Evaluation of the second Born amplitude as a twodimensional integral for H+ + H(1s) + H(1s) + H+

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1 J. Phys. B: At. Mol. Phys. 14 (1981) L767-L771. Printed in Great Britain LETTER TO THE EDITOR Evaluation of the second Born amplitude as a twodimensional integral for H+ + H(1s) + H(1s) + H+ J M Wadehrat, Robin Shakeshaft.$ and J H MacekO t Physics Department, Wayne State University, Detroit, Michigan 48202, USA $ Physics Department, University of Southern California, Los Angeles, California 90007, USA P Physics Department, University of Nebraska, Lincoln, Nebraska 68588, USA Received 21 September 1981 Abstract. The second Born amplitude (with the exact propagator replaced by the free particle propagator) for the electron capture reaction H+ + H( 1s) + H( 1s) + H+ is reduced to a two-dimensional integral. The integrand has several singularities, which are dealt with. Results of the numerical integration of the cross section for various energies in the kev range are presented. Recently Miraglia etal(l981) and Simony and McGuire (1981) evaluated by numerical integration the second Born amplitude, with the free particle propagator, for the electron capture process H+ + H( 1s) -* H( 1s) + H+. They did this by first reducing the amplitude to a three-dimensional integral. The main purpose of the present letter is to show that this amplitude can be reduced to a two-dimensional integral. After circumventing certain difficulties caused by singularities in the integrand, as explained below, we evaluated this two-dimensional integral numerically for various projectile energies in the kev range. While the second Born approximation is very inaccurate in this energy range, we take this opportunity to present our results since they can provide a standard for comparison. Simony and McGuire have carried out calculations at energies up to 50 MeV, where the second Born approximation for reaction (1) becomes accurate. As a slight generalisation of reaction (l), we let the target and projectile nuclei be bare ions with unspecified atomic numbers ZT and Zp, respectively, and with masses MT and Mp, respectively. We work throughout in atomic units so that h and the electron charge and mass are unity. We introduce the average momentum transfer vectors K and J defined by (1) where, in the centre of mass frame of all three particles, Ki is the initial momentum of the projectile ion and -Kf is the final momentum of the target ion. Corrections of order l/mp and l/mt are hereafter neglected. Since the integrated cross section is unaffected by the internuclear potential up to corrections of order l/mp and ~/MT (see, for example, Wilets and Wallace 1968) this potential is omitted. (In the second Born /81/ The Institute of Physics L767

2 L768 Letter to the Editor calculations of Kramer (1972) the internuclear potential was included.) If t, is the velocity of the projectile relative to the target ion we have J+K+u=O. (3) The total cross section, U, for capture is where [= (1/2v)(vZ+2+-2;) (5) and where, within the second Born approximation (with the full propagator approximated by the free particle propagator), d = dl +d2; dl is the first-order Born amplitude dl = -25.rr(zpzT)5/2(~2 + z2 P) -3 * (6) d2 is the second-order Born amplitude which, after Fourier transformation, can be written as (see, for example, McDowell and Coleman 1970) y2 = 2Q U +2; -it where 7 is positive but infinitesimal and where a is to be set equal to Zp after the differentiation has been performed. Note that the imaginary part of d2 is negative; this provides a check on the numerical integration. The integrand of equation (76) has singularities at points where any one of the following equations is satisfied Q=O P+K=O (7c) (8b) (P - Q)'+ y2 = 0. (8c) Equation (8c) can be satisfied only if y2 s 0. It is convenient to transform from the variables P and Q to variables p and q defined by p=p-q q=q-j. (9) By making use of the Feynman identity the integration over p (in integral I) may be done directly. Decomposing q into cylindrical components as (p, 4, qz), with the z-axis parallel to U, I can be expressed as

3 letter to the Editor L769 where A is the positive square root of the right-hand side of the equation (lob), J, + U = -K, =.$ and J, and K, are the components of J and K, respectively, perpendicular to U. Equation (3) implies that J, = -Kx. The integral I can be evaluated in closed form by first performing the integration over using the formula dx to give, after writing l /u =p2+b2 with b2=qq +Z$, I = lr lo where llbz d u ( U* Bu-l+y )[-( 1 + Fl U + G1 U )-l/ + y ( 1 + F2 U + G2 U )- 1 ( 11 a ) B = c1- c2 - (1 - y )( c1- b 2, The integral over U in equation (lla) can be evaluated using the formulae of section (2.26) of Gradshteyn and Ryzhik (1965). Therefore I can be reduced to a twodimensional integral over y and qz. We will not give the complete closed form expressions for I and di /aa since they are rather complicated. The singularities listed in equations (8) are contained in the integrand I /A of equation (loa). The singularity (8b) appears as a singularity in the integrand I /A at y = 0, where A = 0. This singularity can easily be removed by transforming from y to Y, where y = Y2. However, since any uniform mesh of points in Y space is transformed into a mesh in y space that is sparse near y = 1, we found it expedient to divide the range of y integration into the intervals (0, 0.64) and (0.64, 1) and to transform from y to Y only in the former interval. The singularity (8a) appears as a singularity in the integrand I /A at qz = - Jz, where I is infinite. The divergence of I at qz = - J, arises from the vanishing of the denominator p2 + 2Jxp COS 4 + C1 in the integrand of equation (loc) at q = -J, To obtain the form of I for qz near -Jz, we may put q = -J in the two non-vanishing denominators of equation (loc) and obtain (for q, = -J,) -2lr If- (J2+Z$)2[(1-y)2K2+(A +ZP)~] lnlq,+jzi.

4 L770 Letter to the Editor The resulting singularity in the integrand of equation (loa) can therefore be removed by subtracting from I the term on the right-hand side of equation (12) over a finite interval centred at q2 = -J, ; this term can be added back and the implied integral over ql evaluated in closed form. The singularity (8c) appears as a branch point of I where y = 0; this occurs at q2 = Qg, where QB=-(2JZu +22,)/(2~). (13) This branch point can be removed by transforming from qz to Z where 2 is defined by In terms of 2 we have q2 =ZIZI/(~U)+QB. (14) y2 = Z/ZI-iq. (15) Note, however, that the derivative with respect to 2 of I is discontinuous at 2 = 0. (In particular, Im I = 0 for Z > 0, that is, for y real.) No difficulty arises if the range of 2 integration is divided into the two intervals Z L 0 and 2 G 0, with Z = 0 treated as an end point. After taking care of the singularities in the fashion just described, we evaluated the two-dimensional integral of equation (loa) using the nine-point sixth-order multidimensional integration rule given by formula ( ) of Abramowitz and Stegun (1970). This rule is probably not the best one for our purpose. However, our main concern was to demonstrate that we had overcome all of the troublesome features of the integrand, and we made no attempt to determine the best integration rule. Our results for reaction (1) at energies in the kev range are shown in table 1. We have also Table 1. Integrated cross sections in units of TU: for H++H(ls) + H(1s) + H+ for various projectile energies in the lab frame. Here uborn uborn and ucs are the cross sections obtained in the first Born, the second Born, and the coupled-state (Shakeshaft 1978) approximations, respectively. EidkeV) VBarn 1 uborn 2 u shown for comparison the (presumably fairly accurate) results of the 2 X 34 coupledstates calculation of Shakeshaft (1978), where these results are available. It is apparent that the second Born approximation is very inaccurate in the kev energy range. Our result for (+Born2 at 100 kev differs (in the second place) from the value 1.6 TU: for (+Born2 calculated by Miraglia et a1 (1981); the source of this discrepancy is not known. As a further numerical check, at 25 kev we evaluated ai/aa by numerical differentiation of I, and obtained the same results as we obtained from analytic differentiation. At energies in the MeV range, the integrand of equation (loa) peaks fairly sharply in some regions of the y - qz plane and it might be helpful (we have not investigated this matter) to perform the integrations over these regions approximately in closed form.

5 Letter to the Editor L77 1 This work was supported by the National Science Foundation under grants PHY and PHY and by a Wayne State University Research Award. References Abramowitz M and Stegun I A 1970 Handbook of Mathematical Functions (New York: Dover) Gradshteyn I S and Ryzhik I W 1965 Tables of Integrals, Series, and Products 4th edn, ed A Jeffrey (New York: Academic) Kramer P J 1972 Phys. Rev. A McDowell M R C and Coleman J P 1970 Introduction to the Theory of Ion-Atom Collisions (London: North-Holland) Miraglia J, Piacentini R D, Rivarola R D and Salin A 1981 J. Phys. B: At. Mol. Phys. 14 L Shakeshaft R 1978 Phys. Rev. A Simony P R and McGuire J H 1981 J. Phys. B: At. Mol. Phys. 14 L Wilets L and Wallace S J 1968 Phys. Rev

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