Mean Square Amplitudes Of Vibration Research Articles

X-ray absorption Debye-Waller factors fromab initiomolecular dynamics

An ab initio equation of motion method is introduced to calculate the temperature-dependent mean-square vibrational amplitudes ${\ensuremath{\sigma}}^{2}$ which appear in the Debye-Waller factors in x-ray absorption, x-ray scattering, and related spectra. The approach avoids explicit calculations of phonon modes, and is based instead on calculations of the displacement-displacement time correlation function from ab initio density functional theory molecular dynamics simulations. The method also yields the vibrational density of states and thermal quantities such as the lattice free energy. Illustrations of the method are presented for a number of systems and compared with other methods and experiment.

Alcohol induced structural and dynamic changes in β-lactoglobulin in aqueous solution: A neutron scattering study

Structural and dynamic properties of β-lactoglobulin (β-LG) were revealed as a function of alcohol concentration in ethanol- and trifluoroethanol(TFE)-water mixtures with circular dichroism (CD), small-angle neutron scattering (SANS) and quasi-elastic neutron scattering (QENS). The CD spectra showed that an increase in TFE concentration promotes the formation of the β-sheet structure of β-LG. The SANS-intensities were fitted using form factors for two attached spheres for the native and native-like states of the protein. At higher alcohol concentrations, where aggregation takes place, a form factor modelling diffusion limited colloidal aggregation (DLCA) was employed. The QENS-data were analyzed in terms of internal motions for all alcohol concentrations. While low concentrations of TFE (10% (v/v)) lead to an increase of the mean square amplitudes of vibrations <u(2)> and a retention of a native-like structure - but not to an increase of the characteristic radius of proton diffusion processes a. Addition of 20% (v/v) of TFE induces aggregation, going along with a further increase of <u(2)>. Further increase of TFE concentration to 30% (v/v) changes the nanoscale structure of the oligomeric nucleate, but induces no further significant changes in <u(2)>. The present study underlines the necessity of methods sensitive to the dynamics of a system to obtain a complete picture of a molecular process.

A study of thermal parameters of Ag 50Pd 50 alloy using X-ray diffraction

Abstract Thermal parameters of Ag 50 Pd 50 alloy were studied using the in-situ X-ray diffraction (XRD) technique. Diffraction experiments in the temperature range 308–1178 K showed that the alloy forms a face centered cubic (fcc; A1-type) structure. Temperature dependence of the lattice parameter was investigated using the Bragg line displacement method. Results show that the lattice parameter for the alloy increases with increase in temperature. The temperature dependent XRD data was utilized to determine the mean linear thermal expansion (MLTE (%)) and coefficient of thermal expansion (CTE) ‘ α ’ in the mentioned temperature range. It was observed that the values of CTE vary between 9.3 and 14.6 [K −1 ][10 −6 ]. The same data was also utilized to determine the Debye temperatures Θ D and mean square amplitude of vibration (MSAV) ( u 2 ( T )) for this alloy. Values of thermal parameters as determined in this investigation are in good agreement with the estimated values as well as with those for other alloys of noble metals.

Diffraction Measurements and Equilibrium Parameters

Structural studies are largely performed without taking into account vibrational effects or with incorrectly taking them into account. The paper presents a first‐order perturbation theory analysis of the problem. It is shown that vibrational effects introduce errors on the order of 0.02 Å or larger (sometimes, up to 0.1‐0.2 Å) into the results of diffraction measurements. Methods for calculating the mean rotational constants, mean‐square vibrational amplitudes, vibrational corrections to internuclear distances, and asymmetry parameters are described. Problems related to low‐frequency motions, including torsional motions that transform into free rotation at low excitation levels, are discussed. The algorithms described are implemented in the program available from the author (free).

Rumpling of LiF(001) surface from fast atom diffraction

Quantum diffraction of fast atoms scattered from the topmost layer of surfaces under grazing angles of incidence can be employed for the analysis of detailed structural properties of insulator surfaces. From comparison of measured and calculated diffraction patterns we deduce the rumpling of the topmost surface layer of LiF(001) (i.e., an inward shift of Li${}^{+}$ ions with respect to ${\mathrm{F}}^{\ensuremath{-}}$ ions). The effect of thermal vibrations on the measurement of rumpling is accounted for by ab initio calculations of the mean-square vibrational amplitudes of surface ions. At room temperature this leads to a reduction of the apparent rumpling by 0.008 \AA{}. We then obtain a rumpling of ($0.05\ifmmode\pm\else\textpm\fi{}0.04$) \AA{}, which improves its accuracy achieved in previous work.

Ablation yield and angular distribution of ablated particles from laser-irradiated metals: The most fundamental determining factor

Abstract Five metals (Zn, Cu, Ni, Ti, and Mo) were irradiated with 150 shots of a Q-switched Nd:YAG pulsed laser in a vacuum of 10 −3 torr. The ions projected out of the laser-produced plasma (LPP) plume were detected by CR-39 detectors positioned at −15°, 0°, 30°, 60°, and 90° with respect to the target-surface normal at a distance of 5 cm from the target in each case. The angular distribution of LPP ions, which is characterized by the exponent n of cos n θ distribution, is given by n = 2.5–11 for the five target metals. The value of the exponent n has no systematic correlation with the square-root of atomic mass of the target metals but exhibits systematic dependence on the room temperature Debye–Waller's thermal parameter B or the mean-square amplitude of atomic vibrations 〈 u 2 〉. Likewise, the ablation yield (atoms/shot) of the twelve target metals investigated by Thestrup et al. (2002) [8] under identical irradiation conditions is a function of the room temperature B -factor or 〈 u 2 〉.

Mean square amplitudes of vibration and associated Debye temperatures of rhenium, osmium and thallium

Abstract The directional mean square amplitudes of vibration, Debye–Waller factors and associated Debye temperatures of hexagonal rhenium, osmium and thallium have been obtained from X-ray integrated intensities. The integrated intensities have been measured with a Philips 3020 powder diffractometer fitted with a proportional counter using filtered CuKα radiation at room temperature and have been corrected for thermal diffuse scattering. Within the limits of experimental errors, the anisotropy observed in these parameters is negligible. The experimental values of directional Debye temperature have been compared with corresponding values obtained from theoretical calculations. Vacancy formation energies of Re, Os and Tl have been estimated using a relation connecting it with the X-ray Debye temperature.

Studies on multiphased mixed crystals of NaCl, KCl and KI

AbstractMultiphased mixed crystals of NaCl, KCl and KI were grown by the melt method, for the first time. Densities and refractive indices of all the grown crystals were determined and used for the estimation of the composition in the crystal. Atomic absorption spectroscopic measurements were done to estimate the metal atom contents in the crystal. Lattice parameters and thermal parameters (Debye‐Waller factor, mean square amplitude of vibration, Debye temperature and Debye frequency) were determined from the X‐ray powder diffraction data. DC and AC electrical measurements were done at various temperatures ranging from 40 to 150°C. Activation energies were also estimated. The observed lattice parameters showed that the system exhibits three phases each nearly corresponds to NaCl, KCl and KI. The thermal and electrical parameters show a highly nonlinear bulk composition dependence. Results are reported. (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Ab-initio theoretical analysis of thermal expansivity, thermal vibrations and melting of thorium

The thermal expansivity and mean-square thermal vibrational amplitudes were determined as a function of temperature for thorium from first principles total energy calculations using the full potential linearized augmented plane wave (FP-LAPW) method. The coefficient of thermal expansion determined from variation of theoretical potential energy with interatomic separation is ∼1.427 × 10−5 K−1 as compared to an experimental value of 1.23 × 10−5 K−1. The mean-square thermal vibrational amplitudes determined as a function of temperature from the theoretically calculated volume-dependent Debye temperature agree well with experimental data derived form neutron diffraction measurements. The melting temperature of thorium, determined theoretically from the mean-square thermal vibrations and the Lindemann rule for melting, is 2234 K, as compared to the reported experimental value of 2021 K.

Thermal parameters of MgSO 4·7H 2O and NiSO 4·7H 2O crystals added with urea and thiourea

Pure and impurity-added (with urea and thiourea) MgSO 4 ·7H 2 O and NiSO 4 ·7H 2 O single crystals were grown by the free-evaporation method from aqueous solutions. Density was measured by the flotation method. X-ray diffraction data were collected for powder samples and used for the estimation of thermal parameters like Debye–Waller factor, mean-square amplitude of vibration, Debye temperature and Debye frequency. The thermal parameters do not vary in a particular order with respect to impurity concentration, which could be attributed to the random disturbance created by the impurity molecules in the hydrogen-bonding system of the MgSO 4 ·7H 2 O and NiSO 4 ·7H 2 O lattices.

Atomic dynamics of liquid lithium and lithium-hydrogen melt investigated by inelastic neutron scattering

From the experimental data on inelastic neutron scattering the frequency spectrum of lithium atomic vibrations is extracted at the temperature of 22, 227, 397 and 557°C. The frequency spectrum for lithium-hydrogen melt (98 at.% Li7, 2 at.% H) has been obtained at 557°C. Based on the frequency spectrum, the temperature dependencies of the mean-square of the lithium atoms displacements in time, the mean-square amplitude of atomic vibrations, velocity autocorrelation function of atoms are calculated. A generalized frequency spectrum of hydrogen atomic vibrations in lithium-hydrogen melt is obtained.

X-ray diffraction studies on (NaCl) x(NaBr) y−x(NaI) 1−y crystals

Abstract We have grown (NaCl)x(NaBr)y−x(NaI)1−y crystals by the melt method for various values of x and y. Thermal parameters like Debye–Waller factor, Debye temperature, mean square amplitude of vibration and Debye frequency were determined using the X-ray powder diffraction intensity data. The results obtained are reported here.

Studies on multiphased mixed crystals grown from NaBr and KCl

Abstract We have grown multiphased binary and ternary mixed crystals by the melt method using the miscible alkali halides, viz. NaBr and KCl and physically characterized. Thermal parameters like Debye–Waller factor, Debye temperature, Debye frequency and mean square amplitude of vibration were determined using the X-ray powder diffraction intensity data. DC and AC electrical measurements were carried out by using the parallel plate capacitor method at various temperatures. Activation energies (DC and AC), mean jump frequency, compressibility and mean sound velocity were also determined. The results obtained are reported here.

Theoretical x-ray absorption Debye-Waller factors

An approach is presented for theoretical calculations of the Debye-Waller factors in x-ray absorption and related spectra. These damping factors are represented in terms of the cumulant expansion up to third order. They account respectively for the net thermal expansion ${\ensuremath{\sigma}}^{(1)}(T)$, the mean-square relative displacements ${\ensuremath{\sigma}}^{2}(T)$, and the asymmetry of the pair distribution function ${\ensuremath{\sigma}}^{(3)}(T)$. Similarly, we obtain Debye-Waller factors for x-ray and neutron scattering in terms of the mean-square vibrational amplitudes ${u}^{2}(T)$. Our method is based on density functional theory calculations of the dynamical matrix, together with an efficient Lanczos algorithm for projected phonon spectra within the quasiharmonic approximation. Because of the anharmonicity in the interatomic forces, the results are highly sensitive to variations in the equilibrium lattice constants, and hence to the choice of exchange-correlation potential. In order to treat this sensitivity, we introduce two prescriptions: one based on the local density approximation, and a second based on a modified generalized gradient approximation. Illustrative results for the leading cumulants are presented for several materials and compared with experiment and with correlated Einstein and Debye models. We also obtain Born-von Karman parameters and corrections due to perpendicular vibrations.

Investigations of nuclear dynamics of LaI3 molecule. I. Potential functions of normal vibrations in harmonic and anharmonic approximations

The potential energy surface of the LaI3 molecule is scanned along the normal coordinate by a B3LYP/SDD, SDD method. It is shown that a nonplanar ν2(A2″) vibrational potential function is most anharmonic. The effect of anharmonicity on the root mean square amplitudes of vibrations and the vibrational molecule spectrum is stated.

Study of the microdynamics of liquid lithium and lithium-hydrogen melt by inelastic neutron scattering

The frequency spectra of vibrations of Li atoms at temperatures of 22, 227, 397, and 557°C and the lithium-hydrogen melt (98 at % 7Li, 2 at % H) at 557°C have been obtained from the experimental neutron inelastic scattering data. On the basis of the frequency spectra, the temperature dependences of the mean-square displacement of Li atoms, the mean-square amplitudes of atomic vibrations, and the velocity autocorrelation function of atoms have been calculated. The speed of sound in liquid lithium has been estimated within the Debye model. The frequency spectra of lithium-hydrogen melt and solid lithium hydride are compared. A generalized frequency spectrum of vibrations of hydrogen atoms in lithium-hydrogen melt is obtained.

Tensor LEED study of the temperature dependent dynamics of the NaCl(100) single crystal surface

The structure and dynamics of the NaCl(100) single crystal surface were investigated at 25, 100, 160, and $230\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ by means of low-energy electron diffraction (LEED). At each temperature, diffraction peak intensities of four inequivalent beam orders were recorded as a function of electron energy. The experimental $I(V)$ data were analyzed using the tensor LEED approach. Compared to the unrelaxed bulk geometry, the ${\mathrm{Na}}^{+}$ ions in the topmost layer are displaced inwards by $0.09\ifmmode\pm\else\textpm\fi{}0.03\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ at all investigated temperatures. Moreover, the root-mean-square vibrational amplitudes of ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{\ensuremath{-}}$ ions in the topmost layer are enhanced significantly in the investigated temperature range. The amplitudes of the ions in the second layer are close to the bulk values. While the amplitudes of the cations in the second layer are larger than those of the anions, ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{\ensuremath{-}}$ in the topmost layer have nearly equal vibrational amplitudes at temperatures above $100\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. In this temperature range, the mean-square amplitudes of vibrations of ions at the surface and in the bulk are in good agreement with previously reported lattice dynamics studies.

Studies on (NaCl) x (KBr) y-x (KI)1-y solid solutions: 1. Lattice and thermal parameters

Solid solutions of NaCl, KBr and KI were prepared by the melt method for the first time. Densities and refractive indices of all the prepared solid solutions were determined and also used for the estimation of the bulk composition in the crystal. Lattice parameters and thermal parameters like Debye–Waller factor, mean square amplitude of vibration, Debye temperature and Debye frequency were determined from the X-ray powder diffraction data. The observed lattice parameters showed the existence of three phases in solid solutions each nearly corresponds to NaCl, KBr and KI. The thermal parameters show a highly non-linear bulk composition dependence.

X-ray diffraction study of the structure and thermal parameters of the ternary Au–Ag–Pd alloys

In situ X-ray diffraction experiments were performed on six samples of the ternary Au–Ag–Pd alloys (with A1 structure) having different compositions using a Cu-target. The integrated intensity data obtained in the temperature range of 373–1200 K was utilized to determine the lattice parameters and the thermal parameters like Einstein's temperatures ( Θ E ), the mean-square amplitudes ( u 2 ¯ ( T ) ) and the coefficients of thermal expansion with a high accuracy. The lattice parameter showed a small negative deviation from the Vegard's rule. It is also found that the mean-square amplitudes are independent of the static displacements. The mean-square amplitudes of vibration and the linear thermal expansion follow the classical Grüneisen relationship in these alloys.

X‐ray study of Bi, Sb and Bi1–x–Sbx alloys

AbstractThe mean square amplitudes of vibration, Debye–Waller factors and Debye temperatures of bismuth (Bi), antimony (Sb) and Bi1–x –Sbx alloys have been obtained from X‐ray intensities. The intensities have been measured with Philips 3020 diffractometer fitted with a proportional counter using filtered CuKα radiation at room temperature and have been corrected for thermal diffuse scattering. The X‐ray Debye temperatures obtained for Bi and Sb have been compared with corresponding values obtained from specific heats and elastic constants. The values of Debye temperature obtained for Bi1–x –Sbx alloys have been compared with the values calculated from Kopp–Neumann relation. Vacancy formation energies of Bi, Sb and Bi1–x –Sbx alloys have been estimated using a relation connecting it with the X‐ray Debye temperature. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)