The Scatman: an approximate method for fast wide-angle scattering simulations

Abstract

Single-shot coherent diffraction imaging (CDI) is a powerful approach to characterize the structure and dynamics of isolated nanoscale objects such as single viruses, aerosols, nanocrystals and droplets. Using X-ray wavelengths, the diffraction images in CDI experiments usually cover only small scattering angles of a few degrees. These small-angle patterns represent the magnitude of the Fourier transform of the 2D projection of the sample's electron density, which can be reconstructed efficiently but lacks any depth information. In cases where the diffracted signal can be measured up to scattering angles exceeding ∼10°, i.e. in the wide-angle regime, some 3D morphological information of the target is contained in a single-shot diffraction pattern. However, the extraction of the 3D structural information is no longer straightforward and defines the key challenge in wide-angle CDI. So far, the most convenient approach relies on iterative forward fitting of the scattering pattern using scattering simulations. Here the Scatman is presented, an approximate and fast numerical tool for the simulation and iterative fitting of wide-angle scattering images of isolated samples. Furthermore, the open-source software implementation of the Scatman algorithm, PyScatman, is published and described in detail. The Scatman approach, which has already been applied in previous work for forward-fitting-based shape retrieval, adopts the multi-slice Fourier transform method. The effects of optical properties are partially included, yielding quantitative results for small, isolated and weakly interacting samples. PyScatman is capable of computing wide-angle scattering patterns in a few milliseconds even on consumer-level computing hardware, potentially enabling new data analysis schemes for wide-angle coherent diffraction experiments.

Three-dimensional femtosecond snapshots of isolated faceted nanostructures

“…The imaging procedure optimizes the plane positions, which are randomly initialized as shown in Fig. 1B , by forward-fitting the experimental diffraction patterns via 3D simulations ( 44 ). Further details on the experimental data and on the imaging procedure are provided in Materials and Methods.…”

“…Any optimization strategy requires several evaluations of the optimization target, each of which implies a full wide-angle scattering simulation. Because of the relatively small variation from unity of the silver’s refractive index at 243 eV, the simulated pattern I sim as a function of parameter p→ can be obtained through a fast approximate simulation method, called Scatman ( 44 ), capable of providing a simulated pattern in few milliseconds. Nevertheless, the optimization of Efalse(p→false) is highly challenging, due to the high amount of free parameters involved and the still relevant computational cost of the error evaluation.…”

“…Such a model can describe any sample that (i) does not have concavities, (ii) is homogeneous, and (iii) has sharp boundaries. While these conditions fit well to the structural properties of silver nanocrystals ( 11 ), they are ill suited for many other types of morphologies, such as distorted droplets, for which different parametrization basis sets may be chosen, e.g., expansion in spherical harmonics as described in ( 44 ). For further considerations about the sample model, see the Supplementary Materials.…”

“…Our imaging approach is based on the so-called forward fitting, where the free parameters are tuned to optimize the similarity between the simulated and the experimental pattern. The latter is downscaled through an 8 × 8 pixel binning, to speed up the simulation time ( 44 ) without having impact on the reconstruction quality, as long as the oversampling condition is met ( 26 , 53 ). In our case, the similarity is encoded in the optimization target, which has to be minimized via a suitable algorithm.…”

“…Here, we introduce a generic approach to the 3D reconstruction of faceted samples from single-shot wide-angle coherent diffraction patterns, based on a fast simulation method ( 44 ), a generic model for the sample shapes, and an efficient optimization strategy. We retrieve the 3D shapes of individual silver nanoparticles imaged with FEL pulses at a wavelength of 5.1 nm.…”