Atomic Displacement Patterns Research Articles

Molecular dynamics study and normal mode analysis of diffuse scattering in quartz

X-ray scattering intensities in quartz were investigated in molecular dynamics (MD) simulation at different temperatures with the aid of normal mode analyses. The MD-simulated structure produced diffuse streaks most remarkably along a hexagon joining symmetry-equivalent points of index 400 in the hk0 plane, extending in ± c*. In addition, most Bragg spots were found to be associated by radial diffuse scattering in the six a* directions. The normal mode analysis and calculations of first-order scattering intensities showed that the lowest-lying branch in Γ-M is of transversal acoustic characters, and is responsible for the diffuse scattering radiating from Bragg reflections in the a*-directions. The transversal acoustic characters of atomic displacements in these modes provide the scattering in phase at points on the fourth hexagon. Phonon dispersion relations of the MD crystal were examined in view of symmetry-adapted atomic displacement patterns, and a possibility was suggested for two curves in Γ-M, optic soft branch and transversal acoustic branch polarized in the z axis (the second-lowest branch), to exchange the eigenvectors in the middle range of q (0.2–0.3).

Charge and orbital ordered states in [formula omitted

The charge and orbital ordered states in the single-layered manganite crystals, Nd 1−x Sr 1+x MnO 4 (x=0.67,0.75) , have been investigated. We have observed the superlattice spots with modulation wave vector, q=( δ, δ,0), δ= 1 8 , 1 4 [ ∼ 1 6 , 1 3 ] in x=0.75[0.67] by resonant and non-resonant X-ray scattering. These results show that not only the atomic displacement pattern but also the charge-orbital structure has the unit cell of 4 2 a× 2 a×c [3 2 a× 2 a×c] .

What we can learn about fast chemical processes from slow diffraction experiments

The potential energy surface is an important determinant of a chemical reaction. Three ways of deducing non-trivial properties of such surfaces from the results of crystal structure analyses are discussed and illustrated with examples. (1) The mapping approach brings together structures of the same molecular fragment from different environments to outline reaction coordinates and vibrations. (2) Correlations between molecular structures and activation energies for a given reaction type reveal general and quantitative relations between seemingly independent entities such as ground state structure, force constants, reaction path length, activation energy and catalysis. (3) The evolution of atomic mean square amplitudes (displacement parameters) with temperature uncovers frequencies and atomic displacement patterns of large-amplitude vibrations in molecular crystals. Examples include the vibrations of molecular zeolite building blocks, the crankshaft motion of stilbenes, the dynamic coupling between pyramidal deformation of the amide NH2 group and hydrogen bonding, the bowl inversion of corannulenes and nucleophilic addition/elimination reactions.

An analytical solution for the zero frequency hyperbolic RUM modes of distortion of SiO 2-tridymite

The constraint equations controlling the existence of soft, Rigid Unit Mode (RUM) modes of distortion of the ideal tridymite tetrahedral framework structure are derived and used to obtain an analytical solution for the potential soft phonon mode wave-vectors and their associated displacement patterns. One of two types of RUM known to exist are characterized by modulation wave-vectors perpendicular to 〈110〉 and atomic displacement patterns involving tetrahedral edge rotation while a second type of RUM is found to be soft on a hyperbolic reciprocal space distribution and to involve tetrahedral rotation about in-plane rotation axes. The analytical form of this hyperbolic diffuse distribution is derived as is an analytical expression for the associated RUM eigenvector of each point on the diffuse distribution.

Surface phonons of S:Si(001)-(1×1)

We present results of ab initio calculations of vibrational properties of the Si(001) surface covered by a monolayer of sulfur. The S:Si(001) system is a prototype of chalcogen adlayers on (001) surfaces of Si or Ge. Our results highlight all features characteristic for the whole class of such adsorption systems. We find a number of pronounced localized surface phonon modes as well as resonances part of which are characteristic for the adlayer while others are directly related to the substrate surface. The physical origin and nature of these modes is discussed employing surface phonon dispersions, atomic displacement patterns, respective charge density perturbations, and spectral phonon densities. In our calculations, the surface is modeled within the supercell approach and we use pseudopotentials together with a plane wave basis set. The surface phonon properties are calculated within density functional perturbation theory.

The inherent displacive flexibility of the hexacelsian tetrahedral framework and its relationship to polymorphism in Ba-hexacelsian

The results of a rigid unit mode (RUM) analysis of the inherent displacive structural flexibility of the tetrahedral framework of the ideal hexacelsian structure type are presented. One of the three types of RUM found to exist is characterized by modulation wave vectors perpendicular to and atomic displacement patterns involving tetrahedral rotation around the parent c axis while a second type of RUM is found to be soft at any modulation wave vector and to involve tetrahedral rotation about in-plane rotation axes. It is shown how a combination of these two types of RUM motion associated with the same q=1/2 * modulation wave vector enables the outstanding crystal chemical problems and apparently mutually contradictory results as regards polymorphism in Ba-hexacelsian to be resolved.

A Note on the Long Period Commensurate Structures

The long period commensurate structures are discussed in the framework of discrete atomic models. It is pointed out that usually there are two kinds of atomic displacement patterns in equilibrium in discrete models. A simple but reliable method to guess qualitatively the long period commensurate structures will be proposed.

Atomic geometry, electronic structure, and vibrational properties of the Ge(001)(2×1) surface

Results of ab initio pseudopotential calculations for atomic geometry, bonding, and electronic band structure of the Ge(001)(2$\ifmmode\times\else\texttimes\fi{}$1) surface are presented and discussed. These are then used as input for calculations of the vibrational properties of this surface within the adiabatic bond-charge model. The phonon spectrum thus obtained is compared in detail with that of the Si(001)(2$\ifmmode\times\else\texttimes\fi{}$1) surface. The spectra and atomic displacement patterns are found to show marked qualitative similarities, while quantitative differences are attributed to the mass difference between Si and Ge atoms.

Local Arrangements in Fe-31% at. Pd above Ms

The diffuse x-ray scattering from an Fe -31 at. pct. Pd single crystal quenched from 1173 K was analyzed to obtain information about the local atomic environment around each atom including the local chemical order and the average atomic displacements from the normal lattice positions. To better reveal the meaning of these average quantities, computer simulations were employed to provide the structure using these values. The results indicate an interconnected distribution of ordered and clustered regions, no more than a few atoms in dimension, and only slightly more developed than for a random arrangement. The pattern of atomic displacements from the average lattice is revealed to be small embryos of a displacement array similar to the FCT martensitic phase and aligned in directions, which may be the premartensitic phase that has been predicted by several authors.

A TEM and Group Theoretical Study of α-PbO and Its Low-Temperature Improper Ferroelastic Phase Transition

The results of a detailed TEM and group theoretical study of α-PbO and its low-temperature, improper ferroelastic phase transition are reported. Ferroelastic twinning of the low-temperature, incommensurately modulated phase is observed and typically found to be on a scale rather less than 1 μm. Some unusual features of the reciprocal space of this low temperature phase are reported including the coexistence of sharp incommensurate satellite reflections and a closely related diffuse intensity distribution as well as the presence of "bands" of diffuse intensity absence along {hkh}* directions of reciprocal space. Single domain electron diffraction patterns of the low-temperature incommensurately modulated phase are found to be incompatible with a previously reported structure refinement and to require a 4-dimensional superspace group symmetry of P : Cmma : s, -1, 1. The most general possible atomic displacement pattern compatible with this symmetry is derived from group theoretical considerations.

Comparison of normal mode analyses on a small globular protein in dihedral angle space and Cartesian coordinate space

Normal mode analyses on the protein, bovine pancreatic trypsin inhibitor, in dihedral angle space and Cartesian coordinate space are compared. In Cartesian coordinate space it is found that modes of frequencies lower than 30 cm −1 contribute 80% of the total mean-square fluctuation and are represented almost completely by motions in the dihedral angles. Bond angle and length fluctuations dominate in modes above 200 cm −1, but contribute less than 2% to the total mean-square fluctuation. In the low-frequency modes a good correspondence between patterns of atomic displacements was found, but on average the root-mean-square fluctuations of the Cartesian coordinate modes are 13% greater than their dihedral angle counterparts. The main effect of fluctuations in the bond angles and lengths, therefore, is to allow the dihedral angles to become more flexible. As the important subspaces determined from the two methods overlap considerably, dihedral angle space analysis can be applied to proteins too large for Cartesian coordinate space analysis

DISORDER EFFECTS ON SMALL-POLARON FORMATION AND HOPPING

It was noted some time ago that many properties of amorphous semiconductors could be explained if the charge carriers self-trapped to form small polarons. Self trapping occurs when an electronic carrier lingers at a site long enough to permit surrounding atoms to displace in response to the presence of the carrier. The electronic carrier then becomes severely localized within a potential well produced by the atomic displacements. This self-trapped carrier cannot move unless the atoms alter their positions. The localized carrier together with the atomic displacement pattern that confines it is termed a small polaron. The adjective small denotes the severe localization of the electronic state. The energy of a small polaron in a covalent solid is lower than that of a static electron by E{sub b} {triple_bond} F{sup 2}/2k, where F is the force between the carrier and atoms adjacent to it, and k is the material`s stiffness constant. As a result of its severe localization, a small polaron generally moves by phonon-assisted hopping. Small-polarons will only form in covalent crystals whose electronic halfbandwidths are sufficiently narrow, E{sub b} > W. The absence of small polaronic carriers in most covalent crystals presumably indicates that E{sub b} < W inmore » these instances. However, evidence of small polarons is commonly found in disordered materials despite the estimates of E{sub b} and W not being significantly different from those of crystals. It is found that only modest energetic disorder is required to induce small-polaron formation. Here the author succinctly describes essential elements of this work. Second, the author addresses the role of disorder on the adiabatic hopping motion of small polarons.« less

Conformational dynamics of polypeptides and proteins in the dihedral angle space and in the cartesian coordinate space: Normal mode analysis of deca‐alanine

AbstractEffects of different treatments of the degrees of freedom of bond length stretching and bond angle bending in computational analysis of conformational dynamics of proteins and polypeptides are assessed. More specifically, the normal mode analysis of conformational dynamics of α‐helix of deca‐alanine has been carried out both in the dihedral angle space (DAS) and in the Cartesian coordinate space (CCS). Almost perfect one‐to‐one correspondence has been found between normal modes in the CCS with frequencies less than 128 cm−1 and those in the DAS with frequencies less than 164 cm−1. Patterns of atomic displacements in the corresponding modes are very similar. This indicates that the effects of fixing degrees of freedom of bond length stretching and bond angle bending on the very‐low‐frequency normal modes in the CCS with frequencies less than 128 cm−1 are almost solely to increase the frequencies by about 20%. The conclusion indicates that the different treatment of these degree does not lead to qualitatively different results as long as low‐frequency motions are concerned. Based on the results of calculation, mechanical property of the α‐helix of deca‐alanine is discussed.

Influence of bulk-phonon-branch dispersion on displacement patterns and the intermixing of interface and confined optical phonons in superlattices.

An envelope-function formalism is developed to treat zone-center superlattice phonons with inclusion of effects due to the full bulk-phonon-branch dispersion and the long-range macroscopic electric field. Starting from a rigid-ion model, differential equations for the envelopes of the atomic displacement patterns are derived. It is shown that both the transverse component of the electric field and the longitudinal component of the electric displacement field that appear automatically satisfy the boundary conditions of macroscopic electrodynamics. For optical and acoustical phonons, the same additional mechanical boundary conditions for the envelopes themselves are extracted from the underlying microscopic equations of motion. They therefore depend on the atomic structure of the interfaces. Explicit results are given for (GaAs${)}_{\mathit{N}1}$(AlAs${)}_{\mathit{N}2}$ (001) superlattices. In the case of p-polarized symmetrical modes they show a strong anisotropy and intermixing of nominal confined and interface optical phonons. The considered material combination with nearly the same bulk acoustic branches and well-separated optical-phonon bands allows some approximations giving physical insights into the correct solutions of the dispersionless macroscopic continuum theory. Starting from the concept of effective layer thicknesses or a band-edge expansion, one obtains analytical results for frequencies, confinement wave numbers, and displacement patterns, which represent generalizations of the macroscopic continuum theory with the influence of the bulk-phonon-branch dispersion taken into account.

Validity of the continuum approach to optical phonons in short-period superlattices

The authors report a strict comparison of long-wavelength optical lattice vibrations in a superlattice obtained with the macroscopic continuum model and a closely parallel microscopic model. It is shown that the proper envelopes of the atomic displacement patterns can be found if the uncertainties inherent in any dispersionless theory such as the continuum model are appropriately treated. Apart from small discrepancies due to the discrete nature of the matter and effective material layer thicknesses the main failure of the continuum theory is the neglect of bulk phonon dispersion.

Relationship of Microscopic and Macroscopic Theories for Long‐Wavelength Optical Phonons in GaAs‐AlAs Superlattices

AbstractA critical comparison is made of the lattice‐vibrational properties resulting within the macroscopic continuum model and a closely parallel microscopic force‐constant model. The elastic and electric forces are treated in the same manner. However, the atomic structure of matter and the bulk phonon dispersion are neglected in the continuum approach. For that reason it cannot explain the optical frequency spectrum of a superlattice even not in the long‐wavelength limit. On the other hand, the principal line shape of the atomic displacement patterns for optical phonons can be satisfactorily described by envelopes, at least, if the arbitrariness inherent any theory neglecting dispersion is utilized in an appropriate way. This holds particularly for modes which are not accompanied by long‐range electric fields. For the other modes — the p‐polarized symmetrical phonons propagating not parallel to the superlattice axis — the problem of identification appears additionally since the interface character restricted to four modes in the dispersionless continuum approach is distributed among all symmetrical phonons within the microscopic theory.

Modulated Phases in the Ba2SiO4-Ca2SiO4System of A2BX4, K2SO4-related Structures

Abstract The A2BX4 family of K2SO4-related structures have long been of interest to crystal chemists, largely due to the numerous different polymorphs and complicated sequences of phase transitions they sometimes exhibit as a function of temperature. For most such A2BX4 compounds, there are essentially only two distinct structure types or parent structures — a high-temperature, ‚hexagonal‘ form isomorphous to α-K2SO4 and a lower-temperature, orthorhombic form isomorphous to β-K2SO4. In addition to these two prototype structures, however, there often exist weakly distorted, or modulated, variants. In the case of the Ba2-xCaxSiO4 system, five such modulated variants have been found via an electron diffraction study and characterized. The characteristic satellite extinction conditions associated with the weak satellite reflections have been used to determine displacement eigenvectors and atomic displacement patterns associated with each of the observed modulation wave-vectors. The widespread occurrence of mo...

An X-ray study of coupling between acoustic and optic modes at the ferroelastic phase transition in As2O5

The intensity of the 006 reflection of As2O5 has been measured as a function of temperature by single-crystal X-ray diffraction across the proper ferroelastic phase transition at Tc=578 K. The intensity is proportional to mod Tc-T mod in the low-temperature (P212121) phase corresponding to second-order Landau behaviour, and vanishes above Tc confirming the previously postulated high-temperature symmetry P41212. The structure factor of the 006 reflection has been expressed in terms of the amplitude of a cooperative atomic displacement pattern arising from coupling between the critical acoustic instability and optic phonons.

Dynamics of atoms in YBa 2Cu 3O 7−x

In order to get insight into the role played by the phonons in the superconducting phase transition, we have performed infrared reflectivity and neutron scattering experiments on single phase YBCO. The results, together with published Raman data, are used to determine the parameters of a rigid-ion model; the model reproduces qualitatively well the experimental well the experimental data. On the basis of our model we calculate the phonon dispersion relations and the density of states with contributions of individual sublattices. In addition patterns of atomic displacements of some important modes are discussed.

Dynamics of atoms in the high-temperature superconductor YBa2Cu3O7??

In order to obtain insight into the role played by the phonons in the superconducting phase transition, we have performed infrared reflectivity and inelastic neutron scattering experiments on single phase YBa2Cu3O7−δ(δ≊0.1,T c =94 K, ΔT 10–90=1 K). These experimental results, together with published Raman scattering and specific heat data are used to determine the parameters of a rigid-ion model; the model is able to reproduce qualitatively the experimental data. On the basis of our model we derive and discuss the phonon density of states and the pattern of atomic displacements of some important vibrational modes. In particular, the model reveals the existence of high-frequency oxygen breathing modes and of low-frequency copper dimerization modes in both the CuO3-chains and the CuO2-planes. Furthermore, Jahn-Teller like oxygen modes exist in the chains. The possible role of these modes with respect to the pairing mechanism is discussed. In addition there exist low-frequency “chain-degrading” modes which favour jumps of oxygen atoms out of the CuO3-chains into neighbouring vacant sites in the basal plane.