X-ray Echo Spectroscopy

X-ray echo spectroscopy, a counterpart of neutron spin echo, is being introduced here to overcome limitations in spectral resolution and weak signals of the traditional inelastic x-ray scattering (IXS) probes. An image of a pointlike x-ray source is defocused by a dispersing system comprised of asymmetrically cut specially arranged Bragg diffracting crystals. The defocused image is refocused into a point (echo) in a time-reversal dispersing system. If the defocused beam is inelastically scattered from a sample, the echo signal acquires a spatial distribution, which is a map of the inelastic scattering spectrum. The spectral resolution of the echo spectroscopy does not rely on the monochromaticity of the x rays, ensuring strong signals along with a very high spectral resolution. Particular schemes of x-ray echo spectrometers for 0.1–0.02 meV ultrahigh-resolution IXS applications (resolving power $>{10}^8$) with broadband $\simeq 5–13\mathrm{meV}$ dispersing systems are introduced featuring more than ${10}^3$ signal enhancement. The technique is general, applicable in different photon frequency domains.

Figure 1

Graphical presentation of the echo spectroscopy principles. (a) Photons from a source with a linear size $\mathrm{\Delta }{x}_0$ in the reference source plane 0 are focused onto a spot $\mathrm{\Delta }{x}_1$ in the intermediate image plane 1 by a focusing-dispersing system ${\widehat{O}}_D$. Each spectral component, indicated by different color, is focused at a different location due to dispersion in ${\widehat{O}}_D$. All spectral components of the x rays are refocused by a consecutive time-reversal focusing-dispersing system ${\widehat{O}}_R$ onto the same spot $\mathrm{\Delta }{x}_2$ (echo) in the image plane 2. (b) Inelastic x-ray scattering with an energy transfer $ϵ$ (indicated by changed color) from a sample in the reference plane 1 results in a lateral shift $G_Rϵ$ of the echo signal equal for all spectral components.

Figure 2

Optical scheme of an x-ray echo spectrometer, comprised of the defocusing ${\widehat{O}}_D$ and refocusing ${\widehat{O}}_R$ dispersing systems; the x-ray source in reference plane 0; the sample in 1; and the position-sensitive detector in 2. The spectrometer is shown in the vertical dispersion plane for elastic ($v_e$) and inelastic ($v_i$) scattering, and in the horizontal scattering plane ($h$) with the refocusing system at a scattering angle $\mathrm{\Phi }$. See text for details.

Figure 3

Dispersing element $D_D$ (a) of the defocusing system ${\widehat{O}}_D$ (see Fig. 2) and its spectral transmittance function (b). $D_D$ is an example of an in-line four-crystal CDDW-type dispersing optic, comprised of collimating ($C$), dispersing ($D_1$, $D_2$), and wavelength-selecting ($W$) crystals in a $(\pi +,0-,0+,0-)$ scattering configuration. With the crystal parameters provided in the Supplemental Material [15], the dispersing element $D_D$ features a spectral transmission function with a $\mathrm{\Delta }{E}_{\cup }=8.8\mathrm{meV}$ bandwidth (b), a cumulative angular dispersion rate ${\mathcal{D}}_{{\cup }_D}=-25\mu \mathrm{rad}/\mathrm{meV}$, and a cumulative asymmetry factor $b_{{\cup }_D}=1.4$. The sharp line in (b) presents the 0.1 meV spectral resolution of an x-ray echo spectrometer, designed for use with 9.1 keV photons.

Figure 4

Dispersing element $D_R$ (a) of the refocusing system ${\widehat{O}}_R$ (see Fig. 2) and its spectral transmittance function (b). Similar to $D_D$ in Fig. 3, $D_R$ is an example of the in-line four-crystal CDDW-type dispersing optic, but, in a $(\pi +,\pi +,\pi -,0-)$ scattering configuration. With the crystal parameters provided in [15], the dispersing element $D_D$ features a spectral transmission function with a $\mathrm{\Delta }{E}_{\cup }=5.5\mathrm{meV}$ bandwidth (b), a cumulative angular dispersion rate ${\mathcal{D}}_{{\cup }_D}=-43\mu \mathrm{rad}/\mathrm{meV}$, a cumulative asymmetry factor $b_{{\cup }_D}=0.36$, and ${\mathcal{D}}_{{\cup }_D}/b_{{\cup }_D}=-120\mu \mathrm{rad}/\mathrm{meV}$