Cross dipole approximation

Equation (A3.13.17) is a simple, usefiil fomuila relating the integrated cross section and the electric dipole transition moment as dimensionless quantities, in the electric dipole approximation [10, 100] ... 

If the isotropic coefficient is specified to be unity, a is just the total (integrated) cross-section. In Appendix A, an alternative quantum mechanical expression for this cross-section is obtained in the electric dipole approximation. By comparing the two expressions, it can be seen that the Legendre polynomial coefficients in Eq. (11) may be obtained from the inner summation terms in Eq. (A.15). Hence, the Legendre polynomial coefficients are... 

We consider the expression of the lab frame photoelectron angular distribution for a randomly oriented molecular sample. The frozen core, electric dipole approximation for the differential cross-section for electron emission into a solid angle about a direction k can be written as... 

Within the semiclassical, perturbational treatment of the interaction of radiation with matter [77,78] and within the dipole approximation [79], the total energy absorption cross section may be written in the form [11,12,20,80]... 

The electronic optical cross section a(hv) can be expressed in the dipole approximation as... 

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... 

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... 

However, because of the integration over all directions of the emitted photoelectron, the cross terms must vanish, and only the diagonal terms remain due to the following symmetry arguments. Within the dipole approximation and for incident linearly polarized light, the convenient quantization axis is the direction of the electric field vector. For randomly oriented atoms in the initial state this electric field vector is then the only direction of preference in the initial system (atom plus photon). Since the observation of the final system (ion, photoelectron and Auger electron) is made only for one constituent (the ejected Auger electron),... 

Now we need to determine which excited states are dipole allowed. To this end, we want to find the value for the crossing transition probability. In the dipole approximation the interatomic interaction is described with an operator V ... 

Quantum-mechanically, the process of energy loss by electrons can be considered as formally equivalent to the absorption of photons, so that we may express the differential cross-section for inelastic scattering (energy loss) under the dipole approximation as ... 

Following this theory, the expression of the photoabsorbing cross section for a cluster of atoms with light polarized in the vector e direction is given, in the dipole approximation by... 

For the sake of comparison with the ESCA spectrum of Nj, and to provide a more critical test of the EOM method, we calculate the peak intensities as well as the peak positions. The details of the intensity calculations are left to the Appendix. The dipole approximation is invoked, as the calculation is based on Fermi s golden rule. A plane wave approximation is employed for the outgoing electron. The cross section for the ejected electron is averaged over all possible molecular orientations and polarizations of the incident photon as described by Ellison. ... 

The XSW field established inside the crystal and above the crystal surface induces photoelectron emission from atoms within the field. The excited atoms (ions), in turn, emit characteristic fluorescence X-rays and Auger electrons. In the dipole approximation, the photoelectric effect cross section is proportional to the E-field intensity at the center of the atom. (It is necessary to consider higher-order multi-pole terms in the photoelectric cross-section under special conditions, as discussed by Fischer et al. (1998) and Schreiber et al. (2001). For this review, we will assume the dipole approximation.) Therefore, with the XSW intensity from Equation (6), the normalized X-ray fluorescence yield is defined as... 

We have mentioned that the small-K limit is unrealistic when discussing inelastic events because the minimum value of k is always finite in such experiments. A useful guide to the K-dependence of the cross-section is obtained from the so-called dipole approximation to the magnetic interaction operator. Experience shows that this approximation is the next best thing to use beyond the small-K limit when building up a picture of a class of scattering events. To implement this approximation for lanthanide ions, there is next to no extra work involved beyond what we have discussed already. 

The modification to the cross-section, brought about by employing the dipole approximation, is more or less obvious. For an inelastic event in which a transition occurs between states J = J the matrix element of J is zero, i.e.,... 

We conclude that the dipole approximation to the cross-section is obtained from our first approximation by multiplying the corresponding expression for the cross-section (21) by ((/q) -... 

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 oscillatory photoelectric cross section written in the dipole approximation of radiative transitions is... 

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