Dynamics of resonant x-ray and Auger scattering

An overview of both experimental and theoretical results in the field of resonant scattering of tunable soft and hard x-ray radiation is presented, with a main focus on the closely related processes of resonant inelastic x-ray scattering (RIXS) and resonant Auger scattering (RAS). The review starts with an overview of fundamental dynamical aspects of RIXS illustrated for different systems. A detailed analysis of case studies with increasing complexity, considering both gas-phase and condensed matter (liquids and solids) applications, is given. In the review, the most important achievements in investigations of coupled electron-nuclear dynamics and structural aspects in studies of liquids and solids over the last two decades are outlined. To give a perspective on the insights from RIXS and RAS, the x-ray results are discussed against the background of complementary experimental techniques like vibrational infrared absorption and Raman spectroscopy, as well as small-angle x-ray and neutron scattering. Finally, recent achievements in time-resolved studies based on x-ray free-electron lasers are described.

Figure 1

Illustration of resonant elastic and inelastic x-ray scattering (photon in and photon out) and resonant Auger (photon-in and electron-out) scattering; 0, $i$, and $f$ refer to the ground, intermediate core-excited, and final electronic states, respectively.

Figure 2

Ratio ${\sigma }_u/{\sigma }_g=15j_2(qR)/(qR{)}^2$ [Eq. (13)] for the Cl $1s$ RIXS of the ${\mathrm{Cl}}_2$ molecule vs the scattering angle $\mathrm{\Theta }$. ${\mathbf{e}}^{\prime }\parallel \mathbf{e}$.

Figure 3

Soft x-ray RIXS spectra of the ${\mathrm{N}}_2$ ($1{\sigma }_g^2,1{\sigma }_u^2,2{\sigma }_g^2,2{\sigma }_u^2,1{\pi }_u^4,3{\sigma }_g^2$) molecule shows that the scatterings to ungerade final states $3{\sigma }_g^{-1}3p{\pi }_u^1$, ($1{\pi }_u^{-1}3s{\sigma }_g^1$, $2{\sigma }_u^{-1}3s{\sigma }_g^1$), and ($1{\pi }_u^{-1}1{\pi }_g^1$, $2{\sigma }_u^{-1}1{\pi }_g^1$) are forbidden under core excitation to ungerade core-excited states $1{\sigma }_g^{-1}3p{\pi }_u^1$, $1{\sigma }_u^{-1}3s{\sigma }_g^1$, and $1{\sigma }_u^{-1}1{\pi }_g^1$, respectively. These selection rules are absent for nonresonant x-ray emission due to degeneracy of the final continuum states with opposite parities ([125]). Circles and solid lines show experimental and theoretical spectra, respectively. From [152].

Figure 4

RIXS in the hard x-ray region of the ${\mathrm{Cl}}_2$ molecule shows the symmetry-forbidden resonances ${C_{1,2}}^1{\mathrm{\Sigma }}_u^{+}$ and $A^1{\mathrm{\Pi }}_u$ [$\varrho <1$; see Eq. (13)]. The experiment was performed for $\angle ({\mathbf{k}}^{\prime },\mathbf{k})=90°$ and two orientations of polarizations of initial and scattered photons $\angle ({\mathbf{e}}^{\prime },\mathbf{e})=90°$ (squares) and 0° (circles). From [293].

Figure 5

Physical picture of the evolution with detuning of RIXS resonance in a crystal. (a) Momentum transfer $\mathbf{q}$ is negligible for soft x rays. For this reason the final state consists of a vertical VB-to-CB excitation, whose crystal momentum is selected by the energy of the incoming photon as indicated. A showcase of this is the RIXS of graphite ([51]). (b) RIXS transitions cease to be vertical in the hard x-ray region.

Figure 6

The LVI line shape of the normal Auger spectral component $1^3{\mathrm{\Delta }}_u$ of the ${\mathrm{O}}_2$ molecule. The thick line with squares is the total LVI line shape $\sigma (E,\omega )$, the line with triangles is calculated using the direct term ${\sigma }_{\mathrm{dir}}(E,\omega )$ in the LVI formula (22), and the line with circles shows the LVI interference term ${\sigma }_{\mathrm{int}}(E,\omega )$. From [17].

Figure 7

(a) Irreversible [$\exp (-\mathrm{\Gamma }t)$] and reversible [$\cos (\mathrm{\Omega }t)$] quenching of scattering amplitude $F$. (b) Scattering duration $|\tau |=1/\sqrt{{\mathrm{\Omega }}^2+{\mathrm{\Gamma }}^2}$ ([383]; [59]; [138]).

Figure 8

Dynamical interference of the $Q$ and $D$ core-excited states of ${\mathrm{O}}_2$ results in the quantum beats of the RIXS cross section. $\mathrm{\Delta }\omega ={\omega }^{\prime }-\omega$. From [334].

Figure 9

(a) Experimental and (b) theoretical RIXS spectra of ${\mathrm{O}}_2$ for $\chi =90°$ for excitation energies $\omega$ shown by arrows in the XAS spectrum. $\mathrm{\Delta }\omega ={\omega }^{\prime }-\omega$. Adapted from [334].

Figure 10

Detuning from the absorption resonance leads to a collapse of the vibrational structure ([144]) in RIXS of liquid water ([434]).

Figure 11

Collapse of vibrational structure in RIXS. Full width at half maximum (FWHM) of the I $L_3\text{−}{N}_{4,5}$ of ${\mathrm{CH}}_3\mathrm{I}$ (black) and Cl $K$-$L$ of ${\mathrm{CH}}_3\mathrm{Cl}$ (green or gray) emission lines as a function of the incident photon energy detuning with respect to the resonance energy. Adapted from [280].

Figure 12

During the emission transition, a nuclear wave packet of an intermediate state, being projected onto the ground state, allows one to obtain a slice of the potential surface of the ground state along the direction of propagation of the wave packet.

Figure 13

Physical picture of the formation of fast (molecular) and slow (atomic) scattering channels. In the limit of fast scattering, RIXS corresponds to a sudden transition directly to the final state $0\to f$ [Eq. (34)].

Figure 14

RAS spectra of HCl and DCl. The spectral hole seen in HCl for $\mathrm{\Omega }=1.9\mathrm{eV}$ has completely disappeared for DCl for the same detuning value, which is in full agreement with the theoretical prediction ([357]). From [114].

Figure 15

Doppler splitting of the atomic peak in RAS from the ${\mathrm{SF}}_6$ molecule. ${\mathrm{L}}_{\parallel }$ and ${\mathrm{L}}_{\perp }$ mark parallel $(\mathbf{e}\parallel \mathbf{k})$ and perpendicular $(\mathbf{e}\perp \mathbf{k})$ geometries for linear polarization. C, circular polarization. Circles show the experimental data. The theoretical simulations are indicated by sold lines. From [420].

Figure 16

Analogy of the YDSE experiment with macroscopic slits and RAS implementation of the YDSE experiment with fixed-in-space oxygen molecules. The RAS experiment was performed with oriented molecules by simultaneously detecting the electron and the cation ${\mathrm{O}}^{+}$ using the coincidence technique. From [265].

Figure 17

Both (a) experiment and (b) theory display the interference pattern for the molecular band. (c),(d) The absence of the interference structure of the atomic peak upon dissociation in the $|1{\sigma }_g^{-1}3{\sigma }_u⟩$ state, in contrast to the “rigidly bonded” atoms in the bound $|1{\sigma }_g^{-1}4p{\sigma }_u⟩$ condition, is consistent with Bohr’s conclusion. Here $\beta$ is the angle between the Auger momenta of the electron and the cation ${\mathrm{O}}^{+}$. Adapted from [265].

Figure 18

Restoration of the selection rules in RIXS of the ${\mathrm{C}}_2{\mathrm{H}}_4$ molecule for $\mathrm{C}1s\to 1b_{2g}$ core excitation ([181], [182]). $\mathrm{\Delta }\omega ={\omega }^{\prime }-\omega$. RIXS to the final (forbidden by symmetry) ungerade states $|1b_{2u}^{-1}1b_{2g}^1⟩$ and $|1b_{1u}^{-1}1b_{2g}^1⟩$ becomes forbidden upon shortening the scattering duration $|\tau |$ by increasing the detuning $|\mathrm{\Omega }|$. From [181].

Figure 19

A decrease in the relative intensity of the symmetry-forbidden peak $|1{\pi }_g^{-1}2{\pi }_u^1⟩$ in the RIXS of the ${\mathrm{CO}}_2$ molecule. Upper panel: symmetry breaking in the course of the vibration. Adapted from [383].

Figure 20

Potential energy curves of the ground, core-excited $\mathrm{O}\hspace{0.17em}1s^{-1}2{\pi }^1$ and final valence-excited states of CO. Right panel: XAS and RIXS spectra. Adapted from [77].

Figure 21

The experimental and theoretical RIXS spectra of CO vs energy loss $\omega -{\omega }^{\prime }$. Lower panel: RIXS profile computed without accounting for the VC between the final states. Adapted from [77].

Figure 22

The interference of the Thomson and resonance scattering channels in ${\mathrm{O}}_2$ leads to a different polarization dependence of the vibrational resonances $\nu =0$ and $\nu =1$. Adapted from [405].

Figure 23

The interference of the direct and resonant scattering channels leads to a different polarization dependence of the vibrational resonances (the VSA effect) in RAS from the acetylene molecule ${\mathrm{C}}_2{\mathrm{H}}_2$. Adapted from [294].

Figure 24

Phonon dispersion along the lines of high symmetry of a superconductor ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.6}$: $\mathrm{\Gamma }\text{−}\mathrm{Y}$, $\text{Y-}T$ $T\text{−}Z$ ($T\approx 300\mathrm{K}$). White symbols, momentum resolved RIXS experiment; color map and lines, density-functional theory (DFT) calculation. Adapted from [248].

Figure 25

Photoelectron spectrum of the ${\mathrm{H}}_2$ molecule for $\omega =0.02$, 5, 10, and 20 keV. Binding energy $BE=E-\omega$, $E_{\mathrm{rec}}^{\mathrm{CG}}$ is the recoil energy of the center of gravity. Green (gray) and black lines correspond to the angles between momenta of the photon and photoelectron: 90° and 145°, respectively. The vertical dashed line marks the dissociation threshold. Adapted from [260].

Figure 26

(a) Initial Boltzmann distribution. Distributions over (b) angular velocities and (c) velocities induced by recoil from an ejected fast photoelectron. Adapted from [58].

Figure 27

Recoil-induced Doppler splitting of an Auger resonance in a Ne atom. From [380].

Figure 28

The delay of Auger decay with respect to ionization leads to a change in the Doppler shift with time [Eq. (48)]. This makes the (a) Doppler doublet (b),(c) asymmetric. From [58].

Figure 29

Transition scheme and experimental RIXS spectrum of liquid acetone. $\omega =529.9$ (D), 530.2 (C), 530.5 (B), and 530.9 eV (A). Adapted from [404].

Figure 30

The potential energy surfaces of the three most active modes of the acetone molecule. (a) CO stretching mode. (b) CO out-of-plane wagging mode. (c) ${\mathrm{CH}}_3$ torsion mode. The ground (lower panels), core-excited (upper panels), and final (middle panels) state potentials are shown; in (b) and (c) potentials were extracted from experimental UV data ([13]).

Figure 31

The broadening mechanisms of $n^{-1}{{\pi }^{*}}^1$ RIXS of the acetone spectrum. Left panel: spectra of the “direct transition” $0\to f$ for (a) the CO out-of-plane wagging mode and the torsion mode at (b) $T=300\mathrm{K}$ and (c) $T=0\mathrm{K}$. The blue (thick gray) line in (b) displays a spectrum taking into account both modes for $\gamma =0.025\mathrm{eV}$. The spectra are computed using potential energy curves shown in Fig. 30. Right panel: distribution of transition energy ${\omega }_{f0}$ for (a) pure liquid acetone and (b) aqueous acetone from MD simulations. The blue arrow corresponds to the vertical transition energy for gas acetone. From [404].

Figure 32

The experimental and theoretical $n^{-1}{{\pi }^{*}}^1$ RIXS profiles ([404]) of liquid acetone for excitation frequencies from Fig. 29. $\gamma =0.025\mathrm{eV}$. $\angle ({\mathbf{k}}^{\prime },\mathbf{e})=90°$. $\mathrm{\Delta }\omega ={\omega }^{\prime }-\omega$. From [404].

Figure 33

X-ray absorption spectrum and potential energy surfaces of gas-phase water. (a) Simulated XAS spectrum for the three lowest core-excited states of water. (b) Potential energy curves (1D) of the bending vibrational mode for the ground states (GSs) and core-excited states. (c) Stretching potential energy surfaces (2D) as a function of bond lengths for the core-excited states. The color bars represent the energy range of the surfaces in eV relative to the bottom of the GS potential. $Q_s$ is the symmetric stretching coordinate. (d) Squared integral wave packet $|\mathrm{\Psi }{|}^2$ for each of the core-excited states for 2D stretching motion in 2D stretching potential energy surfaces of the ground electronic state. From [78].

Figure 34

(a) Quasielastic RIXS spectra for the $|1s_0^{-1}2b_1^1⟩$, $|1s_0^{-1}2b_2^1⟩$, and $|1s_0^{-1}4a_1^1⟩$ core-excited states of a water molecule. (b) Fine structure of RIXS caused by overtones. Adapted from [78].

Figure 35

Dynamical origin of the splitting of the $1b_1$ peak. (a),(d) Corresponding excitation energies in the XAS preedge region for liquid- and gas-phase water. Comparison of the dispersion of the split components in experimental RIXS spectra of (b) liquid- and (c) gas-phase water at the preedge region. Solid lines in (b) and (c) show theoretical spectra. Both (b) liquid- and (c) gas-phase spectra display a nondispersive component (pseudoatomic peak) and a molecular band following the Raman dispersion law. (e) Schematic illustration showing how the pseudoatomic peak is formed near equilibrium ($R=3\mathrm{a}\mathrm{.}\mathrm{u}\mathrm{.}$) as the PECs of the core-excited [$E_i(R)$] and final [$E_f(R)$] states become almost parallel: $\mathrm{\Delta }U=E_i(R)-E_f(R)\approx \text{const}$. The splitting between molecular and pseudoatomic peaks on top of the XAS resonance is approximately 0.45 eV. Adapted from [110], and [434].

Figure 36

Isotope effect for $2a^{\prime \prime }$ RIXS in gas-phase methanol: theory and experiment. From [435].

Figure 37

Quasielastic RIXS of water in the gas and liquid phases under $4a_1$ preedge core excitation. Theory takes into account only OH vibrations. From [434].

Figure 38

Comparison of RIXS under preedge core excitations and IR spectra of water in the gas and liquid phases. From [434].

Figure 39

Quasielastic RIXS of acetic acid and water. Top panels: gas phase. Bottom panels: liquid. From [360].

Figure 40

Calculated XAS spectra of (a) the acetic acid monomer, (b) the regular dimer, and (c) the inverted dimer, showed in comparison to the measured XAS spectra of (d) acetic acid liquid given by [418] and (e) the gas phase given by [343]. The calculated oscillator strengths are shown as bar graphs, where each bar is colored according to the atom of origin depicted in the molecular structure next to each spectrum. Adapted from [360].

Figure 41

The typical energy scale of elementary excitations in solids is exemplified for the RIXS spectrum at the Cu $L_3$ edge of the cuprate antiferromagnet ${\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_4{\mathrm{Cl}}_2$; see also [349]. The elastic peak is at energy loss $\omega -{\omega }^{\prime }=0$. From [113].

Figure 42

Sketch of the Zaanen-Sawatzky-Allen ([462]) classification of transition-metal compounds into (a) a $d\text{−}d$ character Mott-Hubbard insulator and (b) a character charge-transfer insulator. Filled areas show the occupied $d$ states of metal and ligand states, while hollow areas depict electron affinity states $d^{n+1}$. ${\mathrm{\Delta }}_{\mathrm{CT}}$ is the charge-transfer energy, while $U$ is the Coulomb repulsion energy of $D$ electrons at the same site. Adapted from [60].

Figure 43

Experimental picture of the MH-to-CT transition for $3d$-TM difluorides ($M{\mathrm{F}}_2$) and oxides ($M{\text{O}}_2$). $M=\mathrm{Cr}$ or Zn. The values for ${\mathrm{\Delta }}_{\mathrm{CT}}$ were decreased by 3 eV for oxides to compensate for the difference in electronegativity between F and O. Adapted from [325].

Figure 44

Spin, charge, and orbital ordering pattern of the charge-exchange antiferromagnetic type. From [356].

Figure 45

Azimuthal-angle dependence of the intensity of the orbital ordering reflections $(3,0,0)$. From [305].

Figure 46

Temperature dependence of the normalized intensity of the orbital ordering reflection $(3,0,0)$ at $\omega =6.555\mathrm{keV}$. Empty symbols correspond to results obtained at an azimuthal angle $\psi =127°$ and cooling with a Displex cryostat; filled symbols were obtained for $\psi =90°$ by heating with an oven. The two datasets were scaled to be equal at $T=300\mathrm{K}$. From [305].

Figure 47

Longitudinal scans of the $(0,2,0)$ Bragg reflection, the $(0,1,0)$ charge order peak, and the $(0,2.5,0)$ orbital-order peak. The orbital-order peak is significantly broadened. From [464].

Figure 48

Kitaev interactions on a honeycomb lattice indicating the three kinds of bonds ($x$, $y$, and $z$) and spins sitting on the vertices of a lattice.

Figure 49

Measured spin-orbital separation process in an antiferromagnetic spin chain, emerging after exciting a ground-state copper $3d_{x^2-y^2}$ orbital to an excited copper $3d_{xy}$ or $3d_{xz}$ orbital. RIXS intensity map of the dispersing spin and orbital excitations in quasi-one-dimensional Mott insulator ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$ as functions of photon momentum transfer along the chains and photon energy transfer. Adapted from [362].

Figure 50

(a) Sketch of double spin-flip RIXS process in a Cu-O-Cu chain. (b) Two- [${\omega }_{2s}(q)$] and four- [${\omega }_{4s}(q)$] spinon continua: $(1/2)|\sin (qa)|\le {\omega }_{2s}(q)/\pi J\le \sin (qa/2)$, $(1/2)|\sin (qa)|\le {\omega }_{4s}(q)/\pi J\le \sqrt{2[1+\cos (qa/2)]}$ ([57]). The dispersion law of the single spinon ${\omega }_{1s}(q)=(1/2)|\sin (qa)|$ is depicted by the red line.

Figure 51

Pump energy dependence of the SRIXS signal from Ne: measured number of photons in the emission line as a function of the incoming pulse energy for $\omega >870\mathrm{eV}$ (average $\overline{\omega }=876\mathrm{eV}$) and $\omega <870\mathrm{eV}$ (average $\overline{\omega }=868\mathrm{eV}$) along with the theoretical results. Adapted from [442], and [443].

Figure 52

(a) Experimental setup of the stimulated x-ray emission from a $5M$ solution of a ${\mathrm{MnCl}}_2$ complex. The incoming XFEL pulses are focused onto the liquid jet. (b) Sketch of the stimulated x-ray emission. The arrows represent incoming (green) and emitted (red) photons, while the circles indicate the excited (filled) and nonexcited (open) Mn ions, respectively. (c) Detected number of photons in the Mn $K_{{\alpha }_1}$ region (5 eV integration window) as a function of the nominal incoming XFEL pulse energy for the $5M$ ${\mathrm{MnCl}}_2$ solution. The actual pulse energy on the target is $\sim 20\%$ of the nominal pulse energy shown. The 50 strongest shots in each 0.1 mJ interval are shown in orange; all other shots are shown in black. From [237].

Figure 53

Higher number densities lead to larger peaks with shorter delays, a hallmark of superfluorescence. From [176].

Figure 54

Experimentally observed change in the transmitted intensity as a function of the incident intensity (solid blue circles and line). Each data point is an average over multiple shots. The open blue circles correspond to the simulated shot-by-shot response. Adapted from [67].

Figure 55

(a) tr-RIXS spectra showing magnetic excitations (0–200 meV) and orbital excitations ($\sim 600\mathrm{meV}$) in equilibrium states of 50 ps before photoexcitation (labeled Equil.) and 2 ps after photoexcitation. (b) Intensity difference spectra $\mathrm{\Delta }I$ between the equilibrium state and the 2 ps transient state and between the equilibrium state and 10 ps. This shows a depletion of approximately 20% of the magnetic spectral weight of $\sim 100\mathrm{meV}$ and additional spectral intensity appearing at low energy. Adapted from [83].

Figure 56

Two distinct phase transitions are visible in the electronic structure of silicon. (a) In the false color map of the band gap region (higher intensities are brighter) with smooth white lines of equal intensity, the transient occupation of the conduction band and the transitions to the LDL phase in the first picosecond and to the HDL phase around 6 ps are obvious. (b) Hot electrons appear in time with laser excitation in the conduction band. Highly excited electrons decay faster than thermalized electrons trapped at the conduction band bottom. (c) The phase transitions shift the valence band edge. The transition to the LDL phase starts with the electronic excitation. The plateau between 2 and 4 ps followed by a step is the footprint of the first-order phase transition to the HDL state. (d) The structure of stronger bound states changes alongside and is prominently represented by the changing ratio of the $L_1$ and $L_2$ peaks in the x-ray emission. From [25].

Figure 57

Schemes and results of the time-resolved RIXS experiment on $\mathrm{Fe}(\mathrm{CO}{)}_5$ in an ethanolic solution. (a) Optical-laser pump and soft x-ray probe setup with pump-probe time delay $\mathrm{\Delta }t$. (b) RIXS and valence transitions in $\mathrm{Fe}(\mathrm{CO}{)}_5$. (c) RIXS intensities measured at the Fe $L_3$ edge. Top panel: ground state $\mathrm{Fe}(\mathrm{CO}{)}_5$ (probed before pump). Middle and bottom panels: intensities for transient species at delay intervals $\mathrm{\Delta }t=0–700\mathrm{fs}$ and 0.7–3.5 ps, respectively. From [446].