Mechanisms of electron impact ionization at threshold

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Recently, it has been shown (Rost et a/ 1991a, b) that accurate two-electron ... resonances can be explained in terms of MO quantum numbers (Rost and Briggs ...
I. Phys. B: At. Mol. Opt. Phys. 24 (1991) L393-L396. Printed in the UK

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LEITER TO THE EPITOR

Mechanisms of electron impact ionization at threshold J M Rostt5 and J S BriggstB t Department of Chemistry, Harvard

University, Cambridge, MA, USA

% Instituta de Fisica, Laboratorio de Cuernavaca, Cuernavaca, Mexico Received 3 June 1991

Abstract. We extend previous analyses of the six-dimensional wavefunction describing threshold ionization and show that the nodal structure near the Wannier saddle can be compadly described by the nodal structure of three-dimensional molecular orbital wavefunctions and their associated quantum numbers. Independent of the Wannier assumption a mechanism is proposed by which only cenain states o f LSr are expected to dominate at threshold.

Electron impact ionization at threshold (we will consider the simplest case of the hydrogen atom as an example) has been much discussed in terms of the Wannier mechanism (Wannier 1953). In this mechanism it is assumed that the region r 2 = - r , (where I , , r, are the electron coordinates with respect to the nucleus) is the important region for two particles to emerge at threshold. After establishment of energy threshold laws (Wannier 1953, Rau 1971, Peterkop 1971, Klar and Schlecht 1976, Klar 1982, Feagin 1984) attention was given to the angular distribution, in terms of the most likely states of given LSTI to appear at threshold. These analyses were made of the sixdimensional wavefunction ' P L s n ( r l r, 2 )by considering the nodal structure in the Wannier region via expansion in the six spherical coordinates of r I , r2 and their associated one-electron quantum numbers. In particular, following the first analysis by Rau and Greene (1982), Stauffer (1982) made important statements concerning the nodal structure of given LSn states. However, again he used the full six dimensions and arrived at his conclusions by considering properties of Clebsch-Gordan coefficients and two-electron radial wavefunctions composed of single-electron radial wavefunctions. Here we will show by a transformation to molecular coordinates r, R, where r = f (r, + r 2 ) and R = r I - r,, that all of the symmetry properties with respect to the Wannier region can be expressed in terms of the three-dimensional I coordinate describing the motion of the electronic centre-of-mass (ECM) with respect to the nucleus. Not only that, we can extend previous results and show that the nodal structure appears directly due to the separation in prolate spheroidal coordinates A, p, 4 (Feagin and Briggs 1988). That is, the nodal structure in the Wannier region can be expressed in terms of the 'three-dimensional' quantum numbers n,, nr, m. Recently, it has been shown (Rost et a/ 1991a, b) that accurate two-electron wavefunctions for doubly-excited states have well defined n,, n,, m molecular orbital (MO) nodal structure. Furthermore the decay widths of symmetric doubly-excited 5 Permanent address: Fakultit fsr Physik, Alben-Ludwigs-Univer~it~t, Hermann-Herder-Strasse 3. D-7800 Freiburg. Federal Republic of Germany.

0953.4075/91/160393+04$03.50 @ 1991 IOP Publishing Ltd

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resonances can be explained in terms of MO quantum numbers (Rost and Briggs 1990). Since ionization is the inverse process of decay to the ground state here we assume the validity of the MO model for this process also. This allows us to predict the dominant states contributing to threshold ionization quite independently of the Wannier assumption. Nevertheless we show that the major conclusions of the Wannier analysis also emerge from the MO picture. The angular distribution in the continuum is described by a linear combination of wavefunctions of fixed L:M: S and parity v (Feagin 1988, Selles er a1 1987). More importantly (Feagin and Briggs 1988) they are eigenstates of the product operator of parity P and permutation P , , P P 1 2 v L s r ( rR, ) = v L s , ( - r , R I = ( - l ) ' * L s , ( p , R) (1) where (-1)' = v(-I)'. The states I = 0 are called 'gerade' and the states I = 1 'ungerade'. A state of fixed LMSI or equivalently L M S v can be expanded (Feagin and Briggs 1488)

v L d r . R ) =C

i, m

L+S+,+M [Dhm+(-l)

D & - m l R - ' f f ( R ) @ : ( rR, )

(2)

where i denotes the quantum numbers (n,,, n,, m) of the MO wavefunction @:. Note that r=n,+m(mod2). In the MO picture the Wannier point r2= - r I is expressed by the condition r=O. (3) For fixed R this origin of ECM coordinates coincides with a saddlepoint of the two-centre potential (Rost and Briggs 1991). Condition (3) is expressible as two separate conditions

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r2 = rl @,U = 0

(4)

(ii) p2= -PI e A = 1 (5) i.e. by the origin in prolate spheroidal coordinates. This is important since p motion is along the unstable saddle direction and connected to decay whilst A motion is in the stable direction. The expression of the Wannier saddle according to (3) allows us to derive nodal properties from a consideration of the three internal dimensions r only. For example from (I) we 8% immediate!y that at the Wannier point [l+(-l)'t']'P(O, R ) =O. (6) This is the first result; it shows that only gerade states with f even or TI = (-1)' can be finite at the Wannier saddle (Greene and Rau 1982). What is new however is that we see from (2) that the origin of this rule is the MO g, U parity of the adiabatic function @ l ( r ,R). The condition r, = r I , given by (4), can also be expressed in MO quantum numbers since = 0 at p = 0 when nr is odd. More precisely the wavefunction @! has a node o r antinode at r, = r2 according as (-1)"" = -1 or +I. The importance of the quantum number A=(-l)". for decay of resonant states has been demonstrated (Rost and Briggs 1990). Here we can generalize the result of Klar and Schlecht (1976) to state that states built solely on A = + I or solely on A = -1 have the Wannier exponent 1.127 or 3.881 respectively. From (2) it is readily seen that 's'states are built solely on uu MO and 'P" states solely on TI^ MO. Both have A = -1 which explains the origin of the higher exponent for's' and 'Pc states (Stauffer 1982). All states of L a 2 can be built on MO of A = + I and therefore can have the lower exponent (Feagin and Briggs 1988).

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Letter to the Editor

L395

The origin of the other spheroidal coordinate A = 1 expresses the condition t2= -i, or equivalently Q,, = 180", where Q,, is the interelectronic angle. Since only m = 0 MO are finite at A = 1 this leads to the new result that all states (regardless of LST) built on MO with m > 0 are zero at e,, = 180". This result can also h e extracted from the results of Feagin (1984) who expanded the total wavefunction in cylindrical coordinates about the point r = O . At this point the stable part of the ECM motion is represented by a two-dimensional oscillator (Rost s n A Drinnr 1 0 0 1 \

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to vanishing 'vibrational' angular momentum, i.e. to m =O. A similar conclusion is also reached by Selles et a / (1990). However, these authors used a body-fixed frame with z-axis along i,, This has the important consequence that experiments measuring electrons at relative angle 180" measure only m = 0, 1 = 0 body-frame wavefunctions. From (2) one sees th.! &those b.i!t 0" cg, nnly $!.!es (-I),+,+'= T ( - ! ) L = 1 czfi be fi"i!t (Green and Rau 1982) at this point. Finally predictions as to the preferred LSf states to be expected at threshold can be deduced from the MO analysis, independent of any assumptionsconcerning the Wannier mechanism. A more detailed description will be published subsequently. Only a sketch can he given here. The correlation diagram of H- is the same as that of H:. As R decreases from infinity the initial 1s state develops adiabatically into equal 1suz and 2pg. components that propagate independently. The lsu8 MO supports states Is','Po, 'De etc. Transitions into higher MO occur preferentially at avoided crossings. The lsug MO with ( n k , n,,, m)=(O,O,O) is the 'precursor' of an 'in-saddle' ( A = + l ) sequence (0,2N -2,0), N = 1 , 2 , . . . of MO which show avoided crossings at increasing values of R. Diahatic transition through this sequence will lead to threshold ionization at R + m. This route will populate Is',P",ID', etc states of A = +1 and gerade symmetry. finite at e,,=180" and with an antinode at r , = r,. The 2pu. (0,1,0) MO is also the precursor of a saddle sequence ( 0 , 2 N - 1,O) of MO showing avoided crossings and the diabatic ladder through these crossings leads similarly to ionization. However, this is a 'side-saddle' sequence with A = -1 and a node at r , = r,. This means that the crossings are broad and it is expected that the diabatic transversal is made with much lower probability than the uxseries. However, the 2pu, state is degenerate with the 2 p r , at R = 0. This opens up a new channel of excitation very familiar in ion-atom collisions. As R decreases from m a rotational coupling at small R is made from the 2puu to 2 p r , MO. This MO with (0,0, 1) is the precursor of the A = + l (0,2N-4,1) saddle sequence with narrow avoided crossings. The diahatic following of this sequence to threshold should provide an efficient pathway for ionization. Of the states supported by 2pu, only the states 'Po, 3 C l O D , F , etc can be supported also by 2p7-r". All these states have an anti-node at rl = r2 but a node at Q,, = 180". To summarize, we predict predominantly gerade '9, 'PO, ID', etc states from ionization via the ugsequence and predominantly ungerade 'P", ' W . 'F", etc states via the T. sequence. Of the two mechanisms, the latter giving rise to a zero at 0 , 2= 180" is probably more favoured. We have shown that the nodal structure of six-dimensional wavefunctions near the Wannier point r2 = - r , can be expressed entirely in terms of three-dimensional MO nodal surfaces and associated quantum numbers. An analysis of MO ionization mechanisms has allowed us to propose propensity rules forthe LSt states appearing at threshold. The ionization occurs preferentially by promotion of the ECM along saddle sequences;

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Letter to the Editor

a mechanism that is exactly the opposite of the preferred mode of decay of doublyexcited states.

We acknowledge very helpful discussions with Alain Huetz and Jim Feagin. JMR would like to thank Professor D Herschbach for the hospitality of his group. JSB is grateful to the DAAD for a travel grant, to Professor Carmen Cisneros for her hospitality and the Laboratorio de Cuernavaca of UNAM for financial support.

References Feagin J M 1984 J. Phys 8: AI. Mol. Phyr. 17 2433

-1988 Fundamental Processes ofAtomic Dynamics ed J S Briggs, H Kleinpoppen and H 0 Lutz (Nato AS1 8181) (New York: Plenum) p 275 Feagin I M and nriggs I S 1Y88 Phys. R e v A 37 45YY Greene C H and Rau A R P 1982 Phyx. Rev. Lstt. 48 533 Klar H 1982 2. Phys. A 307 75 Klar H and Schlecht W 1976 J. Phys B: Al. Mol. Phys. 9 1699 Peterkop R 1971 J. Phys B: Al.' Mol. Phys. 4 513 Rau A R P 1971 Phys. Rev. A 4 207 Rost J M and Briggs J S 1990 J. Phys. B: Al. Mol. Opt. Phys 23 L339 ,On, I DL... ~.". 0. I . n ., m...- ,A *" I. ^..LI:-L.A vr.. -.. Rost I M, Briggs J S and Feagin J M 1991a Phys. Reo. Lelr. 66 1642 Rost J M, Gersbacher R, Richter K. Briggs J Sand Wintgen D 1991b J. Phys. B: Al. Mol. Opl. Phyr. 24 2455-65 Selles P, Mareau J and Huetz A I987 1. Phys. B: Al. Mol. Phys. 20 5183 -1990 J. Phyx B: Al. Mol. Opl. Phys. 23 2613 Stauffer A D 1982 Phys. LeII. 91A 114 Wannier G H 1953 Phys. Reo. 90 817 -.,,.I.

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