The comparison of experimental and theoretical symmetric charge transfer cross sections. A. V. Phelps, April 10, 2003 I have always thought of symmetric charge transfer as being that aspect of the differential scattering cross section that shows as a peak in the backward direction at high energies. This picture is the result of having been brought up on the model of symmetric charge transfer as the result of nearly straight line particle trajectories appropriate to relatively high collision energies, e.g., Holstein, J. Phys. Chem. 56, 832 (1952). However, this picture fails as the elastic differential scattering cross section becomes more isotropic at low collision energies. Also, when looked at on a fine enough angle scale, the backward peak is generally lower than the forward peak and actually can be a dip (see 7 below) at 180 deg in center of mass. It is important to note that the quantum mechanical expression for the symmetric charge transfer cross section is also obtained by extending the picture appropriate to high energies to all energies. Thus, at high energies the forward and backward components of the scattering function do not interfere and can be evaluated separately. The backward component is identified as giving the charge transfer cross section. The resultant expression in then used under all conditions, e.g., even when the detailed calculations show strong interference effects at all angles. Many authors use this formula. Papers like Sinha, Lin, and Bardsley, J. Phys. B 12, 1613 (1979) and Krstic and Schulz (see below) are faintly critical of the formula, but do not say it is wrong. Thus, quantum mechanics texts give us a potentially questionably quantity to be compared with experiment. Recently, it dawned on me that the symmetric charge transfer cross section results from quantum mechanics can be more accurately compared with experiment at most (all?) energies by measurement or calculation of the integrated scattering into the backward half sphere. This empirical correlation can be seen to predict the right relation between the diffusion cross section and the charge transfer cross section for isotropic scattering, as for very low energies, and for backward scattering, as at high energies. In both of these cases the half-sphere, charge transfer cross section is easily shown to be half the momentum transfer or diffusion cross section. Some points to consider are: 1)This change in the picture of symmetric charge transfer from a backward peak to the backward half sphere allows one "understand" the 2:1 cross section ratio found in several recent quantum mechanical calculations: a) For the scattering of H+ by H at 0.1 eV, Fig. 8 of Krstic and Schulz, J. Phys. B 32, 3485 (1999) shows that it is essentially impossible to assign a backward scattering component to the elastic cross section. Yet Krstic and Schulz show, see Fig. 5, that the symmetric charge transfer cross section has its maximum values near 0.1 eV and is very nearly half the diffusion cross section at all energies. b) Cote and Dalgarno, Phys. Rev. A 62, 012709 (2000) show that the theoretical charge transfer cross section for Na+ + Na is almost exactly half the diffusion (momentum transfer) cross section from 1E-15eV to 10 eV. The departures from 2:1 near 1E-5 eV may be the result of interference effects. At very low energies the scattering is s-wave and the 2:1 ratio is expected. 2) The determination of symmetric charge transfer cross sections from measurements of scattering into the backward half-sphere was used by Hinds and Novick, J. Phys. B 11, 2201 (1978). These authors used a retarding grid set at one quarter of the beam energy to reject ions that have less than half their initial velocity in the beam direction after a collision. These authors claim that their measured cross section is equal to the charge transfer cross section to within a few percent, but do not give details. I think that there is another reference using this approach, but I have misplaced it. 3) Helm, J. Phys. B 10, 3683 (1977) takes the symmetric charge exchange cross section to be half the diffusion (momentum transfer) cross section determined from mobility measurements at all energies. So far, I have not found his argument for doing this at low energies. 4) As far as Monte Carlo calculations of ion transport are concerned, the definition of symmetric charge transfer as scattering into the back half-sphere results in a great deal of uncertainty as to what to assume for the differential cross section. The importance of this uncertainty is significantly reduced by the conclusion of Piscitelli and Pitchford (private communication) that the calculated mobilities of Ne+ in Ne and Xe+ in Xe are independent of whether one uses an isotropic elastic differential cross section or a two component (backward and isotropic) elastic differential scattering cross section with the same diffusion cross section. 5) A potentially significant loss to the simplicity of some ion transport models from this definition of symmetric charge transfer is breakdown of the idea that zero energy ions are produced by the collision with a "cold", parent gas atom. The idea is still useful at "high" energies because of the detailed behavior of the differential cross section within the backward half sphere. 6) The definition of the experimental symmetric charge transfer cross section as the integrated scattering in the backward half sphere considerably simplifies the interpretation of many beam measurements that make use of the collection of slow ions using a transverse electric field. This is because the details of the angular distribution of product ions is unimportant provided the ions are concentrated in the backward and forward directions in CM. 7) A point that is related to the above is that for the case of symmetrical ion scattering, the differential cross section should not be symmetrical about 90 deg CM and that there should be a minimum in the differential cross section at 180 deg CM. Using classical trajectories, these effects are the result of impact parameters for which the distance of closest approach is outside the range of effective charge transfer and the corresponding scattering angle is too small for charge transfer to be effective. The result is a peak in the forward scattering at small angles that is not reflected about 90 deg and a corresponding deficiency of scattered particles at 180 deg CM. Calculations show that this structure is often not obscured by ion diffraction effects. These structures in the differential cross section can be seen in the theoretical results of Krstic and Schulz for H+ + H for 0.1 and 1 eV (Fig. 8). In the case of He+ + He, this effect shows as a lower scattering peak in the backward direction than in the forward direction - Sinha et al (1979). 8) Sinha et al (1979) point out that the "formulation of transport theory does not require the definition of a cross section for charge transfer". 9)The discussion of the results of the theory of charge transfer in this note point to the importance of using terminology other than the charge transfer cross cross section when referring to what I have been calling the backward scattered component of the differential scattering cross section for symmetric ion atom scattering. See J. Appl. Phys. 76, 747 (1994) for Ar+ + Ar and Phys. Rev. E 68, 046408 (2003) for Ne+ +Ne and Xe+ +Xe. By using the correct terminology one avoids confusion caused by the discrepancy at low energies between the magnitude of my backward scattered component and the magnitude of the charge transfer cross section from quantum theory as discussed above. This discrepancy increases as the collision energy is reduced. The backward scattering and charge transfer cross sections are equal only at high collision energies.