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OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 329-342

doi:10.1088/0953-4075/40/2/006

Linear magnetic and alignment dichroism in Auger-photoelectron coincidence spectroscopy
F Da Pieve1, S Fritzsche2, G Stefani1 and N M Kabachnik3,4
1 2 3 4

CNISM Institut f Е Fakultat Institute

and Dipartimento di Fisica Universita' Roma Tre, I-00146 Roma, Italy Е Е ur Physik, Universitat Kassel, D-34132 Kassel, Germany Е Е fur Physik, Universitat Bielefeld, D-33615 Bielefeld, Germany of Nuclear Physics, Moscow State University, Moscow 119992, Russia

Received 20 September 2006, in final form 24 November 2006 Published 3 January 2007 Online at stacks.iop.org/JPhysB/40/329 Abstract Auger-photoelectron coincidence measurements for polarized targets are theoretically considered. We present a general expression for the cross section of sequential double photoionization of polarized atoms within the two-step model. Linear magnetic and alignment dichroism in coincidence experiments is discussed in more detail. For illustration, we made ab initio calculations based on the multiconfigurational Dirac-Fock theory for the 4d photoionization of atomic Sn followed by N-OO Auger decay.

1. Introduction Photoelectron-Auger electron coincidence spectrometry is an established method for studying the structure and dynamics of atoms, molecules and solids [1]. In atomic physics, for example, it was used to realize a so-called complete experiment in the photoionization of atoms [2-4]or to explore the tiny effects of coherence and post-collision interaction in photoinduced Auger decay [5-10]. Moreover, similar measurements were used for detailed investigation of the photoinduced resonant Auger cascades [11-16]. In molecular physics, on the other hand, these electron-electron coincidence measurements of core-level ionization [4] enable one to disentangle the contribution of the different vibrational levels and of the atoms in different sites, to the molecular Auger spectrum. In solid state physics the technique of Auger-photoelectron coincidence spectroscopy (APECS) has proved to be successful in studying complicated Auger spectra [17-19]. APECS measurements for solid targets are site-specific and show an increased surface sensitivity in comparison with conventional one-electron experiments as well as a strong reduction of secondary electron background. These capabilities make this technique a powerful tool to study systems with reduced symmetry, like thin films or adsorbates [20-22]. Recently, angle-resolved APECS has been used for studying the electronic alignment of ions in solids [23]. An overview of the application of APECS for solid-state systems may be found in [24]. In all these experiments the underlying physical process is a so-called resonant or sequential photo-double-ionization (PDI) in which the absorption of a photon
0953-4075/07/020329+14$30.00 ї 2007 IOP Publishing Ltd Printed in the UK 329


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results in the photoionization of an inner-shell electron followed by the emission of an Auger electron. In experiments both electrons are detected in coincidence and the energy and angular distributions are measured. Thus the resonant process is a particular case of PDI with one photon IN and two electrons OUT. In contrast to direct PDI, however, the emission of the two electrons proceeds through the formation of an intermediate ionic state, and thus can often be considered within a two-step model. Until the present time the investigations of sequential PDI have been performed with unpolarized targets. However, especially for studying the magnetic properties of solids, it would be very important to apply APECS to polarized (oriented and/or aligned) targets. In this way one could obtain information about site and element specific magnetization. Similar measurements in single photoelectron (or Auger electron) studies are very informative. Widespread investigations of magnetic circular dichroism in photoemission from solids are a good example of application of electron spectroscopy of polarized targets to studying the magnetic properties of solids (see, for example, [25] and references therein). In the gas phase, moreover, experiments with polarized targets provide important information about the electronic structure and correlations in atoms (see, for example, [26-28] and references therein) including complete information about the photoionization amplitudes (complete experiment) [29-31]. Although coincidence measurements from polarized targets are more difficult than conventional single-electron detection, such experiments are quite feasible with modern synchrotron radiation sources and modern equipment [32]. In order to design a coincidence experiment with a polarized target it is necessary to establish the theory of the process which enables one to determine the most effective geometry of the experiment and to predict the cross sections and counting rates to be expected. This task is fulfilled by the present work which is devoted to the theoretical analysis of the angular distribution of electrons produced by photoionization and decay of polarized atoms, using the formalism of density matrix and statistical tensors [33-35]. Since the general form of the differential cross section for a polarized target is very complicated and not convenient for analysis, usually different kinds of dichroism, i.e. the differences in the cross sections with regard to a different polarization of the incoming light or a different magnetization of the target, are analysed [36]. Following this trend, we use the derived general expression for the cross section of the sequential PDI and analyse the dichroism for different scenarios of the coincidence measurements. Conventionally, the dichroism is related to a change in the light polarization for one and the same state of the target polarization [37]. Following the experiment of Baumgarten et al [38], the dichroism due to the difference in the target polarization (for one and the same light polarization) has also been investigated. In this paper, we consider only the dichroism which arises due to changes of the target atom polarization. In particular, we concentrate on sequential PDI by linearly polarized synchrotron light for reversed (antiparallel) directions of the atomic orientation and for a mutually perpendicular alignment of the target, i.e. we discuss only linear magnetic dichroism and linear alignment dichroism, respectively. The formalism for sequential PDI is similar to that developed recently for direct PDI [39]. In contrast to the direct case, however, the existence of a well-defined intermediate state in the resonant PDI imposes several additional constrains on the angular correlation function and on the considered dichroism. Therefore, resonant PDI deserves a special analysis which is presented below. As an illustration of the resonant PDI we have calculated the cross sections and dichroism for the photoionization of 4d subshell of Sn atoms followed by a N4,5 -O2,3 O2,3 Auger decay. The ground state of the Sn atom has the following configuration: (36 Kr) 4d10 5s2 5p2 3 P0 where (36 Kr) is the electronic configuration of the Kr atom. Since the total angular momentum of the


Linear dichroism in Auger-photoelectron coincidence spectroscopy

331

ground state is zero, it cannot be polarized. However, close to the ground state there are two excited states of the same multiplet, the 3 P1 (0.210 eV) and 3 P2 (0.425 eV) states [40] which can be pumped by lasers and which become either aligned (by using a linearly polarized laser) or oriented (by a circularly polarized laser). This is a well-developed technique of laser excitation and polarization of atoms (see, for example, [26, 41, 42]). Below, we suppose that the excited and oriented and/or aligned Sn atoms in the initial 3 P1 or 3 P2 levels are further ionized in the 4d inner subshell by linearly polarized synchrotron radiation and the emitted photoelectron and the following Auger electron are analysed in coincidence. Atomic tin has been chosen, among other reasons, because the N-OO Auger spectrum of Sn has been recently studied experimentally [43] and the theoretical calculations of the photoionization cross section and of the Auger rates for all three components of the 3 P multiplet are available [43]. In addition, APECS measurements with an unpolarized solid Sn target have been reported recently [44] demonstrating the feasibility of such experiments. Thus Sn could well be a good candidate for the first experiment with a magnetized (oriented) target. Since inner-shell photoemission from solids in the first approximation can be considered as atomic photoemission due to the local character of the inner-shell electrons we believe that the atomic-like calculations may be useful for planning experiments also from solids. Atomic units are used throughout unless otherwise indicated. 2. Cross section for resonant photo-double-ionization of polarized atoms We consider the process of sequential (resonant) PDI within the conventional two-step model. In the first step the atom is photoionized in the inner shell and the intermediate highly excited ionic state is formed, + A(0 ,J0 ) A+ (1 ,J1 ) + e1 (l1 ,j1 ). (1)

Here J0 and J1 are the total angular momenta of the initial atomic and intermediate ionic states, respectively, 0 and 1 are all other quantum numbers which are necessary for characterization of these states, l1 ,j1 are the orbital and total angular momenta of the emitted photoelectron. In the second step an Auger decay of the intermediate state occurs to the final state with the quantum numbers 2 ,J2 A+ (1 ,J1 ) A2+ (2 ,J2 ) + e2 (l2 ,j2 ) (2)

and with emission of the Auger electron, characterized by the orbital and total angular momenta l2 ,j2 , respectively. The polarization state of the initial atom is characterized by the statistical tensors k0 0 (0 J0 ) [34, 35] which are defined with respect to the axis of the atomic polarization ? ^ a = (a ,a ). While the odd-rank tensors k0 = 1, 3,..., are known to characterize the orientation of the target atom, the even-rank tensors k0 = 2, 4,..., describe its alignment. 2J0 . In the laboratory The maximal rank of these tensors is limited by the condition k0 frame the statistical tensors are expressed by [35] k
0 q0

(0 J0 ) =

4 Y 2k0 +1

k0 q0

(a ,a )k0 0 (0 J0 ) ?

(3)

where Ykq ( , ) are the spherical harmonics. The photon beam is described by the photon statistical tensors k q which depend on the Stokes parameters characterizing the polarization properties of the light [35]. In the dipole approximation the rank of the tensor k is limited to k 2.


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In the sequential PDI of polarized targets there are three relevant directions: the directions ^ ^ of emission of the photoelectron n1 = (1 ,1 ) and of the Auger electron n2 = (2 ,2 ) and ^ the direction of the target polarization a. In this case the cross section may be expanded in terms of tripolar spherical harmonics which are defined as [45] ^ ^ ^ Yk0 (a) Yk1 (n1 ) Yk2 (n2 ) k k q =
q1 q2 q0 q kq

C

kq k0 q0 kq

C

kq k1 q1 k2 q2 Yk0 q0

(a ,a )Y

k1 q1

(1 ,1 )Y

k2 q2

(2 ,2 )

(4)

where Ck1 q1 k2 q2 are the Clebsch-Gordan coefficients. Using the standard technique of the density matrix and statistical tensor theory [35] one can present the triple differential cross section (TDCS) in the following form (a similar expression but for the case of direct PDI has been obtained in [39]) 1 d3 kk ^? J -1 k 0 (0 J0 )Bk01kr2k (J0 ,J1 )Ak2 (J1 ,J2 )Fkk01kkr2k . = (5) d 1 d 2 dE1 /2 k k k k k 1 0
r012

Here is the fine-structure constant, is the photon energy, is the total width of the ^ intermediate level J1 , J 2J + 1, and E1 is the energy of the photoelectron. In equation (5) we suppose that the energy conservation is implicitly taken into account. Fkk01kkr2k is a kinematical factor which depends on the geometry of the experiment and the beam polarization, but which is independent of any dynamics of the photo and Auger processes. It is given by the scalar product of the tripolar spherical harmonics and the photon statistical tensor: ^ ^ ^ 4 Yk0 (a) Yk1 (n1 ) Yk2 (n2 ) kr k q k q . (6) Fkk01kkr2k =
q

The coefficients B (J0 ,J1 ) and Ak2 (J1 ,J2 ) are related to the first and the second steps of the process, respectively, and contain the corresponding matrix elements. In particular, the kk coefficient Bk01kr2k (J0 ,J1 ) describes the photoionization of the target by the incident radiation: B
k1 k2 k0 kr k


k1 k2 k0 kr k

(J0 ,J1 ) =
l1 l1 j1 j1 Jr Jr

(-1)

j1 +1/2+kr -k



^ ^ll ^ ^^ J r J r ^1 ^1 j 1 j 1 kr C

k1 0 l1 0l1 0

1 j1 l 1 2 l 1 j1 k 1

J1 j1 Jr J0 1 Jr J0 1 Jr DJ1 j1 Jr J0 DJ1 j Jr J0 Ч J1 j1 Jr (7) 1 k2 k1 kr k0 k kr where we use the standard notations for the Wigner 6j and 9j symbols and where the notation DJ1 j1 Jr J0 = J1 ,j1 : Jr D J0 has been introduced to denote the reduced matrix element of photoionization. The coefficient Ak2 (J1 ,J2 ) in equation (5) is relevant to the subsequent Auger decay of the inner-shell vacancy: 1 J1 j2 J2 l2 j2 2 V ll ^ ^ ^ (-1)J2 +J1 +k2 -1/2 ^2 ^2 j 2 j 2 J 1 Clk2200l 0 V (8) Ak2 (J1 ,J2 ) = 2 j2 J1 k2 j2 l2 k2 J2 j2 J1 J2 j2 J1
l2 l2 j2 j2

where VJ2 j2 J1 = J2 ,j2 : J1 V J1 designates the matrix element of the interelectronic (Coulomb) interaction which is responsible for the decay. The cross section (5) is normalized in such a way that the total cross section (integrated over all emission angles and energy) for the unpolarized atom is equal to J0 J1 J1 J2 where J0 J1 is the total cross section for the photoionization transition to the ionic state J1 : 4 2 2 DJ1 j1 Jr J0 , J0 J1 = (9) 3(2J0 +1) l j J
11r

and

J1 J2

is the partial width for the Auger decay to the state J2 :


Linear dichroism in Auger-photoelectron coincidence spectroscopy
J1 J2

333

=

2 2J1 +1

VJ2
l2 j2

2 j2 J1

.

(10)

Note that within the two-step model, the emission of the two electrons is assumed to proceed through a well-defined intermediate ionic state with a certain angular momentum and parity. Consequently, the partial waves of the photoelectron l1 and l1 should be of the same parity and k1 should be only even due to the properties of the Clebsch-Gordan coefficients with zero projections in expression (7). The same is valid for the Auger decay: l2 and l2 should be of the same parity thus k2 is also even. Inspection of the symmetry properties of the kk Bk01kr2k coefficients with respect to the exchange of primed and non-primed angular momenta
kk then leads to the conclusion that Bk01kr2k are either real or imaginary for k0 + k even or odd, respectively. Expression (5) together with expressions (7) and (8) describes the angular correlation function for two-electron emission as measured in the coincidence experiment. It is useful to present also single-electron cross sections using the same definitions and notations. If only a photoelectron is detected then by integrating equation (5) over the angles of Auger electron emission and by summing over all possible final states one should obtain the known expression for the differential cross section of photoemission from a polarized atom (see, for example, [46]). Indeed, by integrating the kinematical factor (6) over the angles of the Auger electron one gets

F

k1 k2 k0 kr k



d

2

= 4
q0 q1 q

C

k q k0 q0 kr qr

C

kr qr k1 q1 00 Yk0 q0

(a ,a )Y

k1 q1

(1 ,1 )k
k 2 0 k 1 k
r


q

= 4
q

Yk0 (a ,a ) Yk1 (1 ,1 )

k q

k

q

=F

k0 k1 k





k 2 0 k 1 k

r

(11)

where Fk0 k1 k is the kinematical factor introduced by Baier et al [46]. Taking this equation into account and integrating the general formula (5) over Auger electron angle and energy, one can obtain the angle-differential cross section for photoionization of polarized atoms in exactly the same form as in [46]: d ^ = (3J 0 )-1 k0 0 (0 J0 )Bk0 k1 k (J1 )Fk0 k1 k ? (12) d1 kkk
01

^ ^ k0 where Bk0 k1 k (J1 ) = 3J 0 J 1 Bk01k1 k (J0 ,J1 ). If only an Auger electron is detected, one should integrate equation (5) over the angles of photoelectron emission. Integration of the kinematical factor gives F
k1 k2 k0 kr k


d

1

= 4k1 0

k2 k

r

C
q0 q2 q

k q k0 q0 k2 q2 Yk0 q0

(a ,a )Y

k2 q2

(2 ,2 )k


q

.

(13)

Taking this equation into account one can present the result of integration of equation (5), i.e. the angular distribution of the Auger electrons following the inner-shell photoionization of a polarized target, in a standard form [35]: d = d2 d
1

d3 d d 2 dE1 ^J J 1 k21
k2 q2

1

dE1 = J0 k
2 q2

J1

J1 J2


J2

1 4 (2 ,2 ) (14)

Ч

(1 J1 )

4 Y ^ k2

k2 q2

J where k21 J2 = Ak2 (J1 ,J2 ) are the intrinsic anisotropy coefficients for l2 j2 VJ2 j2 J1 the Auger decay [35], and we have introduced the statistical tensors of photoions produced in

2 -1


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ionization of the polarized atom [35, 47]: -1 k
2 q2

^0 (1 J1 ) = 3J 2
l1 j1 Jr

D

2 J1 j1 Jr J0


k0 k


^^ k0 k (-1)
k2 q2 k0 q0 k q

k2 +k



C k

q

ЧB

0k2 k0 k2 k



(J0 ,J1 )k0 0 (0 J0 ) ?
q0 q

4 Y ^ k0

k0 q0

(a ,a ).

(15)

We note that due to integration over all directions of the electron emission, the partial waves of the photoelectron enter incoherently in this expression, thus there is no interference between them. Equations (14) and (15) coincide with those given in [48]. The general expression (5) for the TDCS of a sequential PDI is very complicated and difficult to use in the analysis of experimental data. In practice, it is more convenient to analyse different kinds of dichroism which is a difference between cross sections [36]. In this case the large number of terms in equation (5) cancel each other and only some of these terms will finally remain. Thus in measuring the dichroism one can selectively study the different parts of the cross section. If, for instance, the ionizing radiation is circularly polarized one can study either the circular dichroism or the circular magnetic dichroism. These two types of measurements for the case of a coincidence experiment have been discussed in detail in [39, 49]. In the present paper, instead, we concentrate on the case of linearly polarized ionizing radiation. Here one can study linear magnetic dichroism by flipping the target orientation or linear alignment dichroism by comparing targets aligned along orthogonal directions. Below we consider these types of dichroism in more detail. 3. Linear magnetic dichroism Linear magnetic dichroism in angular distribution (LMD)5 is defined as the difference between the cross sections for two directions of the atomic orientation antiparallel to each other when the target is ionized by means of linearly polarized synchrotron radiation [27, 36]. LMD = d
1

d3 d 2 dE1



-

d

1

d3 d 2 dE1

.


(16)

For this type of measurement, one has first to produce an orientation of the target. One of the ways to achieve atomic orientation is laser pumping by circularly polarized light [26]. Using this technique, LMD is then measured by changing the helicity of the laser radiation. Let us consider some particular experimental setups typical for measurements with polarized targets. Let us choose z-axis parallel to the synchrotron light polarization, and y -axis along the photon beam. In such a reference frame only two components of the photon tensors 00 and 20 are non-zero [35]. Suppose that the atomic orientation lies in the direction of y -axis and is then changed from +y to -y (see the inset in figure 1(a)). In an experiment with laser pumping this corresponds to collinear laser and synchrotron beams. When considering the LMD, only terms with odd-rank tensors of the target may remain from taking the difference (16), since these terms are not invariant under reflection through a plane perpendicular to the axis of the atomic polarization. If, for the sake of simplicity, we restrict our consideration to the case of J0 = 1, then only the term with k0 = 1 will contribute. Moreover, the sum over projections of the orientation tensor is limited to q0 = +1 under the considered conditions. Inspecting the k1 kinematical factor F1krkk2 we note that for the atomic orientation along the photon beam, the
5 In our paper only angular distributions will be considered and we shall use the acronym LMD instead of the conventional LMDAD to shorten notations.


Linear dichroism in Auger-photoelectron coincidence spectroscopy
z
|| z

335

e1 || z , e2 variable e2 || z , e1 variable
0.005

]

scanning plane to the beam
O || - y x O || y O || z e1 (e2 ) || z

0.004 0.003 0.002
(-y)-(y)

y

0.001 0.000 -0.001 -0.002 -0.003 -0.004 -0.005
0 60 120 180 240 300 360
z
|| z

LMD

(a)

scanning angle (deg)

e1 || y , e2 variable e2 || y , e1 variable
0.00004

]

scanning plane to the light polarization
x

O || z

e 1 (e2) || y O || - z

0.00003 0.00002 0.00001
z-(-z)

y

LMD

0.00000 -0.00001 -0.00002

(b)
-0.00003 -0.00004
0 60 120 180 240 300 360

scanning angle (deg)

Figure 1. (a) LMD signal, namely, the difference between the TDCS's for atomic orientation antiparallel and parallel to the photon beam direction, normalized to the total cross section. (b) The same as above but for atomic orientation antiparallel and parallel to the photon polarization direction.

LMD can be written as follows: LMD B
k1 k2 1kr k y -(-y)

=

1^ J /2

-1 1

i 6? 10 (0 1)
k1 k2 kr k




(

1,J1 )Ak2 (J1 ,J2 )k 0 q0

C

k 0 1-q0 kr q0

(17)
kr q0

^ ^ Yk1 (n1 ) Yk2 (n2 )

.

Thus the linear magnetic dichroism is proportional to orientation of the atomic target and contains bipolar spherical harmonics (of both odd and even ranks) of the two emitted electrons. The summations in (17) are restricted by the condition that k = 0, 2. When k = 0 then


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kr = 1 and the two electrons are described by spherical harmonics of the same rank k1 = k2 (we recall that k1 and k2 should be even). When k = 2, then the ranks of spherical harmonics describing outgoing electrons can be either equal or differ by 2: k2 = k1 ,k1 + 2. Since k1 2 k0 + k = od d , the coefficients B1krkk in equation (17) are imaginary. So far, we have specified the direction of the atomic polarization with respect to the incoming synchrotron radiation. We still can choose a particular geometry for the detection of the two electrons, i.e. for a particular cut through the distribution function (17) by defining a proper reaction plane. The maximal dichroic effect can be expected if both electrons are detected in the plane perpendicular to the photon beam. In contrast, in any plane containing the beam the magnetic dichroism is zero in the considered case. Let both electrons be detected in the plane perpendicular to the photon beam, i.e. 1 = 2 = 0 and suppose that the Auger electron is detected along the light polarization while the photoelectron scans the xz plane ^ ^ (see the inset in figure 1(a)). Then 2 = 0 and thus Yk2 q2 (n2 ) = q2 0 k2 / 4 , i.e., q2 = 0. In this case the LMD is proportional only to the polar part of the spherical harmonics (associated Legendre polynomials) describing the scanning electron and can be written in the form
nmax

LMD

y -(-y)

=
n=1

an sin (2n1 ).

(18)

A similar expression applies also for the angular dependence of the LMD if the photoelectron is detected along the light polarization and the Auger electron scans the xz plane. In both cases, the complexity of the LMDy -(-y) pattern (nmax ) is determined by the maximal angular momentum of either the photo or the Auger electron, respectively. If one considers the orientation of the target along the polarization of the synchrotron radiation (this corresponds to pumping by the circularly polarized laser beam perpendicular to the ionizing beam) then only q0 = 0 components of the orientation tensor will contribute. Inspection of the kinematical functions leads to LMD
z-(-z)

=

1^ J /2

-1 1



3? 10 (0 1)
k1 k2 kr k


B

k1 k2 1kr k



(1,J1 ) (19)

Ч Ak2 (J1 ,J2 )k 0 C



k 0 10kr 0

^ ^ Yk1 (n1 ) Yk2 (n2 )

kr 0

where only bipolar spherical harmonics of odd rank contribute. For this scenario, the dichroic effect should be maximal if both electrons are detected in the xy plane, i.e., perpendicular to the photon polarization (see the inset in figure 1(b)). In the particular case when the photoelectron is detected along the y -axis and the Auger electron scans the xy plane the angular dependence of the LMD can be presented as
nmax

LMD

z-(-z)

=
n=1

an sin(2n2 )

(20)

max max where nmax = 1/2 min k1 ,k2 . In the reversed case when the Auger electron is detected along the y -axis and the photoelectron scans the xy plane, the LMD has the same value but reversed sign. Finally, it is easy to verify that the LMD disappears if the two electrons are detected in the xz plane, since no sense of rotation perpendicular to the z-axis is impressed. In dependence on the relative directions of the photon beam and the orientation of the target, the measurement of Auger-photoelectron coincidences can be sensitive to different projections of the orientation tensor. For the case of a surface, for instance, this means that one can study the spatial distribution of the magnetization within the surface.


Linear dichroism in Auger-photoelectron coincidence spectroscopy

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4. Linear alignment dichroism Linear alignment dichroism (LAD) is the difference between cross sections for two perpendicular atomic alignments and for linearly polarized ionizing radiation [27]. LAD = d
1

d3 d 2 dE1

-

d

1

d3 d 2 dE1

.


(21)

If target atoms are prepared using optical pumping with linearly polarized laser radiation then they can only be aligned, i.e. only the statistical tensors k0 0 (0 J0 ) with even k0 contribute to ? the cross section. If we restrict ourselves again to the case J0 = 1, we note that only terms containing 20 will contribute to the LAD. Let us first consider the case in which the alignment ? is parallel to the x and then to the y -axis (alignment in the plane perpendicular to the photon polarization). The difference between these two angular distributions contains only the terms in equation (5) with the projections q0 = +2 since the diagonal elements of the density matrix related to alignment cancel. The result can be written as 1 ^ -1 k1 2 J 2 3020 (0 1) ? B2krkk (1,J1 ) LADx -y = /2 1 kkkk
12r

Ч Ak2 (J1 ,J2 )C

k 0 22kr -2

^ ^ Re Yk1 (n1 ) Yk2 (n2 )
k1 k2 2kr k

kr -2 k 0

.

(22)

Since k0 + k = ev en, the coefficients B are all real. When in particular one of the electrons is detected along the z-axis, i.e. along the light polarization, the angular dependence of the LAD (22) can be presented as
nmax

LAD

x -y

=
n=0

bn cos (2n ) cos 2

(23)

where and are the emission angles of the second electron which can be scanned through. The analysis of the LAD signal measured for the target aligned along the other two perpendicular directions, for example, along the z and y -axis is more complicated since both projections q0 = 0 and q0 = +2 contribute. In a particular case when one of the electrons is detected along the light polarization (along the z-axis) the angular dependence of the LAD (22) reduces to the following expression:
nmax

LAD

z-y

=
n=0

( ( ( cn1) + cn2) cos 2 + cn3) sin 2 cos (2n ).

(24)

Here, as in the previous case, and are the emission angles of the second scanning electron. Finally, we note that the complete information about the density matrix of the polarized target state can be obtained by combining the LMD and LAD studies for different experimental setups and conditions. 5. Example: Sn 4d photoionization and subsequent N4,5-O2,3O2,3 Auger decay For the application of the above-derived theoretical formulae and in order to estimate the expected value of the dichroic effects we have performed ab initio calculations for the 4d photoionization of atomic tin and its subsequent N4,5 -O2,3 O2,3 Auger decay. For the tin atom, the lowest multiplet of the ground-state configuration is 5s2 5p2 3 P. In the following we suppose that the tin atoms are first excited from the 3 P0 ground state into the state 3 P1 by using a laser which provides also for orientation and/or alignment of the atom. Then the target atoms are further ionized in the 4d subshell by means of synchrotron radiation. The produced


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photoions Sn+ (4d-1 ) finally decay by Auger emission to doubly ionized ions Sn2+ in the 5s2 1 S0 state. 5.1. Computation We have calculated the 4d photoelectron spectra for the photoionization of the 3 P1 excited state. For all intermediate ionic states with configuration 4d-1 5s2 5p2 we have also computed the Auger rates. The wavefunctions of the initial, intermediate and final states of the transitions have been computed within the multiconfigurational Dirac-Fock (MCDF) method by using the GRASP92 code [50]. In this method the wavefunctions are obtained as linear combinations of jj coupled configuration state functions (CSF) of the same symmetry, i.e., the same total angular momentum and parity. The CSF are optimized on the basis of the many-electron Dirac- Coulomb Hamiltonian while some important relativistic corrections to the electron-electron interaction are added later by diagonalizing the Dirac-Coulomb-Breit Hamiltonian matrix. The basis set of CSF includes all configurations corresponding to the nominal non-relativistic configurations of the states involved. The continuum wavefunctions of the photoelectron and of the Auger electron have been calculated in the field of the singly charged ion and doubly charged ion, respectively. With the wavefunctions obtained, the photoionization and Auger amplitudes have been computed using the program package RATIP [51]. The computed cross sections and Auger rates are in good agreement with the results by Huttula et al [43]. In contrast to the computations in [43], however, the photoionization cross sections have been obtained here by evaluating the dipole matrix elements for bound-free transitions. This calculation is expected to be somewhat more accurate than those in [43] where the photoionization probability was derived using the weight of the ground-state parent configurations in the final states of photoionization. A good qualitative agreement between the two calculations confirms the validity of the simple model used in [43]. Below, one particular line from the Sn 4d-1 5p2 , 5s2 and 5s5p Auger spectrum has been selected to demonstrate the effects of the atomic orientation on the electron-electron 3 coincidence cross sections. We have chosen the transition 5p2 P2 4d-/1 (J = 3/2) 52 5p-2 S0 which corresponds to line A12 in the nomenclature of the paper [43]. This line is strongly excited in ionization from the 3 P1 excited state of neutral Sn, but it is weak if the photoionization occurs from the ground 3 P0 state of atomic Sn. This transition is therefore appropriate for a future experiment on the dichroism from a polarized target, since it avoids strong background due to transitions from the non-oriented ground state. Moreover, this line appears rather strong in the overall Auger spectrum. For this Auger line the chain of transitions under investigation is the following 5p2 P0 5p2 P1 5p2 P2 4d-/1 (J = 3/2) 5p-2 S0 . 52
3 3 3 1 1

(25)

Since the polarized state has J0 = 1 its orientation is determined by the only statistical tensor ? 10 (0 1) while its alignment is characterized by the only tensor 20 (0 1). For the particular ? Auger transition that we have chosen, the Auger matrix elements do not influence the angular correlation function. Indeed, the angular momentum of the final state is zero, therefore, only one partial wave d3/2 contributes to the Auger transition J1 = 3/2 J2 = 0; in this case the single Auger amplitude gives rise to just the overall factor to the cross section. Apart from this factor the value Ak2 contains only coefficients of vector coupling and may be easily computed. Besides, since the intermediate state has J1 = 3/2 the possible values for the rank k2 are limited to 0 and 2. All these constraints limit the complexity of the angular correlation function in Auger-photoelectron coincidence measurements and of the dichroism patterns. The results of the computations for the particular transition (25) are presented and discussed


Linear dichroism in Auger-photoelectron coincidence spectroscopy
LADy-x LADy-x LADy-x (1,1)=(0 ,0 ), (2=0 , 2=variable) (1=0 , 1=variable) , (a,a)=(0 ,0 ) (1,1)=(0 ,0 ), (2=90 , 2=variable) (1=90 , 1=variable) , (2,2)=(0 ,0 )
o o o o o o o o o o o o

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0.006

LADy-x

] ]

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to the beam to the lightpolarization

0.004

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-0.004

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LADz-y LADz-y LADz-y LADz-y (1,1)=(0 ,0 ), 2=0 , 2=variable
1=0 , 1=variable , (2,2)=(0 ,0 )
o o o o o o

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(1,1)=(0 ,0 ), 2=90 , 2=variable
1=90 , 1=variable , (2,2)=(0 ,0 )
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0.010 0.008 0.006 0.004

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0.002 0.000 -0.002 -0.004 -50 0 50 100 150 200 250 300 350 400

scanning angle (deg)
Figure 2. (a) LAD signal, namely, the difference between the TDCS's for atomic alignment parallel to the y -axis (the photon beam direction) and to the x-axis, normalized to the total cross section. (b) The same as above but for atomic alignment parallel to the photon polarization and parallel to the photon beam direction. In both cases (a) and (b) the calculations have been performed for scanning through the plane perpendicular to the photon beam and through the plane perpendicular to the light polarization.

in the next section. Since the target orientation, 10 (0 1), and the alignment, 20 (0 1), are ? ? determined by the conditions of a particular experiment which are not specified here, we set both these parameters equal to unity. All results are normalized to the total cross section integrated over both emission angles.


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5.2. Results and discussion First, let us consider the LMD in the case when the atomic polarization is taken parallel and antiparallel to the photon propagation direction and when both electrons are detected in a plane perpendicular to the beam. Two results are shown in figure 1(a), one with the photoelectron detected along the light polarization and the Auger electron scanning the xz plane (solid curve), and the other with the Auger electron fixed along the light polarization and the photoelectron scanning the xz plane (dashed curve). In both cases, the distribution of the scanned electron is described by expression (18). It can be seen that the dichroism signal obtained when the direction of the Auger emission is fixed is larger than that obtained by fixing the photoelectron direction. The former is also less symmetric than the latter. This is due to constraints imposed by the angular momenta involved in the considered transition. From expression (7) it follows that for the rank k1 describing the properties of the photoion and the anisotropy of the photoelectron angular distribution, the values k1 = 0, 2, 4, 6 are possible. In contrast the anisotropy of the Auger electron emission is characterized by only two values, k2 = 0, 2. This suggests that the angular distribution of the photoelectrons at a fixed Auger emission angle is expected to be more complicated than in the case where the Auger electron is measured for a fixed direction of the photoelectron emission. In other words, the measurement of the Auger-photoelectron coincidences provides more information if the Auger electron is observed under a fixed direction while the angle of the photoelectron is scanned through. In figure 1(b), we show the calculated LMD for the case of the atomic polarization parallel and antiparallel to the light polarization. Both curves correspond to scans in the xy plane where the dichroic effect should be maximal. One of the two electrons is detected along the photon beam direction. The LMD dependence on the azimuthal angle of the second electron is described by equation (20). The LMD effect is small in this case and, as expected, the value of the effect is independent of the choice of the electron which is scanning the plane except the overall phase of the LMD curve. Figure 2(a) displays the results of the calculations for the linear alignment dichroism, LADy -x , i.e. the difference between the cross sections when the atomic alignment is along the y -axis (parallel to the photon beam) and when it is along the x-axis. The two curves in this figure (solid and dashed lines) correspond to the scan of the second electron in the plane which is perpendicular to the beam. The other two curves (dotted and dash-dotted lines) correspond to the scan in the plane which is perpendicular to the polarization of the incident light. For each scanning plane, one curve corresponds to the case when the photoelectron is fixed along light polarization and the scan is performed by changing the Auger electron direction. Another curve is calculated when the role of the electrons is interchanged. One can see that, when the scanning plane is perpendicular to the photon beam, the angular dependence of the dichroic signal shows a higher complexity if the emission direction is fixed for the Auger electron but changed for the photoelectron, when compared with the reversed situation in which the photoelectron is observed in a fixed direction and the scan is over the directions of the subsequent Auger electron emission. The position of the main minima is the same in both curves but the scan over the photoelectron has additional structures. This can be easily explained using expression (23). For the plane perpendicular to the beam = 0. The complexity of the curve is determined by nmax which is larger when the photoelectron detector is rotated rather than the Auger electron detector, as was explained above. If the scanning plane is perpendicular to the light polarization, the two LAD signals (dotted and dash-dotted curves) have quite the same behaviour, only the anisotropy is different. In this case, we have = 90 in equation (23) and, hence, the angular dependence of the LAD is described by the function cos 2 . The only parameter which can be determined


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from the experiment is the amplitude of the variation of the LAD signal. This suggests that measurements in the plane perpendicular to the beam are more sensitive to the details of the photoelectron and the Auger electron emission. For our example, however, the experiments with a fixed direction of the Auger electron emission and scanning the photoelectron direction are more informative than for the reversed case, since more parameters may be extracted from the experiment. In figure 2(b) the results of calculations for LADz-y are presented. Again, when the scanning plane is perpendicular to the photon beam ( = 0 in expression (24)), scanning over the photoelectron emission direction at the fixed Auger emission direction gives a more complicated angular dependence of the LAD (dashed curve) than for scanning the Auger electron (solid curve). Therefore, the former measurements are more informative than the latter. For measurements in the plane perpendicular to the light polarization ( = 90 in expression (24)), the two curves corresponding to the fixed photoelectron direction (dotted curve) and to the fixed Auger electron direction (dash-dotted curve) have similar simple behaviour and again only the anisotropy is different. Therefore such measurements are less informative. In addition, the value of the LAD signal is smaller in the latter case. 6. Conclusions We have presented and analysed a general expression for the angular correlations between two emitted electrons in Auger-photoelectron coincidence experiments on polarized atoms. Two types of dichroism, linear magnetic dichroism and liner alignment dichroism, have been considered in detail. Ab initio computations for the particular case of a 4d photoionization of polarized (i.e. oriented or/and aligned) tin atoms and the subsequent Auger decay show that the dichroic effects depend significantly on the geometry chosen in the experiment. The value of the dichroism as well as the complexity of the dichroic pattern strongly vary under different experimental conditions. We believe that the APECS measurements on magnetized targets will provide important information about site-specific local magnetic properties of surfaces and solids. Acknowledgments NMK is grateful to Bielefeld University for hospitality and acknowledges financial support from the Volkswagen Foundation. SF acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) under the contract FR 1251/13-1. References
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