Oscillator Strength and Sum Rules

FIGURE 4.4 Schematic of threshold behavior of ionization processes. Under ideal conditions, one expects a step function for photoionization, a linear variation with energy under electron impact, and a parabolic dependence for double ionization by electron impact. 

Closely related to the absorption coefficient is the concept of the oscillator strength for the transition, or the/-number. It is given by 

The dipole oscillator strength is the dominant factor in dipole-allowed transitions, as in photoabsorption. Bethe (1930) showed that for charged-particle impact, the transition probability is proportional to the matrix elements of the operator exp(ik r), where ftk is the momentum transfer. Thus, in collision with fast charged particles where k r is small, the process is again controlled by dipole oscillator strength (see Sects. 2.3.4 and 4.5). 

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). 

It is not easy to calculate oscillator strengths from first principles except in some very simple cases. On the other hand, the oscillator strength distribution must fulfill certain sum rules, which in some cases help to unravel their character. Referring the (dipole) oscillator strength for the transition from the ground state with excitation energy n to state n as fn, a sum may be defined by 

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