Oscillations inNa+-Na Charge-Transfer Cross Sections

Abstract

Total charge-transfer cross sections of fast sodium ions on sodium atoms are calculated by an impact-parameter method using an analytical Hartree-Fock wave function. The results show oscillations in the cross section as a function of the velocity of ions, and they are compared with the experimental result observed by Daley and Perel.Daley and Perel 1 measured the total chargetransfer cross section of sodium ions on sodium atoms as a function of ion energy ranging from 0.5 to 24 keV using a cross-beam technique and phase-sensitive lockin detection similar to that used to measure other alkali-metal cross sections. 1 * 2 The results of the Na + -Na cross-section measurement show oscillations in the cross section as a function of ion energy similar to those observed for different alkali-metal ionatom combinations. 1 ' 2 So far no theoretical results predicting such oscillations in the Na + -Na charge-transfer cross section have been reported. In the present work we have calculated the results of the Na + +Na-Na + Na + cross section by an impact-parameter method using dementi's analytical Hartree-Fock wave function. 3 All quantities are in atomic units. In this formulation, the cross section for charge transfer is given bywhere v is the velocity of the ion along the x axis and with H fi ' = {f\H f \i), H H ' = (i\w \i), and S = (i\f) = {f\i), i and/ denoting the initial and the final states, respectively. The interaction Hamilto-nian iswhere R is the distance between the two nuclei and y f the distance of the active electron from the ion. With the interaction Hamiltonian (3) and using dementi's eight-parameter analytical Hartree-Fock wave function 3 for sodium, the integrand of Eq. (2) is calculated numerically and is represented by XI(R) =e' aR [b + C R +dR 2 +eR? +/R 4 +gR 5 ]with an average error of 0.4% from R = 2.5 to R = 20. The values of the parameters are as follows: a = 0.724 994 88, b = 0.698 023 75, c = -1.239 414 2, d = 0.455 940 90, e =-0.052308 794, / = 0.003 847 385 8, #=-0.486 97075X10" 7 . Now we write Eq. (2) as £(fl 0 )=2X(# 0 ),where X(R 0 ) = 2f~QXI(R)RdR/(R 2 -tf, 2 ) 1 ' 2 .Integral (5) is evaluated in terms of modified Bessel functions of the third kind. The results 583