Debye–Waller factor and the Lindemann parameter of some hexagonal close-packed metals

1980 ◽  
Vol 58 (3) ◽  
pp. 384-387 ◽  
Author(s):  
A. Ramanand ◽  
R. Ramji Rao
Keyword(s):  
Distribution Function ◽  
Harmonic Approximation ◽  
Normal Modes ◽  
Waller Factor ◽  
Hexagonal Close Packed ◽  
Debye Waller Factor ◽  
Anisotropic Temperature ◽  
Modes Of Vibration ◽  
The Mean ◽  
Temperature Factors

The Debye–Waller factor has been calculated as a function of temperature for the four hexagonal close-packed (hcp) metals cobalt, ruthenium, erbium, and scandium, using a lattice-dynamical model to evaluate the normal mode frequencies and eigenvectors in the harmonic approximation. The calculation of the anisotropic temperature factors for these metals requires a knowledge of the eigenvectors for the various normal modes of vibration. The frequency distribution function is also used to calculate the mean-square amplitude of displacement of the atoms, in the cubic approximation. The first and second negative moments of the distribution function are used to calculate the low- and high-temperature limits of [Formula: see text], respectively. The value of the Lindemann parameter obtained from the present calculations is consistent with the value quoted by Gschneidner.

In Situ Observation of the Oxygen Nucleation in Silicon with X-Ray Single Crystal Diffraction

2011 ◽  
Vol 178-179 ◽  
pp. 353-359 ◽  
Author(s):  
Johannes Will ◽  
Alexander Gröschel ◽  
Christoph Bergmann ◽  
Andreas Magerl
Keyword(s):  
Single Crystal ◽  
Waller Factor ◽  
Dynamical Theory ◽  
Mean Value ◽  
Single Crystal Diffraction ◽  
X Ray ◽  
Float Zone ◽  
Debye Waller Factor ◽  
The Mean

The measurement of Pendellösungs oscillations was used to observe the time dependent nucleation of oxygen in a Czochralski grown single crystal at 750°C. It is shown, that the theoretical approach of the statistical dynamical theory describes the data well. Within the framework of this theory it is possible to determine the static Debye-Waller-factor as a function of the annealing time by evaluating the mean value of the Bragg intensity and the period length. The temperature influence on the Pendellösungs distance was corrected for by measurement of a Float-zone sample at the same temperature.

scholarly journals Effect of the non-ideal axial ratio c/a on anharmonic EXAFS oscillation of h.c.p. crystals

2021 ◽  
Vol 28 (5) ◽  
Author(s):  
Tong Sy Tien
Keyword(s):  
Axial Ratio ◽  
Debye Model ◽  
Waller Factor ◽  
Calculation Model ◽  
Near Neighbor ◽  
Perturbation Approach ◽  
Many Body ◽  
Hexagonal Close Packed ◽  
Debye Waller Factor ◽  
The Many

The temperature and wavenumber dependence of the extended X-ray absorption fine-structure (EXAFS) oscillation of hexagonal close-packed (h.c.p.) crystals have been calculated and analyzed under the effect of the non-ideal axial ratio c/a. The anharmonic EXAFS oscillation is presented in terms of the Debye–Waller factor using the cumulant expansion approach up to the fourth order. An effective calculation model is expanded and developed from the many-body perturbation approach and correlated Debye model using the anharmonic effective potential. This potential, depending on the non-ideal axial ratio c/a, is obtained from the first-shell near-neighbor contribution approach. A suitable analysis procedure is performed by evaluating the influence of EXAFS cumulants on the phase shift and amplitude reduction of the anharmonic EXAFS oscillation. The numerical results for crystalline zinc are found to be in good agreement with those obtained from experiments and other theoretical methods at various temperatures. The obtained results show that the present theoretical model is essential and effective in improving the accuracy for analyzing the experimental data of anharmonic EXAFS signals of h.c.p. crystals with a non-ideal axial ratio c/a.

Temperature-dependent atomic B factor: an ab initio calculation

2019 ◽  
Vol 75 (4) ◽  
pp. 624-632 ◽  
Author(s):  
Cristiano Malica ◽  
Andrea Dal Corso
Keyword(s):  
Ab Initio ◽  
Harmonic Approximation ◽  
Debye Model ◽  
Waller Factor ◽  
Temperature Dependent ◽  
Atomic Displacements ◽  
Debye Waller Factor ◽  
Diffraction Peaks ◽  
Effects Of Temperature ◽  
Quantum Espresso

The Debye–Waller factor explains the temperature dependence of the intensities of X-ray or neutron diffraction peaks. It is defined in terms of the B matrix whose elements B αβ are mean-square atomic displacements in different directions. These quantities, introduced in several contexts, account for the effects of temperature and quantum fluctuations on the lattice dynamics. This paper presents an implementation of the B factor (8π2 B αβ) in the thermo_pw software, a driver of Quantum ESPRESSO routines that provides several thermodynamic properties of materials. The B factor can be calculated from the ab initio phonon frequencies and displacements or can be estimated, although less accurately, from the elastic constants, using the Debye model. The B factors are computed for a few elemental crystals: silicon, ruthenium, magnesium and cadmium; the harmonic approximation at fixed geometry is compared with the quasi-harmonic approximation where the B factors are calculated accounting for thermal expansion. The results are compared with the available experimental data.

EFFECT OF LATTICE DISTORTION ON THE DEBYE–WALLER FACTOR: I. EXTENDED INTERNAL DEFECTS

1967 ◽  
Vol 45 (8) ◽  
pp. 2651-2660 ◽  
Author(s):  
J. Vail
Keyword(s):  
Nearest Neighbor ◽  
Harmonic Approximation ◽  
Waller Factor ◽  
Potential Interaction ◽  
Coupling Constants ◽  
Extended Defects ◽  
Extended Defect ◽  
Cubic Crystal ◽  
Perfect Lattice ◽  
Debye Waller Factor

A model is introduced in which a Mössbauer atom in an extended internal defect is replaced by a point defect in a perfect lattice, with coupling equal to that in the extended defect. Lattice distortions are considered which are typical for extended defects, with 5 and 10% dilatation and compression, effective coupling constants in nearest-neighbor harmonic approximation are estimated for these cases for a monatomic cubic crystal with Morse potential interaction, and Visscher's data are then used to estimate the fractional change in the Debye–Waller factor, e−2W. Decreases of 14 and 38% are found in e−2W for 5 and 10%, respectively, of lattice dilatation in extended defects, using parameters that are typical of monatomic metals.

scholarly journals Atomic Structure of Decagonal Al-Cu-Rh Quasicrystal–Revisited: New Correction for Phonons

Crystals ◽  
2019 ◽  
Vol 9 (2) ◽  
pp. 78 ◽  
Author(s):  
Radoslaw Strzalka ◽  
Ireneusz Buganski ◽  
Pawel Kuczera ◽  
Lucjan Pytlik ◽  
Janusz Wolny
Keyword(s):  
Distribution Function ◽  
Diffraction Data ◽  
Structure Refinement ◽  
Waller Factor ◽  
Correction Function ◽  
Phonon Modes ◽  
Atomic Displacements ◽  
Alternative Description ◽  
Debye Waller Factor ◽  
Gaussian Distribution Function

The standard approach applies the Gaussian distribution function to estimate atomic displacements due to thermal vibrations in periodic and aperiodic systems, which is used in a form of the Debye–Waller factor during the structure refinement. Acoustic phonons provide the largest contribution to the Gaussian correction although the character of other phonon modes remains relatively unclear. In this paper, we provide an alternative description of localized and dispersionless phonons based on an assumption of the harmonic displacement distribution function, which was recently proposed for model quasicrystals, and apply this approach for a decagonal Al-Cu-Rh quasicrystal that was previously studied by Kuczera et al. in 2012. We used the same X-ray diffraction data and the statistical method of structural analysis of the aperiodic systems. The correction function for phonons takes the form of a Bessel function instead of a conventional (Gaussian) Debye–Waller factor. This allowed us to achieve R-factor of 7.2% compared to 7.9% reported in the original paper. A significant improvement of the calculated atomic composition towards experimentally obtained and minor positional changes is also reported compared to the original paper. The results show the usefulness of investigating different corrective terms for diffraction data during a structure refinement.

Monte Carlo simulation of an anharmonic Debye–Waller factor to theT4order

2017 ◽  
Vol 73 (2) ◽  
pp. 151-156 ◽  
Author(s):  
Kun-lun Wang ◽  
Xian-bin Huang ◽  
Jing Li ◽  
Qiang Xu ◽  
Jia-kun Dan ◽  
...  
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Fourth Order ◽  
Taylor Expansion ◽  
Harmonic Approximation ◽  
Perturbation Expansion ◽  
Waller Factor ◽  
Bragg Diffraction ◽  
Atomic Interaction ◽  
Debye Waller Factor

In an increasing number of cases the harmonic approximation is incommensurate with the quality of Bragg diffraction data, while results of the anharmonic Debye–Waller factor are not typically available. This paper presents a Monte Carlo computation of a Taylor expansion of an anharmonic Debye–Waller factor with respect to temperature up to the fourth order, where the lattice was a face-centred cubic lattice and the atomic interaction was described by the Lennard–Jones potential. The anharmonic Debye–Waller factor was interpreted in terms of cumulants. The results revealed three significant points. Firstly, the leading term of anharmonicity had a negative contribution to the Debye–Waller factor, which was confirmed by Green's function method. Secondly, the fourth-order cumulants indicated a non-spherical probability density function. Thirdly, up to the melting point of two different densities, the cumulants up to the fourth order were well fitted by the Taylor expansion up toT4, which suggested that the Debye–Waller factor may be calculated by perturbation expansion up to the corresponding terms. In conclusion, Monte Carlo simulation is a useful approach for calculating the Debye–Waller factor.

Diffuse X-ray scattering by 2,3-dimethylnaphthalene

1999 ◽  
Vol 32 (2) ◽  
pp. 309-321 ◽  
Author(s):  
R. Schreier ◽  
J. Kalus
Keyword(s):  
Correlation Coefficients ◽  
Waller Factor ◽  
Mutual Orientation ◽  
X Rays ◽  
Static Disorder ◽  
Diffuse Intensity ◽  
X Ray Scattering ◽  
Debye Waller Factor ◽  
Isotropic Displacement Parameter ◽  
The Mean

The diffuse intensity distribution of X-rays scattered from a dipolar disordered single crystal of 2,3-dimethylnaphthalene (2,3-DMN) has been measured and was found to be nearly independent of temperature in some restricted regions of the reciprocal space. This behaviour was related to a static disorder of the crystal. Lattice parameters were determined between room temperature and 9 K and phase transitions at 227 K and 112 K were observed. The results of the data analysis at 298 K are: the equivalent isotropic displacement parameterUeqfor the thermal Debye–Waller factor is (0.18 Å)2; the displacement parameters describing the static disorder are U\raise.15em{_{xx}}\kern-.65em^C\le(0.05 Å)2, U\raise.15em{_{yy}}\kern-.65em^C=[0.22\,(2) Å]2and U_{zz}^C= [0.35\,(3) Å]2. The molecules show an orientational Gaussian distribution around thexaxis with a width of 5.0 (8)°. All these results refer to the molecular inertia system. Correlation coefficients characterizing the mean mutual orientation of neighbouring molecules were determined. The highest correlation coefficient was found along [010]: C=-0.231\pm 0.004.

scholarly journals Calculation of X-ray absorption structure above K-edge of laser shock-compressed aluminum

1990 ◽  
Vol 8 (1-2) ◽  
pp. 319-325 ◽  
Author(s):  
A. Djaoui ◽  
T. A. Hall ◽  
R. C. Albers ◽  
J. J. Rehr ◽  
J. Mustre
Keyword(s):  
Mean Free Path ◽  
Waller Factor ◽  
Normal Density ◽  
Structure Model ◽  
Exchange Correlation ◽  
Debye Waller Factor ◽  
Correlation Potential ◽  
The Mean ◽  
X Ray Absorption ◽  
Band Structure Model

Aluminum sandwich foils can be compressed to more than twice the normal density by the use of shock waves generated by laser beams. We present calculations of the K-edge absorption structure of two fold compressed aluminum, and compare the results with experimental data. The atomic potential is calculated using a spherical band-structure model with Fermi-Dirac statistics for the electrons. The structure above the K-edge is calculated according to the Müller-Shaich formula for the propagating photo-electron. The mean free path of the electron is taken into account through a complex exchange correlation potential. An appropriate Debye-Waller factor is used to represent the effect of the relatively high temperature encountered in these experiments.

The crystal and molecular structure of tetraphenylarsonium Tetrakis(benzenethiolato)oxomolybdate(V)

1978 ◽  
Vol 31 (11) ◽  
pp. 2423 ◽  
Author(s):  
JR Bradbury ◽  
MF Mackay ◽  
AG Wedd
Keyword(s):  
Molecular Structure ◽  
Title Compound ◽  
Basal Plane ◽  
Crystal And Molecular Structure ◽  
Triclinic Space Group ◽  
R Factor ◽  
Anisotropic Temperature ◽  
Cell Dimensions ◽  
The Mean ◽  
Temperature Factors

Crystals of the title compound, Ph4As+ [MoO(SPh)4]-, belong to the triclinic space group Pī with cell dimensions a 12.623(8), b 14.295(7), c 14.301(9) Ǻ, α 94.43(4), β 111.43(5), γ 65.70(4)° and Z2. The structure has been refined with diffractometer data measured with Mo Kα radiation to give a conventional R factor of 0.056 for 3215 unique terms. The Mo, As, S and O atoms were refined with anisotropic temperature factors and the C atoms with isotropic factors. The square- pyrimidal [MoO(SPh)4]- anion has the Mo atom lying 0.82 Ǻ from the basal plane of the four S atoms with Mo-O 1.699(9) Ǻ, the mean Mo-S bond lengths 2.403(5) Ǻ and the mean S...S distance 3.20(3) Ǻ. The geometry about the As atom in the Ph4As+ cation is regular tetrahedral, the mean As-C bond length being 1.91(1) Ǻ.

Crystal Equilibrium and Lattice Dynamical Study of α-Iron and Chromium

1977 ◽  
Vol 32 (12) ◽  
pp. 1495-1502 ◽  
Author(s):  
O. P. Gupta ◽  
M. P. Hemkar
Keyword(s):  
Short Range ◽  
Phonon Dispersion ◽  
Mean Square Displacement ◽  
Waller Factor ◽  
Dynamical Study ◽  
Ion Interactions ◽  
Debye Waller Factor ◽  
Screening Parameter ◽  
The Mean ◽  
Ionic Lattice

Abstract Studies of the phonon dispersion, the temperature dependence of Debye-Waller factor, the Debye temperature, and the mean square displacement of α-iron and chromium have been made by means of a lattice dynamical model proposed by the authors. The model considers short range pairwise forces effective up to the second neighbours and electron-ion interactions on the lines of Bhatia using an appropriate value of the screening parameter. The volume force is aver­ aged over the whole Wigner-Seitz sphere. The ionic lattice is in equilibrium in a medium of electrons. The computed results compare well with experimental data.

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