that even higher spectral resolution $\Delta {\varepsilon}\lesssim 0.02$ meV can be achieved with x-ray echo spectrometers by increasing the dispersion rates ${{{\mathcal D}}_{\cup_{{_{}}}}}$ in the dispersing elements. This, however, will result in their narrower transmission bandwidths $\Delta E_{{_{\cup}}}$. Still, an approximately constant ratio $\Delta E_{{_{\cup}}}/\Delta {\varepsilon}$ holds. Alternatively, the spectral resolution can be improved by increasing the focal length $f_{{_{1}}}$ in the refocusing system, see Eq. .

The essential feature of the echo spectrometers is that the signal strength, which is proportional to the product of the bandwidths of the photons on the sample and on the detector, is enhanced by $(\Delta E_{{_{\cup}}}/\Delta
{\varepsilon})^2 \simeq 10^3-10^4$ compared to what is possible with the standard scanning-IXS-spectrometer approach.

Conclusions
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In conclusion, x-ray echo spectroscopy, a counterpart of neutron spin-echo, is introduced here to overcome limitations in spectral resolution and weak signals of the traditional inelastic hard x-ray scattering (IXS) probes. Operational principles, refocusing conditions, and spectral resolutions of echo spectrometers are substantiated by an analytical ray-transfer-matrix approach. A principle optical scheme for a hard x-ray echo spectrometer is proposed with multi-crystal arrangements as dispersing elements. Concrete schemes are discussed with 5–13-meV transmission bandwidths, a spectral resolution of 0.1-meV (extension to 0.02-meV 