spectrometers [@SSS14; @Shvydko15], including echo spectrometers, see Eq. (6). This is achieved, first, by using strongly asymmetric Bragg reflections close to backscattering [@Shvydko-SB; @SLK06], and, second, by enhancing the single-reflection dispersion rate considerably by subsequent asymmetric Bragg reflections from crystals in special arrangements [@SSM13] exemplified below. In the following two steps, we will show how the principle scheme of a generic echo spectrometer presented above, can be realized in the hard x-ray regime by using multi-crystal arrangements as dispersing elements.

![Principle optical scheme of an x-ray echo spectrometer, comprised of the defocusing $\hat{O}_{{_{\mathrm {{D}}}}}$ and refocusing $\hat{O}_{{_{\mathrm {{R}}}}}$ dispersing systems; the x-ray source in reference plane $0$; the sample in plane $1$; and the position-sensitive detector in $2$. The spectrometer is shown in the vertical dispersion plane for elastic (v$_{{_{\mathrm {e}}}}$) and inelastic (v$_{{_{\mathrm {i}}}}$) scattering, and in the horizontal scattering plane (h) with the refocusing system at a scattering angle $\Phi$. The defocusing system $\hat{O}_{{_{\mathrm {{D}}}}}$ consists of a dispersing Bragg diffracting (multi)crystal element D$_{{_{\mathrm {{D}}}}}$, and of a focusing element $f$. The refocusing system $\hat{O}_{{_{\mathrm {{R}}}}}$ is of a spectrograph type comprising collimating element $f_{{_{1}}}$; a dispersing Bragg diffracting (multi)crystal element D$_{{_{\mathrm {{R}}}}}$; and a focusing element $f_{{_{2}}}$. 