Dynamical X-Ray Diffraction in Crystalline Structures

Dynamical X-ray diffraction describes the full interaction of incident and scattered wavefields within a crystal, accounting for multiple scattering events that the simpler kinematical theory neglects. In crystalline materials of perfect or near-perfect quality, these interferences lead to phenomena such as Pendellösung fringes, total reflection under Bragg conditions and anomalous absorption effects. The dynamical regime requires solution of coupled wave equations—most notably the Takagi–Taupin equations—to predict the amplitude and phase of transmitted and diffracted beams as they traverse a lattice. This theory underpins high-resolution rocking-curve analysis, reciprocal-space mapping and phase-contrast imaging, enabling precise characterisation of strain fields, point defects and interfacial roughness at the nanoscale. Recent advances in algorithmic integration, beam-shaping optics and coherent diffraction imaging have extended its reach into three-dimensional defect tomography and the design of bent-crystal focussing elements. Applications span semiconductor device fabrication, materials science under extreme conditions and the development of next-generation synchrotron and free-electron laser beamlines.

Research from Nature Portfolio

Recent studies have demonstrated the power of dynamical diffraction tomography to recover three-dimensional displacement-field functions of point defects in silicon. By employing a semi-kinematical solution of the Takagi–Taupin equations alongside iterative optimisation algorithms, it is now possible to reconstruct a Coulomb-type defect from a single two-dimensional projection, achieving high accuracy even in the presence of noise. Separately, a deterministic approach to Bragg coherent diffraction imaging in imperfect crystals has been introduced, offering closed-form reconstruction of structure factors and displacement fields directly from three-dimensional intensity distributions. This method bypasses iterative phase retrieval in its kinematical approximation and remains robust against realistic noise levels and departures from ideal reference volumes, opening avenues for rapid defect mapping in micro- and nanocrystals.

Research from all publishers

A new finite-difference integration scheme for the Takagi–Taupin equations has been developed on arbitrary orthogonal grids, eliminating the need for sheared computational meshes and improving convergence for slab-shaped samples. By implicitly exploiting Fourier interpolation, this method achieves second-order accuracy and comparable error rates to traditional approaches while simplifying grid geometry. In the realm of beam optics, the crystal-lens equation has been extended to account for dynamical diffraction in Laue geometries, clarifying focal-position shifts in bent-crystal monochromators and enabling polychromatic focussing with minimal aberration. Analytical and numerical results guide the design of bent crystals for high-flux X-ray beamlines. Meanwhile, analytical solutions of the Takagi–Taupin equations under weak lattice deformation have shed light on the evolution of rocking-curve widths and peak intensities as a function of thermal-gradient–induced strain, providing a theoretical basis for strain mapping in thermal-field studies.

Dynamical X-Ray Diffraction in Crystalline Structures publication trend

The graph below shows the total number of articles in dynamical x-ray diffraction in crystalline structures across all publications each year (not limited to Nature Index journals).

Technical terms

Dynamical X-ray diffraction: A theory accounting for multiple scattering of X-rays within a crystal, predicting anomalous transmission and interference effects.

Bragg geometry: A diffraction configuration where incident and diffracted beams emerge from the same crystal surface, satisfying Bragg’s law.

Laue geometry: A diffraction configuration in which the incident beam enters one face of the crystal and the diffracted beam exits through the opposite face.

Takagi–Taupin equations: Coupled differential equations describing the amplitudes of transmitted and reflected X-ray waves in a deformed crystal lattice.

Rocking curve: The intensity profile of a diffracted beam as the crystal or incident angle is varied around the Bragg angle.

Reciprocal lattice: A mathematical construct in which the periodicity of a crystal lattice is represented by points in reciprocal (momentum) space, used to index diffraction conditions.

References

1. X-Ray Diffraction Tomography Recovery of the 3D Displacement-Field Function of the Coulomb-Type Point Defect in a Crystal. Scientific Reports (2019).
2. Deterministic Bragg Coherent Diffraction Imaging. Scientific Reports (2017).
3. A finite difference scheme for integrating the Takagi–Taupin equations on an arbitrary orthogonal grid. Acta Crystallographica Section A: Foundations and advances (2022).
4. X-ray focusing by bent crystals: focal positions as predicted by the crystal lens equation and the dynamical diffraction theory. Journal of Synchrotron Radiation (2022).
5. Dynamic diffraction of planar X-rays in a crystal lattice with a weak deformation field. Journal of Physics Conference Series (2023).