Final state radial wave functions

Figure 13 shows a pictorial view of the final-state radial wave functions relevant to core transitions in a molecule. Core transitions take place in an effective molecular potential seen by the excited photoelectron. Whereas for E < E0, that is, below the continuum threshold (where E0 is the energy of the core ionization potential), discrete transitions occur to the unoccupied valence states, photoelectrons excited in the photoionization process with... 

Fig. 5. Schematic representation (after [13]) of the numerical calculation of the spatial part of the matrix element Mspace in the p + p—> d c u, reaction. The top part shows the potential well of depth Vo and nuclear radius R of deuterium with binding energy of —2.22 MeV. The next part shows the radius dependence of the deuterium radial wave function Xd(r)- The wave-function extends far outside the nuclear radius with appreciable amplitude due to the loose binding of deuterium ground state. The p-p wave-function XppM which comprise the U = 0 initial state has small amplitude inside the final nuclear radius. The radial part of the integrand entering into the calculation of Mspac is a convolution of both Xd and Xpp in the second and third panels and is given with the hatched shading in the bottom panel. It has the major contribution far outside the nuclear radius...

It is assumed that the radial wave functions in the initial and final states are constant within a sphere of radius R, that is R ij r) = const. 0 if r < 1 and zero if 1 < r. This is a rather crude approximation, not taking into account the real shapes of the radial wave functions, but it simplifies the calculations considerably and it is very useful in practice. The reduced transition probability is evaluated for / = L 1/2 initial and J = 1/2 final states. 

Figure S.2.2.3 Radial wave functions for (a) the occupied orbitals of atomic carbon and (b-e) the final-state wave functions resulting from C 2p excitation at different photon energies as indicated. Final-state wave functions for both allowed emission channels are shown (2p -r d-wave and 2p —r s-wave according to A/ = 1). Near the nucleus, the final states are spherical Bessel...

Ionization at a given photon energy may proceed in several channels. For example, the dipole selection rule, A l- 1, permits an initial electronic state of angular momentum / to decay into two degenerate ionization channels, the / +1 and / -I channels in which the photoelectrons have angular momenta (/ + 1) h and (/ - 1 )h. Since the parameters a and P contain the radial matrix elements for ionization into the two channels, and since these elements are proportional to the overlap of the electronic wavefunctions for the initial and final states of the ionization process, it follows that a and P are functions of these overlaps. Secondly, since the two photoelectron waves have different phase and nodal structures, they may interfere this interference is also determinative of o and p values. For atomic photoionization and LS coupling, one finds ... 

S-wave scattering is the only practical outcome since P-wave final neutron states are not accessible to thermal neutrons, because these wave functions have negligible amplitude at the small radial values that are typical of atomic nuclei. It is convenient to rewrite the equation as a dynamical structure factor (or Scattering Law), which emphasises the dynamics of the sample. 

Finally, note that an ns orbital has n 1 radial nodes a radial node is a spherical surface about the nucleus on which tf/ and are 0. These spherical surfaces are the analogues of the nodal planes in the wave functions for a particle in a cubic box (Figure 4.28). The more numerous the nodes in an orbital, the higher the energy of the corresponding quantum state of the atom. Just as for the particle in a box, the energies of orbitals increase as the number of nodes increases. 

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