Anharmonic Temperature Factors Research Articles

Anharmonic Thermal Motion Modelling in the Experimental XRD Charge Density Determination of 1-Methyluracil at T = 23 K.

The experimental electron density distribution (EDD) of 1-methyluracil (1-MUR) was obtained by single crystal X-ray diffraction (XRD) experiments at 23 K. Four different structural models fitting an extensive set of XRD data to a resolution of (sinθ/λ)max = 1.143 Å−1 are compared. Two of the models include anharmonic temperature factors, whose inclusion is supported by the Hamilton test at a 99.95% level of confidence. Positive Fourier residuals up to 0.5 eÅ–3 in magnitude were found close to the methyl group and in the region of hydrogen bonds. Residual density analysis (RDA) and molecular dynamics simulations in the solid-state demonstrate that these residuals can be likely attributed to unresolved disorder, possibly dynamical and long–range in nature. Atomic volumes and charges, molecular moments up to hexadecapoles, as well as maps of the molecular electrostatic potential were obtained from distributed multipole analysis of the EDD. The derived electrostatic properties neither depend on the details of the multipole model, nor are significantly affected by the explicit inclusion of anharmonicity in the least–squares model. The distribution of atomic charges in 1-MUR is not affected by the crystal environment in a significant way. The quality of experimental findings is discussed in light of in-crystal and gas-phase quantum simulations.

Structural fluctuation of Pb(Mg1/3Nb2/3)O3 in the cubic phase

To investigate the structural fluctuations of relaxor Pb(Mg1/3Nb2/3)O3 (PMN) in the cubic phase, the probability density function of each atom was determined by high-energy synchrotron radiation X-ray powder diffraction measurements and a Rietveld analysis with anharmonic temperature factors. A rotator-like behavior of the Pb ion was observed in the cubic phase. The off-center behavior along the 〈1 1 1〉 directions became pronounced with decreasing temperature. The ferroelectricity of PMN was attributed to a combination of the freezing of the 〈1 1 1〉-preferred rotator-like disorder and the ferroelectric distortion of the B–O6 octahedron.

Structure fluctuation in Gd- and Mg-substituted BaTiO3 with cubic structure

Gd- and Mg-substituted barium titanate crystals (Ba1−xGdx)(Ti1−x/2Mgx/2)O3 with the Gd composition range of x > 0.1 show no ferroelectric phase transition. To discuss atomic thermal fluctuations in such unusual barium titanate crystals with cubic symmetry, accurate crystal structure analyses were performed by the maximum entropy method (MEM)/Rietveld method with the split-atom model and anharmonic temperature factor model using high-energy synchrotron radiation X-ray diffraction data. The off-centered positions of Gd ions at the Ba site in the probability density function were clearly observed at a low temperature but smeared in the high-temperature region by the over-barrier motion of Gd ions.

Anharmonicity of potentials of atoms in potassium hydrogensulfide (KDS) determined by neutron single-crystal diffraction

Potassium hydrogensulfide (KHS) is an ionic compound with an anionic molecular group HS(-). The fast reorientational disorder of the anions was determined for the ambient temperature modification [R3;m; Jeffrey (1974). Can. J. Phys. 52, 2370-2378]. Single crystals are available now as protonated or deuterated specimens. With neutron single-crystal diffraction at room temperature, a considerable anharmonicity of the atom potential of the H or D atoms was observed. Even the thermal motions of K and S atoms show small deviations from an isotropic probability density function, which can be modelled using anharmonic temperature factors. The temperature factors of the atoms were expanded into a Gram-Charlier series [Kuhs (1992). Acta Cryst. A48, 80-98] in order to evaluate the anharmonicity quantitatively. Parameters up to a fourth-order approximation are relevant for the D atoms. Results from neutron single-crystal diffraction are compared with split-atom models extracted from neutron powder diffraction patterns of fully deuterated samples.

Core and Valence Thermal Vibrations in Diamond, Silicon, and Germanium

AbstractAn analysis is made using published X‐ray data, for the thermal vibrations in diamond, silicon, and germanium. The overall thermal vibration is split into core and valence contributions, because valence electrons can oscillate with different phase and amplitude than core electrons due to bond polarizability. Using indigenously developed computer programs the published data are analyzed. In the first phase, the overall Debye‐Waller factor of diamond, Si, and Ge together with scaling and extinction factors are refined using the method of least squares. In the second phase, the core and valence contributions of the harmonic temperature factor are evaluated. Finally, in the third phase, the core and valence contributions of the anharmonic temperature factor are evaluated. The error in the fitting procedure is less than 1% for Si and Ge and about 3% for diamond.

The thermal vibrations and the fluorine ionic conductivity in LaF3

Abstract The LaF3-structure has been reinvestigated using diffraction data measured with twinned crystals. LaF3 is a fluorine ion conductor with essentially 3-dimensional conductivity. The conduction perpendicular to the c-axis is shown to occur on pathways which include all three sets of fluorine positions in the structure. These pathways connect symmetrically non-equivalent nearest neighbour positions only. This conclusion is supported by the determination of anharmonic temperature factors. The temperature dependence of the harmonic Debye-Waller factors indicates that thermal vibrations are described rather than static discorder. The magnitude of the temperature parameters of fluorine is consistent with an intrinsic activation energy of 0.8 eV for fluorine conduction.

Neutron powder investigation of tetragonal and cubic stabilized zirconia, TZP and CSZ, at temperatures up to 1400 K

A comparative study of tetragonal (Y-TZP, 3 mol% Y 2 O 3 ) and cubic (CSZ, 15 mol% CaO) zirconia powder samples was carried out to learn about a possible structural origin of the superior ionic conductivity of TZP at low temperatures. Structure refinements including anharmonic temperature factors and oxygen occupancies as well as a qualitative analysis of the diffuse background revealed large differences especially of the anisotropy of thermal motion and the underlying disorder in both samples

The Identification of Phonon-Phonon Interactions in Anharmonic Crystals by X-Ray Diffraction Data

Abstract The effect of anharmonic atomic vibrations on the elastic scattering of X-rays by a crystal is treated by time-independent perturbation theory. The complete expression for the anharmonic temperature factor (TF) in the high-temperature limit is presented for atoms in general positions. The phonon-phonon interactions in an anharmonic crystal may be identified with X-ray diffraction data by using various temperature and scattering vector q dependencies of different anharmonic contri-butions to the TF. The expression for the intensity of elastic scattering by an anharmonic crystal contains an additional factor -the anharmonic cross-term. This term has to be taken into account for obtaining the unbiased thermal parameters of fourth order in | q | from X-ray diffraction data.

Anharmonic temperature factors, anomalous-dispersion effects and bonding charges in gallium arsenide

The introduction of dispersion corrections in the generalized structure-factor expression for the zinc blende structure leads to the breakdown of Friedel's law for all even-index reflections, owing to bonding or anharmonicity in these crystals. A systematic and precise X-ray diffraction intensity measurement has been carried out using a spherical single-crystal of gallium arsenide at 300 K. Least-squares refinement of the experimental data yields the Debye–Waller factors BGa = 0.62 (2), BAs + 0.49 (2) A2 for the harmonic model and BGa = 0.670 (3), BAs = 0.470 (2) A2 and individual anharmonic thermal factors BGa = 3.15 (59) × 10−18 and BAs= 8.92 (182) × 10−18 J A−3 for the anharmonic model. The residual index for the discrepancy between measured and calculated structure factors in the present work is R = 1.237% for the harmonic model and R = 1.229% for the anharmonic model. The present work reveals excellent agreement in both sign and magnitude of the Bijvoet inequalities for a large number of reflections. Covalent charge transfer has also been deduced from the experimental measurements on a few quasi-forbidden Bragg reflections showing a net transfer of charge from Ga to As.

Investigation of thermally induced Li+ ion disorder in Li2O using neutron diffraction

Neutron diffraction measurements have been performed on a single crystal of 7Li2O to investigate the time-averaged positions of the nuclei at seven temperatures between 293 and 1603 K. At temperatures up to 873 K the observed Bragg peak intensities can be well fitted using the regular antifluorite structure, including harmonic and anharmonic temperature factors to describe the thermal vibrations. At higher temperatures an improved fit to the experimental data is obtained by allowing a fraction of cations to leave their regular lattice sites. This is indicative of a transition to a 'fast-ion' phase, analogous to that observed in the fluorite-structured compounds at elevated temperatures. The model of the time-averaged disordered structure which best fits the data consists of interstitial lithium ions occupying mean positions very close to the midpoint between two nearest-neighbour Li+ ions. The fraction of Li+ ions occupying these sites increases rapidly above 1200 K, reaching approximately 16% at 1603 K. There is some evidence that the presence of the interstitial cations produces slight relaxations of the two nearest-neighbour Li+ ions in (111) directions, away from the interstitial.

Neutron powder investigation of CuI*

CuI is known to undergo two phase transitions at T1 = 367°C and T2 = 407°C. The structural changes involve a rearrangement of the anions and disordered distributions of the cations. The low temperature γ-phase has zincblende structure, space group F[unk]3m [a = 6.0337(3) Å, Z = 4]. The intermediate β-phase belongs to the hexagonal system with a hcp anion lattice [a = 4.289(9) Å), c = 7.189(5) Å, Z = 2], space group P[unk]m2 (coexisting with γ- or α-phase). The high temperature α-phase is cubic again, the anions form a fcc lattice and the cations are randomly distributed over all tetrahedral sites, space group Fm[unk]m [a = 6.148(1) Å, Z = 4]. A modified powder profile refinement program, which includes anharmonic temperature factors, was used for the refinement of CuI at different temperatures and probability density functions were calculated to elucidate the diffusion paths of this high temperature superionic conductor. Different models were tried with anharmonic coefficients upto the fourth order and split models with anisotropic temperature factors. With increasing temperature the density of Cu cations begins to spread from the center of the tetrahedra towards the vacant octahedral interstices. Aproaching the α-β transition from above the lobes near the tetrahedral faces become more pronounced indicating a transition from a diffusion to a more static (jump) behaviour near the phase transition.

Anharmonic thermal motion of Ag in AgCrSe 2: A high-temperature single-crystal X-ray diffraction study

The thermal motion of Ag in the two-dimensional silver ion conductor AgCrSe 2 (at 295 K; a = 3.6798(2) Å, c = 21.225(2) Å; Z = 3; spacegroup R3 m) is studied as a function of temperature by single-crystal X-ray diffraction. At room temperature the Ag atoms occupy only one-half of the available tetrahedral holes between the CrSe 2 sandwiches. Disordering of the silver sublattice over all tetrahedral sites proceeds second order with T c = 475 K: above T c the spacegroup is R 3m . Refinements were performed with data measured at six different temperatures in the range 295–673 K. Anharmonic temperature factors of Ag based on the Gram-Charlier expansion have been refined up to fourth order. R w values range from 0.022 to 0.049, the number of variables from 17 to 22. The refinements confirm the second-order nature of the order-disorder transition. Below T c the largest thermal vibrations are directed to the octahedral sites between the CrSe 2 sandwiches. The plane between the α- and β-lattice plays a crucial role. The joint probability density function (jpdf) of Ag is high in regions surrounding the projected α- and β-sites and the octahedral site; at these sites the jpdf shows a minimum. The diffusion of Ag from an α- to a β-site proceeds therefore via this intermediate plane.

Structural investigations up to 1300 K and ionic conductivity in Ca-stabilized zirconia

Cubic Ca-stabilized zirconia (CSZ) of composition Zr 0.85 Ca 0.15 O 1.85 was investigated by neutron elastic Bragg scattering technique. Data were collected at 1170, 1370 and 1770 K and analysed in terms of split atom models and/or by an anharmonic temperature factor formalism for the oxygens. There are distinct deviations from the ideal fluorite structure at all the temperatures. At 1170 and 1370 K the derived split-atom positions along , indicate a “tetragonal” distortion at least over a few cells. At 1770 K an analysis with anharmonic temperature factors and interstitital sites either along or at gives the best results. Significant nuclear scattering density is found away from the ideal positions, which may be interpreted in terms of a mobile oxygen substructure. From the probability density function, V eff values are calculated, Low-lying barriers of 0.5–0.6 eV indicate the most probable diffusion pathways. In comparison with thermal energies these findings may be closely correlated with the enhanced ionic conductivity of CSZ observed at elevated temperatures.

Negative volumes and central moments of anharmonic probability density functions

The suitability of different functions used for anharmonic temperature factors in describing anharmonicity in structures is studied from the point of view of whether the respective probability density functions (p.d.f.'s) (isolated-atom-potential expansions, Gram-Charlier series, p.d.f. of the cumulant expansion of the temperature factor) are non-negative for the final values of the parameters. Certain central moments of the p.d.f. should also not become negative. Analytical formulae have been derived for the central moments of several (different) p.d.f.'s. 20 p.d.f.'s of atoms in published structures have been examined, their negative volumes determined by numerical integration and their central moments calculated. For Al(4) in the structure of VAl10.42 at 293 K (strong anharmonicity) the negative volume was -0.238 and several moments were negative. Here the temperature factor and p.d.f. used are not acceptable. For the remaining structures (weak anharmonicity) the negative volumes found were <|-0.03| and the respective central moments were positive. Temperature factors and p.d.f.'s proved to be acceptable except for the cumulant expansion. In some cases its p.d.f. is not accessible, and, if it is, its negative volumes were found to be relatively large. A p.d.f. is proposed which is non-negative for all values of the parameters and whose Fourier transform can be derived in analytical form. An explicit formula is given for the cubic site symmetries.

Studying Ionic Conductivity with Diffraction Techniques an Application to laF3

AbstractThe LaF3-structure has been reinvestigated using diffraction data measured with twinned crystals.LaF3 is fluoride ion conductor with essentially 3-dimensional conductivity. The conduction perpendicular to the c-axis is shown to occur on paths which include all three sets of fluorine positions in the structure and which connect symmetrically non-equivalent nearest neighbor positions only. This is confirmed by determination of anharmonic temperature factors. The temperature dependence of the harmonic thermal parameters indicates that thermal vibrations are described rather than static disorder. The magnitude of the temperature factors of fluorine is consistent with an intrinsic activation energy of 0.8 eV for fluorine conduction.

Single crystal x-ray diffraction studies of materials at high temperature

The thermal motion of the atoms influences the intensities of the Bragg reflections. This influence is expressed by a temperature factor. Static disorder generates similar effects, which may be expressed by a pseudo temperature factor. The displacements of the atoms from their mean positions can be described by a probability density function (pdf). The temperature factors and the probability density functions are connected with each other by a Fourier transformation. An effective one particle potential can be calculated from the pdf. The so-called anisotropic temperature factors, frequently calculated in standard structure determinations, assumes a harmonic (parabolic) potential. A more general temperature factor formalism has to take into account anharmonic potentials and the corresponding anharmonic temperature factors. This article describes the mathematical basis of a general temperature factor formalism and an application of this formalism.

Positional parameters obtained with anharmonic temperature factors

Comparisons of distances between the positions obtained from harmonic and anharmonic refinements lead to the conclusion that the positional parameters may have different physical meanings for the two cases. The mean positions are obtained if the temperature factor (t.f.) has no first-order terms in the reciprocal-lattice vector Q (harmonic t.f., Fourier transforms of the Gram-Charlier series); if there are first-order terms in Q, other positions will be obtained whose meaning needs to be established. The advantages associated with the mean positions are described, and the disadvantages associated with other positions are illustrated with an example from the literature. A procedure is described in which the physical meaning (if there is any) of a non-mean position can be established and the mean position calculated. The problem of parameter bias is analyzed and numerical results are discussed for three structures. Provided the anharmonic model produces a significantly better fit to the experimental data, it is concluded that better estimates of the mean positions are obtained with anharmonic than with harmonic t.f.'s

Anharmonic temperature factors: the limitations of perturbation theory expressions. Erratum

The error in the temperature factor arising from the use of anharmonic perturbation theory is investigated for a one-dimensional one-particle potential by comparison with an exact evaluation of the Fourier transform of the Boltzmann probability density. Results are presented for a range of values of cubic and quartic coefficients for temperature factors derived from moment or cumulant expansions about the harmonic probability density function. It is found that either expansion provides an adequate approximation to the anharmonic temperature factor for moderately small anharmonicity but that both expansions become increasingly Inadequate as the anharmonicity gets larger.

Anharmonic temperature factors: the limitations of perturbation-theory expressions

Anharmonic temperature factors: the limitations of perturbation-theory expressions

A deficiency of the cumulant expansion of the anharmonic temperature factor

It is shown that the cumulant expansion of the anharmonic temperature factor is a function whose inverse Fourier transform either does not exist or has negative regions. Since the probability density function for an atom should always be non-negative, the inverse Fourier transform of the cumulant expansion may be a poor approximation to the true probability density function. Correspondingly, the cumulant expansion may be an inadequate tool for describing anharmonic motions. Five examples from the literature are quoted where the cumulant expansion gave worse results than other anharmonic expansions.