I. COLLISIONS PROCESSES NOT INVOLVING CHEMICAL REACTIONS Resonance Effects in Atom-atom Collisions BY D. R. BATES Dept. of Applied Mathematics, Queen’s University, Belfast, Northern Ireland Received 1st February, 1962 Neglecting the diagonal matrix elements and making other simplifications, an approximate formula is obtained for the cross-section of an excitation or charge-transfer collision. This formula is evaluated in some representative cases. The resonance peak is found to be very sharp at thermal energies. However, the effect of the diagonal matrix elements is likely to be important. In some cases it increases and in others it decreases the sharpness of the resonance peak. This paper is concerned with the influence of the closeness of the energy balance on the cross section for atom-atom collisions involving excitation transfer A+ B*-+ A* + B, (1) or charge transfer A+B++A++B.If the energy balance is exact the process is said to be in resonance. The resonance is called symmetrical if A and B are the same species of atom and asymmetrical (or accidental) if A and B are different species. In the calculations to be described the nuclei are treated as classical particles. States other than the initial state p and the final state q are ignored. Atomic units are used except where otherwise specified. GENERAL THEORY Neglecting the effect of the inter-atomic force on the relative motion, let one of the nuclei be located at the origin of a fixed co-ordinate system and let the other travel with constant speed t, parallel to and at a distance p from the Z-axis.On the two- state approximation adopted expansion coefficients cp(Z,p) and cq(Z,p) associated with the states p and q occupied by the atomic systems on the left and right of (1) or (2) satisfy the coupled differential equations, and the Y’s (which are functions of 2 and p) being the indicated matrix elements of the interaction potential and the E’S the indicated internal energies of the separated systems.l 78 RESONANCE EFFECTS I N ATOM-ATOM COLLISIONS The boundary conditions corresponding to the colliding systems being initially in state p are The probability that the systems are Snally in state q is I C p ( - %P) I = 1 9 cq(- cv) = 0. (6) P@) = I Cq(mYP) f (7) and the cross-section for process (1) or (2) is f* NEGLECT OF DIAGONAL MATRIX ELEMENTS If the difference * between the diagonal matrix elements Vpp and V& is neglected (3) and (4) reduce to and iac, = Tc v exp (-- iaz).az v qp Taking V& and Vgp to be real and equal, and putting VPqdZ = a@) Km and V, exp (iaZ)dZ = A(p9v), s:* Rosen and Zener 2 have conjectured that the required solution to these equations is such that the transfer probability A2 o2 9 = - sin2 { +}. The conjecture is true if the interaction is weak enough for the first Born approxi- mation to be valid or has a form of a certain class (e.g.¶ case (iii) below). It is also true if the energy balance is exact; and recent calculations by Skinner 3 show that it provides at least a fair approximation if the cancellation within the integral of (1 3) is not so severe as to make A smaller than S2 by a large factor.? For present purposes (13) is adequate.To evaluate the transfer cross section from (8) in the low velocity region defined by 1 1) 2’ -Q(O) > It * This difference vanishes in the case of symmetrical resonance but does not in general do so in the case of asymmetrical resonance. t When the cancellation is severe, 9 decreases veryrapidly as a is increased. Instead of being regarded as involving a considerable displacement along the 9-axis the error in the Rosen-Zener approximation to the 9 against a curve may be regarded as involving only an unimportant displace- ment along the a-axis.D. R . BATES 9 it is sufficient to use a slight generalization of a remarkably accurate simple approxi- mation due to Firsov.4 According to this where 3 = A2/2Q2 andp* is the greatest root of the equation 1 1 -Q(p*) = -.V n Using (15) the expressions given below were obtained for the transfer cross- sections associated with three spherically symmetrical, but otherwise representative, forms of the transition matrix element. As usual &(x) is the nth order modified Bessel function of the third kind. CASE (i) (EXCITATION TRANSFER, DIPOLE TRANSITIONS) If the transition matrix element V,, = A/(a2 + p2 + Z2)>5 the transfer cross-section is Q, = na2E*p(a2 +p2)Kf[a(aZ +p2)*]dp (19) 2 A / ~ ( a ~ + p * ~ ) = 1/n. (20) with At resonance (19) becomes Qi = n[,--i;]; nA a2 ' a = 0; while if the energy balance is not close it yields that (22) 32 25 1 Qi-G[(aa)2+-Jaa)+-+ n2 7 . . . exp (-2aa); a a a l ; p*%a.CASE (ii) (EXCITATION TRANSFER, QUADRUPOLE TRANSITIONS) If Vpq = B/(b2+p2+Z2)0, the transfer cross-section is with 4B/3v(b2 + p*2)2 = l/n. At resonance (24) becomes Qii=n[-$} * -.]: b2 a = 0 ; while if the energy balance is not close it yields that (27) 1 Q i i - ~ [ ( b u ) 4 + $ b u ) 3 + ~ ( b a ) 2 + 1 2 g ( b a ) + n2 23 2079 . . . exp (-2ba); ba$l; p * $ h10 RESONANCE EFFECTS IN ATOM-ATOM COLLISIONS CASE (iii) (EXCITATION TRANSFER, AN s--s OR SPIN-CHANGE TRANSITION; CHARGE TRANSFER) If VP, = (++ CC exp [Y{1- (c2 + p2)+ }Isech [(c27;2)*]9 the transfer cross-section is with and with At resonance (27) becomes (nC/yv) exp [y(1- (c" +p*")+/c)] = l/n, (30) x1 = na/2y, x2 = na(c2 +p*2)+/2yc. (3 1) while if the energy balance is not close it yields As already mentioned the Rosen-Zener formula is here exact.Stueckelberg 5 has carried through the very complicated mathematical analysis which arises in the wave treatment. Part of his investigation was, like the present, concerned with transfer in the absence of a pseudo-crossing of the initial and final potential energy surfaces. Account was not taken of matrix elements other than the transition matrix element V& which was assumed for simplicity to be spherically symmetrical and to be of the inverse power law type. The formulae displayed above are in quite satisfactory accord with the corres- ponding formulae of Stuckelberg if the energy balance is not close. Moreover, they correcly predict that the transfer cross-section is greatest at resonance, whereas the formulae of Stueckelberg wrongly predict that the transfer cross-section is zero at resonance unless Vpq falls off at least as slowly as the inverse cube of the internuclear distance .Fig. 1, 2 and 3 show computed Q against A& curves at selected values of v (ex- pressed for convenience as the corresponding kinetic energy E of a system of mass 10 on the chemical scale) the parameters occurring in expressions (18), (23) and (28) for Vpg being assigned the following values : a = b =2, c = y = 1, (34) (35) 1 A = 3, 1 or 2, B = 2,4 or 8, c = l 1 169 5 Or (which make the maximum of V& in each of the three cases 1.7 eV, 3.4 eV or 6.8 eV).D. R. BATES 11 The cross-sections at resonance are very large in case (i), and are quite large even in cases (ii) and (iii).They fall off with increasing rapidity as I A& I is increased from zero. As would be expected the fall off is especially sharp if Eis low or if the range -7 E = lOeV A I I A I 1 I I 10 16' L3 Id ' 16' lCj2 10' lo3 I 2 change in internal energy l d ~ l in electron volts FIG. 1.-Q against A& curves for case (i) with parameters as in (34) and (35) of text. Note that the horizontal scale is broken and that the vertical axes on the left (each with three points marked) change in internal energy ld&I in electron volts FIG. 2.-Q against A& curves for case (ii) with parameters as in (34) and (35) of text. Note that the horizontal scale is broken and that the vertical axes on the left (each with three points marked) are for exact energy balance.of Vpp is long. In the region away from the resonance peak the cross-sections are insensitive to A, B and C, that is, are insensitive to the magnitude of a transition matrix element of given form. Examination of (15) provides the explanation : thus the dependence of Q on A, B or C arises only thronghp" ; and in the region concerned12 RESONANCE EFFECTS IN ATOM-ATOM COLLISIONS the oscillatory nature of the integrand on the left of (12) makes is as great as this upper limit. negligible before p INCLUSION OF DIAGONAL MATRIX ELEMENTS As already noted Vpp and Vqq are equal in the case of symmetrical resonance so that they disappear from eqn. (3) and (4) owing to cancellation. It does not follow that Q is independent of these diagonal matrix elements since (3) and (4) ignore the change in the relative motion arising from them. The effect which this change has on Q is likely to be appreciable if the value of the diagonal matrix elements at inter- nuclear distance p* is comparable with the energy of relative motion.Buckingham and Dalgarno 6 have shown that excitation transfer between metastable and normal helium atoms is greatly inhibited at energies below about 0-3 eV. FIG. 3.-Q against A& curves for case (iii) with parameters as in (34) and (35) of text. Note that the horizontal scale is broken and that the vertical axes on the left (each with three points marked) are for exact energy balance. If the transfer is not of the symmetrical resonance type the change in the relative motion is, in general, much less important than the presence of the diagonal matrix elements in the exponents on the right of (3) and (4).Consider the conditions for the two effects to be negligible. They are respectively, where E is the energy of relative motion, M is the reduced mass and R, is an ill-defined effective collision radius. Expressing Vpp, VQQ and E in eV and ex- pressing M on the chemical scale, but keeping Rc in atomic units, (37) may be written It may be seen that (38) is a more severe condition than (36) unless Vpp and V& are almost equal. If the energy balance is not close, the transfer cross-section determined from the simplified equations, (9) and (lo), is moderate or small at thermal energies (cf. fig. 1, 2 and 3). The effective collision radius R, cannot be large and condition (38) is unlikely to be satisfied.Eqn. (3) and (4) should therefore be used. If fVpp- Vqg] and [ E ~ - E ~ ] have the same sign it is apparent that these equations tend to yield lower 1 Vpp(Rc) - bq(R3 1 / E .e 1mM+@l* (33)D. R. BATES 13 cross-sections than do (9) and (10) ; that is, the distortion due to the diagonal matrix elements tends to increase the sharpness of the resonance peak. The position is different if [Vpp- V&] and [ E ~ - E ~ ] have opposite signs. In this circumstance the tendency is for the distortion to make the resonance peak less sharp (irrespective of whether or not there is a pseudo-crossing of the initial and final potential-energy surfaces). The rather complicated effect of the diagonal matrix elements should be borne in mind in any attempt at arranging measured transfer cross-sections into a pattern showing the influence of the closeness of the energy balance. This work was supported by the U.S. Office of Naval Research under Contract N 62558-2637. 1 cf. Bates, Quantum Theory I Elements (Academic Press, New York, 1961), chap. 8. 2 Rosen and Zener, Physic. Rev., 1932,40, 502. 3 Skinner, Proc. Physic. SOC., 1961, 77, 551. 4 Firsov, J. Expt. Theor. Physics (U.S.S.R.), 1951,21, 1001. 5 Stueckelberg, HeZv. phys. Acta, 1932, 5, 370 ; cf. Mott and Massey, Theory of Atomic Col- 6 Buckingham and Dalgarno, Proc. Roy. SOC. A, 1952, 213, 506. lisions (Oxford University Press, London, 2nd ed., 1949), chap. 12.