Generalized oscillator strength

The Bom approximation for the differential cross section provides the basis for the interpretation of many experimental observations. The discussion is often couched in temis of the generalized oscillator strength. 

Assuming the validity of the Bom approximation, an effective generalized oscillator strength can be derived in temis of experimentally accessible quantities ... 

A particularly important property of the generalized oscillator strength is that, for high-energy, small-angle... 

A succinct picture of the nature of high-energy electron scattering is provided by the Bethe surface [4], a tlnee-dimensional plot of the generalized oscillator strength as a fiinction of the logaritlnn of the square of the... 

B2.2.5.5 ATOMIC FORM FACTOR AND GENERALIZED OSCILLATOR STRENGTH... 

In temis of the fonn factor the generalized oscillator strength is defined as... 

The Bethe Sum Rule and Basis Set Selection in the Calculation of Generalized Oscillator Strengths... 

It is a pleasure for us who are friends, colleagues, and collaborators, to offer this contribution to a volume published in honor of Yngve Ohm s bS " birthday. For most of his career, Yngve has been interested in response properties of various systems to various probes, and we offer this contribution in that spirit. The Generalized Oscillator Strength, the subject of this paper, is the materials property that describes the response of a medium to swift particle, and thus, perhaps, an appropriate subject for this volume. Mostly, we are happy to take this opportunity to thank Yngve for his help, inspiration, and friendship over the years. 

Inelastic collisions of swift, charged particles with matter are completely described by the distribution of generalized oscillator strengths (GOS s) characterizing the collision. These quantities, characteristic of excitation in the N-electron target (or, in fact, of a dressed projectile as well [1]) from some initial state 0) to a final state n) and concomitant momentum transfer, can be written... 

Here Z is the charge of the projectile with velocity v. In order to calculate stopping powers for atomic and molecular targets with reliability, however, one must choose a one-electron basis set appropriate for calculation of the generalized oscillator strength distribution (GOSD). The development of reasonable criteria for the choice of a reliable basis for such calculations is the concern of this paper. 

Thus the Bethe sum rule is fulfilled exactly in the RPA at all values of the momentum transferred, provided that a complete basis set is used. Therefore, as in the case of the TRK sum rule when optical transition properties (q = 0) are considered, we expect that the BSR sum rule will be useful in evaluating basis set completeness when generalized oscillator strength distributions are calculated, for example for use in calculating stopping cross sections. It should be noted [12] that the completeness of the computational basis set is dependent on q, and thus care needs be taken to evaluate the BSR at various values of q. 

In this contribution, we have shown that the Bethe sum rule, like the Thomas-Reiche-Kuhn sum rule, is satisfied exactly in the random phase approximation for a complete basis. Thus, in calculations that are related to the generalized oscillator strengths of a system, the Bethe sum rule may be used as an indicator of completeness of the basis set, much as the Thomas-Reiche-Kuhn... 

Bethe (1930) defined the generalized oscillator strength in terms of the form factor as... 

To further reduce of the cross section formula (4.11), we note that it is proportional to the area of the curve of Fn(K)/en plotted against In (Kag)2 between the maximum and minimum momentum transfers. Since T is large and the generalized oscillator strength falls rapidly with the momentum transfer, the upper limit may be extended to infinity. In addition, the minimum momentum transfer decreases with T in such a manner that the limit Fn(K) may be replaced by /, the dipole oscillator strength for the same energy loss. This implies that a mean momentum transfer can be defined independently of T such that the relevant area of the curve of Fn(K)/ n is equal to (// ) [ (In Kag)2 - (In Ka0)2]. Thus, by definition (Bethe, 1930 Inokuti, 1971),... 

Therefore, fast-charged-particle impact resembles optical transition to some extent. The oscillator strength introduced in Chapter 2 corresponds to this kind of transition, whereas that for the entire operator exp(ik r) is called the generalized oscillator strength, which also has some interesting properties (Inokuti, 1971). 

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