X-ray backscattering diffraction （θ_B≅"90" °）exhibits an ultra-narrow intrinsic energy bandwidth at the meV scale together with a relatively large angular acceptance on the mrad scale, making it essential for high-resolution X-ray optics such as monochromators and X-ray free-electron laser oscillator (XFELO) cavities. However, conventional descriptions based on one-dimensional rocking curves or two-dimensional DuMond diagrams fail to capture the intrinsic coupling between photon energy and angular variables under this extreme diffraction condition,thus limiting a comprehensive understanding of the diffraction behavior.
In this work, a three-dimensional (3D) DuMond diagram for X-ray backscattering is systematically established and experimentally validated using a Si(800) crystal as a model system. Starting from the geometric representation of dispersion surfaces in reciprocal space, a modified deviation parameter is derived within the framework of dynamical diffraction theory. This formulation explicitly incorporates both two-dimensional angular deviations and photon energy, enabling a unified and quantitative description of the diffraction process in a three-dimensional phase space. Based on this theoretical model, numerical simulations are performed to reconstruct the full diffraction intensity distribution, revealing that the 3D DuMond diagram forms an inverted paraboloid of revolution in the coupled energy-angle phase space.
To verify the validity of the theoretical model, experiments were carried out at the BL09B beamline of the Shanghai Synchrotron Radiation Facility (SSRF). By scanning rocking curves under different conditions, the characteristic projections of the three-dimensional DuMond diagram in both the transverse (Θ,Φ) and longitudinal (ΔE,Θ) phase spaces were reconstructed.The results show that: (i) the longitudinal projection in the energy-angle plane exhibits a clear parabolic profile, directly confirming the quadratic coupling relationship between energy deviation and angular deviation; (ii) the transverse projection in the two-dimensional angular space undergoes a distinct topological evolution, evolving from a filled disk to an expanding annulus as the incident energy increases away from the backscattering energy. The experimentally measured phase-space distributions and their full width at half maximum (FWHM) are in good quantitative agreement with theoretical simulations.
This work provides a three-dimensional phase-space framework for describing X-ray backscattering, revealing the intrinsic energy-angle coupling mechanism and offering a clear physical picture of the diffraction process. It establishes a solid theoretical and experimental foundation for precision metrology and the design of high-resolution backscattering-based X-ray optical systems.