Statistical tensors

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... 

Making use of the restrictions brought in by the statistical tensors of the incident light, equ. (4.65) can be rewritten as... 

Derivation of angular distributions by means of statistical tensors... 

The last equation shows that the statistical tensors obey well-defined rotational properties which are similar to those of the angular momentum functions, except for the complex-conjugated rotation matrix elements which describe the inverse rotation x", y", z" - x, y, z, and that no physical significance is attached to the coefficients k and k (they are convenient summation indices only). 

As a consequence of the Wigner-Eckart theorem, relations between statistical tensors which can be derived from purely vector coupling procedures will be supplemented for transitions by introducing the corresponding reduced matrix elements Dy or Cy for the process of photoionization or Auger decay, respectively (see equs. (8.102) and (8.103b)). 

From the abundance of relations which have been worked out for statistical tensors, only those needed to derive expressions for photon-induced electron emission from closed-shell atoms will be quoted here. (In this context it should be mentioned that many different conventions are in use which are not always compatible and which lead to different phases and/or numerical values. Here the convention of [Fer65] is employed.) The necessary expressions are ... 

The detector response is comprised in the statistical tensors pkK(j) of the spin subsystem, for which one has... 

With these efficiency tensors and the corresponding statistical tensors the transition rate P then follows as... 

Introducing the couplings of the statistical tensors towards the initial state also, one derives for the transition rate P ... 

It is also possible to introduce the angle-dependent statistical tensor pkK(J,J, ,< > ) of the photoionized state which is defined for k < 2/2max, ki < 2/ lmax, with both k and k1 even, and is given by [Kab92]... 

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 ... 

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