Polarisation potential

This result is remarkably simple as compared to the usual methods. For a spin-polarised potential V, Kraft, Oppeneer, Antonov and Eschrig (1995) used the elimination method and found the corrections as a sum of 9 terms, which is equivalent to our Eq.(ll). They notice that three terms of their sum have a known physical meaning (spin-orbit, Darwin and mass-velocity corrections), but the other terms have no special name . 

It should be emphasised that potential-pH diagrams can also be constructed from experimental E -I curves, where E is the polarised potential and / the current. These diagrams, which are of more direct practical significance than the equilibrium potential-pH equilibrium diagrams constructed from thermodynamic data, show how a metal in a natural environment (e.g. iron in water of given chloride ion concentration) may give rise... 

If now the resistance in the external circuit is decreased slightly the reaction will proceed at a finite rate, and the electrodes constituting the cell will become mutually polarised and displaced from their equilibrium values, i.e. the polarised potential of the anode (Zn /Zn) will become more positive, whilst that of the cathode (2H /H2> will become more negative (Fig. 1.23). 

The Schwerdtfeger polarisation break and the polarisation resistance methods have been studied by Jones and Lowe " in relation to their effectiveness in evaluating corrosion rates of buried metals. A Holler bridge circuit was used to remove IR contributions during the measurement of the polarised potential. Jones and Lowe, on the basis of their studies of buried steel and aluminium specimens, concluded that the polarisation resistance was the most useful, and that the polarisation break had the serious limitation that it was difficult to identify the breaks in the curve. 

If the reaction is proceeding at a finite rate, the Es will be the polarised potentials Fp, and since... 

Morrison, M.A., Gibson, T.L. and Austin, D. (1984). Polarisation potentials for positron-molecule collisions positron- scattering. J. Phys. B At. Mol. Phys. 17 2725-2745. 

The effluent from the column is mixed with hydrogen and the mixture burnt at a small in a flow of air. Above the is the collector electrode (a wire or ring around the j ) and a polarising. potential of about 150 volts is applied betweoi the and the electrode. Whoi a component elutes from the column it is burnt in the flame and the resulting ions carry a current between the electrodes which provides the signal. 

Table 5.3. One-electron separation energies for the lower-energy states of sodium (units eV). Experimental data (EXP) are from Moore (1949). The calculations are FCHE, frozen-core Hartree—Fock and POL, frozen-core Hartree—Fock with the phenomenological core-polarisation potential (5.82)...

The coupled-channels-optical equations are formally analogous to the Lippmann—Schwinger equivalent of (7.29) in which the coupling potential includes the potential V (7.40) and a polarisation potential that describes the real (on-shell) and virtual (off-shell) excitation of the complementary channel space, called Q space. The total coupling potential is the optical potential... 

The polarisation potential is complex and nonlocal. The imaginary part is due to on-shell amplitudes for the excitation of Q space from P space. At long range the potential is real. We will show its relationship for large r, where it is due to virtual dipole excitations, to the classical dipole potential where a is the polarisability. 

Our first task in constructing the reduced matrix elements of the polarisation potential is to find the reduced matrix elements of the Q-projected Green s function (7.111). Using the notation (7.48) we have... 

The reduced matrix element of the two-electron polarisation potential (7.115,7.116) is... 

The result (7.153) was first derived by O Malley, Spruch and Rosenberg (1961). It is identical to the first Born approximation T-matrix element for scattering from the local polarisation potential... 

We may consider long-range elastic scattering to be due to a polarisation potential that is strictly nonlocal, but is equivalent to (7.155) in the sense that it gives the same T-matrix element or phase shift. 

The Q-projected Green s function (7.111) in the polarisation potential (7.115) contains the collision Hamiltonian H, but practical implementation of it involves the weak-coupling approximation (7.132). A second-Born calculation is equivalent to a calculation of the polarisation potential with Q=1 and Vopul being the reduced matrix element of U. This has... 

Here we have summed the partial-wave series whose terms are represented by (7.139). The first term of (7.159) is the polarisation potential due to the discrete part Q of Q space. The second term is due to the ionisation space Q, and T( )(q, q)) is the exact solution of the Schrodinger equation for an ionised channel. 

Computation is too laborious for the necessary range of k, k, so a further approximation is necessary. The polarisation potential matrix element is calculated only at about 10 points in the variable K, where... 

Fig. 8.2. Differential cross section for the elastic scattering of electrons on hydrogen. Circles, Williams (1975) solid curve, coupled-channels-optical calculation long-dashed curve, one channel with discrete polarisation potential only short-dashed curve, one channel without polarisation potential. Adapted from Bray et al. (1991h).

Fig. 8.8 shows that the coupled-channels-optical method with the equivalent-local polarisation potential (McCarthy and Shang, 1992) gives a good semiquantitative description of the experimental data of Williams (1976 ) for elastic differential cross sections below the n—2 threshold. At energies just below the n=3 threshold the resonances affect the n=2 excitations. Fig. 8.9 shows the energy dependence of the integrated cross sections for the 2s and 2p channels. Since a resonance is a property of the compound system, not the channel, the resonances observed in... 

Relative differential cross sections for the 3s and 3p channels at several energies have been measured by different groups. These are shown in figs. 8.10 and 8.11 in comparison with a coupled-channels-optical calculation for which P space consists of the 3s, 3p and 3d channels and the polarisation potential treats all Q space channels to convergence. A 3s, 3p, 3d coupled-channels calculation has been included to assess the effect of Q space. 

Fig. 8.12 shows the example of 50 eV electron scattering to the three lowest singlet states (McCarthy, Ratnavelu and Zhou, 1991). In the calculation P space consisted of ten channels 1,2,3 S 2,3 S 2,3 P 2,3 P 3 D. Polarisation potentials for ionisation were included for all couplings in the n=l and 2 subspace. The basis used in the configuration-... 

The equivalent-local form of the coupled-channels-optical method does not give a satisfactory description of the excitation of triplet states (Brun-ger et al, 1990). Here only the exchange part of the polarisation potential contributes. The equivalent-local approximation to this is not sufficiently accurate. It is necessary to check the overall validity of the treatment of the complete target space by comparing calculated total cross sections with experiment. This is done in table 8.8. The experiments of Nickel et al. (1985) were done by a beam-transmission technique (section 2.1.3). The calculation overestimates total cross sections by about 20%, due to an overestimate of the total ionisation cross section. However, an error of this magnitude in the (second-order) polarisation potential does not invalidate the coupled-channels-optical calculation for low-lying discrete channels. 

In the absence of independent measurements of the total cross section the total ionisation cross section gives an estimate of the validity of the equivalent-local polarisation potential used for the coupled-channels-optical calculation of fig. 8.13. The calculated value at 40 eV is 5.2 nal, compared with 4.66+0.47 nal measured by Karstensen and Schneider (1975). 

The coupled-channels-optical calculation converges at 15 channels in P space with polarisation potentials for the continuum included for all couplings in the first six channels. The effect of the inclusion of the continuum is shown by the 15-state coupled-channels calculation. The distorted-wave... 

Essentially-complete agreement with experiment is achieved by the coupled-channels-optical calculation. We can therefore ask if scattering is so sensitive to the structure details in the calculation that it constitutes a sensitive probe for structure. The coupled-channels calculations in fig. 9.3 included the polarisation potential (5.82) in addition to the frozen-core Hartree—Fock potential. Fig. 9.4 shows that addition of the polarisation potential has a large effect on the elastic asymmetry at 1.6 eV, bringing it into agreement with experiment. However, in general the probe is not very sensitive to this level of detail. 

It is useful to test approximations for the total ionisation cross section of helium, since it is a common target for the scattering and ionisation reactions treated in chapters 8, 10 and 11. Fig. 10.15 compares the data reported as the experimental average by de Heer and Jansen (1977) with the distorted-wave Born approximation and the coupled-channels-optical calculation using the equivalent-local polarisation potential. Cross sections... 

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