Optical calculation

We gratefully acknowledge the invaluable assistance of Dr. D. E. Lobb, (University of Victoria) who carried out the ion optics calculations for the new injection system. 

The term Qsh is the net solar radiant energy absorption rate on the basin bottom. It is equivalent to total radiation incident on the basin cover minus reflection from the cover, the water surface, and the basin bottom, and minus loss due to structural shadowing. Its determination from Weather Bureau records of total daily radiation on a horizontal surface is complicated by many factors such as variation in angle of incidence, and resulting transmissivity of cover, hourly and seasonally, intensity change due to cloudiness, and different properties of direct and diffuse radiations. Detailed explanation of these meteorological and optical calculations is beyond the scope of this paper, but may be found in the literature (6). 

It is apparent that for A3, = 0, the electric field component does not contain a product of potential terms. In general the vanishing of this term occurs if there are no longitudinal electric field components. Within the framework of most quantum electrodynamic, or quantum optical, calculations this is often the case. The B(3) field then is a Fourier sum over modes with operators a qaq. The B(3 ) field is then directed orthogonal to the plane defined by A1 and A2. The fourdimensional dual to this term is defined on a time-like surface that has the interpretation, under dyad-vector duality in three dimensions as, as an electric... 

There exist two main methods for implementing nonlinear optical calculations into a given computational technique coupled methods and uncoupled methods. Coupled methods [sometimes called finite field (FF) or coupled-perturbed Hartree-Fock (CPHF) methods] include the effect of the perturbing field into the Hamiltonian. The energy (e) of the system in the field E... 

Yee TK, Guststafson. Diagrammatic analysis of the density operator for nonlinear optical calculations. Phys Rev A 1978 18 1597. 

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).

It is perhaps as interesting to compare the approximate calculations with the convergent-close-coupling calculation as with experiment. The one that takes all channels into account most completely is the coupled-channels-optical calculation (Bray, Konovalov and McCarthy, 1991c) in which P space consists of the n=l, 2 and 3 channels. It agrees closely, but not completely, with the convergent calculation and similarly with... 

There are only isolated measurements of integrated cross sections, but there are absolute measurements of differential cross sections. We adopt the procedure of using the coupled-channels-optical calculation of Bray et al. (1991c) to interpolate and extrapolate these measurements, since it agrees quite well with differential cross sections in figs. 8.4 and 8.5. 

Table 8.4 shows the state multipoles in comparison with the coupled-channels-optical calculation (Bray, Madison and McCarthy, 1990) and the pseudostate calculation of van Wyngaarden and Walters (1986). 

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. 

Table 8.7. The integrated cross section a p for the 3p channel of electron—sodium scattering and the total cross section ot (ICr cm ). EXP (a-ip), interpolation in the data of Enemark and Gallagher (1972) EXP (oj), Kwan et al. (1991) CCO, coupled-channels-optical calculation (Bray et al, 199 Id)...

Absolute measurements of total cross sections have been made by beam-transmission techniques. The results of Kwan et al. (1991) are compared with the coupled-channels-optical calculation in table 8.7. In most cases the coupled-channels-optical cross section is within one standard deviation of the experimental result. 

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. 

Williams and Trajmar (1978) for the 3 S and 3 P channels. The figure gives a good idea of the present state of theory and experiment for all but the most-detailed investigations. There is a need for more experimental data and for a full coupled-channels-optical calculation. 

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... 

The need for inclusion of the continuum or for full coupling is not very obvious for the triplet reactions, which dominate the spin-independent data. It is the asymmetry that provides the critical test of theory. Very good agreement with experiment is obtained by the full coupled-channels-optical calculation, but the other two calculations are qualitatively incorrect, even giving the opposite sign for the 3p asymmetry. These conclusions hold for experimental—theoretical comparisons at 1.0, 1.6, 4.1, 12.1 and 40 eV (Bray and McCarthy, 1992). 

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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