Dipole approximation photoionization

Less widely appreciated is the fact that the angular distribution functions Eqs. (2 and 3) are actually subcases of a more general form. It was first proposed by Ritchie [34] that, even in the pure electric-dipole approximation, another term was required for completeness, and that hence the general photoionization angular distribution function, normalized over the surface of a unit sphere, should be written as [35] ... 

In non-relativistic dipole approximation the photoionization of closed electronic shell is described by the formula... 

Calculations show that cross-sections obtained in the Hartree-Fock approximation utilizing length and velocity forms of the appropriate operator, may essentially differ from each other for transitions between neighbouring outer shells, particularly with the same n. However, they are usually close to each other in the case of photoionization or excitation from an inner shell whose wave function is almost orthogonal with the relevant function of the outer open shell. In dipole approximation an electron from a shell lN may be excited to V = l + 1, but the channel /— / + prevails. For configurations ni/f1 n2l 2 an important role is... 

Formally, alignment and orientation follow from the general symmetry properties of statistical tensors pkK introduced in Section 8.4. Spherical symmetry leads to k = k = 0, axial symmetry to k — 0, and alignment requires k = even, and orientation k = odd. Since the dipole approximation in the photoionization process restricts k to k < 2, a photoionized axially symmetric state can only have Poo, p10, and p20 Poo describes isotropy, and the alignment is given by [BKa77]... 

This expression disentangles the properties of the light polarization (coefficients /9, ( 1)), the geometry of two-electron emission (coefficients B t2(Ka, Kb)) and the dynamical parameters of the double photoionization process (coefficients A(ku k2, k)). The pkq(El) are the statistical tensors of the incident light which describe its polarization properties in the electric dipole approximation represented by 1. For linearly polarized light in which the electric field vector defines the z-axis of the coordinate frame, one has only two non-vanishing components given by (see equ. (8.99b))t... 

Angular distribution parameters ft of photoelectrons Is photoionization in helium / = 2.0 (in dipole approximation)... 

The previous formulation for the photoionization process provides the starting point for theoretical calculations. For simplicity, and because the conditions are well fulfilled, in many applications the dipole approximation is often used. (For extensions and derivations, relevant in the present context of photoionization studies with synchrotron radiation, see [KJG95] and references therein.) This approximation is based on a special property of the matrix element ... 

A compilation of data relevant for a discussion of the dipole approximation for photoionization processes in neon is given in Table 8.1. Since , should be small compared to reference, it can be seen that the approximation is well justified for excess photon energies of the order of the binding energy of the active nAelectron, and this statement also holds for other systems (see, however, [KJG95]). 

Finally, the differential cross section for photoionization, dff/dfl, will be given explicitly for the dipole approximation and the length form of the matrix element by collecting all the individual steps. This cross section is related to the transition rate w by... 

Within the dipole approximation (jphoton = 1) and for the example of photoionizing an wp-electron from a closed-shell atom (J = 0), the photoelectron angular momentum is either t = 0 or 2 and, hence,... 

In the present context the two transition operators of relevance are that for photoionization which is given in the dipole approximation, and within the length form, by (see equ. (1.28a))... 

The next application is photoionization of He atom including autoionization features by the CBF method and CCl (CISD) method. In this subsection, the outline of this method is described, but for more details, the reader is referred to references [8-10]. In the dipole approximation, the total photoionization cross section imaginary part of the frequency-dependent polarizability. 

The statement that there is good agreement between theoretical and experimental data is conventionally the conclusion drawn from the results presented. However, a deeper insight can be obtained when the dipole matrix elements involved are determined experimentally, and compared with results from different theoretical approximations of the influences of electron-electron interactions. In order to make such a comparison, the experimental matrix elements will be derived first for the case of 2p photoionization in magnesium at 80 eV photon energy. From equs. (5.14), (5.15), and (5.16) one can calculate DS, Dd and A from the measured observables. The results are listed in Table 5.1, together with data from several theoretical calculations. 

In the context of the different forms of the dipole matrix elements a comment should be added. A good calculation for observables of photoionization which takes into account most of the important electron correlation effects should yield approximately the same values, whatever the form used for the transition matrix element. However, this necessary condition must not be a sufficient one. The most... 

The dynamical parameters 11(1, k2, k3) are fixed by the photoionization matrix elements, e.g., in the LSJ-coupling approximation by the radial dipole integrals es,2P and Pcd 2p and the relative phase A, as follows ... 

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