Mean Square Relative Displacement Research Articles

Ab Initio XAFS Debye-Waller Factors

An efficient, finite temperature, equation of motion method is introduced for calculations of the mean-square relative displacements σ 2 in XAFS Debye-Waller factors. Instead of calculating projected densities of modes, the calculations are based on the displacement-displacement correlation function and done in real time, over a few vibration cycles. Results for the Debye-Waller factors are found to be accurate to within about 15% percent at all temperatures. We also discuss how force constants used in the calculations can be obtained from ab initio Car-Parrinello/molecular dynamics calculations. Tests of the method are presented based on the Wiggs-Jonsson plane-wave pseudopotential code [Comp. Phys. Comm. 87, 319 (1995)]. Application to XAFS Debye-Waller factors in biological molecules is discussed.

The Mean-Square Relative Displacement and Displacement Correlation Functions in Tetrahedrally and Octahedrally Coordinated ANB8-N Crystals

The Debye-Waller factors of atoms in Ge, GaN, ZnO, CdS, CdSe, CuBr, CuCl, AgI, CdO, AgBr, KBr and RbCl in order of Phillips ionicity (f i) are studied by both EXAFS and the diffraction method. The displacement correlation functions (DCF) are derived from the mean-square relative displacement (MSRD) and the mean-square displacement (MSD) at room temperature. The magnitudes of the MSRD and MSD correlate well with the coordination number and ionicity. The MSRDs of tetrahedrally coordinated compounds show a gradual approach to those of octahedrally coordinated ones as the ionicity increases. A divergent-like curve toward f i=0.785 is observed for the MSD in the tetrahedrally coordinated compounds, which indicates the behavior of lattice instability. In the tetrahedrally coordinated covalent materials, the MSRD is nearly half the MSD cation or MSD anion and the ratio of the DCF to MSD given by 2DCF/(MSD cation+MSD anion) is about 80%. In the octahedrally coordinated ionic materials, the MSRD is comparable to the MSD and 2DCF/(MSD cation+MSD anion)=50%. An anharmonic contribution to the Debye-Waller factor determined by EXAFS appears pronouncedly when the magnitude of σ(2) is greater than 0.01 Å2. The MSRD for the tetrahedrally coordinated compounds of silver and copper halides obeys the systematic ionicity dependence and pronounced specificity of anharmonicity was not observed: AgI, CuBr and CuCl of which the high-temperature forms are typical superionic conductors have a broad interatomic potential which is similar to that of the octahedrally coordinated compounds and strongly correlated displacement in thermal vibration between cation and anion.

Microbeam XAFS Investigations on Fluid Inclusions

AbstractIn this paper, we discuss the use of x-ray absorption fine structure (XAFS) techniques for the determination of structure of metal complexes in fluid inclusions heated to elevated temperatures. Analysis of Zn K-edge XAFS spectra measured from a single hypersaline fluid inclusion in quartz shows that the ZnCl42- complex is dominant at all temperatures up to and including 430 °C. The Zn-Cl bond length was found to decrease uniformly with temperature, up to nearly 2% at 430 °C in comparison to the value at 25 °C. The Zn-Cl mean-square relative displacement increased linearly with temperature, from a value of 0.0043 Å2 at 25 °C to 0.0089 Å2 at 430 °C.

CALCULATIONS OF THE MEAN SQUARE RELATIVE ATOMIC DISPLACEMENTS IN A SIMPLE MODEL OF AN ANHARMONIC CRYSTAL

We have used the correlative method of unsymmetrized self-consistent field to calculate the mean square relative displacements (MSRD) of atoms in the two-dimensional model of the anharmonic crystal with square lattice. The longitudinal and transversal components of MSRD between the nearest, second, third and next neighbors have been computed. An influence of the anharmonicity on various components as well as peculiarities of the lattice considered are discussed.

Structure of Au ultrafine particles in silica glass by x-ray absorption fine structure spectroscopy

The structure of ultrafine gold particles embedded in silica glass by ion implantation has been studied by x-ray absorption fine structure spectroscopy. It is found that the Au–Au interatomic distance in the particles is very similar to that in bulk gold within 0.01 Å. Mean-square relative displacement in the particles is 1.2–1.3 times larger than that in bulk gold.

Local Lattice Anomalies in Epitaxial YBa2Cu3O7-δThin Film Studied by X-Ray Absorption Fine Structure

The temperature dependence of the local structure in epitaxial YBa 2 Cu 3 O 7-δ thin film has been studied by polarized extended X-ray absorption fine structure (EXAFS) on the Cu K-edge. The results suggest that the local lattice anomalies occur not only at T c but also well above T c . The mean-square relative displacements for the two out-of-plane Cu–O bonds (Cu1–O4, Cu2–O4) s i m u l t a n e o u s l y increased at T c . The in-plane Cu–O bonds showed the anomalous decrease of dynamical disorder below 120 K ∼1.5 T c followed by a sharp increase at T c which saturates at lower temperature, consistent with other lattice anomalies or dynamical spin susceptibilities observed at ∼1.5 T c . These suggest that the anomaly in in-plane oxygen atom vibrations below 1.5 T c is caused by the coupling between lattice distortions and magnetic excitations. The correlated displacement of apical oxygen atoms may induce a bilayer coupling associated with a charge transfer.

Anharmonicity effects on the extended x-ray-absorption fine structure: The case of beta -AgI.

Anharmonicity effects on the extended x-ray-absorption fine structure (EXAFS) of \ensuremath{\beta}-AgI have been studied within the temperature range 77--410 K through the cumulant analysis of the first-shell contribution. The distributions of I-Ag distances and the corresponding pair potentials have been reconstructed directly from cumulants. The shape of the EXAFS pair potential is temperature independent, while its equilibrium position shifts to lower values with increasing temperature. The I-Ag thermal expansion is influenced not only by the EXAFS pair-potential anharmonicity but also by its shift, so that \ensuremath{\Delta}${\mathit{C}}_{1}$ is not equal to the ratio ${\mathit{C}}_{3}$/2${\mathit{C}}_{2}$. Thermal expansion measured by EXAFS is significantly larger than the one measured by diffraction; the difference is due to vibrational motion normal to the I-Ag bond direction, responsible also for peculiar correlation effects on the mean-square relative displacement (MSRD). The distribution of I-Ag distances sampled by EXAFS is consistent not only with the excluded-volume model but also with vibrational models leading to strong distortions of the first-shell tetrahedral cage. Harmonic and anharmonic contributions to the I-Ag MSRD have been singled out, the anharmonic fraction amounting to 20% at 300 K.

Vibrational correlation functions of hydrogen-terminated C(111)-(1 x 1) and Si(111)-(1 x 1) surfaces.

We report lattice-dynamical calculations of surface and subsurface displacement-correlation functions of hydrogen-terminated (111) surfaces of diamond and Si. These are evaluated employing a semiempirical total-energy scheme together with a 38-layer slab geometry. The total-energy scheme has recently been shown to yield a very accurate description of phonons at these surfaces in excellent agreement with available high-resolution electron-energy-loss spectroscopy data. The calculated vibrational correlation functions, such as mean-square displacements or mean-square relative displacements, to be reported in this paper show very pronounced orientational anisotropies and they are distinctly different on different layers at or near the surface. Even at the fourth layer, convergence to the respective bulk values is not yet reached. The significance of our results for an interpretation of surface-scattering experiments is assessed and it is shown that a fully quantitative interpretation of surface-scattering data cannot be achieved by using isotropic bulk displacement-correlation functions or the results of simple phenomenological models.

The effect of temperature on the radial distribution function for iron in native horse spleen ferritin

Iron K-edge X-ray absorption spectra were recorded for a sample of freeze-dried horse spleen ferritin over a range of temperatures from 40 to 300 K. Gaussian-type radial distribution functions were fitted to the data from all temperatures simultaneously and the obtained mean-square relative displacements σ 2 (EXAFS Debye-Waller factors) were fitted using an Einstein model. For iron atoms in the second and/or third coordination shell an Einstein temperature of 330 ± 20 K was obtained. If oxygen was assumed in the second or third shell, its Einstein temperature was 460 ± 20 K. This indicates that the core of ferritin may have a somewhat more rigid structure than those of previously studied ferritin analogues (polysaccharide iron complexes).

An EXAFS study of thermal disorder in GaAs

This paper presents the experimental results of a cumulant analysis of the extended X-ray absorption fine structure amplitude of GaAs at the K edges of Ga and As. The second and fourth cumulants of the distance distributions of the near neighbours of Ga and As have been obtained as a function of temperature in the range 77-450 K. The fourth cumulant of Ga-As nearest-neighbour distance distributions is zero in the examined temperature range; a non-Gaussian distribution of the second and third coordination shell distances of Ga and As has been revealed for the first time. Temperature dependences of mean square relative displacements (MSRDS) of the first three shells of Ga and As have been compared with Einstein and correlated Debye models. The MSRD of the first-shell Ga-As distance is connected to the optical modes at 7.6-8 THz of the phonon spectrum of GaAs.

Determination of bond lengths, atomic mean-square relative displacements, and local thermal expansion by means of soft-x-ray photoabsorption.

Temperature-dependent extended x-ray-absorption fine-structure (EXAFS) measurements at the oxygen and fluorine K edges of CuO, ${\mathrm{Cu}}_{2}$O, ZnO, ${\mathrm{CaF}}_{2}$, and LiF have been performed. We present an EXAFS analysis of bulk samples in the soft-x-ray region of h\ensuremath{\nu}\ensuremath{\le}1500 eV determining the moments of the radial pair distribution function (RDF) of the oxygen and fluorine nearest-neighbor bonds by use of the conventional cumulant expansion method, i.e., coordination numbers, bond lengths, atomic mean-square, and mean-cubic relative displacements of the RDF. It is shown that high-quality K\ensuremath{\alpha}-fluorescence-yield measurements, analyzed in combination with theoretical standards, allow a determination of nearest-neighbor distances within 0.015 \AA{} and of coordination numbers with 10--20 % accuracy. Using quantum-mechanical models for the description of the atomic motions, the EXAFS Debye and Einstein temperatures, as well as the local thermal expansion of the bond under consideration, are obtained. In particular, these quantities for ${\mathrm{CaF}}_{2}$ are found to be in good agreement with those measured by other techniques. In contrast to the fluorides, no thermal expansion could be observed up to room temperature for the transition-metal oxides, which confirms a recent finding of enhanced anharmonicity in the low-Z adsorbate-surface interaction. A detailed compilation is given of the majority of EXAFS studies from the literature where moments of the RDF higher than the second one are reported. For these compounds the local thermal expansion is quantum mechanically calculated in contrast to previous calculations that were performed in the classical limit. Debye temperatures and the local thermal expansion measured by EXAFS and other techniques agree well for fcc metals. For binary compounds like alkali halides or superionic conductors a deviation up to 100% can be found.

Erratum: Temperature dependence of the Pd K-edge extended x-ray-absorption fine structure of PdCx (x~0.13)

Pd K-edge extended x-ray-absorption fine-structure (EXAFS) and x-ray-absorption near-edge-structure (XANES) measurements were performed on a Pd carbide phase, ${\mathrm{PdC}}_{\mathit{x}}$ (with x\ensuremath{\sim}0.13), and metallic Pd powder at temperatures between 40 and 423 K. The data were analyzed by nonlinear least-squares fitting using an ab initio theoretical standard. The average lattice expansion of the carbide phase, 2.7\ifmmode\pm\else\textpm\fi{}0.1% relative to Pd metal, agrees well with previous x-ray- and neutron-diffraction results. The temperature dependence of the Pd-Pd nearest-neighbor mean-squared relative displacement (${\mathrm{\ensuremath{\sigma}}}^{2}$) yields a Debye temperature, ${\mathrm{\ensuremath{\Theta}}}_{\mathit{D}}$=295\ifmmode\pm\else\textpm\fi{}10 K for Pd powder. There is an additional static-disorder contribution in ${\mathrm{\ensuremath{\sigma}}}^{2}$ of 0.0009\ifmmode\pm\else\textpm\fi{}0.0002 A${\mathrm{\r{}}}^{2}$, relative to bulk Pd metal. The Debye temperature of ${\mathrm{PdC}}_{0.13}$ powder is also 295\ifmmode\pm\else\textpm\fi{}10 K. There is, however, an additional static-disorder contribution to ${\mathrm{\ensuremath{\sigma}}}^{2}$ of 0.0012 A${\mathrm{\r{}}}^{2}$, relative to Pd powder, probably the result of local distortion of the Pd lattice by interstitial C. The Pd-C coordination is not directly observed in a Fourier transform of the EXAFS. The location of C in octahedral interstices is evident by its effect, through multiple scattering, on the magnitude of the second Pd-Pd peak in the Fourier transform magnitude of the EXAFS data, which is greatly reduced. The Pd K-edge XANES spectrum of ${\mathrm{PdC}}_{0.13}$ is nearly identical to that of Pd metal. There is no significant chemical shift (\ensuremath{\le}0.5 eV) of the Pd K edge of ${\mathrm{PdC}}_{\mathrm{\ensuremath{\sim}}0.13}$.

Application of the Bayesian Estimation Method to the EXAFS Data Analysis

The Bayesian estimation method which is based on mathematical statistics combines experimental signals and pre-information in a common loss function. We have improved the original version of the Bayesian method and can take into account implicit constraints between the parameters of the EXAFS function, that must be fulfilled only within chosen error bounds. These linear relations connect occupation probabilities or phase fractions, coordination numbers, interatomic distances and mean-square relative displacements of the atoms.

Extended x-ray-absorption fine-structure investigation of short-range order in a-Ge1-xSnx alloys.

The local environments of germanium and tin atoms in a-Ge 1-x Sn x alloys (0.0≤x≤0.2) have been investigated by measuring the extended x-ray-absorption fine structure (EXAFS) at the Ge K edge and at the Sn L III edge. The measured EXAFS at both edges could be described only by assuming the presence of a mixed first coordination shell composed of both Ge and Sn atoms. Coordination numbers, bond lengths, and relative mean-square displacement values characterizing the first coordination shell around Ge and Sn atoms were obtained

Temperature dependence of EXAFS Debye-Waller factors in beta - and gamma -AgI

EXAFS at the iodine L3 edge in beta - and gamma -AgI has been studied. The EXAFS Debye-Waller factors of the first and second coordination shells of iodine have been extracted by a standard analysis based on the harmonic approximation. No difference has been found between the temperature dependence of mean square relative displacements (MSRD) of the two beta and gamma phases. The harmonic approximation has been checked against more complete anharmonic EXAFS and X-ray diffraction analyses; it has been found adequate to interpret the MSRDs of AgI for temperatures lower than 300 K. The experimental MSRDs for the two distances I-Ag and I-I have been compared with MSRDs calculated on the grounds of the available lattice dynamics models. The sensitivity of EXAFS to phonon polarization properties has been made evident.

Thermal variation of the mean-square relative displacement for the Pt-Pt pair in the Ni10Pt90 random solid-solution alloy.

The Pt ${\mathit{L}}_{3}$-edge extended x-ray-absorption fine-structure measurements were made on the ${\mathrm{Ni}}_{10}$${\mathrm{Pt}}_{90}$ random solid-solution alloy, as well as on pure Pt, at four different temperatures (in the range 7 K\ensuremath{\lesssim}T\ensuremath{\lesssim}300 K). The data have been analyzed with the ratio fit, to obtain the relative values of the nearest-neighbor mean-square relative displacement (MSRD) for the Pt-Pt pair in each temperature case with respect to the MSRD value for the Pt-Pt pair in pure Pt at 7 K. For the ${\mathrm{Ni}}_{10}$${\mathrm{Pt}}_{90}$-alloy cases, only a high-k region above \ensuremath{\sim}12 \AA{} $^{\mathrm{\ensuremath{-}}1}$ was used for the ratio fit. In this approach, the contribution of Ni-neighbor backscattering is assumed to be negligibly small in the high-k region where the Pt backscattering overwhelms the Ni backscattering, since about 90% of the nearest neighbors are Pt atoms. The intercept values of the ratio fits agree well with the expected values, indicating that this approximation is valid. Subsequently the MSRD values were determined by Einstein-model fits. The Debye-model fit was also tried for the pure-Pt data, and the Debye temperature obtained from the fit is 240 K, showing the good quality of the data. The results of the ratio fit for the ${\mathrm{Ni}}_{10}$${\mathrm{Pt}}_{90}$ data are in good agreement with those of two-shell fits.

Temperature effects on orientation-dependent Exelfs

Recently, investigations have begun to explore the effects of Mean-Square-Relative-Displacements (MSRD), or thermal damping, on the EXELFS oscillations above electron absorption edges. These types of analysis can be extended to include orientation effects in anisotropic crystal systems. Furthermore, as this technique is a direct probe of the electronic structure of a crystal, this information could be combined with the crystal symmetry and Cauchy relations to examine various physical properties. One such property is the elastic compliance quadric.With the development of the parallel detection EELS spectrometer, EXELFS type studies have become practical. This makes accessible a large field of analysis, previously explored only by EXAFS. Thus, we are able to combine a wealth of experimental and data analysis algorithms from that technique with the high spatial resolution and crystallographic information afforded by the modern analytical TEM. We have developed computer simulation programs which demonstrate the feasibility of this hybrid analysis.

Temperature dependence of EXAFS spectra of supported small metal particles

Temperature dependence of EXAFS was systematically investigated on small metal (Rh or Ru) particles on three different supports, SiO2, NaY zeolite and TiO2. The observed EXAFS oscillations were analysed by the ratio method combined with the cumulant expansion method. The results of these analyses of EXAFS data revealed the existence of a considerable static disorder. It is also shown that the Debye temperature characteristic of particle surface can be determined from analysis of the temperature dependence of the mean-square relative displacement derived from EXAFS. In Rh/TiO2, the surface Debye temperature was found to be ca. 100 K higher in its strong metal–support interaction (SMSI) state in comparison with the normal state, supporting the decoration model for the SMSI state.

Mean-square displacements of surface atoms: c(2 x 2) overlayers on Ni(100).

We present lattice-dynamical calculations of the mean-square displacements of the adsorbate and first-layer substrate atoms for c(2\ifmmode\times\else\texttimes\fi{}2) S and O overlayers on Ni(100). To be consistent with available structural information and to reproduce the observed surface-phonon dispersion data on these systems, surface force constants between Ni atoms are chosen to be different from the value in the bulk. It is found that for sulfur atoms the mean-square displacement in the plane parallel to the surface is larger than that along the surface normal. The opposite is the case for the oxygen atoms. The underlying Ni atoms in the two systems also display dissimilar anisotropies in the three orthogonal directions. We have examined the relative strengths of the mean-square relative displacements between the adsorbate and the first- and second-layer substrate atoms for both systems and explored the effect of the variation of the surface force constants on these quantities. These results have been brought in contact with available surface extended x-ray-absorption fine-structure data on these systems.

Temperature dependence of EXAFS spectra of rhodium and palladium catalysts in the strong metal-support interaction state

The temperature dependence of EXAFS spectrum was investigated on the normal and SMSI states of Rh/TiO{sub 2} and Pd/TiO{sub 2} catalysts. The mean square relative displacement (MSRD) of metal-metal bondings of supported metal was derived by the analysis of EXAFS data with the cumulant expansion technique, and the Debye temperature was estimated from the temperature dependence of MSRD. The Debye temperature or Rh-Rh bond of Rh/TiO{sub 2} catalyst was found to be higher by 38 K in its SMSI state in comparison with the normal state. This difference was shown to be due to the significant increase of the surface Debye temperature of Rh metal particles. A similar results was obtained for Pd/TiO{sub 2} catalyst. These facts were concluded to support the decoration model for the SMSI state.