Coulomb phase shift

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

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. 

Figure 3 shows the variation of Coulomb phase shifts a . Changes between successive shifts become less as both and the energy increase. The variation of the functions cos(ct — (J1-2) and sin(a/ — expected to feamre in respe-... 

Figure 3. Coulomb phase shifts plotted for various kinetic energies (Rydbergs) (a) ct relative to the = 0 wave (b) cos(ct - 0 2) (c) sin(a -...

This formula for the screened Coulomb field has been used for many applications it gives, for instance, a fairly satisfactory description of the residual resistance due to a small concentration of Zn, Ga, etc. in monovalent metals (Mott and Jones 1936, p. 293). Friedel (1956) points out, however, that the use of the Bom approximation gives too large a value of the scattering, and better values are obtained if one calculates the phase shifts exactly. 

For Na or any other atom, the r— 0 boundary condition of H is replaced by the requirement that at r r0 the wavefunction is a wave phase shifted from the coulomb wave by r ... 

In a Rydberg state of any atom but H, when the distance, r, of the Rydberg electron from the ion core is greater than a core radius, rc, the potential is a coulomb potential, but for r < rc the potential is usually deeper than a coulomb potential. The effect of the deeper potential is that if an incoming coulomb wave scatters from the ion core, the reflected wave has a phase shift of nfi compared to what it would have if it scattered from a proton. In other words the standing wavefunction for all r > rc is given by... 

The wavefunctions of the normal scattering modes are the standing waves produced by a linear combination of incoming coulomb wavefunctions which is reflected from the ionic core with only a phase shift. The composition of the linear combination is not altered by scattering from the ionic core. These normal modes are usually called the a channels, and have wavefunctions in the region rc[Pg.418]

If the energy is raised above the second limit there are two open channels. In Fig. 20.2 at an energy WB for r > rB the wavefunction is composed of a linear combination of 0, and 02. If we put a radial box of radius rB around the ionic core we can again ask, What are the normal modes for electron scattering from the contents of the box In other words, what linear combinations of incoming coulomb wavefunctions will suffer at most a phase shift when scattering from the contents of the box There are two wavefunctions, labelled by p = 1,2. They are linear combinations of 0, and 02, given by... 

Camus et a/.34 explained their observations by a picture which has sometimes been called the frozen planet model. Qualitatively, the relatively slowly moving outer electron produces a quasi-static field at the inner electron given by l/rc2, and this field leads to the Stark effect in Ba+. The field allows the transitions to the n >n0Z and ,f 0 states and leads to shifts of the ionic energies. The presence of the njpn0f and n in0t resonances in the spectrum of Fig. 23.12 is quite evident. Camus et al. compared the shifts to those calculated in a fashion similar to a Bom-Oppenheimer calculation. With the outer electron frozen in place at ra they calculated the Ba+ energies, W,(rQ), and wavefunctions. They then added the energy W0(r0) to the normal screened coulomb potential seen by the outer electron. This procedure leads to a phase shift in the outer electron wavefunction... 

It is not necessary for the physicist to know how to compute the Coulomb functions. They are found in subroutine libraries, for example Barnett et al. (1974). A sufficient idea of their form is obtained by putting j = L = 0 in (4.62), when they are seen to be sinp and cosp respectively. The potential terms dilate or compress the sine and cosine waves, resulting in an overall phase shift at long range. 

Notice that the term rj n2p in the large-r form is a phase that depends on r at all distances. The large-r form is not valid for any range relevant to computation. The quantity (Tl is the Coulomb phase shift, defined by... 

If the potential V(r) is a pure Coulomb potential the asymptotic partial wave is given by the regular Coulomb function (4.64), apart from a constant phase factor. We strictly have no incident plane wave since the Coulomb potential modifies the wave function everywhere. We make the normalisation of the Coulomb distorted wave t/j,j(k,r) analogous to that of (4.83) by choosing the phase factor to be the Coulomb phase shift [Pg.95]

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