Relative Atomic Displacements Research Articles

On the Mean Square Relative Atomic Displacements in Strongly Anharmonic Crystals

AbstractThe correlative method of unsymmetrized self‐consistent field is used to study the mean square relative atomic displacements in crystals taking into account the strong anharmonicity and interatomic correlations. General expressions for these quantities in arbitrary crystal lattices are obtained. As an illustration, the mean square relative displacements between nearest, second, and third neighbours in a strongly anharmonic linear chain are calculated. The results are compared with those obtained with more rough approximations (harmonic, quasi‐harmonic, etc.). The influence of anharmonic effects is discussed.

Calculation of the XAFS Debye-Waller Factor with Spherical-Wave Corrections

To calculate XAFS spherical-wave corrections and multiple scattering contributions have to be included improving the single scattering plane-wave approximation. A suitable method is described to include such corrections. The temperature dependence of the spectra is determined by a Debye-Waller factor in the plane-wave approximation. An additional temperature dependence arises in a spherical-wave description. An approximative expression is developed, which is valid for the exact angular momentum expansion and any spherical-wave approximation. Due to the thermal movement of the atoms the elements of the t-matrix tl have to be replaced by tlTD=tl+k2 σ102 \ ildetl where \ ildet is a modified t-matrix. σ102 is the average relative atomic displacement between the emitter and the scatterer. t and \ ildet have a different k-dependence. The consequence is as shift in the peaks of the absolut value of the fouriertransformed spectrum. For Ni and NiO we get a shift up to 0.1 atomic units for the nearest and next nearest neighbour.

The Effect of Pressure on the Thermodynamic Properties of Crystals near the Lattice Instability Point

AbstractUsing the self‐consistent phonon theory based on the thermodynamic double‐time Green's function method the thermodynamical properties of the cubic crystals in the vicinity of the instability point are investigated. The results of calculations of the pressure dependence of the relative displacement of atoms, internal and free energy, free Gibbs energy obtained with the help of the generalized form of the Lennard‐Jones (n, m) renormalized potential are given and compared with experimental and other theoretical data for a number of cubic metals.

Experimental and theoretical evidence for a strong anisotropy of the surface Debye-Waller factor as determined for a monolayer of cobalt on copper (111) by surface extended x-ray-absorption fine structure.

We present here the first determination of the anisotropy of the surface Debye-Waller factor involved in a surface extended x-ray-absorption fine-structure (SEXAFS) experiment. It is deduced by use of relative temperature-dependent amplitude functions for two different polarizations. The mean square relative displacements of atoms are calculated with an elaborate lattice-dynamical model, treated by a continued-fraction technique. The results are in good agreement which shows the ability of SEXAFS as a valuable tool for the study of surface vibrations.

Dynamics and Thermodynamics of Anharmonic Crystals near the Instability Point

AbstractUsing the self‐consistent phonon theory based on the thermodynamic double‐time Green's function method the dynamics of crystal lattice and the thermodynamics of the anharmonic cubic crystals in the vicinity of the instability point are investigated. The results of calculations for the instability temperature Ts, relative displacements of atoms, anharmonic contributions to the free energy, and free Gibbs energy as well as the isothermal elastic constants are given and compared with experimental and other theoretical data for potassium, sodium, and rubidium.

Localized optical modes induced by point and extended defects in ionic crystals

It is shown that the vector character of the relative displacement of atoms in ionic crystals causes a nonexponential (power) law decay of localized optical mode amplitudes at large distances from point or linear defects.

Mean-square relative displacements of nearest-neighbour atoms in Ge

EXAFS spectra of crystalline and amorphous Ge have been used to determine the mean-square relative displacements sigma 2 of nearest- and next-nearest-neighbour atoms in the temperature range 80-400K. A strong correlation of the nearest-neighbour motion is observed for crystalline Ge, but this correlation decreases in the amorphous material. The results are compared with calculations performed within the Debye approximation. The structural parameters of amorphous Ge are consistent with a continuous random network approach.

Temperature dependence of the mean square relative displacements of nearest-neighbour atoms derived from EXAFS spectra

An investigation of the dependence on temperature of the extended X-ray absorption fine structure (EXAFS) above the K-edges of Cu and Co in the metals and above the K-edges of Rb and Sr in the compounds RbCl, SrS, SrF2 and SrCl2 has been performed. For Cu, Co, RbCl and SrS, the values of the correlated mean square relative displacements of nearest-neighbour atoms derived from EXAFS spectra show a good agreement with those calculated from a Debye model. The experimental data obtained are also compared with available data for mean square displacements of the individual atoms calculated from more-refined lattice vibrational models and with Debye-Waller factors used in X-ray diffraction experiments.

(110) surface atomic structures of covalent and ionic semiconductors

A total-energy-minimization approach is applied to the determination of the (110) surface atomic geometries of Si, Ge, GaAs, InP, InSb, ZnSe, and ZnTe. The accuracy of the model employed in the energy minimization was tested by comparing calculated values of bulk elastic coefficients and phonon frequencies to experimental data. Generally good agreement between the calculated and experimental results were obtained. The relative displacements of nearest-neighbor surface atoms, when scaled for differences in lattice constants, are found to be very similar for all the semiconductors studied. The displacements of individual atoms from unrelaxed bulk terminated positions are, however, appreciably different and ionicity dependent. The bond rotation or tilt angle at the surface which provides a partial characterization of surface relaxation is nearly constant varying only from 30\ifmmode^\circ\else\textdegree\fi{} in Si to 25.6\ifmmode^\circ\else\textdegree\fi{} in ZnSe. The reduction in total energy resulting from surface relaxation decreases from approximately -0.55 eV (per surface atom) in Si to -0.30 eV in ZnSe and ZnTe. Subsurface relaxations are generally found to make a very small (\ensuremath{\approx} 0.02 eV) contribution to the lowering of the total energy.

ChemInform Abstract: MATRIX‐ISOLATION INFRARED AND COMPUTER SIMULATION SPECTRA OF M(CH3)NCL4‐N (M = SI, GE AND SN) TYPE COMPOUNDS IN THE M‐C AND M‐CL STRETCHING REGIONS

Abstract The IR spectra of the M(CH3)nCl4−n (M = Si, Ge and Sn; n = 4,3,2 and 1) type compounds have been measured in argon and nitrogen matrices at ~ 15 K. The isotope structures have been observed for most of the M-C and M-Cl stretching bands, and assigned by using the computer simulation technique. Relative displacements of individual atoms have been calculated from the observed isotope shifts.

Matrix-isolation infrared and computer simulation spectra of M(CH 3) nCl 4− n (M = Si, Ge and Sn) type compounds in the M-C and M-Cl stretching regions

The IR spectra of the M(CH 3) n Cl 4− n (M = Si, Ge and Sn; n = 4,3,2 and 1) type compounds have been measured in argon and nitrogen matrices at ~ 15 K. The isotope structures have been observed for most of the M-C and M-Cl stretching bands, and assigned by using the computer simulation technique. Relative displacements of individual atoms have been calculated from the observed isotope shifts.

A neutron diffraction study of room-temperature monoclinic Kd 2PO 4

The crystal structure of the room-temperature monoclinic phase of high-deuteration DKDP has been determined by neutron diffraction measurements on a single crystal. The arrangement of K, P and O atoms is essentially that determined previously by X-ray diffraction: the D atoms have been located. It is confirmed that the monoclinic-tetragonal transition is a reconstructive one involving large (> 1 Å) relative atomic displacements.

A comparison of the structures of low and high pigeonite

These listings are available in microfiche. Order from the American Geophysical Union, 1707 L Street, N.W., Washington, D.C. 20036. Document J72-006; $1.00. Payment must accompany order. The crystal structure of high pigeonite has been determined from single-crystal intensity data taken on a pigeonite crystal (Wo9En39Fs52) from the Isle of Mull, Scotland, at 960°C (∼30°C above the low- to high-pigeonite transition for this composition). Anisotropic refinement in space group C2/c using 183 nonequivalent reflections resulted in an unweighted R of 0.035. Refinement in space group P21/c yielded positional and thermal parameters not significantly different from those consistent with C2/c symmetry. A difference Fourier calculated with the final coordinates of the C2/c refinement showed no anomalies indicative of a lower symmetry. (C2 or Cc). Analysis of the relative atomic displacements and the apparent rms thermal displacements provides evidence that the high-temperature polymorph is truly C-centered and not a space-average artefact. The structure of low pigeonite was anisotropically refined to an unweighted R factor of 0.057 using 1126 nonequivalent intensities collected at 24°C with the same crystal prior to the high-temperature experiment. Comparison of the low and high structures reveals two major structural changes: (1) extension of the B silicate chain with increasing temperature until it becomes crystallographically equivalent to the A chain at the transition (O3-O3′-O3” = 173.4°) and (2) a concomitant change in the primary coordination sphere of the M2 cation. The average bond length expansions of the SiO4 tetrahedron, the M1O6 octahedron, and the M2O8 polyhedron during the 24°–960°C temperature increase are 0.06, 1.2, and 1.4%, respectively; the average polyhedral volume expansions are 0.2, 3.5, and 5.9%, respectively. The apparent anisotropic rms atomic displacements at 960°C are approximately double those at 24°C. Refinement of Fe/Mg distributions in both the 24° and 960°C refinements resulted in KD values of 0.045 and 0.134, respectively. The temperature at which the Fe/Mg distribution equilibrated is roughly estimated to be ∼520°C. The fact that heat treament of Mull pigeonite at 940°C for 50 hours resulted in a decrease in class b peak diffusiveness relative to class a peaks suggests an increase in domain size.

Neutron scattering study of the soft modes in cubic potassium tantalate-niobate

Measurements have been made of the phonon dispersion relation for the transverse optic and acoustic branches of lowest energy in cubic KTN (KTa0.63Nb0.37O3). As in KNbO3 and BaTiO3 one optic branch is soft and overdarnped for all wavevectors q in 〈100〉 directions. The dispersion relation for this branch Js very anisotropic, the energy increasing rapidly with increasing q in 〈110〉 directions for polarization e parallel to 〈110〉 directions. The eigenvectors have been determined at |q| =0.2 a* by measuring the neutron scattering cross section in various Brillouin zones for both the TO and TA branches. The pattern of atomic displacement in the soft mode is found to be primarily of the “Slater” (g1) and “Last” (22) types and does not contain in g3 component. In this respect it resembles the corresponding modes in KTaO3 and SrTiO3 more closely than those in KNbO3 and BaTiO3. At||q|| =0.2 a* the relative atomic displacements in the soft mode are ζK = 3ζTa_Nb,) ζor = ζon = ζom = -2ζTa-Nb The mode contains an acou...

Atomic Displacements in Perovskite Strontium Titanate

AbstractThe relative atomic displacements of the sublattices of perovskite strontium titanate associated with optical frequency mode are calculated using all possible interionic couplings. The couplings are estimated from the observed optical frequencies of SrTiO3. The results of the calculation are plotted as a function of a parameter Φ(∼T – Tc). It is found that two sets of solutions exist which correspond to the proposals of SLATER and LAST.

Electric-Field-Induced Infrared Absorption in Diamond

In the presence of an external electric field, certain Raman-active vibration modes in diamond-type crystals also become infrared active, giving rise to an absorption proportional to the first-order change in the electronic polarizability per unit cell with the relative displacement of atoms, $\frac{\ensuremath{\partial}\ensuremath{\alpha}}{\ensuremath{\partial}u}$. We have observed such a transmission minimum and from its characteristics we have determined the electric-field-induced effective ionic charge and $\frac{\ensuremath{\partial}\ensuremath{\alpha}}{\ensuremath{\partial}u}$.

A method for determining the magnitude of the Raman scattering matrix element for diamond-type crystals

Abstract : A discussion is presented of the mechanisms for the electric field induced first order infrared absorption band in diamond-type crystals. The strength of the induced band is determined by the dependence of the electronic polarization on relative atomic displacements in the unit cell. A measurement of the absorption constant of the induced band should therefore provide quantitative information about the first order Raman scattering matrix elements. The phenomenon of field induced infrared absorption bands should also exist for Raman active modes in other centrosymmetric crystal structures, and for Raman active impurity modes. (Author)

磁鐵鑛の酸化並に還元に關するX線的研究

Magnetite, when oxidized, becomes hematite of a different crystal structure, and when reduced, it becomes at first FeO and then Fe. In the present experiment, the orientations of these products as regard to the, parent Magnetite were studied by means of X-rays. Natural martite exists in an octahedral crystal form, the composition of which is chiefly Fe2O3. It was found that the orientation of the crystal of Fe2O3 contained in this substance is similar to that of magnetite heated in air. That is, the plane (111) of Fe2O3 is parallel to the plane (111) of the parent magnetite and the direction [112] in this plane is also parallel to the direction [110] of magnetite. Therefore, it may be concluded that natural martite is a product of oxidized magnetite. In the mechanism of oxidation which was formerly proposed by the author a few years ago, the plane (111) was found to be parallel to each other, but the direction [112] of hematite was not parallel to [110] of magnetite. Hence in the present communication, the former mechanism has been partly revised. The orientation of the crystals of FeO as obtained by the reduction of magnetite agrees with that of the parent magnetite; so it may be concluded that the mechanism of this change is the same as that which is reverse to the case of oxidation. Three kinds of the orientations of iron have been obtained from the reduction of FeO: the first agrees with the parent substance; the second is such that the plane (100) of Fe is parallel to the plane (100), and the direction [011] of Fe is also, parallel to [010] of the parent crystal. The last is that the plane (110) of Fe is parallel to the plane (111) of the original FeO and the direction [011] of FeO shows a rotation of 9° against the direction [211] of FeO in the same plane. From the above results the following mechanisms may be given: the first kind is produced by the contraction in the direction [111] and by the expansion in the directions perpendicular to each other in the plane (111), just as in the case of a solid body undergoing a compression. The second kind of orientations is explained just in the same way as the mechanism proposed by E. C. Bain and K. Honda and S. Sekito in the Ar3 transformation of steel. As for the third, it may be explained by the similar mechanisms to those proposed by G. Kurdjumow, G. Sachs and Z. Nishiyama for that transformation. That is, it is obtained by rotating as a whole the second kind of orientation by 9°44' about its axis of [100], and then by 9° about the axis of [011]. Hence, the mechanism of the lattice change of the second and third kinds of orientation, agrees with each other, if the rotation as a whole is not taken into account. In any of the three-kinds of orientation in FeO, those planes which are of the greatest atomic density for iron, becomes the planes (111) or (011) of Fe by the least relative displacement of atoms.