proposed in the paper for use as dispersing elements D$_{{_{\mathrm {D}}}}$, D$_{{_{\mathrm {R}}}}$ of the defocusing $\hat{O}_{{_{\mathrm {D}}}}$ and refocusing systems $\hat{O}_{{_{\mathrm {{R}}}}}$ of the echo spectrometer, respectively. The in-line four-crystal CDDW-type dispersing optic (see schematics in Figs. 3-4 and Figs. \[fig00399\]-\[fig0049\]), comprises collimating (C), dispersing (D$_{{_{1}}}$, D$_{{_{2}}}$), and wavelength-selecting (W) crystals, which can be arranged in different scattering configurations. In a general case, the scattering configuration is defined as $(\phi_{{_{1}}}s_{{_{1}}},\phi_{{_{2}}}s_{{_{2}}},\phi_{{_{3}}}s_{{_{3}}},\phi_{{_{4}}}s_{{_{4}}})$. Here, for each crystal $n=1,2,3,4$ (1=C, 2=D$_{{_{1}}}$, 3=D$_{{_{2}}}$, 4=W) the value $\phi_{{_{n}}}=0$ corresponds to the grazing reflection, see Fig. \[fig005\](a)-(b); while $\phi_{{_{n}}}=\pi$ corresponds to the grazing incidence, see Fig. \[fig005\](c)-(d). The sign $s_{{_{n}}}=+1$ corresponds to a reflection in the counterclockwise direction, see Figs. \[fig005\](a),(c); while $s_{{_{n}}}=-1$ means the clockwise direction, see Figs. \[fig005\](b),(d). Those scattering geometries have been selected for use as dispersing elements of the x-ray echo spectrometer in the paper, which feature the largest cumulative dispersion rates ${{{\mathcal D}}_{\cup_{{_{n}}}}}$.

The cumulative dispersion rate ${{{\mathcal D}}_{\cup_{{_{4}}}}}$ in a four-crystal system is given in a general case by $${{{\mathcal D}}_{\cup_{{_{4}}}}} =
b_{{_{4}}}b_{{_{3}}}b_{{_{2}}}{s}_{{_{1}}}{{\mathcal D}}_{{_{1}}}+b_{{_{4}}}b_{{_{3}}}{s}_{{_{2}}}{{\mathcal D}}_{{_{2}}}+b_{{_{4}}}{s}_{{_{3}}}{{\mathcal D}}_{{_{3}}}+{s}_{{_{4}}}{{\mathcal D}}_{{_{4}}},$$ with the asymmetry parameters $b$ and dispersion rates ${{\mathcal D}}$ defined in Table \[tab2\], rows (3) and (4). For the CDDW-type dispersing elements 