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Physics Reports 451 (2007) 155 ­ 233 www.elsevier.com/locate/physrep

Coherence and correlations in photoinduced Auger and fluorescence cascades in atoms
N.M. Kabachnika, b , S. Fritzschec
, d ,

, A.N. Grum-Grzhimailoa , M. Meyere , K. Uedaf

a Institute of Nuclear Physics, Moscow State University, Moscow 119991, Russia b FakultÄt fÝr Physik, UniversitÄt Bielefeld, D­33615 Bielefeld, Germany c Max­Planck­Institut fÝr Kernphysik, D­69029 Heidelberg, Germany d Gesellschaft fÝr Schwerionenforschung (GSI), D­64291 Darmstadt, Germany e LIXAM (UMR 8624), Centre Universitaire Paris-Sud, BÁtiment 350, F­91405 Orsay Cedex, France f Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan

Accepted 20 July 2007 Available online 10 August 2007 editor: J. Eichler

Abstract An overview of the recent experimental and theoretical works on photoinduced cascades of Auger and fluorescence transitions in atoms is presented. We concentrate on the angular correlation and polarization phenomena in such cascades and how they are related to the alignment and orientation of the core-excited or core-ionized atomic states produced in the photoabsorption. During the decay, the alignment and orientation are then transferred to the energetically lower atomic states, causing an anisotropy and polarization of the subsequent Auger or fluorescence emission. Special attention is paid to the practically important case of overlapping resonances and intermediate states. Coherent excitation of the resonances and the coherence transfer in the decay strongly influence the angular and polarization characteristics of the decay products. The studies on the angular correlations and polarization of the emitted particles provide detailed spectroscopic and dynamic information about the transitions involved. In some cases, moreover, a so-called `complete' experiment, i.e. the experimental determination of the transition amplitudes including their relative phases, becomes possible. © 2007 Elsevier B.V. All rights reserved.
PACS: 32.80.Hd; 32.80.Fb Keywords: Alignment; Auger transition; Auger electron spectroscopy; Coherence transfer; Coincidence experiments; Complete experiment; Fluorescence spectroscopy; Orientation; Photoexcitation and photoionization; Polarization; Statistical tensors; Transition cascades

Contents
1. Introduction ......................................................................................................... 2. Experiment ......................................................................................................... 2.1. Angle resolved Auger electron spectroscopy ......................................................................... 2.2. Spin polarization measurements for Auger electrons .................................................................. 2.3. Fluorescence spectroscopy ........................................................................................
Corresponding author. Gesellschaft fÝr Schwerionenforschung (GSI), D­64291 Darmstadt, Germany.

156 158 159 160 161

E-mail address: s.fritzsche@gsi.de (S. Fritzsche). 0370-1573/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2007.07.005


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2.4. Coincidence experiments ......................................................................................... 2.4.1. Electron­electron coincidences ............................................................................. 2.4.2. Electron­photon coincidences .............................................................................. 3. Basic theory ........................................................................................................ 3.1. General formalism for correlation and polarization studies ............................................................. 3.1.1. Characterization of a polarized atomic state ................................................................... 3.1.2. Polarization of photoinduced Auger states .................................................................... 3.1.3. General equations for angular correlations in cascades .......................................................... 3.1.4. Orientation and alignment transfer in Auger and fluorescence cascades ............................................ 3.1.5. Angular distribution of Auger electrons ....................................................................... 3.1.6. Spin polarization of Auger electrons ......................................................................... 3.1.7. Polarization and angular distribution of fluorescence ............................................................ 3.1.8. Interference between intermediate states ...................................................................... 3.2. Multiconfiguration Dirac­Fock computations of Auger cascades ........................................................ 3.2.1. Multiconfiguration expansions: The challenge of open shells ..................................................... 3.2.2. Evaluation of photoionization and Auger amplitudes ............................................................ 3.2.3. Description of coherence transfer through the computation of the density matrix .................................... 3.3. On the problem of a complete experiment ........................................................................... 3.4. Beyond the stepwise model of the Auger decay ...................................................................... 4. Auger cascades ...................................................................................................... 4.1. Angular distribution of the resonant (first step) Auger transitions ........................................................ 4.2. Effects of coherent excitation of resonances ......................................................................... 4.3. Spin polarization of Auger electrons in the resonant Auger transitions .................................................... 4.4. Interference effects in the alignment of intermediate states and angular distribution of the second step Auger electrons ........... 4.5. Coincidence studies of Auger cascades ............................................................................. 4.6. Realization of a complete experiment in resonant Auger cascade ........................................................ 4.7. Auger cascades following the PCI induced electron recapture in the near-threshold photoionization ........................... 5. Auger­fluorescence cascades .......................................................................................... 5.1. Radiative cascades in the VUV and visible region .................................................................... 5.2. Polarization analysis and depolarization mechanisms ................................................................. 5.3. Contribution of fluorescence studies to complete experiments .......................................................... 5.4. Scanning across Auger resonances ................................................................................. 5.5. Auger electron­fluorescence coincidence ........................................................................... 6. Related processes .................................................................................................... 6.1. Direct double Auger decay ........................................................................................ 6.1.1. Energy distribution of emitted electrons ...................................................................... 6.1.2. Angular correlation of emitted electrons ...................................................................... 6.2. Photoinduced Auger process with initially polarized atoms ............................................................. 6.3. Auger decay in molecules ........................................................................................ 6.3.1. General theoretical considerations ........................................................................... 6.3.2. Linear molecules ......................................................................................... 6.4. Alignment and orientation in the direct photoionization of atoms ........................................................ 7. Concluding remarks .................................................................................................. Acknowledgement ....................................................................................................... References .............................................................................................................

1. Introduction The study of core-excited atomic states is a well developed part of atomic spectroscopy. During the last two decades, it has undergone dramatic progress mainly due to the improvement of the experimental facilities and instrumentations, such as vacuum ultraviolet and soft X-ray beam-lines at modern synchrotron radiation sources as well as different types of electron or photon analyzers. Nowadays, the synchrotron sources can provide not only tunable and highly monochromatic radiation but also high fluxes and photons with well defined polarization. The interest in core-excited states and their relaxation stems partly from the fact that electron­electron correlations play a decisive role in the course of their excitation and decay. For example, the selective excitation of one of the core electrons to an unoccupied Rydberg orbital results in the formation of a highly excited, strongly correlated atomic state which usually decays


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via electron emission, called resonant Auger transitions. The resonant Auger decay, especially in rare gas atoms, has been studied intensively by means of electron spectroscopy during the last two decades (see review articles by Aksela et al. (1996a), Mehlhorn (1998, 2000), Armen et al. (2000), Piancastelli (2000), Kabachnik and Ueda (2004), and references therein). In particular, the low-energy part of the resonant Auger spectra has attracted much recent interest, because the majority of these transitions still leads to highly excited ionic states which decay further via some second-step Auger (Coster­Kronig) electron emission, forming an Auger cascade. Of course, the excited ionic state may emit also photons and, hence, Auger electron­fluorescence cascades have been observed. Both of these cascades proceed through some intermediate ionic states with two or more open subshells. Owing to such a complex shell structure, most of these states are strongly correlated and their quantitative description has remained a challenge for modern atomic theory. On the other hand, the spectroscopy of such states of positively charged ions has been found important for many applications, including the modeling of stellar atmospheres, laboratory plasma or even for the design of fusion reactors, because the investigation of these ions by other methods is an experimentally difficult task (cf. the review by West (2001)). The electron spectra from core-excited resonances are usually very complex. They consist of a large number of partly overlapping lines which arise not only from the so-called spectator Auger transitions, where the outer electron excited from the core stays within one and the same orbital during the subsequent electron emission, but also from shake-modified transitions as well as from numerous configuration satellites (Aksela et al., 1996a). In the case of the shake-modified transitions, for instance, the outer electron changes its principal quantum number owing to a change in the atomic potential. Frequently, moreover, the low-energy part of such spectra appears to be even more complex, since the first-step resonant Auger lines may overlap with the second-step cascade Auger transitions. For a proper identification of the observed lines, additional information is therefore required and can be obtained by measuring the angular distributions and the spin polarization of the emitted electrons. In addition, further spectroscopic information about the intermediate states in the cascade can be obtained from measuring the fluorescence. Especially studies of the angular distribution and polarization of the fluorescence lines are found important for the proper assignment of the intermediate and final states. The origin of the anisotropy of the Auger electron or fluorescence emission is a non-statistical population of the magnetic sublevels of the excited or ionized states which in turn is caused by the anisotropy of the photon­atom interaction (Mehlhorn, 1968; FlÝgge et al., 1972). For non-polarized or linearly polarized photon beams the excited atom can be aligned along the direction of the photon momentum or the photon polarization, respectively. For an aligned atom with total angular momentum J , the population of the substates with different projections MJ is different but independent of the sign of the projection. If the photon beam is circularly polarized, the excited atomic state can be also oriented along the beam direction. The orientation is often characterized by a predominant population of one of the magnetic substates. A rigorous definition of the alignment parameter, A20 , and the orientation parameter, A10 , is given in Section 3. The decay of an aligned state shows an anisotropic angular distribution of the emitted electrons as well as anisotropy and polarization of fluorescence (Mehlhorn, 1968; Eichler and Fritsch, 1976). Due to the spin­orbit interaction, moreover, the emitted Auger electrons can be spin-polarized. This phenomenon is known in the literature as dynamical spin polarization (Klar, 1980; Kabachnik, 1981; Huang, 1982). In addition, an orientation of the decaying state often leads to either the circular polarization of the fluorescence light or the spin polarization of the emitted electrons (polarization transfer). In these cases, the primary alignment and orientation of the photoproduced coreexcited state is partly transferred to the intermediate state following the first-step transition of the cascade (Kabachnik et al., 1999). Obviously, this may lead in turn to an anisotropy and polarization of the subsequent second-step Auger and fluorescence transitions, and of possible further decay steps. Therefore, experimental studies on the angular distributions and polarization of the emitted radiation help to identify individual transitions and yield information about the alignment and orientation of the intermediate states as well as the dynamics of the Auger decay. Up to now, we briefly described the pure case of a cascade decay of an isolated resonance through a single intermediate state. In practice, this is a rather rare case since often in experiments, several overlapping resonances are excited which decay subsequently through a number of (overlapping) intermediate ionic states. Excitation of the overlapping resonances occurs coherently leading to an interference of the decay channels which often drastically affects the angular anisotropy and the polarization of the emitted radiation. In addition, the coherence of the excitation is further transferred to the excitation of overlapping intermediate states. Therefore also the angular distributions of radiation emitted in the second step decay will be affected. An adequate theoretical description of the cascade should take into account the coherence of the excitation of the atomic states and the coherence transfer through the cascade.


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In complicated cases of many overlapping states, the powerful method of coincidence spectroscopy can be used for a proper identification of the lines and for studying the dynamics of Auger cascades. In these measurements, a certain decay channel is separated and identified by detecting two electrons in coincidence (von Raven et al., 1990; von Raven, 1992; Alkemper et al., 1997). Even more detailed spectroscopic information on the initial- and final-state quantum numbers can be obtained by measuring the angular correlations between the successively emitted electrons in the Auger cascade or Auger electron­fluorescence cascade (Kabachnik, 1997; Kabachnik et al., 1999; Kabachnik and Ueda, 1995). In contrast to the rather common non-coincidence measurements, the coincidence angle resolved experiments, which involve two Auger electrons from a cascade, are still rather scarce (Becker and Viefhaus, 1996; Wehlitz et al., 1999; Turri et al., 2001; Ueda et al., 1999a, 2001, 2003b; Da Pieve et al., 2005). Furthermore, only a few experiments were reported in which an emitted Auger electron was detected in coincidence with polarized fluorescence photons (Beyer et al., 1995, 1996; West et al., 1996, 1998; Ueda et al., 1998). Not only the spectroscopy, but also the dynamics of relaxation of the core-excited states can be studied by measuring the angular correlations of the emitted electrons as well as the angular distributions and polarization of the fluorescence radiation. In some cases these measurements permit one to realize a so-called `complete' or `perfect' experiment on the Auger decay, i.e. the experimental determination of the decay amplitudes including their relative phases (Kabachnik and Sazhina, 1990). In quantum mechanics, such measurements are of fundamental importance. They give most detailed information about the process up to the degree allowed by quantum mechanics, thus providing the most comprehensive test for any quantum­mechanical model. Being determined from the experiment, these amplitudes permit one to predict all other characteristics of the process even though some of them are impossible to measure by means of present-day experimental facilities. First attempts to realize the complete experiment for the resonant Auger process have been discussed and published recently (Ueda et al., 1999a; O'Keeffe et al., 2003, 2004). Dynamically, the near-threshold photoionization, including the resonant Auger decay, has been found a complicated process which can be considered only in a rather `rough approach' as a two-step process, with first an excitation and a subsequent decay of a (intermediate) resonance state. A more elaborate consideration should involve the effects of post-collision interaction (PCI), including the recapture of the photoelectron (Kuchiev and Sheinerman, 1989; Schmidt, 1997). Therefore, studies on the angular correlations in Auger transitions close to the inner-shell threshold will shed new light upon the complicated dynamics of near-threshold photoionization. The present overview is devoted mainly to the recent experimental and theoretical work on cascades of Auger and radiative transitions as induced by the photoexcitation of inner-shell electrons. Thus the first step of the considered processes is in most cases a resonant Auger decay. Below, we focus on angular correlation and polarization measurements and their role in studying the relaxation of core-excited states. In a few cases, if appropriate, we also discuss similar measurements for normal Auger cascades which follow the photoionization of an inner atomic shell. An additional goal is to provide the reader with a detailed list of references to the most important contributions in this field which have been published during the last decade. Following this introduction, we shall describe in Section 2 a few typical experimental installations used in modern investigations as well as advanced methods to perform such measurements. In Section 3, we overview the theoretical methods as applied in the analysis of the experimental data. The formal theory of angular correlation and polarization measurements can be found elsewhere (for example, Devons and Goldfarb, 1957; Ferguson, 1965; Blum, 1996; Balashov et al., 2000). Here we present the formulas for the angular distributions and polarizations which are necessary for the discussion of the experimental results. We discuss also the approximations used for calculating the relevant matrix elements. Since the considered atomic and ionic states contain two or more open shells, configuration interaction plays a very important role in their description. The usual approach to the solution of this problem, based on the multi-configurational Dirac­Fock (MCDF) theory, is discussed. Section 4 is devoted to the discussion of resonant Auger­normal Auger transition cascades. In Section 5, thereafter, we consider another type of cascades, namely resonant Auger­fluorescence cascades, while a few related processes are discussed in Section 6. Finally, a summary and few concluding remarks are given in Section 7. 2. Experiment The study of core excitations in atoms and molecules and, in particular, the investigations of the angular correlations and polarization in subsequent cascade transitions benefit significantly from the developments of the third-generation synchrotron radiation (SR) sources and soft-X-ray monochromators. The present third-generation SR sources such as


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ALS in Berkeley, ELETTRA in Trieste, MAX II in Lund, BESSY II in Berlin, SPring-8 in Hyogo are based on highperformance undulators. All of these centers have specially designed beam-lines dedicated to the gas phase atomic and molecular experiments and equipped with high-resolution monochromators. A concise description of these beam-lines can be found in the recent review by Ueda (2003). Often, the monochromator bandwidth is smaller than the natural lifetime width of the inner-shell excited states. This feature is especially important for the resonant Auger spectroscopy since, if the bandwidth of the exciting photons is smaller than the width of the core-excited state, the spectral widths of the resonant Auger lines is no longer given by the natural width of the inner-shell excited state but is determined instead by the convolution of the monochromator bandwidth, the bandwidth of the electron energy analyzer and the Doppler width due to thermal motion of the sample ° gases (KivimÄki et al., 1993; Aberg and Crasemann, 1994; Armen et al., 2000). This line-narrowing effect, sometimes called the Auger resonant Raman effect, can be used also for spectroscopic investigations of the Auger final ionic states. The activity on this particular topic was one of the main investigation directions at the second-generation SR facilities (see the review by Armen et al. (2000)). Further progress in this field at the third-generation facilities can be seen, for example, in the experiments by Shimizu et al. (2000), Huttula et al. (2001a), Sankari et al. (2001), KivimÄki et al. (2001), Snell et al. (2001) and De Fanis et al. (2001, 2002). Owing to the greatly improved resolution in the spectroscopy of core-excited states, the importance of angular correlation measurements have increased significantly, since they provide additional spectroscopic information as required for the identification of the excited states. Below, we present examples of the experimental set-ups which are used in studying the angular distribution, polarization and angular correlation of the electrons and photons emitted in the resonant Auger processes and the following cascades. Since some of the examples discussed here were obtained still on second-generation SR sources, we give a brief description of the experimental conditions at these sources as well. 2.1. Angle resolved Auger electron spectroscopy Non-coincidence angular distribution measurements are nowadays a rather routine experimental method in the analysis of resonant Auger electrons (Farhat et al., 1999; Huttula et al., 2001a, 2003; De Fanis et al., 2001, 2004a, b, 2005; HeinÄsmÄki, 2001; Kitajima et al., 2001, 2002, 2006; Oura et al., 2004; Sankari et al., 2001; Shimizu et al., 2000; Snell et al., 2000; Ueda et al., 1999a, b, 2000, 2001, 2003a, b; Yoshida et al., 2000, 2005, for earlier references see the reviews by Armen et al. (2000) and by Piancastelli (2000)). As a typical example, we describe the experimental studies of the resonant Auger cascades in Ar, Kr and Xe as performed at the Photon Factory in Tsukuba, Japan (Ueda et al., 1999b, 2000, 2001; Kitajima et al., 2001, 2002). These experiments were carried out by using a 24-m spherical grating monochromator on the BL-16B undulator beam-line (Shigemasa et al., 1998). The apparatus comprises two identical 150 spherical sector electron analyzers with a mean radius of 80 mm (Shimizu et al., 1998) and thus can be used also for electron­electron coincidence experiments (see Section 2.4). For the angular distribution measurement for the resonant Auger and the second-step Auger electrons, only one rotatable analyzer is used in most cases. The rotatable analyzer is mounted on a turntable whose axis of rotation is aligned to coincide with the incident photon beam axis, so that the angular distribution of the Auger electrons can be measured in the plane perpendicular to the beam direction. The incident light was focused onto the source point with two mirrors located after the monochromator. At the source point, the light was merged with an effusive gas beam ejected from an axial cell through eight straight needles. The incident photon flux was monitored with an Au photocathode mounted behind the interaction region. For the angular distribution measurement, the electrons passing through the rotatable analyzer were counted as a function of the electron detection angle relative to the polarization axis of the incident light. In most cases, the Auger electron spectra were recorded only at = 0 and 90 . More precise and accurate angular distribution measurements were done for some selected prominent lines by recording spectra at each 10 degrees between = 0 and 90 . Photon energy bandwidth and the energy resolution of the electron analyzer were typically 0.1 eV. The resulting overall energy resolution was therefore 0.14 eV for recording the first-step resonant Auger spectrum and 0.1 eV for the second-step Auger spectrum. Note that the spectral widths of the second-step Auger lines are independent of the photon energy bandwidth. The angular distributions of the 1s photoelectron of He and the 2s and 2p photoelectrons of Ne, which are well known, were also measured and the degree of linear polarization of the incident light was estimated to be Plin = 100+0 % with -3 the polarization vector in the horizontal plane (Ueda et al., 2001). The angular dependence for the detection efficiency of the analyzer was also determined from the same photoelectron measurements.


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Another approach for measuring the angular anisotropy of the emitted electrons was used in the experiments on the resonant Auger and second-step Auger processes in Ne 1s photoexcitation (Shimizu et al., 2000; De Fanis et al., 2004b, 2005; Yoshida et al., 2005; Kitajima et al., 2006). These measurements were carried out using the high-resolution electron spectroscopy end-station of the soft X-ray photochemistry beam-line 27SU (Ohashi et al., 2001b) at SPring-8, the third generation SR source in Japan. The high-resolution soft X-ray monochromator of this beam-line is a variedline-space plane grating monochromator of Hettrick type (Ohashi et al., 2001a). The heart of the end-station is an SES-2002 hemispherical electron energy analyzer (Gammadata-Scienta AB) (Shimizu et al., 2001). The lens axis of the analyzer is fixed in the horizontal direction, whereas the entrance slit of the analyzer is fixed to be parallel to the photon beam direction. For inserting the gas sample into the source volume, one can use either a gas cell (GC-50, Gammadata-Scienta AB) or a molecular beam source (MBS JD-01, MB Scientific). The whole system, which consists of the analyzer, the gas cell or the molecular beam source, and the differentially pumped vacuum chamber, is mounted on a XY Z stage, so that the focus point of the analyzer can be adjusted easily to the fixed photon beam position. The radiation source of this beam-line is a figure-8 undulator which provides linearly polarized light. The polarization axis is horizontal for the first order harmonic and vertical for the 0.5th order harmonic (Tanaka and Kitamura, 1996). Thus, one can perform angle resolved photoemission studies by switching the direction of the polarization vector from the horizontal to the vertical direction by varying the gap of the undulator. All these adaptations can be carried out without changing the monochromator setting and without the need to rotate the analyzer. As mentioned above, the resolution of modern soft X-ray monochromators is sufficiently high that it enables one to perform resonant photoemission studies with the excitation photon bandwidth smaller than the natural width of the inner-shell excited states. In this situation, the profile of the observed resonant Auger line is given by the convolution of the instrumental line-shape function and the Lorentzian profile with a width which is determined by the lifetime of only the final ionic state and not of the Auger state. Then, in turn, the instrumental function is given by the convolution of the monochromator and analyzer line-shape functions as well as by the Doppler broadening due to thermal motion of the sample gas. With the decrease of the monochromator and analyzer bandwidths, the Doppler broadening eventually governs the instrumental function as long as the gaseous samples are at room temperature in the gas-cell. Recently, a very interesting modification of the experiment was suggested which strongly suppresses the Doppler broadening and, thus, essentially improves the final resolution (Ueda et al., 2003a; De Fanis et al., 2004a). The Doppler energy shift is determined by the scalar product of the momentum vector of the emitted electron and that of the emitter. For this reason, the Doppler effect can be suppressed, if one uses as a target a molecular beam with a negligible transverse velocity component and if the emitted electrons are sampled perpendicular to the molecular beam. Using such a set-up, one can successfully achieve an unprecedented resolving power of more than 10,000 for the energies of the Auger electrons. This resolution enables one to measure the natural widths of the inner-valence excited states (Ueda et al., 2003a; De Fanis et al., 2004a). 2.2. Spin polarization measurements for Auger electrons Even more information about the resonant Auger process can be obtained by means of spin resolved experiments. Starting from the pioneering experiments by Stoppmanns et al. (1992), Kuntze et al. (1993, 1994a) and David et al. (1994, 1998) (see also review papers by Heinzmann (1996a, b)), spin polarization measurements for Auger and resonant Auger electrons have added a new dimension to the investigation of the Auger emission of atoms and provide a very sensitive test-ground for the theoretical description of autoionization processes (Kuntze et al., 1994b; Snell et al., 1996, 1999a, b, 2002; Hergenhahn et al., 1999; Schmidtke et al., 2000a, 2001; Drescher et al., 2003; Lohmann et al., 2005). Moreover, spin resolved experiments have been recognized as a necessary component for performing a complete experiment (see Section 3.3). The technique applied in spin resolved measurements of Auger electrons is quite similar to that utilized in more advanced spin polarization measurements of photoelectrons (Heinzmann and Cherepkov, 1996). Note that the dynamical spin polarization of Auger electrons is usually small (see Section 3.1.6), yet a non-zero spin polarization can arise when the atoms are excited by linearly polarized light. A much stronger spin polarization effect typically appears when the exciting photons are circularly polarized. In the latter case, the Auger electrons are strongly spin-polarized due to the transfer of polarization from the light to the intermediate atomic states and, finally, to the emitted electrons. Modern elliptical undulators provide intense circularly polarized photon beams suitable for spin and angle resolved measurements of Auger and resonant Auger electrons.


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Fig. 1. Experimental set-up for measuring the spin polarization of emitted electrons. From Drescher et al. (2003).

As an example, we consider the experimental set-up (Fig. 1) used at BESSY II SR facility in Berlin (Drescher et al., 2003). A high flux of wavelength-tunable circularly polarized light is provided by the beam-line UE56-PGM. The degree of circular polarization is Pcirc > 0.98. The photon flux of more than 1013 photons/s for a 130 meV bandwidth was achieved in the ionization volume located directly above a gas-inlet tube. The wavelength selection is performed by a planar grating monochromator, while the emitted electrons were energy analyzed in a simulated hemispherical electron analyzer, then guided by the electron lens system (Schmidtke et al., 2000a) and finally spin-analyzed in a spherical retarding-field Mott polarimeter (MÝller et al., 1995), operated at 45 keV electron scattering energy. In order to measure two components of the spin polarization vector, a special vacuum chamber was built that can be rotated independently around two perpendicular axes. The first rotation axis is the photon beam direction; the corresponding azimuthal angle can be varied between -180 + 180 . The rotation of the chamber around the second axis perpendicular to the photon beam (see Fig. 1) is accomplished by bellows allowing a variation of the inclination angle within +50 + 130 . Magnetic fields were controlled by a combination of Helmholtz coils and -metal shields around the analyzer and the electron lens system. For minimizing the asymmetry of the apparatus, the helicity of the undulator radiation was reversed by a longitudinal shift of the magnet arrays. Since the measured asymmetry must reverse its sign, any asymmetry due to misalignment or residual magnetic fields can efficiently be compensated. The spin polarization was measured with the electron analyzer resolution of 350 meV. At background pressure of about 10-4 mbar in the vacuum chamber, the count rate obtained for each detector was typically 20 s-1 , with an electronic background of about 5 s-1 . 2.3. Fluorescence spectroscopy Further information on the photoionization process, and which is complementary to the electron analysis in Auger and resonant Auger processes, can be obtained from fluorescence spectroscopy of the residual photoion. Indeed, fluorescence spectroscopy in the UV/visible as well as in the VUV wavelength regime has been found to be a powerful tool for such studies, provided that the ion is formed in an excited fluorescing state. In a typical experimental set-up for fluorescence spectroscopy used at the SU6 beam-line at SuperACO in Orsay (Meyer et al., 2001a) and at the "Circular Polarization" beam-line at ELETTRA in Trieste (O'Keeffe et al., 2004), the UV/visible photons are emitted from the interaction volume which is given by the intersection of the exciting XUV radiation and the gas target. These photons are collected by a convex lens and refocused outside of the vacuum chamber onto the entrance slit of a spectrometer. In some experiments, the collection efficiency was doubled owing to the installation of a spherical mirror. In this case, up to 15% of the emitted photons can be collected, but the overall efficiency is defined by the acceptance of the spectrometer, i.e. the ratio between the size of the first optical element and the focal distance. A high-resolution spectrometer was used to separate the different emission lines and to allow for a state selected analysis of the fluorescence lines. Spectral resolution of up to 0.08 nm is easily achieved in the UV/visible wavelength regime by means of different dispersion gratings. For the detection of the fluorescence, a liquid nitrogen cooled CCD detector can be utilized in order to facilitate the simultaneous recording of a wide wavelength interval (typically 30 and


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Fig. 2. Experimental geometries to measure the alignment of the photoions by means of fluorescence spectroscopy. Left panel: determination of the angular asymmetry parameter Fl of the radiative emission (from Mentzel et al., 1998). Right panel: determination of the degree of linear polarization of the fluorescence (from O'Keeffe et al., 2004).

200 nm for a 1800 lines/mm and 300 lines/mm grating, respectively), while the noise level is kept extremely low. In this way, fluorescence transitions with count rates of less than 1 photon/s can still be measured with reasonably high statistics. In general, the gas pressure in the interaction volume is kept low in order to avoid collisions and a subsequent depolarization of the observed fluorescence. Typically, the experiments are performed under a background pressure in the experimental chamber of some 10-5 mbar. The local pressure in the interaction volume, i.e. at the outlet of the gas capillary, is estimated to be one or two orders of magnitude higher. After spectral identification of the individual ionic states, the fluorescence spectroscopy can be utilized to provide access to the alignment and orientation of the photoion (see Section 3), following the first-step Auger emission. In fact, the alignment A20 (J ) of the residual ion can be determined either via the angular distribution of the fluorescence or by analyzing its linear polarization. In contrast, the orientation A10 (J ) is accessible only by measuring the degree of circular polarization of the fluorescence. In Fig. 2a, a schematic representation is given for the measurement of the angular asymmetry parameter of the fluorescence (Mentzel et al., 1998). Similar to the experiments on the angular distribution of photoelectrons, the fluorescence intensity is determined hereby at two observation angles with respect to the polarization vector of the exciting linearly polarized synchrotron radiation, typically at = 0 and 90 . The alignment parameter A20 (J ) is then obtained in the standard way by using expression (41) of Section 3 (Berezhko and Kabachnik, 1977; Balashov et al., 2000). This method was applied, for instance, in experiments using SR beam-lines at BESSY I and II in Berlin delivering rotatable linear polarization of the exciting photons, which makes it possible to perform experiments at a fixed spectrometer geometry (Mentzel et al., 1998; Schmoranzer et al., 1997; Ehresmann et al., 1998; Lagutin et al., 2000; Demekhin et al., 2005). The main advantage of this method is that it can be applied not only in the visible, but also in the VUV wavelength region, where the possibility to use transmission filters and optics is strongly reduced. In the UV/visible wavelength region, there is another possibility for determining the alignment of the residual ions. This is given by the analysis of the fluorescence polarization (Fig. 2b) as described, for instance, by O'Keeffe et al. (2004). After the excitation of the target atoms with linearly polarized synchrotron light, the degree of polarization of the fluorescence is measured using a commercial sheet polarizer. The measurements are performed by recording the spectral intensities with the axis of the polarizer placed parallel (Ipara ) and perpendicular (Iperp ) to the electric field vector of the SR. These experiments yield the degree of linear polarization of the fluorescence PL = I I
para para

-I +I

perp perp

,

(1)

which is connected to the alignment of the fluorescing ionic states by expression (42) of Section 3. In the same way, the orientation of the residual ions can be obtained through the analysis of the circular polarization of fluorescence (O'Keeffe et al., 2004) after an excitation with circularly polarized synchrotron radiation (Fig. 3). Here a quarter wave plate is introduced in the fluorescence pathway transforming the circular polarization of the fluorescence light into linear polarization, for which the conventional analysis with sheet polarizers can be applied. The spectral intensities are measured again for two angles of the quarter wave plate, i.e. at =+45 and -45 for right and left


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Fig. 3. Experimental geometry to measure the orientation of the photoions by fluorescence spectroscopy. From O'Keeffe et al. (2004).

handed polarization, respectively. The circular polarization of the fluorescence transition determined in this way, PC = IR - I L , IR + I L (2)

is related to the orientation of the ionic state through expression (45) as discussed below in Section 3. If only one particular (fluorescence) transition is to be measured, the wavelength selection can also be obtained by means of a narrow band path filter (e.g. Schill et al., 2003). While this set-up has a higher overall transmission than the spectrometer, it requires a slightly different geometry, i.e. the fluorescence is detected under a small angle with respect to the direction of the SR beam, in order to avoid direct exposure of the detector to the exciting XUV radiation. The principal interest for this arrangement arises from the possibility to use it for coincidence experiments, since the higher transmission facilitates the use of single photon counting devices, such as conventional photomultipliers. For all experimental geometries and acquisition methods, in general, the purity of the polarization of the exciting photons has to be taken into account in the final analysis of the data. In addition, all possible polarization dependences of the light transmission upon the optical elements have to be determined and adequate corrections to the experimental spectra need to be done. For studies on the resonant photoionization process, which typically results in numerous fluorescence lines in the entire wavelength region under consideration, a direct and convenient way to control the quality of the set-up is given by the analysis of those fluorescence lines, which are not influenced by the polarization of the incident radiation (Meyer et al., 2001a). For example, the initial fluorescing states with total angular momentum 1 J = 2 should not show any alignment and no polarization dependence of the fluorescence intensities. Therefore, all observed intensity variation against any change of linear polarization are then a direct measure of experimental artifacts. 2.4. Coincidence experiments 2.4.1. Electron­electron coincidences The first experiment in which electron­electron coincidence (e­e coincidence) techniques were utilized for studying the cascade Auger transitions has shown the usefulness and advantages of the method for unraveling the complex electron spectrum at low kinetic energies (von Raven et al., 1990). One could therefore expect that coincidence measurements on the angular correlation of two subsequently emitted electrons might provide even further details (Kabachnik, 1997; Kabachnik et al., 1999). The recent development of the e­e coincidence experiments have confirmed these expectations. The angular correlations between two emitted electrons from an Auger cascade have been measured by various groups (Becker and Viefhaus, 1996; Ueda et al., 1999a, 2001, 2003b; Wehlitz et al., 1999; Turri et al., 2001; Viefhaus et al., 2005; Da Pieve et al., 2005). Note that we shall not discuss here the experiments in which a direct photoelectron emission is measured in coincidence with an Auger electron. Pioneered by KÄmmerling and Schmidt (1991), these experiments are devoted mainly to the study of photoionization and are beyond the scope of this review. Different experimental groups have used different set-ups for the e­e coincidence measurements. Perhaps, the simplest lay-out was used by Ueda et al. (1999a, 2001, 2003b). Here we describe it in more detail. In the interaction region, the linearly polarized photon beam from the 24-m spherical grating monochromator is focused and merged with an effusive gas beam. This monochromator was installed on the BL-16B undulator beam-line at the Photon Factory in Tsukuba (Shigemasa et al., 1998). Fig. 4 displays schematically the geometry of this experimental set-up which is basically the same as the one discussed in Section 2.1. Two 150 spherical sector analyzers with the radius of 80-mm were used; they were installed perpendicular to the photon beam. While the first analyzer for the detection of the


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x
analyzer 1 (resonant Auger)

E vector


z

h
y
analyzer 2 (2nd step Auger)

Fig. 4. The experimental geometry for the study of the Auger cascades by measuring the angular correlation between two emitted electrons. From Ueda et al. (2001).

Fig. 5. Setup for the e­e coincidence measurements using multiple time-of-flight electron­electron coincidence spectroscopy. The plane of linear polarization was horizontal (parallel to y axis). Adapted from Viefhaus et al. (2004a).

resonant Auger electrons was mounted on a turntable whose axis of rotation was aligned to coincide with the incident light beam, the second analyzer was installed in such a way that it detected the (second-step Auger) electrons in the direction perpendicular to the linear polarization axis of the incident light. The coincidence rate between the first- and second-step electrons was then measured as a function of the detection angle of the first-step resonant Auger electrons, relative to the linear polarization axis of the incident light. The rate of false coincidences, i.e. the detection of two electrons that do not belong to the same ionization event, was in general significantly smaller (typically 10%) than the true coincidence rate. Only the true coincidences were used in the analysis by subtracting the false coincidence background. In order to remove the effects of the fluctuation of the coincidence counting rates as a function of time, the coincidence counts were normalized by the corresponding electron counts as detected by a fixed analyzer. This experimental lay-out was used for the study of the resonant Auger­normal Auger cascades in Ar (Ueda et al., 1999a, 2001) as well as in Kr and Xe (Ueda et al., 2003b). A more elaborated experimental set-up with multiple time-of-flight spectrometers has been developed by the group of U. Becker at BESSY II in Berlin and is shown in Fig. 5 (Becker and Viefhaus, 1996; Viefhaus et al., 2004a, b, 2005). In this set-up, monochromatic synchrotron radiation crosses an effusive beam of the target gas, and the electrons produced in the interaction region are then analyzed by using a combination of six time-of-flight spectrometers, mounted in a plane perpendicular to the incoming photon beam. Two types of analyzers with different drift-tube length ( 450 mm and 150 mm) were utilized. The angles of these analyzers with respect to the horizontal plane were set to =-54.7 , 0 ,22.5 ,45 ,90 , and 180 . The electrons are accelerated before they enter the drift tube and are detected by a chevron stack of microchannel plates. The actual acceleration potential depends on the length of the drift tube as well as on the time structure of the pulses in the storage ring which produces the SR pulses. In all cases, the parameters of the ring were such that electrons having kinetic energies larger than about 0.5 eV could be detected without interference with any electron coming from subsequent pulses. Because of the time-of-flight measurement, the kinetic energy resolution does not have a fixed value for the entire energy range. Typical values for the resolution are about 1­2% for long analyzers and 3­6% for short analyzers, respectively.


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P.M. Tube

Window

Polarising Filter Blue Filter Vaccum Tank Lens h e- Ca Oven

Hemispherical Electron Analyser To Pump
Fig. 6. The experimental equipment for the e­ coincidence measurements of photoinduced autoionization electrons and subsequent fluorescence photons. The primary photon beam is perpendicular to the plane of the figure. From Beyer et al. (1996).

In the set-up of Fig. 5, the data acquisition is such that all time-of-flight electron spectra are recorded in parallel. Apart from the non-coincident spectra, all double coincidence events are stored separately for each pair of analyzers (15 in total). Typical recording times for one photon energy are in the order of 1000 s. Usually a set of 50­200 spectra has been recorded for each case under study. The details of calibration and transmission corrections can be found in the original papers cited above. For the coincidence spectra, the data processing includes a correction for random coincidences. This correction takes into account the calculated probabilities as determined directly from the corresponding singles spectra and by including so-called dead time effects. Typical total rates for the remaining physical (`true') coincidences are in the order of 10 Hz. The described experimental set-up was used for studying the Auger cascades in Kr and Xe as well as for investigation of the double Auger process (Becker and Viefhaus, 1996; Viefhaus et al., 2004a, b, 2005). An even more sophisticated set-up for e­e coincidence measurements has been developed by the Italian group at the Gas Phase Photoemission beam-line at the ELETTRA storage ring in Trieste (Gotter et al., 2001; Turri et al., 2001). The combination of ten hemispherical electrostatic analyzers enables one to study the angular correlations between emitted electrons not only in the plane perpendicular to the photon beam, but also out-of-plane correlations. Such an experimental arrangement was used for studying the angular correlations between resonant Auger and subsequent normal Auger electrons in Ne (Turri et al., 2001; Da Pieve et al., 2005) as well as in several experiments on solid surfaces (Stefani et al., 2002). 2.4.2. Electron­photon coincidences The electron­photon coincidence (e­ coincidence) technique has been used for studying the Auger (autoionization) electron­fluorescence photon cascades in metallic vapors, beginning with the pioneering work by Beyer et al. (1995). For these coincidence measurements, the experimental set-up is shown in Fig. 6, where the incident photon beam propagates perpendicular to the plane of the figure. Photons in the range 20­30 eV with a resolution of 50 meV were provided by a toroidal grating monochromator fitted to the beam-line 3 at the Daresbury Synchrotron Radiation Source. A capillary light guide was used to bring the photons close to the interaction region from the exit slit of the monochromator. The oven shown in Fig. 6 is a compact design published by Ross (1993), and gave a vapor density of 10-3 torr in the interaction region. The electron analyzer, a 150 hemispherical sector of 90 mm mean radius, was fitted with a special shield at its entrance, in order to prevent its entrance slit assembly from being exposed to the metal vapor when moving between = 0 and 90 positions. The asymmetry parameter of the angular distribution was calculated from the two measurements at the angles given above. The electron analyzer was calibrated for variations in its efficiency with electron energy by making a series of measurements with argon gas.


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The fluorescence photons were measured at a fixed direction perpendicular to the incoming light and to its linear polarization. In fact, in order to measure the polarization of fluorescence two measurements were needed, one with the transmission axis of the polarization filter (shown in Fig. 6) parallel to the incoming light beam and one perpendicular to it. This arrangement was used both, for non-coincidence measurements and for those measurements where the autoionization electron and the fluorescence photon were detected in coincidence. Experiments with this set-up have been carried out for the excitation region of the 3p­3d resonances in Ca (Beyer et al., 1995, 1996; West et al., 1996) and 4p­4d resonances in Sr (West et al., 1998; Ueda et al., 1998). It is interesting to mention here another type of the e­ coincidence experiments, in which the angular correlations were measured in the photoinduced cascade of the K X-ray fluorescence and the subsequently emitted L2,3 ­M2,3 M2,3 Auger electrons, following the 1s photoionization in Ar (Arp et al., 1996). In these measurements, the Auger spectrum was observed in coincidence with the K photons by using a cylindrical mirror analyzer mounted with its axis along the principal axis of polarization of the incident X-ray beam from the Brookhaven NSLS storage ring. Photons were collected either along this axis or perpendicular to it. 3. Basic theory To describe angular correlations in Auger and fluorescence cascades which follow the resonant core excitation, one can use the same principal models and approaches as for the description of an usual Auger process. In the simplest case, the excitation and cascade decay of a strong isolated resonance may be treated in a stepwise model, similar to the two-step model commonly applied for describing Auger transitions (Mehlhorn, 1990). In this approach, the photoexcitation and subsequent decay of the atom can be presented as follows: + A( g Jg ) A ( 0 J0 ) A+ ( 1 J1 ) + e
1 2

(3a) (3b) (3c)

A2+ ( 2 J2 ) + e

A+ ( 2 J2 ) + . In step (3a), the photon is absorbed by the atom in the ground state with the total angular momentum Jg , and a resonant state with well defined angular momentum J0 is formed. This resonance state autoionizes then in the step (3b) to some intermediate ionic state | 1 J1 by emitting an electron e1 (first-step decay). For a core-excited resonance A , this step corresponds to the resonant Auger decay. As a result, an excited state of a singly charged ion is produced. In step (3c), thereafter, the intermediate state decays further to the final state | 2 J2 under the emission of either a (second) Auger electron e2 or a fluorescence photon (second-step decay). In Eq. (3) and below, the i denote all other quantum numbers which are necessary for specifying the considered states uniquely. Similar sequence of processes occurs when the core is ionized due to the photoabsorption, i.e. if a photoelectron is emitted in the first step. In this case, the intermediate state belongs to the doubly charged ion: + A( g Jg ) A+ ( 0 J0 ) + e
ph 1 2

(4a) (4b) (4c)

A2+ ( 1 J1 ) + e

A3+ ( 2 J2 ) + e

A2+ ( 2 J2 ) + . In the stepwise model, obviously, the direct (non-resonant) transitions are neglected in the theoretical description. This model is appropriate, if the resonant process dominates in the transitions (3) and (4), i.e. in the region near the center of the corresponding line in the Auger spectrum which gives the main contribution to the integrated Auger yield. On the far wings of the Auger line, in contrast, the cross section of the resonant process drops down rapidly and the contribution from the direct transition, i.e. the double ionization of the ions for the normal Auger decay or the direct single photoionization for the resonant Auger decay, cannot be neglected usually, since they lead to the same final state of the ion. In the following, we often suppose that the stepwise model is appropriate and that the decay cascade proceeds


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through a well defined and isolated intermediate state. If there are several overlapping resonances or/and intermediate states, the simple stepwise model is not applicable. However, the model can be modified to include also the situation of the overlapping states. This case will be considered in Section 3.1.8. In Section 3.4, we discuss the cases where the stepwise model breaks down and should be replaced by more advanced theories which are sometimes referred to as one-step approaches. 3.1. General formalism for correlation and polarization studies Within the stepwise approach as described by the schemes (3) or (4), the angular distribution and polarization of the emitted particles as well as their angular correlation, can be described using the well developed theory of spin density matrix and statistical tensors (Devons and Goldfarb, 1957; Ferguson, 1965; Blum, 1996). Applications of this formalism to particular problems of the (resonant) Auger decay and Auger cascades are presented in detail in the book by Balashov et al. (2000), together with the derivations of all basic formulas. Here, instead, we shall give only the final relations which are needed for the subsequent discussion. Note that the density matrix formalism is very general and can be also used in the case of coherently excited overlapping states. A necessary modification of the angular correlation formula for this case is also presented below. 3.1.1. Characterization of a polarized atomic state Obviously, the angular distribution and polarization of the decay products depend on the polarization state of the decaying atom or ion. Therefore, the knowledge of the polarization of the decaying state is essential for understanding the decay dynamics and for analyzing the cascade transitions. In general, the polarization state of an ion (atom) can be described by the statistical tensors kq ( J, J ) (k is the rank of the tensor, q is its projection) which are determined by the amplitudes of the production process (Blum, 1996; Balashov et al., 2000). By definition, the statistical tensors are related to the spin-density matrix of the system as
kq

( J, J ) =
MM

(-1)

J -M

(J M , J - M | kq ) JM | | J M ,

(5)

where M and M are the projections of the total angular momenta J and J , respectively, and (j1 m1 ,j2 m2 | jm) is the Clebsch­Gordan coefficient. For an isolated state, moreover, it is convenient to introduce the reduced statistical tensors Akq ( J) =
kq 00

( J) ( J)

,

(6)

where we use the abbreviation kq ( J) kq ( J, J). For an atomic state with total angular momentum J , the rank and the projection of the statistical tensor are restricted due to the conditions k 2J and -k q k , respectively. If all substates with different projections M are equally populated, the state is isotropic, and all statistical tensors are equal to zero except 00 ( J). If the population of different magnetic substates of the state J is different but independent of the signs of the projection M , the state is called aligned. In this case, only the statistical tensors with even ranks are non-zero, while the state of an ion is called oriented, if at least one odd rank statistical tensor is non-zero. We remind the reader also that, if the system has axial symmetry and the z axis is taken as the symmetry axis, then only the statistical tensors with q = 0 are non-zero. In this case in particular, an isolated state is characterized by the reduced statistical tensors Ak 0 ( J) from which the most important for the following discussion are the parameters (tensors) of orientation, A10 ( J), and alignment, A20 ( J). 3.1.2. Polarization of photoinduced Auger states Let us consider first the production of the Auger state A+ ( 0 J0 ) by means of photoionization of an initially unpolarized atom (4a), and for cases where the photoelectron is not detected. Then, the polarization state of the photoion A+ ( 0 J0 ) is determined by the reduced statistical tensors (Buúert and Klar, 1983; Balashov et al., 2000, Eq. (2.163)) Akq ( 0 J0 ) = 3 â
lj J J kq

N

-1

^ J0
J +J +j +J0 +Jg +1

(-1)

^^ JJ

J J0

J0 J

j k

1 J

J 1

Jg DJ0 k

lj J

D

J0 lj J

,

(7)


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and where N = lj J |DJ0 lj J |2 denotes a normalization constant. In this and the following equations, we use the standard notation for the Wigner nj -symbols, a 2a + 1, and DJ0 lj J 0 J0 ,lj : J D g Jg to abbreviate the ^ reduced photoionization amplitude, with J referring to the total angular momentum of the system `ion+photoelectron'. The outgoing electron is described by the orbital and total angular momenta l and j , respectively. The statistical tensors of the photon, kq , which are present in Eq. (7), are expressed as usual in terms of the Stokes parameters of the ionizing photon beam. If we choose the quantization axis (z-axis) along the incident radiation beam, and if the dipole approximation is applied, then the non-vanishing statistical tensors may be presented in the form
10

p3 = , 2 1 = p3 , 2

20

1 = , 6

2 ±2

1 =- (p1 ip2 ), 2 i = p2 , 2 1 = (1 - p1 ) 4

(8)

while we have
1±1 20

1 =- (1 + 3p1 ), 26

2±1

2±2

(9)

1 if the z-axis is taken perpendicular to the radiation beam. The zero rank tensor is always 00 = . The Stokes parameters 3 p1 =+1(-1) and p2 =+1(-1) describe radiation which is completely linearly polarized in the direction = 0 , i.e. along x -axis ( = 90 , i.e. along y -axis), and = 45 ( = 135 ), respectively. The Stokes parameter p3 =+1(-1) corresponds to the positive (negative) value of the photon helicity. More details on the statistical tensors of photons can be found in Ferguson (1965) and Grum-Grzhimailo (2003). According to Eq. (7), the statistical tensors of the photoion and the ionizing photon are proportional to each other and, hence, the photoion cannot possess tensors with other ranks and projections than those of the incoming photon. For this reason, the photoion can only be aligned (k = 2) but not oriented, if an unpolarized atom is photoionized by means of unpolarized or linearly polarized light. Similarly, the rank of the tensor cannot be larger than 2 (in dipole approximation). Usually, the alignment A2q ( 0 J0 ) of the photoions is small (about 0.1) for all photon energies except in the near-threshold region and in the region of Cooper minimum (Berezhko et al., 1978a; Kleiman and Lohmann, 2003). In contrast, circularly polarized light can produce also an orientation of the photoion, i.e. can give rise to a non-zero first rank tensors A1q ( 0 J0 ). The orientation tensor is usually large (about unity) at all energies (Kabachnik and Lee, 1989; Kleiman and Lohmann, 2003). A particularly simple expression for the statistical tensors is obtained for a target atom with Jg = 0. In this case, J = J = 1 and only the squares of the dipole photoionization amplitudes determine the statistical tensors:

Akq ( 0 J0 ) = 3

kq

N

-1

^ J0
lj

(-1)

j +J0 +k +1

1 J0

J0 1

j |DJ0 k

lj 1

|2 .

(10)

For the resonant photoexcitation (3a), the reduced statistical tensor of the Auger state is a pure algebraic value and independent of the photoexcitation amplitudes (Balashov et al., 2000, Eq. (2.15)) Akq ( 0 J0 ) = 3(-1)
J0 +Jg +k +1

^J J0 0 1

1 J0

Jg k

kq

.

(11)

Moreover, in the resonant case, the alignment of the photoexcited state is usually rather large (of the order of 1) and, thus, the anisotropy and polarization of the subsequently emitted radiation, electron or photon, is also larger, when compared with the case of photoionization. If the target atom has closed subshells (Jg = 0) and the light is linearly polarized along the z axis, the alignment of the excited state is described by the single component A20 ( 0 J0 = 1) =- 2. 3.1.3. General equations for angular correlations in cascades Next, we consider a simple case of a cascade with two consecutive decays of the atom with either an electron or a photon emission. According to the stepwise model, we assume that the atomic transitions occur in a sequence and that the atomic states involved at each stage have a well-defined total angular momentum and parity. Then, a two-step cascade process can be presented as follows: A0 ( 0 J0 ) A1 ( 1 J1 ) + b1 (j1 ) A2 ( 2 J2 ) + b2 (j2 ). (12a) (12b)


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Here bi (ji ), i = 1, 2 refers to either the electron or photon which carries away the (total) angular momentum ji . The two particles are emitted in directions, which are characterized in the laboratory system by the angles i and i (i = 1, 2), respectively. We suppose, moreover, that the detectors are insensitive to the polarization (spin) of the particles. Then, the expression for the angular correlation function of the two emitted particles can be written in the general form (Kabachnik et al., 1999; Balashov et al., 2000, Eq. (3.67)): W(1 , where
k0 q0 1

; 2 ,

2

)=c
k0 q0 k1 k2

G

k1 k2 k0 k0 q0

( 0 J0 ) Yk1 (1 ,

1

) Yk2 (2 ,

2

)

k0 q0

,

(13)

( 0 J0 ) are the statistical tensors of the initial (Auger) state in the laboratory system, c is a normalization
1

constant, and Yk1 (1 ,

) Yk2 (2 ,

2

)

product of two spherical functions Ykq ( , ). Each term in the sum (13) consists of three factors. The last two factors, the statistical tensors and the bipolar harmonics, are determined by the initial conditions and the geometry of the experiment. Only the first factor, the value Gk1 k2 k0 , is determined by the dynamics of the decay process; it contains the amplitudes of the decay processes and various angular momentum coupling coefficients. At the moment, we neglect here all depolarization effects between the two decay steps. Owing to the stepwise treatment of the decay cascade, each of the coefficients Gk1 k2 k0 can be separated further into two factors, each of them depending only on one of the transitions, either the first- or second-step decay: Gk1
k2 k0

k0 q0

are the bipolar harmonics (Varshalovich et al., 1988), that is, the tensorial

=B

k1 k2 k0

(J0 ,J1 )Ak2 (J1 ,J2 ).
-1

(14)

Explicitly, these factors are given by Bk1
k2 k0

(J0 ,J1 ) = (-1)

k1 -k0

^ J0 j1 j1 k1

^ J1
j1 j
1

C

k1 0

(j1 ,j1 )


â and Ak2 (J1 ,J2 ) = (-1)

J1 J1 k2 ^ J1

J0 J0 k0

J1 ,j1 T J

0

J1 ,j1 T J

0

,

(15)

J1 +J2

-1 j2 j
2

C j2 J1

k2 0

(j2 ,j2 ) J2 ,j2 T J J2 ,j2 T J


â (-1)

j

2

J1 j2

J2 k2

1

1

,

(16)

where Jb ,j T Ja denotes the decay amplitude for the transition of the ion from state Ja to the state Jb under emission of a particle with total angular momentum j ; the Ck 0 (j , j ) refer to the corresponding radiation parameters as defined by Devons and Goldfarb (1957) or Balashov et al. (2000). For Auger electrons, these radiation parameters are given by Ck 0 (lj , l j ) = (-1)j 4
+1/2

j ^^ ^ ^ l l j j (l 0,l 0 | k 0) l

l j

k

1 2

,

(17)

while, for photons (within the dipole approximation), they take the form Ck 0 ( ) = 3 (11, 1 - 1 | k 0), 16 k = 0, 2. (18)

Since all atomic states involved are supposed to have a well-defined parity, and because parity is conserved in the decay processes, it is quite easy to show that only even values of k1 and k2 can occur in Eq. (13). Moreover, the values of k0 and k2 are restricted by the conditions 0 k0 2J0 and 0 k2 2J1 , whereas the summation in Eq. (13) must obey the triangle rule |k0 - k2 | k1 k0 + k2 as seen, for instance, from the angular­momentum coupling in the Clebsch­Gordan expansion of the bipolar spherical harmonics. Nevertheless, the angular correlation function (13) is very complex and quite difficult to analyze in general terms. However, by selecting the initial conditions and the


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geometry of the experiment, it is possible to simplify both, the angular correlation function and its analysis for a given experiment. For example, if both detectors are placed in a plane perpendicular to the symmetry axis of the initial state (z axis), the angular correlation function only depends on the two angles 1 and 2 , and can be expressed in terms of m the associated Legendre polynomials Pn (cos i ), i = 1, 2. 3.1.4. Orientation and alignment transfer in Auger and fluorescence cascades The angular distribution and polarization of the emitted particles are determined by the transition amplitudes, the quantum numbers of the initial and final atomic (or ionic) states as well as the polarization parameters of the initial state. When the initial state of a particular transition is formed already in the course of a prior cascade, the evolution of the polarization characteristics of the atoms produced in the cascade (polarization transfer) is of great importance. If the target atom is initially unpolarized, the polarization (i.e. the alignment and/or orientation) is introduced in the system by the photon. In the first decay step, this polarization causes the anisotropy of the angular distribution (and polarization) of the emitted particle and is partly transferred to the residual ion. The polarization of the ion, in turn, affects the anisotropy of the next-step radiation and is partly transferred further to the final state of the ion. A simple but well-known case for the transfer of polarization is the anisotropy of the second-step decay, see the processes (3c) or (12b) above. The angular distribution in the second-step emission is determined by the statistical tensors of the residual ion A+ ( 1 J1 ) as produced in the first-step transition. For the case of electron emission, if the emission direction is not detected, the reduced statistical tensors of the residual ion can be written in the form (Balashov et al., 2000, Eq. (3.59)) Ak1 q1 ( 1 J1 ) =
k1 k0 q1 q0

A

k0 q0

^^ ( 0 J0 )J0 J1
l1 j1

(-1)

J0 +J1 +j1 +k1

J0 J1

J0 J1
1 j1

k0 j1

1 j1

,

(19)

where we have introduced the relative partial decay width into the channel
1 j1

, (20)

=|VJ1

l1 j1 J0

|2
1

|V
j1

J1 l1 j1 J0

|2 ,

and where the Auger decay amplitudes are abbreviated by VJ1 lj J J1 ,lj T J . In this case, the reduced statistical tensors (19) include only the square moduli of the decay matrix elements and appear insensitive to the phase differences between the amplitudes. If in the decay a photon is emitted in a dipole transition and if its emission direction is not detected, the reduced statistical tensors of the residual atom (ion) are independent of the dipole amplitudes and are given by (see Balashov et al., 2000, Eq. (3.65)) Ak1 q1 ( 1 J1 ) =
k1 k0 q1 q0

A

k0 q0

( 0 J0 )(-1)

J1 +J0 +k1 +1

^^ J J 0 J1 0 J1

J1 J0

1 . k1

(21)

Expressions (19) and (21) illustrate a rather obvious result: namely, if the direction of the emitted particle is not detected, the residual atom (ion) can have only those statistical tensors which were present already in the representation of the initial state. In particular, the residual ion is always characterized by tensors with rank k1 not larger than the rank of the decaying system, and the absolute value of the final state normalized tensor component cannot be larger than the corresponding component of the initial tensor. In other words, the atomic polarization generally decreases in the course of the decay cascade, provided that the direction of the ejected particles is not detected. An interesting general theorem for the polarization transfer in radiation cascades has been proven by Korenman (1975). In the framework of dipole transitions, he showed that if |J0 - J1 |= 1, certain quantities proportional to the statistical tensors k0 q0 ( 0 J0 ) are conserved. This theorem enables one to determine the polarization of the final state of some cascades (without any computations for the intermediate states) provided that the total angular momenta of the atomic states decreases (increases) always by 1. Owing to the variety of atomic pathways, the applicability of this finding in atomic (ionic) cascades is however limited. 3.1.5. Angular distribution of Auger electrons If only Auger electrons are detected by a non-coincidence measurement as, for example, the first-step Auger electron A ( 0 J0 ) - A+ ( 1 J1 ) + e(lj ) (22)


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in the decay of the resonance state A ( 0 J0 ), their angular distribution can be written in the general form (Berezhko et al., 1978b; Balashov et al., 2000, Eq. (3.10)) kmax k W0 1 J1 4 1 + Akq ( 0 J0 )Ykq ( , ) , (23) W 1 J1 (, ) = k 4 2k + 1
k =2,4,... q =-k

where Akq ( 0 J0 ) are the reduced statistical tensors of the decaying state and for the electron emission
k

k

are the intrinsic anisotropy parameters

=N

-1

(-1)

J0 +J1 +k -1/2

^ J0
0

â
ll jj

J ^^ ^ ^ l l j j (l 0,l 0 | k 0) j

j J0

J1 k

l j

j l

k

1 2

VJ1

lj J

0

V

J1 l j J0

.

(24)

In this notation, N = lj |VJ1 lj J 0 |2 and the factor W0 1 J1 denotes the total decay probability for the transition | 0 J0 | 1 J1 , integrated over all angles of the outgoing electron. The anisotropy parameters k in Eq. (24) contain information about the dynamics of the decay, while the tensors Akq ( 0 J0 ) describe the polarization properties of the decaying state. Note that, whenever the initial state is produced by photoabsorption, the maximal rank of the statistical tensors (taken within the dipole approximation) is k = 2 and, therefore, only one term in the first sum of Eq. (23) is present. For some given transition | 0 J0 | 1 J1 , the ejected electron may still carry away (in general) different total angular momenta j and orbital angular momenta l . These different angular momenta are associated with the outgoing electron and refer to different decay channels of the transitions between the two levels | 0 J0 and | 1 J1 with welldefined angular momentum and parity. Due to parity conservation in the decay, however, l and l must be of the same parity and, consequently, k must be even in Eq. (23) because of the property of the Clebsch­Gordan coefficient with zero projections. In addition, the angular distribution (23) occurs to be sensitive to the phase difference between the decay matrix elements, quite in contrast to the integrated probability W0 which depends only on the square moduli of the decay amplitudes. In the sum (24), the coefficients are symmetric with respect to the interchange lj l j . Moreover, the angular distribution (23) is invariant with respect to the inversion: - , + because it contains only the spherical harmonics of even rank. Therefore, the angular distribution is not affected by the odd tensors of the decaying system and only the alignment of the initial state 0 J0 can be important, but not the orientation. Since the maximal complexity of the angular distribution is kmax 2J0 , the decay of all states with J0 < 1 is always isotropic, provided the parity is conserved in the process. When the decaying state is symmetric with respect to the z axis (symmetry axis), i.e. Akq ( 0 J0 ) q 0 , the angular distribution depends only on the angle between the symmetry axis and the linear momentum of the ejected electron (Berezhko and Kabachnik, 1977; Kabachnik and Sazhina, 1984): kmax W0 1 J1 1 + (25) W 1 J1 () = k Ak 0 ( 0 J0 )Pk (cos ) . 4
k =2,4,...

In this case, the angular distribution includes only the Legendre polynomials Pk (cos ) with even k and is always symmetric with respect to a reflection at the plane perpendicular to the symmetry axis: - . For the photoinduced Auger process, the angular distribution (25) simplifies to W
1 J1

() =

W0 4

1 J1

1 + P2 (cos ) =

W0 4

1 J1

[1 +

2

A20 ( 0 J0 )P2 (cos )],

(26)

provided the initial atom was unpolarized, because only the statistical tensors with ranks 0, 1, and 2 can be non-zero in the atomic photoionization or photoexcitation, if described within the dipole approximation. In fact, form (26) of the angular distribution for the photoinduced Auger emission coincides with that commonly applied for photoelectrons, but in the case of an Auger decay, the asymmetry parameter is now a product of two factors: = 2 A20 ( 0 J0 ). This factorization of the asymmetry parameter reflects the two-step model and is valid only for an isolated decaying state. The decay of overlapping resonances will be considered later.


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Fig. 7. The coordinate systems usually used for the description of the Auger electron spin polarization.

Note, that k reduces to a pure algebraic value independent of the Auger amplitudes, if only one decay channel with fixed l and j contributes to the decay:
k

= (-1)

J0 +J1 +k -1/2

J ^^ ^ J0 l 2 j 2 (l 0,l 0 | k 0) 0 j

j J0

J1 k

l j

j l

k

1 2

.

(27)

This case is realized, for example, if the angular momentum of the final state is J1 = 0. Expression (26) has been used widely in the analysis of experimental data. If the alignment of the decaying state is known (as, for example, in the resonant Auger process of initially unpolarized atoms), then Eq. (26) can be utilized for determining the intrinsic anisotropy parameter 2 , which bears information about the decay amplitudes including their phases. Conversely, if the 2 parameter is known (for example, Eq. (27)), then fitting the experimental data by Eq. (26) enables one to determine the alignment parameter A20 ( 0 J0 ) and, hence, to characterize the process of the Auger state production. If the fine-structure of the final states is not resolved by the detector, the angular distribution is obtained as the incoherent sum of the individual angular distributions of the decay (22) into the levels with different angular momenta J1 : W(, ) =
1 J1

W

1 J1

(, ).

(28)

In this case, the angular distribution is described by the same equation (26) but with the average anisotropy parameter ¯ ; this averaged parameter can be written in terms of the individual parameters as 1 J1 ¯=
1 J1 1 1 J1

W
1

1 J1

J1

W

0 J1

.

(29)

0

3.1.6. Spin polarization of Auger electrons The spin polarization of the Auger electrons is described by the polarization vector P. Its three components can be expressed in terms of statistical tensors of the Auger state and the corresponding intrinsic parameters (Klar, 1980; Kabachnik, 1981; Huang, 1982; Lohmann, 1998). Two coordinate systems, S(X Y Z ) and S (X Y Z ), are commonly used for describing the spin polarization, cf. Fig. 7. We restrict ourselves here to the simple case of a resonance (Auger) state, which is produced by a photon beam, starting from an unpolarized target atom. Let us begin with the coordinates associated with the Auger electron; the longitudinal spin component along the direction of the Auger electron emission from the initial state J can be presented as (Klar, 1980; Lohmann, 1998) Pz = A10 ( J) 1 cos , 2 A20 ( J)P2 (cos ) (30)

1+


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173

while the transverse spin component Px , which is laying within the (x - z) reaction plane as defined by the directions of the photon beam and the Auger electron, takes the form Px = 1+ A10 ( J) 1 sin . 2 A20 ( J)P2 (cos ) (31)

The transverse component Py , perpendicular to the reaction plane, is the same as in the S coordinates and given by (Klar, 1980; Kabachnik, 1981) Py = Py = A20 ( J) 2 sin 2 . 1 + 2 A20 ( J)P2 (cos ) (32)

In the S frame, fixed by the plane of the Auger emission in the laboratory, the spin polarization components are (Balashov et al., 2000, Section 4.2.2.2) Pz = A10 ( J)( 1 + 1 P2 (cos )) . 1 + 2 A20 ( J)P2 (cos ) (33)

for the component parallel to the photon beam direction, and Px =
3 4

1+

A10 ( J) 1 sin 2 2 A20 ( J)P2 (cos )

(34)

for the transverse component within the reaction plane. We note that the denominator of Eqs. (30)­(34) is proportional to the angular distribution of Auger electrons. The intrinsic spin-polarization parameters are expressed in terms of the Auger decay amplitudes as1 ^ ^ ^^ ^ -1 (-1)l -J -J1 -j j j l l (l 0,l 0 | 20) 2 = - 3 5JN
lj lj

â

j J

j J

2 J1 l 2
J +J1 +1/2

l

1 2 1 2 1

j j 2 (-1)

I[VJ1

lj J

V

J1 l j J

],

(35)

1

=



^ 2JN

-1

(-1)

l +j +j

^^ jj

ljj 1

J j

J j

1 J1

j 1 2

j 1 2

1 l VJ1
lj J

V

J1 lj J

,

(36)

= (-1)

J +J1 +1/2

^ 2 3JN l

-1 ll jj

(-1) j

l +j -1/2

^ ^ ^^ j j l l (l 0,l 0 | 20)

â

J j

J j

1 J1 l 2

1 2 1 2 1

j 1

VJ1

lj J

V

J1 l j J

.

(37)

Two other intrinsic parameters are the linear combinations: 1 = 1 + 1 and 1 = 1 /2 - 1 , respectively. In photoionization by unpolarized or linearly polarized light the produced ion is aligned along the beam direction or the photon polarization direction, respectively. As follows from Eqs. (30)­(34), the spin of the Auger electron in this case can only be oriented perpendicular to the plane as defined by the alignment axis and the direction of the electron emission. This effect is called the dynamic polarization of the Auger electron (Klar, 1980; Kabachnik, 1981). If the photoionization is produced by circularly polarized light, the ion is not only aligned but in general also oriented. In this case, the spin components of the Auger electrons in the reaction plane are non-vanishing too. This is an example of the polarization transfer process (Klar, 1980).
1 The corresponding Eqs. (4.101) and (4.106) in the book by Balashov et al. (2000) contain misprints.


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3.1.7. Polarization and angular distribution of fluorescence We consider now the angular distribution of photons emitted from an excited atomic state (c.f. processes (3c) and (12b)): A ( J) - A(
f Jf

)+ .

(38)

If the detector is not sensitive to the photon polarization, the angular distribution of photons in the dipole approximation is of the form (Balashov et al., 2000, Eq. (3.35)) 2 4 W0 A2q ( J)Y2q ( , ) , (39) 1+ 2 W f Jf (, ) = 5 4
q =-2

where the intrinsic anisotropy parameters for photoemission are given by
k

=

3^ J(-1) 2

J +Jf +k +1

J 1

J 1

k Jf

.

(40)

In the dipole approximation no statistical tensors of the decaying atom with rank k> 2 can affect the angular distribution. This is true for both, the polarization-sensitive and polarization-insensitive detectors, and is the main qualitative difference between the angular distributions in the radiative and non-radiative decays. The other properties of the angular distribution (39) are quite similar to those of the angular distribution of the ejected electrons. If the decaying state is aligned along some direction (chosen as z axis of the reference frame), the angular distribution of the dipole emission has the simple form W
f Jf

() =

W0 [1 + 4

2

A20 ( J)P2 (cos )].

(41)

For polarization-insensitive detectors, an anisotropy of the angular distribution can occur only if the decaying state is aligned i.e. if it is characterized by non-zero statistical tensors with k = 2. Therefore, the angular distribution (39) is always isotropic for J < 1. An orientation of the decaying state, that is the tensors with odd ranks, does not affect the angular distribution. For the decaying state with axial symmetry, Eq. (41) gives the angular distribution of the type W a + b cos2 , which is axially symmetric with respect to the symmetry axis of the initial state and which is also symmetric with respect to reflection in the plane perpendicular to this axis. The linear polarization of the emitted radiation depends on the alignment tensor A2q ( J) of the decaying state but does not depend on the orientation A1q ( J) of the decaying state. In particular, only the q = 0 component of the alignment tensor is non-zero, if this state is aligned along the z axis. Typically, the degree of linear polarization PL is determined in the direction perpendicular to the axis of the alignment by measuring the intensities for two orientations of the polarimeter axis, i.e. parallel (W ) and perpendicular (W ) to the axis of the alignment. With these definitions, the degree of linear polarization can be expressed as PL 3 2 A20 ( J) W - W . = W + W 2 A20 ( J) - 2 (42)

Comparing this equation with Eq. (41), we see that both, the anisotropy and the linear polarization of the emitted photon, are determined by the same product 2 A20 ( J). Therefore, the measurement of the linear polarization of the emitted radiation is equivalent to the measurement of the angular distribution: combining Eqs. (42) and (41), one can easily express the polarization in terms of the anisotropy and vice versa. An intensity ratio similar to Eq. (42) applies also for the circular polarization of the emitted light if W+1 and W-1 is used to denote the intensity of the fluorescence with helicity +1 and -1, respectively. In the case of circular polarization, in fact, the polarization degree PC coincides with the Stokes parameter P3 (in the convention of quantum electrodynamics, where P3 =+1 is used for the photon with positive helicity) and is defined by the relations W+1 - W-1 q A1q ( J)Y1q ( , ) 14 = . (43) PC = P3 = W+1 + W-1 4 1+ A ( J)Y ( , )
2 5 q 2q 2q


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175

Obviously, the degree of circular polarization depends on the emission angles and , and does not vanish only for an oriented atom. The denominator in Eq. (43) is again a normalized angular distribution for the photon emission (see Eq. (39)). If the initial state has a symmetry axis, then q = 0 and the circular polarization vanishes for all photons emitted perpendicular ( = 90 ) to this axis. However, when the photon is observed along the axis (i.e. = 0 ), Eq. (43) becomes PC = 2 3 1 A10 ( J) 2-
2

A20 ( J)

.

(44)

In some experiments, the fluorescence photons are observed at the angles +45 or -45 ; in this case, according to the general expression (43), the circular polarization of the fluorescence transition is given by PC =
2

3 1 A10 ( J)

A20 ( J) + 1

.

(45)

Finally, let us note that the tensors A2q ( J) and A1q ( J) in (43) should be taken within the same coordinate system, and this is true also for Eqs. (44) and (45). Often, the two atomic alignments A20 ( J) in Eqs. (44) and (42) are different because they are determined in different coordinate systems: the alignment in the system with the z axis along the photon propagation, Acirc ( J), which enters in Eqs. (44) and (45), is related to the alignment in Eq. (42), 20 1 Alin ( J), by Acirc ( J) =- 2 Alin ( J). 20 20 20 3.1.8. Interference between intermediate states Until now, we have discussed a two-step cascade from a single isolated resonant state, which decays via a single intermediate state to one or more final states. In practice, however, several photoexcited resonances often overlap with each other, since their widths is essentially determined by the lifetime of the core hole, which is usually larger than the fine-structure splitting of the excited electron states. Since the resonances overlap, they are coherently excited by the photon (Blum, 1996) and will interfere in the subsequent decay. It is clear that the approximation of an isolated resonance is no longer valid here. The same is true for the intermediate states, if their fine-structure splitting, and sometimes even the splitting of different multiplets, is smaller than their lifetime width. This coherence in excitation will lead to coherence in the decay (coherence transfer) and consequently to interference of different decay channels, which should be taken into account in the theory. In both cases, of overlapping resonances and overlapping intermediate states, the theory of angular correlations discussed above has to be modified. Below, we shall present the adapted expressions for the case of the photoinduced Auger cascade. The other expressions can be modified accordingly. Suppose, we have the process (3a) and the photon excites several overlapping resonances, {|n } {| 0 J0 }. The angular distribution of the resonant Auger electrons, which is given by Eq. (25) for a single resonance, need then to be cast into the form (Kitajima et al., 2001) d c ( )= d 4 Ak (J0 ,J0 ; J1 )
k ,n,n

k0

( 0 J0 ,

0 J0

)

n,n 2 n,n

-i +

n,n 2 n,n

Pk (cos ),

(46)

with k = 0, 2, in order to account for several resonances. In this equation, the statistical tensor k 0 ( 0 J0 , 0 J0 ) describes the coherent excitation of the two overlapping resonant states n and n ; n,n = En - En is the energy 1 splitting, n,n = 2 ( n + n ) with n being the decay width of the resonance |n , and c is a constant irrelevant for the considered angular distribution. Eq. (46) is similar to the time-averaged angular distribution of Auger electrons obtained by Kabachnik et al. (1994) within the two-step model for the particle-impact excitation of resonances; the additional term with i n,n gives a small contribution only, if the phases of the corresponding decay amplitudes VJ1 lj J 0 are almost equal for the considered resonances (which is the case in the examples discussed below).


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The generalized anisotropy coefficients Ak (J0 ,J0 ; J1 ), the first factor in the product term of Eq. (46), can be expressed in terms of the Auger amplitudes VJ1 l1 j1 J0 (Kabachnik et al., 1994): Ak (J0 ,J0 ; J1 ) = (-1) â
J0 +J1 -1/2

^^ l^ l^ j^1 j^1 J0 J0 (l1 0,l1 0 | k 0) 11 j1 J0 J1 VJ1 k
l1 j1 J0 VJ1 l j J 11

l1 j1

j1 l1

k

l1 l1 j1 j1 1 J0 2

j

1

0

.

(47)

Eq. (46) is written here for a certain final ionic state | 1 J1 ; if there are several unresolved final states, one should include an additional summation over 1 J1 in Eq. (46), similar to Eq. (28), because the contributions of different final states are incoherent. The tensors k 0 ( 0 J0 , 0 J0 ) of the resonant states are calculated by using the standard statistical tensor formalism for the photoexcitation of discrete states (Balashov et al., 2000). Generalizing Eq. (2.14) from the book of Balashov et al. (2000) in order to account for the overlapping resonant states, we obtain
k0

( 0 J0 ,

0 J0

) = J^g

-2

(-1)

1+Jg +J0

1 J0

J0 1

Jg k

k0

0 J0

D

g Jg

0 J0

D

g Jg



,

(48)

where the dipole amplitude 0 J0 D g Jg describes the photoexcitation of the resonance. In the particular case of Jg = 0 and J0 = J0 = 1, Eq. (48) simplifies to
k0

( 0 1,

0

1) =

1 3 k0

0

1D

g

0

0

1D

g

0 .

(49)

Using Eqs. (46)­(49), we can therefore calculate the angular anisotropy parameter in Eq. (26), provided that the matrix elements are known for both the dipole excitation and the subsequent Auger decay. The alignment of the resonant core­excited atomic states, that is produced in course of the photoabsorption, is partly transferred to the intermediate ionic states during the first Auger decay. Suppose that there are several overlapping intermediate states {|m } {| 1 J1 }. By analogy to the modification discussed above, one can obtain the following expression for the statistical tensors of the overlapping intermediate states as produced in the resonant photoionization through overlapping resonances (Kitajima et al., 2001):
k0

( 1 J1 ,

1 J1

)=3
n,n l1 j1

(-1)
n,n 2 n,n

J1 +j1 +J0

^^ J0 J0

k0

( 0 J0 , ,

0 J0

)

J1 J0

J0 J1

j1 k (50)

â

-i +

n,n 2 n,n

V

J1 l1 j1 J0 VJ l1 j1 J 1

0

where the statistical tensors of the resonances are determined by Eq. (48) or (49), respectively. Again, the contribution of the imaginary part ( i n,n ) is negligible, if the phases of the matrix elements VJ1 l1 j1 J0 are almost equal, as it is usually the case. Provided that the statistical tensors are known, the angular distribution of the second-step Auger electrons for the transition from the overlapping intermediate states {|m } {| 1 J1 } to the final state | 2 J2 can be calculated by an expression similar to Eq. (46): W( ) = W0 4 Ak (J1 ,J1 ; J2 )
k ,m,m k0

( 1 J1 ,

1 J1

)

m,m 2 m,m

-i +

m,m 2 m,m

Pk (cos ),

(51)

1 with k = 0, 2 and the same notation m,m = Em - Em and m,m = 2 ( m + m ) as used before for the energy splitting and widths of the intermediate states involved. In Eq. (51), the summation includes the overlapping intermediate states, and the generalized anisotropy parameters Ak (J1 ,J1 ; J2 ) are determined by Eq. (47) with an obvious change of the indices 0 1 and 1 2. Apparently, the Auger amplitudes VJ2 l2 j2 J1 describe in this case the emission of the secondstep Auger electron with the angular momenta l2 j2 . If the final states are not resolved, the corresponding angular distribution (51) should be summed, in addition, also over the final states | 2 J2 . Similar to the angular distribution of the first- and second-step electrons, we need to modify also the angular correlation function (13) in the case of several overlapping resonances and/or intermediate states. For this purpose it is convenient to present the angular correlation function in a slightly different form, namely as an angular distribution


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177

^ of the second-step Auger electrons, given that the first-step Auger electron is emitted in the direction n1 {1 , 1 } (Kabachnik, 1992). Then, in the case of overlapping resonances and intermediate states, we have (cf. Eq. (19) from Kabachnik et al. (1999)): W(1 ,
1

; 2 ,

2

)=

c 4 â

Ak2 (J1 ,J1 ; J2 )
k2 q2 mm m,m 2 m,m

k2 q2

( 1 J1 ,

1 J1

^ ; n1 ) ), (52)

-i +

m,m 2 m,m

(4 )

1/2 ^ -1 k2 Yk2 q2

(2 ,

2

where the coefficients Ak2 (J1 ,J1 ; J2 ) are the same as above in Eq. (16), and where we have introduced the differential ^ statistical tensors k2 q2 ( 1 J1 , 1 J1 ; n1 ) for the intermediate states. These statistical tensors describe the polarization ^ of the intermediate states, if the first-step electron is detected in the direction n1 :
k2 q2

( 1 J1 ,

1 J1

^ ; n1 ) =
k0 q0 k1 q1

(4 ) â
n,n 2 n,n

1/2

^ ^ ^- k2 J0 J1 1 (k1 q1 ,k2 q2 | k0 q0 ) Y
k1 q1

k0 q0

( 0 J0 ,

0 J0

) (53)

-i +

n,n 2 n,n

(1 ,

1

)B

k1 k2 k0

(J0 ; J1 ,J1 ).

The factors Bk1 Bk1
k2 k0

k2 k0

(J0 ; J1 ,J1 ) are given by the relation
k1 -k0

(J0 ; J1 ,J1 ) = (-1) â

(4 )

-1

^^ J 0 J1
l1 l1 j1 j
1

(-1) j1 j1 k1 J0 J0 k0

1 j1 + 2 ^ ^ ^ ^ l1 l1 j1 j1 (l1

0,l1 0 | k1 0)

l1 j1

j1 l1

1 2 k1

J1 J1 k2

VJ1

l1 j1 J0 VJ l j J 11 1

0

,

(54)

where the Auger amplitude VJ1 l1 j1 J0 describes the first-step resonant Auger transition from the initial state |J0 to the intermediate state |J1 under the emission of an electron with orbital and total angular momenta l1 and j1 , respectively. 3.2. Multiconfiguration Dirac­Fock computations of Auger cascades The formalism described above gives the general form of the correlation functions and the polarization of the emitted particles in terms of (many-electron) transition amplitudes. For a quantitative prediction of these functions and parameters, these amplitudes need to be calculated for a particular atom and by using a proper model for representing the wave functions. For the cascades above, in particular, we need wave functions for all of the initial, intermediate and final states pertinent to the electron and photon emission. At least the intermediate states hereby refer to highly correlated (many-electron) states, if they are embedded energetically in the continuum of the next higher charge state of the ion. For these resonances, therefore, special care has to be taken in order to calculate the photoexcitation and autoionization/Auger amplitudes. The multiconfiguration Dirac­Fock (MCDF) method has been found as a versatile tool for calculating the many-electron amplitudes, especially if inner-shell electrons or several open shells are involved in the computations. In the following subsections, we shall briefly outline the basic features of this model and discuss which computational challenges often arise in dealing with resonant photoionization and decay processes. 3.2.1. Multiconfiguration expansions: The challenge of open shells Not much need to be said here about the basic concepts of the MCDF method which has been presented at several places elsewhere (Grant, 1988; Parpia et al., 1996; Fritzsche, 2002). Based on the one­electron Dirac Hamiltonian, this computational model enables one to describe the (dominant) effects of relativity and correlations within the same framework. Similar to the well known Hartree-Fock (HF) method from non-relativistic quantum theory (Cowan, 1981), in which the state of a quantum system is approximated by a symmetry­adapted Slater determinant, an atomic state is written in the MCDF model as linear combinations of configuration state functions (CSF) | r PJ M with well adapted


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symmetry, i.e. with the certain parity, P , the total angular momentum, J , and its projection M ,
nc

(P J M ) =
r =1

cr ( )| r PJ M .

(55)

In this ansatz, nc is the number of CSF and {cr ( )} denotes the representation of the atomic state in this basis. In most standard computations, moreover, the set of CSF {| r PJ M } are constructed as antisymmetrized products of a common set of orthonormal orbitals. In contrast to the HF method, however, both the radial one­electron functions as well as the expansion coefficients {cr ( ), r = 1,...,nc } are then optimized simultaneously on the basis of the Dirac­Coulomb Hamiltonian. All further relativistic contributions to the energies and representation of the atomic states, which arise from either the interaction among the electrons or with the radiation field, are usually treated in first­order perturbation theory by diagonalizing the Dirac­Coulomb­Breit Hamiltonian matrix (Grant and Quiney, 1988). The decision about the explicit choice of the atomic Hamiltonian is often based on the nuclear charge and the charge state of the atom or ion under consideration; in practice, this choice may depend also on the particular shell structure and the atomic property which is to be investigated with these wave functions. Further quantum electrodynamical (QED) corrections to the total energies of the atomic levels can be added in various approximations, but are based typically on some hydrogen-like model (Johnson and Soff, 1985; Fritzsche et al., 2002). In fact, there are two dominant corrections for all medium and heavy elements which arise from QED: the vacuum polarization and the self-energy of the electrons. Compared with missing correlation contributions, however, these QED corrections are relevant only if the K- and L-shell electrons of medium and heavy atoms are involved in some decay cascade, and are often negligible otherwise. The latter is especially true for all those transitions at the present level of computational accuracy, that are coupled to the `electron continuum' of the next higher charge state (Fritzsche, 2005). In the following, therefore, we shall not discuss any further aspects of the theory which go beyond the Dirac­Coulomb­Breit Hamiltonian. During the last decades, a number of codes has become available which implement the MCDF method for free atoms and ions, and which provide approximate energies and wave functions for most atomic bound states of interest. The revised version GRASP92 of the former Oxford package (Parpia et al., 1996), in particular, now facilitates large-scale applications and supports wave function expansion of several (ten) thousand CSF in Eq. (55) as required for many open-shell atoms. Less attention in contrast was paid to the computations of atomic properties other than line strengths and hyperfine structures. Therefore, in order to provide a simple access to the computation of (relativistic) atomic transition and ionization properties, including the description of continuum states, we developed the RATIP program (Fritzsche and Froese Fischer, 1997; Fritzsche and Grant, 1997; Fritzsche, 2001a) that was extended and re-designed in several steps during the last years (Fritzsche et al., 2002, 2003a, b; Gaigalas et al., 2004; Nikkinen et al., 2006). This code applies the wave functions from GRASP92 and will be further discussed below. At the first glance, ansatz (55) paves a simple and straightforward way for generating at the least bound­state wave functions. Apparently, all what is needed to obtain the atomic states of interest is the set-up of a proper CSF basis and the diagonalization of the Hamiltonian matrix. In practice, however, the definition of a physically appropriate basis turns out to be less simple. In particular the difficulties in calculating open-shell atoms and ions have long been underrated. The first (successful) structure calculations of a few simple atoms and ions in the sixties and seventies quickly led to the popular fallacy, that it would take only a bit more effort and computational power in order to theoretically predict the structure and properties of atoms in rather arbitrary configurations. By now, we know much better: many atomic properties have proven to depend very sensitive on electron­electron correlations and, hence, on the shell structure of the atoms and ions (Fritzsche, 2002). Although much larger computations are feasible today, most open-shell atoms are yet not well understood. The real challenge when dealing with open-shell structures are the very rapidly growing wave function expansions, if one or several open shells are being involved already within the reference configurations (cf. Fritzsche et al., 2000b). To include the correlation of the bound-state electrons in a systematic fashion, the complete active space method has been found useful for simple shell structures along various isoelectronic sequences (Fritzsche and Grant, 1994; Kohstall et al., 1998; Dong et al., 1999, Fritzsche et al., 2000a). The idea of this method is to account for the excitation of the active electrons from the outer shells into a number of unoccupied orbitals, while the electronic `core' remains fixed usually. For a complete active space, it is supposed that all possible excitations of the valence electrons can be taken into account among the set of active orbitals. In practice, the concept of a complete active space is often not only unfeasible but also rather unnecessary for predicting many atomic properties, since only those CSF with a total


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179

Fig. 8. Size of the CSF expansion (55) for excitations of the 4s and 4p electrons from the reference configuration 3d9 4s2 4p6 5p in dependence of the orbital layers that are included into the active space. The SD approximation (red solid line) is compared with a SDT approximation (green dash­dotted line), including up to triple excitations within the n = 4 and 5 orbitals. See text for further explanations and for the meaning of the blue dashed line.

`excitation energy' of the same order as the reference configurations need to be included into the expansion. Therefore, it has been found useful to divide the virtual excitations of the active electrons into different classes, namely single (S), double (D), triple (T), ... excitations, in dependence of how many electrons are to be replaced with regard to the set of reference configurations. While single and double excitations are typically treated together, owing to Brillouin's theorem for the pure Hartree­Fock approximation (Lindgren and Morrison, 1986), the higher excitations are often less important and, if at all, are incorporated only within the valence shells. To create the CSF basis for a given set of levels (or transitions in a decay cascade), a number of programs are available and can be utilized to manipulate these expansions in a user-friendly way (Parpia et al., 1996; Fritzsche, 2001b). To illustrate the size of these expansions, let us consider the electron configuration 3d9 4s2 4p6 5p that describes the population of the valence shells for the Kr 3d-1 5p J = 1 resonance in Section 4.1. For this (reference) configuration, Fig. 8 displays the number of CSF in the wave function expansion (55), if excitations are taken into account of the 4s and 4p electrons into orbitals of the higher-lying shells up to n = 6. Here, an expansion with single and double excitations (SD approximation; red solid line) is compared with a second one that includes single, double and triple excitations (SDT; green dash­dotted line). Note that the 1s ... 3d `core shells' are fixed in this case, including the 3d hole. Obviously, the size of the CSF expansion increases very rapidly if another `layer' of orbitals is included and especially, if one wishes to incorporate triple excitations. In addition, the blue dashed line in Fig. 8 refers to the SD approximation for the 4su 4pv 5p (u + v = 6) fine-structure levels, which follow the decay of the 3d-1 5p J = 1 resonance, if one starts from the 4s0 4p6 5p, 4s4p5 5p, 4s2 4p4 5p, and 4s2 4p3 4d5p reference configurations in this case. Note that one may easily obtain wave function expansions with several million CSF if higher orbital layers or further triple excitations are taken into account. The dramatic increase of the CSF expansions in Fig. 8 makes it clear why open-shell configurations are still a challenge for present-day atomic structure calculations. A more detailed account on the active space method is given by Fritzsche (2002, 2005). Owing to the symmetry of the atomic Hamiltonian, the size of the CSF basis can be slightly reduced by about a factor 4­10, if the various angular momenta and parities are considered independently. On the other hand, the size of the CSF expansions increase certainly further, if the atomic (bound-state) electrons are coupled to the electron continuum, since the presence of one (or more) electrons in the continuum automatically leads to extra open shells. This `additional' increase of the CSF expansions, which need to be included in the course of the corresponding computations, occurs in the study of all scattering, ionization and/or autoionization processes and is one of the reasons, why such `scattering processes' are much less well understood than the optical spectra of free atoms and ions. For argon, however, it was shown by means of systematically enlarged wave function expansions that the initial- and final-state configuration


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interactions can be taken into account to a large extent and help to bring both, the energy and the angular distributions in better agreement with experiment (Ueda et al., 2001). 3.2.2. Evaluation of photoionization and Auger amplitudes Using the density matrix theory as outlined above in Section 3.1, the (auto-) ionization dynamics of all atoms and ions can be traced back to the coupling of the bound-state resonances, which are embedded into the continuum, to one or several scattering states. If the ionization is caused by light (of proper frequency), the transition from an initial bound state | i =| (Pi Ji Mi ) with total angular momentum Ji ,Mi and parity Pi to the final scattering state | t is given by the dipole amplitude t | D | i , where D is the dipole operator and | t denotes a (total) scattering state with angular momentum Jt ,Mt and parity Pt . This scattering state is obtained by coupling the final state | f =| (Pf Jf Mf ) of the photoion with the partial wave of the outgoing electron with energy and the angular momentum =±(j + 1/2) with l = j ± 1/2.

Thus the transition amplitude for the photoionization of an atom or ion is given by D
Jf lj J
t

(

f

(Pf Jf ),

)Pt Jt D

i

(Pi Ji ) .

(56)

For the autoionization, similarly, the (Auger) transition amplitude can be obtained as VJf
lj J
r

(

f

(Pf Jf ),

)Pr Jr H - E

r

r

(Pr Jr ) ,

(57)

i.e. from the coupling of some embedded resonance | r to the electron continuum of the next higher charge state. If, as in most computations, a common set of orthonormal orbitals is applied for the representation of the intermediate | r and final ionic states | f , the transition operator (H - Er ) V can be replaced by the electron­electron interaction operator. For most light and medium elements, it is hereby sufficient to include the instantaneous Coulomb repulsion between the electrons but to omit the relativistic Breit interaction of the Hamiltonian H which was found important for the Auger emission of highly charged ions (Fritzsche et al., 1991). Indeed, the restriction of the `autoionization' operator (H - E) to the electron­electron interaction in the computation of the Auger amplitudes is standard in most implementations of Auger calculations and has been utilized even, if the orbital functions of the resonant state | r and the final-ionic states | f are not quite orthogonal to each other (Fritzsche, 1993; Fritzsche et al., 2003a, 2007). Using ansatz (55), all the excitation and (auto-) ionization amplitudes from Section 3.1 can be easily reduced to the evaluation of the matrix elements, which are calculated with pairs of CSF from the corresponding initial and final states, respectively. If a common set of orthogonal orbitals is assumed for the representation of the wave functions, these matrix elements can be represented always as a sum of products of an angular coefficient times a radial integral. The proper reduction and evaluation of the many-electron matrix elements for transition operators of various sorts has been the focus in developing the RATIP program during the last decade (Fritzsche et al., 1992, 2000c, 2005b). When compared with the standard `angular reduction' as implemented in the GRASP92 code, the AUGER and PHOTO components of the RATIP program are now faster by a factor3­8owing to the use of an improved angular integration scheme (Gaigalas et al., 2001; Gaigalas and Fritzsche, 2002). Since all inner-shell hole states lie--by its nature--very high up in energy (with regard to the corresponding ground state of the ion), they have to be treated as `resonances' that are embedded in the electron continuum of (at least) the next higher charge state. From scattering theory, several formal approaches are known, such as the R- and Kmatrix approaches (Burke and Taylor, 1975; Takatsuka and McKoy, 1981) or the solution of the Lippmann­Schwinger equation (Byron and Joachain, 1973), in order to treat the interaction among the different resonances and with other decay channels properly. Up to the present, however, neither of these approaches have been implemented in sufficient detail for complex open-shell atoms and ions, and it is yet an open question to which extent such sophisticated techniques will be utilized in the future. In the PHOTO and AUGER components of the RATIP program, in contrast, the continuum spinors are solved within a spherical but level-dependent potential of the final ion, an approach which is equivalent to the optimal level scheme in GRASP92 (Parpia et al., 1996). This scheme includes the exchange interaction of the emitted electron with the bound-state density. The number of the possible scattering states |( f (Pf Jf ), )Pt Jt of the total system `photoion+electron' often increases rapidly as the free electrons may couple in quite different ways to the bound-state electrons. For further details on the computations of the Auger matrix elements and relative intensities, we refer the reader to


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Fritzsche et al. (1992, 2007). Apart from the proper coupling of the continuum electron, a great deal of code development has still to be done, before these computations for the capture or emission of free electrons become comparable in quality with the calculation of optical spectra (Fritzsche, 2002). 3.2.3. Description of coherence transfer through the computation of the density matrix The amplitudes (56) and (57) are the building blocks to form the scattering transition operator and, hence, the density matrix of the system at the individual steps of the cascade. For scattering states with a single electron in the continuum, all these amplitudes can be calculated by means of the RATIP program (Fritzsche, 2001a, 2002) as briefly discussed above. Apart from the efficient calculation of these building blocks, however, one also needs a simple access to these many-electron matrix elements in order to set-up the density matrices in the various steps of the coherence transfer as outlined in Section 3.1. Therefore, in order to facilitate the handling of the amplitudes, they are provided by most of the components of the RATIP code (together with all quantum numbers that are needed for a unique classification) in so-called `transition amplitude files'. This interface allows one to start directly from the amplitudes (of one or several transitions) and to combine them into an appropriate form in order to generate the data of interest. In fact, the design of clear `interface' files helped considerably in the computation of the coherence transfer through the Auger cascades of the noble gases. Apart from the transition amplitudes (56) and (57), one needs in addition also a fast and reliable access to the Wigner nj - symbols and spherical harmonics, and to all the other symbols from the coupling of the angular momenta that appear in the expressions of Section 3.1. To facilitate the computation of these symbols, we continued to develop the RACAH program in a number of steps, based on the computer-algebra environment MAPLE. Since an early version of this code (Fritzsche et al., 1998), the RACAH program has grown considerably and provides today an interactive and user-friendly tool (Fritzsche et al., 2001; Gaigalas et al., 2005; Pagaran et al., 2006). Beside the numerical computations of a wide class of symbols and functions (cf. Table 1 of Fritzsche et al., 2003b), this program supports also the purely algebraic manipulations of typical expressions from the theory of angular momentum and spherical tensor operators, and became now part of a larger project called the `Symmetry Tools' (Fritzsche, 2006). Having available all the components as required by the various expressions in Section 3.1 the computation of the density matrices and angular functions can be summarized as follows: (i) Generation of all necessary (bound-state) wave functions taking into account the correlations in the bound-state density, (ii) evaluation of the many-electron transition amplitudes between the bound states and scattering states in the continuum, and (iii) the set-up and handling of the density matrices, i.e. the `computation' of all the formulas from above. Apart from the coherence transfer in the course of the electron or photon emission, as discussed below for the resonant photoexcitation and subsequent decay of noble gases, the same approach has been found useful also for studying the capture of electrons into highly charged ions (Surzhykov et al., 2002, 2006; Fritzsche et al., 2005a, b), the photoabsorption of the alkaline atoms (Koide et al., 2002), the entanglement between the photoelectron and the remaining ion (Radtke et al., 2006), and at several places elsewhere (Borowska et al., 2006). 3.3. On the problem of a complete experiment One of the aims in performing correlation experiments is to gain the maximum information, that is possible to obtain about the relaxation process of inner-shell excited atoms and molecules. Of course, the ultimate goal would be to determine from the experiment all the amplitudes of the decay. In quantum mechanics, a set of measurements is called a complete or perfect experiment, if it is possible from the measured quantities to obtain unambiguously all the complex amplitudes of the process (for a more detailed discussion of this topic see, for example, Becker and Crowe (2001) or Kleinpoppen et al. (2005)). For the Auger decay of atoms, a concept of a complete experiment was formulated by Kabachnik and Sazhina (1990) within the two-step model (see also a short review by Kabachnik (2004)). It is shown in these works that in the general case of the Auger decay of the initial state 0 J0 to the final state 1 J1 , the number of complex partial amplitudes describing uniquely the decay, is 2J0 + 1 for J1 >J0 (while, in contrast, only 2J1 + 1 amplitudes would be required for J1 J0 , therefore, 4J0 + 1 real parameters need to be determined by some proper set of measurements since one parameter, the overall phase, has no physical meaning. If only the Auger electrons from a particular transition are considered and if one measures their angular distribution and spin polarization, then the total number of intrinsic parameters, k , k and k , (see Sections 3.1.5 and 3.1.6) which may be derived from the measurements, is also 4J0 + 1. Thus, if these parameters were independent,


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a complete experiment would be possible. As found recently, however, the intrinsic parameters as obtained from the measurement of the angular distribution and spin polarization of the Auger electron, are not independent and there exist purely algebraic relations between these parameters which are related to the parity and momentum conservation in the decay (see Kabachnik (2005) and references therein). 1 Let us consider, for example, the Auger transition M4 --N2,3 N2,3 P1 . The selection rules for the angular momentum and parity limit the possible partial waves of the emitted Auger electron to three waves s1/2 , d3/2 and d5/2 . Correspondingly, the transition is described by the three complex (partial) amplitudes Vs1/2 ,Vd3/2 and Vd5/2 , i.e. by five real parameters. If the absolute yield of the Auger electrons is not measured, then the transition is characterized `almost completely' by four real parameters, two ratios of the moduli of the amplitude and two relative phases. It is possible to measure the angular distribution and the spin polarization of the Auger electrons which give rise to the anisotropy parameter 2 and the three spin polarization parameters 1 , 1 and 2 , respectively; compare Eqs. (32)­(34). These four parameters are related to each other by the equation (Schmidtke et al., 2000a)
2

-

3 5 2

2 1

+ (2 2 )2 = (1 +

2

)(5 - 3 5 1 ).

(58)

Therefore, only three of these parameters are independent which is not sufficient in order to determine four characteristics of the decay. Note that similar relation between the asymmetry parameter and the spin-polarization parameters have been first found for photoelectrons (Schmidtke et al., 2000b). Relation (58) is valid only for the above mentioned transition. Similar relations have been found for other particular transitions (Schmidtke et al., 2001; Kabachnik and Grum-Grzhimailo, 2001; Kabachnik and Sazhina, 2001, 2002). More recently, a general relation between the intrinsic parameters has been obtained by Kabachnik (2005), which includes as particular cases all previously found relations. By making a careful analysis it was demonstrated that a complete experiment cannot be realized, if only the parameters of the Auger electrons are measured. The only exception from this rule is an Auger decay to J1 = 1/2 states, for which the number of unknown amplitudes reduces to two. These two amplitudes can be determined by measuring the anisotropy and spin-polarization parameters of only the Auger electrons. In all other cases, some additional information about the polarization state of the residual ion is necessary. One way to gain this information is a measurement of the angular correlations in the cascades, which essentially depend on the polarization of the intermediate states. Another possibility is to measure the angular distributions of either the second step Auger electrons or the angular distribution (or polarization) of the fluorescence light. Sometimes, the realization of a complete experiment in Auger decay is simplified, if the number of unknown amplitudes can be reduced by more or less plausible model assumptions, for example, by assuming the LSJ coupling approximation for the levels involved (Beyer et al., 1996; Grum-Grzhimailo et al., 1999, 2001, Ueda et al., 1999a, b) or by supposing that the relativistic phase difference in the continuum is negligible (Schmidtke et al., 2000a). Some more details about the realization of a complete experiment for the Auger decay will be given in Sections 4.6 and 5.3. 3.4. Beyond the stepwise model of the Auger decay The stepwise model described and used in the above sections is quite adequate for describing the cascade processes in the majority of cases, where one starts from an inner-shell excitation or ionization of the system. As discussed, however, this model becomes inadequate for overlapping resonances, although the extended formalism developed in Section 3.1 is still based on the idea that the excitation and decay of the resonance can be considered separately. In fact, the separation of `individual' steps works well for any decay as long as the lifetime of the decaying state is much longer than the characteristic time of the excitation process. In practice, however, there is a class of phenomena for which this condition is not fulfilled. It is related to the near-threshold inner-shell photoionization of atoms and molecules. At photon energies just above the threshold, the emitted photoelectron is slow and may be still within or `close' to the atom when the photoion decays via the emission of an Auger electron. That is, the second step starts already before the first step has been completed and, hence, a stepwise model will break down. Then, the interaction of the three particles in the final state of the ionization process (the photoelectron, the Auger electron and the residual ion), which is dubbed the post-collision interaction (PCI), results in several effects, such as shift and distortion of the Auger line, recapture of the photoelectron, shake processes, etc. These effects have been studied extensively in the past two decades (see e.g. the reviews by Kuchiev and Sheinerman (1989), Schmidt (1997) and Armen et al. (2000)). A theoretical description of


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these phenomena has to consider both parts of the reaction, the ionization and the decay, as one process, and therefore ° ° such theories are sometimes called `one-step' or `unified' theories (Aberg, 1980, 1992; Aberg and Howat, 1982). Several approaches have been developed in order to describe the PCI phenomena in the near-threshold photoionization: semiclassical models summarized by Russek and Mehlhorn (1986), see also van der Straten et al. (1988); several fully quantum­mechanical theories, such as diagrammatic many-body theory (Amusia et al., 1977; Kuchiev ° and Sheinerman, 1985, 1986, 1994; Sheinerman, 2003, 2005) or the resonant-scattering theory (Aberg, 1980; Tulkki et al., 1987, 1990; Armen et al., 1987, 1996; Armen, 1988; Armen and Levin, 1997). In quantum­mechanical descriptions, the Green's functions are used to describe the propagation of the intermediate state. These functions are expanded in terms of the stationary eigenfunctions of the system Hamiltonian. An alternative approach has been recently suggested by Kazansky and Kabachnik (2005, 2006a, b). This is based on the numerical solution of the non-stationary SchrÆdinger equation that describes the interaction of the atom with the electromagnetic field. Considering the typical Fano­Feschbach problem of one decaying resonance, the authors reduced the complicated many-electron problem to the solution of one non-stationary and non-homogenous equation with an effective complex Hamiltonian: i j
d (r, t ) ^ = H1 (r) - i jt 2
d (r, t ) - d · E(t )
0 (r) exp(-i 0 t), (59)

where
d (r, t ) is the wavefunction of the photoexcited electron (resonant state) and
0 (r) is the function of this electron ^ in the ground state with the binding energy 0 . In Eq. (59), moreover, d is the dipole operator, H1 (r) a single-electron Hamiltonian, which corresponds to the atom before the Auger decay has happened, and = 2 |V |2 characterizes the width of the resonant state. The solution of Eq. (59) describes the development of a wave packet, which is pumped by the electromagnetic field E(t ) and, at the same time, is subject to the Auger decay with the rate 1/ . As shown by Kazansky and Kabachnik (2005), if the Auger electron is not observed, then the probability of populating any Rydberg state of the final ion | n (r) can be calculated as the integral over time of the squared projection of this state onto the wave packet of intermediate state at the moment t : Pn = dt |
n

(r)|
d (r, t ) |2 .

(60)

Similarly, it can be shown that the probability for finding the excited (Rydberg) electron in the continuum with welldefined energy E is given by PE = where
E

dt |

E

(r)|
d (r, t ) |2 ,

(61)

^ (r) is a continuum wave function of the outer electron and an eigenfunction of the ionic Hamiltonian H2 :
E

^ (H2 (r) - E)

(r) = 0.

(62)

More details of the non-stationary approach can be found in papers by Kazansky and Kabachnik (2005, 2006a, b) ° where it was demonstrated also that this approach is equivalent to the resonant scattering theory (Aberg, 1980; ° berg and Howat, 1982). The theory summarized here was used for describing the population of the Rydberg states in the A near-threshold photoionization of Ne(1s), including their population due to the electron recapture process (Hergenhahn et al., 2005, 2006; see also the next section). The advantage of the time-dependent approach in comparison with the stationary formulation is that such approach can be used also for the excitation by short electromagnetic pulses for which the pulse duration is comparable or even smaller than the life-time of the vacancy. 4. Auger cascades In this chapter we consider the cascades of two non-radiative decays. In the majority of the studied cases, so far the first-step decay is a resonant Auger transition, while the second step is a normal Auger (Coster­Kronig) transition. Such cascades were studied in noble-gas atoms, Ne (Yoshida et al., 2000, 2005; HeinÄsmÄki, 2001; Ueda et al., 2003a, De Fanis et al., 2004a, b, 2005; Da Pieve et al., 2005; Kitajima et al., 2006), Ar (Ueda et al., 1999a, b, 2001; Huttula et al., 2001a, b; Nayandin et al., 2001; Sankari et al., 2002), Kr (Mursu et al., 1998; Ueda et al., 2000, 2003b; Kitajima et al., 2001; Huttula et al., 2003) and Xe (Kitajima et al., 2002; Lablanquie et al., 2001, 2002; Ueda et al., 2003b;


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Huttula et al., 2003; Jonauskas et al., 2003; Osmekhin et al., 2005; Partanen et al., 2005; Viefhaus et al., 2005) [for earlier references see Armen et al. (2000)]. The most detailed and accurate information is obtained for the firststep resonant Auger decay where the high-resolution spectra and angular distributions of the emitted electrons are measured. Examples of such experiments will be given in the next Section 4.1. Although in the majority of experiments the lowest well isolated resonances were photoexcited, in some cases the interference of overlapping resonant states plays an important role. An example of such a case is discussed in Section 4.2. For some particular transitions, the spin polarization of the emitted electrons was also studied. Although spin-polarization measurements are more elaborate than the differential cross section measurements, they provide important complementary information. We discuss such experiments in Section 4.3. The investigation of the second-step Auger transitions, which usually have low energies, is more difficult, since as a rule the low-energy part of the Auger-electron spectrum contains many overlapping lines. They belong not only to the second-step cascade transitions, but also to the low-energy first-step transitions from initial resonances to the highly excited states of the residual ions. The description of the angular distribution of the second-step Auger emission almost inevitably involves interference of the intermediate states, which are coherently excited (Section 4.4). Due to complexity of the low-energy Auger spectrum, it is important to measure not only spectra and angular distributions of the emitted electrons, but also to make coincidence measurements in which both first-step and second-step electrons are detected in coincidence. We discuss examples of such studies in Section 4.5. The angular correlations and spin-polarization measurements allow one to plan and realize the complete experiment aiming for the experimental determination of the Auger amplitudes. One of the first of such experiments in the non-radiative cascade will be described in Section 4.6. Finally, Section 4.7 is devoted to a special type of cascades, which arises when the photoionization occurs very close to an inner-shell threshold and highly excited Rydberg states are populated. In this case PCI strongly influences the first stages of the cascade. The stepwise description becomes invalid and more elaborate one-step approaches (see Section 3.4) should be applied.

4.1. Angular distribution of the resonant (first step) Auger transitions The angular distributions of electrons in the first-step resonant Auger transitions in Auger cascades have been measured with high resolution for all noble gas atoms (see references above). As an example we discuss the study of - the cascade decay of the 3d5/1 5p resonance in Kr (Ueda et al., 2000; Kitajima et al., 2001). This is a well studied isolated 2 resonance which mainly decays by the spectator resonant Auger decay to the Kr + states with the configuration 4p-2 5p (Aksela et al., 1996b, c) which in turn cannot decay further by non-radiative transitions. With a small fraction, however, the resonance decays to the highly excited ionic states with configurations 4s-1 4p-1 5p and 4s-2 5p which can autoionize via participator Auger transitions to the low-energy states of Kr 2+ with configurations 4p-2 and 4s-1 4p-1 , respectively (Mursu et al., 1998). The experiment was carried out on the 24-m spherical grating monochromator installed in the soft X-ray undulator beam-line 16B at the Photon Factory in Japan using the angle-resolved electron spectroscopy apparatus described elsewhere (Shimizu et al., 1998), see also Section 2.1. The photon bandwidth and the energy resolution of the electron spectrometer were 0.1 eV. The overall energy resolution was 0.14 eV for recording the first-step resonant Auger spectrum. Further details of the experiment are given in Ueda et al. (2000) and Kitajima et al. (2001). The part of the Auger spectrum which contains the above transitions is shown in Fig. 9 (upper panel). Some of the prominent peaks are grouped and denoted by roman numbers as indicated. Arabic numbers given in the figure correspond to the line numbers as given by Mursu et al. (1998). The lines from the groups I + II and VI correspond to the transitions 3d-1 5p 4s-1 4p-1 5p and 3d-1 5p 4s-2 5p, respectively, sometimes referred to as the "diagram" transitions (Mursu et al., 1998). The lines from group V are identified as correlation satellites of the diagram transitions 3d-1 5p 4s-1 4p-1 5p. They exist due to a strong admixture of the configuration 4s2 4p-3 4d5p. The groups III, IV, and VII are attributed to the so-called shake-modified resonant Auger transitions or shake-up satellites of groups I, II, and VI, respectively. Apart from the strong peaks numbered in the figure, the spectra reveals numerous small peaks which show up almost continuously in the whole energy range. From the spectra taken at different ejection angles, the angular anisotropy parameters have been obtained [see Eq. (26)]. They are shown in the lower part of Fig. 9. In the same panel the results of the MCDF calculations are presented. Both initial state and final ionic state configuration interaction were taken into account in the calculation (Kitajima et al., 2001). The main difficulty was a reasonable description of the intermediate states with 2 and 3 open shells (see discussion in Section 3.2.1). For these states the expansion of the wave function included all CSF with J = 1/2,..., 7/2 from the 4s4p5 5p configuration as well as the


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185

- Fig. 9. A part of the resonant Auger spectrum for 3d5/1 5p excitation in Kr (the photon energy is 91.2 eV) together with the asymmetry parameter 2 for the main peaks of the spectrum. In the upper panel solid and dashed curves show the experimental spectrum measured at 0 and 90 , respectively. In the lower panel, full circles show the experimental values of while open circles show the results of the MCDF calculations. Adapted from Kitajima et al. (2001).

J = 1/2 and 3/2 levels from 4s0 4p6 5p and 4s2 4p3 4d5p configurations. This results in a total of 194 levels from which about 8­10 are strongly connected to the resonances in the first-step decay. The results of the calculations agree rather well with the experiment (see Fig. 9 lower panel). As a second example we have chosen the first-step resonant Auger transitions in Ne 1s-1 3p excitation, which were recently studied with a record resolution permitted to resolve the multiplet structure of the lines corresponding to the final ionic states of the 2s2p5 3p configuration (Ueda et al., 2003a; De Fanis et al., 2004a, b). The very high resolution was achieved by the Doppler-free resonant Raman Auger technique (see Section 2.1). The experiment was carried out on the c-branch of beam-line 27SU (Ohashi et al., 2001a, b) at the Spring-8 SR facility in Japan. The "Doppler-free" electron spectroscopy system consisted of a hemispherical electron spectrometer of mean radius 200 mm (Gammadata-Scienta SES-2002) and a molecular beam source (MB Scientific MBS JD-01). A detailed description of the experimental set-up is given by De Fanis et al. (2004a, b). The resonant Raman Auger spectra for the transitions to the Ne+ 2p4 (1 D)3p 2 P, 2 D and 2 F states and to the Ne+ 2s2p5 (1,3 P)3p 2 S, 2 P and 2 D states via 1s 3p excitation were recorded sequentially and repeatedly. The ionic states with the configuration 2s2p5 (1,3 P)np belong to the class of the so-called inner-valence excitations (Becker et al., 1989, 1993; Becker and Wehlitz, 1994). They have been observed as satellites in the Ne 2s photoionization spectrum (see, for example, Svensson et al., 1988). The strongest satellites, which belong to the 2s2p5 (1,3 P)np 2 S series, are rather broad, the lowest members of the series being the broadest. In early calculations (Armen and Larkins, 1991) four other series of inner-valence excitations with a np outer electron have been predicted, namely 2s2p5 (1,3 P)np 2 P and 2s2p5 (1,3 P)np 2 D . Their calculated widths are, in general, one order of magnitude smaller than for the 2 S states; the expected widths of these states are 20­60 meV (Armen and Larkins, 1991). These levels have never been seen in the photoionization spectra since they are extremely weakly coupled to the direct photoionization channel. The experiment described was the first spectroscopic observation of the lowest levels of the 2s2p5 (1,3 P)np 2 P and 2s2p5 (1,3 P)np 2 D series in Ne+ and the measurements of their lifetime widths, using Doppler-free resonant Raman Auger spectroscopy. The population of the 2s2p5 (1,3 P)np 2 P and 2 D states in the resonant Auger process is large as compared with the 2s-photoionization due to different selection rules and due to the resonant enhancement. The resonant Auger transitions to these states have already been observed earlier (Aksela et al., 1989; Viefhaus, 1997; Yoshida et al., 2000; KivimÄki et al., 2001) as well as their decay to the doubly charged Ne2+ (von Raven, 1992; Yoshida et al., 2000). However, because of the poor overall resolution of the early experiment, the multiplet structure of these states could not be resolved. The results of the Doppler-free experiment are shown in Fig. 10.


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N.M. Kabachnik et al. / Physics Reports 451 (2007) 155 ­ 233
2s 2p ( P) 3p
1 51

10 8 6

2

0 deg

3 2 1

2s 2p ( P) 3p

1

53

6

0 deg

4 5

Intensity (arb. units)

4 2 0 10 8 6 4 2 0 778.0

1

3 90 deg
0 3 2 1 0

90 deg 5 6

1

2

778.5

779.0

779.5

788.0

788.5

789.0

Kinetic energy (eV)

Fig. 10. Part of electron spectra of resonant Auger transitions to Ne+ 2s2p5 (1 P)3p Ne+ 2s2p5 (3 P)3p 2 S , 2 P , and 2 D states (lines 4, 5, and 6, respectively) via Ne horizontal (upper panels) and vertical (lower panels) polarizations of the SR. The individual components. From Ueda et al. (2003a).

2 P , 2 D , and 2 S states (lines 1, 2, and 3, respectively) and to 1s-1 3p state at a photon energy of 867.12 eV measured with thick lines are the results of the fit, and the thin lines are the

Table 1 Measured and calculated transition energies and anisotropy parameters states of Ne+ .
Final state Ee (eV) Expt. ( ( ( ( ( (
1 1 1 3 3 3

for the resonant Auger decay to the 2s2p5 (1 P, 3 P)3p 2 S , 2 P , and 2 D

Theor. 776.40 776.66 776.43 786.51 787.52 787.64

Expt. -0.94 ± 0.06 0.15 ± 0.06 +0.0 2.0-0.2 1.8 ± 0.2 -0.8 ± 0.1 0.15 ± 0.05

Theor. -0.996 0.200 2.000 1.998 -0.928 0.156

P P P P P P

) ) ) ) ) )

3 3 3 3 3 3

p p p p p p

2 2 2 2 2 2

P D S S P D

778.55 778.81 778.83 788.17 788.90 789.02

Adapted from Ueda et al. (2003a).

The left panel of Fig. 10 shows a portion of the electron spectra corresponding to the transitions to Ne+ 2s2p5 (1 P)3p via the Ne 1s 3p excitation at a photon energy 867.12 eV. In this energy region, two lines 1 and 2 are clearly resolved in the 90 spectrum. In the 0 spectrum, line 1 becomes weak and furthermore one notices one more broad line 3 at the foot of line 2. The right panel of Fig. 10 shows a portion of the electron spectra corresponding to the transitions to Ne+ 2s2p5 (3 P)3p via the Ne 1s 3p excitation. In this energy region, lines 5 and 6 are resolved in the 90 spectrum, whereas line 5 is suppressed and a line 4 appears in the 0 spectrum. These transitions were observed previously with the instrumental width of 1eV (Yoshida et al., 2000). At that resolution, none of the structures shown in Fig. 10 were separated. In order to extract the transition energies, branching ratios, the anisotropy parameters , and the natural widths of these lines, a fitting procedure has been carried out assuming that the line profiles are described by a convolution of the Lorentzian function and the instrumental function. Values of transition energies, anisotropy parameters , and branching ratios obtained from the fitting are summarized in Table 1. In order to assign the observed lines, the measured energies, the values, and the branching ratios were compared with the calculated ones. To this end, a series of ab-initio computations using the MCDF approach has been carried out. Both the initial and the final ionic state configuration interaction are taken into account. In the initial state there are two strongly mixed resonant states in this case with the total angular momentum J = 1 and with dominant configuration 1s-1 3p, which may be photoexcited from the ground state of the Ne atom. According to the calculations, however, the dipole excitation strength for one of them is 500 times larger than that for the other. Therefore, the second one was ignored and the decay of only one resonance to different channels was considered. The resonant state wave function expansion


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187

- Fig. 11. A part of the resonant Auger spectrum for 3d3/1 5p excitation in Kr at a photon energy of 92.425 eV (upper panel) and the asymmetry 2 parameter for the main peaks of the spectrum (lower panel). In the upper panel solid and dashed curves show the experimental spectrum measured at 0 and 90 , respectively. In the lower panel full circles show the experimental values of while open circles show the results of the MCDF calculations. Adapted from Kitajima et al. (2001).

consisted of 422 configuration state functions (CSF), which included all possible single and double excitations within the configuration space involving all orbitals up to n = 3. For the description of the final ionic states (J = 1/2,..., 7/2) with the dominant configuration 2s2p5 3p, all single and double excitations within the configuration space of 2p, 3s, 3p, 3d, and 4p non-relativistic orbitals were taken into account, leading to a total of 433 CSF. The continuum wave functions for the Auger electron were calculated in the potential of the final ion, the exchange interaction being properly taken into account. The results of these calculations are shown in Table 1. Comparing the measured values with the calculated ones averaged over the unresolved final J states for each multiplet, the lines 1, 2, and 3 have been attributed to the transitions to the 2s2p5 (1 P)3p 2 P , 2 D , and 2 S states, respectively, while the lines 4, 5, and 6 have been attributed to the transition to the 2s2p5 (3 P)3p 2 S , 2 P , and 2 D states. The measurements are in good agreement with the calculations for both the values and the branching ratios. 4.2. Effects of coherent excitation of resonances As we discussed earlier, a simple stepwise model is inapplicable in the case of overlapping resonances and should be modified according to prescriptions of Section 3.1.8. As an example of such a case we consider the cascade decay of the - - 3d3/1 5p resonance in Kr which was studied by Kitajima et al. (2001). In contrast to the case of the 3d5/1 5p resonance 2 2 considered above, there are now two excited states with the configuration 3d
-1 3/2 -1 3/2 -1 3/2

5p and total angular momentum J = 1.

In j­j coupling they correspond to the 3d 5p1/2 and 3d 5p3/2 states. Both states may be excited by the photon. Since their lifetime width is larger than the energy splitting, they both are excited coherently. - The experiment was carried out in the same way as described above for the 3d5/1 5p resonance, except that the photon 2 energy was chosen to be 92.425 eV to fit the position of the resonance. The experimental spectrum of the first-step Auger decay of this resonance is shown in Fig. 11 (upper panel). The measured values for some of the strongest resonances are shown in the lower panel. - Let us consider the anisotropy of resonant Auger transitions from the resonance 3d3/1 5p in further detail. As men2 tioned, this resonance consists of two strongly overlapping atomic states. The calculations show that one of the states is excited by photoabsorption much stronger than the other. The photoexcitation strength ratio is 20:1. On this ground the second state is usually ignored. This is justified in the calculations of the branching ratios. However the angular


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Fig. 12. Relative intensity and transferred spin polarization of electrons from Auger decay of intermediate 5p Rydberg states resonantly excited from the 3d5/2 (a) and 3d3/2 (b) fine-structure subshells of krypton atoms. Full circles: experiment; open squares: MCDF calculations. From Drescher et al. (2003).

distributions are reproduced rather poorly in this case (Tulkki et al., 1994). As it was shown (Kitajima et al., 2001), in the angular distribution calculations it is necessary to take into account the interference of the resonances. In fact, the two overlapping resonances are excited coherently and therefore the interference term may change considerably the anisotropy. The results of the computations according to formulas (46)­(49) within the MCDF theory (Kitajima et al., 2001) show that the interference effect is indeed very strong. The calculations with both interfering resonances taken into account compare rather well with the experiment (see Fig. 11 lower panel). 4.3. Spin polarization of Auger electrons in the resonant Auger transitions Modern experimental facilities permit one to study not only the angular anisotropy, but also the spin-polarization of the Auger electrons, at least for the first-step resonant Auger transitions (Hergenhahn et al., 1999; Drescher et al., 2003; Lohmann et al., 2005). As an example of such measurements, we show the results for the transferred spin polarization (see Section 3.1.6, expressions (33) and (34)) in the Kr 3d-1 5p resonances (Drescher et al., 2003). The experiment was performed at the circularly polarized SR beam-line UE56/2-PGM at BESSY II, Berlin. The details of the experiment may be found in the original paper by Drescher et al. (2003) (see also Section 2.2). The results of the measurements are shown in Fig. 12, which presents the experimental spectra and the measured spin-polarization transfer for the resonant


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189

- - Auger electrons from the decay of the 3d5/1 5p (Fig. 12a) and 3d3/1 5p (Fig. 12b) resonances. Due to the necessity to 2 2 maximize the throughput of the electron spectrometer for achieving reasonable count rates, the energy resolution in this type of experiment is lower than in the previous measurements of the angular distributions by Mursu et al. (1998) and Kitajima et al. (2001) (compare with Figs. 9 and 11 above). However, the gross structure of the intensity spectra is very close to those observed in other experiments. As expected from theoretical consideration (Cherepkov, 1973), the emission from different fine-structure components results in opposite signs of the electron spin polarization. The absolute value of the polarization is as large as 0.6­0.8. The most striking feature is the small variation of Ptrans , which hardly differ from line to line in spite of their totally different nature. Indeed, lines II and VI are 4s-1 4p-1 (1 P)5p and 4s-2 (1 S)5p spectator Auger lines, respectively, while line IV is a 4s-1 4p-1 (1 P)6p shake-up satellite of line II, and line V corresponds to a 4p-3 4d5p configuration satellite. This behavior of Ptrans was qualitatively explained using a simplified model with only one emitted partial wave (Drescher et al., 2003). In general, in any particular Auger transition several partial waves of the emitted electron, satisfying angular momentum and parity selection rules, contribute. The corresponding decay matrix elements determine the intrinsic parameters (35)­(37) and thus the electron spin-polarization. According to the theoretical analysis one of the partial waves strongly dominates in all of the above transitions. In this case, ignoring the contribution of other partial waves, it is possible to calculate the spin polarization parameters without knowing the matrix elements of the decay, since the only significant matrix element cancels out of the expressions for the intrinsic parameters. Thus these parameters become independent of the dynamical nature of the states involved. The values of the transferred spin polarization, calculated in this simple model are almost equal for all of the considered transitions (see Drescher et al., 2003). The exact calculations made within the MCDF theory, taking into account the interference effects for overlapping resonances give excellent agreement with the experiment (see Fig. 12) and confirm the simple interpretation.

4.4. Interference effects in the alignment of intermediate states and angular distribution of the second step Auger electrons After the first resonant Auger decay the atomic system may occur in an excited ionic state with one or two vacancies in the subvalence shells. In many cases it can further autoionize with emission of the second-step Auger electron. The initial polarization of the atom produced by the photon transfers partly to the intermediate ionic state. Therefore, the emission of the second-step electrons may be anisotropic and the emitted electrons may be spin-polarized. The study of the angular distributions of the second step Auger electrons reveals strong interference effects due to coherent excitation of the intermediate states (Ueda et al., 1999a, b, 2001; Kitajima et al., 2001, 2002; HeinÄsmÄki, 2001). In fact, the highly excited intermediate states are often overlapping, thus their population during the first decay is coherent. As it was shown in Section 3.1.8, in this case the simple stepwise model is inapplicable, however a modification introduced by expressions (50) and (51) permits one to treat the coherent polarization transfer and angular distributions of the Auger electrons emitted from coherently populated states. To illustrate the interference effect in the decay of overlapping intermediate states we consider the resonance excitation 2p 4s in Ar (Ueda et al., 1999a, b) and the following cascade decay process. First, a 2p innershell electron of Ar is excited by photoabsorption to the 4s orbital (jl-coupling is used for the description of the resonant state): Ar 2p6 3s2 3p
61

S0 + h Ar 2p5 3s2 3p6 (2 Pj0 )4s[K0 ]1 ,

(63)

where j0 = 1/2, 3/2 and K0 = j0 . Then the spectator Auger decay which creates 3s and 3p holes (i.e. L23 M1 M23 ) takes place: Ar 2p5 3s2 3p6 (2 Pj0 )4s[K0 ]1 Ar
+

2p6 3s3p5 (1 P1 )4s 2 P

1/2,3/2

+ e- (l1 j1 )

(1st decay).

(64)

The energy splitting between the resonant Auger final spin­orbit states 2 P1/2,3/2 is much smaller than the natural widths of these states and thus these two states are populated coherently via the resonant Auger decay (64). The configuration of the resonant Auger final state, 2p6 3s3p5 4s, should be considered only as a label; this state is a mixture of this configuration with 2p6 3s2 3p3 3d4s. Finally, the second-step Auger decay takes place: Ar
+

2p6 3s3p5 (1 P1 )4s 2 P

1/2,3/2

Ar

++

2p6 3s2 3p

43

P

0,1,2

or 1 D2 + e- (l2 j2 )

(2nd decay).

(65)


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Table 2 Second-step Auger electron anisotropy parameters for the transition 3s1 3p5 (1 P1 )4s 2 P1/2,3/2 3p4 3 P0,1,2 after 2p1/2 4s and 2p3/2 4s excitation, respectively, in Ar, as predicted with and without lifetime interference (LI) and as measured at HASYLAB and at the Photon Factory
Excitation (theoretical) with LI 2p 2p
1/2 3/2

(experimental) without LI -0.444 -0.056 HASYLAB -0.07 ± 0.15 -0.29 ± 0.15 Photon Factory 0.12 ± 0.07 -0.36 ± 0.03

4s 4s

0.0 -0.5

From Ueda et al. (1999b).

This cascade process was investigated by means of the electron­electron coincidence experiment (von Raven et al., 1990; von Raven, 1992), the angular distributions and angular correlations between emitted electrons have been measured by Ueda et al. (1999a, b, 2001). To demonstrate the role of the lifetime interference, the Auger matrix elements for the two subsequent processes (64) and (65) have been calculated with the use of simple but rather realistic models. For the first-step Auger decay the spectator model was applied, assuming that the outermost 4s electron does not participate in the transition (Hergenhahn et al., 1991). Moreover, it was assumed that from two possible partial waves of the Auger electron, s and d, the s-wave strongly dominates (Tulkki et al., 1993). The second-step Auger decay can be well described within the non-relativistic LSJ-approximation. For the particular transitions (65) only one partial wave of the Auger electron is possible (p-wave) according to selection rules. In this case all matrix elements in the generalized anisotropy coefficients (47) can be easily evaluated up to the common factor (Kabachnik and Sazhina, 1984). As it was mentioned in the case considered above, the intermediate levels strongly overlap, J1 J1 ? J1 J1 . If one implies, in addition, that J1 J1 = where is the total width which is the same for both states of the 2 P intermediate doublet, the joint factor 1/ can be taken out of the sum (51) (see Section 3.1.8). Using the described models and summing over all possible fine-structure states J2 of the final multiplet 3 P , one eventually obtains the following expression for the angular distribution of the second-step Auger electrons (65) following the photoexcitation of the 2p3/2 4s transition: W( ) = C 4
1

1+ T

int

-

1 P2 (cos ) , 18

(66)

where the constant C1 includes all common factors, and the term due to interference is Tint = -4/9. Without this interference term, the value of the angular distribution asymmetry parameter is = -0.056 as predicted previously by Kabachnik et al. (1999). Due to the interference term, however, the anisotropy is expected to be strongly enhanced, resulting in =-0.5. In a similar way one obtains the expression for the angular distribution of the second-step Auger electron emissions (65) at the 2p1/2 4s excitation: W( ) = C 4
2

1+ T

int

-

4 P2 (cos ) , 9

(67)

where Tint = 4/9. In this case the anisotropy is completely suppressed ( = 0) due to interference, although relatively large anisotropy ( =-0.444) is predicted without interference (Kabachnik et al., 1999). The experiment to confirm the predicted enhancement and suppression of the anisotropy for the Auger emission was carried out at HASYLAB in Germany and at the Photon Factory in Japan. The details of the experiment can be found in papers (Ueda et al., 1999a, b, 2001). Table 2 compares the results of both measurements with the predictions of the above described simple theory. We note that a more elaborate calculations within the MCDF theory (Ueda et al., 2001) confirm the results obtained with the simple model. The interference effects connected with the coherent excitations of the intermediate states were observed also in other cascades photoexcited in noble gas atoms (Kitajima et al., 2001, 2002; HeinÄsmÄki, 2001; Yoshida et al., 2005).


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191

4.5. Coincidence studies of Auger cascades The detailed information about the cascade Auger decay can, in principle, be obtained from the angular correlation experiments by detecting both emitted electrons in coincidence. Unfortunately, to make these measurements feasible, the experimentalist has to sacrifice the energy resolution. Nevertheless, even with comparatively poor resolution, the coincidence experiments provide very important, both spectroscopic and dynamic, information about cascades. The first coincidence measurements of the cascades were angle averaged and had the aim to determine the most probable paths of the cascade (von Raven et al., 1990; von Raven, 1992, Alkemper et al., 1997). Quite recently first angular correlation measurements have been carried out for some particular cascades in Ne (Turri et al., 2001; Da Pieve et al., 2005), Ar (Ueda et al., 1999a, b, 2001), Kr and Xe (Ueda et al., 2003b; Viefhaus et al., 2005). We have selected two typical examples of the correlation measurements. One is the study of the angular correlation in cascade decays following the Ne(1s-1 3p) excitation. The experiment has been performed at the Gas Phase Photoemission beam-line at ELETTRA, the SR facility in Trieste. Description of the undulator radiation source and of the apparatus, dedicated to angle resolved photoemission and multicoincidence spectroscopy, have been reported elsewhere (Gotter et al., 2001; Turri et al., 2001). The linearly polarized photons of the beam-line, operating the monochromator at moderate resolution of about 250 meV FWHM, were used to excite the resonance. The angular distribution of the high-energy resonant electrons have been measured in coincidence with the low-energy Coster­Kronig electrons, detected at fixed angles with respect to the polarization direction of the radiation. Both electrons were detected in the plane perpendicular to the photon beam. In Fig. 13 the angular distributions are shown for the cascade Auger transitions + Ne Ne 1s
-1

3p Ne

+

2s

-1

2p 2p

-1 1

( P)3p 2 D, 2 P, 2 S + e P + e2 .

1

Ne

++

-2 3

The measurements have been performed for three angles of the emitted second-step electron 0 ,30 and 60 . Since the multiplet structure of the intermediate state Ne+ 2s-1 2p-1 (1 P)3p was not resolved in the experiment, the contribution of all the multiplets was coherently summed in the corresponding calculations (see Eqs. (52)­(54)). In the described experimental conditions, with the detection plane perpendicular to the photon beam, the angular correlation function can be presented as the following expansion (cf. Eq. (13)):
nmax

W( 1 ,

2

)=
n=0,even

[An ( 2 ) cos(n 1 ) + Bn ( 2 ) sin(n 1 )],

(68)

where 1 and 2 are the polar angles of the first and the second Auger emission, and nmax = 6, 4, 2 for the 2 D, 2 P, 2 S multiplet terms, respectively. The coefficients An and Bn depend both on the geometry of the experiment and on the matrix elements of the two subsequent Auger decays. These coefficients are used as fitting parameters in describing the experimental data. On the other hand, they are evaluated theoretically using Eqs. (52)­(54). The calculations have been done within the MCDF theory as is described in Section 3.2.1. The results of the fitting as well as of the calculations are shown in Fig. 13. In general, the agreement is quite good, although for the angle 2 = 60 the experimental and theoretical patterns are slightly shifted. - Another interesting example is provided by the cascade decay of the 3d5/1 5p resonance in Kr. The measurements 2 were carried out at the Photon Factory, a 2.5-GeV SR facility in Japan. Details of the experimental setup and procedure were described by Ueda et al. (2001) (see also Section 2.4.1). The linearly polarized photon beam with energy of 91.2 eV was used to excite the resonance. Both emitted cascade electrons were detected in the plane perpendicular to the beam. The results of the measurements are shown in Fig. 14 for different cascades. Also shown are the theoretical predictions calculated in two models, the simplified LSJ coupling model and full scale MCDF calculations (for details see Ueda et al., 2003b). The most interesting result of this experiment is that the angular correlation patterns are completely different for different decay channels. Compare, for example, the patterns in Fig. 14(a) and (b). Here the cascades differ only by the second-step transition, since the first step is identical. Nevertheless, the patterns are strikingly different. This shows that the coincidence measurements are very sensitive to the intrinsic anisotropy parameters of the second-step transitions, in contrast to the non-coincidence measurements. As it was shown (Kitajima et al., 2001)inthe non-coincidence experiments, the angular distributions of the second-step decays are practically isotropic due to very


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N.M. Kabachnik et al. / Physics Reports 451 (2007) 155 ­ 233
90 0.32 120 60

2 = 0°
0.24 Intensity (counts)

exp theory fit

150

30

0.16

180

0

0.08

210

330

0.00 120 150 180 210 240 270 300 330

240 270

300

1 (deg)
90 120 0.32 60

2 = 30°
150 30

0.24 Intensity (counts)

180 0.16

0

0.08

210

330

240 0.00 120 150 180 210 240 270 300 330 270

300

1 (deg)
0.32

2 = 60°

90 120 60

0.24 Intensity (counts)

150

30

0.16

180

0

0.08 210 0.00 330

120

150

180

210

240

270

300

330

240 270

300

1 (deg)

Fig. 13. Angular distribution of the resonant Auger electrons in the cascade Ne 1s-1 3p Ne+ 2s-1 2p-1 (1 P)3p 2 D, 2 P, 2 S Ne++ 2p-2 (3 P) at the fixed emission angle of the Coster­Kronig electron of 2 = 0 , 30 and 60 . Solid and dashed lines are the theoretical predictions and the best fitting to the experimental data, respectively. From Da Pieve et al. (2005).


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193

90 120 150 60 30 150 120

90 60 30

180

0 180

0

210 240 270 90 120 150 60 300

330

210 240 270 300

330

e
120 30 150

90 60 30

180

0 180

0

210 240 270 300

330

210 240 270 90 300

330

120 150

60 30

180

0

210 240 270 300

330

Fig. 14. Angular correlation between the - 3d5/1 5p (J0 = 1) 4s1 4p5 (1 P) 5p 4s 2 - 4s1 4p5 (1 P) 5p 4s2 4p4 3 P1 ; (d) 3d5/1 5p 2

2 4p4 1 D ; (b) 3d-1 5p (J = 1) 4s1 4p5 (1 P) 5p 4s2 4p4 3 P ; (c) 3d-1 5p (J = 1) 0 0 2 2 5/2 5/2 - (J0 = 1) 4s2 4p3 4d (1 P) 5p 4s2 4p4 1 D2 ; (e) 3d5/1 5p (J0 = 1) 4s0 4p6 (1 S) 5p 2

resonant Auger emission and the second-step Auger emission for the Kr transitions (a)

4s1 4p5 3 P2 . Coincidence rates are plotted as a function of the angle of the direction of first step Auger electron relative to the polarization vector whereas the second step Auger electron is detected in the direction of 270 (see Fig. 4). The solid curves represent the results of the MCDF calculation. The dashed and dotted curves correspond to the calculations with only one partial wave in the first-step emission and MCDF and LSJ coupling approximation for the second-step transitions, respectively. Theoretical curves are normalized to the same area limited by the curves. From Ueda et al. (2003b).

small alignment transfer in the first Auger decay. Thus it is practically impossible to extract the intrinsic anisotropy parameter for the second-step decay from such measurements. 4.6. Realization of a complete experiment in resonant Auger cascade As it was discussed in Section 3.3, in general, the complete experiment for the Auger decay cannot be realized by measuring the properties of only the Auger electrons. This is valid in the general relativistic case. Measurements of the angular correlations in the Auger­electron cascades provide additional information about the polarization parameters of the photoion, making complete experiment in some cases possible. However, even if the number of measured anisotropy parameters is sufficient, practical determination of the amplitudes from the experimental data is very difficult. This is


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the reason why in the first realizations of the complete experiment the non-relativistic approximation was used, when the decaying state and the final atomic system are described in the LS coupling scheme. This reduces substantially the number of unknown amplitudes. As an example, we discuss one of the first realizations of the complete experiment for the resonant Auger decay of the 2p-1 4s photoexcited state of argon (Ueda et al., 1999a). In this experiment the following cascade in Ar photoexcited to the 2p-1 4s state by linearly polarized light was studied: 3/2 Ar 1 S0 + h Ar 2p5 3s2 3p6 (2 P Ar Ar
+ ++ 3/2

)4s,

J0 = 1,
J1

(69) + e- (l1 j1 ), (70) (71)

2p6 3s3p5 (1 P1 )4s2 P 2p6 3s2 3p
43

P

J2

+ e- (l2 j2 ),

where J1 = 1/2, 3/2 and J2 = 0, 1, 2. The corresponding Auger lines at a kinetic energy 194 eV [transition (70)] and 7 eV [transition (71))] were observed in an electron­electron coincidence experiment (von Raven et al., 1990). The configuration of the intermediate state, 3s-1 3p-1 4s, in (70) should be considered only as a label because of the strong mixture of this configuration with 3p-3 3d4s (von Raven et al., 1990; Hansen and Persson, 1987). The measurements were carried out on the 24-m spherical grating monochromator installed in the BL-16B undulator beam-line at the Photon Factory in Japan (Shigemasa et al., 1998), using an apparatus described elsewhere (Shimizu et al., 1998). The photon energy was tuned to the Ar 2p3/2 4s excitation at 244.4 eV and the photon band pass was 0.4 eV. The incident light was focused onto the interaction region, where light was merged with an effusive gas beam. The degree of linear polarization Plin of the incident light was 100+0 % in the horizontal plane. -3 The angular distributions of both the first-step resonant Auger emission (70) and the second-step Auger emission (71) were independently measured as well as the angular correlation between the two electrons. In the measurement of the angular correlation in the plane perpendicular to the photon beam, two identical 150 spherical sector electron spectrometers with a mean radius of 80 mm were used. For a description of the experimental set-up see Section 2.4.1. The first spectrometer, mounted on a turntable, detected the resonant Auger electrons at kinetic energy of 194 eV ejected in the first decay (70). The second spectrometer was set in such a way that it detected Auger electrons at a kinetic energy of 7 eV ejected in the second decay (71), perpendicular to the linear polarization axis of the incident light. Further details of the experiment may be found in Ueda et al. (1999a, 2001). As follows from the general expression (13) in Section 3.1.3, the angular correlation function in this particular case can be presented by the expression: I( ) = A0 + A2 cos 2 + A4 cos 4 , (72)

where is the angle between the light polarization vector and the emission direction of the resonant Auger electron. By fitting this expression to the experimental points the ratios of the coefficients A2 /A0 and A4 /A0 were obtained. The angular distributions of the electrons measured independently are described by Eq. (26) and yield other two parameters (1) and (2) , the angular distribution parameters for the first- and second-step Auger decay, respectively. Thus four parameters were experimentally determined. Now we discuss how the parameters (i ) and Ai /A0 (i = 1, 2) are related to the resonant Auger amplitudes and their phase difference. In a general (relativistic) case, according to the angular momentum and parity selection rules, the first-step decay is described by two (for the 2 P1/2 final state) and by three (for the 2 P3/2 state) complex partial amplitudes, VJ1 ,j , corresponding to the s1/2 and d3/2,5/2 partial waves of the ejected electron. Five complex amplitudes are described by 9 real parameters (5 moduli and 4 relative phases), thus 9 independent observables should be measured in order to realize the complete experiment in this case. However, in the non-relativistic approximation, with some additional model assumptions discussed below, the number of independent amplitudes describing the decay can be considerably reduced. The resonant state 2p-1 4s, i.e., the initial state for the first Auger decay, is aligned along the photon polarization (z3/2 axis). Using the conventional two-step ansatz, the angular distribution parameter (1) for the resonant Auger electrons can be expressed as a product of two terms (see Eq. (26)), the alignment parameter A20 of the initial state and the intrinsic anisotropy parameter 2 for the Auger decay (Kabachnik and Sazhina, 1984):
(1)

=A

20 2

.

(73)


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195

The superscript is added to indicate that (1) is the average asymmetry parameter for two experimentally unresolved transitions to the 2 P1/2,3/2 intermediate states in decay (70). In fact, they almost completely overlap, and in the following the energy splitting of the two states is ignored. The alignment parameter is A20 =- 2 for a state excited by completely linearly polarized light from the ground state of a closed-shell atom. In order to evaluate 2 , it is assumed that in transition (70) the intermediate state is well described within the LSJ approximation, i.e. the total orbital momentum L and total spin S are good quantum numbers. Note that the LSJ coupling scheme does not exclude strong configuration mixing, which is important in the proper description of this state. Besides the LSJ approximation for the intermediate state, it was assumed that the spin­orbit interaction in the continuum can be neglected. Then all of the five amplitudes VJ1 ,j can be expressed in terms of two Auger amplitudes Vs and Vd for the s and d waves and their phase difference . By expanding the wave function of the 2p-1 4s initial state in terms of LSJ functions and transforming the Auger matrix 3/2 elements to the LSJ coupling scheme (for details see Kabachnik and Sazhina, 1984), one obtains the expression for the angular distribution parameter (1) for the resonant Auger electrons in terms of three parameters Vs , Vd , and :
(1)

=

2 Vd - 2 2Vs Vd cos
2 2(Vs2 + Vd )

.

(74)

Using the same approximations as those employed for the first-step Auger decay, one can obtain an expression for the angular distribution parameter (2) of the second-step Auger electrons (assuming the first-step electron is not detected in coincidence). Here the interference of the two overlapping intermediate states should be taken into account. To do that, the general equation (46) is used. The alignment transferred to the intermediate state, Eq. (50), is also determined by the first-step Auger amplitudes Vs and Vd . Note also that only the p wave contributes to the second decay 2 P1/2,3/2 3 P0,1,2 (see (71)), within the non-relativistic LSJ approximation. Using the above approximations, calculating the alignment transfer as outlined by Ueda et al. (1999a), and summing over the unresolved states of the final multiplet (3 P0,1,2 ), one obtains an expression for (2) for the second-step Auger electrons:
(2)

=-

Vs2 + 0.1V 2(Vs2 + V

2 d 2) d

.

(75)

The angular correlation between the two consecutively emitted electrons can be obtained using the approach described by Kabachnik et al. (1999) generalized to account for the overlapping intermediate states, Eqs. (52)­(54). It has the form of Eq. (72) with coefficients Ai depending on the Auger decay matrix elements. Using the non-relativistic LSJ-approximation for both first-step and second-step Auger transitions, it is possible to eventually express the Ai coefficients in terms of the same parameters Vs , Vd , and . One obtains A2 /A0 = 2 48Vd - 96 2Vs Vd cos , 2 80Vs2 - 16 2Vs Vd cos + 61Vd
2 27Vd 80Vs2 - 16 2Vs Vd cos

(76)

A4 /A0 =

+ 61V

2 d

.

(77)

Thus, within this model, one can express all the anisotropy coefficients for both non-coincidence and coincidence measurements in terms of only two independent parameters: the ratio of the decay amplitudes Vd /Vs and the phase difference (cos ). To determine these values from the experiment it is sufficient to measure only two observables. For example, one can use the results of the coincidence experiment only: this variant is analogous to the complete experiment in photoionization made by KÄmmerling and Schmidt (1991). Alternatively one can use the values (1) and (2) as obtained from the non-coincidence measurements: this variant is analogous to the complete experiment in photoionization made by Hausmann et al. (1988). The most reliable values of Vd /Vs and cos , however, can be obtained from the least squares method treating Vd /Vs and cos as fitting parameters and four values (1) , (2) , A2 /A0 , and A4 /A0 as data points to be fitted using Eqs. (74)­(77). The values thus obtained are Vd /Vs = 0.33 ± 0.10 and cos = 0.02 ± 0.03.


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4.7. Auger cascades following the PCI induced electron recapture in the near-threshold photoionization Until quite recently the resonant Auger cascades have mainly been studied after promotion of an inner-shell electron to one of the lowest Rydberg orbitals. Examples of such cascades have been discussed in previous sections. In recent paper by De Fanis et al. (2004b), a method was suggested for populating effectively a large number of highly excited Rydberg ionic states. The cascade decay of these states produces a rich spectrum of Auger transition series converging to various levels of the doubly charged ion. The method is based on the phenomenon of photoelectron recapture due to a post-collision interaction (PCI). If an atom is photoionized just above the inner-shell ionization threshold, the following Auger decay can occur when the photoelectron is still close to the atom. The fast Auger electron quickly overtakes the slow photoelectron. The resulting change of the screening (i.e., the main part of the PCI) will lead to a decreasing energy of the photoelectron or even to its recapture into a discrete Rydberg state of the ion. This phenomenon is well studied both experimentally (van der Wiel et al., 1976; Eberhardt et al., 1988; Samson et al., 1996, 1997; Lu et al., 1998; LeBrun et al., 1999; Hentges et al., 2004) and theoretically (Amusia et al., 1977; Tulkki et al., 1990; Armen et al., 1996, 1997; Sheinerman, 2003). It is known that in general the photoionization and the PCI-induced recapture of the photoelectron should be considered as a one step process leading to the valence-excited ionic states. Recently, an accurate high-resolution study of the recapture process in Ne near-threshold 1s photoionization have been published (Hergenhahn et al., 2005, 2006). The considered process is + Ne Ne 1s
-1 - (n, )p Ne 2p4 (1 D2 )(n , )p + eA .

(78)

The notation (n, ) indicates a Rydberg or photoelectron. Dependent on the values of n or and its primed counterparts, the reaction (78) describes resonant Auger decay, shake-off, photoelectron recapture or sequential photo-double ionization. In the case of photoelectron recapture in the final state, the singly charged ion with configuration Ne 2p4 (1 D2 )n p is energetically located above the double ionization threshold of Ne for n 5 (Becker and Wehlitz, 1994). By that, these states will autoionize to the 2p4 (3 P) manifold by emission of a low kinetic energy Auger electron (valence intermultiplet Auger transition). These second-step Auger electrons, with kinetic energies below 3.1 eV, were recently observed by De Fanis et al. (2004b, 2005) and for other intermediate states by Hentges et al. (2004). Due to their low energy, they allow one to characterize the energy positions of the Ne+ n p Rydberg series with unprecedented detail. An example of the results obtained in these studies will be given below. But before this, we discuss the spectra - of the electrons emitted in the first step of the Auger cascade [eA in Eq. (78)]. These have kinetic energies on the order of 806 eV. The experiments (Hergenhahn et al., 2005, 2006) have been performed at the c-branch of the beam-line 27SU at the SPring-8 SR source in Japan. This beam-line is equipped with a figure-8 undulator (Tanaka et al., 1999), which is capable of producing linearly polarized light. In the c-branch, the photons from this insertion device are monochromatized by a Hettrick-type varied line space plane grating monochromator (Ohashi et al., 2001a). For the described measurements the photon resolution was set to about 95 meV. The target consisted of a gas cell and electrons were detected by a hemispherical electrostatic analyzer (Gammadata-Scienta SES2002) mounted horizontally and perpendicular to the photon beam. The pass energy was set to 100 eV, giving an analyzer resolution of about 66 meV. Examples of the Auger electron spectra are shown in Fig. 15. Similar spectra were taken for several other excess energies from -0.6 to 0.6 eV. Contributions of resonant Auger decay or recapture, respectively, into final states up to n = 12 can be resolved. In order to compare the experimental spectra with calculations, a smooth continuation of the Auger part of the spectrum into the region below threshold is employed. This problem has been discussed in the theory of atomic oscillator strength (Fano and Cooper, 1968), and the gray boxes are drawn accordingly, taking into account the density of the final states. The area of the boxes represents the experimental intensity of the excitation for the n = 4 - 12 states. Experimental results are compared with two types of calculations. One is based on a semiclassical approach and is an extension of the model by Russek and Mehlhorn (1986). Another is based on the quantum mechanical non-stationary theory of near-threshold photoionization (Kazansky and Kabachnik, 2005). For positive and negative excess energies, the gross behavior of the spectrum is in very good agreement with the semi-classically calculated spectra (solid curve). The quantum­mechanical calculations give very similar results (Hergenhahn et al., 2005). The excited ionic states populated in the recapture process can autoionize to the ground state of the doubly ionized Ne2+ . Thus this process can be used for spectroscopic study of the inner-valence Auger transitions in ions (secondstep Auger transitions). As an example we discuss the experiment performed by De Fanis et al. (2004b, 2005). In the experiment the electron spectrum in the kinetic energy region 0.1­3.3 eV generated by photoionization of Ne atoms by


N.M. Kabachnik et al. / Physics Reports 451 (2007) 155 ­ 233
Relative Energy (eV) 0 2.0 E 1.5
exc

197

1

2

3

4 = 0.6 eV + Exp. Th. (classical)

5

6

Intensity (arb. units)

1.0

0.5

n' = 5

n' = 4

0.0 2.0 E 1.5
exc

= 0.0 eV + Exp. Th. (classical)

Intensity (arb. units)

1.0

0.5

n' = 5

n' = 4

0.0 2.0 E 1.5
exc

= -0.4 eV + Exp. Th. (classical)

Intensity (arb. units)

1.0

0.5

n' = 5

n' = 4

0.0 804

805

806

807

808

809

810

Kinetic Energy (eV)

Fig. 15. Part of the Ne Auger spectrum recorded at three different excitation energies Eexc =+0.6, 0.0, -0.4 eV, relative to Ne 1s ionization threshold at 870.17 eV: +, experimental data points after subtraction of constant background; solid lines, theoretical Auger electron profiles calculated using semiclassical model. Areas of the gray shaded boxes represent the experimental intensity of recapture into the n = 3­12 states. From Hergenhahn et al. (2006).

photons of energy 100 meV above the 1s ionization threshold of 870.17 eV have been measured. The angle resolved measurements have been done for two mutually perpendicular directions of linear polarization, so that the anisotropy of angular distribution of emitted electrons, as well as intensity, have been determined. The measured low energy electron spectrum contains several series of well resolved transitions. In Fig. 16, we show two parts of the spectrum which correspond to the lowest members of the series for the transitions 2p4 (1 D)np 2p4 3 P for n = 5 (Fig. 16a), n = 6 (Fig. 16b). Black dots show the experimental results obtained for 0 emission with respect to


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N.M. Kabachnik et al. / Physics Reports 451 (2007) 155 ­ 233

2

6

F
3

3

P0

3

P1

P2
2

Intensity (arb. units)

D

4

3

P0

3

3

P1

P2

2

0 0.20 0.25 0.30 0.35 0.40

15
3

2

F
3 2

P0
3

3

P1 D
3

P2
3

Intensity (arb. units)

10

P0

P1

P2

5

0 1.15

1.20

1.25

1.30

1.35

Kinetic energy (eV)

Fig. 16. Parts of the low energy electron spectrum recorded at an angle of 0 (full circles) and 90 (open circles) relative to the polarization vector. The 2p4 (1 D)np 2 F,2 D 2p4 3 P autoionization decay of inner-valence excited ionic Rydberg states of Ne, with n = 5, 6 are displayed in panels (a) and (b), respectively. The solid curves are the results of a least-squares fit described in the text. From De Fanis et al. (2004b).

the photon polarization direction. Open circles show the spectrum for the 90 emission. The lines represent the results of a least-square fit to the spectra with Gaussian profiles of width (FWHM) of 11 meV. Further details see in the papers by De Fanis et al. (2004b, 2005). The kinetic energy resolution of the experiment is sufficiently high to resolve not only the transitions to the different fine-structure levels of the final 3 P0,1,2 state, but also the multiplet structure of the excited Rydberg states. In order to identify the transitions, ab initio multi-configuration Dirac­Fock (MCDF) calculations of the energy positions, the relative intensities and the anisotropy parameter of the angular distribution for the considered transitions have been made. Both initial and final ionic state configuration interactions were taken into account in a configuration space, which included 1s2 2s2 2p4 np non-relativistic configurations (n = 3 - 7) in the initial state and 1s2 2s2 2p4 configuration in the final state. Initial and final states were separately optimized using the atomic structure code GRASP92 (Parpia et al., 1996) and the Auger matrix elements were calculated using the RATIP code (Fritzsche, 2001a, b) described in Section 3.2. The intensity and parameter for each Auger line were evaluated by assuming that the initial state population is statistical and the alignment transfer is not distorted by the PCI. Using the calculated values, one can identify the lines in the spectra shown in Fig. 16. The peaks correspond to the transitions 2p4 (1 D)np 2 F 2p4 3 P0,1,2 and 2p4 (1 D)np 2 D 2p4 3 P0,1,2 (n = 5, 6). The measured term values T and the measured and calculated relative


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199

Table 3 Term values, intensity ratios and angular anisotropy parameters for the two lowest observed members of the series 2p4 (1 D)np 2 F 2p4 3 P0,1,2 and 2p4 (1 D)np 2 D 2p4 3 P0,1,2 n = 5, 6 of intermultiplet Auger transitions. Intensities have been normalized to the strongest transition for the respective n. Adapted from De Fanis et al. (2004b)
Initial state Final state T (eV) Intensity ratio Expt. 5p 5p 5p 5p 5p 5p 6p 6p 6p 6p 6p 6p
2 2 2 2 2 2 2 2 2 2 2 2

Theor. 0.18 0.12 0.58 0.40 1.00 0.80 0.18 0.10 0.58 0.37 1.00 0.80

Expt. 0.55(11) -0.56(10) 0.45(6) -0.57(6) 0.28(5) -0.46(5) 0.55(10) -0.64(10) 0.40(6) -0.55(5) 0.24(5) -0.42(5)

Theor. 0.65 -0.70 0.55 -0.60 0.36 -0.34 0.65 -0.70 0.53 -0.57 0.37 -0.35

F D F D F D F D F D F D

3 3 3

P P P P P P

0 1 2

0.212 0.260 0.246 0.294 0.326 0.374 1.191 1.216 1.225 1.250 1.305 1.330

0.19(1) 0.16(1) 0.59(1) 0.44(1) 1.00(1) 0.71(1) 0.15(1) 0.13(1) 0.55(1) 0.46(1) 1.00(1) 0.75(1)

3 3 3

0 1 2

intensities and -values are shown in Table 3. Good agreement between theory and experiment confirms the assignment of the levels. Interesting to note that the transitions 2p4 (1 D)np 2 P 2p4 3 P0,1,2 are not seen in the experiment. A possible reason for this is that the lifetime of the 2p4 (1 D)np 2 P states is much shorter than that of the 2 F and 2 D states, and due to their large width it might be impossible to separate them from the background. In the calculations the width of 2 P comes out two orders of magnitude larger than that of 2 F and 2 D , which is still not sufficient to explain the apparently complete absence of transitions originating from the 2 P state. However, as it was demonstrated earlier (Sinanis et al., 1995), accurate calculations of these widths should include electron correlations on a much larger basis than used in the discussed work. Thus the reason for the absence of the 2p4 (1 D)np 2 P series remains an open question. 5. Auger­fluorescence cascades When the final state of the Auger decay is a bound ionic state, a further decay by electron emission is energetically forbidden and the deexcitation proceeds via one or several radiative transitions. Such fluorescence cascades are considered in this chapter. The ion fluorescence after the resonant Auger/autoionization decay has been first observed with the use of SR sources by Kronast et al. (1984, 1986) and Goodman et al. (1985) in Cd atom. Already in these first papers, the polarization of the fluorescence was measured, giving the access to the alignment of the ionic states. With the advent of new SR sources these studies were extended, especially for the noble gas atoms (for example, O'Keeffe et al., 2004; Schartner et al., 2005 and references therein). The angular correlations between the ejected electron and the polarized fluorescence, measured in the 1990s (Beyer et al., 1995, 1996; West et al., 1996, 1998; Ueda et al., 1998), provided a method of obtaining a very rich or sometimes even complete information on the Auger decay. The radiative deexcitation cascade gives rise to complicated fluorescence spectra in the wavelength regions ranging from VUV to IR. Some general features of the fluorescence spectra and their difference and complementarity to the Auger electron spectra are discussed in the next section. Angular distribution and polarization of the fluorescence carry information on the orientation and alignment parameters of the initial state of the radiative transition formed after the Auger decay. A procedure of extracting these parameters is discussed in Section 5.2 with special attention to depolarization effects. The method of ion fluorescence polarimetry provides complementary information on the Auger decay, which is usually studied by electron spectroscopy. Combination of the two methods enables one to extend the data on the decay amplitudes up to the degree of the complete description of the Auger decay. This important contribution of the fluorescence polarimetry is discussed in Section 5.3. When resolution and brightness of the incoming photon beam are high enough, the Auger resonance can be scanned up to its far wings, where the direct photoionization is no longer negligible and the two-step regime is violated. Section 5.4 describes the phenomenon of a `universal scaling'


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Fig. 17. Left part: schematic representation of the transitions in the resonant Auger process upon the 2p 3d excitation in Ar indicating main decay pathways. Right part: upper panel, part of the electron Auger spectrum (Mursu et al., 1996); middle panel, the fluorescence spectrum in the VUV (Meyer et al., 2001b); lower panel, part of the fluorescence spectrum in the UV/Visible (Meyer et al., 2002).

in the behavior of correlation and polarization parameters as function of the excitation energy in the region of an Auger resonance. This is exemplified by the fluorescence polarimetry data. The influence of neighboring Auger states is also discussed. Finally, in Section 5.5 we give a brief outlook of Auger/autoionization electron­fluorescence coincidence studies. Such studies may be considered as providing the most detailed information about the Auger decay of an intermediate resonance to the bound ionic states. 5.1. Radiative cascades in the VUV and visible region Observation and spectral analysis of fluorescence photons have some particular experimental advantages with respect to the spectroscopy of Auger electrons. This is mainly because the fine structure of the residual ion is much easier resolved by the high resolving power of an optical spectrometer than by that of an electron analyzer. In addition, the fluorescence spectra are free from spectral broadening introduced by the exciting photon beam, since its energy spread is transferred to the photoelectron and not to the photoion. An illustrative example for the fluorescence transitions is given in Fig. 17 showing the general scheme of the resonant Auger process in Ar, when the 2p-1 3d (J = 1) state is 3/2 photoexcited, as well as the corresponding fluorescence spectra in the visible and the VUV wavelength regions. For a complete illustration of all processes also the Auger spectrum is presented in the figure. While in the Auger spectrum a particular ionic state appears only as a single line, several fluorescence transitions from the same ionic state are often observed, providing good possibilities for a cross checking of the measured alignment or orientation of this state, but leading also to a more complicated spectrum. Radiative transitions to the ionic ground state give rise to fluorescence emission in the VUV wavelength region. The relative intensities of the fluorescence lines originating from different ionic states are governed by the relative population of these states and are correlated with the strengths of the corresponding Auger transitions. Therefore, in some cases the fluorescence spectrum represents almost a mirror type image of the electron decay. For example, fluorescence lines attributed to decays of the 3d levels are much stronger than those connected with the 5d levels (cf. Fig. 17), in accordance with the relative strength of the 3d and 5d Auger lines in the electron spectra (Meyer et al., 1991; Mursu et al., 1996). But for a more detailed analysis, dipole selection rules, preventing for example the decay of states with high angular momentum to the 3p5 2 P3/2 or, even more restricted, to the 2 P1/2 levels of the ground state, and radiative cascades have explicitly to be taken into


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201

account. For the higher lying states also visible transitions become possible (Fig. 17), which can be measured with very high spectral resolution independent of the exciting photon energy. Even for the excitation of core electrons at rather high photon energies, about 240 eV in the case of resonant Ar 2p excitation, the fluorescence spectrum is recorded with the same spectral precision as after valence ionization at much lower excitation energies. For example, the spectrum in Fig. 17 is recorded with a spectral resolution of 0.2 nm, which would correspond to an energy resolution of only 2 meV in the electron spectrum. In the most recent studies of the resonant Auger decay in Ar (Osmekhin et al., 2006) the high brilliance of the MAX II third generation SR source and its high photon energy resolution were used. This allows studies under extreme resonant Raman conditions, i.e. with an overall energy resolution of about Ekin = 40 meV. Therefore, the fluorescence spectroscopy is in many cases the unique method to resolve a particular final ionic state and the combined analysis in the visible as well as in the VUV region provides an almost complete picture of the complex relaxation pathways of the ionic states populated by Auger decay. But the fact that intense visible fluorescence can be analyzed represents at the same time also the main drawback of fluorescence studies on excited ions produced upon innershell excitation. It is the direct signature of radiation cascades, which distort the population and polarization of the initial states of the measured optical transitions (Hamdy et al., 1991; JimÈnez-Mier et al., 1993). These cascades might develop in many steps via different intermediate states and can therefore lead to the emission of several photons. In the example of Ar ions produced upon resonant 2p 3d excitation (Fig. 17), the Auger decay leads not only to the Ar II 3p4 3d states, but also to the higher lying states, mainly to the levels 3p4 4d (about 50%) and 3p4 5d (about 10%) caused by shake-up processes. These states decay not only by VUV but also by visible fluorescence. For example, in the related spectral region between 300 and 500 nm strong fluorescence lines corresponding to transitions of the type 3p4 4d 3p4 4p and 3p4 4p 3p4 4s and 3d have been observed (Meyer et al., 2002). These cascading effects have, in general, to be taken into account in the final analysis of the fluorescence spectra in order to extract reliable values, e.g. for the alignment and orientation of the ionic states due to the direct population from the Auger decay. 5.2. Polarization analysis and depolarization mechanisms An important further step of the fluorescence analysis, bringing additional information, is to deduce the orientation A10 ( J) and/or the alignment A20 ( J) transferred to the ionic state J during the Auger process. Fig. 18 illustrates the orientation transfer in the Auger decay from a completely oriented state J = 1- to the ionic states with the total angular 5 1 momentum J = 2 or J = 2 and negative parity through channels with different orbital and total angular momenta of the Auger electrons. The actual transfer is a weighted average of all channels. The quantities A10 ( J) and A20 ( J) are expressed in terms of relative partial decay widths according to Eq. (19). Investigations on the angular distribution (41) or linear polarization (42) of the cascading fluorescence in Eq. (38) should give equivalent information on the alignment A20 ( J) of the initial state of the radiation transition, while the circular polarization (43)­(45) gives information on its orientation A10 ( J). Angular distribution (Lagutin et al., 2000; Demekhin et al., 2005) and polarization (Zimmermann et al., 2000; Meyer et al., 2001a, b, 2002; O'Keeffe et al., 2003, 2004; Schill et al., 2003; Lagutin et al., 2003a, b; Grum-Grzhimailo et al., 2005; Schartner et al., 2005) of the ion fluorescence after the Auger decay was extensively studied during the last years for noble gas atoms. In order to illustrate the polarization analysis we take the resonant Auger process in Xe, - when the 4d5/1 6p J = 1 state is photoexcited at h = 65.1 eV. The corresponding decay pathways and the fluorescence 2 spectrum in the visible wavelength region are shown in Fig. 19 together with the measured values of PC and PL after the Auger decay. We concentrate on the fluorescence from the Xe II 5p4 6p states to the lower lying 6s and 5d levels. All the 21 Xe II 5p4 6p fine-structure states are easily distinguished by dispersed fluorescence spectroscopy. By means of electron spectroscopy they could be resolved only using the high brilliance of the ALS third generation SR source under extreme resonant Raman conditions (æhrwall et al., 1999). The circular and linear polarization measured for transitions from the ionic state to different final states vary in magnitude and even in sign (Table 4). This is due to the fact that the degree of polarization of the fluorescence depends also on the total angular momentum Jf of the final state of the radiative transition via the coefficient k (40). Values of the orientation and alignment parameters are presented in the last columns of Table 4. These values were extracted from the measured values of PL and PC for some of the - Xe II 5p4 6p ionic states, populated via the resonance Auger decay of Xe 4d5/1 6p(J = 1). The depolarization of ionic 2 states due to the radiative cascade and hyperfine interactions has been taken into account (see below). Although the


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Fig. 18. Orientation transfer during the Auger process. Right circularly polarized light prepares the Auger state with J = 1 in a single magnetic 1 sublevel, MJ =+1. The decay to a J = 2 state via either the emission of a s1/2 or d3/2 Auger electron leaves the resulting ion oriented as shown 1 1 5 in (a) where the bars indicate the population of the MJ =- 2 and + 2 sublevels. The same information is given for the decay to a J = 2 state in (b). Average projection of the resulting ion angular momentum along the incoming photon beam Jz = J(J + 1)/3A10 (J ) is indicated for each decay mode. From O'Keeffe et al. (2003).
Table 4 Values of the orientation (A10 ) and alignment (A20 ) parameters, extracted from the measured degree of linear (PL ) and circular (PC ) polarization - of the fluorescence from the Xe II 5p4 6p states produced upon excitation of the Xe 4d5/1 6p (J = 1) resonance with linearly and circularly polarized 2 synchrotron radiation, respectively
Initial state ( P2 )6p[2]
3 3 3/2 5/2

Final state ( P2 )6s[2]
3 3 5/2

(fluo) 533.933 529.222 603.620 605.115 541.915 553.107 571.961 537.239 594.553 484.433 547.261

PC +0.121(20) +0.071(12) +0.079(54) -0.116(51) +0.275(15) -0.174(22) +0.300(18) -0.133(11) +0.292(92) +0.166(8) +0.057(61)

P

L

A

10

A

20

+0.017(15) +0.255(10) +0.270(40) -0.092(20) -0.354(12) -0.113(20) -0.337(31) +0.009(30) +0.015(80) +0.082(25) -0.168(73)

-0.39(14) +0.32(6) +0.35(24) +0.30(11) +0.46(9) +0.36(8) +0.49(9) +0.44(3) +0.49(16) +0.31(9) +0.57(63)

-0.29(27) +0.78(9) +0.83(16) +0.80(18) +0.88(16) +0.85(21) +0.84(17)

( P2 )6p[2]

( P2 )6s[2]5/2 (3 P2 )5d[2]5/2 (3 P2 )5d[3]7/2 (3 P2 )6s[2]3/2 (3 P2 )5d[3]7/2 (3 P2 )5d[2]3/2 (3 P2 )6s[2]3/2 (3 P2 )5d[1]1/2 (3 P2 )6s[2]5/2 (3 P2 )5d[3]7/2

(3 P2 )6p[3]

5/2

(3 P2 )6p[1] (3 P2 )6p[3]

1/2

7/2

-0.36(15) -0.52(25)

From O'Keeffe et al. (2004).

values of A10 and A20 , extracted from different fluorescent lines are consistent, the final selection for the values of the alignment and orientation (indicated by bold characters) is guided by the intensity of the lines and the possibility to separate them completely from other close-lying transitions. Unfortunately, the comparison between the measurements and theoretical predictions for alignment and orientation of


N.M. Kabachnik et al. / Physics Reports 451 (2007) 155 ­ 233
-1

203

4d5/26p j = 1

e cascade lines 7s 6s 7p 6p 5d observed lines 5s5p 5p
5 6

threshhold Xe++ 6d 4f 5p4nl

Xe 5p6 0.6
L

Xe+

P
C

0 -0.6 0.6 0 -0.6

P Intensity / a.u

100

0 400

450

500 550 Fluorescence Wavelength / nm

600

- Fig. 19. Dispersed fluorescence spectrum obtained following excitation of the Xe 4d5/1 6p resonance. The degree of circular PC and linear PL 2 polarization for each line is shown as histogram (O'Keeffe et al., 2004). The upper part gives the scheme of transitions including radiative cascades in Xe II. Adapted from Grum-Grzhimailo and Meyer (2005).

the residual ions due to the Auger decay is not straightforward, since effects of depolarization of the fine-structure ionic states taking place during their lifetime have to be taken into account. With respect to the conventional experimental conditions, there are mainly two depolarization effects, which have to be included for the determination of the alignment and orientation: the hyperfine interactions (for isotopes with non-vanishing nuclear spin) (Fano and Macek, 1973; Greene


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and Zare, 1982) and the fluorescence cascades (JimÈnez-Mier et al., 1993). Spurious magnetic fields and collisions in the interaction volume can be other sources for depolarization, but are neglected here. Since the Auger decay width is much larger than the hyperfine splitting, the hyperfine structure levels are populated coherently during the Auger decay. The nuclear spin is unpolarized immediately after the Auger process and its polarization is not observed. Therefore, to account for the depolarization due to a precession of the angular momentum J of the electronic shell of the ion about the total angular momentum F = J + I, depolarization factors Gk (J ) can be introduced similar to the cases of direct photoionization and electron-impact excitation (Fano and Macek, 1973; Greene and Zare, 1982; Blum, 1996; Balashov et al., 2000) Ah0 (J ) = Gk (J )Ak 0 (J ). k (79)

Here Ah0 (J ) is the alignment (k = 2) or orientation (k = 1) reduced by the hyperfine interactions, Ak 0 (J ) is their k value before the depolarization is taken into account. The Gk (J ) are generally expressed in terms of the hyperfine level separations and natural widths of the hyperfine levels. In a limiting case, when the widths are much smaller than the hyperfine splitting, the depolarization factor takes the simple form Gk (J ) = (2I + 1)
-1 F

(2F + 1)

2

F J

F J

k I

2

,

(80)

where the summation runs over all possible values F for a given fine-structure level with the total angular momentum J of the electronic shell. For the isotope mixture the depolarization factor given in (80) should be weighted according to the abundances of the isotopes. The factors (80) decrease (i.e. the depolarization becomes stronger) with decreasing J . As an example, for the 5p4 6p states of Xe II, the widths are determined by their radiation lifetime of 5­10 ns (for example, Hansen and Persson, 1987; BrostrÆm et al., 1994), and therefore are in the range 15­30 MHz, which is much smaller than the hyperfine level separation, which is of the order of 102 ­103 MHz (BrostrÆm et al., 1996 and references therein). Taking particular values of the angular momenta I and J , and the natural isotope mixture of Xe, one can 1 3 5 determine the depolarization factors for the Xe II fine-structure states: G1 ( 2 ) = 0.74, G1 ( 2 ) = 0.86, G1 ( 2 ) = 0.93, 7 3 5 7 G1 ( 2 ) = 0.96, G2 ( 2 ) = 0.75, G2 ( 2 ) = 0.83, G2 ( 2 ) = 0.89. These values show that the hyperfine interactions have a stronger effect on the alignment than on the orientation. Depolarization effects due to the radiation cascade from higher lying levels is very difficult to analyze, because usually many pathways are possible and not all of them are completely known with respect to their transition probabilities as well as to the population and polarization of the initial states of the cascades. Empirical or semi-empirical cascade corrections can be used, otherwise the depolarization due to the cascade is completely neglected. Provided initial polarizations, populations, and transition probabilities are available, the quantitative treatment implies solving equations of the polarized cascade. We are not aware of the treatments of such polarized cascades after the Auger process, while the numerical analysis of the population cascade is available (Meyer et al., 2001a; Kochur et al., 2000). Considerable simplification follows from an "isotropic" model of the cascade (Meyer et al., 2001a). This model implies that the magnetic substates of the very last levels, which contribute directly to the population of the radiating level, are equally populated. An indication for such a situation could be, e.g. a vanishing polarization of the fluorescence lines attributed to the final transitions in a cascade. The vanishing small polarization of the above lines can be caused by a combination of few factors: a loss of ionic alignment due to a sharing of polarization between the unobserved photon and the residual ion in the first steps of the cascades (e.g. Balashov et al., 2000, p. 130); depolarization due to the hyperfine interactions existing in each ionic state involved in the cascades; mutual compensation of the polarizations introduced by several radiation transitions to the same fine-structure levels from different higher fine-structure states; already small or even zero initial polarization of particular states contributing to the population of the lower states by the radiation transitions. Introducing the isotropic model simplifies the description of the depolarization effect in two respects. Firstly, the depolarization of a given fine-structure level can be described by an identical increase of population for all its magnetic substates. This leads to the simple relation Ac 0 (J ) = D(J )Ak 0 (J ) k (81)


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205

with the depolarization cascade factor D(J ) = W A (J ) . W A (J ) + W c (J ) (82)

Here Ac 0 (J ) is the alignment or orientation reduced by the cascade, W A (J ) and W c (J ) stand for the population k probabilities of the fine-structure level formed directly by the Auger decay and by the fluorescence cascade, respectively. Secondly, the two depolarization mechanisms, due to the hyperfine interactions and due to cascades, are completely independent, because they affect the statistical tensors of different ranks, i.e. the additional isotropic population of the final levels arising from the cascade changes only the zero rank tensor, while the hyperfine interactions affect only tensors with non-zero ranks. It follows from Eqs. (79) and (81) for the observed alignment/orientation Ao0 (J ) = D(J )Gk (J )Ak 0 (J ), k (83)

where now the depolarization factors D(J ), Eq. (82), have to be calculated in an appropriate model. The depolarization factors D(J ) are individual for each ionic fine-structure level. In the case of Xe ions produced upon resonant 4d 6p excitation, the Auger decay leads not only to the Xe II 5p4 6p states, but also to higher lying states: mainly to 5p4 7p levels caused by the shake-up process, and also quite efficiently to some of the 5p4 7s and 5p4 6d levels formed by the conjugate shake-up process (Aksela et al., 1995). Up to 8.5% of the relative population of the configuration 5p4 6p arises from the cascade decay of the configuration 5p4 7p according to a configuration-average estimate (Lagutin et al., 2000). For the majority of the Xe II 5p4 6p fine-structure - states the depolarization factors (82) in the resonant decay of the Xe 4d5/1 6pJ = 1 state were found by Meyer et al. 2 (2001a) and O'Keeffe et al. (2003) to be D(J ) 0.8, while in a few cases it drops down to D(J ) 0.5. 5.3. Contribution of fluorescence studies to complete experiments Observation of only the angular distribution and spin polarization of the Auger electron is generally not enough for a complete characterization of the Auger process (see Section 3.3) and a combination of experimental techniques is required to furnish the complete information on the Auger decay amplitudes (Snell et al., 2001; O'Keeffe et al., 2003). The additional parameters accessible in observing fluorescence are the alignment A20 ( J) and orientation A10 ( J) of the ion after the Auger decay. These quantities carry information on the transition probabilities into different continuum channels, i.e. absolute ratios of the Auger decay amplitudes (see Eq. (19)). This gives complementary information to the angular distribution and spin polarization of the Auger electrons, which normally contain interferences between the decay amplitudes, as presented by Eqs. (24) and (35)­(37). First complete experiments combining photoelectron and fluorescence data were realized by combining data from Hamdy et al. (1991) and Ueda et al. (1993) within the LS-coupling approximation and neglecting the cascade for resonant photoionization of Sr in the region of the 4p­4d giant resonance. The resonant Auger decay investigated by this method differs from a few other complete experiments on the Auger decay (Grum-Grzhimailo et al., 1999, 2001; Hergenhahn et al., 1999; Ueda et al., 1999a) in the fact that the polarization of the residual ion state needs to be accessed in a different manner, namely by the fluorescence, because this is the only relaxation pathway for the excited ionic states. The full power of the method combining data from Auger electron spectroscopy and fluorescence polarimetry was - demonstrated by O'Keeffe et al. (2003, 2004) for the resonant Auger decay of the Xe 4d5/1 6pJ = 1 state to the states of 2 1 the 5p4 6p manifold. For the final ionic states with J = 2 only two amplitudes, Vs1/2 and Vd3/2 , contribute to the Auger decay. Then, two real independent relative parameters have to be determined: the absolute ratio, R = |Vs1/2 |/|Vd3/2 |, and the phase difference, = arg(Vs1/2 /Vd3/2 ) = s1/2 - d3/2 , provided the absolute yield is not measured. An ion with 1 J = 2 can only be oriented and not aligned, and so only measurements on the residual ion using circularly polarized excitation and detection analysis provides useful information. The absolute ratio of the partial Auger decay amplitudes was extracted by O'Keeffe et al. (2003) from the ion orientation parameter A10 ( J), after taking into account the 1 depolarization effect. For the Auger decay to ionic states with J > 2 , four real independent relative parameters are sufficient, within a relativistic approach, to completely characterize the Auger decay step providing two absolute ratios for the amplitudes of the three allowed channels and two phase differences. These ionic states can also be aligned. The ion alignment gives then a second equation for the relative partial decay widths, therefore the two quantities, A10 ( J) and A20 ( J) can be extracted by the fluorescence polarimetry (O'Keeffe et al., 2003, 2004; Schill et al., 2003;


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Table 5 - Comparison of experimental results and calculations for partial decay widths from the Xe 4d5/1 6p(J = 1) resonance 2
Final state s ( P2 )6p[2]
3 3/2 j 1/2

/ ,% d
3/2

d

5/2

g

7/2

g

9/2

14(13) 0.2 0.3

(3 P2 )6p[2] (3 P2 )6p[3] (3 P2 )6p[1] (3 P2 )6p[3]

5/2

5/2

1/2

63(7) 44.1 30.1

2(21) 1.4 1.2 19(2) 3.3 3.2 1(3) 2.2 2.3 37(7) 55.9 69.9

84(20) 98.4 98.5 79(8) 95.9 96.0 103(15) 96.6 96.6

2(9) 0.8 0.8 -4(15) 1.1 1.2

7/2

57(12) 28.9 25.4

12(10) 1.8 1.5

31(8) 69.3 73.1

1st line: experiment (O'Keeffe et al., 2004); 2nd line: theory (O'Keeffe et al., 2004); 3rd line: theory by Lagutin et al. (2000).

Schartner et al., 2005). Table 5 shows selected data for the partial decay widths extracted from the fluorescence polarization data. The general agreement between the experimental and the theoretical results is quite good. Only for 1 7 the j = 2 and the 2 state the comparison points to an overestimation by theory of the channels with higher angular momentum, d3/2 and g9/2 , respectively. Further, more detailed studies are therefore needed here to resolve the remaining discrepancies. The combination of the fluorescence data with spin polarization (Hergenhahn et al., 1999) and angular distribution (Aksela et al., 1996d; Langer et al., 1996) measurements of the Auger electron for the same process provides the phase 1 shift between the two amplitudes for the decay to states with J = 2 . Fig. 20 presents data for the absolute ratio and phase difference between the Auger decay amplitudes. The figure illustrates the accuracy inherent to different experimental methods of extracting the Auger decay amplitudes. Although experiments detecting spin-polarized Auger electrons suffer from poor energy resolution and counting rate, it is sufficient to determine the sign of and thereby to obtain a complete characterization of the Auger decay. The data demonstrate a high selectivity with respect to theoretical models for calculating the Auger decay amplitudes. Only highly sophisticated models can give an adequate description of the full set of data. 1 The complete experiment is more difficult for the states with J > 2 . Nevertheless the two additional parameters from fluorescence polarimetry studies can substantially reduce the parameter space even in the case when only the angular anisotropy parameter of the angular distribution of the Auger electrons 2 is known in addition to A10 ( J) and A20 ( J). Fig. 21 contains examples of such a reduced parameter space for the relative phases of the Auger decay amplitudes with the corresponding theoretical results. Addition of the spin-polarization parameters would allow to further reduce the parametric space up to particular values of the parameters, although the error bars are expected to be rather large. Nevertheless, a further step can be performed without additional measurements (Grum-Grzhimailo and Meyer, 2005), assuming that the relativistic splitting of the photoelectron continuum wavefunctions is negligible. This assumption is confirmed by Dirac­Fock calculations (O'Keeffe et al., 2004), which show that the relative phase of continuum electrons with the same , but different j , is less than 5 in all cases. Setting this difference to zero, the remaining three parameters can be found from the experiment. Some selected results are shown in Table 6. For the 3 ionic states with J = 2 the calculations give satisfactory results for relative intensities and other parameters. Although 7 for the J = 2 state the relative phase of the amplitudes is almost within the error bars, the data on the line intensity and the branching ratios indicate a strong overestimate of the contribution from the g9/2 channel. This points to an effect not included in the theoretical calculations and demonstrates that only the complete data set can asses the validity of a theoretical model.


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207

3

5p4(3P0)6p[1]1/2 AD

a

2 R

1

FL

0 3 5p4(1D2)6p[1]1/2 SP 2 R AD 1 FL

b

0 -3 -2 -1 0 (rad)
Fig. 20. (a) Parametric plot R( ) for the electron angular distribution (AD) data (Aksela et al., 1996d; Langer et al., 1996) together with the value of R determined by fluorescence polarimetry (FL) studies (O'Keeffe et al., 2003) for the final state 5p4 (3 P0 )6p[1]1/2 . (b) The equivalent data for the final state 5p4 (1 D2 )6p[1]1/2 along with the parametric plot for the spin-polarization (SP) data of Hergenhahn et al. (1999). The shaded areas show the error bars. Theoretical results obtained by Lohmann and Kleiman (2001), cited in Meyer et al. (2001a) (·); Lagutin et al. (2000) (filled triangle down) and Hergenhahn et al. (1999), Hergenhahn and Becker (1995) (filled triangle up) are indicated. Calculations of Lagutin et al. (2000) give a negligibly small value of R in (a). From O'Keeffe et al. (2003).

1

2

3

5.4. Scanning across Auger resonances Third generation SR sources producing high-brilliance VUV and soft X-ray photon beams with high energy resolution provide the possibility to investigate polarization and correlation phenomena by scanning the excitation energy over an extended region around and across the resonances, even in cases, where the direct photoionization and therefore the off-resonance signal are extremely weak. The latter is a characteristic of the Auger resonances. As was shown recently (Grum-Grzhimailo et al., 2005), the Fano-like behavior in different correlation parameters for the same resonance possesses some universal features. Generally, in the region of an isolated resonance, the integral photoionization cross section to a particular ionic state f Jf is expressed by the parametric formula
f Jf

( ) =

0
f Jf

1+

2 C1 + C 1 + 2

2

,

(84)

where 0f Jf is the cross section without the resonance and C1 , C2 are the profile parameters introduced by Starace (1977). The parameter C1 describes the asymmetry of the profile, while the parameter C2 determines the integral yield of the resonance into the cross section. Any angle-independent correlation parameter T , like spin-polarization parameters


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Fig. 21. Reduced parameter space 2 = d3/2 - d5/2 vs. 1 = s1/2 - d5/2 found from a combination of the fluorescence polarization data and 3 the Auger electron angular anisotropy parameter 2 (Aksela et al., 1996d; Langer et al., 1996) for the final ionic Xe II 5p4 6p states with J = 2 . Multiconfiguration relativistic calculations (O'Keeffe et al., 2004) are indicated by (red) dots.
Table 6 - Dynamic parameters for the Auger decay of the Xe 4d5/1 6p(J = 1) state to a few selected final ionic states. The branching ratios 2 in %; the relative line strengths Irel are normalized to 100 when summed over the entire 5p4 6p manifold
Final ionic state ( P2 )6p[1]
3 3/2 3/2 1/2

j

:

are given

:

3/2

:

5/2

:

cos(

s

-

d

)

Irel 15.2a ; 15.05 12.45 6.79a ; 5.87b 6.07
b

(1 D2 )6p[2]

Exp Th Exp Th

6(3) 0.3 7(5) 1.7
5/2

16(3) 19.3 49(7) 46.2 :
7/2

78(5) 80.4 44(7) 52.1 :
9/2

- -0.65 0.07+30 -60 -0.66 : cos( 0.43 0.88
d

0.55+13 -16

-

g

)

Irel 2.44a ; 3.66 17.07
b

( D2 )6p[3]
1

7/2

Exp Th

12(9) 2.3

13(10) 1.7

75(10) 96.0

+34 -23

From Grum-Grzhimailo and Meyer (2005). a Aksela et al. (1995). b Langer et al. (1996).

(35)­(37), anisotropy parameters (24), alignment and orientation of photoions (19), and others, not presented in Section 3, are expressed in terms of a bilinear combination of the dipole photoionization amplitudes in the form T=
jJ j J

t

jJ, j J

D

jJ

D



jJ jJ

|D

jJ

|2 ,

(85)

where t j J , j J are the angular coupling coefficients specific for each parameter T . We skipped the fixed index Jf in the dipole amplitude: D j J DJf j J . Note that the equations in Section 3 for the above parameters were written within the two-step approximation, therefore, the photoexcitation part was factorized and only the decay amplitudes VJ1 j J remained. Including the direct photoionization goes beyond the two-step approximation and requires a more general form of the photoionization matrix element. The dipole amplitude in the region of the resonance is written in the form (Kabachnik and Sazhina, 1976; Starace, 1977) q +i 2 D j J = D 0j J + V jJ , (86) V j J D 0 j J -i
jJ

where D 0j J is the photoionization amplitude in the absence of the resonance, q is the Fano profile parameter of the photoabsorption cross section in the region of the Auger state, is the decay width of the Auger state, and = (E - Er )/( /2), with E and Er being the energy of the photon and the Auger state, respectively. After substituting Eq. (86), the parameter (85) can be reduced to the Fano form (Fano, 1961; Fano and Cooper, 1965) T=
T b

+

T a

(q T + )2 ~ , 1 + 2 ~

(87)


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209

15 10 X 5 0 10 5 0
C

200 150 100 -10 -15 0 50
C
2

-5

1

Fig. 22. Scaling factor

as function of the parameters C1 and C2 . The region with

= 0 corresponds to forbidden (

f Jf

< 0) values of C1 and

2 C2 . The parabolic boundary C1 = 1 + C2 between the regions with f Jf > 0 and f Jf < 0 corresponds to a single photoionization channel and, therefore, to the zero cross section (84) in the minimum ('window') of the resonance profile. From Grum-Grzhimailo and Meyer (2006).

where the parameters T , T and q T are specific quantities for each individual T . In contrast, the scaled energy a b ~ dependence = (E - Er )/( ~ /2), which is given in terms of a scaled width ~
2 ~ =[1 + C2 - C1 ]1 /2



(88)

and the shifted resonance energy C1 ~ Er = Er - 2 Er + (89)

are the same for all parameters T . Therefore the Fano profiles for all correlation parameters T of the form (85) and related to a particular final ion state after the resonant Auger decay, possess similar width ~ and similar shift with respect to the energy of the Auger state. Particular cases of this general regularity have been found already much earlier by Grum-Grzhimailo and Zhadamba (1987) and Grum-Grzhimailo et al. (1991). Both quantities, ~ and , are proportional to the decay width of the Auger state and the corresponding proportionality factors depend only on the two parameters of the resonance profile in the integral cross section, C1 and C2 : the scaling factor depends on both parameters (Fig. 22) and the shift only on one, C1 . The 3D plot in Fig. 22 shows regions of C1 and C2 corresponding to small and large values of . Large scaling factors , i.e. strong `broadening' of the resonance structures in the 2 correlation parameters in comparison with the natural width, correspond to C2 ?1, C2 ?C1 . This broadening is related to the relative strength of the resonance with respect to the direct ionization. For a strong resonance, as in the case of the resonant Auger state, the resonance channel dominates even in the regions outside the natural width, leading in these regions to values of the parameters close to their resonant values; the values characteristic for the direct photoionization are reached only in the far wings of the resonance in the cross section. The above described universal scaling phenomenon leads to important conclusions. For investigations on resonant Auger states, experimentalists and theorists can each study a correlation parameter (85), which is the most convenient for them, i.e. not necessarily the same. The comparison can nevertheless be made in terms of the scaled width ~ (88) and the energy shift (89). Moreover, measuring these two quantities gives the profile parameters C1 and C2 of the resonance profile in the integral cross section and vise versa, provided the decay width is known. Thus many studies, considered so far to be independent, turn out to be closely linked. Fluorescence polarimetry was used for checking the scaling phenomenon. The alignment, A20 , and the orientation, A10 , of a final ionic state after the resonant Auger decay of the Xe 4d5/2 6p(J = 1) resonance were chosen as the two different T -parameters. Displayed in Fig. 23 are A20 and A10 as functions of the incident photon energy for one of the final ionic states. For comparison, the profile of the total ion yield is also shown. As expected, the values ~ and extracted from the alignment and orientation of the ionic state are identical within the error bars, despite the completely different shapes of the corresponding resonance profiles. The resonance features in the parameters A20 and A10 are


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- Fig. 23. Resonance Xe 4d5/1 6p(J = 1) in the total ion yield, alignment and orientation parameters of the photoion in the 5p4 (3 P2 )6p[1]3/2 state. 2 The energy dependence of the orientation and alignment parameters is fitted by Eq. (87), providing ~ = 1070 ± 75 meV, =+52 ± 23 for A10 and ~ = 1010 ± 140 meV, =+45 ± 40 for A20 . From Grum-Grzhimailo and Meyer (2006).

- orientation parameters in the region of this isolated Kr 3d5/1 5p3/2 J = 1 Auger state. The influence of the interference 2 with the direct photoionization is pronounced, when the energy dependence is considered for the case of the decay 3 to the final ionic state with J = 2 (Fig. 24d,e), where the conjugate shake up satellite in the direct photoionization 7 is much stronger than in the case of the final ionic state with J = 2 (Fig. 24b, c). In the latter case (Fig. 24b, c) the interference between the resonances is more important. Naturally, the interference effects are more pronounced for the weaker Auger resonances at h = 92.2­92.6eV.

very broad, pointing to scaling factors of around 10, in accordance with the above discussion. The parameters C1 and C2 can be determined from the data (Grum-Grzhimailo et al., 2005), since the decay width of the Auger state, = 106.3(5) meV, is known (Masiu et al., 1995). Recently, the scaling phenomenon was observed also by MÝller et al. (2006) in the photoelectron spin components for the 4d­4f resonant photoemission from magnetized Gd. As follows from the above discussion, even when treating Auger resonances, which are separated much more than their natural width, the resonance structures in the ionic orientation and alignment, as well as in other correlation parameters, can nevertheless strongly overlap. Although the experimental observation of the effect is not straightforward due to the small cross section between the resonances, the appearance of such phenomena can be revealed in combination with theoretical calculations. Such an example is provided by Lagutin et al. (2003a, b) and Schartner et al. (2005, 2007) for alignment and orientation of the Kr II 4p4 5p states following Kr 3d-1 5p/6p resonance excitation. Displayed in Fig. 24 are the alignment and orientation parameters of some of the Kr II 4p4 5p ionic states, when the energy of the exciting photon scans the region h = 91­93 eV, where four Auger states are located. The bandwidth in the experiment (10 meV) was smaller than the natural width of the resonances ( 80 meV) and therefore the - excitation/ionization proceeds in the Raman regime. The strong resonance Kr 3d5/1 5p3/2 (J = 1) at h = 91.2eV 2 is well isolated in the spectrum and the influence of the closest resonances on the cross section is negligible. The background of the direct ionization is very small in comparison with the resonant ionization. Similar to the case of the - Xe 4d5/1 6p(J = 1) resonance, the calculations show a large `broadening' of the resonance features in the alignment and 2

5.5. Auger electron­fluorescence coincidence Studies of the fluorescence cascade by coincidences between the fluorescence and the Auger electron have important advantages in comparison with recording only the fluorescence, because the influence of the cascading population of


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211

- Fig. 24. Dependence of the alignment and orientation parameters on the excitation energy in the region of the Kr (J = 1) Auger states: 4d5/1 5p3/2 (R1), 2 -1 -1 4d5/2 6p3/2 (R4), and perceptible mixtures of the 4d5/2 5p1/2,3/2 states (R2, R3). Open circles, experimental data from Lagutin et al. (2003a) and Schartner et al. (2005). Horizontal bars with lengths equal to the natural widths (the resonance R3 is negligible) indicate parameters computed within the two-step model for isolated resonance. (a) calculated partial cross section for the 5p4 (1 D)5p2 F7/2 ionic state; (b) and (c): thin lines, calculation without taking into account interference between the resonances and between the resonance and direct amplitudes; thick lines, calculations with accounting for the interference terms. (d) and (e): solid and dashed curves, calculation with and without direct photoionization amplitude, respectively. Adapted from Schartner et al. (2005).

the final ionic state is suppressed. Besides, more information is available on the decay amplitudes from the coincidence experiments. Observations of fluorescence in coincidence with the Auger or autoionizing electron can constitute a complete experiment in certain cases (Beyer et al., 1995). A theoretical description of the angular correlations between the photo/Auger electron and polarized fluorescence photons has been developed by Kabachnik and Ueda (1995). So far, there are only a few coincident electron­fluorescence studies with SR beams, which have been performed on Ca and Sr in 1995­1998 and which were reviewed recently by Lohmann et al. (2003). Therefore only a very brief overlook is given here. In the coincidence studies, the photon energies of the incoming beam corresponded to the excitation of the Ca 3p 3d (Beyer et al., 1995, 1996; West et al., 1996) and Sr 4p 4d (West et al., 1998; Ueda et al., 1998) resonances, respectively, and the polarized fluorescence lines from the transitions Ca+ 4p 4s and Sr + 5p 5s were analyzed. The excited ionic 4p and 5p states, respectively, are populated via a decay of the autoionizing resonances with the excitation energies of the order of 30 eV. The conjugate shake up in the direct photoionization with excitation is small. Variation of the linear polarization of the fluorescence emitted under the fixed angle of 90 to the synchrotron radiation


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beam was analyzed as function of the electron emission angle. For Sr, within the resonant model, the combination of the coincidence data with the data on the angular distributions of photo/Auger electrons provided enough parameters to perform a complete experiment even in the relativistic case, i.e. assuming different ionization amplitudes for different total angular momentum of the ejected electron with the same . For Ca, the LS-coupling scheme was assumed to be valid. Experiments with circular polarized incoming photons and observation of circular dichroism in the coincident fluorescence polarization (West et al., 1996) allowed to fix the sign of the phase difference between the outgoing s and d waves of the ejected electrons. To our knowledge, so far this experiment on Ca is the only coincident ejected electron­fluorescence experiment with circularly polarized synchrotron radiation. The great potential of the coincidence Auger electron­fluorescence studies up to the moment is still kept on the sidelines. 6. Related processes In the preceding sections, we have discussed the studies of the Auger and fluorescence cascades in atoms by methods, which involve the angular correlation and polarization measurements for emitted electrons or photons. Naturally, this is only a small part of the works devoted to the study of atomic and molecular relaxation processes which occur after core excitation or ionization. In the following sections, we discuss some selected topics which are closely related to the cascade processes. The experimental and theoretical methods used in their investigation are also similar. The choice of the topics is determined mainly by the interests of the authors and, in a certain sense, is rather arbitrary. Nevertheless, we consider it useful to discuss them in this review, since to our opinion they add some new colours to the picture of polarization and correlation studies for relaxation processes. 6.1. Direct double Auger decay The Auger cascades discussed in Section 4 represent a usual way of deexcitation of the core-excited or ionized states, at least in light and medium atoms. In the simplest cascade two Auger electrons are emitted sequentially, their energy being determined by the energy difference between initial and intermediate ionic states and intermediate and final states, respectively. However, sometimes two electrons are emitted simultaneously, while the ion charge increases by two. Such direct process, first observed by Carlson and Krause (1965), was called double Auger decay. Instead of having discrete kinetic energies (as in a cascade), the two electrons share the total excess energy, which is the energy difference between the initial and the final ionic states. The ratio of the electron energies may be, however, arbitrary and thus the electron energy distribution is continuous. This constitutes the main difficulty of detecting the double Auger process, since it is difficult to separate these decay electrons from the background. Experimentally, the double Auger decay has been studied mainly by measuring the yield of multiply charged ions in some cases in coincidence with the emitted photoelectrons or the Auger electrons. References to the early experimental works can be found in some recent papers (Viefhaus et al., 2004a, b, 2005). It is clear that the double Auger decay and a cascade Auger decay are competing processes, since they can lead to the same final state. To separate these two processes, it is necessary to measure the spectra of the emitted electrons. Recent studies on Ar(2p) and Ne(1s) decay (Viefhaus, 2003; Viefhaus et al., 2004a, b) as well as on Kr(3d), and Xe (3d) and (4d) (Viefhaus et al., 2005) have shown that electron­electron time-of-flight coincidence spectroscopy is well suited to disentangle between Auger cascades and double Auger process. These investigations have shown that in Ar and Ne, the double Auger process is the dominant decay route to triply charged final states producing up to one-fifth of the total Auger intensity in the case of Ar(2p) (Viefhaus et al., 2004a). In contrast, the relaxation of the 3d core holes in Xe and Kr as well as the 4d core hole in Xe is dominated by cascade Auger process. Theoretically, the double Auger effect was first considered in the shake-off model (a sudden perturbation approximation) (Krause and Carlson, 1967; Parilis, 1969; Kochur et al., 1995). However, only the total yield of the double Auger process was calculated. First calculations based on the many-body perturbation theory, which included also energy and angular differential cross sections have been done by Amusia et al. (1992) for the Ne K­LLL process and for the resonant Kr M­NNN process (Amusia et al., 1993). Similar calculations for direct double Auger decay in Kr have been published by Kilin et al. (1997). The general expression for the angular correlations between two emitted electrons has been discussed by Grum-Grzhimailo and Kabachnik (2004). A particular problem of the post-collision interaction in double Auger decay was considered by Sheinerman (1998).


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213

Fig. 25. Two-dimensional electron­electron coincidence spectrum of Ne taken at h = 889 eV along with two corresponding non-coincidence spectra. Structures due to normal Auger decay are marked by Ne2+ . Diagonal stripes in which the sum of the kinetic energies is constant are caused by the double Auger electrons. Two features which are due to the residual background gas (H2 O) are marked by small arrows. Adapted from Viefhaus et al. (2004b).

As an example, we discuss below the experimental investigation of the double Auger process in Ne (Viefhaus et al., 2004b) paying special attention to the study of the angular correlation between the emitted electrons. The experiments were performed both at the storage ring DORIS III at DESY (Hamburg) and at BESSY II (Berlin) using the undulator beam-lines BW3-SX700 and UE56/2-PGM-1, respectively. The experimental setup is shown in Fig. 5. A short description of the experiment is given in Section 2.4.1, more details can be found in papers by Viefhaus et al. (2004b, 2005). The spectra of electrons were measured by a number of time-of-flight spectrometers and recorded in parallel. Besides the single event spectra, all double coincidence events are stored separately for each analyzer combination (15 in total). A compilation of coincidence spectra after Ne 1s photoionization is shown in Fig. 25 together with two noncoincident spectra. The two-dimensional coincidence map is obtained after a summation over all of the 15 coincidence combinations. It should therefore reflect the spectroscopic information disregarding effects of the angular distribution. The most prominent features are coincidences between photoelectrons and normal Auger electrons. The underlying processes yield discrete lines in the spectra and consequently produces intensive isolated spots in the upper left and lower right corners of the coincidence map. Also in coincidence with the photoelectron, the double Auger intensity was found which extends these features towards the lowest kinetic energies detected. These processes are therefore responsible for the `cross' in the map of Fig. 25. Of main interest for the double Auger effect however is the `diagonal' feature, which is due to electrons having a constant sum of kinetic energies. This is a direct observation of the two Auger electrons in coincidence. Intense cascade-like decay would yield to discrete `spots' on this diagonal feature. As these discrete structures are absent here, one can conclude already at this stage that cascade processes must be weak. Discrete structure observed in the energy region above the diagonal is due to the background gas (< 1 â 10-7 hPa). A careful inspection of the double Auger continuum shows that the diagonal consists of three features which can be attributed to different final states of the triply charged ion. However, the energy resolution (especially of the shorter time-of-flight spectrometers) is unable to completely resolve these final states. 6.1.1. Energy distribution of emitted electrons Concerning the energy partitioning among the two Auger electrons one can see a concentration of the intensity at the edges of the diagonal feature in Fig. 25. A more quantitative picture is obtained if we select a suitable range of kinetic


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4

3x10

Coincidence events

Ne KLLL Double Auger h = 889 eV 2x10
4

1x10

4

0x10

4

0

100

200

300

400

500

600

Kinetic energy e0 (eV)
Fig. 26. Measured coincident energy distribution of the Ne K­LLL double Auger taken at h = 889 eV along with the calculated values by Amusia et al. (1992). The gray shaded area marks the region which is omitted in the data analysis as here the double Auger intensity interferes with the Ne 1s photo line. Adapted from Viefhaus et al. (2004b).

energies, which covers the area of the diagonal feature in Fig. 25 completely, and then plot this intensity versus the kinetic energy of one of the two electrons. The corresponding distribution is shown in Fig. 26 which includes all final states of the triply charged ion. Keeping in mind that the data are integral over all triply charged final states, we see a quite satisfactory agreement of the experiment with the calculations by Amusia et al. (1992) which are performed for the Ne 2s-2 2p-1 final states only. However, no information on the intensity sharing of the two double Auger electrons can be obtained in the region where the photoelectron interferes with the double Auger continuum (gray area in Fig. 26). Nevertheless, we see a pronounced preference for asymmetric energy sharing and no strong evidence for cascade decay. For the strongly asymmetric energy sharing the main contribution comes from the shake-off mechanism of double Auger decay, in which the fast electron is emitted in a normal Auger process and the slow electron is shaken off by a sudden change in the ionic potential. 6.1.2. Angular correlation of emitted electrons So far we discussed the experimental results integrated over all angular settings. With the data sets obtained, it was possible however to select a certain energy sharing range of a particular final state (or a group of them) and to plot the coincidence intensity with respect to the relative angle between the two double Auger electrons. Before presenting the experimental results it is useful to consider the theoretical predictions for the shape of the angular correlation patterns (Grum-Grzhimailo and Kabachnik, 2004) for the considered case of the Ne(1s-1 ) double Auger decay. On the theoretical side, the description of the angular correlation patterns in the double Auger decay is based on a two-step approach, in which the direct triple photoionization is assumed to be negligible and where the amplitude of the process is treated as a product of the hole-creation and the double Auger decay amplitudes. Within this model, the angular correlation pattern between the two outgoing Auger electrons in the decay of the Ne 1s-1 2 S e vacancy state is presented as an incoherent sum of partial probabilities W SL ( ) corresponding to different total orbital angular momentum L, total spin S and parity of the outgoing pair of Auger electrons (Grum-Grzhimailo and Kabachnik, 2004): W( ) =
SL

W

SL

( ),

(90)

where is the angle between two emission directions. The double Auger decay of the Ne 1s-1 2 S e state is possible to three groups of final states in the triply charged ions Ne3+ : 2s-2 2p-1 2 P o , 2s-1 2p-2 2 S e , 2 D e , 2,4 P e and 2p-3 2 P o , 2 D o . Due to the angular momentum and parity conservation, the two Auger electrons have the same L and as the final Ne3+ states. As has been shown (Grum-Grzhimailo and Kabachnik, 2004), many properties of the partial angular functions W SL ( ) in the double Auger emission are similar to those of the double photoionization process, discussed already in nu-


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215

[330.350]

[60.90]

[90.120] [120.150] [150.180]

[180.210] [210.240] [240.270] [270.300] [300.330] [330.360]

[330.375]

Fig. 27. Angular correlation patterns summed over all Ne3+ final ionic states (total kinetic energy Ekin = 720 ± 40 eV). The slower electron ea is emitted in x -direction (indicated by an arrow). The numbers in brackets give the corresponding kinetic energy range of the slower electron. Adapted from Viefhaus et al. (2004b).

merous papers (see, for example, the review by Briggs and Schmidt (2000) and references therein). In particular, all selection rules derived by Maulbetsch and Briggs (1995) from the symmetry properties of the two-electron wavefunction are fulfilled for the double Auger decay. Convenient parametrizations for the functions W SL ( ) in the double Auger decay can be derived (Grum-Grzhimailo and Kabachnik, 2004), which separate kinematical and dynamical correlation factors. For example, for the Ne3+ 2s-2 2p-1 2 P o final state, only the W1P o ( ) and W3P o ( ) partial angular functions of the form W1Po ( ) =|ag 0 ( )|2 (1 + cos ) +|au0 ( )|2 (1 - cos ), W3Po ( ) =|ag 1 ( )| (1 - cos ) +|au1 ( )| (1 + cos )
2 2

(91) (92)

contribute into the sum (90). Here agS ( ) and auS ( ) are gerade and ungerade (with respect to the interchange of the two outgoing electrons) amplitudes, respectively. For symmetric energy sharing between the outgoing electrons, the ungerade amplitudes vanish. The dynamical correlation factors, represented by |agS ( )|2 and |auS ( )|2 , reflect primarily the electron­electron repulsion. Similar to the correlation factors for the double photoionization (Cvejanovic and Reddish, ´ 2000; Briggs and Schmidt, 2000), they are approximated by Gaussians as functions of with the maximum at = 180 . The products of the kinematical factors, containing all kinematical selection rules specific for each set LS , and the dynamical factors, determine the main features of the functions W SL ( ). Due to unresolved final Ne3+ ionic states in the described experiment, the sum in (90) contains terms with ten types of symmetry: 1,3 P o , 1,3 D o , 1,3 S e , 1,3 P e , 1,3 D e . At equal energy sharing five of them, 1P o , 1,3 D o , 3 P e , 3 D e show the node at = 180 and a twofold lobe around the node (Grum-Grzhimailo and Kabachnik, 2004). Experimental results for Ne K­LLL electrons are displayed in Fig. 27 for various energy sharing values. They show a similar general pattern of a twofold lobe structure having a minimum in the back-to-back emission for the case of near-equal energy sharing. This general pattern is an indication of the above mentioned features of the functions W SL ( ). Generally, the measured patterns resemble the case of Ar L­MMM (Viefhaus et al., 2004a) and are very similar to the angular correlation patterns of He double photoionization. 6.2. Photoinduced Auger process with initially polarized atoms A bulk of studies on coherence and correlations in the Auger decay in free atoms were performed for closed shell targets with zero angular momentum. For the open shell atoms new aspects of the process appear related to polarization


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of the initial atomic state. Additionally, a selective population of its fine structure levels brings effects, which do not occur in the case of their statistical population (Grum-Grzhimailo and Dorn, 1995). Polarization of the atomic target can be achieved e.g. by the laser optical pumping, by applying the magnetic field or by single-photon absorption. In the first case, laser-excited polarized atomic states can be produced with high density, allowing to observe their subsequent ionization by the synchrotron radiation. This field has been reviewed recently by Wuilleumier and Meyer (2006), therefore we outline here only some aspects related to the resonant Auger/autoionization studies and to the most recent developments. Photoemission from polarized atoms is a very informative method of studying the dynamics of photoionization. For example, it has been established (Klar and Kleinpoppen, 1982) that measurements of the angular distribution of photoelectrons from a polarized atom can constitute a complete experiment for polarizable one-electron atoms and many other atoms without a spin polarization analysis of the photoelectron. A similar statement is true for the process of resonant Auger decay. Experimentally, a weak point of the method is that often the values of the initial polarization parameters, such as the alignment and the orientation of the initial target, are not well known, although they are generally needed for the data analysis. Furthermore, since normally the laser-prepared states are polarized (anisotropic), it is a non-trivial task to extract the intensity proportional to the ionization cross section for the isotropic target, which is a very important characteristic of the ionization process. Auger decay, resonant and normal, of polarized atoms in the gas phase is closely related to the Auger decay in magnetic solids. When studying the magnetic properties of thin films, surfaces or multilayers, the outgoing electron carries information about site and element specific magnetization. Since the inner-shell photoionization in solids is strongly influenced by local interactions, atomic models can be successfully used in many cases as a first step for a qualitative understanding of the photoelectron spectra of atoms bound to a surface. On another side, the atomic data can be very useful as a reference to disentangle intra- and inter-atomic effects. Expressions for the angular distribution and spin polarization of the Auger electrons, as well as the characteristics of the products of the Auger and fluorescence cascades and the angular correlations between the outgoing particles (radiation) can be derived within the standard density matrix and statistical tensor formalism. In comparison with Eqs. (7) and (11) of Section 3, the new feature is a set of additional statistical tensors of the initial atomic state with non-zero rank contributing to the statistical tensors of the photoinduced Auger state. Analogues of these equations for the case of polarized atoms (Buúert and Klar, 1983) can be found e.g. in Balashov et al. (2000) (Eqs. (2.161) and (2.12), respectively). In particular, the statistical tensors of the Auger state with ranks higher than two can generally be achieved, when photoproducing the Auger state from polarized atoms (Kabachnik, 1996). As a result, more dynamical parameters govern the process and more detailed information on the dynamics of the Auger decay can be obtained than in the case of unpolarized initial atoms. General formulas for the angular distribution of the ejected electrons in the resonant Auger/autoionization process with polarized target atom was considered in detail by Baier et al. (1994a). They presented the angular distribution in the form d d
e

=

^ (3J0 )

-1 k0 kk

k0 0 Bk0 kk

F

k0 kk

,

(93)

where is the frequency of the ionizing photon and denotes the fine-structure constant. The statistical tensors k0 0 with rank k0 = 0, 1,..., 2J0 describe the atomic polarization of the initial state in the coordinate system with z axis along the polarization axis of the target atom. The geometrical factors F
k0 kk

=
q

4 {Yk0 ( a ,
a ) Yk ( e ,
e )}k

q

kq

(p1 ,p2 ,p3 )

(94)

contain the direction of the atomic polarization ( a ,
a ) and the direction of the photoelectron emission ( e ,
e ) in the laboratory frame, together with the statistical tensors k q of the photon which depend on the Stokes parameters p1 ,p2 ,p3 . The coefficients Bk0 kk describe the dynamics of the resonant photoionization in terms of the dipole amplitudes. The vast majority of experimental results on the angle-resolved photoemission from the laser-pumped polarized atoms have been analyzed by using Eqs. (93) and (94) with their consequences, which are valid for the direct photoionization too. Other formulations are given by Cherepkov and Kuznetsov (1989), Cherepkov et al. (1995) and Manakov et al. (1996). The latter paper includes also the case of spin-sensitive photoelectron detection.


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217

Fig. 28. Scheme of the typical experimental set-ups applied for studies of the photoionization of laser-pumped atoms using: (a) a cylindrical mirror analyzer (CMA-180) and (b) a rotatable electron spectrometer.

Experimental studies of the photoinduced Auger process with initially polarized atoms concerned so far mostly resonance photoionization via excitation of the subvalence atomic shells. In this case, it is more appropriate to speak about excitation of autoionizing resonances. First measurements of the electron spectra and angular distributions of the autoionization electrons after the photoexcitation by synchrotron radiation from polarized targets were realized for Li, Na, and Ca atoms, prepared in the first aligned excited states by the laser optical pumping: Li 1s2 2p (Meyer et al., 1987; Pahler et al., 1992), Na 2p6 3p (Baier et al., 1994a, b), Ca 3p6 4s4p (Baier et al., 1992; Wedowski et al., 1995, 1997). Typical geometries of the set-ups used in experiments on the photoionization of laser-prepared atoms are shown in Fig. 28. The above papers, especially Baier et al. (1994a) and Wedowski et al. (1997), include also extensive theoretical developments, pioneered by Balashov et al. (1986, 1988). The first contributions into the field demonstrated a great potential of the method in investigations of resonances and photoionization channels, optically not connected to the ground atomic state, in a broad VUV energy range. The experiments have shown angular patterns in accordance with the symmetry of the autoionizing state, which allow one to assign the resonances; different ionization channels are distinguished in the complex photoelectron spectra and tentative assignments to the resonances are given. A `phase tilt' method was suggested, which turned out to be a convenient supplementary tool for performing the complete experiments (Wedowski et al., 1997). Since general forms of the differential observable quantities for a polarized target, for example, the angular distribution of ejected electrons, are rather complicated and not convenient for analysis, usually different kind of dichroism, i.e. differences in the cross sections (intensities) for two different directions either of the target atom polarization or of the polarization of the ionizing photons, are analyzed (Cherepkov et al., 1995). Some dichroisms possess special names, for example: · Circular dichroism in the angular distribution (CDAD) CDAD = d d - d d ,
e -

(95)

e

+


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N.M. Kabachnik et al. / Physics Reports 451 (2007) 155 ­ 233

where + and - denote right and left circularly polarized ionizing radiation; · Linear dichroism in the angular distribution (LDAD) LDAD = d d - d d ,
e

(96)

e

where and denote two perpendicular directions of the linear polarization of the ionizing radiation; · Circular magnetic dichroism in the angular distribution (CMDAD) CMDAD = d d -
a

e

d d

,
e -a

(97)

where the ionizing radiation is implied to be circularly polarized and the indices a and -a denote opposite directions of the target atom orientation; · Linear magnetic dichroism in the angular distribution (LMDAD) is also defined by Eq. (97), but the ionizing radiation is implied to be linearly polarized; · Linear alignment dichroism in the angular distributions (LADAD) LADAD = d d - d d , (98)

e

e

-90



where the ionizing radiation is implied to be linearly polarized and and - 90 denote two perpendicular directions of the initial atomic alignment. This observable depends on the angle between the atomic axis and polarization of the ionizing radiation ( = a in Fig. 28). The LADADs for different angles are generally dependent. For the two independent quantities one can take = 0 and 45 . Other names are also used for this kind of dichroism in the literature. Being integrated over the angles of the photoelectron emission, the CMDAD transforms into the circular magnetic dichroism (CMD), the LMDAD transforms into the linear magnetic dichroism (LMD), the CDAD transforms into the circular dichroism (CD), the LADAD transforms into the linear alignment dichroism (LAD). Experimentally, to avoid the normalization problems, it is convenient to study a dichroism, normalized to the cross section. In this case, the same name for the dichroism is used. For the direct photoionization, the dichroic signal as function of the fine-structure state of the residual ion, and therefore as function of the kinetic energy of the ejected electron, gives certain standard spectral patterns characteristic for each type of dichroism (Verweyen et al., 1999; Wernet et al., 2001). Being integrated over the ion multiplet these patterns obey `sum rules'. In the case of the resonant excitation, these regularities are generally broken due to filtering out one of the possible values of the angular momentum in the final (ion + electron) state (Verweyen et al., 1999). This feature was nicely demonstrated by Schulz et al. (2003) in studies of the LAD of the 4f photoemission in the giant resonance transitions 4d10 4f 7 6s2 8 S7/2 4d9 4f 8 6s2 8 PJ in Eu, which mainly autoionize into the (4d10 4f 6 6s2 7 FJf + )8 PJ continua. The rare earth atoms and the transition metal atoms are very interesting objects for the dichroism studies. Understanding of their complex electronic structure and magnetic properties is important for many fields in physics and technology. As an example of such a study, Fig. 29 compares the CMD in the 2p photoelectron spectra of atomic chromium after the + 2p6 3d5 4s 2p5 (3d6 4s + 3d5 4s2 ) photoexcitation and of a Cr submonolayer film deposited on Fe. The excited resonances decay mainly to the 3p-2 and 3d-2 Auger channels. The two curves show striking similarity. As discussed in detail by Wernet et al. (2000), this similarity of the two dichroism patterns points to the localized character of the 3d orbital in the bound Cr atoms and a crucial role of the final intra-atomic interactions in the final ion for the description of dichroism. Usually the dichroism is explained within another model, by a Zeemanlike splitting of 2p1/2 and 2p3/2 into the magnetic sublevels in the initial ground state of the metal (for example, Ebert et al., 1991; van der Laan, 1995; Cherepkov and Kuznetsov, 1996). The latter explanation implies an itinerant character of the 3d bandlike spin-polarized 3d valence electrons. For a further discussion of these two models see Bethke et al. (2005). First observation of the LAD in the photoinduced normal Auger decay from aligned atoms was reported by von dem Borne et al. (1998), who studied the Auger spectra in chromium due to the 3p5 3d5 4s 8 P 3p6 3d4 5 D + ( p, f)


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219

Fig. 29. Comparison of the CMD in the resonant Auger decay in a magnetized submonolayer film of Cr on Fe and in free oriented Cr atoms. The data from O'Brien et al. (1994) (upper curve) were shifted to match the position of the CMD found in the gas phase (lower curve). From PrÝmper et al. (2003).

transitions after photoionization of the 3p electron from the aligned initial state: + 3p6 3d5 4s 7 S3 3p5 3d5 4s 8 P + ( s, d). The initial state was aligned by laser pumping the Cr 3p6 3d5 4s 7 S3 3p6 3d5 4p 7 P2 transition with linearly polarized radiation. In the geometry displayed in Fig. 28a, they observed the normalized difference between the Auger spectra recorded for either parallel or perpendicular orientation of the laser polarization (and therefore the atomic alignment) with respect to the polarization of the undulator radiation. The LAD in the Auger lines reaches 10%. In this particular case, it turned out that the LAD is much more sensitive to the dynamics of the photoionization step than to the dynamics of the Auger decay. Furthermore, due to the strong s photoionization channel, the LAD in the Auger decay and in the direct photoionization are similar. Noteworthy, the magnetic dichroism occurs also in fluorescence after the radiative deexcitation of an Auger state. Although it has been already observed in the CMD (see, for example, Kotani and Shin (2001) and references therein), for the LMD, up to our knowledge, there is only a theoretical prediction so far (Grum-Grzhimailo and Kabachnik, 1999). A promising field, quite feasible with modern SR sources and equipment, is the Auger-photoelectron coincidence spectroscopy (APECS) with polarized atoms, although no experimental results have been presented in the literature yet. Theoretical background for treating such a process in the two-step model (photoionization + Auger decay) and particular predictions for the LMD and the LAD for the Sn 4d photoionization and subsequent N4,5 ­O2,3 O2,3 Auger decay have been published recently by Da Pieve et al. (2007). Calculations show that the value of the dichroism as well as the complexity of the dichroic pattern can strongly vary under different experimental conditions. 6.3. Auger decay in molecules Physics of the Auger process in molecules is more complicated than in atoms due to additional degrees of freedom, reacher symmetry, and the possibility of molecular fragmentation. Three regimes of the Auger decay can be distinguished: (a) the Auger relaxation proceeds faster than the molecular dissociation; (b) the dissociation is faster than the Auger decay (Morin and Nenner, 1986), i.e. the hole relaxation proceeds already in the fragment; (c) an intermediate regime, with an interplay between the Auger decay and the molecular fragmentation (for example, Menzel et al., 1996). Here we will discuss the first case, where the two-step model of excitation and decay is a good first approximation, and there is a close analogy with the theory of atomic Auger decay considered in Section 3. A measurement of the angular distributions of ejected electrons in the molecular Auger decay (`angle-dependent spectra') is a routine procedure, while the recently introduced Auger electron­molecular fragment coincidence technique (Guillemin et al., 2000, 2001; Weber et al., 2003; Liu et al., 2005) brought these studies on a new level of sophistication and called for further development of the theory. Theory for the angular distribution of Auger electrons was considered for both, chaotically oriented molecules (Dill et al., 1980; ZÄhringer et al., 1992; Chandra and Chakraborty, 1993) and `fixed-in-space' molecules (Kuznetsov and Cherepkov, 1996). We especially note here a theoretical approach, which has been developed by K. Blum and coauthors (Bonhoff et al., 1996, 1997, 1998, 1999; Lehmann and Blum, 1997). Based on their studies, we emphasize that similar as for atoms, the density matrix formalism is well suited for description of the angular correlations and polarization in molecular Auger decay. A difference from atoms is that due to


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Fig. 30. Coordinate systems for the description of molecular Auger decay.

the lack of spherical symmetry, the reduction to statistical tensors (e.g. orientation and alignment) is not efficient for the description of molecular states themselves. Instead, an ensemble of molecules is characterized by `order parameters', which characterize the spacial distribution of the molecular axis. Below, we outline the main points of this promising approach and present some key expressions. 6.3.1. General theoretical considerations Assume that the molecule does not change its orientation during the Auger decay. Let us choose a molecular frame X m Y m Z m , strictly bound to the molecule, and define Z m as a `molecular axis' (Fig. 30). The probability to find a molecule after the Auger decay with the Z m axis along n and at the same time for the Auger electron to possess linear momentum p can be written in the form I(p,n) = (Tr =
f

f

)
f

n; p; |
f

f

|

f

n; p;
f

=
f

n; p;

(-)

|V | n

n; p;

(-)

|V |

n



n|

i

|

n.

(99)

Here , f are the sets of quantum numbers of the molecule and the molecular ion, respectively, V is the interaction causing the Auger decay, is the (unobserved) spin component of the Auger electron, i and f are the density operators for the decaying state and for the final state `molecular ion + Auger electron', respectively. An experiment fixes the directions of the Auger emission p and of the molecular axis in a laboratory frame XY Z , while the Auger decay amplitudes are calculated in the molecular frame. The partial wave expansion of the Auger electron wave function and transformation of the decay matrix elements to the molecular frame (V is invariant under rotations) give after standard algebra I(p,n) =
kq q KQQ JM M

(k q , K Q | J M )(k q ,K Q | JM ) â ^ K 8
2 2 K QQ

D

(

) A ( kq

)

4 Y 2k + 1

kq

( ,
)D

J MM

(
n ,

n

,

n

),

(100)

where the rotation, described by the Euler angles R ={
n , n , n }, brings the laboratory system XY Z in coincidence with the molecular system X m Y m Z m . In Eq. (100) we defined the anisotropy parameters Akq ( )= 1 4 â (-1)m ^ ^ ( 0, 0 | k 0)( m,
mm f
f

- m | k - q) (101)

;p m , |V |



f

; p m, | V |


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221

with the property Akq (
K DQ ( Q

) = (-1)q A dR n|
i

k -q

(

) and a so-called `order parameter'
K QQ

)

| nD

(
n ,

n

,

n

).

(102)

The order parameters (102) do not depend on the angles
n , n , n and describe the molecular ensemble itself. Properties of the order parameters, related to the symmetry of the molecular ensemble, are discussed by Blum (1996). The molecular axis distribution is called oriented, if at least one order parameter with K odd is different from zero, and aligned, if only order parameters with K even contribute. The order parameters (102) contain information on the first step, production of the Auger state, and are completely determined by the excitation mechanism and symmetry/structure of the molecule. In this sense they are similar to the statistical tensors (e.g. orientation and alignment) of the decaying atomic state. The anisotropy parameters (101) describe the second step (Auger decay), similar to the decay parameters (24) in atoms. The angular distribution of Auger electrons in the laboratory frame follows after integrating (100) over dR: Ilab (p) =
kq q

D

k qq

(

) A ( kq

)

4 Y 2k + 1

kq

( ,
).

(103)

Eq. (103) can be compared to Eq. (23). The distribution of the molecular orientation follows after integrating (100) over the directions of the Auger emission: I(n) =
KQQ

^ K 2

2

D

K QQ

(

) A00 (

)D

K QQ

(
n ,

n

,

n

).

(104)

One can check that the integral intensity is given by: I0 =
0 2

d
D(
0 0 00

sin d I(p) =
0

2

d


2 n

sin
0

n

d

n 0

d

n

I(n) (105)

=4

) A00 (

).

Considering Eq. (99) in the molecular frame, one obtains after the partial-wave expansion and transformations the angular distribution of the Auger electrons in the molecular frame: Imol (p) =
kq

|

i

|

mol

A ( kq

)

4 Ykq ( ,
). 2k + 1

(106)

Here information on the decaying state is contained in its density matrix, which is given in the molecular frame by n| i | n | i | mol , and the angles ,
in Eq. (106) are counted in the molecular frame, too. Although (106) looks similar to Eq. (103), there is a substantial difference. Generally there are no formal restrictions on the number of spherical harmonics in (106) (a `complexity' of the angular distribution), except those contained in the decay anisotropy parameters (101). The latter gives the restriction kmax = 2 max , where max is the maximum orbital angular momentum of the Auger electron. This number is completely `uncoupled' from the excitation mode. In contrast, for the angular distribution in the laboratory frame (103), kmax is related to the excitation mode directly. The situation is similar to direct photoionization of molecules: angular distributions of photoelectrons from `fixed-in-space' molecules are much reacher than angular distributions from chaotically oriented molecules, where only the second Legendre polynomial contributes (in the dipole approximation) (Kaplan and Markin, 1969; Dill, 1976; Dill et al., 1976; Davenport, 1976). Similar to the alignment or polarization transfer in the Auger decay in atoms, one can consider a transfer of the order parameters in the Auger decay of molecules. For example, an analogue of Eq. (19) for the order parameters of the molecular ion is of the form
K DQ Q ( f f

) =N

-1 m

f

; p m, | V |

f

; p m, | V |



D

K QQ

(

),

(107)

where N is a normalization factor.


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To conclude, in order to describe the molecular Auger decay within the two-step approximation, one needs to find (a) the order parameters of the decaying state (102) and (b) the decay parameters (101). This is similar to the Auger decay in atoms, where, correspondingly, the statistical tensors of the decaying state and the intrinsic decay parameters should be found for this purpose. A theory of angular distributions and angular correlations of ejected electrons in the cascades can be further developed in a close correspondence to the atomic case. Up to the moment we considered a general case of arbitrary excitation mode for molecules with arbitrary symmetry. In the next section we specify the main expressions for resonant Auger decay in linear molecules. 6.3.2. Linear molecules For linear molecules, the set includes the projection of the total angular momentum of the decaying state on the molecular axis: = { }, where are other quantum numbers (omitted below for brevity), which specify the molecular state. Invariance with respect to rotation around the molecular axis gives Q = - . In Eq. (100), the angle n becomes redundant, therefore M = 0. To define unambiguously the molecular frame we fix n = 0, i.e. the Z axis belongs to the Z m X m plane (see Fig. 30). Furthermore, for a symmetric ensemble of decaying molecules (axial symmetry of the excitation process), Z axis can be directed along this axis of symmetry. The angle
n in Eq. (102) K ) remain non-vanishing. also becomes redundant, i.e. Q = 0, and only the order parameters D0 - ( Eqs. (100) and (103) correspondingly reduce to I(p,n) = 1 8 â and Ilab (p) = I 4
0 2 mm kK J K D0 - (

^^ (-1)k K k ) A ( , k

-1

(k

- ,K

-

| J 0)
K0

){Yk ( ,
) YJ ( n ,
n )}

(108)

1+
k>0

Auger Pk k

(cos ) , ) Ak (, ) and

(109)

where I0 is given by Eq. (105), and the following definitions are used: Ak ( ,
Auger k

-

=
Auger

D

k 0- 0 D00 (

(

) Ak ( , ) ) A0 ( , )

.

(110)

Generally, the k parameter depends on both, the excitation and the decay, which are coupled even for the considered model of pure two-step mechanism (in contrast to the spherically symmetric atomic case, see a discussion in Bonhoff et al. (1996)). Eq. (104) takes the form I(n) = where
ion K

I 8

0 2

1+
K>0

ion K PK

(cos

n

),

(111)

=

^ K

2

D D

K 00 0 00 (

(

) A0 ( , ) ) A0 ( , )

.

(112)

For a sharp molecular decaying state, when the projection takes two values, ± , the decay part factors out and one has
ion K

=

^ K2 D D

K 00 0 00 (

( )

)

.

(113)

Eqs. (111)­(113) show that the angular distribution of the molecular axis after the Auger decay is determined only by the excitation branch of the Auger state, in particular, it is not sensitive to the final states of molecules and Auger electrons after the Auger decay.


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223

The order parameters for molecular photoexcitation and photoionization have been treated by Blum (1996) and by Bonhoff et al. (1998). For excitation of a molecular state with symmetry by a linearly polarized beam (Z axis along the polarization), up to an irrelevant numerical factor, one obtains
k D0 -

(

)=

82 (-1) 2k + 1

1+k -

(1 , 1 -

|k -

)

|d |0

|d |0



k0

,

(114)

1 where the non-zero statistical tensors of the radiation in the electric dipole (E1) approximation are 00 = , 20 =- 2 . 3 3 Hence, due to the dipole approximation only terms with k = 2 (or K = 2) contribute into Eq. (109) (or Eq. (111)). Using the symmetry relation for the dipole transition matrix elements +1 | d+1 | 0 = -1 | d-1 | 0 and abbreviating a 2 0 0 general factor G = 815 | +1 | d+1 | 0 |2 , one obtains for non-zero order parameters: D00 (11) = D00 (-1 - 1) = 5G, 2 2 2 2 D00 (11) = D00 (-1 - 1) = -G, D02 (1 - 1) = D0-2 (-11) = - 6G. Substituting this into Eq. (110), one obtains (Bonhoff et al., 1998) Auger 2

=-

1 5

A2 (1, 1) A2 (-1, 1) +6 . A0 (1, 1) A0 (1, 1)

(115)

Due to - = even, the coefficients Ak in Eq. (115) are real numbers. For the recoil ion anisotropy, one immediately obtains from Eq. (113) a natural result
ion 2

=

2 5 D00 (11) 0 D00 (11)

=-1

(116)

without calculation of the decay amplitudes. Numerical calculations have been performed for the anisotropy parameters Ak ( , ) of the Auger decay to different final states of resonantly excited CO (Bonhoff et al., 1999) and HF (Bonhoff et al., 1998) and of the molecular ion HF+ with a 1 hole (Bonhoff et al., 1997). Fig. 31 illustrates the angle dependence of the molecular Auger spectrum with good agreement between calculations and experiment. 6.4. Alignment and orientation in the direct photoionization of atoms Anisotropy and polarization of the cascade products originate from the anisotropy of the interaction of the primary photons with electrons in the atom or molecule. Photoexcitation and photoionization by arbitrarily polarized photon lead to orientation and/or alignment of the excited or ionized state, which determine the correlation and polarization properties in the following cascade. In the former case of photoexcitation, the alignment and orientation of the excited state can be easily obtained since they are determined by the angular momentum conservation law (see Section 3.1.2, Eq. (11)) and no dynamical amplitudes are needed for their calculation. However, in the case of photoionization the problem is not so trivial, since orientation and alignment depend on the relative values of the photoionization amplitudes (Eqs. (7) and (10)) and therefore depend on the photon energy. After the pioneering theoretical papers by FlÝgge et al. (1972) and by Jacobs (1972), and the first measurements of the alignment in photoionization by Caldwell and Zare (1977) and by Southworth et al. (1981, 1983) many experimental and theoretical investigations were published mainly for the alignment in photoionization of closed subshell atoms. A comprehensive list of references can be found in the papers by Yamaoka et al. (2002) and by Kleiman and Lohmann (2003). Experimental values of the alignment are usually obtained from the angular distributions of Auger electrons (see, for example, Snell et al. (2000) and references therein) or from the angular distribution and linear polarization of fluorescence (see, for example, Yamaoka et al., 2003). The orientation is experimentally determined from the circular polarization of fluorescence or spin-polarization of Auger electrons (Snell et al., 1996, 1999a, b, 2002; Schmidtke et al., 2000a, 2001). Experimental data and systematic numerical calculations of the alignment of ions in photoionization of different atomic subshells (Berezhko et al., 1978a; Kleiman and Lohmann, 2003; Kleiman and Becker, 2005) have shown that in general the alignment is small, A20 0.1. For comparison, we remind that in photoexcitation of a closed-shell atom the absolute value of the alignment of the resonant excited state is much larger A20 =- 2 (see Section 3.1.2). However, the alignment in photoionization may be much larger in the near-threshold region as well as near the so-called Cooper minimum (Berezhko et al., 1978a). In both these cases, the usually dominant matrix element of the transition


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Fig. 31. Auger spectrum for the transitions of the 1 (2 -1 2 1 ) vacancy of CO; the vacancy is created by absorption of linearly polarized photons (h = 287.4 eV) and the angle of ejection is counted from the direction of the photon polarization. The labels correspond to different final states: (a) 2 + (5 -1 ); (b) 2 (1 -1 ); (c) 2 + (4 -1 ); (d) 2 - (1 -1 5 -1 2 1 ); (e) 2 - (1 -1 5 -1 2 1 ); (f) 2 ( -2 2 1 ); (g) 2 + (1 -1 5 -1 2 1 ). Experimental spectrum by Hemmers et al. (1993). Adapted from Fig. 4 of Bonhoff et al. (1999).

nl l + 1 is suppressed and a large alignment is determined by the weaker transition nl l - 1. Note that the alignment is large at those energies where the cross section is small. The orientation parameter A10 is usually large |A10 | 0.5 - 1.0 in a broad range of photon energies (Kabachnik and Lee, 1989; Kleiman and Lohmann, 2003). Typical results of the experimental study of the alignment are shown in Fig. 32 for the photoionization of the 4d5/2 subshell in xenon together with the theoretical calculations in the Hartree­Slater (HS) model (Berezhko et al., 1978a) and in the Hartree­Fock model with relativistic corrections (HFR) (Kleiman and Lohmann, 2003). Similar results are shown in Fig. 33 but for the orientation in of Kr + ion with a vacancy in the 3d subshell. In these and in many other cases (see Kleiman and Lohmann, 2003) the agreement between theory and experiment is rather good. A somewhat contradictory situation occurred in the study of the alignment in the 2p photoionization of heavy atoms. Here the alignment is usually deduced from the measurements of the angular distribution of the emitted X-rays. Theoretical calculations predicted a very small alignment for the 2p vacancies (Berezhko et al., 1978a; Kleiman and hyperlinkbib191Lohmann, 2003). Until quite recently the sufficiently intense SR sources were not available for such large energies (several keV). Several groups published the results of measurements of the angular anisotropy of X-ray fluorescence, in which the 2p vacancies in heavy atoms have been created by radiation from radionuclides (see Santra et al., 2007 and references therein). Unusually large anisotropy as well as large forward­backward asymmetry have been reported in some of those experiments (see discussion in Yamaoka et al., 2002, 2003; Santra et al., 2007 and references therein) in drastic disagreement with the theoretical predictions. Recent, more precise and accurate measurements at SR sources, showed a very small anisotropy of L X-ray lines (KÝst et al., 2003; Yamaoka et al., 2003, 2006; Barrea et al., 2005) in complete agreement with the theoretical predictions. As it was suspected, the measurements with the radionuclides, which were interpreted as showing large alignment, contained some large systematic errors probably connected with self-absorption of the produced X-rays in the solid target.


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225

0.40
-1 (4d5/2) 20

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 kinetic energy of the photoelectron [Ry]

Fig. 32. Alignment parameter A20 of photoionized xenon with a vacancy in the 4d5/2 subshell: solid line HFR calculation, dotted line HS calculation; experiments­circles (Snell et al., 2000), open squares (KÄmmerling et al., 1990), filled squares (Whitfield et al., 1992), open triangles (Schaphorst et al., 1997), filled triangles (KÄmmerling and Schmidt, 1993). From Kleiman and Lohmann (2003).

alignment parameter (3d-1)
10

1.2 1.0 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0 2 4 6 8 10 12 14 16 18 kinetic energy of the photoelectron [Ry]
Fig. 33. Orientation parameter A10 of photoionized krypton with a vacancy in the 3d shell: solid line HFR calculation for 3d3/2 , dashed line HFR calculation for 3d5/2 ; experiments--open circles 3d5/2 , closed circles 3d3/2 (Schmidtke et al., 2001), filled squares 3d mean value (Snell et al., 2002). From Kleiman and Lohmann (2003).

7. Concluding remarks In summary, we have presented a survey of various investigations of the cascade relaxation of core-photoexcited and photoionized atoms. Special attention has been paid to the studies of angular correlation and polarization of the decay products: Auger (autoionizing) electrons and fluorescence, i.e. to the phenomena in which coherence of the excited states plays a significant role. Such kind of studies have become particularly feasible with the advent of third generation SR sources and stimulated development of a great deal of refined experimental techniques partly presented in this review. As we have demonstrated, the angular correlation and the polarization are often more sensitive than the total cross sections (populations) to the intricate details of the description of the states involved and of the relaxation mechanism. In addition, the cascades usually involve atomic states with two and more open (sub)shells, therefore multi-electron correlations play a very important role in their description. These features make the angle and/or the polarization resolved study of Auger and fluorescence cascades particularly attractive to the experimentalists. From the other side, their description is a real challenge for theoreticians. We hope to have shown that the study of the photoinduced cascades of the Auger and fluorescence transitions in atoms is a lively and very versatile field of research

orientation parameter


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on the frontier of modern studies of photon-matter interaction with complicated response of many-electron systems to the core excitation. We hope that the chosen examples illustrate the variety of problems, which have been encountered and successfully solved by using ingenious experimental technique supported by the sophisticated theoretical investigations. The reader understands that the presented examples constitute only the top of the iceberg of numerous experimental data, which were accumulated and analyzed in the past decade. To help the reader to find the necessary data, we collected a more or less comprehensive list of references. We apologize to those authors or groups, whose relevant works are accidentally not included in the list. The ideas and the methods, which have been developed for studying the core-excited resonances in atoms and their cascade relaxation, in particular the coincidence technique with angle (and sometimes polarization) resolved measurements, are now used for investigations of molecules, clusters, nano-structures and surfaces. In this respect, the studies of atoms should be considered not only as having their own importance but also as a necessary stage for developing the experimental and theoretical methods, which can be further used in studying more complex objects. Expansion of the described methods to other fields goes hand in hand with the appearance of new directions or better to say of new dimensions of investigation. In particular, we have in mind time-resolved studies of the inner atomic shell relaxation using attosecond pump-probe or equivalent time-resolved technique. Attosecond electromagnetic pulses have duration comparable with typical time of the inner-shell processes and therefore may be used for time-resolved experiments (Agostini and DiMauro, 2004; Reider, 2004). First application of the XUV attosecond pulses to the timeresolved study of the Auger process (Drescher et al., 2002; Uiberacker et al., 2007) has demonstrated the feasibility and great potential of such experiments. The time-resolved investigations will certainly give a new boost to the studies of the cascade relaxation in atoms and other atomic systems. In conclusion, we express our hopes that the deep understanding of the cascade processes for rather simple systems (noble gas atoms), demonstrated in this review, opens the field for many exciting applications and for investigating other more complex systems, which are to be expected in near future. Acknowledgement We gratefully appreciate many useful discussions of various physical problems related to this review with our colleagues and coauthors H. Aksela, S. Aksela, V.V. Balashov, U. Becker, N. Cherepkov, A. Dorn, M. Drescher, A. Ehresmann, S. HeinÄsmÄki, U. Heinzmann, U. Hergenhahn, A. Kazansky, M. Kitajima, H. Kleinpoppen, W. Mehlhorn, J. Nikkinen, P. O'Keeffe, V. Schmidt, B. Sonntag, H. Tanaka, J. Viefhaus, J. West, H. Yoshida and P. Zimmermann. We remember with pleasure and gratitude all our other colleagues, who participated in our works, which constitute the basis of this review. Our special thanks come to M. Drescher, M. Kitajima and U. Kleiman for sending us the figures from their publications. A.N.G. acknowledges the financial support of the CNRS and UniversitÈ Paris-Sud and hospitality of the LURE and LIXAM laboratories in Orsay. N.M.K. is grateful to Bielefeld University for hospitality and financial support via Sonderforschungsbereich SFB 613 of the Deutsche Forschungsgemeinschaft (DFG). He also acknowledges the financial support from Russian Foundation for Fundamental Researches via Grant 06-02-16289. Part of the work on this review was done during the visit of N.M.K. and A.N.G. to IMRAM (Tohoku University) in Sendai. Hospitality of the Tohoku University extended to them during this visit and the financial support from the Japan Society for the Promotion of Science (JSPS) are gratefully acknowledged. S.F. acknowledges support by the DFG under the Project no. FR 1251/13. K.U. acknowledges JSPS for support in the form of Grants-in-aid for Scientific Research. References
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