Anharmonic Thermal Vibrations Research Articles

Investigations of Bijvoet Differences and Anharmonicity in Gallium Phosphide

AbstractPrecise X‐ray diffraction intensity data are collected at 300 K using a perfect single crystal sphere of gallium phosphide. The data are firstly analysed assuming a harmonic thermal vibration of atoms. The refined Debye‐Waller factors for this harmonic model are B(Ga) = 0.446(0.010)× 10−2 nm2 and B(P) = 0.507(0.001) × 10−2 nm2 with the residual index R = 1.2715%. Then anharmonic thermal vibration effects are included in the structure factor expressions. In the anharmonic model, two additional parameters, the individual anharmonic thermal parameters, are also included in the refinement process. The refined values are β(Ga) = −1.201(0.371) × 10−13 J nm−3 and β(P) = 2.002(0.274) × 10−13 J nm−3 with R = 1.2646%. The effect of anomalous dispersion on the structure factor is also analysed. It leads to Bijvoet differences between all odd index inverse reflections. The inclusion of anharmonic effects leads to Bijvoet differences between all even index inverse reflections. The measured Bijvoet ratios are in good agreement with the calculated Bijvoet ratios both in sign and magnitude. The individual Debye‐Waller factors, the Bijvoet ratios, observed and calculated both, for the harmonic and anharmonic models, are reported.

Disorder and thermal motion of Ag in Ag 0.6NbS 2

Refinement of single crystal X-ray diffraction data of the two-dimensional silver conductor Ag 0.6NbS 2, measured in the temperature range 295–695 K, shows significant anharmonicity in the thermal motion of Ag. The plane intermediate between the two sublattices of Ag (α and β), forming together a puckered honeycomb lattice, plays an important role in the conduction process. The α- and β-sites are equally occupied for 0.30 already at room temperature. The diffusion path between the approximately tetrahedrally coordinated α- and β-sites goes via the octahedron formed by sulfur atoms of neighbouring sandwiches NbS 2, but the center of this octahedron is avoided. The behaviour of the thermal expansion is anisotropic, but disagrees with data known for comparable 2-dimensional ionic conductors. The expansion along the unique axis is much larger than perpendicular to it, and seems not related to anharmonic thermal vibrations in that direction.

Thermal expansion of Y 2Ba 4Cu 7O 15 in the temperature range 10–290 K, and comparison with YBa 2Cu 3O 7 and YBa 2Cu 4O 8

The orthorhombic lattice parameters of superconducting Y 2Ba 4Cu 7O 15.28 ( T c = 85.5 K) are measured as a function of temperature (10–290 K) to a precision of 1 × 10 -4 on a low-temperature Guinier X-ray powder diffractometer by using an internal standard. The thermal expansion is anisotropic (( a 297 K− a 10 K)/ a 297 K = 17(2) × 10 -4, ( b 297 K− b 10 K)/ b 297 K = 11(2) × 10 -4, ( c 297 K− c 10 K)/ c 297 K = 34(2) × 10 -4) and intermediate between that of structurally related YBa 2Cu 3O 7 and YBa 2Cu 4O 8. It is attributed to anharmonic thermal vibrations of the CuO 2-planes and CuO single- and double-chains in the directions of vacant oxygen sites. No anomalous changes as a function of temperature are found within experimental resolution.

Anomalous dispersion effects, anharmonic thermal vibrations and bonding charges in indium antimonide

Upon the inclusion of dispersion corrections in the structure factor expressions for the ZnS structure, Friedel's law is violated for all odd and all even index reflections due to bonding effects in the low angle regions and anharmonicity in the high angle regions in this type of crystals. Systematic and precise X-ray diffraction intensity measurements were carried out using a spherical single crystal of indium antimonide at 300 K. From the least squares refinement of the experimental data we obtain the Debye-Waller factors for indium and antimony as B(In) = 1.432(0.003) A ̊ 2 and B(Sb) = 0.865(0.002) A ̊ 2 with the residual index R = 2.24%. A search for anharmonic thermal vibrations was made for the present experimental data. On including the individual anharmonic thermal factors in the refinement process we obtain the anharmonic thermal parameters for indium and antimony as β(In) = −0.57(0.34) 10 −16 J A ̊ −3 and β(Sb)= −2.21(1.18) 10 −16 J A ̊ −3 , the Debye-Waller factors for the anharmonic model being B(In) = 1.333(0.003) A ̊ 2 and B(Sb) = 0.886(0.002) A ̊ 2 with R = 2.22%. The present work reveals excellent agreement, both in sign and magnitude, with the Bijvoet inequalities for a large number of reflections. In the ZnS structure, the weak quasi-forbidden reflections ( h + k + l = 4 n + 2) contain information about the covalent bond charges which was also deduced using a few 4 n + 2 type reflections. A net transfer of charge from In to Sb is observed in the present work.

EXAFS studies on anharmonic thermal vibrations in AgI

The thermal behaviour of atoms in β- and α-AgI has been investigated by means of the EXAFS method in the temperature range 73–815 K. The EXAFS data were analyzed using a formula expressed by the cumulant expansion up to fourth-order terms. The experimental results can be explained satisfactorily by the anharmonic model. The position of each peak in the radial structure function around Ag ions observed in both β- and α-AgI is significantly shorter than that of the simulation of a harmonic model, and shifts apparently to a shorter distance with increasing temperature due to the contributions of third-order terms. The third and fourth cumulants are significant even at 73 K. The temperature dependence for the values of cumulants are different between β- and α-AgI.

Study of the bipyramidal site in magnetoplumbite-like compounds, Sr M12O 19 ( M = Al, Fe, Ga)

Structures of magnetoplumbite-type crystals Sr M 12O 19 ( M = Al, Fe, Ga) have been refined by the single-crystal X-ray diffraction method. The space group is P6 3/ mmc with Z = 2. The cell dimensions are a = 5.5666(2) and c = 22.0018(8) Å for SrAl 12O 19, a = 5.8836(1) and c = 23.0376(9)Å for SrFe 12O 19, and a = 5.7929(1) and c = 22.8123(7) Å for SrGa 12O 19 at 22°C. Four structural models were tested with respect to the states of M atoms at trigonal bipyramidal sites: central atom and split atom models with and without anharmonic thermal vibrations. The simple split atom model gave lower R w values than the central atom model with anharmonic thermal vibrations for SrAl 12O 19 and SrGa 12O 19, whereas it gave an R w value approximately equal to that given by the central atom model with anharmonic vibration for SrFe 12O 19. The present study suggested that the potential around the M atom with trigonal bipyramidal coordination has double minima in all the magnetoplumbite-type crystals at lower temperatures.

Direct methods in surface crystallography

Conventional methods of determining surface structure from LEED intensity measurements face severe problems when confronted with complex or unconventional surface structures forcing a restricted search of parameter space, possibly leading to errors in the determination. Direct Methods sidestep this problem. In this paper we discuss several possible implementations of the Direct Methods and illustrate one of them with the example of p(2 × 2)O Rh(100) . The output from a Direct Methods calculation is the probability distribution of the surface atoms and the paper concludes with a discussion of the possibility for study of anharmonic thermal vibrations, and of translational motion which this opens up.

A time-of-flight neutron diffraction study of anharmonic thermal vibrations in SrF2, at the spallation neutron source ISIS

Measurements have been made, at wavelengths in the range 0.3-6.2 A, of hhl reflections of SrF2 on the single-crystal diffractometer (SXD) at the ISIS Spallation Neutron Source. After application of a variable-wavelength extinction correction to the derived I F,,k,I values, a refinement of the anharmonicity parameter /3F of the fluorine atoms was carried out, yielding a value of-4.19 (30) x 10-19 J ~-3.

Anharmonic thermal vibration in a crystal of lithium(I) tantalum(V) trioxide, LiTaO3.

The electron-density distribution in a crystal of LiTaO3 has been investigated with the single-crystal X-ray diffraction method. The crystal has the cell dimensions a=5.15329(7) and c=13.7806(5) A with the space group R3c and Z=6. Residual electron densities due to anharmonic thermal vibration were observed around the Ta5+ ion. Refinement with anharmonic thermal parameters improved the coincidence of the observed and calculated structure factors significantly. The C333 (the coefficient of Δz3) value of Ta5+ is most significant among the third and fourth order anharmonic potential parameters. It has revealed that significant errors are sometimes introduced into positional parameters if contributions of odd order terms of anharmonic potentials are ignored.

Composition dependence of thermal vibrations in Hg 1−xZn xTe solid solutions determined by X-Ray diffraction

We report on the composition dependence of thermal vibrations in Hg[in1- xZn x Te solid solutions ( x = 0.13,0.20, 0.50, 0.75) determined by X-ray diffraction at room temperature (297 K) and in some cases ( x = 0.50, 0.75) at 143 K. Although some ambiguity on the results obtained for x ≅ 0.50 cannot be avoided as being intrinsic to the problem, it is shown, by taking into account the results obtained by Cooper, Rouse and Fuess Acta crystallogr. A29, 1973 on ZnTe ( x = 1), that the conventional harmonic B factor ( B = 8 π 2〈 u 2〉 where 〈 u 2〉 is the mean-square displacement of the atom in any direction) of the average (HgZn) cation decreases almost linearly from B Hg- Zn = 2.82 ( σ = 0.09) Å2 to B jn = 1.296 ( σ = 0.034) Å 2 when x varies from x = 0.13 to x = 1. This is interpreted as resulting from the almost composition independent B Hg ≅ 2.7 Å 2 and B Hg ≌ 1.3 Å 2. By contrast the B factor of the Te anion exhibits a more complex behaviour varying in a non-monotonic way from 2.03 (σ = 0.07) Å 2 for x = 0.13 to 0.76 ( σ = 0.01) Å 2 for x = 1. In the alloys the B factors do not vary linearly with T between 297 and 143 K. It is shown that this is not attributable to quantum effects (zero point motion) but is due to a static structural disorder which affects the metal as well as the anion sublattices, the average atomic distortion being about 0.08 Å. The experimental data also show contributions arising from third-order anharmonic thermal vibrations of the atoms.

ChemInform Abstract: Ferroelectric Structure and Related Properties of Pb5W3O9F10.

AbstractThe new ferroelectric title material crystallizes in the tetragonal system (I4, Z=4).

Experimental electron-density-distribution study of potassium iron disulfide, a low-dimensional material.

The experimental electron-density distribution of potassium iron disulfide, ${\mathrm{KFeS}}_{2}$, has been determined from high-resolution single-crystal x-ray intensity measurements collected with Ag K\ensuremath{\alpha} radiation to a resolution of (sin\ensuremath{\theta})/\ensuremath{\lambda}=1.30 A${\r{}}^{\mathrm{\ensuremath{-}}1}$. The structure of ${\mathrm{KFeS}}_{2}$ consists of parallel chains of edge-sharing ${\mathrm{FeS}}_{4}$ tetrahedra extending along the c-axis. In addition to the conventional structural parameters, multipole deformation functions and third- and fourth-cumulant anharmonic thermal parameters were included in some least-squares refinements. In all cases, inclusion of anharmonic thermal vibration parameters yielded significantly better fits to the x-ray data. From the experimental multipole population parameters of the iron atom, apparent d-orbital occupancies and the number of unpaired 3${d}_{\ensuremath{\uparrow}}$ electrons have been estimated within the crystal-field approximation and the assumption that a minimal basis set of atomic d orbitals is adequate for the d electrons. The values obtained for the number of 3${d}_{\ensuremath{\uparrow}}$ electrons from various refinements of the x-ray data agree well with values deduced from previous magnetic-susceptibility measurements and from theoretical calculations. The crystal-field splitting of the iron d orbitals and the ensuing asphericity is clearly evident in the experimental electron-density maps. The electric field gradient at the iron nucleus, calculated from the x-ray multipole parameters due to the slightly distorted (${D}_{2d}$) tetrahedral crystal field, predicts a quadrupole splitting of \ensuremath{\Delta}${E}_{\mathrm{QS}=3.3(10}$) mm/sec, somewhat larger than observed in M\ossbauer experiments (\ensuremath{\Delta}${E}_{\mathrm{QS}=0.53}$ mm/sec). The major contribution to the electric field gradient comes from the valence-electron distribution, which is consistent with the observed asphericity of electron density around iron.

Ferroelectric structure and related properties of Pb5W3O9F1

Lead tungsten oxyfluoride, Pb5W3O9F10, is a new ferroelectric material that crystallizes in the tetragonal system with space group I4 and four formulas in the unit cell. The lattice constants at 295 K are a=14.583(3) and c=7.365(1) Å for λ(MoKα1,2)=0.709 30, 0.713 59 Å. The integrated intensities of all 10 757 reflections within a reciprocal hemisphere of radius (sin θ)/λ ≤0.91 Å−1 were measured on a CAD-4 diffractometer with MoKα radiation and resulted in 7080 F2m ≥2σF2m. Correction for linear anisotropic radiation damage, followed by averaging the symmetry equivalent reflections, gave 2933 independent 〈 F2m〉≥2σ〈 F2m〉 with internal agreement factor Rint〈Fm〉 =0.0289. The crystal structure was solved from the Patterson function and difference Fourier series and refined on w〈 F2m〉 by the method of least squares. The final agreement factors between measurement and model were R=0.0382 and wR=0.0329 on 〈 Fm〉. The as-grown crystal contained equal volumes of ±Z domains. The three independent tungsten atoms form WO6, WO3F3, and WO2F4 octahedra, and the three independent lead atoms form two different PbO4F5 and a PbO4F4 polyhedron. The major local dipole contribution to the spontaneous polarization is associated with a tungsten atom that is displaced 0.172(43) Å from the center of its octahedron. The corresponding predicted Curie temperature is 590(260) K, in agreement with the experimental value of 785(15) K. All six metal atoms undergo anharmonic thermal vibrations, which are fitted either to fifth or sixth order in a Gram–Charlier expansion of the probability density function. The average interatomic distances are W–F=1.798(8) Å, W–O=1.944(18) Å, Pb–F=2.68(19) Å, and Pb–O=2.72(21) Å: standard deviations are from Bessel’s relationship. The atomic rearrangement required in forming Pb5W3O9F10 from PbF2 and WO3 is discussed, and similarities and contrasts between the related families for which Pb5W3O9F10, SrAlF5, and Sr3(FeF6)2 provide the archetype structures are considered. All three families exhibit ferroelectric behavior.

Anomalous dispersion effects, thermal vibrations and bonding charges in gallium arsenide

Accurate X-ray diffraction intensity data have been obtained using a spherical single crystal of GaAs at ambient temperature, 298 °K. Least squares refinement of this data yielded the Debye-Waller factors,BGa=0.68±0.05 A2 andBAs=0.42±0.03 A2. The residual index for the discrepancy between measured and calculated structure factors in the present work is quite small, beingR=0.0068. Anharmonic thermal vibration was sought for and found to be negligible. The inclusion of dispersion corrections in the structure factor expressions gives rise to Bijvoet inequality between inverse reflections with the indices all odd. The present work reveals excellent agreement, both in sign and magnitude, of the Bijvoet inequalities for a large number of such reflections. Covalent charge transfer has been deduced from the experimental measurements on a few quasi-forbidden Bragg reflections.

Electron densities in VN. I. High-precision x-ray-diffraction determination of the valence-electron density distribution and atomic displacement parameters.

High-precision, high-resolution x-ray-diffraction measurements were performed on a nearly stoichiometric, spherical single crystal of vanadium nitride [NaCl structure, a=4.137 11(4) A\r{}] using Ag K\ensuremath{\alpha} radiation. Structure factors with an average accuracy of 0.05e were obtained for 90 unique reflections from 12 807 intensity profiles measured at room temperature. Effects due to anisotropic extinction and crystal inhomogeneity were modeled and refined together with a scale factor, metal-atom 4s and 3d and nonmetal-atom 2s and 2p orbital-occupancy factors, \ensuremath{\kappa} expansion-contraction parameters, atomic site occupation, and anharmonic thermal parameters. At convergence, the reliability factors were R=0.0068 (before) and R=0.0031 (after) averaging symmetry-equivalent reflections. The charge distribution around the metal sites showed a positive deformation along [111], corresponding to preferential occupancy of ${t}_{2g}$ orbitals. A comparison with isostructural ${\mathrm{TiC}}_{0.94}$ and ${\mathrm{TiN}}_{0.99}$ showed that the shape of the metal atoms as expressed by the orbital-occupancy ratio p(${t}_{2g}$)/p(${e}_{g}$) changes as a function of valence-electron concentration: p(${t}_{2g}$)/p(${e}_{g}$)=3.3 in ${\mathrm{VN}}_{1.00}$, 1.9 in ${\mathrm{TiN}}_{0.99}$, and 1.0 in ${\mathrm{TiC}}_{0.94}$, in agreement with theoretical predictions.The charge distribution around the nonmetal sites is spherically symmetric. No buildup of charge density occurred between atomic sites. The charge transfer from the metal to the nonmetal site [1.2(1)e] was similar to those in ${\mathrm{TiC}}_{0\mathrm{e}.94}$ [2.1(4)e] and ${\mathrm{TiN}}_{0.99}$ [1.9(4)e]. The occupancy factor of the metal site, ${\mathrm{scrp}}_{\mathrm{V}=0.9986(8)}$, was consistent with chemical analysis (${\mathrm{VN}}_{0.996(3)}$). The mean-square displacements of the metal atoms, 〈${u}^{2}$${〉}_{\mathrm{V}}$=0.006 11(3) A${\mathit{\r{}}}^{2}$, were larger than those of the nonmetal atoms, 〈${u}^{2}$${〉}_{\mathrm{N}}$=0.004 33(7) A${\mathit{\r{}}}^{2}$. No evidence for static atomic displacements and anharmonic thermal vibrations was found.

X-ray diffraction study of anharmonic thermal vibrations in β-AgI

Intensity data were collected at room temperature from an extended-face crystal of AgI using Mo Kα radiation. Least-squares refinement indicates that the thermal motion of the Ag+ ions is large with a considerable anharmonic component. The wurtzite positional parameter, u, was found to be 0.3748(2). The proportion of point defects was determined to be no greater than 0.9%. The proportion of random staking faults was determined by two experimental techniques to be no greater than 2%.

An X-ray analysis of anomalous dispersion effects, anharmonicity and thermal vibrations in gallium phosphide

Thermal vibrations of gallium and phosphorus atoms in gallium phosphide have been determined from the integrated intensities of Bragg reflections measured at ambient temperature. The thermal parameter values, obtained from the least-squares refinement, are B Ga = 0.66 ± 0.06 Å 2 and B p = 0.94 ± 0.10 Å 2 . Anharmonic thermal vibration effect is found to be small. The inclusion of dispersion corrections and anharmonicity in the structure factor expressions for GaP is found to give rise to the breakdown of Friedel's law for all even index reflections also. The observed Bijvoet ratios for several reflections are in general agreement with the calculated values.

Treatment of Anharmonic Thermal Vibration by Using Transformation of Scattering Vector

A treatment to get the thermal factor T j ( Q ) at site j for a harmonic and anharmonic potential field was discussed. A transformation of the scattering vector Q j = R j -1 Q 1 was introduced in the treatment where R j is the rotational part of a symmetry operation which transforms the atomic positions from r 1 to r j . This method can be applied to a general potential form including the 1st, 2nd, 3rd and higher order components. An application of the treatment to solid electrolytes α-AgI, β-AgI 3 SI and β-Ag 3 SBr was studied.

The anharmonic thermal vibration in ZnX (X = S, Se, Te) and its dependence on the chemical-bond characters

The X-ray diffraction study of anharmonic thermal vibration was carded out with respect to ZnX (X = S, Se, Te), having a zinc-blende structure, whose latticedynamical properties such as force constant and effective charge were determined by Raman spectroscopy. Difference Fourier syntheses based on harmonic refinements after correction of the diffraction intensities for thermal diffuse scattering exhibit a tetrapod shape of residual electron density around the Zn atoms, implying an anisotropy of the anhar

An X-ray Study of Thermal Atomic Vibrations in Oxygen-Octahedra Compounds

An X-ray diffraction study of ferro-, antiferro- and paraelectrics with perovskite, elpasolite and pyrochlore type structures has been carried out. The influence of anharmonic thermal atomic vibration on temperature dependences of temperature factors of individual sublattices, Debye temperatures and logarithms of relative integrated intensities has been studied. The potential parameters of individual sublattices αk and γk have been calculated. The conclusion has been made that in perovskites ABO3 (A=Pb, B=Ti, Zr), elpasolites A2Ca2+B6+O6 (A=Ba, B6+=Re, Te) and pyrochlores A2B2O6X (A=Cd, B=Nb, X=O, S), A-sublattice should be considered as ferroelectrically active one, and that deviations from harmonic behaviour for A-cations seem to be appreciable.