Non-coincident case

In the case of a non-coincident angular distribution of Auger electrons it is the angular distribution parameter / A, introduced in equ. (3.29) as 

The population numbers a(JMj) can be related directly to the photoionization process, because they are equal to the partial cross section which depends on the 

For the alignment, which is of interest in the present context, one obtains 

Obviously, different expressions are obtained for the alignment parameters, because these quantities are defined with respect to different quantization axes. However, it is possible to express the alignment properties in a unified manner, e.g., j/2oliin the coordinate frame with the quantization axis along the photon beam direction. Since the alignment tensor s/2k is defined in connection with statistical tensors, equ. (8.115c), one can use the rotation properties in equ. (8.82) to change the reference axis for the representation from the z-axis to the x-axis  

This representation gives the following components of the alignment tensor (the index z-axis will be omitted from now on, and quantities referred to the x-axis will be characterized by a tilde) 

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Although symmetry considerations often permit g- and hyperfine matrix principal axes to be non-coincident, there are relatively few cases of such noncoincidence reported in the literature. Most of the examples discussed by Pilbrow and Lowrey in their 1980 review36 cite cases of transition metal ions doped into a host lattice at sites of low symmetry. This is not to say that matrix axis non-coincidence is rare but that the effects have only rarely been recognized. 

In these cases, the g-matrix is nearly isotropic, but the principal axes of the two 59Co hyperfine matrices are non-coincident. The largest hyperfine matrix component (ay = 66.0 G in the case of the Co-Co-Fe-S cluster) results in 15 features, evenly spaced (apart from small second-order shifts). Another series of features, less widely spaced, shows some variation in spacing and, in a few cases, resolution into components. This behavior can be understood as follows Suppose that the hyperfine matrix y-axes are coincident and consider molecular orientations with the magnetic field in the vz-plane. To first order, the resonant field then is ... 

It is relatively easy to understand the significance of the non-coincident matrix axes in these cases. For the Co2C2 cluster, the C2v molecular symmetry permits a specific prediction of the possible matrix axis orientations. The g-matrix principal axes must be coincident with the molecular symmetry axes. The two cobalt nuclei are located in a reflection plane (which we label xz) so that symmetry requires the y-axis to be a principal axis for all three matrices. The other two axes may be rotated, relative to the molecular x- and z-axes, by /J. (Since the two nuclei are symmetrically equivalent, the rotations must be equal and opposite.)... 

The dynamics are contained in the photoionization cross section <7ph(2p3/2), in the Auger yield oja(L3-M1M1) and in the angular distribution parameters j ph(2p3/2) and j9A(L3-M1M1) of the non-coincident photo- and Auger electrons (see Section 5.2). For convenience, a ratio r is used in equ. (4.86b) which, however, is not a new parameter since in the present case it depends on j9A(L3-M1M1) (see equ. (8.133g)) ... 

As a special application of the two-step model the non-coincident observation of photon-induced Auger electron emission will be considered further. In this case one has to integrate the transition rate P of equ. (8.66a) over dKa, because the photoelectron is not observed, i.e.,... 

For a correct analysis of photoionization processes studied by electron spectrometry, convolution procedures are essential because of the combined influence of several distinct energy distribution functions which enter the response signal of the electron spectrometer. In the following such a convolution procedure will be formulated for the general case of photon-induced two-electron emission needed for electron-electron coincidence measurements. As a special application, the convolution results for the non-coincident observation of photoelectrons or Auger electrons, and for photoelectrons in coincidence with subsequent Auger electrons are worked out. Finally, the convolutions of two Gaussian and of two Lorentzian functions are treated. 

Coincidence techniques have also been used for Compton interference reduction in the use of large volume Ge(Li) detectors together with plastic scintillator anticoincidence shields 70), In some cases it might be desirable to use the coincidence electronics to gate the multichannel analyzer to accept only non-coincident pulses. In 14 MeV neutron activation procedures the annihilation radiation resulting from the decay of 13N produced indirectly from the carbon in the plastic irradiation unit may be discriminated against by gating the analyzer to accept only non-coincident events. 

These results mean that, once the GME coinciding with the CTRW has been built up, we cannot look at it as a fundamental law of nature. If this GME were the expression of a law of nature, it would be possible to use it to study the response to external perturbations. The linear response theory is based on this fundamental assumption and its impressive success is an indirect confirmation that ordinary quantum and statistical mechanics are indeed a fair representation of the laws of nature. But, as proved by the authors of Ref. 104, this is no longer true in the non-Poisson case discussed in this review. 

If the specimen is thicker, domains are likely to overlap, and we must be concerned about whether extinction can ever occur when superimposed birefringent domains are viewed between crossed polars. By way of a simple example, we can consider the case of two domains with non-coincident vibration directions. The formula for transmitted intensity can be derived by analogy to the analysis leading to Equation 1 ... 

We investigated the special sp structures, in planar as well as non-planar configuration, for c = 10 and 14. Their energies are quite favourable. Of course, the non-planar one is the most favourable structure, but the difference between the values for planar and non-planar case decreases when going from c = 10 to nc = 14. For nc = 18 they coincide and they are both identical with the Deh structure. However, the bonds are symmetry inequivalent for these structures, even in the case of D h symmetry, with the only exception of... 

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