Electron-Impact Double Ionization of the H 2

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1 I R A P 6(), Dec. 5, pp. 9- Electron-Impact Double Ionzaton of the H olecule Internatonal Scence Press ISSN: 9-59 Electron-Impact Double Ionzaton of the H olecule. S. PINDZOLA AND J. COLGAN Department of Physcs, Auburn Unversty, Auburn, AL Theoretcal Dvson, Los Alamos Natonal Laboratory, Los Alamos, N ABSTRACT: A tme-dependent close-couplng method n sphercal polar coordnates s developed to calculate the electron-mpact double onzaton of the H molecule. The full wavefuncton s represented by an expanson n products of sx-dmensonal radal-angular numercal functons and analytc rotatonal functons. A test calculaton fnds good agreement between the new method and a prevous frozen core method for the sngle onzaton of H for the = l = partal wave and an mpact energy of. ev. A test calculaton s also made for the double onzaton of H for the same partal wave and mpact energy. I. INTRODUCTION A tme-dependent close-couplng (TDCC) method was orgnally developed to calculate the electron-mpact sngle onzaton of H + []. The full wavefuncton was represented by an expanson n products of four-dmensonal radalangular numercal functons and analytc rotatonal functons. When the close-couplng results for low angular momentum are combned wth dstorted-wave results for hgh angular momentum, the total cross secton was found to be n excellent agreement wth experment []. A frozen-core TDCC method was then used to calculate the electron-mpact sngle onzaton of H []. The total cross secton was agan found to be n excellent agreement wth experment [4]. The frozen core TDCC method has also been used to calculate the electron-mpact sngle onzaton of L [5]. In ths artcle we develop a tme-dependent close-couplng method to calculate the electron-mpact double onzaton of H. We note that a TDCC method for atoms has been prevously appled to calculate the electronmpact double onzaton of He [6], [7], g [8], Be [9], and B + []. For H the full wavefuncton s represented by an expanson n products of sx-dmensonal radal-angular numercal functons and analytc rotatonal functons. Test calculatons are made on a relatvely small numercal lattce for one partal wave and one ncdent energy. Detals of the TDCC method of H are presented n Secton II, test calculatons are presented n Secton III, and a bref summary of future plans s gven n Secton IV. Unless otherwse stated, all quanttes are gven n atomc unts. II. THEORY A. Relaxaton to the Ground State The sx-dmensonal wavefuncton for the ground state of H s obtaned by relaxaton of the tme-dependent Schrodnger equaton n magnary tme (): ( r, r, ) ( ) (,, ) (,, ), () V r r r r r r r Internatonal Revew of Atomc and olecular Physcs, 6 (), July-December 5 9

2 . S. Pndzola and J. Colgan where V ( r ) s a sngle partcle nteracton wth the target nucle. The wavefuncton s represented by an expanson n smple products of four-dmensonal radal-angular functons Pm m ( r,, r,, ) and rotatonal functons: P ( r,, r,, ) (,, ) ( ) ( ), m m r r m m m m r r sn sn () m e where m( ) and m + m =. The angular reducton of the tme-dependent Schrodnger equaton n magnary tme yelds a set of close-couplng equatons gven by: Pm m ( r,, r,, ) T ( r, ) P ( r,, r,, ) m m m m m V ( r,, r, ) P ( r,, r,, ). m m, m m m m () The sngle partcle operator n the close-couplng equatons s gven by: T ( r, ) K ( r ) K ( r, ) A ( r, ) N( r, ), (4) m m where K(r) and K(r, ) are netc energy operators []. The axal angular momentum operator s gven by: m Am ( r, ). r sn (5) The nuclear nteracton operator for H s gven by: N( r, ), r R rrcos r R rr cos (6) 4 4 where R s the nternuclear separaton, whch s algned along the z axs. The two partcle operator n the closecouplng equatons s gven by: r ( q ) V P (cos ) P (cos ) q q m m j, m mj j r q ( q ) ( m, m ) e ( m, m ), q( j ) j j (7) where q (cos ) s an assocated Legendre functon. P At tme = the radal-angular functons are gven by: P ( r,, r,, ) P ( r, ) P ( r, ), (8) m m s s m, m, where the radal-angular orbtal, P s (r, ), s obtaned by matrx dagonalzaton of the Hamltonan, T m = (r,). Upon relaxaton n magnary tme of Eq.(), an accurate wavefuncton for the ground state of H s obtaned. 94 Internatonal Revew of Atomc and olecular Physcs, 6 (), July-December 5

3 B. Propagaton of the Scatterng State Electron-Impact Double Ionzaton of the H olecule The nne-dmensonal wave functon for electron onzaton of the ground state of H s obtaned by solvng the tme-dependent Schrodnger equaton: ( r, r, r, t) ( ) (,,, ) (,,, ). (9) V r r r r t r r r t t j r rj The wavefuncton for a gven symmetry s represented by an expanson n smple products of sx- dmensonal radal-angular functons P ( r,, r,, r,, t) and rotatonal functons: m m m ( r, r, r, t) m m m P ( r,, r,, r,, t) m m m r r r sn sn sn ( ) ( ) ( ), () m m m where = m + m + m. The angular reducton of the tme-dependent Schrodnger equaton yelds a set of tmedependent close-couplng equatons gven by: Pm m m ( r,, r,, r,, t) t T ( r, ) P ( r,, r,, r,, t) m m m m m m V ( r,, r, ) P ( r,, r,, r,, t) m m m m, m m m m m V ( r,, r, ) P ( r,, r,, r,, t) m m, m m m m m V ( r,, r, ) P ( r,, r,, r,, t). m m m m, m m m m m () At tme t = the radal-angular functons are gven by: P ( r,, r,, r,, t ) m m m The Gaussan wavepacet s gven by: Pm m ( r,, r,, ) G ( lm r, ) m,. () m m ( ra) w e ( rl / ) (, ) sn (, ), lm lm 4 G r e Y ( w ) () where a s the localzaton radus, w s the pacet wdth, l s the ncdent angular momentum, and the ncdent energy equals /. Followng propagaton n real tme of Eq. (), momentum space ampltudes are calculated usng: Internatonal Revew of Atomc and olecular Physcs, 6 (), July-December 5 95

4 . S. Pndzola and J. Colgan A (, ) lmlm A (, ) l ml m dr d dr d dr d P ( r, ) P ( r, ) P ( r, ) * * * s l m l m P ( r,, r,, r,, t ), (4) mm m A (, ) lm lm dr d dr d dr d P ( r, ) P ( r, ) P ( r, ) * * * l m s lm P ( r,, r,, r,, t ), (5) mm m B (,, ) l m l m l m dr d dr d dr d P ( r, ) P ( r, ) P ( r, ) * * * l m l m s P ( r,, r,, r,, t ), (6) mm m dr d dr d dr d P ( r, ) P ( r, ) P ( r, ) * * * l m lm l m P ( r,, r,, r,, t ), (7) mm m where the radal-angular orbtals, P lm (r, ), are obtaned by matrx dagonalzaton of the Hamltonan, T m (r, ). In addton, the radal-angular orbta ls, P ( r, ), are obtaned by matrx dagonalzaton of the Hamltonan, T ( r, ) V ( r, ), where V HS (r, ) s the Hartree-Slater potental []. m HS C. Cross Sectons The total sngle onzaton cross secton leavng H + n the ground state s gven by: lm (, ). (8) d d j Al ml jm j j l j l m l jm j The total double onzaton cross secton s gven by: d d d B l m l m l m l lm lm lm (,,. (9) 96 Internatonal Revew of Atomc and olecular Physcs, 6 (), July-December 5

5 Electron-Impact Double Ionzaton of the H olecule The energy dfferental double onzaton cross secton s gven by: d d d d d d l lm lm lm arctan arctan B (,, ), l ml mlm () where s an angle n the (, ) hypersphercal plane and s an angle n the plane perpendcular to the (, ) hypersphercal plane, both defned from to /. The energy and angle dfferental double onzaton cross secton s gven by: d ddd d d d d d lm l m l m l * arctan arctan Yl (, ) e e l l l l l l l ( ) e B (,, ) l m l m l m lm lm lm m m m, Y ˆ Y ˆ Y ˆ, () where the ncomng electron beam s orented at angles ( e, e ) wth respect to the z axs, Y lm (, ) s a sphercal harmonc, and l s the Coulomb phase shft. III. RESULTS As a smple numercal test of the theory, we use a radal-angular grd of r =.4 wth N r = 7 and =.5 wth N = 8. The nternuclear separaton s R =.4. Bound and contnuum radal-angular orbtals for H are found upon matrx dagonalzaton of T m (r,). For m = we obtaned 9 bound states, begnnng wth P s (r, ) at -5.8 ev, and contnuum states rangng from.6 ev to 48.9 ev. For m = we obtaned bound states, begnnng wth P p (r, ) at -.5 ev, and 9 contnuum states rangng from.9 ev to 47. ev. Bound and contnuum radal-angular orbtals for H are found upon matrx dagonalzaton of T m (r, ) + V HS (r, ). For m = we obtaned 6 bound states, begnnng wth P (, ) r at -5.4 ev, and 4 contnuum states rangng from. ev to 49.9 ev. For m = we obtaned bound states, begnnng wth P (, ) r at -.8 ev, and 4 contnuum states rangng from.5 ev to 47.5 ev. The choce of the parameter n the local exchange potental allows adjustment of the s bndng energy to be near the expermental value. For relaxaton to the ground state, we use a numercal lattce of (7 8) ponts parttoned over 4 parallel computer cores and the coupled channels found n Table I. At tme = the radal-angular functons of Eq.(8) s p Internatonal Revew of Atomc and olecular Physcs, 6 (), July-December 5 97

6 . S. Pndzola and J. Colgan yeld an energy of -49. ev. Followng tme steps at =. the radal-angular functons of Eq.() yeld an energy of -5.5 ev. For propagaton of the scatterng state, we use a numercal lattce of (7 8) ponts parttoned over 58 parallel computer cores and the 7 coupled channels found n Table II. At tme t = we choose a Gaussan wavepacet of Eq. () wth a localzaton radus a = 4.4, a pacet wdth =.6, an ncdent angular momentum l =, and an ncdent energy of E =. ev. Followng 5 tme steps at t =. the radal angular functons of Eq. () are used to calculate the (48) momentum space ampltudes of Eqs.(4)-(6) and the (459) momentum space ampltudes of Eq. (7). The total sngle onzaton cross secton leavng H n the ground state from Eq. (8) s found to be.5 barns for = l = at. ev ncdent energy. The total double onzaton cross secton from Eq.(9) s found to be 9.5 Kbarns for = l = at. ev ncdent energy. To chec our total sngle onzaton cross secton, we carred out frozen-core TDCC calculatons [], as outlned n the Appendx. The ntal state s the bound radal-angular orbtal P (, ) s r at -5.4 ev. For propagaton of the scatterng state, we use a numercal lattce of (7 8) ponts parttoned over 4 parallel computer cores and the coupled channels found n Table I. Followng 5 tme steps at t =. the radal-angular functons of Eq. (4) are used to calculate the (48) momentum space ampltudes of Eq.(6). The total sngle onzaton cross secton from Eq. (7) s found to be. barns for = l = at. ev ncdent energy, ncludng both the S = snglet and S = trplet contrbutons. IV. SUARY In the future we plan to apply the TDCC method to a full calculaton of the electron-mpact double onzaton of H. We wll choose an mpact energy above the full breaup energy of 5.5 ev and the number of l partal waves wll nclude and l 6. We plan on usng a radal-angular grd of r =. wth N r = 44 and =.8 wth N =. The numercal lattce of (44 8) ponts wll be parttoned over 46,656 parallel computer cores. The number of coupled channels for relaxaton and propagaton wll also be ncreased to at least nclude m = ±. Acnowledgments Ths wor was supported n part by grants from the US Department of Energy and the US Natonal Aeronautcs and Space Admnstraton. Computatonal wor was carred out at the Natonal Energy Research Scentfc Computng Center (NERSC) n Bereley, Calforna and the Hgh Performance Computng Center (HLRS) n Stuttgart, Germany. APPENDIX The sx-dmensonal wavefuncton for electron onzaton of one actve electron n the ground state of H s obtaned by solvng the tme-dependent Schrodnger equaton: ( r, r, t) ( ) (,, ) (,, ). () V r r r t r r t t r r The wavefuncton for a gven symmetry s represented by an expanson n smple products of four-dmensonal radal-angular functons P ( r,, r,, t) and rotatonal functons: m m P ( r,, r,, t) (,, ) ( ) ( ), mm r r t m m m m r r sn sn () where = m + m. The angular reducton of the tme-dependent Schrodnger equaton yelds a set of tme-dependent close-couplng equatons gven by: 98 Internatonal Revew of Atomc and olecular Physcs, 6 (), July-December 5

7 Electron-Impact Double Ionzaton of the H olecule Pm m ( r,, r,, t) t m HS m m T ( r, ) V ( r, ) P ( r,, r,, t) V ( r,, r, ) P ( r,, r,, t). m m mm, m m m m At tme t = the radal-angular functons are gven by: P ( r,, r,, t ) m m / [ P ( r, ) G ( r, ) s lm m, m, S ( ) G ( r, ) P ( r, ) ] lm s m, m, (5) omentum space ampltudes are calculated usng: B (, ) lm lm (, ) (, ) * * dr d dr d P l m r P lm r P ( r,, r,, t ). m m The total sngle onzaton cross secton s gven by: w (S ) t d d B l ml m l 4 S l m lm (, ), (6) (7) where the subshell occupaton number w t = for the ground state of H. Table I 4D Coupled Channels m m - - Table II 6D Coupled Channels m m m References []. S. Pndzola, F. Robcheaux, and J. Colgan, J. Phys. B 8, L85 (5). [] B. Peart and K. T. Dolder, J. Phys. B 6, 49 (97). Internatonal Revew of Atomc and olecular Physcs, 6 (), July-December 5 99

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