Electrons differential cross section

The END trajectories for the simultaneous dynamics of classical nuclei and quantum electrons will yield deflection functions. For collision processes with nonspherical targets and projectiles, one obtains one deflection function per orientation, which in turn yields the semiclassical phase shift and thus the scattering amplitude and the semiclassical differential cross-section... 

The differential cross-section, averaged over the initial electron spin states and summed over the final spin states for the potential... 

We consider the expression of the lab frame photoelectron angular distribution for a randomly oriented molecular sample. The frozen core, electric dipole approximation for the differential cross-section for electron emission into a solid angle about a direction k can be written as... 

Application of the formalism of the impulse approximation to the double differential cross section in terms of the dielectric response (Equation 12), that is, using free-electron-like final states E = p+q 2/2m in the calculation ofU(p+q, E +7ko)... 

At high enough energies the free electrons can be described by plane waves, and the differential cross section is given by [2]... 

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... 

By further integrations over the energy or angle of the emitted electron we obtain the single differential cross section as a function of the angle and energy of the emitted electron, respectively ... 

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 ... 

The longitudinal momentum projectile transfer da / dp P obtained by consideration of Eq. (66) is expressed as a function of the singly differential cross section of the emitted electron from the projectile ion by... 

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 10. Double differential cross sections (ddcs = Avj

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],...

The projectile scattering angle QP has always proved to be extremely difficult to measure experimentally due to the small deflection of the projectile. However, the singly differential cross section as a function of fIp contains a wealth of information on binary collisions between the projectile and the target electrons. This singly differential cross section is given by... 

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... 

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