Differential cross section ionisation

Brauner, M., Briggs, J.S. and Klar, H. (1989). Triply-differential cross sections for ionisation of hydrogen atoms by electrons and positrons. J. Phys. B At. Mol. Opt. Phys. 22 2265-2287. 

Fig. 2.7. Schematic representation of the single-differential cross section. The threshold energy for ionisation is given by eo, the ground-state separation energy.

Because our description of differential cross sections for momentum transfer in a reaction initiated by an electron beam depends on our ability to describe both the structure and the reaction mechanism, scattering provides much more information about bound states. This is even more true of ionisation. The information is less accurate than from photon spectroscopy and is obtained only after a thorough understanding of reactions, the subject of this book, is achieved. The understanding of structure and reactions is of course achieved iteratively. A theoretical description of a reaction is completely tested only when we know the structure of the relevant target states with accuracy that is at least commensurate with that of the reaction calculation. The hydrogen atom is the prototype... 

In general Q space does not have a very large effect on cross sections. This is true also of later calculations (illustrated in chapter 9) where P space includes discrete channels up to convergence. Differential cross sections do not critically test the need to include the ionisation continuum... 

Up to now there has been no calculation of differential cross sections by a method that is generally valid. We use a formulation due to Konovalov (1993). Understanding of ionisation has advanced by an iterative process involving experiments and calculations that emphasise different aspects of the reaction. Kinematic regions have been found that are completely understood in the sense that absolute differential cross sections in detailed agreement with experiment can be calculated. These form the basis of a structure probe, electron momentum spectroscopy, that is extremely sensitive to one-electron and electron-correlation properties of the target ground state and observed states of the residual ion. It forms a test of unprecedented scope and sensitivity for structure calculations that is described in chapter 11. 

In a kinematically-complete ionisation experiment for an incident beam of momentum ko, the differential cross section is normally measured for a range of a single variable determining the momenta k/ and k of the faster and slower final-state electrons. The kinematic variables are the kinetic energies /, Eg, the polar angles Of, 6s, measured from ko, and the relative azimuthal angle... 

The differential cross section (6.60) is sometimes called the triple differential cross section because it is differential in two solid angles and one energy. In the absence of spin analysis it provides the most-detailed information about the ionisation mechanism, but it is impracticable to study it over the full kinematic range available to a three-body final state. It is more informative to study it as a function of one variable in restricted kinematic regions. 

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. 

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).

Fig. 10.6. Relative differential cross section for the ionisation of helium (Pan and Starace, 1991). Ef = Eg = 2 dV, (f> = 0, the polar angle is Of = n+Og. Open circles, Schlemmer et al. (1989) solid triangles, Selles, Huetz and Mazeau (1987) full curve, distorted-wave calculation including the screening effect of the final-state electron-electron interaction broken curve, distorted-wave Born approximation.

The differential cross section for resonant ionisation has been calculated by McCarthy and Shang (1993). The approximation treats the T-matrix element as a coherent superposition of two amplitudes. One describes direct ionisation and is analogous to the amplitudes of section 10.1. The other contains the momentum kj in a resonant amplitude that has an entirely different structure. One can find values of 9s for which the direct amplitude is small (see fig. 10.1 for an analogous reaction). Here the resonant amplitude dominates the T-matrix element. At angles 6s where the direct amplitude is large, interference between the direct and resonant amplitudes is observed. 

Fig. 10.12. Primary-electron double differential cross section for electron-helium ionisation. Experimental data are due to Muller-Fiedler et al. (1986) (open circles) and Avaldi et al. (1987a) (full circles). Full curves, distorted-wave Born approximation (McCarthy and Zhang, 1989). Cases illustrated are (a) Eq = 100 eV, Ef = 73.4 eV(A), 71.4 eV(B), 55.4 eV(C) (b) Eq = 300 eV, Ef = 235.4 eV (cross section multiplied by 100) (A), 271.4 eV(B) (c) Eq = 500 eV, Ef = 471.4 eV(A), 435.4 eV(B). From McCarthy and Zhang (1989).

Scattering experiments are usually not very sensitive to structure. On the other hand the differential cross section for ionisation in a kinematic region where the plane-wave impulse approximation is valid gives a direct representation (10.31) of the structure of simple targets in the form of the momentum-space orbital of a target electron. 

Electron momentum spectroscopy (McCarthy and Weigold, 1991) is based on ionisation experiments at incident energies of the order of 1000 eV, where the plane-wave impulse approximation is roughly valid. The differential cross section is measured for each ion state over a range of ion recoil momentum p from about 0 to 2.5 a.u. Noncoplanar-symmetric kinematics is the usual mode. In such experiments the distorted-wave impulse approximation turns out to be a sufficiently-refined theory. Checks of this based on a generally-valid sum rule will be described. 

Fig. 11.2. Relative differential cross section at = 10° for the 400 eV noncoplanar-symmetric ionisation of argon (Weigold et ai, 1973). The arrows indicate known energy levels of Ar+.

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