Recent technological breakthroughs in synchrotron and x-ray free electron laser facilities have revolutionized nanoscale structural and dynamic analyses in condensed matter systems. This review provides a comprehensive overview of the advancements in coherent scattering and diffractive imaging techniques, which are now at the forefront of exploring materials science complexities. These techniques, notably Bragg coherent diffractive imaging and x-ray photon correlation spectroscopy, x-ray magnetic dichroism, and x-ray correlation analysis leverage beam coherence to achieve volumetric three-dimensional imaging at unprecedented sub-nanometer resolutions and explore dynamic phenomena within sub-millisecond timeframes. Such capabilities are critical in understanding and developing advanced materials and technologies. Simultaneously, the emergence of chiral crystals—characterized by their unique absence of standard inversion, mirror, or other roto-inversion symmetries—presents both challenges and opportunities. These materials exhibit distinctive interactions with light, leading to phenomena such as molecular optical activity, chiral photonic waveguides, and valley-specific light emissions, which are pivotal in the burgeoning fields of photonic and spintronic devices. This review elucidates how novel x-ray probes can be leveraged to unravel these properties and their implications for future technological applications. A significant focus of this review is the exploration of new avenues in research, particularly the shift from conventional methods to more innovative approaches in studying these chiral materials. Inspired by structured optical beams, the potential of coherent scattering techniques utilizing twisted x-ray beams is examined. This promising direction not only offers higher spatial resolution but also opens the door to previously unattainable insights in materials science. By contextualizing these advancements within the broader scientific landscape and highlighting their practical applications, this review aims to chart a course for future research in this rapidly evolving field.

Recent years have witnessed remarkable advancements in the field of condensed matter physics, particularly in the domain of nanoscale structural and dynamic analysis. The development and refinement of synchrotron and x-ray free electron laser facilities1 have propelled coherent scattering and diffractive imaging techniques to the forefront of this research area.2–8 These techniques have opened new vistas for exploring and understanding the intricate phenomena inherent in materials science, enabling volumetric three-dimensional (3D) imaging at sub-nanometer resolutions and capturing dynamic processes over sub-millisecond timeframes.

The emergence of chiral crystals and chirality in ferroic9,10 order parameter space11 (Fig. 1), characterized by their unique optical and electronic properties due to the absence of standard inversion, mirror, or other roto-inversion symmetries, has introduced both novel challenges and opportunities.9,11,13,14,17 These materials exhibit fascinating phenomena such as molecular optical activity,18 chiral photonic waveguides,19,20 valley-specific light emissions and absorption,21,22 and the magnetochiral effect.23 Understanding these properties is not only fundamental to condensed matter physics but also crucial for the development of next-generation photonic and spintronic devices.24,25

FIG. 1.

Characterizing order parameter chirality using Twisted X-rays. Schematic depicting the ability of the use of twisted X-rays to characterize and access chirality in order parameter.12 T. Lottermoser and D. Meier, Phys. Sci. Rev. 6, 20200032 (2021). Copyright 2021 Lottermoser and Meier, licensed under a Creative Commons Attribution (CC BY) license space including but not limited to strain.13 Yang et al., Nature Commun., 12, 5292 (2021). Copyright 2021 Yang et al. licensed under a Creative Commons Attribution (CC BY) license, i.e., lattice strain and lattice chirality, polarization.14 P. Shafer, Proc. Natl. Acad. Sci., 115, 1711652115 (2018). Copyright 2018 Shafer licensed under a Creative Commons Attribution (CC BY) license, i.e., right and left-handed vortices, magnetization.15 Shao et al., Nat. Commun., 14, 1355 (2023). Copyright 2023 Shao et al. licensed under a Creative Commons Attribution (CC BY) license i.e., right and left-handed skyrmions and toroidization.11,16 A. Zimmermann, Nat. Commun., 5, 4796 (2014). Copyright 2014 Zimmermann licensed under a Creative Commons Attribution (CC BY) license, N. A. Spaldin, J. Phys.: Condens. Matter, 20, 434203 (2008). Copyright 2008 Spaldin licensed under a Creative Commons Attribution (CC BY) license.

FIG. 1.

Characterizing order parameter chirality using Twisted X-rays. Schematic depicting the ability of the use of twisted X-rays to characterize and access chirality in order parameter.12 T. Lottermoser and D. Meier, Phys. Sci. Rev. 6, 20200032 (2021). Copyright 2021 Lottermoser and Meier, licensed under a Creative Commons Attribution (CC BY) license space including but not limited to strain.13 Yang et al., Nature Commun., 12, 5292 (2021). Copyright 2021 Yang et al. licensed under a Creative Commons Attribution (CC BY) license, i.e., lattice strain and lattice chirality, polarization.14 P. Shafer, Proc. Natl. Acad. Sci., 115, 1711652115 (2018). Copyright 2018 Shafer licensed under a Creative Commons Attribution (CC BY) license, i.e., right and left-handed vortices, magnetization.15 Shao et al., Nat. Commun., 14, 1355 (2023). Copyright 2023 Shao et al. licensed under a Creative Commons Attribution (CC BY) license i.e., right and left-handed skyrmions and toroidization.11,16 A. Zimmermann, Nat. Commun., 5, 4796 (2014). Copyright 2014 Zimmermann licensed under a Creative Commons Attribution (CC BY) license, N. A. Spaldin, J. Phys.: Condens. Matter, 20, 434203 (2008). Copyright 2008 Spaldin licensed under a Creative Commons Attribution (CC BY) license.

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This review article aims to highlight the recent progress in coherent scattering and diffractive imaging techniques, particularly focusing on their application in the study of chiral materials. We begin by discussing the fundamentals of coherent x-ray scattering and diffraction (Sec. II A) and the basic concepts of twisted beams (Sec. II B). We then explore the applications and opportunities for twisted X-rays in unraveling chirality (Sec. III), followed by a review of some important techniques that can be employed using twisted X-rays. These include, but are not limited to, phase retrieval (Sec. IV A), Bragg coherent diffractive imaging (Sec. IV A 2), ptychography (Sec. IV A 3), x-ray photon correlation spectroscopy (XPCS) (Sec. IV B), and x-ray cross correlation analysis (Sec. IV C). Furthermore, we discuss the theoretical development of twisted x-ray diffraction (Sec. V A) and twisted x-ray dichroism (Sec. V B). Finally, we conclude with a future perspective (Sec. VI), positing the potential of using twisted x-ray beams in coherent scattering techniques as a key to unlocking higher spatial resolutions and new insights in materials science.

Coherent diffractive imaging (CDI), a sophisticated imaging method, has seen rapid advancements over the last 20 years, propelled by the evolution of high-brightness, coherent x-ray light sources. Initially conceptualized by David Sayre in 1952,26 CDI's foundational idea was that capturing diffraction intensity information beyond Bragg peaks could yield enough data for unique crystal structure reconstruction. Sayre expanded on CDI theory in 1980,27 demonstrating that diffraction patterns in isolated, non-periodic samples are not confined to Bragg peaks but exhibit a weak, continuous nature. This revelation expanded x-ray crystallography's scope to include non-periodic sample imaging. However, transitioning from theory to practice took nearly two decades, fraught with challenges in both hardware and software domains. Conducting a CDI experiment Fig. 2 necessitates a high-brightness, coherent x-ray source,29–31 essential due to the weak diffraction patterns non-periodic samples produce, which require high-sensitivity, broad dynamic range, and low-noise x-ray detectors. CDI, an indirect imaging technique, relies on phase retrieval algorithms to reconstruct sample structures from these patterns. The synthesis of third-generation synchrotron radiation facilities, silicon-based high-sensitivity detectors, and advancements in computer technology eventually culminated in the first successful CDI experiment.2 

FIG. 2.

Experimental configurations for x-ray coherent diffractive imaging. (a) Plane-wave CDI, in which a coherent planar beam of X-rays is incident on the sample. (b) Fresnel CDI, in which a coherent phase-curved beam created by a Fresnel zone plate is incident on the sample. An order-sorting aperture eliminates the unwanted diffracted orders. A beam stop prevents the undiffracted order from passing through the order-sorting aperature. (c) Bragg CDI, in which a nanocrystal is illuminated and the detailed structure in the Bragg diffraction spots is used to recover information about the shape and strain distribution within the crystal. (d) Scanning diffraction microscopy, in which a finite beam probe is scanned across the sample and the diffraction pattern observed at each beam position. The finite probe may be formed using a focusing optic such as a Fresnel zone plate. Reproduced with permission from Chapman et al.28 H. N. Chapman, Nature Photonics, 4, 240 (2010). Copyright 2010 Chapman, licensed under a Creative Commons Attribution (CC BY) license.

FIG. 2.

Experimental configurations for x-ray coherent diffractive imaging. (a) Plane-wave CDI, in which a coherent planar beam of X-rays is incident on the sample. (b) Fresnel CDI, in which a coherent phase-curved beam created by a Fresnel zone plate is incident on the sample. An order-sorting aperture eliminates the unwanted diffracted orders. A beam stop prevents the undiffracted order from passing through the order-sorting aperature. (c) Bragg CDI, in which a nanocrystal is illuminated and the detailed structure in the Bragg diffraction spots is used to recover information about the shape and strain distribution within the crystal. (d) Scanning diffraction microscopy, in which a finite beam probe is scanned across the sample and the diffraction pattern observed at each beam position. The finite probe may be formed using a focusing optic such as a Fresnel zone plate. Reproduced with permission from Chapman et al.28 H. N. Chapman, Nature Photonics, 4, 240 (2010). Copyright 2010 Chapman, licensed under a Creative Commons Attribution (CC BY) license.

Close modal

Photons, exhibiting wave-particle duality, are pivotal in applications like sensing, imaging, and quantum communication. Their ability to traverse long distances with minimal loss and their immunity to decoherence in free space make them particularly valuable.32 A unique characteristic of photons is their orbital angular momentum (OAM).33 In quantum terms, OAM manifests in discrete increments of , where is an integer that, in principle, can reach unlimited values. This allows for the encoding of high-dimensional quantum information within the OAM of single photons.

The electromagnetic field of a Laguerre–Gauss (LG) mode, denoted as LG p, aptly represents a photon endowed with an OAM of . Figure 1 displays experimental captures of single photons with varying OAM magnitudes. LG modes are distinguished by their spatial phase patterns, specifically, a helical phase that encircles the propagation axis times, leading to a phase singularity at the beam's heart. Consequently, the intensity profile of an LG mode exhibits a characteristic donut shape. For single photons, this intensity distribution reflects the probability of photon detection at specific locations.

Utilizing a triggered single-photon camera to capture numerous heralded photons,34 the intensity profile of the corresponding LG mode becomes apparent, as illustrated in the filmstrips of Fig. 3. A twisted wave function, represented as ψ = exp ( i ϕ ), where ϕ is the azimuthal angle in cylindrical coordinates, serves as an eigenstate of the orbital angular momentum (OAM) operator L z = i ϕ, possessing an eigenvalue . Consequently, a particle's spiraling wavefront endows it with an OAM of per photon around its propagation axis. The vortex beam is characterized by its topological charge , defined as = 1 2 π C θ ( r ) · d r, where θ symbolizes a phase distribution and C denotes a closed loop encircling the propagation axis. The emergence of vortex beams in visible light generated in free space35,36 has spurred extensive interest across various research domains, including super-resolution microscopy,37 optical tweezers,37 telecommunications,38 and quantum information processes.34 Our focus here is on the potential of twisted x-ray photons produced via refractive and diffractive optical elements, aiming to pioneer novel coherent twisted x-ray diffraction imaging methodologies. Notably, groups in Berkeley39 and other institutions have achieved significant strides in the development of soft x-ray vortex beams.

FIG. 3.

Orbital angular momentum (OAM) of twisted photons. The helical phase structure e i l ϕ imparts a quantized l amount of OAM. (a) Propagating phase profile of a Twisted beam carrying a charge of + l and the modeled intensity profile for a twisted optical beam. (b) Propagating phase profile of a twisted beam carrying a charge of l .

FIG. 3.

Orbital angular momentum (OAM) of twisted photons. The helical phase structure e i l ϕ imparts a quantized l amount of OAM. (a) Propagating phase profile of a Twisted beam carrying a charge of + l and the modeled intensity profile for a twisted optical beam. (b) Propagating phase profile of a twisted beam carrying a charge of l .

Close modal

1. Laguerre–Gaussian modes of twisted light and the orbital angular momentum

The foundation of Laguerre–Gaussian (LG) modes of twisted light and the notion of orbital angular momentum (OAM) in light is revisited herein. Maxwell's equations describe electromagnetic (EM) waves, leading to the wave equation in three dimensions as
(1)
where 2 signifies the Laplace operator, k represents the wave number, and ψ ( r ) can be interpreted as either the electric or magnetic field component when multiplied by a polarization unit vector. Within cylindrical coordinates ( ρ , ϕ , z ), ρ refers to the radial distance, ϕ is the azimuthal angle, and z indicates the longitudinal axis.
Considering the propagation of an x-ray LG wave along the z-axis, as depicted in Fig. 1 where k z ̂, the wave function ψ ( r ) is expressed as u ( ρ ) exp ( ikz ). Thus, we reformulate Eq. (1) to accommodate propagation along the z-axis
(2)
where T 2 = 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 ϕ 2 is the Laplacian's transverse part. Under the paraxial approximation,
(3)
the term 2 u z 2 in Eq. (1) can be neglected, yielding the simplified form of the LG mode as follows:
(4)
with w ( z ) = w 0 1 + ( z z R ) 2 as the beam's width, w0 being the waist, and zR is the Rayleigh length; R(z) is the curvature radius of the wavefronts, defined by z ( 1 + ( z R z ) 2 ). The phase term known as the Gouy phase, denoted as ψ ( z ), and the normalized coefficient Clm linked with the Laguerre function. The intensity profile | u ( ρ , ϕ , z ) | 2 exhibits circular symmetry and is primarily a function of two integer parameters l and m, yet it remains unaffected by the polarization vector E . Also, for this film approximation we assume small z and ψ ( z ) = 0. These assumptions simplify the mode profile to
(5)
where the polarization vector of circularly-polarized beam is defined as ε = ε x + i ε y, allowing us to describe the electric field E ( r , t ) at the sample plane as
(6)
where ε x and ε y are polarization vectors along the x- and y-axes, corresponding to the [100] and [010] pseudo-cubic (p.c.) directions, respectively. The parameters m ϕ and σ define the phase twist of the field and the handedness of the polarization, symbolizing orbital and spin angular momentum; E0, ω, and w are the field's amplitude, the light's angular frequency, and the beam's radius.
We also draw a parallel between the angular momentum of classical objects, L = r × p , and the total angular momentum of an electromagnetic (EM) wave, expressed as
(7)
with the adoption of the Coulomb gauge · A = 0. The total angular momentum L tot comprises two components: the spin angular momentum (SAM) L s associated with the polarization vector ε , represented by L s = σ for left- or right-polarized light, and the orbital angular momentum (OAM) L o. For monochromatic waves under paraxial approximation, L o is given by
(8)
Focusing on cylindrical coordinates, the z-component of L o becomes
(9)
which is the quantification of OAM for light. Specifically for an l = 0 mode with a wave profile ψ ( ρ , ϕ , z ) = u ( ρ , ϕ ) e i m ϕ, the OAM is
(10)

As we delve deeper into materials science and condensed matter research, the merits of traditional imaging techniques, such as coherent x-ray diffraction (CXD), are becoming increasingly overshadowed by their limitations. One particularly vexing challenge is the inability of traditional x-ray diffraction (XRD) techniques to distinguish between randomly-oriented enantiomers. This limitation was historically circumvented by either combining XRD data with chiral-sensitive signals like circular dichroism (CD) or relying on oriented samples.

The advent of ultrafast coherent x-ray pulses in synchrotrons and XFELs has, however, paved the way for innovative solutions. These advancements enable the generation and manipulation of chiral X-rays within a coherent diffraction framework. By leveraging the three-dimensional and nanoscale spatial resolution capacities of CXD techniques, such as Bragg CDI and ptychography, we can now explore the volumetric chirality in materials, unveiling insights never before accessible.

The intricacies of chirality are foundational in both organic and inorganic realms. For instance, nature's inherent selectivity is evident when molecules comprised exclusively of l-amino acids metabolize only d-glucose. This inherent chirality transcends biological systems. In functional materials, chirality manifests in ferromagnets, ferroelectrics, and the molecular orientation within liquid crystals. Designer metamaterials have even exhibited phenomena like the photonic spin Hall effects, illustrating engineered structures that display handed responses to optical stimuli.

Yet, with chiral crystals (ref Fig. 4) and their multifaceted interactions with light, traditional coherent imaging tools falter. This is where the twisted nature of x-ray beams presents a promising avenue. Drawing from the potential of structured optical beams, twisted x-rays offer the possibility to navigate the complexities of chiral materials and other elusive phenomena.

FIG. 4.

The four variants of Japanese twins of quartz that could carry both R and L cells in one crystal. Adapted from Ref. 40. Such domain structure (or twin structure) can modify the physical properties (e.g., optical activity) of the crystal significantly.

FIG. 4.

The four variants of Japanese twins of quartz that could carry both R and L cells in one crystal. Adapted from Ref. 40. Such domain structure (or twin structure) can modify the physical properties (e.g., optical activity) of the crystal significantly.

Close modal

This shift toward twisted x-rays is not merely experimental but an essential evolution in our scientific toolkit. As the domains of materials science and condensed matter physics continue to expand, our investigative tools must adapt in tandem. Harnessing twisted x-ray beams could represent the next scientific frontier, driving breakthroughs in our understanding of complex systems and propelling the advancement of cutting-edge technologies.

The diverse electronic characteristics inherent to chiral structures and crystals have attracted significant attention. For instance, skyrmions are supported by chiral magnets, a phenomenon elucidated by Mühlbauer et al.41 Simultaneously, the exploration of strongly correlated physics and the discovery of superconductivity has been enabled by twisted moiré lattices in 2D homo and hetero structures.42,43

Chiral metals have been observed to exhibit non-local and non-reciprocal electron transport, while chiral semiconductors display a range of phenomena including optical activity, magnetochiral dichroism, the hosting of Kramers–Weyl fermions, and a quantized circular photo galvanic effect.44 

These unique properties have inspired the conceptualization of new electronic devices, such as chiral spintronics and twistronics. These are poised to tackle the challenges anticipated for silicon-based devices. The complexity of chiral materials is further amplified by the presence of enantiomeric pairs, adding an extra layer of complexity to their structures. For instance, it has been found that a singular piece of chiral sample often incorporates vast domain structures, inclusive of the chiral structure and its mirrored version.

Furthermore, improper ferroelectricity (or trimerization) in magnetic oxides, particularly hexagonal manganites, culminates in a network of interlinked structural and chiral magnetic vortices and skyrmions. These can incite domain wall magnetizations, which are absent in the bulk. In the context of twisted moiré lattice, spatial inconsistencies in twisted angles have been identified as contributing to variable device performance.45 

Figure 5 illustrates various manifestations of chirality in various order parameter space in ferroics: namely, structural lattice distortions, ferroelectric polarization texture, and multiferroic. The intricate textures and structures revealed in these systems are indicative of the complex underlying physical processes that give rise to multiferroic behavior. Chirality, a lack of mirror symmetry, is a critical feature in these materials and can lead to novel physical properties. Twisted X-rays, or X-rays with orbital angular momentum, can provide a unique investigative tool for probing these chiral structures. Due to their helical phase fronts, twisted X-rays interact differently with materials depending on their chirality. This interaction can be used to distinguish between left-handed and right-handed structures in the material, which is not possible with conventional X-rays. Such sensitivity to chirality makes twisted X-rays an excellent candidate for exploring the chiral properties of multiferroics. They can be used to map out the electric polarization textures, structural distortions, and spin magnetic tubes with unprecedented detail, potentially unveiling new insight into the coupling mechanisms between the electric and magnetic orders in these materials.

FIG. 5.

Chirality in Multiferroics. This figure presents a self-organized array of electric polarization textures forming chiral polar vortices in ferroelectrics.46,47 Z. Liu, Phys. Rev. Appl., 8, 034014 (2017). Copyright 2017 Liu, licensed under a Creative Commons Attribution (CC BY) license, S. Tang, Nature, 568, 368 (2019). Copyright 2019 Tang, licensed under a Creative Commons Attribution (CC BY) license, structural distortions and trimerization observed in multiferroics.48 Karpov, Phys. Rev. B, 100, 054432 (2019). Copyright 2019 Karpov, licensed under a Creative Commons Attribution (CC BY) license, and spin magnetic tube textures in multiferroics.49,50 M. Birch, Nat. Commun., 11, 1726 (2020); licensed under a Creative Commons Attribution (CC BY) license, Y. Tokura and N. Kanazawa, Chem. Rev., 121, 33164494 (2021); licensed under a Creative Commons Attribution (CC BY) license.

FIG. 5.

Chirality in Multiferroics. This figure presents a self-organized array of electric polarization textures forming chiral polar vortices in ferroelectrics.46,47 Z. Liu, Phys. Rev. Appl., 8, 034014 (2017). Copyright 2017 Liu, licensed under a Creative Commons Attribution (CC BY) license, S. Tang, Nature, 568, 368 (2019). Copyright 2019 Tang, licensed under a Creative Commons Attribution (CC BY) license, structural distortions and trimerization observed in multiferroics.48 Karpov, Phys. Rev. B, 100, 054432 (2019). Copyright 2019 Karpov, licensed under a Creative Commons Attribution (CC BY) license, and spin magnetic tube textures in multiferroics.49,50 M. Birch, Nat. Commun., 11, 1726 (2020); licensed under a Creative Commons Attribution (CC BY) license, Y. Tokura and N. Kanazawa, Chem. Rev., 121, 33164494 (2021); licensed under a Creative Commons Attribution (CC BY) license.

Close modal

As a consequence, understanding the local structures and chemical specifics of chiral crystals and structures becomes critical to the establishment of structure-chemistry-property relations within chiral systems. This underscores the necessity for developing high-resolution, nanoscale probes capable of direct and noninvasive investigation of these chiral crystals across spatial, energy, momentum, and time-domain.

Recent experiments have uncovered the presence of helical dichroism51 in the visible light regime using twisted photons, specifically dichroism occurring when light possesses orbital angular momentum (OAM) but lacks spin angular momentum (SAM). Recent simulations modeling the helical dichroism for x-rays in a chiral system show promising differential scattering in the said chiral samples giving a non-zero asymmetry ratio (see Fig. 6). This finding implies the possibility of a comparable effect in x-ray photons. Indeed, recent theoretical studies have accounted for multipole contributions to all orders, predicting the potential for helical dichroism to manifest in chiral organic molecules, such as cysteine, linked to core electron levels at the sulfur K-edge.51 

FIG. 6.

Theoretical prediction of helical x-ray dichroism. (a) When s = 0 and = 1 , 2 , or 3, the non-zero value of the asymmetry ratio indicates the presence of x-ray helical dichroism in a chiral system. (b) When s = 1 and = 1 , 2 , or 3, circular helical dichroism is either intensified or diminished. Adapted from Ref. 52. Reprinted (adapted) with permission from Ye et al., “Probing molecular chirality by orbital-angular-momentum-carrying x-ray pulses,” J. Chem. Theory Computat. 15, 4180–4186 (2019). Copyright 2019 American Chemical Society.

FIG. 6.

Theoretical prediction of helical x-ray dichroism. (a) When s = 0 and = 1 , 2 , or 3, the non-zero value of the asymmetry ratio indicates the presence of x-ray helical dichroism in a chiral system. (b) When s = 1 and = 1 , 2 , or 3, circular helical dichroism is either intensified or diminished. Adapted from Ref. 52. Reprinted (adapted) with permission from Ye et al., “Probing molecular chirality by orbital-angular-momentum-carrying x-ray pulses,” J. Chem. Theory Computat. 15, 4180–4186 (2019). Copyright 2019 American Chemical Society.

Close modal

Helical dichroism, already established in the visible regime, has thus intensified the demand for similar probes in the x-ray regime. Such a probe, when coupled with powerful x-ray techniques like Bragg coherent diffraction imaging (BCDI), could potentially lead to a more profound understanding of light-matter interactions at the nanoscale, along with insights into chiral electronic and structural properties of chiral systems. Certain groups, such as Rouxe et al.53 have demonstrated helical dichroism in the hard x-ray regime. One of the main challenges in advancing structured x-ray scattering and imaging techniques lies in the development of refractive and diffractive optical elements required to produce structured x-ray beams. It is noteworthy that groups in Berkeley39 and other facilities53 have made substantial progress in developing these optical elements and alternative means to produce vortex beams.53–58 

Materials exhibiting chirality in their structural or electronic phases pique scientific interest due to the diverse materials physics they feature, such as the emergence of superconductivity and the occurrence of strongly correlated physics in moiré lattices. These systems hold significant technological potential, influencing a variety of fields such as functional electric and magnetic devices, spintronics, and twistronics.

The intriguing phenomena of superconductivity and correlated electron physics, for example, can be examined through the unique platform provided by moiré patterns formed from bilayer graphene or other similar two-dimensional materials. The emergence of flat bands at so-called “magic” angles in twisted bilayer graphene is heavily influenced by strain, defects, and structural relaxation.59 

In a moiré superlattice, the displacement and distortion fields characterize the relative shift between the two layers with respect to a reference point. In an ideal scenario, pure twist and stretch moiré patterns would exhibit displacement fields linearly proportional to the distance from the origin and maintain constant distortion tensor components. However, in real materials, instances of pure twists or stretches are scarce. If allowed, local internal relaxations can cause atomic positions to deviate from these idealized fields.

The application of twisted x-ray beams could potentially discern between left-handed and right-handed moiré twists, while also offering the possibility to spatially resolve twist strain with 3D atomic precision. By incorporating OAM x-ray probes, current x-ray scattering techniques used in microscopy, diffraction, and spectroscopy could potentially experience a revolutionary shift. Yet, the imperative need to develop a thorough understanding of both elastic and inelastic scattering theory in the context of twisted x-ray remains.

The crux of CXDI lies in its complex phase retrieval algorithms that serve to convert the coherent diffraction intensities present in reciprocal-space into a complex wavefunction in the specimen's real-space, in adherence to a strict oversampling criterion. Phase retrieval from coherent x-ray diffraction (CXD) patterns is a complex and challenging task but has great importance in various scientific and engineering fields, especially in physics, materials science, and imaging.

Coherent diffraction imaging (CDI) is a lensless imaging technique. Instead of traditional optics, it relies on the principles of diffraction and the properties of coherently scattered waves:

  • Coherent Illumination: The specimen must be illuminated by a spatially coherent beam ensuring the wavefront consistency across the sample.

  • Far-Field Diffraction: The pattern is a far-field diffraction pattern, recorded on a 2D detector where the diffracted waves are approximately parallel.

  • Phase Retrieval: CDI uses iterative algorithms to retrieve the lost phase information from the measured diffraction pattern intensities. The iterative process involves forward and inverse Fourier transforms to optimize the phase.

  • Oversampling Criterion: The diffraction pattern should be recorded with spatial resolution finer than half the pixel size of the real-space image, ensuring there's sufficient information for phase retrieval.

  • Uniqueness and Constraints: To resolve potential ambiguities in phase retrieval, additional constraints are applied, such as setting the object to be real and positive or utilizing a support constraint.

  • Exit Wave Reconstruction: The reconstructed exit wave gives the image of the sample.

  • Limitations and Extensions: CDI generally requires isolated samples. Extensions like ptychography allow imaging of extended objects by using multiple overlapping diffraction patterns.

In CXD, the intensity of the diffracted X-rays is measured, but the phase information is lost. Recovering this phase is vital for the real-space image reconstruction of the sample. The phase retrieval processes aim to recover this missing information from the intensity measurements, addressing the so-called phase problem of crystallography. This problem refers to determining the wave's phase from a diffraction pattern when only the intensity is known. Solving the phase problem necessitates the fulfillment of the oversampling condition.60,61 Given an N × N pixel sampled amplitude in reciprocal space, a set of N2 equations is obtained, and both the amplitude and phase must be found.

1. Holography

Holography62 is a technique that uses the interference pattern between a reference wave and a scattered wave to recover the phase information. It often requires a known reference wave or a separate measurement of the reference wave.

2. Bragg coherent diffractive imaging

The foundation of BCDI lies in the understanding of x-ray diffraction. In 1912, Max von Laue first demonstrated that crystals diffract x-rays.63 This led to the formulation of Bragg's Law by Bragg and Bragg.64 With the advent of the 21st century, advanced x-ray sources, such as synchrotrons, emerged, producing highly coherent beams vital for new imaging techniques, including BCDI.65 

BCDI operates on the principle that a spatially coherent x-ray beam illuminating a specimen produces information-rich diffraction patterns that can be translated into real-space images. This methodology is an offshoot of the overarching principles of coherent diffractive imaging and the iterative phase retrieval process, pioneered by Sayre in 195326 and subsequently enhanced by Miao in 1999.31 

Recorded diffraction patterns in reciprocal space, particularly in BCDI experimental setups, are primarily devoid of content, paving the way for in-depth analysis of individual nanoparticles and grains. Over the past decade, BCDI's evolution has positioned it as a powerful tool for delving into complex questions in materials research and condensed matter physics.

  • Iterative Phasing Algorithms: In this context, iterative phasing algorithms are computational methods used to solve the phase problem. These are iterative methods that gradually update the phase and intensity information until they converge to a solution. Iterative phasing algorithms operate by iteratively refining an initial guess for the phases, through alternating between the real and reciprocal spaces. They apply some constraints in each iteration to gradually improve the phase estimates. These constraints are based on the physical properties of the structure, such as its non-negativity (electron density cannot be negative), support constraint (electron density is zero outside the molecule), and the measured magnitudes of the diffracted beams. Various iterative phasing algorithms have been developed over the years. Among them are:

    • Gerchberg–Saxton algorithm:66 This is one of the earliest iterative phase retrieval algorithms. It iteratively updates the phases in Fourier space and applies constraints in real space.

    • Fienup's algorithms:67–71 These include hybrid input-output (HIO) and error reduction (ER) algorithms. They are improvements upon the Gerchberg–Saxton algorithm with additional feedback mechanisms for improving convergence.

    • Charge flipping algorithm: This algorithm, used in crystallography, involves flipping the sign of the electron density in regions where it is below a certain threshold.

    • Difference map algorithm:72 This algorithm uses two projection operations and a parameter that controls the balance between them to find a solution.

    • Shrinkwrap algorithm: This improves upon other methods by dynamically updating the support constraint.

    • Hybrid methods: Some methods combine experimental measurements with prior information or constraints about the sample to aid in phase retrieval. Examples include the difference map algorithm and the relaxed averaged alternating reflections (RAAR) algorithm.73–75 

  • Regularization techniques: Regularization methods involve introducing additional constraints or regularization terms to stabilize the phase retrieval process76,77 and prevent overfitting. Total variation (TV) regularization and Tikhonov regularization are examples of regularization techniques used in phase retrieval.

The BCDI technique is showcased using a BaTiO3 nanocrystal from a recent study on topological vortex dynamics (refer to Fig. 7).3,78 Notably, the physical density of the crystal was majorly uniform, with deviations in areas exhibiting topological defects as projected by phase field simulations. An observable imaginary component, largely tied to an internal ferroelectric displacement field, was also present. The ensuing figures contrast the reconstructed ferroelectric displacement fields with a theoretical model rooted in Landau theory.3 Discrepancies observed in the tetragonal structural phase underline BCDI's transformative role in solving intricate challenges in materials science and condensed matter physics.

FIG. 7.

3D reconstructions of ferroelectric polarization. (a), (b), and (c) depict the domain distribution and the ferroelectric polarization within a single nanocrystal under varying applied electric fields. (d) Phase field simulation reveals the structural phases within the nanocrystal. (e) Simulated behavior of the toroidal moment and axial polarization under different electric fields. The scale bars correspond to 60 nm.3 Reprinted with permission from Karpov, Nat. Commun., 8(1) 2017. Copyright 2017 licensed under a Creative Commons Attribution (CC BY) license.

FIG. 7.

3D reconstructions of ferroelectric polarization. (a), (b), and (c) depict the domain distribution and the ferroelectric polarization within a single nanocrystal under varying applied electric fields. (d) Phase field simulation reveals the structural phases within the nanocrystal. (e) Simulated behavior of the toroidal moment and axial polarization under different electric fields. The scale bars correspond to 60 nm.3 Reprinted with permission from Karpov, Nat. Commun., 8(1) 2017. Copyright 2017 licensed under a Creative Commons Attribution (CC BY) license.

Close modal

A comprehensive review of significant iterative reconstruction algorithms proposed for phase retrieval is discussed in Ref. 79. Among these, the most adopted and successful is the HIO + ER-meta algorithm, which marries NHIO repetitions of the hybrid input output (HIO) algorithm with NER repetitions of the error reduction (ER) algorithm as a follow-up.68–70 

3. Ptychography

Ptychography80–83 is an advanced coherent diffractive imaging technique that involves scanning a beam across the sample in overlapping regions. This approach can provide high-resolution phase and amplitude information of the sample. The two widely used examples of ptychography are:

  • Forward geometry ptychography: Ptychography, a technique was initially proposed to address the phase problem through the recording of overlapping Bragg reflections in electron diffraction experiments utilizing convergent beams. Following a decade of coherent X-ray diffraction imaging (CXDI) method developments, Rodenburg and his team84 integrated scanning transmission X-ray microscopy (STXM) and CXDI for the examination of large samples, resulting in the modern form of ptychography. The “overlap constraint”81,84 was transferred to real-space, forming a key part of ptychographic iterative algorithms. Various references provide comprehensive explanations of X-ray coherence's mathematical and experimental notions and the function of overlap and modulus constraints in ptychography.

    The initial batch of iterative algorithms implemented in ptychography included the ptychographic iterative engine (PIE) and extended-PIE categories. The principles underlying PIE algorithms are quite straightforward, with updates in real-space occurring sequentially. These algorithms are particularly effective when the probe remains stable and unchanged. Another set of ptychographic algorithms, incorporating the original difference-map method,82 enable simultaneous updates of the object and probe. Consequently, these algorithms provide a more mathematically rigorous approach and can aggressively achieve data inversion convergence.

    The relaxed averaged alternating reflections (RAAR) algorithms offer a more sophisticated real-space overlap constraint compared to the difference-map, thereby maximizing the rate of data reconstruction convergence in theory. Ptychography's ability to handle data redundancies allows its algorithms to consider experimental imperfections like varying diffraction intensities, background noise, and positional inaccuracies.

    For instance, Maximum-likelihood, an extension of the original PIE and difference-map, assimilates real experimental features, such as Gaussian or Poisson photon counting and associated photon noise, into the ptychographic iterative algorithms to enhance data inversion quality. Additionally, a sophisticated class of algorithms, termed Augmented projections ptychographic algorithms, accommodate fluctuating intensities, positioning mistakes, poor calibration due to multiplexed illumination, and unknown background offsets in experimental data. This method shows promise for high-quality ptychographic reconstructions, especially when dealing with experimental data with undesirably low-counting statistics and data acquisition defects.

  • Bragg ptychography: Ptychography, when applied in Bragg geometry, has shown promise for potential applications in both compact and extended single crystalline nano-structures. This technique particularly emphasizes surface sciences and thin film samples, and it enables the reconstruction of three-dimensional atomic displacement fields within a single crystal object.

    This approach extends from the widely recognized method known as Bragg Coherent Diffraction Imaging (Bragg CDI). Current state-of-the-art instrumentation at globally accessible beamlines, such as 34-ID-C at the Advanced Photon Source (APS), ID-01 and ID-13 at the European synchrotron radiation facilities, and I13 at diamond light source, have facilitated the study of isolated single crystalline structures using Bragg CDI.

    To further progress this technique, Bragg-geometry ptychography is viewed as superior to the traditional Bragg CDI. This is due to the integration of overlapping scan positions into the coherent diffraction intensities in two- or three-dimensional datasets. Given that the experimental setups are sufficiently stable, a field-of-view of any size can be measured for extended samples. However, Bragg ptychography does have a limitation—the structures of the probes need to be known beforehand, as the current method does not support simultaneous reconstructions of both the sample and the probe.

    Bragg-geometry ptychography is considered more robust for resolving highly strained structures due to the inherent overlapping in real-space during data acquisition and algorithm execution. While numerous attempts have provided some solutions to these issues in traditional Bragg CDI, these challenges still persist, as highlighted in recent studies.85 

It is important to note that phase retrieval is an ill-posed problem, meaning there could be multiple solutions that fit the intensity data equally well. Therefore, the choice of the method and the quality of the results depend on the specifics of the experimental setup, the noise level in the measurements, and the prior information available about the sample. A prospective solution to this inherent issue could be realized through the integration of scanning transmission x-ray microscopy (STXM) and CXDI.

X ray photon correlation spectroscopy (XPCS)86,87 is an x-ray method that evaluates the dynamism of a specimen by gauging its time correlation functions. This technique harnesses the coherence of the X-rays generated by synchrotron radiation facilities, forming speckle patterns as a result of the interaction of the coherent x-ray beam with the sample,collected over a time period. With a time resolution that spans from 1 s to 1  μs, the intriguing characteristics of the specimens, such as diffusion, dynamical disparities, and structural relaxation processes, can be deduced by scrutinizing the XPCS speckle patterns. Depending on the structural and dynamic features of the specimen, XPCS can compute two-time or even higher-order correlation functions. To date, this method has been employed to investigate Brownian motion in gold colloids suspended in glycerol,88 slow dynamics89 and capillary wave dynamics in polymers,90 nanoscale domain wall fluctuations in antiferromagnetic crystals,5 domain wall dynamics in ferroelectric crystals,91 atomic diffusion in metallic alloys,92 among other things. Historically, the application of XPCS was predominantly focused on understanding the slow relaxation dynamics often observed in glasses. These dynamics exhibit time scales that can span from several milliseconds up to extended durations like minutes or even hours. A substantial factor contributing to this limitation has been the modest coherent flux provided by third-generation synchrotron sources. Concurrently, the technology underpinning area x-ray detectors often imposed a bottleneck due to slow readout durations, although some recent technological strides in detectors and data compression have offered a respite.93–99 An effective strategy to sidestep these two-dimensional detector constraints involves employing point (0D) detectors. In specific contexts where strong scattering is evident, like within liquid crystal dynamics or liquid surface studies, point detectors have achieved a remarkable temporal resolution on the microsecond scale. Interestingly, this timescale aligns closely with the typical repetition rates observed in many synchrotron storage rings, usually ranging between 10 and 100 MHz. Consequently, the intricate physics and materials science phenomena occurring under the sub-millisecond regime have largely remained elusive to XPCS. This encompasses pivotal queries in the realm of magnetism, where spin reorientation events occur within time frames as short as tens of picoseconds, and also involves lattice dynamics, which cover activities in ferroelectrics and structural phase transitions that have time scales stretching from nanoseconds to microseconds. A significant portion of these collective behaviors, which include phase separation, domain formulation, and their associated fluctuations, likely exhibit slower dynamics compared to the time scales observed at the granular level, such as those of individual spins or unit cells. This generates a notable gap in our understanding, particularly in the timescale window of 1 ps to 1 ms, which is challenging to probe with several scattering techniques, be they time-domain oriented like XPCS or rooted in the frequency domain, akin to inelastic x-ray scattering. Unveiling the mysteries of these nanoscale events is an aspiration that seems attainable with the forthcoming wave of synchrotron light sources and possible instrumentation relying providing coherent twisted x-ray photons.

The technique known as x-ray cross correlation analysis (XCCA) utilizes coherent diffractive imaging (CDI) to examine the speckle patterns. This method facilitates the exploration of structures within single particles, including those within disordered and semi-ordered systems, such as alloys and colloidal glasses. As depicted in Fig. 8(a), a typical XCCA experiment involves the examination of the scattered field intensity in reciprocal space, denoted as I(q, t). This intensity is analyzed for two distinct scattering vectors q at separate time points t1 and t2,
(11)
where C ( q 1 , q 2 , t 1 , t 2 ) represents the cross-correlation function of intensity.
FIG. 8.

Cross-correlation analysis: (a) Analysis of the cross-correlation function C ( q , Δ ϕ ) parameters. (b) Application of XCCA to a hypothetical system consisting of five hexagonally clustered spherical entities, with rnn indicating the inter-particle separation, denoted by an arrow. (c) Deduction of the coherent scattering profile from the proposed structure. (d) Variation of the Fourier coefficients C l ( q m n ) with respect to l values, computed for q m n = 2 π r m n. Reproduced with permission from Sheyfer,100 Sheyfer, Ph. D. thesis, 2017. Copyright 2017 licensed under a Creative Commons Attribution (CC BY) license.

FIG. 8.

Cross-correlation analysis: (a) Analysis of the cross-correlation function C ( q , Δ ϕ ) parameters. (b) Application of XCCA to a hypothetical system consisting of five hexagonally clustered spherical entities, with rnn indicating the inter-particle separation, denoted by an arrow. (c) Deduction of the coherent scattering profile from the proposed structure. (d) Variation of the Fourier coefficients C l ( q m n ) with respect to l values, computed for q m n = 2 π r m n. Reproduced with permission from Sheyfer,100 Sheyfer, Ph. D. thesis, 2017. Copyright 2017 licensed under a Creative Commons Attribution (CC BY) license.

Close modal

Kam first introduced the concept of XCCA, also known as fluctuation scattering, in 1977.101 It was originally developed as a method to deduce the structure of macromolecules in a solution using visible light. This approach assumes that the distribution of particles is static over the duration of the scattering of light by identical, unoriented particles. The time frame for this scattering is less than that required for the particles to reorient themselves. By correlating the changes in angular intensity detected, and complementing this with radial direction variations analysis (akin to small-angle x-ray scattering), it is feasible to deduce information about the structure of particles. Following this methodology, Clark advanced the technique to analyze the local order within dense phases of two-dimensional systems, such as particle monolayers approximately 230 nm in size,102 and the spatial correlation of particles in three-dimensional colloidal systems.103 Advancements in x-ray and synchrotron radiation (SR) technology have facilitated the application of cross correlation methodologies to coherent x-ray scattering experiments. Initial studies employing XCCA have demonstrated its capability to uncover underlying symmetries, which are not readily apparent in the disordered structure of amorphous samples, such as densely packed colloidal glasses.104 The foundational principles and computational algorithms relevant to XCCA have been thoroughly articulated in the literature.105–107 

XCCA research primarily progresses along two investigative paths: (i) the examination of structural characteristics within partially ordered systems and (ii) the elucidation of single particle structures.

For systems lacking order, XCCA has been instrumental in probing the local organization and structural patterns of particle assemblies within both two-dimensional and three-dimensional colloidal matrices. This includes, but is not limited to, assessing the effects of pressure on such systems,108 investigating the interplay of particle interactions in thin colloidal layers via small-angle x-ray scatterometry,109 quantifying order parameters and molecular linkages within liquid crystals,110 examining the structure of nanocrystalline superlattices,111 identifying structural irregularities within polymeric materials,112 and exploring the local ordering and symmetry within magnetic domains.113 The versatility of x-ray cross correlation analysis (XCCA) is evident in its application to both advanced synchrotron radiation (SR) facilities and x-ray free-electron lasers (XFEL). The XFEL, in particular, emits ultrashort x-ray pulses of exceptional brightness, capturing transient “snapshots” of randomly oriented particles.

In one methodology, the cross correlation function is employed as a constraint within the iterative algorithm for phase retrieval. Utilizing this technique, the structural details of individual polystyrene dimers were successfully determined at the XFEL LCLS,114 and the arrangement of 2D gold nanoparticle arrays was elucidated using the SR source SLS.115 An alternative strategy involves directly approximating the real-structure model by integrating the cross correlation function to align with experimentally observed scattering patterns.

XCCA enables the execution of time-resolved measurements critical for deciphering structure formation processes and the dynamics of phase transitions. The specific form of the cross correlation function is selected based on the study's objective and the parameters that need determination. For instance, in the investigation of static order within colloidal dispersions, the function is computed using angular intensity correlations in polar coordinates with a consistent modulus of the wave vector q where | q 1 | = | q 2 | = q
(12)

Here, q is the scattering vector situated in a plane perpendicular to the incident beam, and Δ ϕ represents the angular differential between the vectors q1 and q2, ranging from 0 to 2 π. The angular average ϕ denotes averaging over the angle ϕ along a ring with radius q.

Employing a Fourier series expansion of the correlation function C ( q , Δ ϕ ) enhances the analysis of local orientational order within a sample. This detailed study is facilitated by the mathematical properties of the function
(13)
where the Fourier coefficients are determined by
(14)

The interplay between these Fourier coefficients C l ( q ) and the structural orientational order is exemplified through the study of a two-dimensional system composed of hexagonal clusters of spherical entities. The scattering pattern derived from such a system, depicted in Fig. 8(c), allows the Fourier coefficients C l ( q ) of the correlation function C ( q , Δ ϕ ), as extracted from the scattering data, to be plotted against the l components as shown in Fig. 8(d). It is observed that coefficients with l = 6 and l = 12 are markedly prominent, mirroring the hexagonal symmetry inherent in the system's structure.

Here, we look into the detailed analysis of the coherent elastic scattering of x-ray beams, specifically those known as Bessel and Laguerre Gaussian beams, when directed at two contrasting structures: a non-chiral body-centered cubic (BCC) configuration and a chiral crystal, composed of Tellurium (Te). This exploration is set against a comparative backdrop, where the outcomes are compared with results derived from an incident plane wave radiation scenario. The primary objective here is to establish a theoretical framework indicating that coherent elastic scattering, when combined with twisted x-rays, could potentially cause a discernible alteration in the scattering cross section. This outcome is particularly significant for nanocrystals whose dimensions approach few nanometers. The implications of this theory might open up new avenues in the field of x-ray scattering and crystallography. We start our analysis with the scattering of plane wave x-ray photons from a nanocrystalline material. This is considered in the context of the form-factor approximation.116 This approximation is relevant for low-Z atoms and photon energies that are considerably greater than binding.117 In this approximation, the scattering amplitude for a crystal can be generally expressed as116 
(15)
where k i , f and ε ̂ i , f are the wave and polarization vectors of the circularly polarized incident and outgoing photons, respectively. b n indicates the position of the nth atom, and f n ( k i , k f ) is the atomic form factor, defined as117,118
(16)
Here, ρ ( r ) signifies a spherically symmetric charge distribution. As we are modeling the atom under the single-electron assumption, it allows for the decomposition of the charge distribution into a sum of individual electron contributions. To delve further, we employ the expansion of a photon plane wave with respect to spherical Bessel functions119 
(17)
where the magnitude of the photon wave vector is k = ω / c. Utilizing the above expansion in the form factor Eq. (2) and integrating over the angles, we deduce that
(18)
where the symbol ∑e represents the summation over all bound electrons and the function f ̃ l m ( q k i , k f ) is defined as
(19)

Here, f l ( r ) and g l ( r ) are, correspondingly, the large and small radial components of the Dirac wave function.

Our analysis is particularly focused on the body-centered cubic (bcc) structure, adopted by substances like chiral Tellurium, or non-chiral Lithium at room temperature. The bcc structure is a cubic lattice hosting two atoms per unit cell, essentially forming two interpenetrating simple cubic sublattices specified by a set of vectors of the form120 
(20)
where a 1 , a 2, and a 3 are the lattice vectors, whereas n1, n2, and n3 are integers. On assuming that all of the atoms in the unit cell are identical, the scattering amplitude for plane waves becomes
(21)
If we take the direction of the incident plane wave to be along the z axis, the differential scattering cross section, averaged over final polarizations, may be expressed in terms of the scattering amplitude as
(22)
where r0 is the classical electron radius and the relation | ε ̂ i · ε ̂ f | 2 = 1 + cos 2 θ k 2 for the polarization vectors can be utilized. From these expressions, it is evident that the differential cross section (8) can be effectively used to analyze the angular properties of the scattered x-ray.

After addressing the nature of x-ray scattering by plane waves, we now have framework to address the scattering by twisted x-rays from crystalline materials.

Let us quickly revisit the twisted x-rays (Bessel and Laguare Gauas) beam, characterized by a well-defined longitudinal momentum kzi, modulus of the transverse momentum ξ, photon energy ω = c k = c k z i 2 + ξ 2, as well as the projection m γ of the total angular momentum (TAM) upon its propagation (z) direction. Such a Bessel beam is uniquely identified by its vector potential, which is defined by121,122
(23)
where the amplitude a m i g ( k i ) is expressed as
(24)

From Eqs. (23) and (24), a Bessel beam can be interpreted as a superposition of plane waves with wave vectors k i, situated on a cone with an opening angle θ k i = arctan ( ξ / k z i ). For simplicity, we only consider cases where the transverse momentum of the photon is smaller than its longitudinal momentum, i.e., ξ k z i. Within this paraxial approximation, the opening angle θki is also minuscule, and the TAM projection m γ of the Bessel beam is merely the sum of the orbital angular momentum (OAM) and the helicity.122  ϕ k i represents the azimuthal angle of the plane wave vector k i, and ξ is the modulus of the transverse momentum of the twisted x-ray beam.

Since we have established that the Bessel beam can be represented as an integration of standard plane wave components with the function a m i g ( k i ), we infer that the scattering amplitude for twisted x-rays can be obtained by integrating the plane wave amplitude.116 This is given by
(25)
Performing integration over the transverse momentum using the delta function reveals that k i = ξ i. The scattering amplitude can be further simplified with the observation that e i · e f 0.5 ( 1 + i f cos θ k i ) for sufficiently small opening angles θki, with the helicities i , f of the incident and scattered photons. Integrating over the azimuthal angle ϕ k i demonstrates that the primary contribution to the amplitude Eq. (23) originates from the terms in Eq. (18) with m = m γ i, assuming the crystal to be centered on the beam axis. Eventually, we can express the scattering amplitude for twisted x-rays as
(26)
With this result in mind, when the scattered photons are detected with a polarization-insensitive x-ray detector, the differential scattering cross section from a crystalline material for twisted light is given by
(27)

In the following, we shall discuss in detail how this cross section and the scattering amplitude or structure factor depends on the helicity of twisted x-rays.

Consider an examination of plane wave scattering by a crystal. It is a well-established fact that for a specific crystalline material, the lattice sum in the scattering amplitude is maximized when the momentum transfer q, given by the difference k i k f, aligns with a reciprocal lattice vector. This phenomenon, often referred to as the Laue condition, is pivotal for observing x-ray diffraction.31 This condition is fulfilled when all phase contributions to the lattice sum, n exp [ i ( k i k f ) · b ], become integral multiples of 2 π, which results in the sum equaling a substantial count of terms.

Let us examine a Tellurium crystal as an illustrative example. We shall initially demonstrate numerically that handedness could be established unambiguously, by probing single crystals consisting only of elemental tellurium using twisted x-rays. Chirality in tellurium crystals is described by the symmetry group P 3 1 21 (a right-handed screw) or P 3 2 21 (a left-handed screw). Since there are three tellurium atoms in each unit cell, the difference between space group P 3 1 21 and P 3 2 21 is simply the stacking sequence of atomic planes along the c axis, which is opposite in each case.

Tellurium crystal has an ideal lattice parameters: a = 4.495 Å, b = 3.74 Å, c = 5.912 Å and x-ray beam of photon energy 8.04 keV, corresponding to the 1.54 Å wavelength of the Kα line in Cu, often used in laboratory settings. Storage rings available at synchrorons and XFELs have been employed to experimentally generate such twisted X-rays, using either a spiral zone plate82 or fork gatings.83 In this specific system, the Laue or Bragg condition is satisfied when the crystal is oriented at Euler angles of ϕ, θ, ψ (values to be provided) with regard to the scattering plane, defined by the vectors ki and kf.

In this configuration, diffraction conditions mandate that the plane wave differential cross section exhibits a sharp peak about the scattering angle qkf for the (002) reflection. This behavior is demonstrated by numerous studies on x-ray diffraction.123 Our numerical calculations provide considerable support for this trend, as displayed in Fig. 9.

FIG. 9.

Simulated scattering amplitudes for different handedness of Terrilium crystal. Scattering amplitude M tw P 3 1 21 ( q ) for the P 3 1 21 unit cell and M tw P 3 2 21 ( q ) for the P 3 2 21 unit cell as a function of the momentum transfer q, with different OAM indices . (a) for  = 1,2. (b) for  = 3, 10.

FIG. 9.

Simulated scattering amplitudes for different handedness of Terrilium crystal. Scattering amplitude M tw P 3 1 21 ( q ) for the P 3 1 21 unit cell and M tw P 3 2 21 ( q ) for the P 3 2 21 unit cell as a function of the momentum transfer q, with different OAM indices . (a) for  = 1,2. (b) for  = 3, 10.

Close modal

Let us further delve into the scattering characteristics of twisted x-rays. We shall examine cases where the Te nanocrystal dimension is relatively small, roughly around 2.5 nm per side. This corresponds to crystals constructed from roughly 302 atoms. Scattering amplitudes for twisted x-rays (Bessel beams) with transverse angular momentum (TAM) projections m γ = 2 and topological charge “ = 0,” “ = 1,” and “ = + 1” (displayed in Fig. 9) differ markedly from the results obtained from plane waves, = 0. These differences are particularly pronounced in the multi-peak structure of the scattering, wherein the position of peaks is highly sensitive to the OAM of the light [refer to Figs. 9(a) and 9(b) for more details].

Furthermore, the prominence of these peaks undergoes a rapid decline with the augmentation of the photon's TAM m γ. This is largely attributable to the swift reduction of twisted light intensity at the beam center, which is precisely where the crystal is situated. The dispersion of the opening angle qki scarcely affects the scattering pattern. In order to understand the dependence of the opposite handedness on the scattering amplitude, we take helical dichroism plots w.r.t q for different s (as shown in Fig. 10). There is a differential scattering for both = ± 1 where there is more scattering with = 1 having a periodic scattering in q whereas = 1 shows a higher scattering at periodic q values.

FIG. 10.

Simulated helical dichroism for Te nano-crystals for = ± 1 and = 0.

FIG. 10.

Simulated helical dichroism for Te nano-crystals for = ± 1 and = 0.

Close modal

On the other hand, when the crystal size increases substantially (approximately 14 nm or more), we observe only one pronounced maximum for twisted beams, coinciding with the scattering angle for plane waves. This particular observation is highlighted in Fig. 10, which also demonstrates the general disregard of the angular distribution of the radiated emission to the phase structure of a singular twisted beam when employing macroscopic atomic ensembles. This similar emission pattern for paraxial Bessel and plane wave incident radiation arises due to the dominating role of the lattice sum in the amplitude for a substantial number N of atoms in the crystal. This dominance leads to the suppression of all scattering cross section peaks (apart from the one dictated by the Laue condition31) by a factor of approximately 1 / N.

It is worth emphasizing that a crystal displacement of approximately 3/ξ ( 1.0 nm for the considered opening angle and energy) for m γ i = 2, or 2/ξ ( 0.7 nm) for m γ i = 1,from the central dark spot to its bright ring of the twisted beam can lead to results that resemble plane wave scattering. Such a displacement can arise from imperfections in the positioning of the crystal, which typically vary by about 20 nm in our current numerical x-ray diffraction experiments utilizing optical trapping techniques.124 In such scenarios, the influence of orbital angular momentum (OAM) on the atomic electrons becomes negligible.

Our current understanding of the atomic structure in condensed matter primarily derives from x-ray diffraction studies, focusing on the interaction between electric fields and the electron's electric charge.125 In magnetic materials, specific electrons exhibit a magnetic moment due to their spin and angular momentum, hinting at possible magnetic interactions in addition to the standard charge interactions. The concept of magnetic interactions was theoretically established in the early 20th century but only empirically demonstrated in 1981 through the pioneering work126,127 of de Bergevin and Brunel using conventional x-ray tubes.116 This groundbreaking research not only demonstrated the magnetic diffraction phenomenon but also detailed the process of interaction, delineating the polarization dependence and providing experimental data on magnetic compounds.

Despite these advancements, the relatively weak nature of magnetic scattering—significantly less intense than charge scattering—initially confined it to being a niche area in the field. This perception changed with the seminal experiments by Gibbs128,129 and colleagues on Holmium, utilizing the high brilliance of synchrotron radiation sources. These experiments highlighted the unique properties of synchrotron radiation, such as polarization characteristics and tunability, paving the way for new directions in magnetic x-ray investigations. Following studies have expanded on this foundation,130 particularly in terms of resonant enhancement of magnetic signals and the distinct separation of spin and angular momentum contributions.

Today, the use of synchrotron radiation techniques in the analysis of magnetic materials is well-established.116 These methods predominantly employ incoherent probes to measure an aggregate of local magnetic properties. Techniques such as Kerr microscopy, Faraday effect measurements, and x-ray magnetic dichroism130 are particularly noteworthy. Both Kerr and Faraday effects involve the alteration of the polarization plane of electromagnetic waves in interaction with magnetic substances, whereas magnetic circular dichroism differentiates the absorption of right- and left-circularly polarized X-rays in magnetic materials. These techniques collectively enable the assessment of both orbital and spin contributions to magnetic moments with specificity to elements and sites. Absorption methods become more localized microscopic probes when aligned with spin-resolved x-ray absorption fine structure analysis, providing insights into the local environments with an emphasis on magnetic interactions. Furthermore, coherent probes like magnetic x-ray diffraction and nuclear resonant scattering have been instrumental in offering detailed information about magnetic structures, phase transitions, and correlation lengths. Such techniques have significantly contributed to our understanding of magnetic thin films and hyperfine fields.

In the subsequent subsections, we will delve deeper into magnetic x-ray diffraction, focusing particularly on the nuances between plane and twisted x-ray interaction. We will introduce the cross section for non-resonant and resonant magnetic x-ray scattering, discuss experimental considerations, present case studies from various material types, and summarize key features of magnetic x-ray scattering. This will include a comparison and analysis of the differences between plane wave and twisted x-ray magnetic scattering.

1. Cross section for magnetic x-ray scattering

The calculation of the cross section for x-ray scattering, which includes the magnetic terms, can be derived from a quasi-relativistic Hamiltonian for electrons in a quantized electromagnetic field. This is typically done within second-order perturbation theory as delineated by Blume and others. Platzman and Tzoar, as well as de Bergevin and Brunel, initiated their approach from the Dirac equation, then simplified the relativistic framework using a Foldy–Wouthuysen transformation to a form comparable to that derived from the non-relativistic Hamiltonian. The subsequent expansion of this quasi-non-relativistic Hamiltonian in terms of the ratio of photon energy to electron rest mass energy facilitates the portrayal of the magnetic scattering process. Further extensions to the Foldy–Wouthuysen transformation were made to include second-order terms in ( 2 m c ω ) 2. Our exposition follows the methodology presented by Blume based on a non-relativistic approach in second-order perturbation theory. We begin with the Hamiltonian for electrons in a quantized electromagnetic field
(28)
where P j is the momentum operator, A is the vector potential, E is the electric field, S j is the spin operator, and a k λ , a k λ are the creation and annihilation operators for the photons with wavevector k and polarization λ. The terms represent, respectively, the kinetic energy, potential energy, magnetic moment interaction, spin–orbit coupling, and the quantized electromagnetic field energy.
Here, the first term corresponds to the kinetic energy of the electrons in the electromagnetic field, represented by the vector potential A ( r ), the second term corresponds to the Coulomb interaction between the electrons, the third term to the Zeeman energy μ · H of the electrons with spin s j, the fourth term to the spin–orbit coupling, and the final term to the self-energy of the electromagnetic field. From the form of Eq. (28), we can infer that the cross section and polarization dependence of the scattering of an electromagnetic wave from magnetic materials is more complex than the corresponding cross section for neutron scattering—at least if we consider only the two main interaction potentials for nuclear scattering and magnetic dipole scattering. In the case of neutron scattering, only the magnetic dipole interaction of the neutron spin with the magnetic field of the electrons gives rise to magnetic scattering. For x-rays, we have multiple interaction terms, including those between the spin of the electrons and the electromagnetic field, as well as between the orbital momentum and the magnetic field. Additionally, photons are spin-1 particles, unlike spin-1/2 neutrons, leading to a more complex polarization dependence. For a plane wave, the vector potential A ( r ) in (1) is linear in photon creation and annihilation operators a k λ and a k λ and is given in a plane wave expansion by
(29)

In Eq. (29), V is the quantization volume, and ε ( q σ ) is the unit polarization vector corresponding to a wave with wavevector q and polarization state σ. For photons, two polarization states σ = 1 , 2 must be distinguished. We can choose a basis consisting of either two perpendicular directions of linear polarization or left and right circular polarization. For plane waves, since A ( r ) is linear in the creation and annihilation operators, scattering occurs in second order for terms linear in A and in first order for terms quadratic in A.

To understand twisted x-ray interaction with magnetic materials before delving into x-ray magnetic circular dichroism (XMCD) and XMLD, we need to derive the vector potential for twisted x-rays.

2. Twisted x-ray vector potential and interaction Hamiltonian

The vector potential for twisted (vortex beams) photons differs from that of plane waves due to the incorporation of orbital angular momentum. As earlier discussed, twisted photons are characterized by carrying quantized orbital angular momentum of per photon, with being the topological charge, an integer value. The vector potential A ( r ) for twisted waves is more complex, often involving Bessel functions and phase factors that depend on the azimuthal angle ϕ and the topological charge .

For brevity, we can invoke a common representation of the vector potential for a twisted wave is
(30)
where A ( r ) could be a function representing the radial dependence, such as a Bessel function of the first kind J ( k r ), and e i ϕ is the azimuthal phase factor that imparts the “twist” to the wave. The precise form of A ( r ) and subsequently Eq. (30) will depend on the experimental geometry, sample type, details of the twisted wave, for example, whether it is a Bessel beam, a Laguerre–Gaussian beam, or another type of structured light.
Such a twisted vector potential should capture the unique spatial structure and phase factors that are characteristic of the orbital angular momentum of the twisted x-ray photons. For brevity, using cylindrical coordinates ( r , ϕ , z ) and incorporating both the Bessel function J for the radial part and the azimuthal phase factor e i ϕ for the orbital angular momentum, Eq. (30) can be written in a similar form as that for plane waves in the form
(31)

In Eq. (31), σ represents the polarization state of the twisted x-rays. b σ and b σ represent the annihilation and creation operators for twisted photons. is the orbital angular momentum quantum number or topological charge representing the twistedness of the beam. k and kz are the components of the wavevector perpendicular and parallel to the direction of propagation, respectively. J ( k r ) is the Bessel function of the first kind, dictating the radial distribution of the wave. The phase factors e i ϕ and e i ϕ correspond to the azimuthal variation, while e i k z z and e i k z z relate to the propagation along the z-axis.

This representation, while indicative, should be tailored to the specifics of the physical system and experimental conditions in question. The precise configuration of the vector potential for twisted x-rays will depend on the particular theoretical framework and empirical setup under consideration.

To derive the Hamiltonian for a system under the influence of twisted X-rays, we need to modify the standard Hamiltonian in Eq. (28) to incorporate the effects of the twisted electromagnetic fields. This involves substituting the standard vector potential A ( r j ) with the twisted vector potential A twisted ( r j ). The resulting Hamiltonian will describe the dynamics of particles (electron charge and spin) interacting with twisted X-rays. The Hamiltonian of the system in the presence of twisted X-rays can thus be written as
(32)

In this Hamiltonian, H twisted, the interaction of particles with the structured light fields of the twisted X-rays is represented, reflecting the complex dynamics introduced by these specialized electromagnetic fields.

3. Charge and magnetic scattering by plane waves: Selection rules

The Hamiltonian, for plane waves [Eq. (28)], can be segmented into
(33)
where H0 encapsulates the degrees of freedom of the electron system, Hf characterizes the Hamiltonian for the quantized electromagnetic field, and H delineates the interaction between electrons and the plane waves. Scattering cross sections are computed under the assumption of an initial state | a , which is an eigenstate of H0 with energy Ea, and the presence of a singular photon. We then assess the probability of a transition triggered by the interaction Hamiltonian H to a final state | b with a photon k ν . For elastic scattering, the final state | b is identical to the initial state | a . This transition likelihood per unit time is estimated using the golden rule to the second order perturbation theory. The fact that we have terms linear in A intimates that apart from the standard non-resonant magnetic x-ray scattering, resonance phenomena will emerge due to the energy denominator identified in the second-order perturbation theory (comparable to the Breit-Wigner131 formula for resonant scattering of the neutron from a nucleus). We will now cite the final result of this computation: for a high enough x-ray energy and far from all absorption edges of the elements in the sample, the elastic cross section for the scattering of photons (plane waves) with initial polarization ε into a state of final polarization ε is expressed as
(34)
where R e = e 2 / m c 2 is the classical electron radius and λ C = / m c is the Compton wavelength of an electron. The scattering amplitudes f | c ε · c ε | i and f | M ε · c ε | i are matrix elements which describe the polarization-dependent components of charge and magnetic scattering, respectively. The case of linear polarization is contemplated here, characterized by unit vectors perpendicular to the wave vectors of incident and scattered photons, k and k . The σ-polarization correlates with the basis vector perpendicular to the scattering plane, while π-polarization is aligned with the vectors in the k , k plane (ref to Fig. 11). The basis vectors for the components of the magnetic moment of the sample and for the polarization states are defined as follows:
(35)
FIG. 11.

Illustration of the coordinate system and the basis vectors used to describe the polarization dependence of plane wave x-ray scattering.116 

FIG. 11.

Illustration of the coordinate system and the basis vectors used to describe the polarization dependence of plane wave x-ray scattering.116 

Close modal
In the established basis from Eq. (35), the matrices in Eq. (34) can be expressed for magnetic, f | M | i ,
(36)
and similarly for charge, f | c | i scattering components
(37)

Here, S i = S ( Q ) and L i = L ( Q ) (for i = 1 , 2 , 3) denote the Fourier components of the magnetization density due to the spin and orbital angular momentum, respectively. ρ ( Q ) symbolizes the Fourier transform of the electronic charge density distribution with the magnetic material.

As elucidated from Eq. (34), magnetic plane wave magnetic scattering is a relativistic correction to charge scattering. In the context of coherent elastic Bragg scattering, the magnetic and charge scattering amplitudes are modulated by the momentum transfer, thus the magnetic contribution in the cross section is signified by λ C / d, highlighting that for a given Bragg reflection, the ratio between magnetic and charge scattering remains essentially constant, irrespective of photon energy, as it adheres to the approximations leading to Eq. (34).

Equation (34) has the following distinct terms: pure Thomson-scattering, pure magnetic scattering, and an interference term. The latter becomes significant when both charge and magnetic scattering occur concurrently. The prefix “i” in front of the magnetic scattering amplitude implies that if both f | c | i and f | M | i are real, the interference term vanishes. Interference is observable only if one of the amplitudes contains an imaginary part, such as in the case of non-centrosymmetric structures, or when the energy is close to an absorption edge for charge scattering or in the presence of circularly polarized x-rays. The relevance of the interference term in ferromagnetic scattering becomes evident when we consider the ratio between the magnetic and charge scattering amplitudes for plane waves. The ratio can be estimated by
(38)

Here, N M and f M denote the number and form factor of the magnetic electrons, respectively, while S is the expectation value of the spin quantum number. Employing suitable values for the parameters in Eq. (38), it was found130 that the amplitude for magnetic scattering is typically several orders of magnitude smaller than that of charge scattering, resulting in an intensity ratio of 10 6 between pure magnetic and pure charge scattering. This intensity discrepancy is impractical for direct measurement, making the interference term crucial for ferromagnets, where charge and magnetic scattering coincide in reciprocal space. This interference term, akin to flipping-ratio measurements in neutron scattering,132 can change sign by altering the direction of the magnetization or the polarization of the incident photons, thereby isolating the contribution from pure charge scattering.

Equations (34) and (38) prescribe that magnetic scattering can be distinguished from charge scattering by a polarization analysis experiment, where the off diagonal terms of the scattering matrix are measured. Specifically, terms that switch the initial polarization state from σ to π or vice versa are discernible.

A derivation of the non-resonant magnetic scattering cross section from non-relativistic quantum mechanics using perturbation theory has been outlined, reflecting classical calculations. This agreement between classical and quantum mechanical methods reaffirms the perturbation theory's validity, as evidenced by previous studies.

In charge scattering, the polarization of the incident wave is conserved, which is immediately apparent from the diagonal nature of the classical matrix.35 However, the cos 2 θ factor for π π scattering is an explicit indication of the matrix's dependence on the scattering angle, contrasting with the charge scattering matrix 36 which does not vary with direction. This distinction in polarization dependence is fundamental for the interpretation of experimental results, as it allows for a clear differentiation between magnetic and charge scattering phenomena (ref to Fig. 12).

FIG. 12.

Illustration of the processes leading to the scattering of x-rays by the charge (top) and the spin moment (bottom three) of the electron in a classical picture. Adapted from Ref. 116.

FIG. 12.

Illustration of the processes leading to the scattering of x-rays by the charge (top) and the spin moment (bottom three) of the electron in a classical picture. Adapted from Ref. 116.

Close modal
Examining the high energy limit of the purely magnetic differential cross section is instructive. Derived from Eq. (37), this cross section for plane waves becomes essentially independent of polarization analysis at energies around 100 keV, focusing on the Fourier component perpendicular to the scattering plane
(39)

At these photon energies, the pure spin density distribution is accessible without polarization analysis, unlike neutron diffraction which integrates both spin S and orbital L contributions.

Tuning the x-ray energy to the absorption edges of magnetic elements invokes resonance effects described by the second-order perturbation theory
(40)
In this context, | c represents an intermediate excited state with energy Ec, ω is the photon energy, and Γ characterizes the width of the excited state, which is inversely proportional to its lifetime ( Γ · τ ). The operator O ( k ) is expressed by
(41)

Equation (40) introduces anomalous dispersion, meaning there is an energy dependence of the charge scattering, as well as resonance magnetic scattering. Operator given in Eq. (41) can be expanded in a multipole series. It is noted that in the x-ray regime, spin and orbital contributions are often negligible, with only the electric multipole terms being significant.

These electric multipole (predominantly dipole and quadrupole) operators induce virtual transitions between core levels and unoccupied states above the Fermi energy with subsequent reemission of a photon. These processes become sensitive to the magnetic state in exchange split bands due to the difference in occupation of minority and majority bands leading to so-called resonance elastic x-ray scattering (REXS) using plane waves133 as illustrated schematically in Fig. 13.

FIG. 13.

Schematic illustration of the second-order perturbation process leading to REXS in the case of a lanthanide metal, e.g., a Gd3+ ion. Adapted from Ref. 116.

FIG. 13.

Schematic illustration of the second-order perturbation process leading to REXS in the case of a lanthanide metal, e.g., a Gd3+ ion. Adapted from Ref. 116.

Close modal

The resonance denominator in Eq. (40) leads to resonance enhancements at the absorption edges of magnetic elements in REXS. The intensification of these enhancements is contingent on three major factors:

  1. The magnitude of the transition matrix element is critical. Dipole transitions between states with a change in orbital angular momentum quantum number ( Δ L = 1) are typically more pronounced than quadrupolar transitions ( Δ L = 2). This is because the overlap of wave functions between the initial and intermediate states affects the size of the matrix elements. In contrast, transitions from “s” core levels to “p” or “d” excited states are less likely to exhibit large resonance enhancements owing to the minimal overlap of their wave functions.

  2. The energy density of states above the Fermi level also plays a role, particularly for minority and majority spin states. In lanthanide metals, the density of empty states is polarized due to magnetic 4f states. However, the exchange splitting in the 5d band is not as strong as in the 4f states, and thus, dipolar transitions from 2p to 5d can be stronger than those leading to 4f.

  3. The strength of spin-orbit coupling in the ground and excited states influences the sensitivity of dipole transitions to the spin magnetism. This coupling causes electric multipole transitions to become more sensitive to the magnetic properties of the system.

With these considerations, the possible transitions can be qualitatively classified based on the magnitude of their resonance enhancements, which is represented in Table I. The term “resonance enhancement” here refers to the increase in the intensity of magnetic Bragg peaks at the resonance peak compared to the intensity of non-resonant magnetic scattering.

TABLE I.

Magnitude of the resonance enhancements for REXS for some elements relevant for magnetism. Only order of magnitude estimates are given with “weak” corresponding to a factor of about 10°, “medium” to about 101, and “strong” to greater than 102.

Elements Edge Transition Energy range (keV) Resonance strength Comment
3d  1 s 4 p  5–9  Weak  Small overlap 
3d  LI  2 s 3 d  0.5–1.2  Weak  Small overlap 
3d  LII, LIII  2 p 3 d  0.4–1.0  Strong  Dipolar, large overlap, high spin polarization of 3d 
4f  1 s 5 p  40–63  Weak  Small overlap 
4f  LI  2 s 5 d  6.5–11  Weak  Small overlap 
4f  LII, LIII  2 p 5 d  6–10  Medium  Dipolar 
4f  LII, LIII  2 p 4 f  ⋯  ⋯  Quadrupolar 
4f  MI  3 s 5 p  1.4–2.5  Weak  Small overlap 
4f  MII, MIII  3 p 5 d  1.3–2.2  Medium  Dipolar 
4f  MII, MIII  3 p 4 f  ⋯  ⋯  Quadrupolar 
4f  M I V , M V  3 d 4 f  0.9–1.6  Strong  Dipolar, large overlap, high spin polarization of 4f 
5f  M I V , M I  3 d 5 f  3.3–3.9  Strong  Dipolar, large overlap, high spin polarization of 5f 
Elements Edge Transition Energy range (keV) Resonance strength Comment
3d  1 s 4 p  5–9  Weak  Small overlap 
3d  LI  2 s 3 d  0.5–1.2  Weak  Small overlap 
3d  LII, LIII  2 p 3 d  0.4–1.0  Strong  Dipolar, large overlap, high spin polarization of 3d 
4f  1 s 5 p  40–63  Weak  Small overlap 
4f  LI  2 s 5 d  6.5–11  Weak  Small overlap 
4f  LII, LIII  2 p 5 d  6–10  Medium  Dipolar 
4f  LII, LIII  2 p 4 f  ⋯  ⋯  Quadrupolar 
4f  MI  3 s 5 p  1.4–2.5  Weak  Small overlap 
4f  MII, MIII  3 p 5 d  1.3–2.2  Medium  Dipolar 
4f  MII, MIII  3 p 4 f  ⋯  ⋯  Quadrupolar 
4f  M I V , M V  3 d 4 f  0.9–1.6  Strong  Dipolar, large overlap, high spin polarization of 4f 
5f  M I V , M I  3 d 5 f  3.3–3.9  Strong  Dipolar, large overlap, high spin polarization of 5f 

Table I presents a selection of prominent examples and highlights that resonance enhancements for 3d transition metal ions are generally negligible in the hard x-ray regime, while they can be significant for soft x-rays. Despite this, high resolution is not attainable at wavelengths of 12–30 Å under standard conditions. However, resonance enhancements are crucial for the LII and LIII edges of transition metals in the study of magnetic thin films and nanostructures. Resonance enhancements for 4f elements can reach two orders of magnitude in the hard x-ray range at the LI and LIII edges. Here, dipolar transitions are typically predominant, though quadrupolar transitions are also notable. The “branching ratio,” or the ratio between the resonance enhancements at the LI edge and LIII edge, shows systematic variation across the rare earth series. This ratio is closer to unity for rare earth ions with seven 4f electrons, while for those with more than seven 4f electrons, the LIII resonance is usually more pronounced, especially when the 4f shell is less than half-filled. In the soft x-ray range, the MIV and MV resonances become important for magnetic nanostructures. At the MIV edge for actinides, the intensity gain from REXS can be as much as seven orders of magnitude. The 4d and 5d transition metal elements, although not listed in Table I, can exhibit such substantial resonance enhancements at the LII and LIII edges that surface magnetic x-ray diffraction is feasible, as demonstrated in materials like Co 3 Pt (111).134 From these observations, it can be inferred that REXS, enabling intense gains for magnetic x-ray scattering, facilitates spectroscopy of the exchange-split empty states above the Fermi level and enhances the sensitivity of magnetic diffraction to specific magnetic species.

Returning to the detailed nature of the cross section, which now includes resonant magnetic scattering, we reference Eq. (40) for the overarching form of the cross section pertinent to anomalous scattering phenomena. Here, we will disregard the spin-dependent portion, focusing instead on electric dipole transitions. Comprehensive explanations and the polarization dependencies, inclusive of electric quadrupole transitions, can be found in the referenced literature. Anomalous scattering gains significance in proximity to the absorption edges. The energy-dependent amplitude for the scattering cross section in dipole approximation is represented as follows:
(42)
where the individual components are given by
(43)
(44)
(45)
The component f0 is independent of the magnetic state, whereas f circ and f lin are connected to forward scattering with circular and linear dichroism, respectively. The energy dependence of these amplitudes is captured in the oscillator strengths
(46)
where ω is the photon energy, ω res is the position of the absorption edge, and Γ is the resonance width. The parameter αM quantifies the resonance amplitude, correlating to the transition matrix elements.

4. Elastic scattering cross section with twisted light

In the context of twisted light, the Hamiltonian in Eq. (33) can be extended to include the quantized field of twisted x-rays photons. The scattering cross section for twisted light, considering an initial state | a and a final state | b , is influenced by the unique properties of the twisted photons. Assuming a high-energy regime where absorption edges can be neglected, and focusing only on electric dipole transitions, the differential cross section is given by
(47)
where T ε , ε ( ) is the transition matrix that now includes the effect of the twisted light's angular momentum. This term is expected to provide greater contrast in the cross section due to the additional degree of freedom in the scattering process. For twisted light, we anticipate that the term related to the magnetic component will be modified as follows:
(48)
In this equation, J ( k r ) represents the Bessel function corresponding to the radial part of the photon's wavefunction, and e i ϕ is the azimuthal phase factor associated with the twisted light. The transition matrix T ε , ε ( ) will likely enhance the scattering contrast due to the structured interference of the twisted beam with the electronic structure of the material
(49)
(50)
(51)

The above equations are adapted for twisted light, integrating the distinctive properties of such beams into the scattering formalism. Consequently, the differential cross section for twisted light is expected to be richer in structure and to reveal additional information about the sample's electronic and magnetic properties, providing a higher level of detail in scattering experiments. Soft x-ray coherent diffractive imaging (CDI) on magnetic systems generally benefits from its fundamental mechanism of 2p -> 3d dipolar electronic transition, which has very strong resonance scattering strength, large overlap, and large spin polarization of 3d orbital. This applies to most transition metals materials in the x-ray energy range of 0.4–1.0 keV. In hard X-rays, rare-earth LII and LIII resonance edges 2p -> 5d or 2p -> 4f are dipolar or quadrupolar electronic transitions, and the scattering strength is moderate (ref Table II).

TABLE II.

Selection rules for x-ray absorption near-edge spectra for OAM beam.

Transition Selection rule Strength K-edge LIII-edge
Dipole  Δ = ± 1  strong  1s -> np  2p -> nd 
Quadrupole  Δ = ± 2  Strong  1s -> nd  2p -> nf 
Transition Selection rule Strength K-edge LIII-edge
Dipole  Δ = ± 1  strong  1s -> np  2p -> nd 
Quadrupole  Δ = ± 2  Strong  1s -> nd  2p -> nf 

5. Selection rules for x-ray absorption near-edge spectra for OAM beam

6. Transitions

Soft coherent x-ray diffractive imaging (CDI) in magnetic systems is primarily effective due to the 2p -> 3d dipolar electronic transition. This transition is characterized by strong resonance scattering strength, significant overlap, and a high degree of spin polarization in the 3d orbital. This mechanism is particularly relevant for most transition metal materials within the x-ray energy range of 0.4–1.0 keV.135–188 

In contrast, hard X-rays typically involve rare-earth LII and LIII resonance edges, where the electronic transitions are either 2p -> 5d or 2p -> 4f. These transitions can be either dipolar or quadrupolar. While the scattering strength in hard X-rays is moderate, they offer advantages in terms of working space and attenuation length. However, it is important to note that magnetic scattering strength in hard X-rays is comparatively lower than in soft X-rays (ref to Table III).

TABLE III.

Expected transitions for OAM beams.

Transition K, LI, MI LII, LIII, MII, MIII MIV, MV
Dipole  ns -> np  np -> nd  nd -> nf 
Quadrupole  ns -> nd  np -> nf  nd -> ng 
Transition K, LI, MI LII, LIII, MII, MIII MIV, MV
Dipole  ns -> np  np -> nd  nd -> nf 
Quadrupole  ns -> nd  np -> nf  nd -> ng 

7. Discussion

Microscopy using dichroism as a contrast mechanism has significantly advanced our understanding of phase ordering, separation, and coexistence in various systems. These materials have an order parameter that scatters light differently based on the direction or helicity of the photon polarization. Polarized X-rays have been ideal for probing buried magnetic structures, owing to the strong resonant enhancement of scattering at electronic transitions influenced by the spin–orbit effect. Being unaffected by magnetic or electric fields, x-ray beams are particularly suited for studying phase transitions as a function of applied fields. Established techniques such as transmission x-ray microscopy and x-ray photoemission electron microscopy, along with novel holographic methods, have enhanced our knowledge of dichroic materials at the nanoscale.

However, these traditional approaches require focusing optics or apertures with precision comparable to the desired spatial resolution, which is a significant limitation. In contrast, resonant x-ray coherent diffractive imaging (CDI) offers an alternative approach. In CDI, the diffraction pattern formed by scattering a coherent x-ray beam from a sample is numerically inverted to form an image. Here, the spatial resolution is not limited by the optical elements but by the highest spatial frequencies measured in the x-ray diffraction pattern. CDI is capable of providing three-dimensional information and offers elemental selectivity near electronic resonances. When combined with ptychographic methods, it allows for imaging large regions quantitatively with high spatial resolution.

The introduction of twisted X-rays with unbounded topological charge opens new possibilities in helical dichroic CDI. Characterized by their helical wavefronts, these twisted X-rays interact uniquely with the magnetic and electronic structures of materials, offering enhanced contrast and sensitivity in magnetic CDI. This is particularly effective for imaging complex magnetic textures and domains.

Magnetic resonant scattering is pivotal in studying magnetic materials. It leverages the interaction of X-rays with the electronic and magnetic structures of a sample. Key to this technique are the dispersion coefficients (often represented as f and f ), resonance conditions, and dispersion corrections, which are vital for data interpretation. These coefficients, dependent on the x-ray energy relative to the absorption edges, influence the phase shift and absorption in the scattering process. Resonance occurs when the x-ray energy matches the absorption edge, enhancing the scattering. Dispersion correction then adjusts the scattering data to account for the energy dependence of these coefficients, allowing for accurate data interpretation.

Twisted X-rays, with their intrinsic orbital angular momentum (OAM), offer a new dimension in the study of material properties. The resonance enhancement in these systems might be particularly pronounced for specific transitions, providing a predictive tool for experimental setups designed to exploit the unique interactions of twisted X-rays with matter. This emerging field combines theoretical predictions with experimental observations, continually evolving to uncover more about the “Twisted x-ray Effect.”

In conclusion, the integration of traditional magnetic resonant scattering techniques with the novel approach of twisted x-ray CDI presents a promising frontier in the study of magnetic materials. It combines the strengths of various methods to provide a more detailed and nuanced view of the magnetic and electronic structures of materials at the nanoscale.

Utilizing twisted x-ray beams, or beams with orbital angular momentum (OAM), STXM-XMCD, XMCD in reflection geometry, CDI, and other techniques discussed earlier presents an innovative frontier in condensed matter and materials science research. By merging the temporal coherence intrinsic to x-ray photon correlation spectroscopy (XPCS) with the unique spatial attributes of twisted x-ray beams, a plethora of new avenues for investigating material dynamics across diverse length scales and geometries emerge.

Here is an overview of potential advantages and insights this combination could yield:

Enhanced Spatial Resolution: The distinct spatial profile of twisted x-ray beams, reminiscent of a vortex, can be leveraged to attain superior spatial resolution in x-ray scattering and imaging experiments. This could pave the way for intricate studies of local dynamics in materials. This is mainly due to the additional degree of freedom that the twisted nature of the beam offers. The phase variations along the azimuthal direction of twisted X-rays add complexity to the interference pattern, which can be harnessed to improve resolution. In techniques like CDI, twisted X-rays can potentially lead to sharper and more detailed reconstructions by leveraging their unique interference pattern.

Angular Momentum Transfer: Analogous to how photons can relay spin angular momentum to materials via polarization, twisted X-rays could transfer OAM to specimens. Such an interaction might evoke unique material responses or dynamics, subsequently probed using x-ray pair distribution function (XPDF), XPCS and pump-probe CDI.

Improved Contrast in Soft Chiral Matter Systems: In systems such as colloids or polymers, DNA the unique spatial configuration of twisted x-ray beams might offer enhanced scattering contrast. This could facilitate the observation of dynamics otherwise elusive with conventional X-rays.

OAM-Based Imaging and Helical Dichroism: The orbital angular momentum (OAM) of twisted x-ray beams could be utilized as a contrast mechanism in imaging, enabling the extraction of novel types of information from samples that are inaccessible to traditional x-ray imaging techniques. In helical dichroism, the absorption of circularly polarized light depends on the match between the chirality of the light and the sample. With twisted X-rays, a similar principle can be applied to BCDI, where the helicity of the X-rays can interact differently with chiral samples. This “helical dichroic BCDI” could potentially outperform anomalous diffraction techniques by providing a new contrast mechanism based on OAM, thus uncovering new information about chiral samples.

Improved Material Characterization: Twisted x-ray beams could deliver more comprehensive information about a material's structure. As these beams interact with the orbital angular momentum of the electrons in a material, they can provide additional insights into its electronic structure. In techniques such as x-ray absorption spectroscopy or crystallography, twisted x-ray beams could provide additional structural information, helping to resolve ambiguities in complex structures or to identify features that would be missed by conventional X-rays.

Novel Interactions with Matter: Twisted x-ray beams can interact with matter in unique ways that are not possible with conventional x-ray beams. For instance, they can induce unique electronic transitions in atoms. In ptychography, the measurement of phase changes in the transmitted beam can provide high-resolution images. With twisted X-rays, the unique interactions with matter could lead to new kinds of phase changes that could be exploited to improve ptychographic reconstructions or to reveal new information about the sample. Furthermore, these novel interactions could potentially be harnessed for a range of innovative applications, from materials science to biological imaging.

Quantum Information: Just like in the optical domain, x-ray beams with OAM can be potentially used in quantum communication and quantum computation setups, although practical applications in this domain are still under exploration. The confluence of OAM physics and condensed matter systems is an area witnessing rapid growth. Integrating with twisted X-rays holds the promise of unearthing new phenomena at the intersection of quantum optics and materials science.

It is noteworthy that the technology for generating and controlling twisted x-ray beams is still under development. Therefore, while the potential benefits of these beams are significant, many challenges remain that must be overcome to fully realize these benefits. These hurdles include technical issues associated with the generation and control of the beams, and theoretical issues related to interpreting the results obtained using these beams.

This review has underscored the significant advancements in the field of coherent scattering and diffractive imaging techniques, with a particular focus on the nanoscale analysis of chiral materials. The advent of synchrotron and x-ray free electron laser facilities has revolutionized our capability to understand and manipulate materials at the atomic scale. These technologies have enabled researchers to achieve sub-nanometer resolutions and explore dynamic processes within sub-millisecond timeframes, marking a quantum leap from traditional techniques to today's state-of-the-art methods.

The exploration of chiral crystals exemplifies the evolving challenges and opportunities in condensed matter physics. These materials, with their complex interactions with light, demand more sophisticated approaches than traditional methods like coherent x-ray diffraction (CXD). In this context, the innovative use of twisted x-ray beams emerges as a promising tool, offering potential breakthroughs in understanding chirality and other complex phenomena.

Looking ahead, the integration of twisted x-ray beams into coherent scattering techniques represents a fertile ground for future research. This novel approach promises not only higher spatial resolutions but also deeper insights into the intricate behaviors of materials. The implications of these techniques extend beyond materials science, encompassing fields such as biomedical imaging and beyond.

As we move forward, it is crucial to recognize that our objective extends beyond the development of advanced imaging tools; it is about enriching our comprehension of the atomic and subatomic worlds. The combination of twisted X-rays and coherent scattering principles paves the way for a new era of discovery, where we can delve deeper into the complexities of matter and its interactions.

Standing at this juncture, we must continue to push the frontiers of both technology and thought. The synergy of twisted X-rays with coherent diffractive imaging could represent the next significant stride in this journey, heralding an era rich with novel insights and groundbreaking innovations.

N.P.N., J.S., and E.F. acknowledge support from the US Department of Energy (DOE), Office of Science, (Grant No. DE-SC0023148). E.F. also acknowledges support from the US Department of Defense, Air Force Office of Scientific Research (AFOSR), (Award No. FA9550-23-1-0325) (Program Manager: Dr. Ali Sayir) for work on probing topological vortices and piezoelectric enhancements. This research used resources of the Advanced Photon Source (APS), a U.S. Department of Energy (DOE) Office of Science User Facility, operated for the DOE Office of Science by Argonne National Laboratory (ANL) (Contract No. DE-AC02-06CH11357).

The authors have no conflicts to disclose.

Nimish P. Nazirkar: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Xiaowen Shi: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal). Jian Shi: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Moussa N'Gom: Methodology (supporting); Supervision (supporting); Validation (equal); Writing – review & editing (equal). Edwin Fohtung: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (lead); Validation (lead); Visualization (equal); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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