X-ray Convergent-Beam Diffraction

• X-ray convergent-beam diffraction is a method that uses a focused, monochromatic convergent beam to produce continuous reciprocal-space sampling and generate Bragg streaks.
• It employs high-numerical-aperture optics like multilayer Laue lenses to achieve high spatial resolution and enable projection topography of crystal structures.
• The technique unifies structure factor extraction, strain mapping, and ultrafast pump–probe measurements, offering detailed insights into crystal morphology and dynamics.

Searching arXiv for papers on X-ray convergent-beam diffraction and closely related work. Search results gathered. Reviewing the most relevant arXiv records for convergent-beam X-ray diffraction, attosecond crystallography, and dynamical scattering. X-ray convergent-beam diffraction (CBXD) is a Bragg diffraction modality in which a monochromatic but convergent X-ray beam, typically focused by high-numerical-aperture multilayer Laue lenses (MLLs), illuminates a crystal so that the union of Ewald spheres for incident rays within the lens pupil fills a continuous volume of reciprocal space. A reciprocal-lattice point is diffracted whenever there exists an incident $\mathbf{k}_i$ and diffracted $\mathbf{k}_f$ such that $\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$ with $|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$, and the continuous distribution of $\mathbf{k}_i$ gives rise to Bragg streaks rather than isolated reflections. In recent formulations, CBXD is extended to convergent-beam attosecond X-ray crystallography, in which dispersive optics encode time into the resulting diffraction pattern with deep sub-femtosecond precision, while the same geometry enables projection topographic imaging and spatially resolved structure determination (Chapman et al., 2024, Li et al., 16 Feb 2026).

1. Reciprocal-space geometry and Bragg conditions

In CBXD a lens of high numerical aperture $\mathrm{NA}$ focuses a quasi-monochromatic beam into the crystal. The semi-convergence angle is

$\alpha=\sin^{-1}(\mathrm{NA}) .$

Equivalently, every plane-wave component has wave vector $\mathbf{k}_i$ satisfying $|\mathbf{k}_i|=1/\lambda$ and incident at angles $(\phi_x,\phi_y)$. In reciprocal space each incident $\mathbf{k}_f$0 defines an Ewald sphere of radius $\mathbf{k}_f$1 centered at $\mathbf{k}_f$2, and the union of these spheres for $\mathbf{k}_f$3 within the lens pupil fills a continuous volume of reciprocal space (Chapman et al., 2024).

A reciprocal-lattice point $\mathbf{k}_f$4 is diffracted whenever

$\mathbf{k}_f$5

For a collimated monochromatic beam the Bragg condition $\mathbf{k}_f$6 yields isolated reflections; in CBXD the continuous distribution of $\mathbf{k}_f$7 gives rise to “Bragg streaks” each formed by the locus of $\mathbf{k}_f$8 satisfying the same scattering equation. In kinematical diffraction this is equivalently written as

$\mathbf{k}_f$9

or

$\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$0

A recurrent point of confusion is the relation between CBXD and Laue diffraction. Laue diffraction uses a polychromatic, collimated beam; CBXD uses a monochromatic but convergent beam. Both sample volumes in reciprocal space via Ewald spheres, but CBXD samples by $\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$1angle instead of $\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$2. The relation between equivalent bandwidth and convergence follows from differentiating Bragg’s law:

$\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$3

For CBXD the reciprocal-space region between the most and least inclined Ewald spheres (semi-angle $\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$4) determines the number of reflections in one exposure (Chapman et al., 2024).

The same geometry can be described through the Kossel cone construction. If one ray $\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$5 satisfies the Bragg condition for a given $\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$6, then every ray obtained by rotating $\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$7 about the axis $\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$8 by an angle $\mathbf{k}_f-\mathbf{k}_i=\mathbf{q}_{hkl}$9 will also diffract by the same Bragg angle $|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$0. The set of all such rays forms the Kossel cone, with parametrisation

$|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$1

where $|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$2. Only those $|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$3 that lie within the lens aperture produce Bragg streaks on the detector; where the corresponding incident ray is extinguished in the crystal one sees a “deficiency line” instead of a bright streak (Li et al., 16 Feb 2026).

2. High-NA optics and multilayer Laue lenses

The optical realization of CBXD in hard X-rays relies on high-NA focusing optics, particularly MLLs. For a linear zone plate, the numerical aperture is related to the outermost zone width $|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$4 by

$|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$5

and the focal length of a single-order diffractive lens is

$|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$6

where $|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$7 is the radius of the outermost zone or half-width of the aperture for a rectangular lens (Chapman et al., 2024).

A 1D MLL is a volume Bragg grating whose layer spacing $|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$8 varies so as to maintain the Bragg condition across its aperture, thereby deflecting all incident rays through the same angle $|\mathbf{k}_i|=|\mathbf{k}_f|=1/\lambda$9 toward a common focus at distance $\mathbf{k}_i$0. In the thin-lens approximation, its transmission function is

$\mathbf{k}_i$1

where $\mathbf{k}_i$2 is the finite aperture window. A pair of orthogonal MLLs gives a two-dimensional focus, and the field just downstream of the lens pair can be written as

$\mathbf{k}_i$3

with

$\mathbf{k}_i$4

The real-space wavefield at the sample plane is then

$\mathbf{k}_i$5

This formalism emphasizes that CBXD is a wave-optical focusing problem as well as a crystallographic sampling problem (Li et al., 16 Feb 2026).

Typical hard-X-ray MLLs are fabricated by ion-beam–deposition or sputtering of alternating high-Z/low-Z materials such as $\mathbf{k}_i$6 with zone thickness grading from a few nanometers to tens of nanometers. Stacks of $\mathbf{k}_i$7–$\mathbf{k}_i$8 bi-layers achieve $\mathbf{k}_i$9 up to $\mathrm{NA}$0–$\mathrm{NA}$1 at $\mathrm{NA}$2, and diffraction efficiency in the $\mathrm{NA}$3st order can reach $\mathrm{NA}$4–$\mathrm{NA}$5 per lens; two perpendicular lenses in series focus in both transverse dimensions (Chapman et al., 2024).

Experimentally, two lens pairs at $\mathrm{NA}$6 were reported: an “off-axis” pair with semi-NA $\mathrm{NA}$7 in each direction and focal lengths $\mathrm{NA}$8, $\mathrm{NA}$9, and an “on-axis” pair with semi-NA $\alpha=\sin^{-1}(\mathrm{NA}) .$0 and focal lengths $\alpha=\sin^{-1}(\mathrm{NA}) .$1, $\alpha=\sin^{-1}(\mathrm{NA}) .$2. The diffraction-limited spot size of an ideal lens is $\alpha=\sin^{-1}(\mathrm{NA}) .$3, although in practice aberrations inflate the focus to tens of nanometers (Li et al., 16 Feb 2026).

3. Bragg streaks, deficiency lines, and projection topographs

The characteristic observable in CBXD is the Bragg streak. In 3D crystals, a given reciprocal lattice point $\alpha=\sin^{-1}(\mathrm{NA}) .$4 is satisfied by all incident $\alpha=\sin^{-1}(\mathrm{NA}) .$5 on a small circle of directions. On a flat 2D detector these circles project into curved or, for small $\alpha=\sin^{-1}(\mathrm{NA}) .$6, straight streaks. The width of each Bragg streak in the rocking direction is set by the crystal thickness $\alpha=\sin^{-1}(\mathrm{NA}) .$7 and obeys

$\alpha=\sin^{-1}(\mathrm{NA}) .$8

This directly links streak width to finite-thickness broadening and the Lorentz factor (Li et al., 16 Feb 2026).

A single “snapshot” CBXD pattern records many Bragg streaks simultaneously. By indexing each streak $\alpha=\sin^{-1}(\mathrm{NA}) .$9 and mapping it back to its incident deficiency line $\mathbf{k}_i$0 in the lens pupil, one obtains a sparse projected map of the crystal’s diffraction strength:

$\mathbf{k}_i$1

where $\mathbf{k}_i$2 is the polarisation factor, $\mathbf{k}_i$3 is the Lorentz factor, $\mathbf{k}_i$4 is the aperture function, and $\mathbf{k}_i$5 is the line integral of the crystal shape function $\mathbf{k}_i$6 along the ray path. By iterating between normalising each streak by an estimate of $\mathbf{k}_i$7 and re-projecting a composite topograph back onto each streak, one can recover both $\mathbf{k}_i$8 and a high-quality projected topograph in a single pattern (Li et al., 16 Feb 2026).

If the crystal is rocked through the Bragg condition for a given $\mathbf{k}_i$9, its deficiency line sweeps across the face, and every detector frame contributes one slice of that reflection’s magnified topograph. Summing over the rocking range produces an undistorted 2D image

$|\mathbf{k}_i|=1/\lambda$0

with an extra sampling factor $|\mathbf{k}_i|=1/\lambda$1. By collecting full topographs for many $|\mathbf{k}_i|=1/\lambda$2 over a large rotation range, such as $|\mathbf{k}_i|=1/\lambda$3 in $|\mathbf{k}_i|=1/\lambda$4 steps, one obtains different projected views of $|\mathbf{k}_i|=1/\lambda$5; a standard algebraic tomographic algorithm such as SIRT then solves for the 3D crystal shape. In the Si “F-cube” test, $|\mathbf{k}_i|=1/\lambda$6 views gave a voxel resolution $|\mathbf{k}_i|=1/\lambda$7, consistent with the Crowther criterion

$|\mathbf{k}_i|=1/\lambda$8

This establishes CBXD as a combined diffraction and projection-topography method rather than solely a reflection-counting method (Li et al., 16 Feb 2026).

4. Time encoding and convergent-beam attosecond crystallography

In convergent-beam attosecond X-ray crystallography, the critical extension is the use of dispersive optics and attosecond hard-X-ray pulses. A dispersive lens such as a diffractive MLL introduces a wavelength dependence of focal length, $|\mathbf{k}_i|=1/\lambda$9, with dispersive power

$(\phi_x,\phi_y)$0

The pulse front lags the phase front by an angle-dependent delay. For an on-axis lens and ray deflection $(\phi_x,\phi_y)$1,

$(\phi_x,\phi_y)$2

If the crystal sits a distance $(\phi_x,\phi_y)$3 downstream of focus,

$(\phi_x,\phi_y)$4

Thus the convergent geometry is not only reciprocal-space sampling; it is also a spatiotemporal encoding geometry (Chapman et al., 2024).

A ray of angle $(\phi_x,\phi_y)$5 intersects the crystal or detector at $(\phi_x,\phi_y)$6, so the arrival time is

$(\phi_x,\phi_y)$7

or, rearranged,

$(\phi_x,\phi_y)$8

Every Bragg streak is thus encoded in time via its position along the streak on the detector. After indexing the pattern, each detector pixel $(\phi_x,\phi_y)$9 is assigned scattering vector components $\mathbf{k}_f$00 and the corresponding incident ray $\mathbf{k}_f$01, and the pump–probe delay is

$\mathbf{k}_f$02

where $\mathbf{k}_f$03 depends on pump geometry, including crossed-beam or tilted-pulse-front schemes (Chapman et al., 2024).

The experimental parameter range reported for this scheme includes photon energy $\mathbf{k}_f$04–$\mathbf{k}_f$05, wavelength $\mathbf{k}_f$06–$\mathbf{k}_f$07, sub-femtosecond XFEL probe pulses, optical or EUV pump pulses of few-fs or shorter duration, repetition rates from $\mathbf{k}_f$08 to $\mathbf{k}_f$09 depending on facility, MLL focal length $\mathbf{k}_f$10, $\mathbf{k}_f$11–$\mathbf{k}_f$12, sample placement in focus $\mathbf{k}_f$13 or out of focus $\mathbf{k}_f$14–$\mathbf{k}_f$15, detector distance typically $\mathbf{k}_f$16–$\mathbf{k}_f$17, and maximum $\mathbf{k}_f$18 covered of approximately $\mathbf{k}_f$19, giving time ranges of tens of femtoseconds. Pixel pitch of $\mathbf{k}_f$20–$\mathbf{k}_f$21 at $\mathbf{k}_f$22 subtends $\mathbf{k}_f$23–$\mathbf{k}_f$24. Typical $\mathbf{k}_f$25–$\mathbf{k}_f$26 yields $\mathbf{k}_f$27–$\mathbf{k}_f$28 (Chapman et al., 2024).

5. Quantitative structure factors, spatial resolution, and example studies

Once the crystal morphology $\mathbf{k}_f$29 is known, CBXD permits extraction of structure factors from full or partial topographs at any location in the crystal. In the simple “one-axis” dataset the crystal is always fully illuminated, so that

$\mathbf{k}_f$30

Important correction factors are the polarisation factor

$\mathbf{k}_f$31

the Lorentz factor $\mathbf{k}_f$32, the sampling factor $\mathbf{k}_f$33, the detector quantum efficiency

$\mathbf{k}_f$34

and the absorption in the crystal

$\mathbf{k}_f$35

for thin samples. In practice one divides each pixel of a topograph by the local projection $\mathbf{k}_f$36 and by $\mathbf{k}_f$37, then takes the mean or median over all pixels to get $\mathbf{k}_f$38, with typical errors of a few percent and $\mathbf{k}_f$39 in the Si test (Li et al., 16 Feb 2026).

The spatial resolution of topographs is set by detector-pixel angular sampling. For a $\mathbf{k}_f$40 pixel at $\mathbf{k}_f$41 one gets $\mathbf{k}_f$42 in the sample plane, and higher magnifications with smaller $\mathbf{k}_f$43 push this below $\mathbf{k}_f$44. Strain or lattice tilt along a deficiency line causes a systematic bend of the Bragg streak in the topograph; from the local deviation $\mathbf{k}_f$45 one can infer $\mathbf{k}_f$46 or $\mathbf{k}_f$47 with sensitivity $\mathbf{k}_f$48 (Li et al., 16 Feb 2026).

Reported example studies show the scope of the method. For a Si lamella with a focussed-ion-beam wedge and hole, $\mathbf{k}_f$49 reflections to $\mathbf{k}_f$50 were recorded, topographs matched kinematical simulations, and structure-factor errors were $\mathbf{k}_f$51. For the cubic Si “F-cube”, $\mathbf{k}_f$52 views produced voxel size $\mathbf{k}_f$53, the carved “F” feature was resolved in the surface slab, and the full set of Si structure factors to $\mathbf{k}_f$54 agreed at $\mathbf{k}_f$55 with tabulated values after $\mathbf{k}_f$56 correction. For vitamin $\mathbf{k}_f$57, on-axis MLLs with $\mathbf{k}_f$58 placed $\mathbf{k}_f$59 downstream yielded $\mathbf{k}_f$60 high-quality topographs, reconstruction of $\mathbf{k}_f$61, extraction of $\mathbf{k}_f$62 to $\mathbf{k}_f$63, and refinement with an $\mathbf{k}_f$64-factor $\mathbf{k}_f$65 overall; separate integration over top and bottom halves of the crystal yielded identical structure factors, demonstrating position-dependent analysis (Li et al., 16 Feb 2026).

Applications stated for CBXD include characterization of strain and defects at high resolution, studies of diffusion and binding in MOFS, protein-drug binding, crystal growth, and the mechanical responses of photo-reactive or thermally driven dynamic crystals, as well as real-time mapping of ultrafast charge transfer, non-adiabatic dynamics in small-molecule crystals, and attosecond quantum crystallography of electronic wave-packet evolution (Chapman et al., 2024, Li et al., 16 Feb 2026). A plausible implication is that CBXD links unit-cell structure determination and crystal-scale morphology in a single experimental framework.

6. Dynamical diffraction, limitations, and boundary conditions of validity

The simplest CBXD analyses are kinematical, but dynamical diffraction may become important in thicker $\mathbf{k}_f$66, perfect crystals, requiring full Takagi–Taupin modelling. This limitation is explicit for Si and other highly perfect specimens, and it becomes central when the crystal thickness exceeds the X-ray extinction depth or when nanoscale and mesoscale features coexist (Li et al., 16 Feb 2026, Pateras et al., 2020).

In the two-beam approximation, the transmitted amplitude $\mathbf{k}_f$67 and diffracted amplitude $\mathbf{k}_f$68 inside a perfect crystal satisfy coupled Darwin or Takagi–Taupin equations,

$\mathbf{k}_f$69

$\mathbf{k}_f$70

with boundary conditions $\mathbf{k}_f$71 and $\mathbf{k}_f$72. The extinction depth is

$\mathbf{k}_f$73

and for GaAs $\mathbf{k}_f$74 at $\mathbf{k}_f$75 one finds $\mathbf{k}_f$76. In the thick-crystal limit $\mathbf{k}_f$77, the diffracted intensity saturates to near unity over the Darwin angular width

$\mathbf{k}_f$78

while the transmitted beam is strongly suppressed by primary extinction (Pateras et al., 2020).

The dynamical-scattering description explains features that a purely kinematical CBXD treatment does not: a sharp intense vertical line at $\mathbf{k}_f$79 due to a thick substrate, saturation of peak reflectivity, thickness fringes from thin layers, interference between equivalent layers, and the sweep of the substrate line into the beam cone with increasing incidence angle. In coherent hard-X-ray nanobeam Bragg diffraction, an optical model combined with Darwin theory accurately reproduced experimental diffraction patterns from a $\mathbf{k}_f$80 epitaxial heterostructure, including primary extinction, multiple scattering, and absorption (Pateras et al., 2020).

Other practical limitations are experimental rather than formal. Incomplete topographs, such as those for $\mathbf{k}_f$81 axes nearly aligned with the rotation axis, must be scaled against the tomographic model of $\mathbf{k}_f$82; multi-axis rotation can alleviate this. Radiation damage limits use of the nm-beam in macromolecular crystals, whereas the out-of-focus, $\mathbf{k}_f$83 beam reduces dose by more than $\mathbf{k}_f$84 for identical fluence in focus. The current angular scan speed of $\mathbf{k}_f$85 steps times $\mathbf{k}_f$86 frames is slow; faster acquisition and larger beam divergence could accelerate data collection by two orders of magnitude (Li et al., 16 Feb 2026).

CBXD therefore occupies a technically specific regime: it is a monochromatic convergent-beam Bragg method that replaces isolated reflections with Bragg streaks, maps those streaks into projection topographs and, with dispersive high-NA optics, can encode pump–probe delay directly into detector position. Its core promise is the unification of high-resolution imaging, structure-factor extraction, and ultrafast diffraction within a single convergent-beam geometry (Chapman et al., 2024).