A simple rule of thumb based on resolution is not adequate to identify the best treatment of atomic displacements in macromolecular structural models. The choice to use isotropic B factors, anisotropic B factors, TLS models or some combination of the three should be validated through statistical analysis of the model refinement.
In choosing and refining any crystallographic structural model, there is tension between the desire to extract the most detailed information possible and the necessity to describe no more than what is justified by the observed data. A more complex model is not necessarily a better model. Thus, it is important to validate the choice of parameters as well as validating their refined values. One recurring task is to choose the best model for describing the displacement of each atom about its mean position. At atomic resolution one has the option of devoting six model parameters (a ‘thermal ellipsoid’) to describe the displacement of each atom. At medium resolution one typically devotes at most one model parameter per atom to describe the same thing (a ‘B factor’). At very low resolution one cannot justify the use of even one parameter per atom. Furthermore, this aspect of the structure may be described better by an explicit model of bulk displacements, the most common of which is the translation/libration/screw (TLS) formalism, rather than by assigning some number of parameters to each atom individually. One can sidestep this choice between atomic displacement parameters and TLS descriptions by including both treatments in the same model, but this is not always statistically justifiable. The choice of which treatment is best for a particular structure refinement at a particular resolution can be guided by general considerations of the ratio of model parameters to the number of observations and by specific statistics such as the Hamilton R-factor ratio test.
atomic displacements; B factors; TLS models; model parameters
In comparison with a previous crystallographic study [Goh et al. (2002 ▶). J. Solid State Chem.
168, 119–125] of the title compound, silver diniobium tris(disulfide) tetrathiophosphate(V), that reports a full occupation of the silver position and isotropic displacement parameters for the atoms, the current redetermination reveals a silver deficiency with a site-occupation factor of 0.88 (1) and reports all atoms with anisotropic displacement parameters. The structure of Ag0.88Nb2PS10 is composed of ∞
1[Nb2PS10] chains, which are built up from pairs of distorted bicapped trigonal-prismatic [NbS8] polyhedra forming [Nb2S12] dimers and of tetrahedral [PS4] groups. These chains are connected via the statistically disordered Ag+ ions, forming double layers. Adjacent layers are stacked solely through van der Waals forces into a three-dimensional structure. Short and long Nb—Nb distances [2.880 (1) and 3.770 (2) Å, respectively] alternate along the chain and S2
2− and S2− anionic species are observed.
Single crystals of warwickite, trimagnesium titanium(IV) dioxide bis(borate), Mg3TiO2(BO3)2, were prepared by slow cooling of the melt. The title compound is isotypic with Co3TiO2(BO3)2. In contrast to the previous refinement of warwickite [Moore & Araki (1974 ▶). Am. Mineral.
59, 985–1004], that reported only isotropic atomic displacement parameters for all atoms, anisotropic displacement parameters of all atoms were refined during the current redetermination. All atoms are situated on special positions (site symmetry .m.). One of the two Mg sites is statistically disordered with Ti atoms (ratio 1:1), while the other is fully occupied by Mg atoms. The occupancy ratio of the Mg and Ti atoms is similar to that reported in the previous study. Metal atoms (M) at the Ti/Mg and Mg sites are coordinated by six O atoms in form of distorted octahedra. Four edge-sharing MO6 octahedra form M
4O18 units, which are connected by common corners into layers parallel to (010). Adjacent layers are linked along  into a framework structure by sharing common edges. The B atoms are located in the triangular prismatic tunnels of the framework.
An extension is proposed to the rigid-bond description of atomic thermal motion in crystals.
The rigid-bond model [Hirshfeld (1976 ▶). Acta Cryst. A32, 239–244] states that the mean-square displacements of two atoms are equal in the direction of the bond joining them. This criterion is widely used for verification (as intended by Hirshfeld) and also as a restraint in structure refinement as suggested by Rollett [Crystallographic Computing (1970 ▶), edited by F. R. Ahmed et al., pp. 167–181. Copenhagen: Munksgaard]. By reformulating this condition, so that the relative motion of the two atoms is required to be perpendicular to the bond, the number of restraints that can be applied per anisotropic atom is increased from about one to about three. Application of this condition to 1,3-distances in addition to the 1,2-distances means that on average just over six restraints can be applied to the six anisotropic displacement parameters of each atom. This concept is tested against very high resolution data of a small peptide and employed as a restraint for protein refinement at more modest resolution (e.g. 1.7 Å).
rigid-bond test; refinement restraints; anisotropic displacement parameters
The application of a new normal-mode-based X-ray crystallographic refinement method to a total of eight structures of moderate resolution is illustrated.
The structural refinement of large complexes at the lower resolution limit is often difficult and inefficient owing to the limited number of reflections and the frequently high-level structural flexibility. A new normal-mode-based X-ray crystallographic refinement method has recently been developed that enables anisotropic B-factor refinement using a drastically smaller number of thermal parameters than even isotropic refinement. Here, the method has been systematically tested on a total of eight systems in the resolution range 3.0–3.9 Å. This series of tests established the most applicable scenarios for the method, the detailed procedures for its application and the degree of structural improvement. The results demonstrated substantial model improvement at the lower resolution limit, especially in cases in which other methods such as the translation–libration–screw (TLS) model were not applicable owing to the poorly converged isotropic B-factor distribution. It is expected that this normal-mode-based method will be a useful tool for structural refinement, in particular at the lower resolution limit, in the field of X-ray crystallography.
conformational flexibility; anisotropic thermal parameters; moderate resolution
Guidelines and specific tests for validating macromolecular crystal structures that include TLS models are introduced. Validation may used to troubleshoot problems during refinement, to confirm the internal consistency of the model as part of deposition into the Protein Data Bank or to assess the plausibility of interpretating the boundary between two TLS groups as indicating a hinge point between structural domains.
The use of TLS (translation/libration/screw) models to describe anisotropic displacement of atoms within a protein crystal structure has become increasingly common. These models may be used purely as an improved methodology for crystallographic refinement or as the basis for analyzing inter-domain and other large-scale motions implied by the crystal structure. In either case it is desirable to validate that the crystallographic model, including the TLS description of anisotropy, conforms to our best understanding of protein structures and their modes of flexibility. A set of validation tests has been implemented that can be integrated into ongoing crystallographic refinement or run afterwards to evaluate a previously refined structure. In either case validation can serve to increase confidence that the model is correct, to highlight aspects of the model that may be improved or to strengthen the evidence supporting specific modes of flexibility inferred from the refined TLS model. Automated validation checks have been added to the PARVATI and TLSMD web servers and incorporated into the CCP4i user interface.
validation; TLS models; anisotropy
A robust method for determining bulk-solvent and anisotropic scaling parameters for macromolecular refinement is described. A maximum-likelihood target function for determination of flat bulk-solvent model parameters and overall anisotropic scale factor is also proposed.
A reliable method for the determination of bulk-solvent model parameters and an overall anisotropic scale factor is of increasing importance as structure determination becomes more automated. Current protocols require the manual inspection of refinement results in order to detect errors in the calculation of these parameters. Here, a robust method for determining bulk-solvent and anisotropic scaling parameters in macromolecular refinement is described. The implementation of a maximum-likelihood target function for determining the same parameters is also discussed. The formulas and corresponding derivatives of the likelihood function with respect to the solvent parameters and the components of anisotropic scale matrix are presented. These algorithms are implemented in the CCTBX bulk-solvent correction and scaling module.
bulk-solvent correction; anisotropic scaling
The structure of human carbonic anhydrase II has been solved with a sulfonamide inhibitor at 0.9 Å resolution. Structural variation and flexibility is seen on the surface of the protein and is consistent with the anisotropic ADPs obtained from refinement. Comparison with 13 other atomic resolution carbonic anhydrase structures shows that surface variation exists even in these highly ordered isomorphous crystals.
Carbonic anhydrase has been well studied structurally and functionally owing to its importance in respiration. A large number of X-ray crystallographic structures of carbonic anhydrase and its inhibitor complexes have been determined, some at atomic resolution. Structure determination of a sulfonamide-containing inhibitor complex has been carried out and the structure was refined at 0.9 Å resolution with anisotropic atomic displacement parameters to an R value of 0.141. The structure is similar to those of other carbonic anhydrase complexes, with the inhibitor providing a fourth nonprotein ligand to the active-site zinc. Comparison of this structure with 13 other atomic resolution (higher than 1.25 Å) isomorphous carbonic anhydrase structures provides a view of the structural similarity and variability in a series of crystal structures. At the center of the protein the structures superpose very well. The metal complexes superpose (with only two exceptions) with standard deviations of 0.01 Å in some zinc–protein and zinc–ligand bond lengths. In contrast, regions of structural variability are found on the protein surface, possibly owing to flexibility and disorder in the individual structures, differences in the chemical and crystalline environments or the different approaches used by different investigators to model weak or complicated electron-density maps. These findings suggest that care must be taken in interpreting structural details on protein surfaces on the basis of individual X-ray structures, even if atomic resolution data are available.
carbonic anhydrase; structure comparison; metalloproteins; atomic resolution
The crystal structure of katayamalite, ideally KLi3Ca7Ti2(SiO3)12(OH)2 (potassium trilithium heptacalcium dititanium dodecasilicate dihydroxide), was previously reported in triclinic symmetry (C-1), with isotropic displacement parameters for all atoms and without the H-atom position [Kato & Murakami (1985 ▶). Mineral. J.
12, 206–217]. The present study redetermines the katayamalite structure with monoclinic symmetry (space group C2/c) based on single-crystal X-ray diffraction data from a sample from the type locality, Iwagi Island, Ehime Prefecture, Japan, with anisotropic displacement parameters for all non-H atoms, and with the H atoms located by difference Fourier analysis. The structure of katayamalite contains a set of six-membered silicate rings interconnected by sheets of Ca atoms on one side and by an ordered mixture of Li, Ti and K atoms on the other side, forming layers which are stacked normal to (001). From the eight different metal sites, three are located on special positions, viz. one K and one Li atom on twofold rotation axes and one Ca atom on an inversion center. The Raman spectrum of kataymalite shows a band at 3678 cm−1, similar to that observed for hydroxyl-amphiboles, indicating no or very weak hydrogen bonding.
We employ a generalized van der Waals-Onsager perturbation theory to construct a free energy functional capable of describing the thermodynamic properties and orientational order of the isotropic and nematic phases of attractive disc particles. The model mesogen is a hard (purely repulsive) cylindrical disc particle decorated with an anisotropic square-well attractive potential placed at the centre of mass. Even for isotropic attractive interactions, the resulting overall inter-particle potential is anisotropic, due to the orientation-dependent excluded volume of the underlying hard core. An algebraic equation of state for attractive disc particles is developed by adopting the Onsager trial function to characterize the orientational order in the nematic phase. The theory is then used to represent the fluid-phase behaviour (vapour-liquid, isotropic-nematic, and nematic-nematic) of the oblate attractive particles for varying values of the molecular aspect ratio and parameters of the attractive potential. When compared to the phase diagram of their athermal analogues, it is seen that the addition of an attractive interaction facilitates the formation of orientationally-ordered phases. Most interestingly, for certain aspect ratios, a coexistence between two anisotropic nematic phases is exhibited by the attractive disc-like fluids.
equation of state; discotics; attractive cylindrical disc; nematic-nematic equilibria; anisotropic square well; phase diagrams
The crystal structure of kovdorskite, ideally Mg2PO4(OH)·3H2O (dimagnesium phosphate hydroxide trihydrate), was reported previously with isotropic displacement paramaters only and without H-atom positions [Ovchinnikov et al. (1980 ▶). Dokl. Akad. Nauk SSSR.
255, 351–354]. In this study, the kovdorskite structure is redetermined based on single-crystal X-ray diffraction data from a sample from the type locality, the Kovdor massif, Kola Peninsula, Russia, with anisotropic displacement parameters for all non-H atoms, with all H-atom located and with higher precision. Moreover, inconsistencies of the previously published structural data with respect to reported and calculated X-ray powder patterns are also discussed. The structure of kovdorskite contains a set of four edge-sharing MgO6 octahedra interconnected by PO4 tetrahedra and O—H⋯O hydrogen bonds, forming columns and channels parallel to . The hydrogen-bonding system in kovdorskite is formed through the water molecules, with the OH− ions contributing little, if any, to the system, as indicated by the long H⋯A distances (>2.50 Å) to the nearest O atoms. The hydrogen-bond lengths determined from the structure refinement agree well with Raman spectroscopic data.
Objective: The purposes of the study were (1) to evaluate the usefulness of the International Standards Organization (ISO) Reference Terminology Model for Nursing Diagnoses as a terminology model for defining nursing diagnostic concepts in the Medical Entities Dictionary (MED) and (2) to create the additional hierarchical structures required for integration of nursing diagnostic concepts into the MED.
Design and Measurements: The authors dissected nursing diagnostic terms from two source terminologies (Home Health Care Classification and the Omaha System) into the semantic categories of the ISO model. Consistent with the ISO model, they selected Focus and Judgment as required semantic categories for creating intensional definitions of nursing diagnostic concepts in the MED. Because the MED does not include Focus and Judgment hierarchies, the authors developed them to define the nursing diagnostic concepts.
Results: The ISO model was sufficient for dissecting the source terminologies into atomic terms. The authors identified 162 unique focus concepts from the 266 nursing diagnosis terms for inclusion in the Focus hierarchy. For the Judgment hierarchy, the authors precoordinated Judgment and Potentiality instead of using Potentiality as a qualifier of Judgment as in the ISO model. Impairment and Alteration were the most frequently occurring judgments.
Conclusions: Nursing care represents a large proportion of health care activities; thus, it is vital that terms used by nurses are integrated into concept-oriented terminologies that provide broad coverage for the domain of health care. This study supports the utility of the ISO Reference Terminology Model for Nursing Diagnoses as a facilitator for the integration process.
The crystal structure of the sweet protein MNEI at 1.15 Å resolution reveals networks of alternate conformations and stably bound negative ions.
The X-ray crystal structure of a single-chain monellin protein (MNEI) has been determined at 1.15 Å resolution. The model was refined to convergence employing anisotropic displacement parameters and riding H atoms to produce a final model with R
work and R
free values of 0.132 and 0.162, respectively. The crystal contains a single MNEI protein in the asymmetric unit and unusually lacks the dimer interface observed in all previous crystal structures of monellin and its single-chain derivatives. The high resolution allowed a more detailed view of MNEI than previously possible, with 38 of the 96 residues modelled with alternative side-chain conformations, including four core residues Thr12, Cys41, Leu62 and Ile75. Four stably bound negative ions were also located, providing new insight into potential electrostatic interactions of MNEI with the largely negatively charged surface of the sweet taste receptor T1R2–T1R3.
monellin; sweet protein; sweet taste; T1R2–T1R3 receptor
This paper presents a generalization of the Lockhart equation for plant cell/organ expansion in the anisotropic case. The intent is to take into account the temporal and spatial variation in the cell wall mechanical properties by considering the wall ‘extensibility’ (Φ), a time- and space-dependent parameter. A dynamic linear differential equation of a second-order tensor is introduced by describing the anisotropic growth process with some key biochemical aspects included. The distortion and expansion of plant cell walls initiated by expansins, a class of proteins known to enhance cell wall ‘extensibility’, is also described. In this approach, expansin proteins are treated as active agents participating in isotropic/anisotropic growth. Two-parameter models and an equation for describing α- and β-expansin proteins are proposed by delineating the extension of isolated wall samples, allowing turgor-driven polymer creep, where expansins weaken the non-covalent binding between wall polysaccharides. We observe that the calculated halftime (t1/2 = εΦ0 log 2) of stress relaxation due to expansin action can be described in mechanical terms.
creep rate; expansins; growth rate; plant cells; wall extensibility; growth equation
It is important to predict the tumor growth so that appropriate treatment can be planned in the early stage. In this letter, we propose a finite-element method (FEM)-based 3-D tumor growth prediction system using longitudinal kidney tumor images. To the best of our knowledge, this is the first kidney tumor growth prediction system. The kidney tissues are classified into three types: renal cortex, renal medulla, and renal pelvis. The reaction–diffusion model is applied as the tumor growth model. Different diffusion properties are considered in the model: the diffusion for renal medulla is considered as anisotropic, while those of renal cortex and renal pelvis are considered as isotropic. The FEM is employed to solve the diffusion model. Themodel parameters are estimated by the optimization of an objective function of overlap accuracy using a hybrid optimization parallel search package. The proposed method was tested on two longitudinal studies with seven time points on five tumors. The average true positive volume fraction and false positive volume fraction on all tumors is 91.4% and 4.0%, respectively. The experimental results showed the feasibility and efficacy of the proposed method.
Finite-element method (FEM); kidney tumor; segmentation; tumor growth prediction
The decision-making algorithms and software used in PDB_REDO to re-refine and rebuild crystallographic protein structures in the PDB are presented and discussed.
Developments of the PDB_REDO procedure that combine re-refinement and rebuilding within a unique decision-making framework to improve structures in the PDB are presented. PDB_REDO uses a variety of existing and custom-built software modules to choose an optimal refinement protocol (e.g. anisotropic, isotropic or overall B-factor refinement, TLS model) and to optimize the geometry versus data-refinement weights. Next, it proceeds to rebuild side chains and peptide planes before a final optimization round. PDB_REDO works fully automatically without the need for intervention by a crystallographic expert. The pipeline was tested on 12 000 PDB entries and the great majority of the test cases improved both in terms of crystallographic criteria such as R
free and in terms of widely accepted geometric validation criteria. It is concluded that PDB_REDO is useful to update the otherwise ‘static’ structures in the PDB to modern crystallographic standards. The publically available PDB_REDO database provides better model statistics and contributes to better refinement and validation targets.
validation; refinement; model building; automation; PDB
The crystal structure of Mg2B25, dimagnesium pentaeicosaboride, was reexamined from single-crystal X-ray diffraction data. The structural model previously reported on the basis of powder X-ray diffraction data [Giunchi et al. (2006 ▶). Solid State Sci.
8, 1202–1208] has been confirmed, although a much higher precision refinement was achieved, leading to much smaller standard uncertainties on bond lengths and refined occupancy factors. Moreover, all atoms were refined with anisotropic displacement parameters. Mg2B25 crystallizes in the β-boron structure type and is isostructural with other rhombohedral compounds of the boron-rich metal boride family. Magnesium atoms are found in interstitial sites on special positions (two with site symmetry .m, one with .2 and one with 3m), all with partial occupancies.
In the framework of a recently proposed method for in vivo lung morphometry, acinar lung airways are considered as a set of randomly oriented cylinders covered by alveolar sleeves. Diffusion of 3He in each airway is anisotropic and can be described by distinct longitudinal and transverse diffusion coefficients. This macroscopically isotropic but microscopically anisotropic model allows estimation of these diffusion coefficients from multi b-value MR experiments despite the airways being too small to be resolved by direct imaging. Herein a Bayesian approach is used for analyzing the uncertainties in the model parameter estimates. The approach allows evaluation of relative errors of the parameter estimates as functions of the “true” values of the parameters, the signal-to noise ratio, the maximum b-value and the total number of b-values used in the experiment. For a given set of the “true” diffusion parameters, the uncertainty in the estimated diffusion coefficients has a minimum as a function of maximum b-value and total number of data points. Choosing the MR pulse sequence parameters corresponding to this minimum optimizes the diffusion MR experiment and gives the best possible estimates of the diffusion coefficients. The mathematical approach presented can be generalized for models containing arbitrary numbers of estimated parameters.
diffusion MRI; hyperpolarized gas; lung airways; Bayesian analysis
Many biological materials are composites composed of a soft matrix reinforced with stiffer fibers. These stiffer fibers may have a tortuous shape and wind through the soft matrix. At small material deformation, these fibers deform in a bending mode and contribute little to the material stiffness; at large material deformation, these fibers deform in a stretching mode and induce a stiffening effect in the material behavior. The transition from bending mode deformation to stretching mode deformation yields a characteristic J-shape stress-strain curve. In addition, the spatial distribution of these fibers may render the composite an anisotropic behavior. In this paper, we present an anisotropic finite-deformation hyperelastic constitutive model for such materials. Here, the matrix is modeled as an isotropic neo-Hookean material. “The behaviors of single tortuous fiber are represented by a crimped fiber model”. The anisotropic behavior is introduced by a structure tensor representing the effective orientation distribution of crimped fibers. Parametric studies show the effect of fiber tortuosity and fiber orientation distribution on the overall stress-strain behaviors of the materials.
Fiber reinforced composites; Constitutive model; Collagen fiber; Soft tissues
The redetermined, low temperature (150 K), structure of tetra-n-butylammonium bromide, (C4H9)4N+·Br−, has been found to be merohedrally twinned via twin law −1 0 0, 0 − 1 0, 1 0 1. The structure was previously determined, with low precision, no inclusion of H atoms and only the bromide ion refined with anisotropic displacement parameters, by Wang et al. (1995 ▶). Mol. Cryst. Liq. Cryst. Sci. Tech. A, 264, 115–129. The redetermined structure has considerably improved precision in all geometrical parameters, has all non-H atoms refined anisotropically, H atoms included, and is isomorphous with the iodide analogue. The structure is otherwise routine, with the shortest cation to anion contacts being between the bromide anion and the CH atoms close to the ammonium nitrogen centre at a distance of ca. 2.98–3.11 Å. Each anion makes eight such contacts to four different anions. The n-butyl chains are fully extended, adopting an all-anti conformation with approximate S
4 point symmetry.
The structure of the hexagonal modification of caesium hexaaquamagnesium phosphate has been redetermined from single-crystal X-ray data. The previous refinement from photographic data [Ferrari, Calvaca & Nardelli (1955 ▶). Gazz. Chim. Ital.
85, 1232–1238] was basically confirmed, but with all H atoms located and with all non H-atoms refined with anisotropic displacement parameters. The structure can be derived from the NiAs structure type: the PO4 tetrahedra (3m. symmetry) are on the Ni positions and the complex [Mg(OH2)6] octahedra (3m. symmetry) are on the As positions. The building units are connected to each other by hydrogen bonds. The Cs+ cations (3m. symmetry) are located in the voids of this arrangement and exhibit a distorted cuboctahedral 12-coordination by the O atoms of the water molecules.
Cigarette smoking is the leading self-inflicted risk factor for cardiovascular diseases; it causes arterial stiffening with serious sequelea including atherosclerosis and abdominal aortic aneurysms. This work presents a new interpretation of arterial stiffening caused by smoking based on data published for rat pulmonary arteries. A structurally motivated “four fiber family” constitutive relation was used to fit the available biaxial data and associated best-fit values of material parameters were estimated using multivariate nonlinear regression. Results suggested that arterial stiffening caused by smoking was reflected by consistent increase in an elastin-associated parameter and moreover by marked increase in the collagen-associated parameters. That is, we suggest that arterial stiffening due to cigarette smoking appears to be isotropic, which may allow simpler phenomenological models to capture these effects using a single stiffening parameter similar to the approach in isotropic continuum damage mechanics. There is a pressing need, however, for more detailed histological information coupled with more complete biaxial mechanical data for a broader range of systemic arteries.
Elastin; Collagen; Anisotropy; Stress; Biomechanics
The crystal structure of the title compound, [Sn(C5H10NS2)4], was originally determined by Harreld & Schlemper [Acta Cryst. (1971), B27, 1964–1969] using intensity data estimated from Weissenberg films. In comparison with the previous refinement, the current redetermination reveals anisotropic displacement parameters for all non-H atoms, localization of the H atoms, and higher precision of lattice parameters and interatomic distances. The complex features a distorted S6 octahedral coordination geometry for tin and a cis disposition of the monodentate dithiocarbamate ligands.
The criterion to determine residue contact is a fundamental problem in deriving knowledge-based mean-force potential energy calculations for protein structures. A frequently used criterion is to require the side chain center-to-center distance or the -to- atom distance to be within a pre-determined cutoff distance. However, the spatially anisotropic nature of the side chain determines that it is challenging to identify the contact pairs. This study compares three side chain contact models: the Atom Distance criteria (ADC) model, the Isotropic Sphere Side chain (ISS) model and the Anisotropic Ellipsoid Side chain (AES) model using 424 high resolution protein structures in the Protein Data Bank. The results indicate that the ADC model is the most accurate and ISS is the worst. The AES model eliminates about 95% of the incorrectly counted contact-pairs in the ISS model. Algorithm analysis shows that AES model is the most computational intensive while ADC model has moderate computational cost. We derived a dataset of the mis-estimated contact pairs by AES model. The most misjudged pairs are Arg-Glu, Arg-Asp and Arg-Tyr. Such a dataset can be useful for developing the improved AES model by incorporating the pair-specific information for the cutoff distance.
The crystal structure of the title compound, DyNi3, was redetermined from single-crystal X-ray diffraction data. In comparison with previous studies based on powder X-ray diffraction data [Lemaire & Paccard (1969 ▶). Bull. Soc. Fr. Minéral. Cristallogr.
92, 9–16; Tsai et al. (1974 ▶). J. Appl. Phys.
45, 3582–3586], the present redetermination revealed refined coordinates and anisotropic displacement parameters for all atoms. The crystal structure of DyNi3 adopts the PuNi3 structure type and can be derived from the CaCu5 structure type as an intergrowth structure. The asymmetric unit contains two Dy sites (site symmetries 3m and -3) and three Ni sites (m, 3m and -3m). The two different coordination polyhedra of Dy are a Frank–Kasper polyhedron formed by four Dy and 12 Ni atoms and a pseudo-Frank–Kasper polyhedron formed by two Dy and 18 Ni atoms. The three different coordination polyhedra of Ni are Frank–Kasper icosahedra formed by five Dy and seven Ni atoms, three Dy and nine Ni atoms, and six Dy and six Ni atoms.