Fine structure constant photoionization

The latter equation assumes a 100% linearly polarized ionizing radiation, a is the fine structure constant, Nni is the number of electrons in a nl subshell, Dni->ei l is a radial dipole photoionization amplitude, fini is the dipole photoelectron angular asymmetry parameter, and A i2 is the electric dipole-quadrupole interference term arising due to the correction term ikr in the above expression for Mab,... 

The fine-structure constant a indicates that first-order perturbation theory has been applied the linear dependence on the photon energy Eph is due to the length form of the dipole operator used in equ. (2.1), and the wavenumber k compensates the 1 /k which appears if the absolute squared value of the continuum wavefunction is used (see equ. (7.29)). The summations over the magnetic quantum numbers M, of the photoion and ms of the photoelectron s spin are necessary because no observation is made with respect to these substates. Due to the closed-shell structure of the initial state with f — 0 and M = 0, the averaging over the magnetic quantum numbers M simply yields unity and is omitted. 

In order finally to derive the differential cross section of photoionization one inserts equ. (8.26) in equ. (8.24) and replaces the number nph of incident photons by nPh = ce0Alo)/2ti (see equs. (8.4b) and (8.8a) and (8.8b)) and the interaction operator by equ. (8.21). Then one removes the factor h2/m0 resulting from the normalization of the continuum function from the matrix element and incorporates it in the final prefactor (see footnote concerning equ. (7.28d)), and one introduces the fine structure constant a using a = el/4ne0hc. This leads to (for the summations over magnetic quantum numbers see below)... 

Here, a is the fine structure constant and U (x) is the dipole operator. Unfortunately, the suggestively denoted quantities, and ip.(x) in (la) cannot be identified with a Koopman s removal energy and Slater determinant wave function as in HF theory. Nonetheless, regarding the solution of (1) as providing a basis set, one can construct an LDA to atomic photoionization using (4),... 

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