Anisotropic Atomic Displacement Parameters Research Articles

Synthesis and in-depth structure determination of a novel metastable high-pressure CrTe3 phase

This study reports the synthesis and crystal structure determination of a novel CrTe3 phase using various experimental and theoretical methods. The average stoichiometry and local phase separation of this quenched high-pressure phase were characterized by ex situ synchrotron powder X-ray diffraction and total scattering. Several structural models were obtained using simulated annealing, but all suffered from an imperfect Rietveld refinement, especially at higher diffraction angles. Finally, a novel stoichiometrically correct crystal structure model was proposed on the basis of electron diffraction data and refined against powder diffraction data using the Rietveld method. Scanning electron microscopy–energy-dispersive X-ray spectrometry (EDX) measurements verified the targeted 1:3 (Cr:Te) average stoichiometry for the starting compound and for the quenched high-pressure phase within experimental errors. Scanning transmission electron microscopy (STEM)–EDX was used to examine minute variations of the Cr-to-Te ratio at the nanoscale. Precession electron diffraction (PED) experiments were applied for the nanoscale structure analysis of the quenched high-pressure phase. The proposed monoclinic model from PED experiments provided an improved fit to the X-ray patterns, especially after introducing atomic anisotropic displacement parameters and partial occupancy of Cr atoms. Atomic resolution STEM and simulations were conducted to identify variations in the Cr-atom site-occupancy factor. No significant variations were observed experimentally for several zone axes. The magnetic properties of the novel CrTe3 phase were investigated through temperature- and field-dependent magnetization measurements. In order to understand these properties, auxiliary theoretical investigations have been performed by first-principles electronic structure calculations and Monte Carlo simulations. The obtained results allow the observed magnetization behavior to be interpreted as the consequence of competition between the applied magnetic field and the Cr–Cr exchange interactions, leading to a decrease of the magnetization towards T = 0 K typical for antiferromagnetic systems, as well as a field-induced enhanced magnetization around the critical temperature due to the high magnetic susceptibility in this region.

Mechanism of Negative Thermal Expansion in Monoclinic Cu2P2O7 from First Principles.

Negative thermal expansion (NTE) materials generally have high-symmetry space groups, large average atomic volumes, and corner-sharing octahedral and tetrahedral coordination structures. By contrast, monoclinic α-Cu2P2O7, which has a small average atomic volume and edge-sharing structure, has been reported to exhibit NTE, the detailed mechanism of which is unclear. In this study, we investigate the A2B2O7 polymorphs and analyze the NTE behavior of α-Cu2P2O7 using first-principles lattice-dynamics calculations. From the polymorphism investigation in 20 A2B2O7 compounds using 6 representative crystal structures, small A and B cationic radii are found to stabilize the α-Cu2P2O7-type structure. We then analyze the NTE behavior of α-Cu2P2O7 using quasi-harmonic approximation. Our calculated thermal expansion coefficients and anisotropic atomic displacement parameters were in good agreement with those of the experimental reports at low temperatures. From the mode-Grüneisen parameter distribution plotted over the entire first-Brillouin zone, we found that the phonon contributing most significantly to NTE emerges not into the special points but between them. In this phonon mode, the O connecting two PO4 tetrahedra rotates, and the Cu and O vibrate perpendicular to the bottom of the CuO5 pyramidal unit, which folds the ac lattice plane. This vibration behavior can explain the experimentally reported anisotropic NTE behavior of α-Cu2P2O7. Our results demonstrate that the most negative mode-Grüneisen parameter contributing to NTE behavior is not always located on high-symmetry special points, indicating the importance of lattice vibration analyses for the entire first-Brillouin zone.

Preparation, crystal structure and Li ion conductivity of Li10.8Ba1.85Al3.5O12F

Polycrystalline ceramics comprising the newly discovered compound Li10.8Ba1.85Al3.5O12F were synthesized by heating a compacted mixture of Li2O, BaO and Al2O3 together with LiF as a fluorine source at 1023 K for 6 h in an Ar atmosphere. Single crystals of this oxide were subsequently obtained from a melt-solidified sample prepared by heating and slow cooling of the compact from 1133 K. Single-crystal X-ray diffraction established that this compound had a trigonal crystal structure (a = 6.03346(10) Å, c =10.13780(17) Å, space group P-3) containing Li split sites around the (001) plane at z = 0 with approximately 1/3 occupancy and large anisotropic atomic displacement parameters. The Li ion conductivities for the samples prepared by reaction sintering at 1023 K (relative density 55%) and by sintering of a powder compact at 1123 K (relative density 87%) were determined to be 1.6(3) × 10−8 and 2.8(2) × 10−8 S·cm−1 at 305 K and 3.9(2) × 10−6 and 1.0(2) × 10−5 S·cm−1 at 423 K, respectively, while the activation energies for conduction of the samples was found to be 0.52(2) and 0.55(3) eV, respectively. Cyclic voltammetry data acquired using Li and Ta electrodes indicated that this compound was stable in contact with Li metal and demonstrated a Li ion transport number >99%.

Synthesis, crystal structure, photoluminescence of Ba3Y2(BO3)4:Er3+ and thermal expansion of Ba3Y2(BO3)4

The Ba3Y2–xErx(BO3)4 (х = 0.01, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3) phosphors were obtained by crystallization from a melt. The Ba3Y2(BO3)4 crystal structure was refined from single crystal X-ray diffraction data to R = 0.037. Its anisotropic atomic displacement parameters for all atoms were refined for the first time. The borate crystallizes in the orthorhombic crystal system, space group Pnma with unit cell parameters a = 7.673(1), b = 16.44(1), c = 8.977(2) Å, V = 1132.3(3) Å3, Z = 4. These phosphors are isotypical to those of the A3M2(BO3)4 (A = Ca, Sr, Ba, M = Ln, Y, Bi) family. The crystal structure contains the isolated BO3 triangles, two general and a special one independent crystallographic sites for large cations, which are disordered over sites. Thermal behavior of Ba3Y2(BO3)4 was investigated by high-temperature X-ray powder diffraction and thermal expansion coefficients are calculated in a wide temperature range. An inflections of temperature dependencies of the unit cell parameters is observed in a range 600–740 °C. Luminescence spectra, excitation and kinetic curves of the Ba3Y2(BO3)4:Er3+ series are reported. A maximum luminescence intensity is observed for the x = 0.1 sample. According to vibrational spectroscopy data no structural changes upon activation of the Ba3Y2(BO3)4 matrix with the Er ions are observed.

Temperature-dependent crystal structure investigation of 4f hybridized thermoelectric clathrate Ba8-xCexAuySi46-y.

Thermoelectric materials allow for conversion of waste heat into electrical energy, and they represent a green solution for improving our energy efficiency. Inclusion of 4f electrons near the Fermi level may boost the Seebeck coefficient, which is essential for high thermoelectric performance. In this study, Ce was successfully substituted for Ba on the guest atom sites in the type-I clathrate Ba8-xCexAuySi46-y and the material was characterized using high-resolution synchrotron powder X-ray diffraction data measured from 100 K to 1000 K to investigate potential structural implications of the inclusion of a 4f element. The thermal expansion and bonding of the host structure are not affected by the presence of Ce, as seen from the linear coefficient of unit-cell thermal expansion of 7.30 (8) × 10-6 K-1 and the average host Debye temperature of 404 (7) K determined from the multi-temperature atomic displacement parameters, both of which are similar to values obtained for pure Ba8AuySi46-y. The anisotropic atomic displacement parameters on the guest atom site in the large clathrate cage populated by Ba surprisingly reveals isotropic behavior, which is different from all other clathrates reported in literature, and thus represents a unique host-guest bonding situation.

Probing Molecular Motions in Metal-Organic Frameworks by Three-Dimensional Electron Diffraction.

Flexible metal–organic frameworks (MOFs) are known for their vast functional diversities and variable pore architectures. Dynamic motions or perturbations are among the highly desired flexibilities, which are key to guest diffusion processes. Therefore, probing such motions, especially at an atomic level, is crucial for revealing the unique properties and identifying the applications of MOFs. Nuclear magnetic resonance (NMR) and single-crystal X-ray diffraction (SCXRD) are the most important techniques to characterize molecular motions but require pure samples or large single crystals (>5 × 5 × 5 μm3), which are often inaccessible for MOF synthesis. Recent developments of three-dimensional electron diffraction (3D ED) have pushed the limits of single-crystal structural analysis. Accurate atomic information can be obtained by 3D ED from nanometer- and submicrometer-sized crystals and samples containing multiple phases. Here, we report the study of molecular motions by using the 3D ED method in MIL-140C and UiO-67, which are obtained as nanosized crystals coexisting in a mixture. In addition to an ab initio determination of their framework structures, we discovered that motions of the linker molecules could be revealed by observing the thermal ellipsoid models and analyzing the atomic anisotropic displacement parameters (ADPs) at room temperature (298 K) and cryogenic temperature (98 K). Interestingly, despite the same type of linker molecule occupying two symmetry-independent positions in MIL-140C, we observed significantly larger motions for the isolated linkers in comparison to those reinforced by π–π stacking. With an accuracy comparable to that of SCXRD, we show for the first time that 3D ED can be a powerful tool to investigate dynamics at an atomic level, which is particularly beneficial for nanocrystalline materials and/or phase mixtures.

Investigation of the Anisotropic Thermal Expansion Mechanism of AgxGaxGe1-xSe2 Crystals.

Quaternary nonlinear optical single crystals AgxGaxGe1-xSe2 (x = 0.250, 0.167) were grown by the Bridgman method in a four-zone furnace. The thermal expansion behavior of AgxGaxGe1-xSe2 (x = 0.25, 0.167) was studied by the method of single-crystal X-ray diffraction from 150 to 295 K and powder X-ray diffraction in the range of 298-773 K. Both results show the crystals have positive linear thermal expansion coefficients in different directions and a positive volume thermal expansion coefficient, and it is observed that they satisfy the relationship of αa > αc > αb and αV ≈ αa + αb + αc for the orthorhombic structure. It is found that the AgxGaxGe1-xSe2 (x = 0.25, 0.167) unit cells varying with temperature were mainly dominated by variations in framework geometry (AgSe4 tetrahedron), and the thermal motion of Ag atoms in the AgSe4 tetrahedron. As it was revealed, according to the powder X-ray diffraction, it is found that the isotropic thermal atomic displacement parameter of the Ag atoms is much larger than those of the Se and Ga(Ge) atoms in the AgSe4 tetrahedron. Furthermore, anisotropic atomic displacement parameters (ADPs) of Ag atoms are extracted from the single-crystal diffraction; the ADPs along the a axis, b axis, and c axis have a significant difference, which means the thermal vibration of Ag atoms is anisotropic. It is of great significance for improving crystal growth technology and understanding the thermal properties of this kind of crystals.

X-ray Crystal Structure Analysis of Magnetoelectric Metal UNi4B

The crystal structure of the magnetoelectric metal UNi4B was determined from high-resolution synchrotron X-ray diffraction experiments using single-crystalline samples. The direct method and the least-square refinement indicate a structural model with the space group Cmcm (no. 63, \(D_{2h}^{17}\)), where U atoms form a pseudo-triangular lattice in the U–Ni–B layers stacking along the a-axis alternately with the Ni layers. The refined values of the structural parameters including anisotropic atomic displacement parameters are provided for all atoms in the unit cell. The obtained structural model is discussed in terms of the magnetoelectric effect reported for this compound.

Crystal Structure of Moganite and Its Anisotropic Atomic Displacement Parameters Determined by Synchrotron X-ray Diffraction and X-ray/Neutron Pair Distribution Function Analyses

The crystal structure of moganite from the Mogán formation on Gran Canaria has been re-investigated using high-resolution synchrotron X-ray diffraction (XRD) and X-ray/neutron pair distribution function (PDF) analyses. Our study for the first time reports the anisotropic atomic displacement parameters (ADPs) of a natural moganite. Rietveld analysis of synchrotron XRD data determined the crystal structure of moganite with the space group I2/a. The refined unit-cell parameters are a = 8.7363(8), b = 4.8688(5), c = 10.7203(9) Å, and β = 90.212(4)°. The ADPs of Si and O in moganite were obtained from X-ray and neutron PDF analyses. The shapes and orientations of the anisotropic ellipsoids determined from X-ray and neutron measurements are similar. The anisotropic ellipsoids for O extend along planes perpendicular to the Si-Si axis of corner-sharing SiO4 tetrahedra, suggesting precession-like movement. Neutron PDF result confirms the occurrence of OH over some of the tetrahedral sites. We postulate that moganite nanomineral is stable with respect to quartz in hypersaline water. The ADPs of moganite show a similar trend as those of quartz determined by single-crystal XRD. In short, the combined methods can provide high-quality structural parameters of moganite nanomineral, including its ADPs and extra OH position at the surface. This approach can be used as an alternative means for solving the structures of crystals that are not large enough for single-crystal XRD measurements, such as fine-grained and nanocrystalline minerals formed in various geological environments.

Multipole electron densities and structural parameters from synchrotron powder X-ray diffraction data obtained with a MYTHEN detector system (OHGI).

Powder X-ray diffraction has some inherent advantages over traditional single-crystal X-ray diffraction in accurately determining electron densities and structural parameters due to the lower requirements for sample crystallinity, simpler corrections and measurement simultaneity. For some simple inorganic materials, it has been shown that these advantages can compensate for disadvantages such as peak overlap and error-prone background subtraction. Although it is challenging to extend powder X-ray diffraction-based electron-density studies to organic materials with significant peak overlap, previous results using a dedicated vacuum diffractometer with a large image-plate camera (AVID) demonstrated that it can be done. However, the vacuum setup with the off-line detector system was found to prohibit a widespread use. Fast microstrip detectors, which have been employed at a number of powder diffraction beamlines, have the potential to facilitate electron-density studies. Nevertheless, no electron-density studies even for materials with slight peak overlap have been performed with microstrip detectors. One of the most critical problems has been a difference in sensitivity between microstrip channels, which substantially defines the dynamic range of a detector. Recently, a robust approach to this problem has been developed and applied to a total scattering measurement system (OHGI) with 15 MYTHEN microstrip modules. In the present study, synchrotron powder X-ray diffraction data obtained with OHGI are evaulated in terms of multipole electron densities and structural parameters (atomic positions and displacement parameters). These results show that, even without a dedicated setup and perfect samples, electron-density modelling can be carried out on high-quality powder X-ray diffraction data. However, it was also found that the required prior information about the sample prohibits widespread use of the method. With the presently obtainable data quality, electron densities of molecular crystals in general are not reliably obtained from powder data, but it is an excellent, possibly superior, alternative to single-crystal measurements for small-unit-cell inorganic solids. If aspherical atomic scattering factors can be obtained from other means (multipole databases, theoretical calculations), then atomic positions (including for hydrogen) and anisotropic atomic displacement parameters (non-hydrogen atoms) of excellent accuracy can be refined from synchrotron powder X-ray diffraction data on organic crystals.

Single-crystal neutron and X-ray diffraction study of garnet-type solid-state electrolyte Li6La3ZrTaO12: an in situ temperature-dependence investigation (2.5 ≤ T ≤ 873 K)

Large single crystals of garnet-type Li6La3ZrTaO12 (LLZTO) were grown by the Czochralski method and analysed using neutron diffraction between 2.5 and 873 K in order to fully characterize the Li atom distribution, and possible Li ion mobility in this class of potential candidates for solid-state electrolyte battery material. LLZTO retains its cubic symmetry (space group Ia 3 d) over the complete temperature range. When compared to other sites, the octahedral sites behave as the most rigid unit and show the smallest increase in atomic displacement parameters and bond length. The La and Li sites show similar thermal expansion in their bond lengths with temperature, and the anisotropic and equivalent atomic displacement parameters exhibit a distinctly larger increase at temperatures above 400 K. Detailed inspection of nuclear densities at the Li1 site reveal a small but significant displacement from the 24d position to the typical 96h position, which cannot, however, be resolved from the single-crystal X-ray diffraction data. The site occupation of LiI ions on Li1 and Li2 sites remains constant, so there is no change in site occupation with temperature.

First principles calculations of the vibrational properties of single and dimer F-type centers in corundum crystals.

The present paper investigates the F-type centers in α-Al2O3 through their electronic and vibrational properties from first principle calculations using a periodic supercell approach, a hybrid functional, and all-electron Gaussian basis sets as implemented in the CRYSTAL17 code. Single F-type and dimer F2-type centers related to oxygen vacancies in various charge states were considered. The defect-induced vibrational modes were identified and found to appear mainly in the low (up to 300 cm-1) and high (above 700 cm-1) frequency regions, depending on the defect charge. The perturbation introduced by the defects to the thermal nuclear motion in the crystal lattice is discussed in terms of atomic anisotropic displacement parameters. The calculated Raman spectra are discussed for the first time for such defects in α-Al2O3, suggesting important information for future experimental and theoretical studies and revealing deeper insight into their behavior.

Redetermination of the crystal structure of R 5Si4 (R = Pr, Nd) from single-crystal X-ray diffraction data

The crystal structures of praseodymium silicide (5/4), Pr5Si4, and neodymium silicide (5/4), Nd5Si4, were redetermined using high-quality single-crystal X-ray diffraction data. The previous structure reports of Pr5Si4 were only based on powder X-ray diffraction data [Smith et al. (1967 ▸). Acta Cryst. 22 940-943; Yang et al. (2002b ▸). J. Alloys Compd. 339, 189-194; Yang et al., (2003 ▸). J. Alloys Compd. 263, 146-153]. On the other hand, the structure of Nd5Si4 has been determined from powder data [neutron; Cadogan et al., (2002 ▸). J. Phys. Condens. Matter, 14, 7191-7200] and X-ray [Smith et al. (1967 ▸). Acta Cryst. 22 940-943; Yang et al. (2002b ▸). J. Alloys Compd. 339, 189-194; Yang et al., (2003 ▸). J. Alloys Compd. 263, 146-153] and single-crystal data with isotropic atomic displacement parameters [Roger et al., (2006 ▸). J. Alloys Compd. 415, 73-84]. In addition, the anisotropic atomic displacement parameters for all atomic sites have been determined for the first time. These compounds are confirmed to have the tetra-gonal Zr5Si4-type structure (space group: P41212), as reported previously (Smith et al., 1967 ▸). The structure is built up by distorted body-centered cubes consisting of Pr(Nd) atoms, which are linked to each other by edge-sharing to form a three-dimensional framework. This framework delimits zigzag channels in which the silicon dimers are situated.

Mobility of water and of protein atoms at the protein-water interface, monitored by anisotropic atomic displacement parameters, are largely uncorrelated.

A non-redundant set of 231 protein crystal structures refined at a resolution better than (or equal to) 1Å was extracted from the Protein Data Bank and the degree of conformational rigidity at the protein-water interface was examined by means of the Hirshfeld test and by comparing the orientations of the anisotropic Us for contacting protein and water atoms. Contacts between protein and water atoms are more rigid that contacts between water atoms and the degree of rigidity increases for shorter contacts and for more hydrogen-bonded atoms. Nevertheless, water and protein atoms are not rigidly held together. On the contrary, they seem to have little influence on their mobility to such an extent that hydration water, different from the protein atoms, cannot be considered to be properly in the solid state.

Using Complementary Methods of Synchrotron Radiation Powder Diffraction and Pair Distribution Function to Refine Crystal Structures with High Quality Parameters—A Review

Determination of the atomic-scale structures of certain fine-grained minerals using single-crystal X-ray diffraction (XRD) has been challenging because they commonly occur as submicron and nanocrystals in the geological environment. Synchrotron powder diffraction and scattering techniques are useful complementary methods for studying this type of minerals. In this review, we discussed three example studies investigated by combined methods of synchrotron radiation XRD and pair distribution function (PDF) techniques: (1) low-temperature cristobalite; (2) kaolinite; and (3) vernadite. Powder XRD is useful to determine the average structure including unit-cell parameters, fractional atomic coordinates, occupancies and isotropic atomic displacement parameters. X-ray/Neutron PDF methods are sensitive to study the local structure with anisotropic atomic displacement parameters (ADP). The results and case studies suggest that the crystal structure and high-quality ADP values can be obtained using the combined methods. The method can be useful to characterize crystals and minerals that are not suitable for single-crystal XRD.

High temperature neutron diffraction investigation of U3Si2

The U-Si system is actively undergoing studies due to its promise as a component of an accident tolerant nuclear fuel. The crystal structure of the U3Si2 compound in the U-Si system was investigated as a function of temperature from room temperature to 1298 K for a sample of approximately 42.02 at.% Si composition and 1373 K for a sample with approximately 42.10 at.% Si using high temperature time-of-flight neutron diffraction on the High-Pressure Preferred Orientation (HIPPO) diffractometer at Los Alamos Neutron Science Center (LANSCE). The simultaneous Rietveld refinement of five histograms from the five HIPPO detector backs (40°, 60°, 90°, 120°, and 145°) provided datasets for the lattice parameters, anisotropic atomic displacement parameters, and atomic positions as a function of temperature.To explore the possibility of a homogeneity range, two sample compositions were analyzed, a 40.02 at.% Si (near stoichiometric) and a potentially hyperstoichiometric (40.10 at.% Si) sample. While minor differences in the anisotropic atomic displacement parameters between the two samples were observed, over the entire investigated temperature range no additional phases were found indicative of a single phase sample and the ability of the U3Si2 compound to accommodate nonstoichiometry, suggesting that the U3Si2 compound is not a line compound confirming previous determinations. However, the differences observed in the average coefficient of linear thermal expansion identified between the two compositions warrant future investigation.SynopsisHigh temperature time-of-flight neutron diffraction measurements were carried out on 2 different compositions of U3Si2 and detailed analyses of the U3Si2 crystal structure as a function of temperature to 1298 K (for a 42.02 at.% Si sample) and 1373 K (for a 42.10 at.% Si sample) were performed. Anisotropic atomic displacement parameters, bond lengths, thermal expansion, coefficient of thermal expansion, and related properties were assessed as a function of temperature as well as the location of excess Si atoms and compared with simulation predictions.

Preparation of Submicron CaCu3Ti4O12 Dispersions and Filled Epoxy Compositions Based on Them

We have developed a technique for the preparation of a dielectric ceramic CaCu3Ti4O12 filler and submicron dispersions of the filler in epoxy. The crystal structure of CaCu3Ti4O12 has been refined using high-resolution neutron diffraction data with anisotropic atomic displacement parameters. We have studied the particle size distribution and rheological properties of CaCu3Ti4O12 dispersions and, using rheological data, identified the surfactants that ensure Newtonian rheology of dispersions, convenient for filling porous templates. Surfactants have been shown to have a significant effect on dielectric characteristics of filled epoxy compositions.

Using powder XRD and pair distribution function to determine anisotropic atomic displacement parameters of orthorhombic tridymite and tetragonal cristobalite.

Determination of the crystal structures of low-temperature tridymite and cristobalite using single-crystal XRD has been challenging because they generally occur as metastable fine-grained crystals in the geological environment. Therefore, using powder diffraction and scattering techniques is critical to study the low-temperature crystals. Synchrotron powder X-ray diffraction (XRD), pair distribution function (PDF) and transmission electron microscopy were used to investigate the structure of orthorhombic tridymite with C2221 symmetry and tetragonal cristobalite with P41212 symmetry, including their anisotropic atomic displacement parameters (ADPs). Rietveld refinement was used to determine the unit-cell parameters, fractional coordinates and isotropic atomic displacement parameters (Uiso) of the tridymite and cristobalite. The PDF method was used to determine ADPs for each atom. The results suggest that the crystal structure with high quality ADP values can be obtained using the combined methods of XRD and PDF analyses. The method can be used for characterizing crystals that are not suitable for single-crystal XRD.

TensorView: A software tool for displaying NMR tensors.

The representation of nuclear magnetic resonance (NMR) tensors as surfaces on three-dimensional molecular models is an information-rich presentation that highlights the geometric relationship between tensor principal components and the underlying molecular and electronic structure. Here, we describe a new computational tool, TensorView, for depicting NMR tensors on the molecular framework. This package makes use of the graphical interface and built-in molecular display functionality present within the Mathematica programming environment and is robust for displaying tensor properties from a broad range of commercial and user-specific computational chemistry packages. Two mathematical forms for representing tensor interaction surfaces are presented, the popular ellipsoidal construct and the more technically correct "ovaloid" form. Examples are provided for chemical shielding and shift tensors, dipole-dipole and quadrupolar couplings, and atomic anisotropic displacement parameters (thermal ellipsoids) derived from NMR crystallography.

From deep TLS validation to ensembles of atomic models built from elemental motions. II. Analysis of TLS refinement results by explicit interpretation.

TLS modelling was developed by Schomaker and Trueblood to describe atomic displacement parameters through concerted (rigid-body) harmonic motions of an atomic group [Schomaker & Trueblood (1968), Acta Cryst. B24, 63-76]. The results of a TLS refinement are T, L and S matrices that provide individual anisotropic atomic displacement parameters (ADPs) for all atoms belonging to the group. These ADPs can be calculated analytically using a formula that relates the elements of the TLS matrices to atomic parameters. Alternatively, ADPs can be obtained numerically from the parameters of concerted atomic motions corresponding to the TLS matrices. Both procedures are expected to produce the same ADP values and therefore can be used to assess the results of TLS refinement. Here, the implementation of this approach in PHENIX is described and several illustrations, including the use of all models from the PDB that have been subjected to TLS refinement, are provided.