4.44). 

Fig. 52. Level spacings (Do) as a function of atomic weight, after Levin and Hughes. The full curve corresponds to Bethe s formula (for 7 Mev) the dashed curve is that of Lang and Le Couteur. Full circles odd-odd nuclei open circles, even-odd nuclei open squares, even-even nuclei.

See ref. 3, p- 246 the measured temperatures are all consistent with S = kElA with kf 9, as predicted by Bethe s formula (57-4), provided only heavy nuclei are considered. 

BETHEsche Formel, level density, Bethe s formula 297. 

Bethe s formula, Term-Dichte, BETHE5c/tg Formel 297. 

Using Bethe s formula - Eq. [10] - in the non-relativistic form and series expansion of the logaritmic term, one can show that to a good approximation... 

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