Coulomb phase interferences for small-angle inelastic scattering from ions

Transcription

1 J. Phys. B: At. Mol. Opt. Phys. 21 (1988) L25-L30. Printed in the UK LETTER TO THE EDITOR Coulomb phase interferences for small-angle inelastic scattering from ions Jim Mitroy Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado , USA Received 22 September 1987, in final form 27 October 1987 Abstract. We investigate the expected behaviour of the small-angle differential cross section for inelastic e--atom and e--ion collisions. The presence of the Coulomb phaseshift in the expression for the scattering amplitude for e--ion collisions leads to destructive interference between the individual partial-wave components at small angles. As a result, forward-angle differential cross sections for ions will be much smaller than for atoms. One consequence of this is that the limiting value of generalised oscillator strength for a dipole transition at zero momentum transfer is not the optical oscillator strength. Calculations for the Be+ 2s-2p transition show a large reduction in the forward-angle differential cross section, and demonstrate that this interference effect is so pronounced that it should be experimentally observable. It is well known that the calculation of electron-ion excitation (and ionisation) cross sections is a subject which is partly, if not primarily, motivated by the necessity of utilising the results in a number of important applications (e.g., the modelling of fusion plasmas). Indeed, there have been over 1000 papers reporting the results of such calculations. However, there have been relatively few calculations and only a couple of experiments which report data other than total excitation cross sections. There have been some recent experiments (Chutjian 1984, Chutjian and Newel1 1982, Chutjian et al 1983, Williams et al 1985, 1986) reporting differential cross sections for three ionic species (Mg+, Zn+ and Cd+). These measurements, which were carried out at the Jet Propulsion Laboratory (JPL), were restricted in both the range of angles (4-18") and the range of incident energies ( ev) studied. While there have been calculations performed on these species (Chutjian et al 1983, Msezane and Henry 1985, 1986, Williams et a1 1985), the differential cross sections were calculated using a program (Brandt et a1 1973) that is inappropriate for ions since the Coulomb phase factors are omitted in the expression of the scattering amplitude (Mitroy 1987). The JPL group used these incorrectly calculated cross sections to put their results on an absolute scale. There have also been calculations of the differential cross section for the 1s-2s and 1s-2p transitions for a number of hydrogenic ions (Deb et al 1983, Mitra and Si1 1978, Walker 1974). These calculations were done within the framework of the Coulomb-Born approximation. Mitra and Si1 (1978) noted that the differential cross section for the 1s-2p transition achieved its peak value at a non-zero angle for a sufficiently large value of 2. Walker (1974) noted that including the Coulomb phase in the scattering /88/ IOP Publishing Ltd L25

2 L26 amplitude resulted in differential cross sections for the 1s-2s transition that were very different from differential cross sections computed without the Coulomb phase. In this letter we demonstrate that the Coulomb phase factor, implicit in the scattering amplitude for ions (Lane and Thomas 1958), will cause the individual partial-wave contributions to the scattering amplitude to sum differently for ions than for atoms. A particular example where Coulomb phase effects are expected to be important is in the differential cross sections for the resonance transitions of alkali-like ions. Specifically, for small-angle scattering, the Coulomb phases will cause the partial-wave contributions to the scattering amplitude to interfere destructively (when otherwise they would interfere constructively), resulting in a differential cross section that is considerably reduced. Indeed, the effect is so pronounced that the strong forward peaking (at 0 = 0') that is a characteristic signature of strongly allowed dipole transitions for neutral species is greatly reduced or even eliminated for ions. The scattering amplitude for an inelastic transition between states i and j with quantum numbers Li, ML,, Si and Ms, and Lj, MLJ, Sj and MsJ (for an electron beam incident in the 2 direction) is where the T matrix is defined by where T=l-S S = (1 +ik)(l -ik)-'. In the above expression, ul, is the Coulomb phaseshift defined by ulz = arg r( li iz/ ki) where z is the asymptotic charge and ki is the incident momentum. The Coulomb phaseshift satisfies the recursion relation where al+' = ul + tan-'( q/ L. + 1)) q = -z/ k. The differential cross section is defined in terms of the scattering amplitude by (2) (3) (4) (5) As the purpose of this letter is to look at the qualitative differences between e--atom and e--ion collisions, this discussion will concentrate on a physical situation that can be adequately described by first-order perturbation theory. The resonance transitions (ns+ np) in alkai-like atoms (ions) can be described by the plane-wave (Coulomb) Born approximation provided fk: >> E, (7)

3 ~ 1/2 L27 where E, is the energy difference between the target initial and final states. A supplementary condition is that the momentum transfer K = Iki - kjl should be small. For a dipole transition on a target with L, = 0 one can set Ii = L, and so the scattering amplitude, equation (l), reduces to fli = -i (-) 1 il-'j exp( ial + ia,) (2L + 1 )"*( 1 MLf4 - MLf I LO) Ti; Y&,,j ( 6) (8) kikj ~.ir 5 where we have omitted the spin quantum numbers since we are interested in a kinematic range where exchange is unimportant. In the above expression the Coulomb phaseshift al is a function of k,. Let us restrict the discussion to forward-angle scattering (i.e., 8 = 0') for didactic reasons. Using asymptotic forms for the Coulomb-Bethe matrix elements (Burgess et a2 1970) and the Clebsch-Gordon coefficient in equation (8) gives where the constant a is and (r) is the radial dipole matrix element. While this expression is simplistic, it will be useful in elucidating those aspects of e--ion collisions that are different from e--atom collisions. For small values of 2Eij/ kf we can use and kj ki( 1 - E,/ kf) (11) 17, - vjl= vi(ezj/kf)* Furthermore, for large L giving f;j=exp(4.rrvie,/k:+2iao) (1 -E,/k:)L exp(2iq In L). (14) L Using the result (1- E,/ k:)l = exp[ L In( 1 -- E,/ k:)] = exp( -LE,/ k:) (15) and changing the sum into an integral gives The integral in equation (16) is the product of an exponentially decaying term with a changing phase factor. Thus, the individual partial-wave contributions to the scattering amplitude will interfere destructively resulting in a scattering amplitude smaller in absolute magnitude. The interference caused by the Coulomb phase will become less important as kf+co. Consider a value of L (e.g., Lo) which is sufficiently large to

4 L28 cause the exponentially decaying term in equation (16) to fall to an arbitrarily small fraction of its initial value, i.e. exp(-loeo/k;) = exp(-m). (17) This means that Lo must satisfy Lo = kf m/ E,. The value of the phase in the integrand of equation (16) becomes &= -(2z/ki) ln(lo) = -(2z/ki) ln(k:m/eij). As kf+ 00, So will go to zero. Therefore at high energies, the partial-wave amplitudes become sufficiently small before the destructive interference caused by the Coulomb phase can begin to take effect. Hence, the partial-wave amplitudes will interfere constructively and the scattering amplitude will tend towards the plane-wave result. Another limiting case of interest occurs when we let kj + ki for a fixed value of the incident energy. This limit corresponds to the limit taken by Lassettre et al (1969) in their demonstration that the generalised oscillator strength (Inokuti 1971) tends towards the optical oscillator strength (00s) as K2+ 0. The generalised oscillator strength (GOS) is defined in terms of the differential cross section by (18) (19) It is well known (Inokuti 1971) that the value of the GOS at K2=0 in the plane-wave Born (PWB) approximation is just the 00s. It is evident from equation (9) that the scattering amplitude diverges as kj + ki. Therefore we consider the ratio of the Coulomb-Bethe result to the plane-wave Bethe result, i.e. R = lim J+m If: dl exp(i2~~ In L)I If: dll Making the transformation u = In L, gives IjrJ du e~p[(l+2ir]~)u]] 1 1 E-= R = lim ~-+m du exp(u)l 11+2i~~l (1+4rli) 2 1/2' While the ratio R tends to 1 as the incident energy increases, it is clear that at any finite energy, the GOS for e--ion excitation does not tend to the 00s as K2+0, even in the Coulomb-Bethe approximation. When the plane-wave Bethe approximation is used, equation (9) reduces to a power series which can be summed to demonstrate that the GOS in this case approaches the 00s as K2+0. Detailed calculations have been done in order to demonstrate how these results will manifest themselves in an e--ion collision. A two-state (2s, 2p) unitarised Coulomb-Born (UCB) calculation for the resonance transition (2s-2p) of Be+ has been completed. Since Be+ is such a light ion, the errors induced by the poor treatment of distortion implicit in UCB are minimised, especially at small angles. In these calculations, the 2s and 2p states were represented by Hartree-Fock wavefunctions and exchange was included for the first 11 partial waves. A large number of partial waves must be included to obtain accurate differential cross sections for this transition (e.g., 140 at 3.0 Ryd and 180 at 4.0 Ryd). The necessary number of partial waves could be determined by comparing PWB differential cross sections computed by partial-wave

5 L29 analysis, or analytically as a function of momentum transfer. The differential cross sections presented in figure 1 have a numerical accuracy of about 1 '/o. The right-hand side of figure 1 shows differential cross sections that have been computed (incorrectly) without the Coulomb phases in equation (1). As expected, the strongly forward peaked cross sections are reminiscent of those typically seen for the resonant transition in alkali atoms. As the energy increases, the differential cross sections become progressively more forward peaked. Interference effects, caused by the Coulomb phase, are very noticeable in the left-hand side of figure 1 where the correctly computed differential cross sections are depicted. The most pronounced feature is the dip in the differential cross section at the forward angles. The dip becomes less pronounced as the energy increases, and has disappeared at an incident energy of 3.0 Ryd. Nevertheless, there is still a large decrease in the 6 = 0" cross section caused by the Coulomb phase even though the incident energy exceeds the threshold energy by a factor of ten. The GOS for Bet 2s-2p transition shown in figure 2 also exhibits features that result from destructive interference caused by the Coulomb phase. At the smallest values of K2, the GOS decreases rapidly and does not appear to be approaching the 00s value in the K2 = 0 limit. The deviation from the PWB curve at small K2 is smaller for the higher energy curve. This is in qualitative agreement with the result contained in equation (22). To summarise, it has been shown that the Coulomb phase causes destructive interference between individual partial-wave contributions to the scattering amplitude. This effect will be strongest for low-energy scattering since the Coulomb phaseshifts are largest at low energies. There is a very pronounced effect for the resonance transition of Be+. The differential cross section is considera5ly reduced at small angles, and the 11 I I I I I I e (deg) 0 (deg) Figure 1. Differential cross sections for the Be' 2s-2p transition (in ai) calculated in the UCB approximation at incident electron energies of 0.5, 1.0, 1.5 and 3.0 Ryd. The left-hand side shows differential cross sections that are correctly computed since the Coulomb phase has been retained in the definition of the scattering amplitude. The Coulomb phase has been omitted in the differential cross sections depicted on the right-hand side, which show the characteristic forward peaking seen in dipole transitions for atoms.

6 L30 I I I K2 Figure 2. GOS for the Be+2s-Zr tran ji:ion in the UCB approximation (full curves) at incident energies of 1.0 and 4.0 Mj. [he COS zi the PWB approximation (broken curve) is also depicted. The value of the ( )S is designated by the open circle. forward peaking that is characteristic of dipole trdxitions does not occur until the incident energy is sufficiently large. This effect should be observable in the current cross-beam experiments (Williams et al 1986), or possibly in the new generation of merged electron-ion beam experiments now under development. The author would like to thank D W Norcross for reading this manuscript and suggesting improvements. This work was supported by a grant from the US Department of Energy (Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research). The calculations performed in this work were done using the VAX 8600 at JILA. References Brandt M A, Truhlar D G and Smith R L 1973 Comput. Phys. Commun Burgess A, Hummer D G and Tully J A 1970 Phil. Trans. R. Soc. A Chutjian A 1984 Phys. Rev. A Chutjian A, Msezane A Z and Henry R J W 1983 Phys. Rev. Lett Chutjian A and Newell W R 1982 Phys. Rev. A Deb N C, Sinha C and Si1 N C 1983 Phys. Rev. A Inokuti M 1971 Rev. Mod. Phys Lane A M and Thomas R G 1958 Rev. Mod. Phys Lassetre E N, Skerbele A and Dillon M A 1969.I. Chem. Phys Mitra C and Si1 N C 1978 Phys. Rev. A Mitroy J 1987 Phys. Rev. A in press Msezane A Z and Henry R J W 1985 Phys. Rev. A Phys. Reo. A Walker D W 1974 J. Phys. B: At. Mol. Phys Williams I D, Chutjian A and Mawhorter R J 1986 J. Phys. B: At. Mol. Phys Williams I D, Chutjian A, Msezane A Z and Henry R J W 1985 Astrophys. J