Phase shift coefficient

The nondimensional attenuation coefficient, a, and the phase shift coefficient, (3, are (Roberts and Valocchi, 1981) ... 

In Eq. (12), l,m are the photoelectron partial wave angular momentum and its projection in the molecular frame and v is the projection of the photon angular momentum on the molecular frame. The presence of an alternative primed set l, m, v signifies interference terms between the primed and unprimed partial waves. The parameter ct is the Coulomb phase shift (see Appendix A). The fi are dipole transition amplitudes to the final-state partial wave I, m and contain dynamical information on the photoionization process. In contrast, the Clebsch-Gordan coefficients (CGC) provide geometric constraints that are consequent upon angular momentum considerations. 

The summation for the coefficient of the P2 term likewise includes phase insensitive contributions where I m = Im, but also now one has terms for which I = I 2, which introduce a partial dependence on relative phase shifts— specifically on the cosine of the relative phase shift. Again, this conclusion has long been recognized for example, by an explicit factor cos(ri j — ri i) in one term of the Cooper-Zare formula for the photoelectron (3 parameter in a central potential model [43]. 

In the more novel case of the P coefficient, only interference terms for which 7 = 1 1 arise, so now all summands contributing to contain phase shift information. This hints that the chiral parameters could be more sensitive to phase differences than traditional (3 parameters, and one would... 

The coefficients a and P, which depend on the size of u, Edis, and to, describe the attenuation of the fluctuations in the aquifer and the corresponding phase shift, respectively. Note the similarity with Eq. 7 of Box 21.3, which describes the periodic disturbance of a box. In Eq. 25-33 the phase shift ( 5x) and the attenuation of the concentration amplitude (e-" ) both depend on the distance from the input, x. The nondimensional version of Eq. 25-33 is ... 

Figure 25.7 Nondimensional attenuation coefficient, a, and 10-1 specific phase shift, P, for sinusoidal input into a saturated groundwater system as a function 10 2 of the Peclet Number Pe. The numbers attributed to the different 1 q 3 lines give the nondimensional angular frequency 2. See text for definitions. 10-4...

The typical attainable nonlinear phase shifts NL(z) for the different absorption processes are given by the second terms in the above equations containing the nonlinear refractive indices n2 and the absorption coefficients a,-. Figures of merit can be defined for each absorption process by demanding a typical nonlinear phase shift (f)NL>2n [17]. 

---