Bethe formula

Fig. 1. Stopping force on oxygen ions in aluminium Comparison of Bohr and Bethe formulae with measurements from numerous laboratories compiled in Ref. [6]. From Ref. [7].

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]... 

By careful inspection of the relationships of the semiclassical close collision approximations and Bethe formula, one can obtain simple and accurate information on ionization cross sections. By the method first proposed by Platzman, and used extensively by others, it is instructive to form the ratio of the differential cross sections [measured c(W,T) or calculated da(W,T)ldW)] to the Rutherford cross section. This ratio, called Y, is mathematically defined as... 

In O Fig. 5.9 the experimentally observed 0 level distances in the transmission resonance groups of Pu (fiiU circles) are compared to calculations of the level density in the first minimum according to different mcxlels. The soUd fine on the left side of the figure corresponds to the level distances in the first minimum calculated in the framework of the back-shifted Fermi gas model in the parameterization by Rauscher et al. (1997). In order to reproduce the experimentally observed 0 level distances, this curve has had to be shifted by 2.25 MeV, corresponding to the ground-state energy in the second minimum of Pu. For comparison, also shown are the level distances in the first minimum calculated with the Bethe formula and within the constant temperature formalism in a parameterization by von Egidy et al. (1988). 

Conversely, for slow collisions the combined system of incoming electron and target molecule has to be considered, leading in the exit channel to a full three-body problem. Quantum-mechanical (approximate) calculations are difficult and have been carried out only for a few selected examples. Therefore, other methods have been developed with the goal of obtaining reasonably accurate cross sections using either classical or semiclassical theories and by devising semiempirical formulae. Some of these concepts are based on the Born-Bethe formula [22] and on the observation that the ejection of an atomic electron with quantum numbers (n,J) is approximately proportional to the mean-square radius of the electron shell (n,J). This leads also to proposed correlations of the ionization cross section with polarizability, dia-... 

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