Direct differential cross section

We have implemented semiclassical corrections via the Schiff approximation [21] for small scattering angles. The direct differential cross section in the Schiff approximation within the END approximation is given hy [22]... 

In other words, the classical direct differential cross section... 

R. Cabrera-Tmjillo, J.R. Sabin, Y. Ohrn and E. Deumens, Direct differential cross section calculations for ion-atom and atom-atom collisions in the keV range, Phys. Rev. A, 61 (2000) 032719. 

With this, the direct differential cross section is obtained as... 

Once the deflection function and the phase shift have been determined for a collisional system, the direct differential cross section is obtained as the classical direct differential cross section when neglecting quantum interference effects [equation (13) with Pfo = 1] or by introducing semiclassical corrections using the Schiff approximation [equation (20)]. 

As an example, in Fig. 3 we show the result for the semiclassical absolute direct differential cross section for neutral helium atoms colliding with neon targets at projectile energies of 0.5, 1.5, and 5.0 keV [12]. Also, for comparison we present the experimental data of Gao et al. [23]. 

The advantage of semiclassical corrections is the inclusion of quantum effects to the differential cross section in the small scattering angle, the so-called forward peak character of the differential cross section. Furthermore, in the particular case of the Schiff approximation, the glory and rainbow angle effects in the interference are accurately represented. This behavior can be observed in Fig. 3, where the absolute direct differential cross section obtained with ENDyne goes through the experimentally determined absolute cross sections. 

Fig. 4. Direct differential cross section for H and H — H2 for H energies from 0.5, 1.5 and 5.0 keV. The experimental data are from Gao et al. [24] for the molecular case. Note that d(r/d/2 for H2 is not twice the value that the one for H.

Although the result follows the experimental data closely, the result does not look as good as the direct differential cross section. We attribute this to the need for a finer average grid for the target orientational average. 

In Fig. 7, we present preliminary results obtained by this procedure for the direct and electron capture differential cross section for protons colliding with helium atoms at 5 keV. The experimental data are from Johnson et al. [30]. Here the quantum effects are evident. The END charge transfer differential cross section is out of phase with the direct differential cross section and in close agreement with the experimental data. 

A unifonn monoenergetic beam of test or projectile particles A with nnmber density and velocity is incident on a single field or target particle B of velocity Vg. The direction of the relative velocity m = v -Vg is along the Z-axis of a Cartesian TTZ frame of reference. The incident current (or intensity) is then = A v, which is tire number of test particles crossing unit area nonnal to the beam in unit time. The differential cross section for scattering of the test particles into unit solid angle dO = d(cos vji) d( ) abont the direction ( )) of the final relative motion is... 

The differential cross section for scattering of both the projectile and target particles into direction 0 is... 

Symmetry oscillations therefore appear in die differential cross sections for femiion-femiion and boson-boson scattering. They originate from the interference between imscattered mcident particles in the forward (0 = 0) direction and backward scattered particles (0 = 7t, 0). A general differential cross section for scattering... 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

The most important observable is the angular distribution of the scattered products with respect to the initial approach direction of the reagents, which is called the state-to-state differential cross-section (DCS). The DCS can be written [57-61]... 

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... 

For the remaining of this chapter we will first describe the basic concept of this new technique, the details of our experimental setup, and the way to invert the measured data directly to the desired center-of-mass differential cross-section. Two types of applications will then be highlighted to illustrate the power of this exceedingly simple technique. We will conclude the chapter by comparing the technique with other contemporary modern techniques. 

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). 

Figure 19(a) shows the QM simulation of the differential cross-section (DCS) in the HF + D channel, over the same extended energy range as in Fig. 5. The agreement with experiment is seen to be qualitatively reasonable. The forward-backward peaking and direct reaction swathe observed in the experiment also occur in the QM calculation, although the relative magnitudes are not consistent. Thus fully quantitative agreement between QM calculations and experiment in all of the reaction attributes must await further refinements of the PES, and/or a more rigorous treatment of the open-shell character of the F(2P) atom.90...

---