Threshold Behaviour of Ionization Cross-Sections
A relationship that is well-known in atomic physics is the Wannier law (3-1)
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Threshold Behaviour of Ionization Cross-Sections F H. Read Schuster Laboratory, University of Manchester, Great Britain
3.1 Introduction A relationship that is well-known in atomic physics is the Wannier law (Jiun 'l. £1.127.
(3-1)
It gives the dependence of the cross-section for the electron-impact ionization process e+A-->A+ +e+e (3-2) or the photo-double-detachment process hv+A--+A+ +e+e
(3-3)
on the amount E by which the energy of the system exceeds the ionization energy of the atom or negative ion. It is a threshold law, since it applies only when E is small, and it was first derived by Wannier (1953) using a treatment based on classical mechanics. It has excited, and continues to excite, considerable interest and some controversy, and has been the subject of many theoretical and experimental studies. A recent experimental result (which will be discussed in Section 3.2.4) is shown in Fig. 3-1. Part of the attraction of the Wannier law is that it concerns a problem that lies at the interface of classical mechanics and quantum mechanics, and so is related to a wider range of problems for which there is, at present, no generally accepted method of solution. These problems are characterised by the fact that they involve interactions between three or more particles over distances that are too large for conventional quantum-mechanical techniques to be tractable. They usually also involve motion in the vicinity of an unstable potential ridge and the existence of an exceptionally high degree of correlation between the particles involved (see for example Fano 1980 a, b, 1983 a, b, Rau 1982). The subject of near-threshold ionization is particularly important since it represents the simplest example of a problem that contains all these features. The hope is therefore that the search for a theoretical technique T. D. Märk et al. (eds.), Electron Impact Ionization © Springer-Verlag Wien 1985
Threshold Behaviour of Ionization Cross-Sections
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a: A(N- 1 )+ + e1 + ... +eN. a When N =2 the volume is proportional to the length of the full line that represents E 1 + E 2 =E. b When N =3 it is proportional to the shaded area on the surface that represents E 1 + £ 2 + £ 3 = E
Threshold Behaviour of Ionization Cross-Sections
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A pictorial representation of these results is given by considering the hypersurface defined by the equation E 1 + E 2 + ... +EN= E in the multidimensional space EI> E 2 ... EN. The integral in equation (3-69) is the area occupied by this surface in the energetically accessible region, defined byE~ Ei ~ 0 for all i. For example when N = 2 the surface is the line shown in Fig. 3-8 a, and when N = 3 it is the twodimensional surface shown in Fig. 3-8 b. We see that the dimensionality of the hypersurface is N-1 and that its area is therefore proportional to EN- 1 • These results are modified when the long-range interactions and correlations between the outgoing electrons are taken into account. When N = 2 the work of Wannier (1953) shows that an instability in the potential
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