Doubly differential cross sections

The conventional differential cross-section1,2 refers to the cross-section per solid angle in the center-of-mass polar coordinate. The desired doubly differential cross-section is then related to the measured quantity by the simple equation... 

To sum up, the basic idea of the Doppler-selected TOF technique is to cast the differential cross-section S ajdv3 in a Cartesian coordinate, and to combine three dispersion techniques with each independently applied along one of the three Cartesian axes. As both the Doppler-shift (vz) and ion velocity (vy) measurements are essentially in the center-of-mass frame, and the (i j-componcnl, associated with the center-of-mass velocity vector can be made small and be largely compensated for by a slight shift in the location of the slit, the measured quantity in the Doppler-selected TOF approach represents directly the center-of-mass differential cross-section in terms of per velocity volume element in a Cartesian coordinate, d3a/dvxdvydvz. As such, the transformation of the raw data to the desired doubly differential cross-section becomes exceedingly simple and direct, Eq. (11). 

It should be noted, however, that gaining a deeper insight into the problem of ionization phenomena is not the only reason for steady interest in the problem. Data on charged particle impact ionization is used both for industrial applications and for fundamental scientific research. For applications it is the collisions rates and total cross sections which are usually the most relevant. But in studies focused on the understanding of collision mechanisms of ionization processes, most of the information is lost in the total cross sections due to the integration over the momenta of the ejected electrons in the exit channel. Therefore it is the singly and doubly differential cross sections which are of... 

More details of the emission of ultralow- and low-energy electrons from fast heavy ion-atom collisions may be seen in the doubly differential cross sections as functions of the longitudinal electron velocity for increasing transverse electron velocity. Examples considered in this chapter include singly ionizing... 

Thirdly we are also interested in the electron spectroscopy method, which allows investigations on the two-center effects that influence electron emission. In particular, the richness of the ionization process lies in the possibility of measuring the doubly differential cross sections as a function of the electron emission angle and energy. This technique of electron emission spectroscopy is... 

Normally these conditions are satisfied in fast highly charged ion-atom collisions. From Eq. (66) we can derive the equations for the singly differential cross sections with respect to the components of the longitudinal momentum distributions for the electron, recoil-ion, and projectile. The longitudinal electron momentum distribution da/dpe for a particular value of p, may be derived by integrating over the doubly differential cross section with respect to the electron energy Ek ... 

After several decades of systematic electron spectroscopy in ion-atom collisions by many groups (for recent reviews see Refs. 13 and 51), there are only two data sets of doubly differential experimental cross sections cfa/dE dfl for the emission of electrons with < 1 eV. It has been only recently that, with entirely new and extremely efficient electron spectrometers combined with recoil-ion momentum spectroscopy [52], doubly differential cross sections for ultralow -and low-energy electrons (1.5 meV < < 100 eV) have been obtained by... 

Figure 11. Doubly differential cross sections (DDCS — 2m> dufdv ) f°r the electrons emitted after the single ionization of helium by 3.6-MeV/amu Au53+ ions, plotted for the electron s longitudinal momentum distributions for increasing transverse momenta. Here only one very small cut has been made in the electron s transverse momenta (pf < 0.04 a.u.). Experimental data and theoretical results are from Schmitt et at. [50],...

Figure 17. Doubly differential cross sections for the ionization of He by 1.5-MeV/amu F9+ impact at an observation angle of 0 — 0" as a function of electron energy. Experimental data are from Lee et al. [12]. Theoretical results CDW results [57], CDW-EIS results [58].

In this section we consider measurements of doubly differential cross sections for electron emission at zero degrees arising from collisions of... 

Calculations using the CDW-EIS model [38] are shown to be in good accord with 40-keV protons incident on molecular hydrogen and helium, and at this energy both theory and experiment show no evidence of any saddle-point enhancement in the doubly differential cross sections. However, for collisions involving 100-keV protons incident on molecular hydrogen and helium the CDW-EIS calculations [39] predict the existence of the saddle-point mechanism, but this is not confirmed by experiment. Recent CDW-EIS calculations and measurement for 80-keV protons on Ne by McSherry et al. [41] find no evidence of the saddle-point electron emission for this collision. 

To summarize this section on saddle-point ionization, we have calculated doubly differential cross sections for the single ionization of He, H2, and Ne by proton... 

In the same manner, one can determine the total ionization cross sections ctj by integration of the doubly differential cross sections over both ejected electron energy and emission angle... 

Figure 4 Doubly differential cross sections for the ionization of water vapor by 1-keV electron impact. (From Ref. 38.)...

Doubly differential cross sections for electron emission following proton impact have been measured for a wide variety of different atomic and molecular target systems (for... 

Figure 14 Doubly differential cross sections for ionization of several low-Z molecular targets by 1-MeV protons plotted for selected ejected electron energies as a function of the emission angle. (From Refs. 47, 48, 56-58.)...

For more complex projectiles and targets, additional features become apparent in the spectra of ejected electrons. Doubly differential cross sections for ionization of CH4 by 0.3 MeV/u ions (3.6 MeV total projectile energy), taken from some of the author s unpublished work, are shown in Fig. 17 to illustrate the increasing complexity of dressed-ion collisions compared to that for bare ions represented in Fig. 9 by proton impact. In Fig. 17, the cross sections are seen to have their maximum at very low energy ejected electrons as was the case for proton impact, and, although their shape is not as sharp and... 

The theoretical basis for an effective charge z f[ E) associated with collisional energy loss has been investigated by several authors (see, e.g.. Refs. 69-72). Within the Born approximation, the doubly differential cross section for ejection of an electron with energy IF by a He ion can be written as... 

Figure 20 Doubly differential cross sections for ejection of electrons of 219 eV (16 Ry) from He by 2-MeV He ions. The points are measured cross sections and the calculated results are line A— projectile ionization, target remains in the ground state line B—projectile ionization with simultaneous target excitation line C— target ionization, projectile remains in the ground state and line D—target ionization with simultaneous projectile excitation. (From Ref. 70.)...

Theoretical CDW-EIS models and computer simulations developed during the last decade have been very successful in reproducing experimental data of doubly differential cross sections as a function of ejected electron energy and angle. These studies have enabled us to understand the main characteristics of electron emission spectra and the nature of two centre effects which may be observed in the double differential cross section spectrum. 

To conclude we have examined the applicability of the continuum-distorted-wave theory to experiments carried out using the reaction microscopy technique for both doubly differential cross sections for longitudinal electron velocities and fully differential cross sections applied to the single ionization of 3.6 MeV amu on neutral target atoms. For the case of the... 

Kover, A., Laricchia, G. and Charlton, M. (1994). Doubly differential cross sections for collisions of 100 eV positrons and electrons with argon atoms. 

Eq. (6) is the most general expression for the doubly-differential cross-section in terms of the convolution between the initial momentum distribution ho and the environment-induced broadening line shape L. 

We stress that condition (7) is essential for avoiding extra-broadening of the scattering line shape, beyond that of pi. Under this condition, the doubly-differential cross-section is related to its standard value [Watson 1996]... 

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