Angular distribution of autoionization and Auger electrons ejected by ...

J. Phys. B: At. Mol. Opt. Phys. 30 (1997) 1269–1291. Printed in the UK

PII: S0953-4075(97)76979-7

Angular distribution of autoionization and Auger electrons ejected by electron impact from laser-excited and polarized atoms V V Balashov, A N Grum-Grzhimailo and N M Kabachnik Institute of Nuclear Physics, Moscow State University, Moscow 119899, Russia Received 8 August 1996, in final form 8 November 1996 Abstract. A general formalism is developed for a description of the angular distribution of electrons ejected after the electron-impact excitation/ionization of the laser-excited and polarized atoms. Properties of the angular distributions are analysed in a general case as well as in some particular geometries of the experiment. Within the two-step approximation the formalism is applied to the resonant single and double ionization with ejection of autoionization and Auger electrons. As an example the angular distributions of autoionization electrons from laserexcited sodium atoms are calculated in the plane-wave and distorted-wave Born approximations. Angular correlations between autoionization and scattered electrons, which can be measured in a coincidence (e, 2e) experiment with polarized targets, are also analysed.

1. Introduction The study of angular distributions of electrons ejected from atoms in electron–atom collisions is a well developed field which gives important information on mechanisms of excitation and decay of atomic quasistationary states (Mehlhorn 1985, 1990). In the majority of the studies, both experimental and theoretical, the electron-impact excitation from the ground states of atoms was investigated. Ten years ago Balashov and Grum-Grzhimailo (1986) suggested the investigation of the electron-impact excitation of atomic autoionizing states in atoms excited by lasers. Theoretical calculations performed by these authors have shown promising perspectives in using this method similarly to the well known autoionization studies with crossed laser and synchrotron radiation beams. In comparison with the usual electronimpact experiments, excitation of the laser-excited atoms is advantageous in investigating the autoionizing states which are not coupled optically with the ground state. On the other hand, strong dependence of the angular distribution of the ejected electrons on the polarization of the laser-excited intermediate state, allows one to obtain more detailed information on excitation of autoionizing states from anisotropic targets. The first high-resolution angle resolved measurements of the autoionization/Auger electron spectra produced by highenergy electrons from the laser-excited sodium atoms (Dorn et al 1993, 1994, 1995a) have confirmed these expectations. Recently, further experimental investigations along this line were performed in Freiburg (Dorn et al 1995b, Grum-Grzhimailo and Dorn 1995). Excited targets with variable polarization and alignment characteristics can be prepared by laser pumping. Similar to the recent extensive studies on inelastic and superelastic electron scattering for transitions between discrete atomic states (Hertel and Stoll 1977, c 1997 IOP Publishing Ltd 0953-4075/97/051269+23$19.50

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Andersen et al 1988, Zetner et al 1992, Sang et al 1994) one can use this advantage of the crossed electron and laser beam technique to obtain the most detailed information on the electron-impact excitation of atomic quasistationary states contained in the amplitudes of the transitions between different magnetic substates. In this paper we present a general formalism connecting these amplitudes as well as statistical tensors of the decaying states with observable parameters characterizing the angular distributions of the autoionization and Auger electrons. In describing the angular distributions we use the formalism of density matrices and statistical tensors of angular momentum (Devons and Goldfarb 1957, Ferguson 1965, Blum 1981, Balashov et al 1984). Ionization by electron impact is closely related to the photoionization. Therefore, the present study is connected with another line of our theoretical investigations (Balashov et al 1988, Baier et al 1994) where a general theory of the angular distributions of photoelectrons in resonant photoionization of laser-excited and polarized atoms has been developed. Some particular results of the present theoretical analysis of the angular distribution of autoionization/Auger electrons from excited atoms in crossed laser and electron beams have been reported earlier (Balashov et al 1996). In section 2 we derive a general expression for the angular distribution of electrons ejected from polarized atoms by electron impact and suggest a parametrization, which is valid for the ionization of an arbitrary multiplicity provided only one of the outgoing electrons is detected. Then the anisotropy of the electron ejection is analysed as a combined contribution of the anisotropy induced by the laser excitation and the anisotropy pertinent to the electron–atom interaction. Particular geometries, which are used in the experiments with crossed laser and electron beams, are considered in more detail. Possibilities of experiments with the electron detector at a fixed position and rotating direction of the linear polarization of the laser beam are specially discussed. Section 3 is devoted mainly to the dynamics of the processes under consideration. A simplification of the general formula is achieved using the two-step approximation introduced to describe the case of resonant ionization via an isolated quasistationary state either in the target atom or in the produced ion when the direct ionization can be neglected. The case of high energy of the incoming electron beam is of special interest. Here the first-order plane-wave Born approximation (PWBA) can be used in calculations since prior- or post-collision electron–atom interactions are negligible. We have transformed the general formulae for the transition amplitudes and for the statistical tensors of the excited autoionizing/Auger states into a special form convenient for usage within the PWBA. In section 4, as an example, we consider an autoionization of sodium atoms induced by electron-impact excitation from the laser-excited state 3p 2 P3/2 . The angular distribution of autoionization electrons is analysed using both the general and simplified formulae. The anisotropy parameters are shown to be sensitive to the details of the dynamics of electron– atom inelastic scattering. It is well known in the physics of autoionizing states that the coincidence experiments, (e, 2e), with non-oriented target atoms are much more informative than the corresponding one-arm experiments. The same is true for the electron-impact ionization of laser-excited oriented atoms as well. Taking into account the growing role of the coincidence (e, 2e) method in autoionization studies (Balashov 1993), in section 5 we analyse the angular correlations between autoionization/Auger electron and the scattered electron in a case of a pure resonant electron-impact ionization of polarized laser-excited atoms. Some illustrative calculation results are presented for the sodium atom. Atomic units are used throughout unless otherwise indicated.

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2. The general formalism 2.1. The general expression for the angular distribution of electrons ionized from polarized atoms by electron impact Consider the process in which an initial unpolarized electron beam, ei , ionizes a polarized atomic target and one of the outgoing electrons, e1 , is detected ei (pi ) + A(γ0 J0 ) → ef (pf ) + e1 (p1 ) + A+ (γ1 J1 ).

(1)

Before the collision the target atom is in a state with the total angular momentum J0 while J1 denotes the total angular momentum of the residual ion, A+ . Sets of other quantum numbers characterizing the atom and the ion are denoted by γ0 and γ1 , respectively. The linear momenta of the electrons involved in process (1) are denoted by pi , pf and p1 . We choose the z-axis of the laboratory frame along the direction of the incident beam pi . Note, that so far we do not specify which of the electrons, ef or e1 , is ‘scattered’ or ‘ejected’. It is essential only that one of the outgoing electrons is detected. However, for convenience, if it does not lead to confusion, we shall refer to the detected electron as ‘ejected’. In the following we suppose that the detector of electrons, positioned at the angles (ϑe , ϕe ) with respect to the laboratory frame, is not sensitive to spin polarization, and that the target atom state is axially symmetric with respect to the axis A which is characterized by two angles (ϑa , ϕa ) in the laboratory frame. The latter means that the angular momentum density matrix of the initial atomic state is diagonal in the coordinate frame with the z-axis directed along the axis A (atomic frame). This is not a strong limitation because this case is realized practically in all experiments with laser pumping. The angular distribution of the electron, e1 , as measured in the energy and angle resolved experiment, is given by the double differential cross section which in the framework of the statistical tensor formalism can be written in the standard form (Ferguson 1965): X d2 σ ∗ (ϑe , ϕe ) = ρkq (γ J, γ 0 J 0 )εkq (γ J, γ 0 J 0 ) dE d γ γ 0J J 0

(2)

kq

where ρkq (γ J, γ 0 J 0 ) is the statistical tensor of the system (e1 +A+ ) characterized by the total angular momentum, J , and other quantum numbers, γ , and εkq (γ J, γ 0 J 0 ) is the efficiency tensor of the detectors. We abbreviate γ = {γ1 J1 , l1 j1 }, where l1 and j1 denote the orbital and total angular momenta of the electron, e1 , respectively. In the case when the ion is not observed and the detector is insensitive to the spin states of the electrons the efficiency tensor εkq (γ J, γ 0 J 0 ) takes the form (Baier et al 1994) εkq (γ J, γ 0 J 0 ) = (4π)−1/2 δγ1 γ10 δJ1 J10 (−1)J +J1 +k−1/2 lˆ1 lˆ10 jˆ1 jˆ10 JˆJˆ 0 kˆ −1  0  
j1 l1 1 J j10 J1 0 2 × l1 0, l1 0|k0 Y∗kq (ϑe , ϕe ) l10 j10 k j1 J k

(3)

√ where Ykq (ϑe , ϕe ) are spherical harmonics and we abbreviate lˆ = 2l + 1. Here and below standard notations are used for the Clebsch–Gordan coefficients, 6j - and 9j -symbols (Edmonds 1957). The statistical tensors ρkq (γ J, γ 0 J 0 ) are expressed in terms of the transition amplitudes and the statistical tensors of the initial atomic state, ρk0 q0 (γ0 J0 ) (we omit for brevity the

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second argument in the diagonal statistical tensors, such as ρk0 q0 (γ0 J0 , γ0 J0 )): X
0 0 (−1)J −M J M, J 0 − M 0 |kq ρkq (γ J, γ 0 J 0 ) = MM 0 M0 M00 µi µf k0 q0


0 ×(−1)J0 −M0 J0 M0 , J0 − M00 |k0 q0 ρk0 q0 (γ0 J0 ) Z ˆ 0 J0 M0 , pi µi ihγ 0 J 0 M 0 , pf µf |R|γ ˆ 0 J0 M 0 , pi µi i∗ . × df hγ J M, pf µf |R|γ 0 (4) ˆ is the transition operator, and µi , µf are the spin projections of the initial and Here R unobserved electrons. We suppose that the initial electron beam is unpolarized. Integration over the solid angle, f , and summation over µf correspond to taking a trace of the total density matrix over the quantum numbers of the unobserved electron, ef . Upon expansion of the initial and final electron wavefunctions in partial waves and straightforward summation over projections of the angular momenta one obtains from (2)–(4) X d2 σ (ϑe , ϕe ) = c ρ k0 0 (γ0 J0 )Ak0 kke {Yk0 (ϑa , ϕa ) ⊗ Yk (ϑe , ϕe )}ke 0 (5) dE d k0 kke where c is a normalization constant, ρ k0 0 (γ0 J0 ) are the statistical tensors of the initial polarized atom in the coordinate frame with the z-axis directed along the A-axis and X0
0 0 (−1)J1 +ji −jf −Jt lˆi lˆi0 jˆi jˆi0 lˆ1 lˆ10 jˆ1 jˆ10 JˆJˆ 0 Jˆt Jˆt0 li 0, li0 0|ke 0 Ak0 kke = (−1)ke +k Jˆ0 lf jf li li0 ji ji0 γ γ 0 J J 0 Jt Jt0

   
ji li 1 j1 l1 12 Jt J jf 2 li0 ji0 ke J 0 Jt0 k l10 j10 k ( )   J0 ji Jt j J J1 × 10 J0 ji0 Jt0 J j10 k k0 ke k ˆ 0 J0 , li ji : Jt ihγ 0 J 0 , lf jf : Jt0 kRkγ ˆ 0 J0 , l 0 j 0 : Jt0 i∗ . ×hγ J, lf jf : Jt kRkγ i i

× l1 0, l10 0|k0

(6)

γ10 J10

A prime at the summation sign indicates that only terms with γ1 J1 = are included in the sum over γ , γ 0 . Equation (5) contains the tensorial products of the spherical harmonics X {Yk0 (ϑa , ϕa ) ⊗ Yk (ϑe , ϕe )}ke qe = (7) (k0 q0 , kq|ke qe )Yk0 q0 (ϑa , ϕa )Ykq (ϑe , ϕe ). q0 q

The reduced matrix elements in (6) and, hence, the parameters Ak0 kke include information on the dynamics of the process and depend on the values of pi , pf and p1 , where pi and p1 are fixed in the experiment and pf depends on the state of the residual ion |γ1 J1 i in accordance with the conservation of energy. The indices k0 , k and ke have a simple physical meaning, namely, k0 is the rank of the tensor characterizing the polarization of the initial state of the atom, k is the rank of the spherical harmonic of the ejected electron, and ke is determined by the ranks of the multipole part of the interaction probing the target atom. These indices are connected by the triangle rule, i.e. |k0 − k| 6 ke 6 k0 + k etc. Equation (5) is very similar to equation (12) of Baier et al (1994) where the photoionization of polarized atoms has been considered. The difference is that here we must use the partial-wave expansion for the incident electron instead of the fixed multipolarity of the incident dipole photon in the paper by Baier et al (1994) and perform a corresponding summation over li , li0 . Therefore, ke can be of high value (ke 6 li + li0 ), not restricted by 2 as in the case of photoionization.

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In the case when the hyperfine interaction is important the general parametrization (5) modifies only slightly. One should then use the statistical tensors ρ k0 0 (γ0 F0 , γ0 F00 ) of the total angular momentum, F0 , of the initial atomic state instead of those of the electronic shell and perform an additional summation over F0 and F00 . The dynamical coefficients Ak0 kke will also depend on the values of F0 and F00 . Equation (5) is the main result of this section. It factorizes the geometrical factor and reduces the problem to the calculation of the dynamical coefficients, Ak0 kke . Note that the above consideration is also valid in a more general case of multiple ionization ei + A(γ0 J0 ) −→ ef + e1 + [A(n+1)+ + e2 + · · · + en+1 ]

(8)

when only one of the electrons, e1 , is detected in the final state. The only difference is that γ1 J1 should be the quantum numbers of the system in square brackets in (8). Therefore, all above equations are valid for the reaction (8) provided the set of quantum numbers, γ1 , γ , is appropriately chosen together with the corresponding angular momenta coupling. 2.2. Analysis of the general expression 2.2.1. Properties of the anisotropy coefficients and angular distributions. The laser-excited state is described by the statistical tensors ρ k0 0 (γ0 J0 ) with even k0 for pumping with linearly polarized laser light (A along the laser polarization vector) and k0 of both parities for pumping with circularly polarized laser light (A along the laser beam). In the triple sum (5) over k0 , k and ke , the range of k0 is limited by the condition 0 6 k0 6 2J0 , whereas the sum over k and ke is in principle infinite. However, in real calculations the sum converges due to diminishing contributions of the higher multipoles. From parity conservation it follows that the indices k and ke must have the same parity. This statement is valid both for process (1) and the more general process (8). It is essential only that a single electron is detected in the final state and, therefore, the identical quantum numbers of unobserved particles are contained in the bra states in (6). When exchanging primed and unprimed momenta the expression under summation in (6) changes to the complex conjugate with an additional phasefactor (−1)k0 . Therefore, Ak0 kke obey the relation Ak0 kke = (−1)k0 A∗k0 kke and the coefficients with even k0 are real whereas those with odd k0 are imaginary. Introducing the reduced statistical tensors of the initial state in the atomic frame Ak0 0 (γ0 J0 ) =

ρ k0 0 (γ0 J0 ) ρ 00 (γ0 J0 )

and defining the generalized anisotropy coefficients  X −1 Ak0 kke 0 2 ˆ = Ak0 kke |hγ J, lf jf : Jt kRkγ0 J0 , li ji : Jt i| βk0 kke = A000 lf jf li ji

(9)

(10)

γ J Jt

we can rewrite the general expression (5) in the form which exposes a dual origin of the angular anisotropy  X (0) 1 dσ (0) d2 σ (ϑe , ϕe ) = 1+ βk Pk (cos ϑe ) dE d 4π dE k>0  X +4π Ak0 0 (γ0 J0 )βk0 kke {Yk0 (ϑa , ϕa ) ⊗ Yk (ϑe , ϕe )}ke 0 . (11) k0 >0 kke

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Here Pk (cos ϑ) is a Legendre polynomial and dσdE is the cross section, integrated over the angle of ejection, which would be calculated if the initial state |γ0 J0 i was randomly oriented. The anisotropy coefficient βk(0) is a particular case of the generalized anisotropy coefficients (10): ˆ 0kk . βk(0) = kβ

(12)

The second term in the parentheses in (11) represents the anisotropy of the angular distribution due to pure collisional mechanism, i.e. the anisotropy from the unpolarized target. The last term includes a combined effect of the two factors: polarization of the laserexcited initial state and the anisotropy induced by the electron impact. The former factor is represented by the reduced statistical tensors Ak0 0 (γ0 J0 ), while the latter is represented by the generalized anisotropy coefficients βk0 kke . Among other things the representation (11) is useful due to the fundamental possibility of experimentally distinguishing the contribution from the three terms in (11). For example, only the last term depends on the polarization of the initial state and its contribution changes as a function of the target density because of depolarization due to the resonance trapping. For the isotropic target (k0 = 0) the last term in (11) vanishes and equation (11) reduces to the usual form describing the angular distribution of particles in reactions with an axis of symmetry (Blatt and Biedenharn 1952, Devons and Goldfarb 1957)   X (0) 1 dσ (0) d2 σ (0) βk Pk (cos ϑe ) . (13) = 1+ dE d 4π dE k>0 The angular distribution (13) is axially symmetric with respect to the initial electron beam direction and in general it shows the forward/backward asymmetry due to the contribution of partial waves of different parity. For the polarized target the axial symmetry is naturally broken, see (11). However, in a particular case of target polarization along the electron beam (ϑa = ϕa = 0) the angular distribution (11) reduces to the axially symmetric form, similar to (13), but with coefficients βk depending on a degree of the target polarization. The energy differential cross section of ionization for polarized target integrated over (0) dσ (ϑa ), does not coincide with dσdE because of the third term in (11), the ejection angle, dE dσ (ϑa ) follows from (11): which contains terms with k = 0. The general expression for the dE   X dσ dσ (0) 0 (ϑa ) = Ak0 0 (γ0 J0 )βk0 Pk0 (cos ϑa ) (14) 1+ dE dE k0 =2,4... where βk0 0 = kˆ0 βk0 0k0 .

(15)

In equation (14) k0 is restricted by the condition k0 6 2J0 . Therefore, for J0 < 1 the angle integrated cross section (14) is independent of ϑa . When the atomic symmetry axis A makes the magic angle with the incident beam, the second term in (14) vanishes for J0 < 2. The cross section (14), as a function of the target polarization direction, is rotationally symmetric about the incident electron beam and possesses reflection symmetry in the plane perpendicular to the beam. 2.2.2. Circular pumping dichroism in the angular distribution (CPDAD). As we will show below, the angular distributions of electrons ejected from the laser-excited state are different for the right and left circular polarized laser. This difference is a kind of circular dichroism and may be called the circular pumping dichroism in the angular distribution (CPDAD).

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To consider CPDAD it is convenient to write down (11) in an alternative form d2 σ (0) d2 σ (1) d2 σ = + dE d dE d dE d

(16)

where the first term on the right of equation (16) is independent of the target polarization and is given by (13). The additional second term is non-zero only for the polarized target. It corresponds to the third term in (11) and can be written in the form d2 σ (1) 1 dσ (0) X Ak 0 (γ0 J0 )βk0 kke = dE d 4π dE k0 >0 0 kke

X

×

q q (−1)q (2 − δ0q )(k0 q, k − q|ke 0)P˜k0 (cos ϑa )P˜k (cos ϑe )

q>0

 ×((1 + (−1) ) cos q(ϕa − ϕe ) + (1 − (−1) )i sin q(ϕa − ϕe )) k0

where

s q P˜k (cos ϑ) =

k0

2k + 1 (k − q)! q P (cos ϑ) 2 (k + q)! k

(17)

(18)

q

with Pk (cos ϑ) being the associated Legendre polynomials. When turning from right to left circular pumping the odd statistical tensors of the target change sign while the even ones remain unchanged. As a result, the difference of angular distributions, the CPDAD, contains only odd statistical tensors (orientation) of the pumped state:  2 +  2 − d σ d σ 1 dσ (0) X − = 8iAk0 0 (γ0 J0 )βk0 kke dE d dE d 4π dE k0 =1,3... ×

X

kke

q q (k0 q, k − q|ke 0)P˜k (cos ϑa )P˜k (cos ϑe ) sin q(ϕa − ϕe ).

(19)

q>0

As is clear from (19), the CPDAD disappears if all three vectors, pi , p1 and A, lie in one plane (ϕa − ϕe = πn; n = 0, ±1). In particular, the CPDAD is zero if any pair of the above vectors are collinear. The angle integrated cross section (14) remains unchanged when turning from right to dσ (ϑa ) does not show this type of circular left circular polarization of the laser beam, i.e. dE pumping dichroism, because only even tensors Ak0 (γ0 J0 ) (alignment) of the target affect the angle integrated cross section (14). 2.2.3. An example of a particular geometry of an experiment: perpendicular laser and electron beams. The general expression (5) can be specified for particular geometries which are of practical interest. For example, in different experiments with crossed laser and particle beams these beams are, as a rule, perpendicular to each other. Consider a linear or circular polarized laser beam directed perpendicular to the electron beam either in the reaction plane (figure 1(a)), or perpendicular to it (figure 1(b)). The reaction plane, xz, is defined by the directions of the incident and detected electrons. The four corresponding

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Figure 1. The coordinate frame and the reaction geometry for perpendicular laser and incident electron beams. The laser beam lies in the reaction plane (a), or is perpendicular to the reaction plane (b). The direction of linear polarization of the laser light EE is shown.

angular distributions have the general form (16) and differ only by the second term:  2 (1) d σ 1 dσ (0) X X = 2Ak0 0 (γ0 J0 )βk0 kke (2 − δ0q ) dE d lin,k 4π dE k0 =2,4... q=0,2... kke



2

d σ dE d

(1)

q q ×(k0 q, k − q|ke 0)P˜k (cos ϑe )(−1)q/2 P˜k0 (cos ϑa ),

= lin,⊥

1 dσ 4π dE

(0)

X X

2Ak0 0 (γ0 J0 )βk0 kke (2 − δ0q )

k0 =2,4... q>0 kke

×(k0 q, k − q|ke 0)P˜k (cos ϑe )(−1)q P˜k0 (cos ϑa ), q





d2 σ dE d

d2 σ dE d

(1)

= circ,k

(1)

1 dσ (0) 4π dE

X

q

X

circ,⊥

(21)

2Ak0 0 (γ0 J0 )βk0 kke (2 − δ0q )

k0 =2,4... q=0,2... kke

q q ×(k0 q, k − q|ke 0)P˜k (cos ϑe )P˜k0 (0),

=

(20)

1 dσ (0) X 4π dE k0 >0 kke

X

(22)

2Ak0 0 (γ0 J0 )βk0 kke (2 − δ0q )

q>0 k0 +q=0,2...

q q ×(k0 q, k − q|ke 0)P˜k (cos ϑe )iq P˜k0 (0).

(23)

The subscripts lin and circ denote a linear or circular polarized laser, while k and ⊥ denote the laser beam in the reaction plane (figure 1(a)) or perpendicular to it (figure 1(b)), respectively. In (21) the result is written for ϕe = ϕa = 0 or π; in the case ϕe = 0(π ), ϕa = π(0) the phase factor (−1)q should be omitted. As discussed above k + ke = even in (20)–(23). The normalized associated Legendre polynomials (18) with zero argument can be found from the relation s (k + q − 1)!!(k − q − 1)!! q (k−q)/2 ˆ . (24) P˜k (0) = k(−1) 2(k + q)!!(k − q)!! Equation (20) has already been used in the analysis of the first experiments by Dorn et al (1994).

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Table 1. The main features of the angular distributions of autoionization and Auger electrons ejected from the laser-excited atom in the general case and in the case of resonance ionizationa for perpendicular laser and incident electron beams (figure 1). Observable (equation)

Variable

General symmetry

d2 σ (0) dE d

(13)

ϑe

Axial symmetry about the z-axis

Reflection symmetry through the xy-plane

Linear pumping  2 (1) d σ (20) dE d

ϑe

Reflection symmetry through the z-axis in the xz-plane

Reflection symmetry through the x-axis in the xz-plane

ϑa

Reflection symmetry through the y- and z-axes in the yz-plane No symmetry

No

lin,k



(1)

d2 σ dE d lin,⊥

(21)

Circular pumping  2 (1) d σ (22) dE d 

circ,k

d2 σ dE d circ,⊥

(23)

Reflection symmetry through the origin in the xz-plane

ϑa

Reflection symmetry through the origin in the xz-plane

No

ϑe

Reflection symmetry through the z-axis in the xz-plane

Reflection symmetry through the x-axis in the xz-plane

ϑe

No symmetry

Reflection symmetry through the origin in the xz-plane

ϑe

No symmetry

Reflection symmetry through the origin in the xz-plane

(1)

CPDAD (25) a

ϑe

Additional symmetry in the resonance process

Additional symmetries can exist for some special values of ϑe or ϑa (see text).

Expression (22) for the laser beam in the reaction plane does not show any CPDAD as the orientation of the target atom (k0 = odd) does not affect the angular distribution. This is in accordance with the above general statement about vanishing CPDAD when the three vectors, pi , p1 and A, lie in one plane. However, as follows from (23) the CPDAD exists, in general, for the case of the laser beam perpendicular to the reaction plane:  2 (+)  2 (−) d σ 1 dσ (0) X X d σ − = 8Ak0 0 (γ0 J0 )βk0 kke dE d ⊥ dE d ⊥ 4π dE k0 =1,3... q=1,3... kke

q q ×(k0 q, k − q|ke 0)P˜k (cos ϑe )iq P˜k0 (0).

(25)

It vanishes at ϑe = 0 or π (pi , p1 and A lie in one plane), as a linear combination of the associated Legendre polynomials with odd projections. Main features of the angular distributions (13), (20)–(23) and the CPDAD (25) are presented in the left column of table 1. All features discussed above in section 2.2, as well as all the equations are valid also for the general process (8) of multiple ionization. 2.2.4. Experiments with fixed electron detector and rotating laser polarization. Equation (5) was considered above as the angular distribution of the electrons ionized from the target at a fixed direction of the polarization of the pumping laser. Alternatively, it describes the dependence of the intensity of the ejected electrons at a fixed detector position (ϑe , ϕe ) on the angles of rotation of the atomic polarization axis (ϑa , ϕa ). In general, the two modes

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of experiments provide different information about the ionization process. In the rotating detector mode, the angular distribution (5) is modulated by the spherical harmonics with the rank up to k which can reach high values in accordance with the triangle rules contained in (6). The effective range of k (as well as of ke ) may be restricted only by dynamical reasons, for example, by decreasing the transition matrix elements with increasing the partial angular momenta l1 or li . In contrast, the angular distribution in the fixed detector mode is modulated by the spherical harmonics with the rank up to k0 6 2J0 which is restricted by the total angular momentum of the initial state independent of the dynamics of ionization. As in the case of photoionization of the polarized targets, experiments in the fixed detector mode can show some important advantages, because they are not sensitive to the possible misalignments of the electron spectrometer with respect to the interaction region and also do not suffer from some other distorting influences (Wedowski et al 1995). 3. Dynamics of the ionization process 3.1. Autoionization in the two-step approximation Consider the resonance ionization process via a single isolated autoionizing state, |γr Jr i, with total angular momentum, Jr , and definite parity: ei + A(γ0 J0 ) → ef + A∗ (γr Jr ) → ef + e1 + A+ (γ1 J1 ).

(26)

Within the two-step approximation the direct (nonresonant) transition to the final state is neglected and the first step (excitation) ei + A(γ0 J0 ) → ef + A∗ (γr Jr )

(27)

as well as the second step (spontaneous decay of the autoionizing state) A∗ (γr Jr ) → e1 + A+ (γ1 J1 )

(28)

are treated as two independent stages of the ionization process. To calculate the angular distribution of the autoionization electrons one can use the general formulae (2)–(6) specifying them for the case under consideration by fixing the quantum numbers J = J 0 = Jr and factorizing, according to the two-step approximation, ˆ the matrix elements of the transition operator R: ˆ 0 J0 , li ji : Jt i hγ Jr , lf jf : Jt kRkγ = hγ1 J1 , l1 j1 : Jr kVˆ kγr Jr ihγr Jr , lf jf : Jt kTˆ kγ0 J0 , li ji : Jt igr (E).

(29)

Here hγ1 J1 , l1 j1 : Jr kVˆ kγr Jr i is the reduced matrix element describing the autoionization of the excited atom A∗ (γr Jr ) into the channel γ Jr = {γ1 J1 , l1 j1 : Jr }, while the second factor is the excitation amplitude for the process (27). The energy dependence of the matrix element is described by the resonant factor gr (E) gr (E) =

1 E − Er + i0r /2

(30)

where Er and 0r are the position and the width of the resonance, respectively. Substituting (29) in (6) and integrating the cross section (5) over the energy we obtain the angular distribution of electrons ejected from the autoionizing state: Z dσ d2 σ (31) (ϑe , ϕe ) = (ϑe , ϕe ) dE. d dE d

Autoionization by electrons from laser-excited atoms

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Within the considered approximation it is convenient to present the general expression for the dσ (ϑe , ϕe ) in an alternative form explicitly demonstrating the two-step angular distribution d character of the ionization process by introducing the statistical tensors ρkq (γr Jr ; ϑa , ϕa ) of the autoionizing state |γr Jr i and electron decay parameters αkJr →J1 characterizing the anisotropy of electron emission (Berezhko et al 1978) √   X σ (ϑa , ϕa ) 4π Jr →J1 X dσ Akq (γr Jr ; ϑa , ϕa )Ykq (ϑe , ϕe ) . (32) (ϑe , ϕe ) = 1+ αk d 4π kˆ q k=2,4... Here σ (ϑa , ϕa ) is the total cross section for the ionization of the target atom to the channel e1 + A+ (γ1 J1 ) via the excitation and decay of the autoionizing state |γr Jr i integrated over the angular distributions of the scattered and ejected electrons, while Akq (γr Jr ; ϑa , ϕa ) =

ρkq (γr Jr ; ϑa , ϕa ) ρ00 (γr Jr ; ϑa , ϕa )

(33)

are the reduced statistical tensors which describe the anisotropy of the autoionizing state. The explicitly shown arguments (ϑa , ϕa ) remind us that in general the cross section and statistical tensors depend on the relative orientation of the incident electron beam and polarization axis of the target atom. The statistical tensors ρkq (γr Jr ; ϑa , ϕa ) are determined by the equation similar to (4) ˆ They may be presented in the following form: where the operator Tˆ is used instead of R. X ρ k0 0 (γ0 J0 )Yk∗0 q (ϑa , ϕa )kˆe (k0 q, ke 0|kq)Bk0 kke , (34) ρkq (γr Jr ; ϑa , ϕa ) = (4π)−1/2 k0 ke

where Bk0 kke = Jˆ0 Jˆr

X lf jf li li0 ji ji0 Jt Jt0

0 0 (−1)jf −ji +Jt +Jr −1/2 lˆi lˆi0 jˆi jˆi0 Jˆt Jˆt0

) ji Jt ji0 Jt0 ke k ∗ ×hγr Jr , lf jf : Jt kTˆ kγ0 J0 , li ji : Jt ihγr Jr , lf jf : Jt0 kTˆ kγ0 J0 , li0 ji0 : Jt0 i . 
ji × li 0, li0 0|ke 0 li0

li ji0

1 2 ke



Jt Jr

Jr Jt0

jf k

( J0 J0 k0

(35)

The total cross section σ (ϑa , ϕa ) for the polarized target is proportional to zero-rank statistical tensor ρ00 (γr Jr ; ϑa , ϕa ) and may be presented in the form similar to (14):   X 0Jr →J1 (0) σ (ϑa , ϕa ) = σJ0 →Jr Ak0 0 (γ0 J0 )δk0 Pk0 (cos ϑa ) . (36) 1+ 0r k0 >0 is the total cross section for the excitation of the autoionizing state, |γr Jr i, Here σJ(0) 0 →Jr from the isotropic initial state, |γ0 J0 i, and δk0 is expressed in terms of coefficients Bk0 kke in a way similar to βk and βk0 , see (10), (12) and (15):  X −1 Bk 0k |hγr Jr , lf jf : Jt kTˆ kγ0 J0 , li ji : Jt i|2 . (37) δk0 = kˆ0 0 0 = kˆ0 Bk0 0k0 B000 lf jf li ji Jt The decay anisotropy parameters, αkJr →J1 , in (32) describe only the second stage of the two-step process and do not depend on anything concerning excitation of the autoionizing state. They are calculated taking into account interference between different channels of the

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final system by the following expression (Berezhko and Kabachnik 1977, Kabachnik and Sazhina 1984)    X
j1 Jr J1 l1 j1 12 αkJr →J1 = (−1)Jr +J1 −1/2 Jˆr lˆ1 lˆ10 jˆ1 jˆ10 l1 0, l10 0|k0 Jr j10 k j10 l10 k 0 0 l1 l1 j1 j1

∗ ×hγ1 J1 , l1 j1 : Jr kVˆ kγr Jr ihγ1 J1 , l10 j10 : Jr kVˆ kγr Jr i X −1 × |hγ1 J1 , l1 j1 : Jr kVˆ kγr Jr i|2 .

(38)

l1 j1

To analyse general properties of the angular distribution in the two-step approximation, note first that the number of parameters Ak0 kke in the general expression (5) as well as the number of terms in sums over k0 , k and ke in (32) and (34) are now finite. This is due to the fact that the intermediate state |γr Jr i has definite angular momentum and therefore the rank of the statistical tensors describing the state is limited by the relation (k 6 2Jr ). This immediately limits the number of possible sets of {k0 kke } according to the triangle rules contained in (35) and (38). Moreover, due to parity conservation in the excitation and decay processes both k and ke are only even. This restriction brings forth specific properties of the angular distributions for the autoionization electrons. Table 1 summarizes the main symmetries in the angular distributions of autoionization and Auger electrons in the general case (left column) and gives new symmetries which appear due to the two-step character of the process (right 2 σ (1) )lin,k in a case of perpendicular electron column). For example, the angular distribution ( dEd d and linearly polarized laser beams (figure 1(a)), being generally symmetric as a function of ϑe relative to the z-axis in the xz-plane, is also symmetric about the x-axis in this plane when the ionization proceeds via a single isolated resonance. Symmetries additional to those indicated in table 1 can appear when the angles ϑa or ϑe take particular values. Consider, as an example, the linear pumping with the laser beam perpendicular to the reaction xz-plane (figure 1(b)) and ϑe = 90◦ . Then the intensity of ejected electrons as a function of ϑa is symmetric in the xz-plane about the x-axis for resonance ionization, while in the general case this symmetry is broken. In addition, there may be restrictions imposed by the model, such as configuration selection rules for the electron-impact excitation of the transitions between atomic states. One such example will be considered in section 4. The symmetry properties which result from the two-step character of the process can serve as a sensitive tool for detecting contributions from final states of different parity: either a contribution from the direct ionization or that from nearly lying resonances of the parity opposite to the parity of the considered autoionizing state. 3.2. Auger decay in the two-step approximation We now consider the angular distribution of Auger electrons, e1 , in the double ionization of an atom by electron impact: ei + A(γ0 J0 ) → ef + A2+ (γ1 J1 ) + e1 + e2

(39)

treating this process as a pure two-step one. The first step is electron-impact ionization of the laser-excited target atom leading to production of the single-charged ion, A+ , in an isolated quasistationary state |γr Jr i: ei + A(γ0 J0 ) → ef + A+∗ (γr Jr ) + e2 .

(40)

Autoionization by electrons from laser-excited atoms

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The second step is the spontaneous electronic decay of this state considered as a process independent of the electron-impact ionization stage: A+∗ (γr Jr ) → A2+ (γ1 J1 ) + e1 .

(41)

Any contribution from the direct double-ionization process is neglected. The concept of the two-step process is the basis for the majority of theoretical investigations of the Auger process induced by fast particles (Mehlhorn 1985, 1990). To find the angular distribution of the electron e1 in the double ionization (39) we can follow the procedure described in section 2.1 taking into account that there are two ejected electrons in continuum one of which is not observed. Having in mind an application of the resulting expression to the case of the two-step Auger process we use the following coupling scheme for the angular momenta: ((J1 + j1 )Jr + j2 )J + jf = Jt . Factorizing the amplitude in a way similar to (29) we obtain the angular distribution of Auger electrons which, naturally, can be written in the same form (32) where the anisotropy coefficient αkJr →J1 (see (38)) relates to the particular Auger transition. As in the case of autoionization both the total yield of Auger electrons and the alignment coefficients determining the angular distribution depend on the orientation of the target polarization axis with respect to the incident electron beam. The statistical tensors of the Auger state A+∗ (γr Jr ) can be obtained from the statistical tensors of the system A+∗ (γr Jr ) + e2 by summation and integration over the variables of the electron e2 . Straightforward calculations of the statistical tensors ρkq (γr Jr ; ϑa , ϕa ) give the same expression (34) as for the autoionizing states but with different dynamical coefficients Bk0 kke : Z X 0 0 0 ˆ ˆ Bk0 kke = J0 Jr dε2 (−1)jf −ji +j2 +J +J +Jt +Jr −1/2 lˆi lˆi0 jˆi jˆi0 JˆJˆ 0 Jˆt Jˆt0 lf jf li li0 ji ji0 j2 J J 0 Jt Jt0

×


ji li0

li 0, li0 0|ke 0

li ji0

1 2 ke



Jt J0

J Jt0

jf k



J Jr

Jr J0

j2 k

( J0 J0 k0

ji ji0 ke

Jt Jt0 k

)

×h(γr Jr , j2 )J, lf jf : Jt kTˆ kγ0 J0 , li ji : Jt i ∗

×h(γr Jr , j2 )J 0 , lf jf : Jt0 kTˆ kγ0 J0 , li0 ji0 : Jt0 i .

(42)

Here h(γr Jr , j2 )J, lf jf : Jt kTˆ kγ0 J0 , li ji : Jt i is the amplitude of ionization, j2 and ε2 are the total angular momentum and energy of the non-observed electron e2 . 3.3. The high-energy limit (PWBA) At high energy of the incoming electrons the dynamics of the inelastic scattering turns out to be much simpler then in the low-energy region because neither prior- nor post-collision electron–atom interactions nor exchange scattering play a role in the excitation or ionization of the target atom. In this subsection we consider the dynamical coefficients Ak0 kke in the general formula (5) for the process (1) in the plane-wave Born approximation (PWBA). The corresponding derivation starting from the general expression (6) is laborious. It is simpler not to use the partial wave expansion for the electrons ei and ef from the very beginning but to substitute plane waves for the wavefunctions of the initial and scattered electrons in the matrix elements of the Coulomb interaction in expression (4) and then expand the exponential factor exp(iQr) in multipoles (Q = pi − pf is a momentum transfer vector

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with the direction ϑQ , ϕQ ). Then the matrix elements in (4) take the form X ˆ 0 J0 M0 , pi µi i = 8π δµi µf hγ J M, pf µf |R|γ iλ Y∗λµ (ϑQ , ϕQ )(J0 M0 , λµ|J M)Mλγ J (Q) 2 ˆ QJ λµ (43) where

P

rn )kγ0 J0 i n jλ (Qrn )Yλ (ˆ

Mλγ J (Q) = hγ J k

(44)

and jλ (x) is the spherical Bessel function. The summation over n in (44) runs over all atomic electrons. The normalization constant in (43) is chosen in such a way that the cross for the transition from the isotropic initial state |γ0 J0 i into a state with fixed section σJ(0) 0 →J γ J is expressed in terms of the PWBA amplitudes (44) as follows Z pf X 1 (0) ˆ 0 J0 M0 , pi µi i|2 σJ0 →J = df |hγ J M, pf µf |R|γ 2(2J0 + 1) pi µi µf =

32π pi2 (2J0

2

+ 1)

Z

M0 M Qmax

Qmin

dQ X |Mλγ J (Q)|2 . Q3 λ

(45)

Substituting the amplitudes (43) into (4) and summing over all projections the following expression for the statistical tensors (4) can be obtained 128π 5/2 X ρ (γ0 J0 )Y∗k0 q (ϑa , ϕa )kˆe (k0 q, ke 0|kq)Ek0 kke (γ J, γ 0 J 0 ) pi pf k0 ke k0 0

ρkq (γ J, γ 0 J 0 ) =

where the part containing the ionization dynamics has ( J0 λ X
0 iλ+λ λˆ λˆ0 λ0, λ0 0|ke 0 J0 λ0 Ek0 kke (γ J, γ 0 J 0 ) = λλ0 k0 ke

the form ) J J 0 Kke (λγ J, λ0 γ 0 J 0 ). k

(46)

(47)

The kernel Kke (λγ J, λ0 γ 0 J 0 ) is given by Z Qmax dQ Mλγ J (Q)Mλ∗0 γ 0 J 0 (Q)Pke (cos ϑQ ) (48) Kke (λγ J, λ0 γ 0 J 0 ) = 3 Qmin Q q with cos ϑQ = (pi2 + Q2 − pf2 )/2pi Q; Qmax = p ± pi2 − 1. The energy loss 1 is given i min by 1 = p12 + (Eγ1 J1 − Eγ0 J0 ) where the term in parentheses is the ionization threshold for the ion state γ1 J1 . The value of pf is determined by the energy conservation as discussed above. Using (46) and (3) one can write down the differential cross section (2) in the form (5) where the dynamical coefficients Ak0 kke which determine the angular distribution of the ionized electrons e1 in the process (1) are of the form X0
(−1)J +J1 +ke −1/2 JˆJˆ 0 lˆ1 lˆ0 1 jˆ1 jˆ0 1 l1 0, l10 0|k0 Ak0 kke = γ γ 0J J 0

 ×

j1 l10

l1 j10

1 2

k



J0 j1

j10 J

 J1 Ek0 kke (γ J, γ 0 J 0 ) k

(49)

while the normalization coefficient in (5) in the PWBA takes the value c = 32π 2 pi−2 . Evaluation of the cross section (5) within the PWBA using (47)–(49) is much simpler than in the general case, when the equation (6) must be used. The two-step approximation further simplifies the calculations of the angular distributions of autoionization and Auger

Autoionization by electrons from laser-excited atoms

1283

electrons. Within the two-step approximation the statistical tensors (34) of an isolated autoionizing state |γr Jr i can be presented in PWBA as follows: 128π 5/2 X ρkq (γr Jr ; ϑa , ϕa ) = ρ (γ0 J0 )Y∗k0 q (ϑa , ϕa )kˆe (k0 q, ke 0|kq)Ek0 kke (γr Jr , γr Jr ) pi pf k0 ke k0 0 (50) where the reduced matrix elements Mλγr Jr (Q), calculated with the wavefunction of the autoionizing state, are used to obtain the kernel Kke (λγr Jr , λ0 γr Jr ), equation (48), and parameters Ek0 kke (γr Jr , γr Jr ), equation (47). Note that now both indices k and ke in (50) are even. Similar simplification takes place in the case of the Auger decay. Here, for the statistical tensors of the Auger state A+∗ (γr Jr ), one can use the same expression (50) but the dynamical part is calculated according to the equation: X
0 0 Ek0 kke (γr Jr , γr Jr ) = (−1)j2 +Jr +J +k iλ+λ λˆ λˆ 0 JˆJˆ0 λ0, λ0 0|ke 0 

J × Jr

λλ0 J J 0 l2 j2

Jr J0

j2 k

( J0 J0 k0

λ λ0 ke

J J0 k

)Z

dε2 Kke (λγ J, λ0 γ 0 J 0 )

(51)

where the kernel, Kke (λγ J, λ0 γ 0 J 0 ), is defined as above by equation (48) but the index, γ = {γr Jr , ε2 l2 j2 }, now includes the characteristics of the ion, A+∗ (γr Jr ), and of the unobserved electron, e2 (ε2 , l2 , j2 ), its energy, orbital and total angular momenta, respectively. The reduced matrix elements, Mλγ J (Q), for the Auger case are determined by P Mλγ J (Q) = hγr Jr , ε2 l2 j2 : J k n jλ (Qrn )Yλ (ˆrn )kγ0 J0 i (52) where in the final state one of the electrons is in continuum. For a particular case J0 = 0 expression (50) with (51) and (52) is identical within the normalization constant with that obtained earlier (Berezhko et al 1978). 4. Electron impact resonance ionization of the laser-excited sodium atom The study of electron scattering from laser excited and aligned sodium atoms (Dorn et al 1994) was the first experiment in which the angular distribution of electrons from atomic autoionizing states excited by crossed electron and laser beams was investigated. We use this case as an example to illustrate the general theoretical considerations made in the preceding sections. Consider the autoionizing states 2p5 3s3p 2 D5/2 (Ee = 28.36 eV) and 2 D3/2 (Ee = 28.31 eV) in a sodium atom excited by electron impact from the intermediate laserpumped discrete state 2p6 3p 2 P3/2 (figure 2). As the contribution from the direct ionization Na(2p6 3p 2 P3/2 ) → Na+ (2p6 1 S0 )+ e1 in the vicinity of the autoionizing states is negligible, we treat the resonant electron-impact ionization as a pure excitation-autoionization two-step process. In the case of excitation of the autoionizing state 2p5 3s3p 2 D5/2 the number of sets {k0 kke } is restricted by the conditions k0 6 3, k 6 4, ke 6 6, where |k0 − k| 6 ke 6 k0 + k and both k and ke are even. Therefore, in general the angular distribution (5) is characterized by 17 coefficients Ak0 kke ; 10 of them with k0 = 0, 2 are real, while seven others with k0 = 1, 3 are imaginary, or, equivalently, by 16 generalized anisotropy parameters βk0 kke (0) and the integrated cross section σ3/2→5/2 . Seven imaginary coefficients affect the angular

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Figure 2. The scheme of autoionization induced by electron impact from the laser-excited sodium atom.

distributions only when the statistical tensors, Ak0 0 (γ0 J0 ), with odd ranks are non-zero, e.g. for circular pumping. For excitation of the state 2p5 3s3p 2 D3/2 the number of sets {k0 kke } is restricted by the conditions k0 6 3, k 6 2, ke 6 4 and in this case the angular distribution (5) is characterized by nine coefficients Ak0 kke ; six of them with k0 = 0, 2 are real, while three others with k0 = 1, 3 are imaginary, or, equivalently, by eight generalized anisotropy (0) . parameters βk0 kke and the integrated cross section σ3/2→3/2 Further arguments related to the dynamics of the process together with structural features of the initial and final atomic states may give additional restrictions for the number of independent dynamical parameters. For example, neglecting a weak octupole contribution to the excitation 2p6 3p 2 P3/2 → 2p5 3s3p 2 D5/2 (this contribution is caused by the admixture of configurations with higher orbital angular momenta of the electrons to the wavefunctions of the initial and autoionizing states) one has ke 6 2. This ‘dipole excitation’ approximation reduces the number of the anisotropy parameters from 16 to 8, namely: β022 , β122 , β220 , β202 , β222 , β242 , β322 , β342 and only five of them with k0 = 0 and two contribute to the angular distribution in the case of linear pumping. In the high-energy (PWBA) limit complete information on the resonance ionization via the excitation 2p6 3p 2 P3/2 → 2p5 3s3p 2 D5/2 is contained in two kernels only, K0 (1, D5/2 ; 1, D5/2 ) ≡ K0 and K2 (1, D5/2 ; 1, D5/2 ) ≡ K2 (see (48)), since the autoionization proceeds via only one channel |ε1 , l1 , j1 i = |ε1 , d5/2 i and the angular distribution of the autoionization electrons does not depend on the decay matrix elements. Therefore, only the ratio κ2 = K2 /K0

(53)

determines the angular distribution. (Note that the quantity κ2 is a well known characteristic of momentum transfer distribution in the collision, which is of interest in the studies of polarization of atomic line radiation excited by electron impact (Heddle 1983 and references therein). In contrast to the studies of radiation from discrete states, measurements of κ2 for excitation of the autoionizing states which decay due to the Coulomb interaction, are free from cascade contributions. The higher-order parameters of the momentum transfer distribution κn with n > 2 are also open for inspection in the autoionization studies.) In the geometry of experiments by Dorn et al (1994), when the linear polarization of the laser light is directed along the incident electron beam, the angular distribution of electrons

Autoionization by electrons from laser-excited atoms

1285

from the autoionizing state 2p5 3s3p 2 D5/2 takes the form σ (ϑa = 0◦ ) dσ (ϑe , ϕe ) = (1 + β2 P2 (cos ϑe ) + β4 P4 (cos ϑe )) d 4π which is a particular case of the general formula (32).

(54)

(a)

(b)

Figure 3. Energy dependence of the anisotropy parameters β2 , β2(0) (a) and β4 , β4(0) (b) for the angular distribution of the ejected electrons from the autoionizing state 2p5 3s3p 2 D5/2 . Full curve, DWBA; long broken curve, DWBA without exchange; short broken curve, PWBA. Experimental results for the polarized target (β2 and β4 ) are from Dorn et al (1994).

Figure 3 shows our earlier PWBA calculations (Dorn et al 1994) of β2 , β2(0) and β4 , for the aligned state with A20 = −0.5 (this initial alignment of the state 2p6 3p 2 P3/2 corresponds to the experiment) and for the unpolarized state together with recent calculations in the distorted-wave Born approximation (DWBA) (Balashov et al 1996). Now, using our general formulation presented for the first time above, we can analyse these calculations together. In the high-energy region the behaviour of the anisotropy parameters as a function of energy is easily explained in the above described ‘dipole excitation’ model, since all the

β4(0)

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parameters are related to each other via the value of κ2 : β2(0) = 45 κ2 β4(0) = 0 β2 = 45 (1 − 15 A20 κ2 )−1 (κ2 − A20 + 27 A20 κ2 )

(55)

β4 = − 36 (1 − 15 A20 κ2 )−1 A20 κ2 . 35 According to (55) and (53) the parameters β2(0) and β4 should cross zero at the same energy, where the kernel, K2 , turns to zero. At this energy the value of β2 is determined only by the alignment A20 of the initial laser-excited atom. The PWBA results show with high accuracy all these properties. This illustrates the validity of the ’dipole excitation’ model for the energies at least higher than several hundreds of eV and a negligible contribution of the octupole part of the interaction. The DWBA results shown in figure 3 illustrate the role of the distortion of the incoming and scattered electron wavefunctions as well as the role of the exchange scattering. The scattering amplitudes including both the direct and exchange terms were calculated in the standard first-order DWBA (Itikawa 1986) with the multiconfiguration intermediate-coupling wavefunctions of the autoionizing state (Zatsarinny and Bandurina 1993). The particular version of the DWBA used in the present work, namely the model with the electron density of the autoionizing state, has been described in detail by Matterstock et al (1995). One can see that at the energy of the incoming electrons higher than 150–200 eV the DWBA and PWBA calculations for the anisotropy coefficients βk (E) are almost indistinguishable (the same is true also for the corresponding total cross sections). At lower energies the distortion and exchange effects included independently are rather strong but they turn out to partly compensate each other for being included together (note, however, that one should consider with caution both the first-order PWBA and DWBA calculations at energies not much higher than the excitation energy of the autoionizing states considered). Balashov et al (1995) considered another version of experiments on excitation and decay of autoionizing states in the sodium atom using crossed laser and electron beams. We show in figure 4 DWBA calculations performed for the geometry with perpendicular electron and laser beams (see figure 1(b)) where the detector of ejected electrons is fixed while the direction of the laser polarization vector is rotated in the reaction plane. Starting from expression (32) one can find that in this case the intensity of the ejected electrons as a function of the angle ϕa = ϑa − ϑe between the polarization vector and the ejected electron beam is modulated according to the formula: dσ⊥ (ϕa ) = W (ϑe )(1 + C(ϑe ) cos 2(ϕa − δ(ϑe ))) d

(56)

where C(ϑe ) and δ(ϑe ) determine the depth of modulation and the position of extrema in the ϕa -dependence of the intensity of the ejected electrons. Figure 4 shows that generally behaviour of these parameters as functions of the ejected angle ϑe is rather stable in the energy range 50–100 eV. Note a large difference in the ϑe dependence of the modulation parameter C(ϑe ) for the autoionizing states 2p5 3s3p 2 D5/2 and 2p5 3s3p 2 D3/2 belonging to one doublet. Such sensitivity of the parameter C(θe ) to the quantum numbers of the resonances can be used for identification of the autoionizing states excited from laser-excited atoms.

Autoionization by electrons from laser-excited atoms

1287

Figure 4. Parameters W (ϑe ), C(ϑe ) and δ(ϑe ) in (56) describing the intensity of the ejected electrons from autoionizing states 2p5 3s3p 2 D5/2 (left column) and 2p5 3s3p 2 D3/2 (right column). The initial alignment of the laser-excited 2p6 3p 2 P3/2 state is A20 = −0.5. Broken curve, the incident electron energy E = 50 eV (Balashov et al 1996); full curve, E = 100 eV (present calculations).

5. Angular correlations between the ejected and scattered electrons The angular distribution of the ‘ejected’ electrons e1 (p1 ) in process (1), measured in coincidence with the ‘scattered’ electrons ef (pf ), is described by the triple differential cross section (TDC) X d3 σ ∗ ρkq (γ J, γ 0 J 0 ; pf )εkq (γ J, γ 0 J 0 ) (57) = df d1 dE 0 0 γγ JJ kq

where the statistical tensors ρkq (γ J, γ 0 J 0 ; pf ) of the system (e1 + A+ ) are ‘tagged’ by the value of the scattered electron momentum pf . They are calculated using an equation similar to (4) for the double differential cross section (DDC) but without an integration

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over the angle of scattering. No changes are to be introduced into the efficiency tensors εkq (γ J, γ 0 J 0 ) as they do not depend on the mechanism of formation of the system (e1 + A+ ). To proceed further in calculating the TDC one can expand again the initial and final electron wavefunctions |pi µi i and |pf µf i in partial waves in order to obtain the statistical tensors ρkq (γ J, γ 0 J 0 ; pf ) in a form of bilinear combination of the matrix elements hγ J, εf lf jf : Jt kRkγ0 J0 , li ji : Jt i of the transition operator R. We have seen above that the polarization properties of the final system (e1 + A+ ) and, hence, the anisotropy of the angular distribution of the ejected autoionization or Auger electrons have their origin in the features of the laser pumping and, on the other hand, in the dynamics of the electron scattering from the polarized atom. Detecting the scattered electron one changes the symmetry condition of formation of this system. As a result a number of selection rules determining the relation between the statistical tensors ρkq (γ J, γ 0 J 0 ) of the final system and the statistical tensors ρk0 q0 (γ0 J0 ) of the laser-excited target atom which play a very important role in the DDC case do not work in the TDC case. Indeed, the integration over pf and the summation over µi and µf in the non-coincidence case mean that the system (e1 + A+ ) is formed in conditions of axial symmetry with respect to the direction of the electron beam. As we saw from (2)–(5), this results in a strong selection rule q = q0 which simplifies the relation of ρkq (γ J, γ 0 J 0 ) and ρk0 q0 (γ0 J0 ). There is no such selection rule in the relation of the statistical tensors ρkq (γ J, γ 0 J 0 , pf ) and ρk0 q0 (γ0 J0 ) (if only pf is not collinear with pi ). It gives an opportunity to use the coincidence (e, 2e) method for more detailed studies of the polarization properties of the decaying state and of the electron-impact excitation amplitudes. To illustrate this statement we take again the transition Na(2p6 3p 2 P3/2 ) → Na(2p5 3s3p 2 D5/2 ) → Na+ (2p6 1 S0 ) + e1 which was considered in the preceding part for the non-coincidence experiment. Figure 5 shows the angular distribution (TDC) of the ejected electrons (57) from the autoioinizing state 2p5 3s3p 2 D5/2 calculated for the coplanar (e, 2e) experiment at E = 150 eV with the laser-excited sodium atom in the state 2p6 3p 2 P3/2 aligned (A20 = −0.5) along the incoming electron beam (left side) and perpendicular to the scattering plane (right side). The upper part of figure 5 corresponds to the scattering angle ϑf = 10◦ (present calculations), while the lower part corresponds to ϑf = 24◦ (Balashov et al 1996). Note a complicated shape of the TDC in the left-hand side case at ϑf = 24◦ which indicates a large contribution of the statistical tensor ρkq (γr Jr ) of the rank k = 4 introduced to the laser-oriented target atom with k0 6 2 by the electron impact. At small angle ϑf = 10◦ the shape of the angular correlation function is about the same when calculated within the PWBA and DWBA, the difference between them rising with increasing scattering angle. The angular correlation function calculated within the DWBA, shown in the right-hand side of figure 5 at ϑf = 24◦ , demonstrates a very well pronounced breaking down of the symmetry with respect to the transferred momentum vector Q due to the distortion effects in the entrance and exit scattering channels of the process Na(2p6 3p 2 P3/2 ) + ei → Na(2p5 3s3p 2 D5/2 ) + ef .

6. Conclusions We have developed a formalism for describing the angular distribution of electrons ejected by electron impact from laser-excited and polarized atoms with a special reference to the single- and double-resonance ionization in crossed laser and fast electron beams. A general parametrization of the angular distribution of autoionization and Auger electrons is suggested which accounts for two sources of anisotropy in the electron emission. One of its parts is of

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Figure 5. The angular distribution of electrons ejected from the autoionizing state 2p5 3s3p 2 D5/2 in coplanar (e, 2e) geometry at E = 150 eV. The laser beam is linear polarized along the incident electron beam (left side) or perpendicular to the scattering plane (right side) (A20 = −0.5). Upper part of the figure, ϑf = 10◦ . Lower part of the figure, ϑf = 24◦ . Full curve, DWBA; broken curve, PWBA. The direction of the momentum transfer, Q, is indicated.

pure collisional nature and is determined by the anisotropy of the electron–atom interaction. Another part is due to a combined effect of the electron-impact induced anisotropy and of the initial target polarization produced by the laser photons. We have used the suggested parametrization for analysis of the possible arrangements of the experiment with the aim to help experimentalists to choose the experimental conditions most sensitive to the particular effect under investigation. In general the number of parameters which determine the angular

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distribution of ejected electrons is large and the distribution itself is complicated. However, in many cases this number can be considerably reduced by the choice of the proper geometry of the experiment. As we have shown, in some particular cases (strong isolated resonance, high-energy initial electrons) the angular distribution of autoionization electrons is virtually determined by a few dynamical parameters which can be easily obtained from the experiment and compared with theoretical calculations. We have demonstrated the usefulness of the suggested formalism in the analysis of the particular experimental data for electron impact ionization of sodium atoms. This and other examples in this paper illustrate, on the other hand, the sensitivity of the considered experiments to the details of the dynamics of the electron–atom interaction. In this respect the proposed coincidence (e, 2e) experiments as well as spin polarization measurements of the autoionization electrons could be even more informative. Detailed analysis of such experiments will be given in the forthcoming publications. Acknowledgments We are grateful to Professor W Mehlhorn and Dr A Dorn for useful discussions and E Golokhov for assistance in performing calculations. The work was supported by the International Science Foundation (grants no M 27000 and M 27300) and the Russian National Programme ‘Universities of Russia’. References Andersen N, Gallagher J W and Hertel I V 1988 Phys. Rep. 165 1 Baier S, Grum-Grzhimailo A N and Kabachnik N M 1994 J. Phys. B: At. Mol. Opt. Phys. 27 3363 Balashov V V 1993 J. Physique Coll. IV 3 C6 59 Balashov V V, Golokhov E I and Grum-Grzhimailo A N 1995 19th Int. Conf. on Physics of Electronic and Atomic Collisions (Whistler, Canada) Abstracts p 216 ——1996 Phys. Lett. 222A 81 Balashov V V and Grum-Grzhimailo A N 1986 Proc. 3rd All-Union Workshop on the Autoionization Phenomena in Atoms (Moscow: Moscow University Press) p 46 Balashov V V, Grum-Grzhimailo A N, Dolinov V K, Korenman G Ya, Krementsova Yu N, Smirnov Yu F and Yudin N P 1984 Theoretical Practicum in Nuclear and Atomic Physics (Moscow: Energoizdat) Balashov V V, Grum-Grzhimailo A N and Zhadamba B 1988 Opt. Spectrosc. (USSR) 65 529 Berezhko E G and Kabachnik N M 1977 J. Phys. B: At. Mol. Phys. 10 2467 Berezhko E G, Kabachnik N M and Sizov V V 1978 J. Phys. B: At. Mol. Phys. 11 1819 Blatt J M and Biedenharn L C 1952 Rev. Mod. Phys. 24 258 Blum K 1981 Density Matrix Theory and Applications (New York: Plenum) Devons S and Goldfarb L J 1957 Handbuch der Physik vol 42 ed S Fl¨ugge (Berlin: Springer) p 362 Dorn A, Nienhaus J, Wetzstein M, Winnewisser C, Mehlhorn W, Balashov V V, Grum-Grzhimailo A N, Kabachnik N M and Zatsarinny O I 1994 J. Phys. B: At. Mol. Opt. Phys. 27 L529 Dorn A, Nienhaus J, Wetzstein M, Winnewisser C, Eichmann U, Sandner W and Mehlhorn W 1995a J. Phys. B: At. Mol. Opt. Phys. 28 L225 Dorn A, Winnewisser C, Eichmann U, Grum-Grzhimailo A N, Kabachnik N M, Sandner W and Mehlhorn W 1993 16th Int. Conf. on X-ray and Inner-shell Processes (Debrecen, Hungary) Abstracts p 137 Dorn A, Winnewisser C, Wetzstein M, Nienhaus J, Grum-Grzhimailo A N, Zatsarinny O I and Mehlhorn W 1995b J. Electron Spectrosc. Relat. Phenom. 76 245 Edmonds A R 1957 Angular Momentum in Quantum Mechanics (Princeton, NJ: Princeton University Press) Ferguson A J 1965 Angular Correlation Methods in Gamma-ray Spectroscopy (Amsterdam: North-Holland) Grum-Grzhimailo A N and Dorn A 1995 J. Phys. B: At. Mol. Opt. Phys. 28 3197 Heddle D W O 1983 J. Phys. B: At. Mol. Phys. 16 275 Hertel I V and Stoll W 1977 Advances in Atomic and Molecular Physics vol 13, ed D R Bates and B Bederson (New York: Academic) p 113 Itikawa Y 1986 Phys. Rep. 143 69

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Kabachnik N M and Sazhina I P 1984 J. Phys. B: At. Mol. Phys. 17 1335 Matterstock B, Huster R, Paripas B, Grum-Grzhimailo A N and Mehlhorn W 1995 J. Phys. B: At. Mol. Opt. Phys. 28 4301 Mehlhorn W 1985 Atomic Inner-Shell Physics ed B Crasemann (New York: Plenum) p 119 ——1990 Proc. 15th Int. Conf. on X-ray and Inner-shell Processes (AIP Conf. Proc. 215) ed T A Carlson, M O Krause, S T Manson (New York: American Institute of Physics) p 465 Sang R T, Farrell P M, Madison D H, MacGillivray W R and Standage M C 1994 J. Phys. B: At. Mol. Opt. Phys. 27 1187 Wedowski M, Godenhusen K, Weisbarth F, Zimmermann P, Dohrmann Th, von dem Borne A, Sonntag B and Grum-Grzhimailo A N 1995 J. Electron Spectrosc. Relat. Phenom. 75 61 Zatsarinny O I and Bandurina L A 1993 J. Phys. B: At. Mol. Opt. Phys. 26 3765 Zetner P W, Li Y and Trajmar S 1992 J. Phys. B: At. Mol. Opt. Phys. 25 3187