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Resonant X-Ray Raman Scattering from Atoms and Molecules P.L. Cowan

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Resonant X-Ray Raman Scattering from Atoms and Molecules P.L. Cowan Resonant x-ray Raman scattering from atoms and molecules P.L. Cowan Physics Division, Argonnc National Laboratory, Argonne IL 60439 Abstract Inelastic x-ray scattering and elastic x-ray scattering are fundamentally related processes. When the x-ray photon energy is near the ionization threshold for an inner shell, the inelastic channel is dominated by resonant x-ray Raman scattering. Studies of this emission not only illuminate the resonant scattering process in general, they also point to new opportunities for spectral studies of electronic structure using x-rays. Atoms in the form of a free gas provide an ideal target for testing the current theoretical understanding of resonant x-ray Raman scattering. In addition, x-ray scattering from molecular gases demonstrates the effect of bonding symmetry on the polarization and angular distribution of the scattered x-rays. Comparisons of experimental data with theory demonstrate both the successes and limitations of simple, single-electron interpretations of the scattering process. 1. Introduction The central focus of this conference proceedings is anomalous scattering, which is an elastic x-ray scattering process. It has been discussed in other papers in these proceedings [1], and will be emphasized in this paper as well, that the elastic and inelastic x-ray scattering channels are intimately related. This fact has been demonstrated to be a direct consequence of the causality condition [2]. X-ray spectroscopy can be thought of as simply a manifestation of inelastic x-ray scattering [3]. Traditional x-ray spectroscopy, in turn has been categorized as either x-ray absorption spectroscopy (XAS) or x-ray emission spectroscopy (XES) [4]. Figure 1 schematically compares these two techniques. Simply put, XAS (fig.la) is a probe of the unoccupied electronic states (above the Fermi energy, £F) of matter, while XES (fig.lb) probes the occupied levels (below the Fermi energy). HjRrJP! h*i been «u!t by a cc^Metof of (the U. S. under cennan Ha. W-21-10»£MG-M. DISTRIBUTION OF THIS DOCUMENT IS UNUMHSJ the |**!*»SI «orm «rt ftit or MIBW eMwrt to do to, »OT . DISCLAIMER < Gom-mo.1. NeitherikeUmted»»t«Gw^nu-wt «* i*ior* ^TS« of 3 p UHted Stole* CtmnmtM. or «ay »ftKy tbcicoT. Figure 1. Schematic energy-level diagrams contrasting three types of spectroscopy. a) XAS - x-ray absorption spectroscopy; b) XES - x-ray emission spectroscopy; c) RXRS - resonant x-ray Raman scattering. There are a number of characteristics of traditional x-ray spectroscopy which can be thought of as either advantages or disadvantages for studies of matter [5]. These characteristics, on the one hand, offer promise of experimental studies of the electronic and atomic structure of matter under a variety of conditions, but on the other hand, present obstacles such as lifetime broadening and satellite contamination. This paper will focus on a particular case of x-ray inelastic-scattering spectroscopy, namely resonant - x-ray Raman scattering (RXRS). Attention will be given to how this process relates to the canonical x-ray spectroscopy, to how RXRS relates to x-ray elastic scattering, and to how the peculiar characteristics of RXRS may be useful in overcoming the limitations of the other x-ray spectroscopic techniques. Figure lc shows a energy-level diagram representing "resonant x-ray Raman scattering". Unlike XAS and XES, x-ray Raman scattering is a two-photon process. The three diagrams of Figure 1 are suggestive of the relationship between each of the first two processes to the third. In the case of RXRS the net transition (broken arrow) is the promotion of an electron from one relatively high-lying occupied level to an unoccupied level. The deep inner-shell hole exists only in the virtual intermediate state. Data illustrating resonant x-ray Raman scattering from argon are shown in figure 2 [6]. These emission spectra are observed near the energy of argon K-beta fluorescence, and are due to the net transition of an Ar 3p electron to the 3j?.1p state*. Note that the energy difference between the RXRS peak and the elastic-scattcied x-ray peak Is constant as the incident energy is changed. This constant energy loss is reminiscent of the Stores peak in optical Raman scattering [7], and is the reason for the choice of term "x-ray Raman scattering". Also note that since this energy loss depends only upon the energy interval of the net transition (fig 1c) the spectral width of the Raman peak will not be dependant upon the hole lifetime of the inner-shell. In other words, the RXRS seed not suffer from the lifetime broadening observed in XAS (fig la) and XES (fig.lb) There have been a number of papers on x-ray Raman scattering, and the particular case of resonant x-ray Raman scattering, most notably the pioneering experimental studies of Sparks [8], but most of this work has involved the more complicated case of scattering from solid samples. In studying the RXRS process it is desirable to begin with the case of scattering from a monatomic gas where the electronic structure is simplest. Results for RXRS from small molecules will also be presented to illustrate a number of additional RXRS effects. Figure 2. Resonant x-ray Raman Figure 3. Schematic drawing cf the scattering from argon. experimental apparatus used for RXRS observations. * The notation xxxyyy has been adopted for electronic configurations, where '«»* indicates the hole states in otherwise filled levels and "yyy" indicates the occupancy of otherwise empty levels. 2. Experimental Method The experimental techniques that have enabled studies of resonant x-ray Raman scattering from gases originated in studies of selective excitation of x-ray emission spectra [5]. This doubly differential x-ray spectroscopy was first demonstrated using laboratory x-ray techniques [10,11], but first become generally applicable with the advent of synchrotron radiation sources and tunable monochromators [12,13]. As will be discussed, observations have become increasingly differential, involving polarization analysis [14] and scattering-angle dependence [15] as well as spectroscopy vs. excitation and emission energies. The measurements of resonant x-ray Raman scattering presented here were performed at the NIST-ANL x-ray beamline, denoted as X-24A [16], at the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory. A schematic of the instrumentation is shown in figure 3. X-rays from a bending magnet in the NSLS x-ray ring are filtered in energy, and their linear polarization is enhanced, by a tunable two-crystal monochromator [17]. The target, a gas confined in a cell for all cases under discussion, is then observed by a curved single-crystal spectrometer [IS]. The secondary- spectrometer crystal disperses the emitted x-rays onto a position-sensitive detector [19]. With the sample placed well inside the Rowland circle of the instrument, the position-sensitive detector simultaneously collects photons within a given spectral region. In some measurements the secondary spectrometer can also perform polarization analysis and/or determine the observation-angle dependence (with respect to incident polarization) of the spectra. The polarization selectivity of the secondary spectrometer stems from the fact that the x-ray equivalent of an optical Brewster*s angle is very close to 45° [20]. Even when die incident angle is several degrees off of 45° sufficient polarization selectivity is obtained to observe the profound polarization effects that occur on sub-threshold resonances [14]. This leeway in Bragg angle permits measurements over a range of emission energies. It is also possible, within some limits, to improve the polarization selectivity at a given energy by one's choice of crystal. For example, Si(lll) crystals have a Bragg angle of 44.5° at 2822 eV (roughly the energy of Chlorine Kf$ fluorescence) while InSb{lll) has a Bragg angle of 42.10<> at 2472 eV (Sulfur K0). 3. Theoretical Background X-ray emission excited near-threshold or on-resonancc can best be described as an x-ray scattering process [3]. A theoretical basis for understanding this process can be found in the Kramers-Heisenbcrg equation, which is derived by treating the electromagnetic-radiation field as a time-dependent perturbation acting upon a set of energy eigenstates of the atom or molecule under study [21]. Under the dipole approximation (i.e. exp(-ikT) "1) the differential cross section for a transition from state A to state B can be written as follows: m. i ErEA-tiatrtrtf2 •"• i Ft- where the summation over I represents all possible intermediate states, €1 and £2 are the polarization vectors of the incident and final state photons, and all other symbols have traditional meanings. This is the so-called modified Kramers-Hcisenberg equation which includes the i!72 term in the denominator of the second term to account for intermediate-state lifetime. It is convenient, both heuristically and computationally to write the integration over continuum states as if it were a sum over discrete states. This is equivalent w assuming the atom is confined in a box. At any time one can take the limit as the energy spacing of the continuum states goes to zero (i.e. the sides of the box g-> to infinity). Also, it should be kept in mind that the sum over intermediate states will in general involve integration over hole-state-alignment and photoelectron-momentum directions as well as energy. Figure 4. Feynman diagrams corresponding to each of the three terms in the Kramers-Heisenberg formula (cq. 1). The three terms in the absolute value brackets can be identified with the three Feynman diagrams [21] shown in figure 4. The first term, sometime called the "Waller term" [22], is associated with the "sea gull" diagram (fig. 4a), and arises from to the A *A term in the interaction Hamiltonian.
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