Total ionisation cross section

The differential cross section for ionisation is given by (6.60). To formulate the T-matrix element we partition the total Hamiltonian H into a channel Hamiltonian K and a short-range potential V and use the distorted-wave representation (6.77). The three-body model is defined as follows. 

To a first approximation angle integrated photoemission measures the density of occupied electronic states, but with the caveat that the contribution of a given state to the spectrum must be weighted by appropriate ionisation cross sections. Comprehensive tabulations of ionisation cross sections calculated within an independent electron framework are available [8]. At X-ray energies cross sections for ionisation of second and third row transition metal d states are often very much greater than for ionisation of O 2p states, so that valence band X-ray photoemission spectra represent not so much the total density of states as the metal d partial density of states [9],... 

Moxom, J., Laricchia, G. and Charlton, M. (1993). Total ionisation cross sections of He, H2 and Ar by positron impact. J. Phys. B At. Mol. Opt. Phys. 26 L367-L372. 

The following tables have been compiled using the formulae given in Sections 3 and 4. The numbers given are in fact the Z coefficients defined in Eq. (4), and are thus the intensities expressed in units of the one-electron cross-section for the orbital which is ionised. Cases where repeated states can arise, as discussed above, are marked with an asterix ( ) in these cases the total intensity given in the table may be distributed among two or more states. 

Total-ionisation cross sections are easiest to obtain by direct measurement (see section 2.2.2). Generally a well-collimated beam of nearly-monoenergetic electrons is passed through a gas or vapour target, and the positive ions formed are essentially all collected. It is necessary to use a thin target to ensure no secondary ionisation is produced. Then the collected positive current is given by (see equns. (2.8) and (2.11)). 

The difficulties associated with making an atomic-hydrogen target have precluded direct measurements of total cross sections for hydrogen. Estimates may be made by adding the best available estimates for the integrated cross sections of particular channels and the total ionisation cross section. 

To estimate the total cross section for hydrogen we use the n=l and 2 estimates above. For n=3 we interpolate in the direct measurements of integrated cross sections by Mahan, Gallagher and Smith (1976). For higher discrete channels we use the roughly-valid rule that integrated cross sections for principal quantum number n are proportional to n. Very accurate measurements of the total ionisation cross section have been made by Shah et al. (1987). These total cross section estimates are shown in table 8.3 in comparison with the three calculations we are considering. 

The comparison of theory and experiment in table 8.3 is somewhat unsatisfactory. The coupled-channels-optical and pseudostate calculations agree with each other and with the convergent-close-coupling calculation within a few percent, yet there are noticeable discrepancies with the experimental estimates. The convergent-close-coupling method calculates total ionisation cross sections in complete agreement with the measurements... 

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 measurement of the total ionisation cross section as a function of total energy gives an important overall check on theoretical methods for describing a collision. Total ionisation cross section experiments have also been performed with spin analysis, yielding the total ionisation asymmetry. 

Fig. 10.3. The differential cross section for electron—helium ionisation at < = 0 in symmetric kinematics, plotted against total energy (van Wingerden et al., 1979). Full curve, distorted-wave impulse approximation broken curve, plane-wave impulse approximation. From McCarthy and Weigold (1988).

The Hamiltonian has radial (Kr) and angular Ka,Ke) kinetic energy operators in addition to the potential V (10.44). By treating these on par with V R,6fs,ot) and by assuming an initial quasi-ergodic distribution in phase space of the escape trajectories as they enter the Coulomb zone, Wannier was able to show that at threshold (small E) the total ionisation cross section was dominated by the instability in the escape trajectories and was given by... 

The total ionisation cross section is a very important quantity in the study of electron—atom collisions. Not only does it give an overall test of theoretical methods for ionisation, but it is an essential check on the treatment of the complete set of target states in a calculation of scattering. 

The total ionisation cross section for hydrogen has been measured by Shah et al. (1987) in a crossed-beam experiment. Slow ions formed as collision products in the interaction region were extracted by a steady transverse electric field. H+ ions were distinguished by time of flight. Relative cross sections were normalised to previously-measured cross sections for hydrogen ionisation by protons of the same velocity. The proton cross sections were normalised to the Born approximation at 1500 keV. 

Fig. 10.14 shows that the convergent-close-coupling method describes the total ionisation cross section within experimental error for the whole energy range above total energy = 4 eV. Just above threshold it underestimates the cross section by up to 30%. 

Fig. 10.14. Total ionisation cross section for hydrogen. Experimental data, Shah et al. (1987) full curve, convergent close coupling (Bray and Stelbovics, 1992fc) plus signs, coupled channels optical (Bray et al., 1991c), crosses, pseudostate method (Callaway and Oza, 1979) long-dashed curve, intermediate-energy R-matrix (Scholz et al., 1990) short-dashed curve, distorted-wave Born approximation.

The distorted-wave Born approximation for ionisation considerably overestimates the total ionisation cross section for hydrogen below about 150 eV. This is a good indication of its lower limit of validity. 

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... 

Fig. 10.15. Total ionisation cross section for helium. Experimental data, de Heer and Jansen (1977) full curve, coupled channels optical (equivalent local) (McCarthy and Stelbovics, 1983a) broken curve, distorted-wave Bom approximation.

It has been measured for hydrogen (Fletcher et al, 1985 Crowe et al, 1990) and for lithium, sodium and potassium (Baum et ai, 1985) at incident energies from threshold to several hundred electron volts. The data were obtained by ionisation of polarised target atoms by polarised electrons. The relative total ionisation cross sections for parallel and antiparallel spins were determined by counting the ions, regardless of the kinematics of the final-state electrons. 

The experimental data for hydrogen are compared with calculations in fig. 10.16. Both the convergent-close-coupling and coupled-channels-optical methods come close to complete agreement with experiment. The total ionisation cross section is a more severe test of theory, since it is an absolute quantity, whereas the asymmetry is a ratio. However, the correct prediction of the asymmetry reinforces the conclusion, reached by comparison with all other available experimental observables, that these methods are valid. 

From 3 to 12 Mev, measurements of the total cross sections in a large number of elements have been made by Nereson and Darden using the neutrons emitted by a fast neutron reactor as a source, and a special energy-selective detector. In that instrument, protons projected in the forward direction from a polythene foil were measured by an ionisation chamber connected in coincidence with a proportional counter located between it and the foil, and in anticoincidence with another counter beyond it. The energy of the protons was measured by the size of the pulses they produced in the chamber the energy resolution was about 10%. 

The advantage of laser ionisation applied to secondary neutral atoms and molecules removed from a surface by stimulating radiation such as an ion or electron beam, as opposed to the one-step LDI, lies in the fact that most of the emitted particles are neutrals, not ions. Hence, the relative variations on the local gas phase populations are less affected by the chemical composition of the local mix. Quantitation is difficult as the total number of emitted particles and the ionisation cross-section are unknown. 

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