Crystal Structure Refinement Research Articles

An idealized model for the crystal structure of intermetallic compounds isostructural with Mg3Cr2Al18

ABSTRACT Mg3Cr2Al18 (abbreviated in this report as MCA) is the parent phase for a large class of intermetallic compounds that belong to the cubic crystal space group, Fd 3 ¯ m . The purpose of this paper is to introduce an ideal, unrelaxed crystal structure for compounds isostructural with MCA. There are five distinct atomic sublattices in MCA compounds, which can be denoted, A, B, C, D, and E. With this, a general description for MCA structures can be written as A 1 8 a B 2 16 c C 2 16 d D 6 48 f E 12 96 g , where the superscripts represent the Wyckoff special equipoints associated with the various sublattices in MCA, and the subscripts indicate the contributions of each sublattice to the stoichiometry of one formula unit in an any given MCA-structured compound. Sublattices D and E are where deviations from ideality occur in real, MCA-like compounds. This paper examines MCA bond lengths, nearest-neighbour polyhedral arrangements, 3-D sublattice crystal structures, 2-D atom tessellation patterns, and crystal chemical effects associated with atomic relaxations on the D and E sublattices. The ideal MCA crystal structure developed in this report provides an appropriate initial structure for use as input to crystal structure refinements of diffraction data for MCA-like phases being examined experimentally, or as input for computational, atomistic simulations of the structures of such compounds.

Mn2+ doped aluminum-rich spinel transparent ceramic phosphor with higher performance for wide color-gamut backlight display

Due to the great potential of narrow-band green-emitting phosphors for wide color-gamut white light-emitting diodes (WLED) backlight displays, there is great interest in finding host materials with the desired luminescent properties and high quantum efficiency. Herein, a series of Mg0.98-xAl2(1+x/3)O4: 0.02Mn2+ (x = 0, 0.125, 0.25) powders were synthesized using high-temperature solid-phase reactions, followed by a combination of pressureless sintering and hot isostatic pressure sintering to fabricate transparent ceramic phosphors. The crystal structure refinement, NMR spectroscopy, and photoluminescence results show tetrahedral coordination of Mn2+ ions and an increase in cation vacancies and disorder with increasing x. These transparent ceramics exhibit narrow-band green emission, which shifts from 523 to 520 nm with increasing x. The Mg0.855Al2.08O4: 0.02Mn2+ ceramic sample has an in-line transmittance of over 80 % above 500 nm and internal quantum efficiencies as high as 96.2 %. The luminous intensity of the Mg0.855Al2.08O4: 0.02Mn2+ ceramic remains above 88.6 % at 425 K compared to room temperature, with a thermal conductivity of 12.4 W·m−1·K−1. A WLED device fabricated using the Mg0.855Al2.08O4: 0.02Mn2+ ceramic, K2SiF6:Mn4+ phosphors, and a blue chip show a wide color gamut of 126 % National Television System Committee standard. The results show that Mn2+-doped aluminum-rich spinel transparent ceramics have great potential for application in wide color gamut WLED backlight devices.

Development of ultrafast four-dimensional precession electron diffraction

Ultrafast electron diffraction/microscopy technique enables us to investigate the nonequilibrium dynamics of crystal structures in the femtosecond-nanosecond time domain. However, the electron diffraction intensities are in general extremely sensitive to the excitation errors (i.e., deviation from the Bragg condition) and the dynamical effects, which had prevented us from quantitatively discussing the crystal structure dynamics particularly in thick samples. Here, we develop a four-dimensional precession electron diffraction (4D-PED) system by which time (t) and electron-incident-angle (ϕ) dependences of electron diffraction patterns (qx,qy) are recorded. Nonequilibrium crystal structure refinement on VTe2 demonstrates that the ultrafast change in the crystal structure can be quantitatively determined from 4D-PED. We further perform the analysis of the ϕ dependence, from which we can qualitatively estimate the change in the reciprocal lattice vector parallel to the optical axis. These results show the capability of the 4D-PED method for the quantitative investigation of ultrafast crystal structural dynamics.

Effect of phase composition on the crystal structure and microwave dielectric properties of ZnMg2TiO5 ceramics

Novel microwave dielectric ceramics ZnMg2TiO5 were synthesized by traditional solid-state method. Crystal structure refinement, transmission electron microscopy (TEM), and Raman spectroscopy reveal the coexistence of a secondary phase, MgO, with the predominant cubic spinel phase (Fd-3m) in ZnMg2TiO5 ceramics. The ceramics with dense microstructures exhibit excellent microwave dielectric properties at 1260 °C: εr = 15.6, Q×f = 92,899GHz (at 10.8GHz), and τf = −56.9 ppm/°C. The content of MgO, driven by thermodynamic factors, influences significantly impact the microstructure and microwave dielectric properties of the ceramics, particularly inducing the generation of localized lattice distortions and the formation of dislocations. The εr follows the same trend as the molecular polarizability of the actual individual crystal cells. Notably, a near-zero τf (−7.4 ppm/°C) was realized by two-phase composites and excellent comprehensive properties were obtained (εr = 17.9, Q×f = 57,546GHz).

Analysis of magnetic structures in JANA2020.

JANA2020 is a program developed for the solution and refinement of regular, twinned, modulated, and composite crystal structures. In addition, JANA2020 also includes a magnetic option for solving magnetic structures from powder and single-crystal neutron diffraction data. This tool uses magnetic space and superspace symmetry to describe commensurate and incommensurate magnetic structures. The basics of the underlying formulation of magnetic structure factors and the use of magnetic symmetry for handling modulated and non-modulated magnetic structures are presented here, together with the general features of the magnetic tool. Examples of structures solved in the magnetic option of JANA2020 are given to illustrate the operation and capabilities of the program.

Nature of Scapolite Color: Ab Initio Calculations, Spectroscopy, and Structural Study

The article describes the results of a comprehensive study of the extra-framework components of scapolites using quantum–chemical calculations, electronic and vibrational spectroscopy, and single-crystal X-ray diffraction and crystal structure refinement. The ab initio calculations were performed using an embedded-cluster approach of extra-framework components in various cation surroundings. As a result, through comparing the experimental and ab initio calculation results, the energies of the electronic and vibrational transitions of various extra-framework components (CO3)2−, (CO3)·−, S3·−, S2·−—as well as the role of these components in the process of the lowering of the symmetry—were determined for scapolites belonging to the marialite–meionite solid–solution series. The nature of the various colors of the scapolites has also been established. Colors from purple to blue are a result of the presence of radiation-induced pairs of defects: carbonate radical anions (CO3)·− and F-centers. However, polysulfide S3·− radical anions are found in some violet scapolites.

The crystal structure of olanzapine form III.

The antipsychotic drug olanzapine is well known for its complex polymorphism. Although widely investigated, the crystal structure of one of its anhydrous polymorphs, form III, is still unknown. Its appearance, always in concomitance with forms II and I, and the impossibility of isolating it from that mixture, have prevented its structure determination so far. The scenario has changed with the emerging field of 3D electron diffraction (3D ED) and its great advantages in the characterization of polyphasic mixtures of nanosized crystals. In this study, we show how the application of 3D ED allows the ab initio structure determination and dynamical refinement of this elusive crystal structure that remained unknown for more than 20 years. Olanzapine form III is monoclinic and shows a similar but shifted packing with respect to form II. It is remarkably different from the lowest-energy structures predicted by the energy-minimization algorithms of crystal structure prediction.

Structural Features of Hydrated Representatives of the Labuntsovite Group: Refinement of Crystal Structures of Intermediate Members of the Tsepinite-Na–“Tsepinite-Ba”–Tsepinite-K Solid Solution

Structural Features of Hydrated Representatives of the Labuntsovite Group: Refinement of Crystal Structures of Intermediate Members of the Tsepinite-Na–“Tsepinite-Ba”–Tsepinite-K Solid Solution

Mn4+-activated NaSr10Y5W4O30 far-red emitting luminescent material for plant growth lighting

A series of Mn4+-doped NaSr10Y5W4O30 (NSYWO) phosphors have been successfully synthesized and further characterized to estimate their potentials for application as a far-red emitting component of pc-LED. An XRD analysis and crystal structure refinement have been carried out to characterize its quality. The doping concentration and temperature (80–500 K) dependent luminescence properties as well as decay kinetics have been studied and analyzed. It has been found that Mn4+ substitutes for W6+ sites resulting in far-red emission at 688 nm. The temperature quenching is attributed to thermal activation of an electron from Mn4+ 2Eg to 4A2g state via crossover, yielding an activation energy of 0.35 eV. Furthermore, to assess practical application of the novel phosphor, a blue-emitting LED chip combined with NSYWO: 0.8 mol% Mn4+has been packaged. The light from both the blue-emitting LED chip and NSYWO: Mn4+ phosphor was shown to be effectively utilized, revealing a good promise for application of the phosphor in the field of plant growth lighting.

Evolution of structure and spectroscopic properties of a new 1,3-diacetylpyrene polymorph with temperature and pressure.

A new polymorph of 1,3-diacetylpyrene has been obtained from its melt and thoroughly characterized using single-crystal X-ray diffraction, steady-state UV-Vis spectroscopy and periodic density functional theory calculations. Experimental studies covered the temperature range from 90 to 390 K and the pressure range from atmospheric to 4.08 GPa. Optimal sample placement in a diamond anvil cell according to our previously presented methodology ensured over 80% data coverage up to 0.8 Å for a monoclinic sample. Unrestrained Hirshfeld atom refinement of the high-pressure crystal structures was successful and anharmonic behavior of carbonyl oxygen atoms was observed. Unlike the previously characterized polymorph, the structure of 2°AP-β is based on infinite π-stacks of antiparallel 2°AP molecules. 2°AP-β displays piezochromism and piezofluorochromism which are directly related to the variation in interplanar distances within the π-stacking. The importance of weak intermolecular interactions is reflected in the substantial negative thermal expansion coefficient of -55.8 (57) MK-1 in the direction of C-H...O interactions.

Redetermination of germacrone type II based on single-crystal X-ray data.

The extraction and purification procedures, crystallization and crystal structure refinement (single-crystal X-ray data) of germacrone type II, C15H22O, are presented. The structural results are compared with a previous powder X-ray synchrotron study [Kaduk et al. (2022 ▸). Powder Diffr. 37, 98-104], revealing significant improvements in terms of accuracy and precision. Hirshfeld atom refinement (HAR), as well as Hirshfeld surface analysis, give insight into the inter-molecular inter-actions of germacrone type II.

Enhancement of superconductivity and phase diagram of Ta-doped Kagome superconductor CsV3Sb5

Kagome superconductors AV3Sb5 (A = K, Rb, and Cs) have attracted enormous interest due to the coexistence of charge density wave (CDW) order, unconventional superconductivity (SC) and anomalous Hall effect (AHE). In this paper, we reported an intensive investigation on Cs(V1−xTax)3Sb5 single crystals with systematic Ta doping. Ta was confirmed to be doped into V-site in the Kagome layer from both single crystal X-ray diffraction structural refinement and scanning transmission electron microscopy observation. The highest Ta doping level was found to be about 16%, which is more than twice as much as 7% in Nb-doped CsV3Sb5. With the increase of Ta doping, CDW order was gradually suppressed and finally vanished when the doping level reached to more than 8%. Meanwhile, superconductivity was enhanced with a maximum critical temperature (Tc) of 5.3 K, which is the highest Tc in the bulk crystal of this Kagome system at ambient pressure so far. The μ0Hc2(T) behavior demonstrates that the system is still a two-band superconductor after Ta doping. Based on the electrical transport measurement, a phase diagram was set up to exhibit the evolution of CDW and SC in the Cs(V1−xTax)3Sb5 system. These findings pave a new way to search for new superconductors with higher Tc in the AV3Sb5 family and establish a new platform for tuning and controlling the multiple orders and superconducting states.

Nanostructured Niobium and Titanium Carbonitrides as Electrocatalyst Supports.

Nanostructured niobium-titanium carbonitrides, (Nb,Ti)C1-xNx, with the cubic-rock salt structure are prepared without the use of reactive gases via thermal treatment (700-1200 °C) under nitrogen of mixtures of guanidine carbonate and ammonium niobate (V) oxalate hydrate, with addition of ammonium titanyl oxalate monohydrate as a titanium source. The bulk structure and chemical composition of the materials are characterized using powder X-ray diffraction (XRD) and powder neutron diffraction, elemental homogeneity is studied using energy dispersive spectroscopy (EDS) mapping using transmission electron microscopy (TEM), and surface chemical analysis is examined using X-ray photoelectron spectroscopy (XPS). Nanoscale crystallites of between 10 and 50 nm are observed by TEM, where EDS reveals the homogeneity of metal distribution for the mixed-metal materials. Titanium carbonitrides are found to be air sensitive, reacting with air under ambient conditions, while titanium-niobium carbonitrides are found to degrade in aqueous sulfuric acid. The niobium carbonitrides, however, show some stability toward acidic solutions. Materials are produced with composition NbC1-xNx with x between 0.35 and 0.45, and more carbon-rich materials (x ≈ 0.35) are found as the synthesis temperature is increased, as proven by Rietveld refinement of crystal structure against powder neutron diffraction data. Despite phase purity seen by diffraction and negligible bulk carbon content, XPS shows a complex surface chemistry for the NbC1-xNx materials, with evidence for Nb2O5-like oxide species in a carbon-rich environment. The NbC1-xNx prepared at 900 °C has a surface area around 50 m2 g-1, making it suitable as a catalyst support. Loading with iridium provides a material active for the oxygen evolution reaction in 0.1 M sulfuric acid, with minimal leaching of either Nb or Ir after 1000 cycles.

Evaluation of self-cleaning performance of TiO2-intermixed and TiO2-coated cement mortars under natural sunlight

Self-cleaning cementitious materials have gained significant attention due to urban environmental pollution. In this study, nano-TiO2 (NT) was first mixed with cement mortar. Second, NTs were coated on fresh cast mortars and on hardened surfaces. Photocatalytic efficiency was evaluated using Rhodamine B under natural sunlight with three types of cement: white Portland (WPC), ordinary Portland (OPC), and Portland pozzolana cement (PPC). Among them, PPC cement mortars exhibited superior photocatalytic performance compared to WPC and OPC even with lower dosages of NT. The order of photocatalytic efficiency was highest for the hard-coat followed by fresh-coat and then intermix sample. Furthermore, Rietveld analysis for crystal structure refinement was used, and crystallography analysis (VESTA software) confirmed the presence of specific atoms and bonds in the crystal structure of PPC with NT. Microstructural analysis using XRD, SEM, and EDAX revealed underlying structural changes and thus quantified the hydration products.

Crystal chemistry, Raman and FTIR spectroscopy, optical absorption, and luminescence study of Fe-dominant sogdianite

Abstract Fe-dominant sogdianite, a cyclosilicate compound with the chemical formula (Fe3+ 0.74Zr0.64Ti0.46 Al0.15)(□1.02Na0.98)K[Li3Si12O30], was studied. The investigation involved a comprehensive analysis of the mineral sample, including crystal-chemical analysis, Raman and FTIR spectroscopy, optical absorption, and luminescence study. Crystallographic site populations were determined through single crystal structure refinement and electron probe microanalysis. The thermoelastic behavior of a powder was studied using in situ high-temperature X-ray diffraction (30–750 °C). Notably, no phase transition was detected; sogdianite exhibited anisotropic thermal expansion. The first time study of vibrational spectra and spectral bands assigning were performed. The electronic transitions in d 5-ion impurities of sogdianite were studied using optical absorption and luminescence spectroscopy. The origin of pink color and luminescence of sogdianite was clarified. The broad spectral bands in the visible UV spectral region are responsible for the pink color exhibited by sogdianite and could be attributed to d–d transitions occurring in Fe3+ ions.

Crystal structures and crystallographic classification of titanium silicophosphates – with a note on structure and composition of silicophosphates “M 3P5SiO19”

Abstract The crystal structures of TiIII 4[Si2O(PO4)6] ( P 3 ‾ $P\overline{3}$ , Z = 3, a = 14.733(1), c = 7.363(1) Å, R1 = 0.040, wR2 = 0.098, 7649 ind. refl., 170 variables), FeII 0.79TiIII 2.42TiIV 0.79[Si2O(PO4)6] ( P 3 ‾ $P\overline{3}$ , Z = 3, a = 14.6534(2), c = 7.3829(1) Å, R1 = 0.036, wR2 = 0.088, 4026 ind. refl., 171 variables), and TiIII 2TiIV 6(PO4)6[Si2O(PO4)6] ( R 3 ‾ $R\overline{3}$ , Z = 1, a = 8.446(2), c = 44.21(2) Å, R1 = 0.047, wR2 = 0.120, 1373 ind. refl., 109 variables) have been refined from single-crystal data. The structures show hexagonal closest packing of phosphate groups with metal cations and [Si2O] groups occupying octahedral voids [□(PO4)6]. The close relationship of these and other silicophosphate structures to the NiAs and β-Fe2(SO4)3 (see also NaZr2(PO4)3 “NASICON”) structure types is rationalized by group/subgroup considerations. This symmetry approach shows that systematic twinning is highly likely in silicophosphates, thus possibly leading to faulty crystal structure refinements. Our investigation strongly suggests that the proper composition of silicophosphates “M III 3P5SiO19” (M = Cr, V, Fe, Mo) reported in literature is actually M III 4-[Si2O(PO4)6]. In the mixed-valent compounds oxidation states were assigned to the cation sites by comparison to Ti2O3, TiP2O7 and FeTiO3. The powder reflectance spectrum of dark-blue FeII 0.79TiIII 2.42TiIV 0.79[Si2O(PO4)6] shows a strong IVCT transition at ν ˜ $\widetilde{\nu }$ = 17,500 cm−1, and magnetic susceptibility data agree very well with the proposed oxidation states.

The OH stretching region in infrared spectra of the apatite OH-Cl binary system

Abstract Polarized Fourier Transform Infrared (FTIR) microspectroscopy of the OH stretching region of hydroxylapatite-chlorapatite solid solutions presents novel problems for the assignment of peaks to specific OH-Cl pairs. Crystal structure refinements of Hughes et al. (2016) identified new positions for column anions in synthetic mixed Cl-OH apatites, with three different column anion arrangements depending on composition. These structural refinements, combined with bond valence calculations, allow for interpretation of the OH stretching region. A peak at 3574 cm-1 is identified as that from end member hydroxylapatite. A second major peak at 3548 cm-1 is only found in mixed chlorapatite-hydroxylapatite solid solutions, as is a third peak at 3592 cm-1. Both represent perturbations of the OH stretching vibration as compared to hydroxylapatite, to lower and higher frequency, respectively. Both of the new peaks are the result of a Clb-OH sequence, with adjacent anions in crystallographically similar positions, both above or both below adjacent mirror planes. One configuration has the hydrogen atom pointed towards the chlorine atom. The second has the hydrogen of the OH group pointed away from the chlorine atom. Both configurations present novel problems. The shift to lower wavenumber at 3548 cm-1 is characteristic of hydrogen bonding in fluorapatite-hydroxylapatite mixtures, yet the distance between O(H) and Clb is too great to allow it. The shift of OH stretching vibrations to lower wavenumber is produced through changes in polarization of intervening Cl-Ca2’ (or Ca2) and Ca2(’)-O3 bonds, which are affected by the presence of the large chlorine atom. Lowering the OH stretching vibration mimics the expected effect of a chlorine on a neighboring OH group in the apatite c-axis column, though without hydrogen bonding. The shift to higher wavenumbers, i.e. higher frequency at 3592 cm-1, is the opposite of that expected for hydrogen bonding between column anions in the apatite mineral group. It is ascribed to interaction between an adjacent Clb and the oxygen end of an adjacent OH dipole. This pairing places an oxygen and a chlorine atom in close proximity. Possible means of accommodation are discussed. A ubiquitous peak at 3498 cm-1 represents hydrogen bonding between an OH and the OHa site, with an interoxygen distance of about 2.9 Å. Published modeling supports the hypothesis that the OHa site is occupied by an O rather than an OH. However, no clear counterpart to this pairing is observed in crystal structure refinements for specimens lacking OHa, although the infrared absorbance is present. The existence of oxyapatite is inferred from studies of plasma-sprayed biomaterials, but the crystallographic details of the substitution have remained elusive. A minor shoulder at 3517 cm-1 does not have a clear counterpart in the structural refinements. Sequences of three columnar anions (e.g. OH-Cl-OH or Cl-OH-OH) can be ruled out, but unequivocal assignment awaits further research.

Synthesis under high pressure: crystal structure and properties of cubic Dy36O11F50[AsO3]12 ∙ H2O.

The multi-anionic compound with the composition Dy36O11F50[AsO3]12 ∙ H2O, which can be described in the non-centrosymmetric cubic space group F c, already shows an unusually large unit cell with an axis of a = 2587.59(14) pm. Its crystal structure exhibits isolated ψ1-tetrahedral [AsO3]3- anions, but both the coordination numbers and the linking schemes of the Dy3+-centered polyhedra differ significantly from the mostly layered structures described so far in literature. (Dy1)3+ is sevenfold coordinated by oxygen atoms and F- anions, forming a capped trigonal prism [(Dy1)O4.333F2.667]8.333-, and the remaining two cations (Dy2)3+ and (Dy3)3+ both reside in an eightfold coordination of anions. In both cases they form slightly distorted square antiprisms, which have the compositions of [(Dy2)O3.667F4.333]8.667- and [(Dy3)O4.667F3.333]9.667-, respectively. Some of the oxygen atoms are not part of ψ1-[AsO3]3- tetrahedra, but occur as O2- anions and one of these even shares a common crystallographic position with fluoride (F-). It is also worth mentioning that the single crystals were obtained as comparatively large cubes with an edge length of several 100µm providing very good data with regard to single-crystal X-ray diffraction. To verify the simultaneous presence of oxygen and fluorine, electron-beam microprobe analysis was carried out, and a single-crystal Raman spectrum ruled out the presence of hydroxide anions or protonated [AsO3]3- groups, but proved the interstitial crystal-water molecules, which could not be determined precisely by the crystal-structure refinement.

Potential of Y2Sn2O7:Eu3+, Dy3+ Inorganic Nanophosphors in Latent Fingermark Detection

In this work, we investigated the potential of Eu3+/Dy3+-codoped Y2Sn2O7 fluorescent nanophosphors to visualize latent fingermarks. We prepared these nanophosphors with various doping concentrations by the conventional coprecipitation reaction. The crystal structure, morphology, luminescence properties, and energy transfer mechanisms were studied. The crystalline phase was characterized by X-ray diffraction and crystal structure refinement using the Rietveld method. XRD measurements showed that the samples crystallized in the pure single pyrochlore phase with few more peaks originated from secondary phases and impurities generated during phosphor production, and that Eu3+ ions occupied D3d symmetry sites. The average crystallite size after mechanical grinding was less than 100 nm for all compositions. The optical characterization showed that, when excited under 532 nm, the Eu3+/Dy3+-codoped Y2Sn2O7 samples’ main intense emission peaks were located at 580–707 nm, corresponding to the 5D0→7Fj (j = l, 2, 3, and 4) transitions of europium. In fact, the 5D0→7F2 hypersensitive transition is strongly dependent on the local environment and was quite weak in Eu3+:Y2Sn2O7 at low Eu3+ doping levels. We found that the presence of Dy3+ as a codopant permitted enhancing the emission from this transition. The calculated PL CIE coordinates for the synthesized nanophosphors were very close to those of the reddish-orange region and only slightly dependent on the doping level. Various surfaces, including difficult ones (wood and ceramic), were successfully tested for latent fingerprint development with the prepared Eu3+/Dy3+-codoped Y2Sn2O7 fluorescent nanophosphor powder. Thanks to the high contrast obtained, fingerprint ridge patterns at all three levels were highlighted: core (level 1) islands, bifurcation, and enclosure (level 2), and even sweat pores (level 3).

Transferable Hirshfeld atom model for rapid evaluation of aspherical atomic form factors.

Form factors based on aspherical models of atomic electron density have brought great improvement in the accuracies of hydrogen atom parameters derived from X-ray crystal structure refinement. Today, two main groups of such models are available, the banks of transferable atomic densities parametrized using the Hansen-Coppens multipole model which allows for rapid evaluation of atomic form factors and Hirshfeld atom refinement (HAR)-related methods which are usually more accurate but also slower. In this work, a model that combines the ideas utilized in the two approaches is tested. It uses atomic electron densities based on Hirshfeld partitions of electron densities, which are precalculated and stored in a databank. This model was also applied during the refinement of the structures of five small molecules. A comparison of the resulting hydrogen atom parameters with those derived from neutron diffraction data indicates that they are more accurate than those obtained with the Hansen-Coppens based databank, and only slightly less accurate than those obtained with a version of HAR that neglects the crystal environment. The advantage of using HAR becomes more noticeable when the effects of the environment are included. To speed up calculations, atomic densities were represented by multipole expansion with spherical harmonics up to l = 7, which used numerical radial functions (a different approach to that applied in the Hansen-Coppens model). Calculations of atomic form factors for the small protein crambin (at 0.73 Å resolution) took only 68 s using 12 CPU cores.