Double-differential cross section

Details of the emission of ultralow- and low-energy electrons can be seen in the cross sections doubly differential in the longitudinal and transverse momenta. The formula for the double differential cross section as a function of the longitudinal electron velocity vf, for various transverse electron velocity ve cuts can be derived from Eq. (66), noting that... 

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)... 

The intensity of the scattered neutrons is given by the double-differential cross section 02er/0Q dE, which is the probability that neutrons are scattered into a solid angle cl l with an energy change 6E = hm. 

For a system containing N atoms the double-differential cross section is... 

If the partially labelled star/solvent system is considered as an incompressible ternary solution, the double differential cross section 02ct/0 20E can be written as... 

Figure 10. Double differential cross sections (ddcs = Avj

Figure 20. Electron emissions at 0 = 0° for 40-keV H+ ion impact in H2. The double differential cross section (DDCS = ifia/dfldE ) is plotted against k/v, where v is the impact velocity, k is the ejected-electron momentum, and dU — 2k sin 0 dd. The filled circles represent the experimental data [38], and the CDW-EIS results are given by the solid line [38].

Figure 22. Electron emissions for 40-keV H+ ion impact on He. A CDW-EIS surface plot [38] for the double differential cross section d2a/dfidEk is plotted against k/v (see the caption of Fig. 20).

In Figs. 24 and 25 we show the measured double differential cross sections for electron emission at zero degrees in collisions of 100-keV protons with He and H2 [39] compared to CDW-EIS predictions [39]. Uncertainties associated with the experimental results vary from 1% near the electron capture to the continuum peak to about 15% near the extreme wings of the distribution. These results have been scaled to provide a best fit with CDW-EIS calculations. In both cases there is satisfactory agreement between the CDW-EIS calculations and experiment, particularly with excellent agreement for electrons with velocities greater than v, where v is the velocity of the projectile. For lower-energy electrons the eikonal description of the initial state may have its limitations, especially for lower-impact parameters. 

Figure 24. Measured double differential cross sections [39] given by SafdQ.dk (where dQ = 2itsin0(i0) for electron emission at zero degrees in collisions of 100-keV protons with He compared to CDW-EIS predictions [39].

Figure 26. Electron emissions for 100-keV H+ ion impact on He. The CDW-EIS calculations [38] now predict a ridge structure from k/v = 0 to k/v 0.6 with a maximum at k/v 0.25. The double differential cross sections are given by d2a/dQ,dEi, where dQ, = 27tsin0d0.

In Fig. 28 the experimental double differential cross sections for electron emission at zero degrees of a neon target in collision with 80-keV protons are compared to theoretical predictions. Again we have quite good agreement between experimental results [41] and the CDW-EIS predictions [41],... 

In general, scattering of thermal neutrons yields information on the sample by measurement and analysis of the double differential cross section ... 

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. 

Figure 1. Double differential cross sections (ddcs = ...

Figure 3. Double differential cross sections fddcs = f—) as a function of...

Example. As a model for two-stage diffusion take i = 1,2 and F as in (7.1). Then 7i,2 = 2 and 72,1 =7i- F°r computing the cross-section for neutron scattering one needs to know the probability density Gs(r, t) that a molecule that, at t = 0, was at r = 0 will, at time t, be at r. The differential cross-section is its double Fourier transform GS(A , co). It is convenient to apply the Fourier transformation in space right away to (7.4) so that both operators Ff reduce to factors,... 

---