Resonant Elastic X-ray Scattering (REXS)

• REXS is a diffraction technique that exploits resonant enhancement at atomic absorption edges to selectively probe charge, spin, orbital, and lattice modulations in materials.
• It employs high-brilliance, tunable X-ray sources and advanced detection systems, enabling precise reciprocal-space mapping and quantitative analysis.
• Its element- and state-specific sensitivity makes REXS a key tool for investigating phase transitions, emergent order parameters, and correlated electron phenomena.

Resonant Elastic X-ray Scattering (REXS) is a photon-in, photon-out diffraction technique in which the incident photon energy is tuned to an absorption edge of an element within the sample, resonantly enhancing scattering from specific atomic species and particular electronic configurations. By exploiting the strong energy dependence and element selectivity near atomic resonances, REXS offers unique sensitivity to spatial modulations of charge, spin, orbital, and lattice degrees of freedom. It is applicable in single crystals, thin films, heterostructures, interfaces, and bulk materials, and has evolved into a principal tool for probing emergent phenomena in correlated electron systems across nanometer to mesoscale length scales.

1. Fundamental Principles of REXS

REXS is based on the interference of X-rays elastically scattered from periodic modulations within the material. Enhancement occurs when the incident photon energy $\hbar\omega$ approaches a dipole-allowed (or quadrupole-allowed) core-level absorption threshold (e.g., the $L_{2,3}$ edge of $3d$ transition metals, $M_{4,5}$ for rare earths), causing the resonant atomic scattering factor $f(\omega, \mathbf{e}, \mathbf{e}')$ to become highly sensitive to the local chemical, electronic, and magnetic environment. The total atomic scattering amplitude is commonly modeled as: $$f(\omega, \mathbf{e}, \mathbf{e}') = f_0 + \Delta f(\omega, \mathbf{e}, \mathbf{e}'),$$ where $f_0$ is the non-resonant (Thomson) term and $\Delta f$ encodes the resonance enhancement (Fink et al., 2012).

In the dipole approximation: $$\Delta f(\omega, \mathbf{e}, \mathbf{e}') = k^2 \sum_{I} \frac{ \langle G | (\mathbf{e}' \cdot \mathbf{D})^{\dagger} | I \rangle \langle I | (\mathbf{e} \cdot \mathbf{D}) | G \rangle }{ E_I - E_G - \hbar\omega - i\Gamma_I/2 }$$ where $|G\rangle$ and $|I\rangle$ are the ground and intermediate states, $E_G$ and $E_I$ their respective energies, $\Gamma_I$ the intermediate-state lifetime broadening, and $\mathbf{e}$, $\mathbf{e}'$ the polarization vectors (Fink et al., 2012). The resonant enhancement typically increases the scattering cross-section by several orders of magnitude and makes REXS acutely sensitive to weak order parameters (e.g., charge stripes, orbital nematicity, or magnetic multipoles).

2. Instrumentation and Methodological Developments

The core requirements for REXS include high-brilliance, tunable photon sources (third-generation synchrotrons, X-ray Free Electron Lasers), UHV-compatible beamlines (to suppress soft X-ray attenuation), advanced polarization control, cryogenic and vector magnetic field environments, and sensitive detector arrays.

A canonical example is the four-circle in-vacuum diffractometer at the REIXS beamline (Hawthorn et al., 2011), which achieves:

• Full $\theta$ and $2\theta$ rotations (–25° to +265°), with $<0.001^\circ$ resolution
• Sample translation (±7.5 mm in $x$, $y$, $z$), vertical detector arm shifts (up to 90 mm), and interchangeable detector systems (photodiode for high dynamic range, single-photon channeltron for weak signals, or a microchannel plate for 2D spatially resolved detection)
• UHV operation at base pressure $2 \times 10^{-10}$ Torr, essential for minimizing soft X-ray absorption
• Sample cooling down to 18 K (ARS DE-210SB closed-cycle cryostat), with continuous temperature monitoring and minimal drift (vertical motion $< 100\,\mu$m)
• Variable slit and filter options, providing angular resolution as fine as $0.1^\circ$ (for 0.5 mm slit at 290 mm distance), stray light and particle suppression, and energy filtering

Such instrumentation allows submicron, sub-millidegree precision, enabling comprehensive reciprocal-space mapping and temperature/field-dependent studies of weakly scattering phenomena (Hawthorn et al., 2011, Fink et al., 2012).

3. Element, Site, and State Sensitivity

The resonant enhancement renders REXS inherently element-, site-, and state-specific. By tuning to the energy of a core-level transition (e.g., Cu $L_3$ edge for cuprates, Mn $L_3$ or Ru $L_3$ edges in oxides, Tb $M_5$ in pyrochlores), scattering is selectively amplified from atoms of the chosen species and can even differentiate between crystallographically distinct sublattices or oxidation states (Hossain et al., 2013, Donnerer et al., 2019).

For instance, impurity-based REXS was used to detect ordering in Mn-doped Sr$_3$Ru$_2$O$_7$ by selectively enhancing the signal at the Mn $L_3$ edge (in dilute concentrations) (Hossain et al., 2013): $$A(\mathbf{Q}) = \sum_j f_{R_j} e^{i \mathbf{Q} \cdot R_j \delta_{R_j,\text{Mn}}},$$ where $\delta_{R_j,\text{Mn}}$ is 1 at Mn sites and 0 elsewhere. This approach is directly analogous to impurity resonance in ESR/NMR/Mössbauer spectroscopy but provides momentum-selective information—a critical distinction for mapping spatial periodicity and coherence lengths (Hossain et al., 2013).

Moreover, polarization and energy analysis at different edges (e.g., Ir $L_3$ vs Tb $M_5$) allows disentanglement of the ordering on different electronic sublattices and the quantification of, e.g., all-in/all-out magnetic versus lattice ATS scattering (Donnerer et al., 2019).

4. Applications: Charge, Spin, and Orbital Modulation

REXS has become the definitive probe for a wide range of emergent modulated phases:

• Stripe and checkerboard charge orders in cuprates: Detection of subtle [H,0,L] superlattice peaks at the Cu $L_3$ edge, measurement of domain correlation lengths, and direct visualization of the charge ordering vector (e.g., H~0.237).
• Commensurate and incommensurate charge density waves (CDW): REXS at hard (Ir $L_3$) and soft (Te $M_{4,5}$) edges in Ir$_{1-x}$Pt$_x$Te$_2$ reveals the shift from Q = (1/5, 0, –1/5) to incommensurate positions with increasing Pt-doping and the coexistence with superconductivity, with energy-dependent lineshape analysis showing the pivotal role of Te $5p$ states (Takubo et al., 2018).
• Skyrmion lattices and spiral phases: REXS at the Cu $L_3$ edge in Cu$_2$OSeO$_3$ uncovers the sixfold symmetric satellite pattern of the skyrmion lattice, distinguishes single-domain from multidomain states via peak splitting, and captures higher harmonics in modulated surface states (Zhang et al., 2016, Mehboodi et al., 20 Dec 2024).
• Antiferromagnetic superstructures: Full linear polarization REXS at the Eu $L_2$ edge in EuPtSi$_3$ identifies cycloidal, conical, fan-like, and commensurate antiferromagnetic structures, with phase selectivity and discrimination of magnetic from structural scattering by Poincaré–Stokes analysis (Simeth et al., 2023).
• Paramagnetic and fluctuating magnetism: Diffuse REXS at the Eu $M_5$ edge in EuCd$_2$As$_2$ demonstrates slow ferromagnetic correlations above the ordering temperature, relevant for the spontaneous formation of Weyl nodes in topological semimetals (Soh et al., 2020).
• Time-resolved order parameter dynamics: Ultrafast tr-REXS at XFEL sources enables direct observation of sub-picosecond CDW melting and recovery in YBa$_2$Cu$_3$O$_{6.73}$, quantifying the time constants and fluence dependence via convolution models with sub-100 fs temporal resolution (Jang et al., 2020).

5. Quantitative Analysis and Theoretical Modeling

The basic intensity in REXS is given by: $$I(\mathbf{Q}, \omega) \propto \left| \sum_{j} f_j(\omega) e^{-i \mathbf{Q} \cdot \mathbf{r}_j} \right|^2$$ with $f_j(\omega)$ the atomic form factor—modulated by local charge/spin/orbital states and their ordering, and $\mathbf{r}_j$ atomic site positions (Fink et al., 2012, Takubo et al., 2018).

Analysis of domain size (from peak width), phase transitions (via thermal and field dependence), and state-resolved contributions (by lineshape and edge-selection) is realized using models including lattice displacements, valence modulation (i.e., $f(\omega, p+\delta p_j)$, with $p$ as orbital occupation), and magnetic structure factors. Specific symmetry-based Jones-matrix formalisms are employed to separate ATS and magnetic channels in forbidden reflection geometries (Donnerer et al., 2019).

In time-resolved REXS, the response function is often fitted to a convolution of an exponential decay with a Gaussian instrument response: $$\Delta I(\Delta t) = \frac{A}{\sqrt{2\pi}\sigma} \int_0^{\infty} dt' \, e^{-t'/\tau} \exp\left[ -\frac{(\Delta t - t')^2}{2\sigma^2} \right],$$ yielding characteristic decay times for the order parameter dynamics (Jang et al., 2020).

6. Limitations, Controversies, and Future Prospects

Limitations remain in the accessible momentum range (set by photon energy), signal strength for extremely dilute or ultra-short periodicities, and instrumental factors such as thermal drift or radiative loss at cryogenic temperatures.

Several debates have arisen concerning the interpretation of satellite peak splitting. In some skyrmion systems, initial reports of sublattice splitting were subsequently reattributed to multidomain formation upon careful REXS domain mapping and XAS checks (Zhang et al., 2016). Similarly, distinctions between electronic versus lattice-driven order have been resolved by edge-selective lineshape analysis and cross-correlation with auxiliary spectroscopy (Takubo et al., 2018).

Future enhancements anticipated include:

• Incorporation of higher-coherence and higher-brilliance sources (soft X-ray FELs), expanding spatial, temporal, and energy resolution (Fink et al., 2012, Jang et al., 2020).
• Improved in-vacuum mechanics and direct integration with complementary techniques (STM/AFM/MBE chambers) for in situ, multi-modal analysis (Hawthorn et al., 2011).
• Systematic, quantitative extraction of order parameter amplitudes, orbital occupations, and depth profiles by combining polarization-, energy-, and momentum-resolved datasets with ab initio modeling.
• Extension to new material classes (organic conductors, soft matter, membranes), ultrafast phase transitions, and topological phases—where element and state specificity are particularly advantageous.

REXS is now positioned as a complementary, and in many cases unique, probe alongside neutron and non-resonant X-ray scattering in the paper of correlated and quantum materials. Its joint structural and electronic sensitivity underpins investigations into complex order parameters, intertwined phases, and emergent topology across an expanding frontier in condensed matter science.