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.

SIGNATURE OF LOCAL STRUCTURE ANOMALY AT Tc IN THE Nb3Ge SUPERCONDUCTOR

2002 ◽  
Vol 16 (11n12) ◽  
pp. 1713-1719 ◽  
Author(s):  
M. FILLIPPI ◽  
N. L. SAINI ◽  
H. OYANAGI ◽  
A. BIANCONI
Keyword(s):  
Local Structure ◽  
Cuprate Superconductors ◽  
Waller Factor ◽  
Superconducting Transition ◽  
Temperature Dependent ◽  
Local Electron ◽  
Atomic Displacements ◽  
Debye Waller Factor ◽  
The One ◽  
X Ray Absorption

We report local structure of Nb3Ge intermetallic superconductor by Ge K-edge extended X-ray absorption fine structure (EXAFS) measurements performed in the temperature range of 6–300 K, with an emphasis to determine the local and instantaneous atomic displacements across the superconducting transition temperature T c . We find that the temperature dependent correlated Debye–Waller factor of the Ge-Nb bonds shows a drop at the T c while cooling the sample, similar to the one observed in the high-T c cuprate superconductors. The results provide a clear indication of an intimacy between the local atomic displacements and the short coherence superconductivity, and suggests that local electron-lattice interaction should be considered to explain the high-T c superconductivity in these materials.

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.

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.

Pressure effects on the EXAFS Debye–Waller factor of iron

2020 ◽  
Vol 27 (5) ◽  
pp. 1372-1375
Author(s):  
Ho Khac Hieu ◽  
Nguyen Ba Duc ◽  
Nguyen Van Hung ◽  
Pham Thi Minh Hanh ◽  
Tran Thi Hai
Keyword(s):  
Relative Displacement ◽  
Debye Model ◽  
Waller Factor ◽  
Pressure Effects ◽  
Displacement Curve ◽  
Mean Square ◽  
Debye Waller Factor ◽  
Debye Frequency ◽  
X Ray Absorption ◽  
Analytical Expressions

The pressure effects on atomic mean-square relative displacement characterizing the extended X-ray absorption fine structure (EXAFS) Debye–Waller factor of iron metal have been investigated based on the Debye model. The analytical expressions of the Debye frequency and EXAFS Debye–Waller factor have been derived as functions of crystal volume compressibility. Based on the well established equation-of-state including the contributions of the anharmonic and electronic thermal pressures, numerical calculations have been performed for iron up to a pressure of 220 GPa and compared with experimental data when possible. These results show that the Debye frequency increases rapidly with compression, and beyond 150 GPa it behaves as a linear function of pressure. Meanwhile the mean-square relative displacement curve drops robustly with pressure, especially at pressures smaller than 100 GPa. This phenomenon causes the enhancement of EXAFS signals at high pressure. Reversely, the increasing of temperature will reduce the amplitude of EXAFS spectra.

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.

Diffusive motions of water molecules near 1 °C

1978 ◽  
Vol 56 (3) ◽  
pp. 373-380 ◽  
Author(s):  
J. D. Irish ◽  
W. G. Graham ◽  
P. A. Egelstaff
Keyword(s):  
Atmospheric Pressure ◽  
Rotational Diffusion ◽  
Waller Factor ◽  
Root Mean Square Amplitude ◽  
Breaking Effect ◽  
Debye Waller Factor ◽  
Effects Of Temperature ◽  
Temperature And Pressure ◽  
Breaking Time ◽  
Structure Breaking

Neutron (quasi-elastic) scattering experiments have been carried out to investigate the effects of temperature and pressure on the motion of molecules in liquid H2O. Spectra were obtained at several temperatures between −10 and +24 °C at atmospheric pressure, and also at +1 °Cand 1.42 × 108 Pa. The root-mean-square amplitude of proton motion in solid and liquid H2O at atmospheric pressure has been inferred from measurements of the Debye–Waller factor and is nearly constant at 0.07 nm for both the solid and liquid. A model in which the translational diffusion and rotational diffusion are based on Brownian motion was fitted to our data using published diffusion coefficient data. The structure breaking time was found to be very small at 1 and 10 °C. Both these conclusions are compared with those reported in several earlier papers, which gave varying results.The spectrum measured at 1.42 × 108 Pa was slightly broader than the corresponding spectrum at atmospheric pressure, probably indicating that the well-known minimum in the viscosity at this pressure is due to a structure breaking effect.

scholarly journals Atomic disorder of Li0.5Ni0.5O thin films caused by Li doping: estimation from X-ray Debye–Waller factors

2015 ◽  
Vol 48 (6) ◽  
pp. 1896-1900 ◽  
Author(s):  
Anli Yang ◽  
Osami Sakata ◽  
Ryosuke Yamauchi ◽  
L. S. R. Kumara ◽  
Chulho Song ◽  
...  
Keyword(s):  
Thin Film ◽  
Thin Films ◽  
Debye Model ◽  
Waller Factor ◽  
Structural Parameter ◽  
X Ray Diffraction ◽  
Atomic Disorder ◽  
X Ray ◽  
Li Doping ◽  
Debye Waller Factor

Cubic type room-temperature (RT) epitaxial Li0.5Ni0.5O and NiO thin films with [111] orientation grown on ultra-smooth sapphire (0001) substrates were examined using synchrotron-based thin-film X-ray diffraction. The 1\overline{1}1 and 2\overline{2}2 rocking curves including six respective equivalent reflections of the Li0.5Ni0.5O and NiO thin films were recorded. The RTB1factor, which appears in the Debye–Waller factor, of a cubic Li0.5Ni0.5O thin film was estimated to be 1.8 (4) Å2from its 1\overline{1}1 and 2\overline{2}2 reflections, even though the Debye model was originally derived on the basis of one cubic element. The corresponding Debye temperature is 281 (39) K. Furthermore, theB2factor in the pseudo-Debye–Waller factor is proposed. This parameter, which is evaluated using one reflection, was also determined for the Li0.5Ni0.5O thin film by treating Li0.5Ni0.5O and NiO as ideal NaCl crystal structures. A structural parameter for the atomic disorder is introduced and evaluated. This parameter includes the combined effects of thermal vibration, interstitial atoms and defects caused by Li doping using the two Debye–Waller factors.

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