Statistical characteristics of radiation spectra

Expressions for a number of main moments of the spectrum may be utilized to develop a new version of the semi-empirical method. Evaluation of the statistical characteristics of spectra with the help of their moments is also useful for studying various statistical peculiarities of the distribution of atomic levels, deviations from normal distribution law, etc. Such a statistical approach is also efficient when considering separate groups of levels in a spectrum (e.g. averaging the energy levels with respect to all quantum numbers but spin), when studying natural widths or lifetimes of excited levels, etc. 

It is convenient to employ the shift of mean energy instead of the average energy of the radiation spectrum (32.43), excluding from E(K —K ) the term equal to the difference of two large numbers (energies of final and initial configurations), i.e. 

Various levels of initial and final configurations take different parts in radiative transitions. Therefore, it is convenient to introduce the concepts of emissive and receptive zones, which characterize the participation of the configurations in particular transitions [298]. The zones as the weighted distributions of the level energies (with the weight of each level equal to the total strength of all lines originating from it) may be expressed in terms of their moments. The initial moments of emissive (em) and receptive (rec) zones are defined, respectively, as 

The moments of zones can be expressed in terms of the averages of products of energy and electron transition operators. Formulas of the kind (32.45) and (32.46) can also be used when K and K represent a number of mixing configurations. 

Having explicit formulas for a number of first moments we can approximately restore the envelope line of the radiation spectrum without its detailed calculations. If lines in the spectrum have one symmetric maximum, then its envelope line is approximated by a normal function whose reconstruction requires only the mean energy and variance of the spectrum. Such an approach is useful for the case of complex spectra consisting of many lines, which, due to low resolutions as well as Doppler and collisional broadening or large natural width, form continuous or quasi-continuous bands. Studies of variation of these statistical characteristics along isoelectronic sequences give a wealth of information on intra-atomic interactions. 

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