Anti-proton and proton impact excitation–ionization of helium including resonant decay

Nuclear Instruments and Methods in Physics Research B 143 (1998) 225±232

Anti-proton and proton impact excitation±ionization of helium including resonant decay Jack C. Straton

a,*

, J.H. McGuire b, A.L. Godunov

c

a

c

Department of Physics, Portland State University, Portland, OR 97207, USA b Department of Physics, Tulane University, New Orleans LA 70118, USA Troitsk Institute for Innovation and Fusion Research, Troitsk, Moscow region, 142092, Russian Federation Received 7 November 1997; received in revised form 3 March 1998

Abstract The cross section for projectile impact excitation±ionization of helium is calculated, including spatial correlation, time-ordering, and interference between autoionizing and continuum states. The computational load is greatly reduced by using con®guration-interaction (CI) helium wave functions and continuum pseudostates, yielding a scattering amplitude that is a sum of analytical terms for the ®rst-order and energy-conserving second- and third-order contributions, and a one-dimensional integral for the energy-nonconserving second-order contributions. Comparison is made with experiment. Ó 1998 Published by Elsevier Science B.V. All rights reserved. PACS: 34.70.+e; 34.10.+x; 03.65.Nk. Keywords: Excitation; Ionization; Helium; Autoionization; Anti-proton

1. Introduction The past decade has seen increasing use of proton, anti-proton, positron, and electron projectiles to probe both charge and mass e€ects in single [1] and multiple [2] electron atoms. In the present paper we calculate charge e€ects in heavy-projectile impact excitation±ionization helium. This is a challenging problem because one must not only account for electron correlation in the initial

* Corresponding author. Tel.: +1 503 72 4227; fax: +1 503 725 3888; e-mail: [email protected].

bound state and the scattering regime [3], but also deal with autoionizing intermediate and ®nal states and their subsequent decay [4], preferentially via the Auger mechanism with the emission of an electron [5]. It is precisely this complexity that makes analysis of this problem interesting. The diculty of the problem may be reduced considerably by ®rst introducing the semi-classical approximation (SCA) [6,7] for these heavy projectiles, in which the projectile of velocity v follows a ^ rectilinear path as a function of time, R ˆ B^i ‡ vtk, with B the impact parameter. The ®rst order SCA has been shown to be the intermediate representation equivalent of (the two-dimensional Fourier

0168-583X/98/$19.00 Ó 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 2 4 9 - 3

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J.C. Straton et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 225±232

transform of) the plane-wave ®rst Born approximation (FBA) of Schr odinger representation [8,9]. This choice ultimately allows us to reduce the energy-conserving part of even the third-order scattering amplitude to a simple form free of numerical integrals. The computational load for including electron correlation in multi-electron bound states is further reduced by representing them by fully correlated con®guration-interaction (CI) wave functions. We also use pseudostates for the intermediate-state propagators [10±12] and the continuum part [13] of the ®nal states because the analytic form for these functions is like that for hydrogenic functions. This allows one to utilize Straton's general-state-to-state transition amplitude for one-electron atoms [14], which is simply a sum of analytic functions involving powers and Bessel functions of impact parameter and transition energy. Thus, the many-electron scattering amplitude may be transformed into a product of transition amplitudes for one-electron atoms whose calculation avoids the computationally intensive task of numerical integration for the ®rst-order and energy-conserving (on-shell) second- and third-order contributions. Replacing the time-ordering operator in the second- and third-order energy-nonconserving (o€-shell) terms by its integral representation gives a one-dimensional integral for these contributions [15]. (The third-order amplitude also contains a cross-term consisting of a two-dimensional integral.) Both the diagonalization of the many-electron Hamiltonian to obtain the CI coecients, Eq. (5) below, and the calculation of the tables of pseudostate expansion coecients may be sequestered from the calculation of the full scattering amplitude to alleviate computation time. This approach is particularly helpful when calculating cross sections for an atom like helium in which the interference between autoionizing and direct ionizing of the ®nal states is marked [16,17]. The autoionizing doubly excited states require the calculation of sums of one-electron transition amplitudes to intermediate-states for which the general analytic form is given by the above process.

Because the ®rst-order direct ionizing amplitude is so much larger than the corresponding o€-shell second-order amplitude, the interference between the two gives little projectile charge-dependence, in contrast to the experimental factorof-two di€erence [18±21]. The present work tests our conjecture that the autoionizing amplitude might lower the former and also interfere with the latter to give a larger splitting. Section 2 presents the general method, Section 3 focuses on autoionizing states, and Section 4 presents the results.

2. Scattering amplitude The probability amplitude for multi-electron transitions from an initial state jii to a ®nal state hf j may be written as [3] 2 3 Z1 afi ˆ hf jT exp 4 ÿ i V …t† dt5 jii ÿ1 …0†

…1†

…2†

ˆ afi ‡ afi ‡ afi . . . ;

…1†

where V is the interaction between the projectile and the electrons in the atom, V …t† ˆ

X

eiH0 t Vj …t†e-iH0 t

…2†

j

in which [22] Vj …t† ˆ ÿ

Zp Zp ‡ ; jR…t† ÿ mrj j R…t†

…3†

where m ˆ Mt =…Mt ‡ me †, and T is the time ordering operator [23]. The ®rst-order amplitude for helium is …1† afi

Z1 ˆ ÿi

dthf jeiH0 t ‰V1 …t† ‡ V2 …t†Še-iH0 t jii:

…4†

ÿ1

For initial and ®nal states given as CI wave functions, anti-symmetrized via Aj ,

J.C. Straton et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 225±232

X jii ˆ Cai 1 a2 Ai ‰ja1 i; ja2 iŠ;

…5†

a1 a2

X Cbf 1 b2 Af ‰hb1 j; hb2 jŠ hf j ˆ

…6†

b1 b2

T -iEn …tÿt0 † 0 ˆ H…t ÿ t0 †e-iEn …tÿt † e 2 Z1 ˆ

0



dX e-iX…tÿt †

ÿ1

one then obtains the fully correlated two-electron amplitude h 0 X X …1† Cai 1 ;a2 Af Ai Abk 1k;a1 …x†hb2 ja2 i afi ˆ…ÿi† Cbf 1 ;b2

227

 1 i 1 : d…X ÿ En † ‡ P 2 2p X ÿ En …11†

Then the second-order amplitude is the sum of terms such as

a1 ;a2

b1 ;b2

0

‡Akb2k;a2 …x†hb1 ja1 i

i

…7†

as a weighted sum of overlap integrals hn0 `0 m0 k0 jn`mki; and one-electron scattering amplitudes, 0 k …x; B; m† Akn0 `0 ÿm0 ;n`m

Z1 ˆ

ixt

0 0

…12†

0 0

dt e hn ` m k jV …t†jn`mki; ÿ1

…8† in which k ˆ Zt =a0 , x ˆ Ef ÿ Ei , and B is the impact parameter. A typical one-electron transition amplitude is [14] p ÿZp 2k4 B kk K2 …BA†; …9† A2s;1s …x† ˆ v p A2 q 2 2 where A ˆ …x=vp † ‡ …3k=2† ; vp is the projectile velocity, and K2 is a modi®ed Bessel function of the second kind. The second-order amplitude, …2† afi

ˆ…ÿi†

2

Z1

Z1 dt ÿ1

dt0

ÿ1

T hf jeiH0 t ‰V1 …t† 2

0

‡ V2 …t†Še-iH0 …tÿt †  ‰V1 …t0 † 0

‡ V2 …t0 †Še-iH0 t jii;

…10†

may be put in the form of products of one-electron transition amplitudes Eq. (8), if the time-ordered intermediate-state propagator is replaced by its integral representation, [24]

where extends over a complete set of correlated two-electron intermediate (bound and continuum) states, whose CI coecients are the mi , and …2ÿoff† a12

i ˆ P p

Z1 ÿ1

i dX h …2ÿon† a12 En !X : X ÿ En

…13†

We have also included the energy-conserving part of the third-order contributions [15].

3. Autoionizing states If some of the intermediate states in the secondorder amplitude are unstable, then additional terms appear in the Hamiltonian leading to Eqs. (12) and (13). In 1971 Girardeau introduced a second-quantized representation [25±27] that provides a very direct path to sorting out such terms since it explicitly distinguishes between bound, decaying, [28] and free electronic states. The two-electron Fock±Tani Hamiltonian previously found [29] may be modi®ed to include doubly excited resonant states. After a single interaction with the projectile, these autoionizing states l decay via an exchange of electronic energy HT , promoting one electron

228

J.C. Straton et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 225±232

to the continuum at the expense of some excitation energy of the remaining (bound) electron. The form of this decay amplitude is …1ÿdec† afi

2

Z1

Z1

ˆ …ÿi†

dt ÿ1

Z1 hlj

dX e ÿ1

-iX…tÿt0 †

dt0 hf jeiEf t ‰HT Š

ÿ1



X

jli

l

1 i 1  d…X ÿ El † ‡ P 2 2p X ÿ El 0

‰V1 …t0 † ‡ V2 …t0 †Še-iEi t jii:



…14†

The superscript label ``1-dec'' indicates that this is a term that is ®rst-order in interactions with the 2 projectile, but it has a coecient of …ÿi† because it arises from (the Fock±Tani equivalent of) the second-order term in Eq. (1). Alternatively, one may use Feshbach projectors [31] within the ``diagonalized approach'', [32] as Godunov et al. [33,34] have done for the equivalent double-excitation problem. This gives a ®nal state resulting from a mixture of direct ionizing and autoionizing doubly excited resonant states, ‡ hf j ˆ hvdec ks j ˆ hvks j ‡ hvks jHT Gres

Eq. (1). Then this amplitude is the sum of terms such as

…15†

an alternative approach [32] to the autoionizing scheme devised by Fano [35]. Substituting this ®nal state for hf j in the ®rst-order Born amplitude Eq. (4) gives exactly the same decay term Eq. (14) as does the second-order resonant-state Fock±Tani Hamiltonian. The intermediate states jli are obtained by diagonalizing the two-electron target Hamiltonian HT in the subspace of the doubly excited states jm1 m2 i; X Cml1 m2 Al ‰jm1 i; jm2 iŠ; …16† jli ˆ

…18† In the present results we have included both the on- and o€-shell (principal value) terms in the propagator for resonant-states jli but only the on-shell part of the term that propagates the bound, (doubly-) excited, and continuum states jni between the two interactions with the projectile. Unlike the P …1=…X1 ÿ El †† term that is signi®cant, the P …1=…X2 ÿ En †† term is likely to be small since it has the same form as the small second-Born o€-shell term that we have included Eq. (13). The second-order Fock±Tani interaction Hamiltonian also has a ®nal-state interaction term representing polarization of the ®nal He‡ state due to an interaction with the outgoing electron,

m1 m2

hljHT jl0 i ˆ El dll0 :

…17†

The third-order Fock±Tani interaction Hamiltonian likewise has a decay term that is second-order in Zp , which is also given by substituting Eq. (15) into the second-order term in

…19† which is ®rst-order in Zp .

J.C. Straton et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 225±232

Finally, if there is a change in the target charge felt by the bound electron between the initial and ®nal states, such as the e€ective (screened) charge of 1.6875 in our initial state and the bare charge of 2 in the helium ion, then conventional theories have nonzero terms zeroth-order in projectile …0† interaction afi . A major motivation for the Fock±Tani transformation is that it rotates the space in which states are de®ned in such nonorthogonal ways to a new, ``ideal'', state space in which second-quantized operators for bound states are elementary and exactly orthogonal to each other and to free electronic states. In consequence, the Fock±Tani interaction Hamiltonian has orthogonalization subtractions for each of the above terms, reducing the interaction potential for free particles so that they do not have enough energy to bind [30], that take the form of projectors onto the bound-states r. At this stage we have included only the ®rst-order orthogonalization term for the bound states of the helium ion, Z1 X …1ÿorth† ˆ ÿ …ÿi† dthf jeiH0 t jri afi ÿ1

r

  1 hrj V1 …t† ‡ V2 …t† e-iH0 t jii: 2

…20†

4. Results and conclusions The phases of all amplitudes are considerably complicated by the fact that the one-electron 0 `0 ‡`ÿmÿm0 k …x; B; m†  …i† , as can amplitude Akn0 `0 ÿm0 ;n`m be seen from the sixth line of equation (41) in [14], with explicit examples given in Appendix B of [36]. Nevertheless, one may sort out the terms for each case. The largest contributions arise when the ionized electron is in an s-state and ®nal bound state has quantum numbers 210 (z-polarization), h 0…1† aKs;210;1s2 ˆ ÿ Zp aKs;210;1s2 0…1ÿorth†

0…2ÿoff†

ÿZp aKs;210;1s2 ‡ Zp2 aKs;210;1s2

229

0…3ÿon†

0…1ÿdecÿoff†

0…1ÿpol†

0…2ÿdecÿon†

ÿZp3 aKs;210;1s2 ÿ Zp aKs;210;1s2 ‡Zp aKs;210;1s2 ÿ Zp2 aKs;210;1s2

i

h 0…1ÿdecÿon† 0…2ÿdecÿoff† ‡ i Zp aKs;210;1s2 ÿ Zp2 aKs;210;1s2 i 0…2ÿon† ‡Zp2 aKs;210;1s2 : 0…n†

…21†

…n†

where afi ˆ jafi j=Zpn . It is clear that our conjecture that the decay terms might reduce the ®rst-order term is armed. The orthogonalization term is, however, the major contributor to this reduction. The combination of these two corrections to the ®rst-born amplitude bring it into the range where it no longer swamps the second-order terms, allowing for the possibility of interference. As can be seen in Fig. 1, this correction also drops our results out of the range of the experimental data for proton and electron projectiles by Pedersen and Folkmann [18] F ulling et al. [19] Schartner et al. [21], and data for electron projectiles only by Forand et al. [20] and corresponding theories by Rudge [39], Raeker et al. [38], and Kuplyauskene and Maknitskas [41]. The present results are similar to the theoretical results for protons and anti-protons by Nagy et al. [37], as well as the electron-only theory by Franz and Altick [40]. As with the calculation by Nagy et al. [37], our second-order terms, including decay and orthogonalization corrections, have a splitting due to charge-dependence that is only about a tenth of the factor-of-two di€erence between experimental cross-sections for positively versus negatively charged projectiles. This e€ect arises from odd-charged cross-terms in the square of the amplitude (the probability). Those corresponding to the 210 ®nal state are h 0…1† 0…2ÿdecÿon† 0…1ÿorth† 0…2ÿdecÿon† ÿaKs;210;1s2 aKs;210;1s2 ‡ aKs;210;1s2 aKs;210;1s2 0…1ÿdecÿon† 0…2ÿdecÿoff†

ÿaKs;210;1s2 aKs;210;1s2 0…1ÿpol†

0…2ÿdecÿon†

0…1ÿdecÿoff† 0…2ÿdecÿon†

‡ aKs;210;1s2 aKs;210;1s2 0…1†

0…2ÿoff†

ÿaKs;210;1s2 aKs;210;1s2 ‡ aKs;210;1s2 aKs;210;1s2

230

J.C. Straton et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 225±232

Fig. 1. 2p Excitation±ionization of helium by protons vs. antiprotons. In the present third-order theory, the protons are represented by a heavy solid line and the anti-protons by a dashed line, with the corresponding second-order theory represented by the thin solid and dashed lines. The ®rst-order direct-ionization term, without orthogonalization or decay corrections, is the dotted line. The open squares and diamonds are experimental points for protons and electrons from Pedersen and Folkmann (Ref. [18]), solid squares and diamonds are from F ulling et al. (Ref. [19]) circles and triangles are from Schartner et al. (Ref. [21]), and 's are electron data from Forand et al. (Ref. [20]). The heavy dot±dash and dot±dot±dash lines are proton and anti-proton second-order curves by Nagy (Ref. [37]). Also shown are electron-only theories by Rudge (Ref. [39]), Franz and Altick (Ref. [40]), and Kuplyauskene and Maknitskas (Ref. [41]) in thin dot±dot±dash, dotted, and dot±dash lines, respectively, and that by Raeker et al. (Ref. [38]) is the light gray line.

0…1ÿdecÿoff† 0…2ÿoff†

0…1ÿorth†

0…2ÿoff†

ÿaKs;210;1s2 aKs;210;1s2 ÿ aKs;210;1s2 aKs;210;1s2 i 0…1ÿpol† 0…2ÿoff† 0…1ÿdecÿon† 0…2ÿon† ‡aKs;210;1s2 aKs;210;1s2 ‡ aKs;210;1s2 aKs;210;1s2 Zp3 h i 0…2ÿdecÿon† 0…3ÿon† 0…2ÿoff† 0…3ÿon† ‡ aKs;210;1s2 aKs;210;1s2 ÿ aKs;210;1s2 aKs;210;1s2 Zp5 : …22† Whereas most of the amplitudes change phase by i as m changes by 1, the phases of the decay ampli-

tudes (both on- and o€-shell) are insensitive to m. Thus, we see that the latter interferes alternately 0…1† 0…2ÿon† with afi or afi , depending on the polarization of the ®nal bound state. Likewise, the polarization amplitude remains real for all m so it, too, inter0…2ÿon† 0…1† for m ˆ 1 and with afi for feres with afi m ˆ 0. There are a number of possible sources for the discrepancy between the modest charge-splitting we calculate and the signi®cant splitting seen by three experimental groups [18±21]. We have checked that cross sections for both projectiles of vary in the same direction by about 3% as the number of terms in the initial-state wave function is increased from the minimum of four to a maximum of 25. Likewise, the variation in the cross sections due to increasing the number of continuum pseudostates from the minimum of two s-states and one p-state to a maximum of six is a few percent. However, the splitting is sensitive to the size of the decay terms. We need to review the approximations that have gone into our derivation of their ®nal form. In particular, the second principal-value term in Eq. (18) was neglected in the present work and can be added without too much diculty. The on-shell part of the third-order amplitude [15], as expected, brings the low-energy theory in better agreement with the data, but splits the cross section in the opposite sense (the proton cross section is larger) in the region near 0.3 MeV. This partially cancels the splitting due to the second-order and decay terms near 1 MeV. Inclusion of the principal-value terms in the third-order amplitude may reverse this cancelation. Also neglected at this point are decay terms akin to Eq. (18) but with the n-sum replaced by another sum over resonant states and V1 …t2 † replaced by HT . In fact that term would be just the second term in an in®nite series of terms involving only one interaction with the projectile ((4) is the zeroth term and (14) is the ®rst). It may even be possible to sum this series analytically. The second-order term at present is calculated using an average-energy approximation. We have checked that cross sections for both projectiles

J.C. Straton et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 225±232

vary in the same direction by about 5% as we doubled or halved our chosen value of this parameter, 79 eV, the double-ionization threshold. Nevertheless, as we replace this with an intermediate pseudostate sum, we will utilize one-electron amplitudes that at present do not appear in the average-energy approximation, and initial indications are that the 2p±2p transitions will give a larger second order result for both on- and o€-shell terms. In this paper we have taken a necessary ®rst step to revealing projectile-charge-dependent e€ects by orthogonalizing initial and ®nal states in order to rid the ®rst-order amplitude of excess probability in the region where traveling waves and bound state waves overlap. We have also taken the initial steps of including decay terms, which interfere with the direct-ionization amplitudes. Full inclusion of such terms shows promise of providing both the magnitudes and the splitting of the cross sections for di€erently charge particles correctly.

Acknowledgements One of the authors JHM was supported by the Division of Chemical Sciences, Oce of Basic Energy Science, Oce of Energy Research, U.S. Department of Energy.

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