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The nature of banded spherulites was further investigated. The principal material used was a polyester, namely, polytrimethylene glutarate, but use was made also of nonpolymeric materials. A variety of observations on extinction effects with the Universal Stage conclusively confirmed the twisted crystal model of banded spherulites. It is shown that banding observed in the form of other than extinction effects need not indicate a zone structure. Such bands can be due to diffraction effects or pseudopleochroism following from the twisted structure, examples of such effects being quoted. In some cases, genuine rhythmic distribution of matter was observed, and it is shown that these effects arise in two-dimensional spherulites only, and can still follow from the twisted morphology under the influence of a substrate. Direct observations of a helicoidal morphology are also quoted.

We address the analytical investigation of nontrivial properties of heat flow, starting from microscopic models of matter. We present an integral representation for the expression of the heat flow, by taking as our crystal model a self-consistent chain of anharmonic oscillators, precisely, a chain of oscillators with harmonic interparticle interactions, anharmonic on-site potentials, thermal reservoirs at the boundaries, and still with some residual inner stochastic baths. We propose an approximative scheme to make the integral formalism analytically treatable: we test the approximations in harmonic models and analyze some anharmonic systems. For the case of graded anharmonic models with weak interparticle interactions, we derive an expression for the thermal conductivity, and show the existence of thermal rectification and also nonnegative differential thermal resistance. The details of our formalism allow us to note the ingredients behind these phenomena, and the generality of our results (i.e., the results will be valid for other interactions and regimes) shows that rectification in graded materials is a ubiquitous property.

The classical criterion for the stability of cubic crystals under constant hydrostatic pressure is distinguished from the notional criterion of stability introduced by Born. Although the latter has been widely used in the literature, theoretical computations demonstrate appreciable quantitative and qualitative divergences between the criteria. The computations are made within the framework of a crystal model that is both mathematically tractable and sufficiently realistic for the intended purpose.

In an introductory mineralogy course, the concepts of crystallography, especially the relationship between spherical and stereographic projections of crystals, are more easily understood by students if they can make a spherical projection of a crystal. This projection is easily made on round, transparent balloons, using a permanent felt-tipped marker to denote the locations of crystal axes, face poles, and symmetry elements. To illustrate this procedure, an isometric crystal model with cubic and octahedral faces is used to make a spherical projection. The stereographic projection of the crystal model can be seen through the balloon. This projection looks very similar to the actual two-dimensional stereogram, making it easier for students to see the relationship between the three-dimensional crystal, the spherical projection, and the two-dimensional stereographic projection.

The small radius potential approximation is employed to construct a semi-infinite crystal model for studying surface states. The energy expression and the existence condition are derived, and solved numerically. It is shown that contrary to other one-dimensional model calculations (for s-states), two Tamm-like surface states can occur in one forbidden energy gap. Schockley-like surface states (for no surface perturbation) are also found.

The Complex-Band Structure Method has been extended to include changes of the pseudopotential form factors caused by short-range disorder. An effective crystal model is introduced to calculate densities of states and optical constants of amorphous tetrahedral semiconductors. Si and GaP are presented as examples.

Stability of long period stacking variants in the hexagonal cell is discussed by determining the structure of lowest energy of the pairwise interaction model. The interlayer interactions extended to several neighbor layers are assumed in the model. Typical stacking variants observed in many cases (Lavesphase alloy, transitiin‐metal carbides, etc.) are obtained as the ground state of the model. Several examples are discussed to show that an experimental observation of the long period stacking variants yields a certain amount of informations about the interactions. Also the possibility of finding new structures is discussed.

Within a model of central and noncentral forces the exact change of the vibrational entropy, ΔS, is calculated, both as function of the degree of order and as a function of the diverse force constants. A linear relationship between ΔS and the short-range order parameter is found. Results of ΔS as a function of the force constants are compared with ΔS as calculated by an empirical formula based on a modified Einstein crystal model.

The phenomena of low temperature persistent impurity photoconductivity in compound semiconductors has attracted considerable recent attention.1,2 For the case of AlxGa1-xAs it involves a non-effective mass like donor-complex labeled a “DX” center. The unoccupied DX center is believed to be resonant with the conduction band, yet sufficiently localized to produce a large lattice relaxation1 (see top of Fig. 1 which shows the configuration coordinate diagram proposed by Lang et al). In this figure the defect activation energy, Eo (∿ 0.1 eV), is small compared to the optical activation energy, Eop (∿ 1.5 eV). When kT

An exact solution of an eight-site crystal model with periodic boundary conditions, a /001/ thin film with face-centered-cubic crystal structure, is presented for the case of a heavy-fermion system. The object is to study modifications of the many-body electronic structure in the presence of surfaces. The model consists of (a) a single, fully symmetric orbital per site, with nearest-neighbor and second-nearest-neighbor hopping; (b) an infinite Coulomb repulsion between electrons on the same site; (c) antiferromagnetic superexchange interactions; and (d) a nearly half-filled band ((7/8 electron per site). Two structures are investigated: a two-layer and a four-layer fcc slab. The nature of the change from bulk to surface is studied by introducing a variable coupling between periodically extended slabs. The many-body spectrum, the one-electron occupancy, and the change in properties from bulk to surface are obtained and discussed.

A three-dimensional crystal model for packing of collagen molecules (type I) in the native fibril has recently been proposed by Hulmes and Miller1. It provides a straightforward explanation of the major features of the X-ray diffraction pattern, and is consistent with measurements of fibril density. However, there is independent evidence for a well defined microfibrillar substructure, which is absent from their model. This evidence, which is derived from electron microscopy2,5 and studies of in vitro assembly6,7, the pattern of covalent crosslinks8 and sequence analysis9,10, is convincing. Therefore, we have searched for a means to reconcile this conflict. We now propose two models which contain five-stranded microfibrils compressed to place molecules (in cross-section) on a pseudo-hexagonal lattice. The unit cells are equivalent or related to the cell proposed by Hulmes and Miller1. In the simplest case, molecules, and thus microfibrils, are straight-tilted. However, it is not ruled out that molecules are supercoiled and microfibrils are straight. Noncrystallographic considerations6,7,9,10 favour supercoiling.

We report on simulations for the molecular-crystal model without dispersion in the adiabatic regime for all lattice dimensionalities. A discrete version of the Feynman path-integral formalism is used to calculate the electron properties. We compare these results with approximate theories and conclude that continuum versions of the molecular-crystal model have little relationship to the discrete lattice model. In addition, the influence of the phonon dispersion on the transition between the “untrapped” and “self-trapped” state of an electron on a one-dimensional lattice is studied within the context of Holstein’s molecular-crystal model. We show that the electron-phonon interaction has a critical value below which self-trapping does not occur. We find that this critical value goes to zero if the optical phonon becomes soft.

The formation of crack-tip blunting with corner under a monotonically increasing tensile load is investigated by use of finite element analysis.

An exact solution of a four-site tetrahedral crystal model with periodic boundary conditions, the smallest face-centered-cubic crystal, is presented in the case of binary and ternary alloy systems. The model consists of (a) a single s orbital per site with nearest-neighbor-only hopping, (b) a Coulomb repulsion between electrons on the same site, and (c) a single electron per atom. The model, which represents Cu, Ag, Au, and their alloys, shows all the segregational, solutional, and compound-forming tendencies of the real systems. Such characteristics are absent in noninteracting independent-electron approaches. Calculations demonstrate extreme sensitivity to the input one-electron parameters, with minor differences resulting in widely different alloying properties.

Thermodynamic behavior of classical many-particle systems such as Ising models, hard sphere gases, fluids etc. can be studied by means of analytical, e.g. high-temperature series, perturbation expansions and renormalization group methods, and computer simulation techniques. There is no doubt that the latter have contributed a lot to the present understanding of the behavior of classical many-body systems [1,2]. Extending the techniques, originally developed for classical-mechanical models, to quantum systems is not straightforward. In this paper a Monte Carlo approach for quantum models at non-zero temperatures (T≠0) is discussed. The technique is complementary to the T = 0 Monte Carlo methods, see for instance CEPERLEY and KALOS [3]. In the following section an outline of the basic concepts and mathematical justification of the method are given. The general methodology is illustrated in subsequent sections by considering explicit examples. Topics treated include the one-, two- and three-dimensional Molecular Crystal Model, a two-level system with non-linear coupling and bipolaron formation.

A self‐consistent procedure for determining the constants that characterize the interionic forces allowed for in a crystal model is described. The procedure allows one to avoid some of the approximations that are commonly made when determining force constants from the experimental values of the bulk modulus and its pressure derivative. The procedure is used in a calculation of the coefficient of thermal expansion of CsBr, and the values obtained are in good agreement with the experimental values.A self‐consistent procedure for determining the constants that characterize the interionic forces allowed for in a crystal model is described. The procedure allows one to avoid some of the approximations that are commonly made when determining force constants from the experimental values of the bulk modulus and its pressure derivative. The procedure is used in a calculation of the coefficient of thermal expansion of CsBr, and the values obtained are in good agreement with the experimental values.

A scheme of calculation of the electronic structure of a solid state surface and chemisorbed molecules is discussed. The method of the Green's function and MO LCAO approximation are used which permits to perform calculations, taking into account the whole crystal but not its fragment only, with the accuracy adopted by quantum chemistry. Results of model calculations are presented: chemisorption of hydrogen-like. atom on the (100) face of the one-band crystal model and dispersion curves for the density of states of nickel (100) face.

An x-ray diffraction method is described for determining thicknesses of thin films grown on single crystal substrates. The equations, based on the kinematical theory of x-ray diffraction and the mosaic crystal model, were developed. The thickness of the thin film was computed from the absorption of the integrated diffracted x-ray intensity from the single crystal substrate by the film. Since the diffracted intensity from the film is not required, the film does not have to be single crystal in nature. Thus, thicknesses of less ordered, polycrystalline or even amorphous films can be measured with high precision by this technique.

Layer structures are characterized by arrays of the c‐ and h‐sequences, so they may be assumed as one dimensional alloys of c and h. The method developed by Kaburagi and Kanamori for the determination of the structure of the ground state is applied on the linear chain alloy. The stable region for each structure has been determined in a four dimensional mathematical space which is defined by the four inter‐layer pair potentials. It has been found that the stacking variants of the Mg base laves phase correspond to successive changes of the apir potentials.

The theory of light absorption by molecular crystals with allowance for lattice vibrations was first developed by Davydov [1] for a one-dimensional crystal model and by Davydov, Rashba, and Lubchenko [2, 3] for three-dimensional crystals. These studies were qualitative and did not take into account the frequency shift caused by exciton-phonon interaction. A theory of light absorption by Wannier excitons with allowance for lattice vibrations was developed by Toyozawa [4]. As was shown in Section 1 of Chapter IV, there is a considerable difference between the interaction of Wannier excitons and the interaction of Frenkel excitons with optical lattice vibrations. Therefore, Toyozawa’s results cannot be carried over directly to molecular crystals.

The vibrational modes localized at the interface between two distinct crystals have been studied for a simple crystal model obeying all of the invariance conditions required for models used in studies of dynamical properties of crystal surfaces, and giving rise to Rayleigh surface waves. The two crystals are assumed to be semi-infinite simple cubic and to have the same lattice parameter a. They differ by their mass (M and MA) and the central force interactions between first (K and Ka) and second nearest neighbors 1 2 K and 1 2 K A . The interface is obtained by coupling the (001) free surfaces of these distinct crystals by central force intractions (K'). We find that the variation of the interaction conditions (K') at the interface and of the ( K M ) ( K A M A ) parameter has the following qualitative effects on the properties of surface and bulk phonons. When (K') increases from zero to a finite value, the frequencies of the surface phonons increase and are splitted in the case of two identical crystals. One can say that the surface phonons are transformed into interface modes. For some values of K ' K and ( K M ) ( K A M A ) parameters these interface phonons may be admixed with bulk phonons and thus become virtual interface states.

A new model for the structure of β-poly- l -alanine is proposed. It consists of interleaved anti-parallel pleated sheets with neighbouring sheets randomly displaced ± 1 2 a (a is the interchain distance). Optimum values of the molecular packing and conformational parameters for this statistical structure have been determined by the linked-atom least-squares method for polymer crystalline fibres (Arnott & Wonacott, 1966a). The crystal model is supported by the infrared data and yields calculated X-ray structure amplitudes in substantially better agreement with the observed diffration than any previous model. The refined values of the standard protein conformation angles, φ, ψ and ω are 41.4 °, −45.3 ° and 1.5 °. The interchain hydrogen-bond length, which was not a fixed feature in the refinement, is 2.83 A.

Crystal lattice deformations can be described microscopically by explicitly accounting for the position of atoms or macroscopically by continuum elasticity. In this work, we report on the description of continuous elastic fields derived from an atomistic representation of crystalline structures that also include features typical of the microscopic scale. Analytic expressions for strain components are obtained from the complex amplitudes of the Fourier modes representing periodic lattice positions, which can be generally provided by atomistic modeling or experiments. The magnitude and phase of these amplitudes, together with the continuous description of strains, are able to characterize crystal rotations, lattice deformations, and dislocations. Moreover, combined with the so-called amplitude expansion of the phase-field crystal model, they provide a suitable tool for bridging microscopic to macroscopic scales. This study enables the in-depth analysis of elasticity effects for macroscale and mesoscale systems taking microscopic details into account.

The effects of dimensionality on the small polaron motion have been studied through a molecular crystal model in which the intermolecular forces are described by a first neighbors pair potential. The crossover temperature [Formula: see text] between band-like and hopping motion is sensibly reduced in low-dimensional systems due to the enhanced importance of the off-diagonal scattering processes. The dispersion in the phonon spectrum is essential for the validity of the model. The results may be of interest in order to understand the anomalous c-axis electrical transport in high-T c superconductors.

We analyse the ground state phase diagrams and thermodynamic properties of charge orderings in narrow band materials using the molecular crystal model in the static limit. We present results for the hypercubic lattices in dimensions d = 2 and d = 1. We focus our study on the problem of phase separations involving charge orderings and the eects of next-nearest-neighbor hopping ( t2) on the charge ordered states in these systems. The ground state phase diagrams are evaluated for a few representative cases. Results for the molecular crystal model are compared with those obtained previously for the spinless fermion model with repulsive intersite interaction W.

In the present work, we present a free energy derivation of the multi-component phase-field crystal model [1] and illustrate the capability to simulate dendritic and eutectic solidification in ternary alloys. Fast free energy minimization by a simulated annealing algorithm of an approximated crystal is compared with the free energy of a fully simulated phase field crystal structure. The calculation of ternary phase diagrams from these free energies is described. Based on the free energies related to the ternary Al–Cu–Mg system, we show phase field crystal simulations of both, ternary dendritic growth as well as lamellar eutectic growth of three distinct solid phases.

A model for an intermediate valence four-site tetrahedral crystal, the smallest face-centered cubic crystal, is presented. It consists of one electron and four spin-orbitals per atom and a variety of interactions. Many-body eigenstates and eigenvalues of the Hamiltonian are calculated exactly with the aid of the symmetry of the crystal.

structure. In this note we describe a similar calculation for the slab with CsCl structure. Consider a thin slab-shaped crystal model composed of point charges distributed over a CsCl structure. Let the (0,O, I) direction or z axis lie perpendicular to the large faces of the slab. We index the (0, 0, 1) planes n = 1, 2, 3, ..., N in such a way that n = 1, n = N refer to the outermost planes and z,,+, < z,,, where z,, is the z coordinate common to all the charges in the nth plane. It is a characteristic of the CsCl structure that each (0, 0, 1) plane contains charges of one sign; thus, to achieve charge neutrality in our system, we must assume that N is even. Let the planes with n odd contain positive charges with position coordinates [2I1a, 2I2a, zn] and the planes with n even contain negative charges with I , I 0 0 4 a O 8 a 12a 1 6 0 2 0 a position coordinates [(211 + ])a, (21, + ])a, z,,]; I,, l2 are positive or negative integers2 and a is z1 z 2 --t& spacing between two adjacent (0, 0, 1) planes FIG. 1 . Graphs of FlC/(q2/4xsoa2) and I~ , ' l / ( q~ / in the CsCl structure, i.e. 4xsOa2) versus (2, z2).

A method is described of measuring the coordinates of balls in a ball-and-spoke crystal model. This is achieved by taking a hologram of the model and making measurements on the reconstructed image.

This paper describes dynamic crystal models which were developed independently at the Worcester Polytech institute and Purdue University.

A simple method in the framework of Koster-Slater Green's function calculation and central cell defect potential (CGDP) approximation is used to calculate the distribution of deep level wavefunctions in Bloch space for Si, GaAs and GaP for the first time. Convergence properties are discussed. It is shown that for definite number of energy bands in the host crystal model, the convergence of deep level wavefunction calculations is faster than that of deep energy calculations.

A structural phase transition in crystal LiKSO4 at Tc = 195 K was confirmed by NMR experiment previously. In this article, the temperature dependence of the electric field gradient tensor at lithium site in the range of 110-300 K is reported, and combined with a simplified ion crystal model the crystal structure is discussed. It is shown that even in the room temperature phase, the lithium position should be at (2/3, 1/3, 0.706) instead of the position at (2/3, 1/3, 0.849) suggested previously by Bradley. The phase transition is attributed to the displacement of lithium ions. The space group of the low temperature phase may be R3.

The mixed crystal system KCl(1-x)Br(x) shows a one-mode type behavior. The variation of the long-wavelength transverse optical (TO) mode frequency at 1 atm pressure can be understood in terms of the virtual crystal model. The variation of the frequency with pressure or the mole fraction x can be expressed in terms of volume change alone. The slopes of the vTO mode vs P are highest for the end numbers, and are less for intermediate values of x, and may be understood in terms of a nonlinear variation of the compressibility with x.