Electron-helium scattering within the S-wave model

Transcription

1 PHYSICAL REVIEW A, VOLUME 65, Electron-helium scattering within the S-wave model Chris Plottke, 1 Igor Bray, 1 Dmitry V. Fursa, 2 and Andris T. Stelbovics 1 1 Centre for Atomic, Molecular and Surface Physics, School of Mathematical and Physical Sciences, Murdoch University, Perth 6150, Australia 2 Electronic Structure of Materials Centre, The Flinders University of South Australia, G.P.O. Box 2100, Adelaide 5001, Australia Received 5 August 2001; published 30 January 2002 The electron-helium scattering system is studied within the S-wave framework with a particular emphasis on one-electron ionization processes. Total ionization and the singly differential cross section for the equalenergy-sharing point are given for three initial states. In addition, discrete excitation cross sections are also given. Whereas generally there is no a priori relationship between cross sections for excitation of corresponding singlet and triplet states, we find that for the case of ionization with equal-energy outgoing electrons the triplet cross section is approximately three times bigger than the corresponding singlet one, irrespective of the initial state or energy. DOI: /PhysRevA PACS numbers: Bm, Dp I. INTRODUCTION The S-wave model, where only states of zero orbital angular momentum are included, plays a central role in the testing of general scattering theories. In the case of the e -H system this model is often referred to as the Temkin-Poet model, which has accurate benchmark solutions for certain cases 1,2. The treatment of the electron exchange and the target continuum may be readily tested against benchmark results 3. The model was also useful for the e -H system, where convergence in the treatment of the atomic and positronium centers could be tested 4. To date most theories have been tested on the hydrogen target. Here we are interested in a systematic study of the e -He system, within the S-wave model. To the best of our knowledge thus far this system has only been considered by Pindzola, Mitnik, and Robicheaux 5. They briefly looked at the case around 200 ev incident energy with a particular interest in the double-ionization cross section. Our interest is in single-ionization processes on a broad range of energy from just above threshold through to several hundred electron volts, and for several initial states. Though presently we study the dominant one-electron processes, we expect that the e -He S-wave model will play a central role in the solution of the full four-body problem, which includes ionization with excitation- and double-ionization processes. II. THEORY A nonrelativistic S-wave model of e -He scattering is assumed. The notation will therefore presume zero orbital, but not spin, angular momenta for target and projectile electrons, and linear momenta will be written as scalars. The total Hamiltonian is then 2 2 Hr 1,r r r r 1 r 2 maxr 1,r 2. 1 The model retains most of the computational difficulties associated with the full model, for which we use the computational approach given by Fursa and Bray 6. Atomic units are assumed throughout, unless stated otherwise. The first step in forming the close-coupling equations is to obtain N target expansion states n (N) with spin s n 0,1. We do this by writing the configuration-interaction CI expansion of the target states as (N) n C (N) n Xs n,, (N) where X(s n ) is the two-electron spin function, C n are the CI coefficients obtained from diagonalizing the He Hamiltonian. The antisymmetry of the two-electron functions n (N) is ensured by the symmetry property of the CI coefficients C N n N 2 1 s nc n. 3 The radial one-electron functions (r) are taken from a Laguerre basis. Since our interest is in one-electron excitation processes we employ the frozen-core approximation. The 1s orbital (1) is replaced with the 1s of He, and only the configurations with either 1, 1, or both are retained see 6 for more details. In(e,2e) calculations the exponential falloff of the Laguerre basis may be adjusted to ensure that one of the states has a desired positive energy 7. The close-coupling equations, for electron-helium scattering at a total spin S and energy E i (N) k i 2 /2 f (N) k f 2 /2, are solved in momentum space for the T matrix k f f (N) T S i (N) k i k f f (N) V S i (N) k i n1 N 0 dk k f (N) f V S (N) n kk (N) n T S (N) i k i. 4 Ei0 (N) n k 2 /2 The discrete excitation amplitude f fi is then simply f (SN) fi k f (N) f T S (N) i k i /2002/653/ /$ The American Physical Society

2 PLOTTKE, BRAY, FURSA, AND STELBOVICS PHYSICAL REVIEW A In the case of ionization the required amplitude f fi (k f,q f )is 7 f (SN) fi k f,q f q () f (N) f k f (N) f T S (N) i k i, where q f () is a true continuum eigenstate of the He Hamiltonian with spin s f and energy q f 2 /2 f (N). The partial spin-weighted cross section for discrete transitions is simply (SN) fi 2S1 22s i k f f (SN) k fi 2. i In the case of S-wave model ionization, the most detailed information available is gained from the singly differential cross section SDCS d (s f SN) de 6 7 e f 2S1 2 4 k f f (SN) 22s i 1 4 q f k fi k f,q f 2, i 8 where e f q f 2 /2. The SDCS definition was chosen so that the total ionization cross section TICS was given by (SN) I s f E d (s f SN) de e 0,1 0 de s f E/2 d (s f SN) de e, 0,1 0 de owing to the fact that k f (N) f T S (N) i k i 0 for k 2 f /2 (N) f. In other words the step-function behavior identified for hydrogen 8 also holds for helium, and hence the equalenergy-sharing amplitudes require multiplication by two. Furthermore, at the step the S1/2 amplitudes should satisfy 9 f (S1/2) s k,k 1 2 1s f (S1/2) s k,k 3 2 f s (S1/2) k,k, 10 where s0,1 is the final target-space spin, and s s. For S 3/2 we require f (S3/2) s1 k,kf (S3/2) s1 k,k These relations arise from the antisymmetry of the total wave function, equivalence of electrons, and the choice of coupling the target spin. We cannot enforce these relations explicitly, but rather can check that they are satisfied approximately following the solution of Eq. 4. For example, in the S-wave e-h model the triplet kq amplitude comes out be near zero 8. Here the same should hold in the case of S 3/2. Furthermore, from Eq. 10 it follows that f (S1/2) s0 k,k 1 3 f s1 (S1/2) k,k. 12 Thus, the qk SDCS for S1/2 arising from excitation of triplet states should be three times bigger than the corresponding singlet one. This is independent of energy, and hence an extraordinary result. Generally, there is no relationship between exciting singlet and corresponding triplet states. For scattering from an initial singlet state, as the incident energy increases excitation of a singlet state dies off slower than excitation of the corresponding triplet state. The latter is an exchange transition and so we expect it to drop off rapidly with increasing energy. However, Eq. 12 says that this should not be so in the case of equal-energy-sharing ionization, where the triplet SDCS should remain three times bigger than the singlet SDCS, irrespective of energy. III. RESULTS Our goal is to provide accurate data for the e -He system within the S-wave model. We restrict the target structure to the frozen-core model which has been shown to be adequate in describing both discrete excitation 6 and ionization 7 processes. We will consider the relaxation of this approximation some time in the future, when ionization with excitation and double-ionization processes will be considered. Accordingly, we are not treating all of the electrons in an equivalent manner. Hence, we cannot expect the relations such as Eq. 12 to be satisfied exactly. For total energy E, we are interested in the SDCS at E/2, and for s f and in TICS, as well as discrete transitions. To yield this SDCS we need to ensure that for every E we have a state with energy f (N) E/2. We achieve this in a reverse way. Rather than taking arbitrary E and then altering Laguerre exponential falloff parameters, we take a single set of 59 states, 30 singlet, and 29 triplet, and solve Eq. 4 for every E2 f (N) 0. Other than for the ground state, such a choice always has a correspondence between a singlet and a triplet state whose energies are similar. For this reason in the following figures the points appear to be in pairs. The separation, in energy, of a pair of points indicates the energy difference of the corresponding singlet and triplet states. Since the cross sections are expected to be generally smooth as a function of energy any deviation from a smooth line of best fit is an indication of the accuracy. Note that we are not able to ensure that E always bisects two pseudothresholds as the underlying energy-integration quadrature rule requires 10. Hence, we expect to see some pseudoresonance phenomena, which we hope to be small due to the relatively large size of the basis. We begin the discussion by examining the results of calculations for scattering from the ground state, presented in Fig. 1. The top panel shows the elastic cross section which is around two orders of magnitude greater than the other cross sections. The fact that all points lie on a smooth curve indicates excellent convergence with the size of the basis. The next panel shows cross sections for exciting the n 2 singlet and triplet states. Here we see that there is no relation between the two cross sections, with the triplet one falling off to zero more rapidly with increasing energy. Once again convergence is good. Excitation of the two n3 states is given in the middle panel. The same conclusions hold here as for the n2 case. We next turn to total ionization, given in the second from

3 ELECTRON-HELIUM SCATTERING WITHIN THE S-... PHYSICAL REVIEW A FIG. 1. S-wave model e -He excitation and ionization cross sections from the ground state. bottom panel. This cross section is simply the sum of excitation cross sections of the positive-energy states. Once more, good convergence is seen with only one point appearing to be out of place. Finally, in the bottom panel we consider the SDCS. The singlet one is multiplied by three to make easier the comparison with the triplet SDCS. The solid and open circles are from calculations where f (N) E/2 for s f 0 and s f 1, respectively. We see that Eq. 12 is indeed approximately satisfied. This is an internal indication that the CCC implementation of e-he excitation and ionization should be quite accurate. Note that the presented SDCS have been obtained after multiplication of the raw CCC results by four, owing to the knowledge that the underlying amplitudes converge to half the required values. The corresponding results for scattering from the 2 1 S are presented in Fig. 2. Similar to the ground state, the total spin takes S1/2 only. In the top panel the dominant elastic cross section is presented. It features a deep minimum at around 4 to 5 ev above the ionization threshold. It probably has a similar origin as do the Ramsauer-Townsend minima. Since in this model problem the large r potential is unphysical owing to the absence of p states we choose not to study such minima. Convergence is very good. The excitation of the 2 3 S also features a minimum, this time at around 2 to 3 ev above the ionization threshold. Past the maximum, at around 10 ev total energy, this cross section falls off rapidly as one would expect of an exchange transition. The excitation of the two n3 states also shows the typical suppression of exchange transitions with increasing energy. Unlike in the case of the 1 1 S initial state the triplet cross section remains below the singlet one at all presented energies. Convergence is good. The total ionization cross section has the classic shape of a rise past threshold to a maximum followed by a gradual decline with increasing energy. Once again convergence is satisfactory. In the bottom panel the SDCS are presented. As for the 1 1 S initial state the singlet cross section has been multiplied by three to yield a curve similar to that of the triplet SDCS. The agreement is not perfect. In fact a slightly smaller factor would yield a better agreement. We are hopeful that when we fully relax the frozen-core model, a computationally immense undertaking, Eq. 12 would be satisfied to even greater accuracy. For the initial 2 3 S state the results are given in Fig. 3 for both total spins S1/2 and 3/2. The top panel gives the elastic cross sections. The two spin components show remarkably different behavior. The S3/2 has a deep minimum around 4 ev above the ionization threshold. The total elastic cross section is the sum of the two spin components since the spin weights have been included. At the higher energies, where electron exchange becomes negligible, we expect the S3/2 cross section to be two times higher than the S1/2 one, due to the spin weights given in Eq. 7. Convergence appears to be quite good. The second from top panel shows cross sections for excitation of the 3 1 S state. This is now an exchange transition and only has contribution from S1/2. It also exhibits a minimum, this time around 6 ev above the ionization threshold. After this it reaches a small local maximum around 20 ev, and proceeds to rapidly decrease with increasing energy. The middle panel shows results for excitation of the 3 3 S state, for both total spins. Interestingly, the magnitudes are commensurate with that of elastic scattering, yet the indi

4 PLOTTKE, BRAY, FURSA, AND STELBOVICS PHYSICAL REVIEW A FIG. 2. S-wave model e -He excitation and ionization cross sections from the 2 1 S state. vidual components show very different qualitative behavior. Here we find no local minima, though as for elastic scattering, at the higher energies the S3/2 cross section converges to be twice that of the S1/2 one. At the higher energies the 3 3 S excitation cross sections are an order of magnitude greater than the corresponding 3 1 S cross sections. The cross sections for total ionization of the 2 3 S state, FIG. 3. S-wave model e -He excitation and ionization cross sections from the 2 3 S state. leaving the He ion in the ground state, is given in the second from bottom panel. The S1/2 component has contributions from exciting both singlet and triplet positive-energy states. The S3/2 component has contributions only from triplet states. For this reason there is no simple spin relation between the two components at the higher energies. Both components show the characteristic maximum several ev af

5 ELECTRON-HELIUM SCATTERING WITHIN THE S-... PHYSICAL REVIEW A ter threshold and then proceed to slowly decrease with increasing energy. Turning our attention to the SDCS at E/2, given in the bottom panel, we first note that the S3/2 component is essentially zero as required by Eq. 11. In addition, once more we observe a factor of 3 difference between the singlet and the triplet S1/2 components. Thus, unlike for discrete excitation, the cross sections for exciting singlet and triplet states with energy E/2 remain related for all E. IV. CONCLUSIONS The S-wave model of e-he scattering has been considered at a broad range of energies above the ionization threshold. In this model only target and projectile states of zero orbital angular momentum are retained. The CCC method of evaluating this model was used with the assumption that the target always has a 1s He inner electron. A rich structure of excitation cross sections has been found, much more so than in the corresponding case of the atomic hydrogen target 3. We believe accurate excitation cross sections have been determined for n f 3 states from initial n i 2 states. In addition, ionization processes have been considered. Accurate total ionization cross sections have been given from initial n i 2 states. Of particular interest has been the study of the equal-energy-sharing singly differential cross sections SDCS at E/2. These have components from excitation of singlet and triplet states with energy E/2. Whereas generally there is no relation between excitation of corresponding singlet and triplet states, we found that 3SDCS(E/2,s0)SDCS(E/2,s1), irrespective of E or initial state. It is our hope that the present paper will be of value to others who are developing e -He scattering and ionization methods. Furthermore, the present results are the first step before generalizing the frozen-core model to allow for the ionization with excitation and double ionization processes. Given the immense success that CCC has enjoyed in dealing with one-electron excitations, the next step is application to two-electron excitation processes. ACKNOWLEDGMENTS Research sponsored in part by the Phillips Laboratory, Air Force Materiel Command, USAF, under cooperative agreement No. F Support of the Australian Research Council is gratefully acknowledged. 1 A. Temkin, Phys. Rev. 126, R. Poet, J. Phys. B 13, I. Bray and A.T. Stelbovics, Phys. Rev. Lett. 69, A.S. Kadyrov and I. Bray, J. Phys. B 33, L M.S. Pindzola, D. Mitnik, and F. Robicheaux, Phys. Rev. A 59, D.V. Fursa and I. Bray, Phys. Rev. A 52, I. Bray and D.V. Fursa, Phys. Rev. A 54, I. Bray, Phys. Rev. Lett. 78, I. Bray, D.V. Fursa, and A.T. Stelbovics, Phys. Rev. A 63, I. Bray and B. Clare, Phys. Rev. A 56, R