Born-Bethe approximations

The Born-Bethe approximation for low-energy electrons requires correction for two reasons. First, the integrals defining the total oscillator strength and the... 

Sharma and Kern [95] have also performed a theoretical analysis, based on the Born-Bethe approximation, of V-R transfer between CO and para-H2. A difference betweenp-H2 and o-H2 in the quenching rate of vibra-tionally excited CO was first observed in 1964 by Millikan and Osburg [96], and more recently over a larger temperature range by Millikan and Switkes [97], The reaction... 

Figure 14 Experimental (solid circles and full line) and theoretical K-shell electron ionization cross section for argon dashed line, Born-Bethe approximation dash-dotted line, Deutsch-Maerk formula open circles, Born-Bethe including relativistic corrections.

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 ... 

Figure 10 Experimental (solid symbols) and theoretical (dashed line, distorted-wave Born approximation full line, Deutsch-Maerk formula and BEB (Binary-Encounter-Bethe) approximation) electron ionization cross section for helium.

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-... 

A review of quantum theories (Born approximation, Bethe approximation, impulse approximation, etc.) as well as information on the classical calculations can be found in [4] (up to 1968) see also [5]. 

In Bethe theory the shell correction ALsheii is conveniently defined as the difference between the stopping number LBom in the Born approximation and the Bethe logarithm LBethe —in (2mv /I). Fano [12] wrote the leading correction in the form... 

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. 

Despite the apparent similarity of the Bohr and the Bethe stopping power formulae, the conditions of their validity are rather complimentary than the same. Bloch [23] pointed out that Born approximation requires the incident particle velocity v ze jh, the speed of a Is electron around the incident electron while the requirement of Bohr s classical theory is exactly the opposite. For heavy, slow particles, for example, fission fragments penetrating light media, Bohr s formula has an inherent advantage, although the typical transition energy has to be taken as an adjustable parameter. 

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

In this condition, Bethe formulated the stopping power for electron according to the Born approximation. Stopping power is a property of irradiated materials and gives the amount of energy deposited per unit path length, —dU/dx. 

A consecutive application of the Born approximation to the problem of calculating the scattering cross sections was first done by Bethe (a detailed discussion of his theory is presented in Refs. 104, 106, 113). Integrating over the coordinates of the incident particle we can obtain simple analytical formulas for the cross sections. 

Fig. 10.4. Factorisation test of the distorted-wave Born approximation for copla-nar asymmetric ionisation from the 2p orbital of neon at = 400 eV, j = 50 eV (Madison, McCarthy and Zhang, 1989). The fast and slow electrons are respectively indicated by the subscripts / and s on the diagram The Bethe ridge condition is df = 20°. Full curve, unfactorised broken curve, factorised.

The validity of the impulse approximation can be tested by factorising the distorted-wave Born approximation in the same way. The differential cross section in the factorised distorted-wave Born approximation, obtained by replacing the two-electron T-matrix element in (10.42) by the potential matrix element (10.36), is compared with that of the full distorted-wave Born approximation in fig. 10.4 for the 2p orbital of neon in coplanar-asymmetric kinematics for =400 eV, s=50 eV. In this case the Bethe-ridge condition is Of = 20°, and p is less than 2 a.u. for 6s between 0° and 120° with this value of 6f. The impulse approximation is verified in Bethe-ridge kinematics for p less than 2 a.u. 

The basic theoretical models to describe the interaction of ionized particles with matter were developed early in the 20th century by Bohr [1,2], Bethe [3] and Bloch [4] (BBB). These models provide the general framework to almost any consideration on the energy loss of swift particles in matter. The first two of these models are based on widely different assumptions, the Bohr description is fully classical, representing the atomic electrons by classical oscillators, while the Bethe model is based on quantum perturbation theory (first-order Born approximation). 

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