Terms Of Lattice Research Articles

Subprojective lattices and projective geometry

The class of lattices we are interested in (subprojective lattices), can be gotten by taking the MacNeille completions of the class of complemented, modular, atomic lattices. McLaughlin showed that subprojective lattices can be represented as the lattices of W-closed subspaces of a vector space U in duality with a vector space W. In this paper, we give a characterization of subprojective lattices in terms of atoms and dual atoms, by means of an incidence space satisfying self-dual axioms. In the finite-dimensional case, a subprojective lattice is projective, and hence our self-dual axioms characterize finite-dimensional projective spaces in terms of points and hyperplanes. No numerical parameters appear explicitly in these axioms. For each subprojective lattice L with at least three elements, we define a projective envelope P ( L) for it. P ( L) is a projective lattice and there is a natural inf-preserving injection of L into P ( L). This injection has other important properties which we take as the definition of a geometric map. In the course of studying geometric maps, we obtain a lattice theoretic proof of Mackey's result that the join of a U-closed subspace of V and a finite-dimensional subspace is U-closed, where ( U, V) form a dual pair of vector spaces over a division ring. Furthermore, we show that if L is a subprojective lattice, P a projective lattice, and ψ: L → P a geometric map, then P is isomorphic to the projective envelope P ( L) of L. The paper presents many other properties of subprojective lattices. It concludes with a characterization of subprojective lattices which are also projective.

On the lattice stability of metals

The success of perturbation calculations of second order for the NFE (“Nearly Free Electron”) metals and that of the two-parameter model of Pettifor for the transition elements show that the lattice-stability of the metals has simple physical reasons. Using the results of Harrison, Heine and Weaire, Deegan, and Pettifor, a model is developed which allows to explain the stability of the three metal lattices in terms of differences in the potentials. Only those potential differences are considered which are caused by the different packing of the lattices. With the aid of the virial theorem the band structure energy is connected with the potential bandstructure energy. The sequence of stability is predicted to be body centered cubic (bcc), hexagonal close packed (hcp), face centered cubic (fcc) with increasing valence electron concentration. The ranges of stability can be expressed in simple numbers. This simple model holds in principle for NFE as well as for transition metals because it contains no assumptions restricted to only one of these metal types. Deviations of the observed lattice stability from the model can be understood from the approximations involved.

Development of butterfly scales. II. Struts, lattices and surface tension.

An investigation of the structure of the iridescent scales of the green hairstreak, Callophrys rubi, reveals an internal lattice which is probably cubic close-packed in form. We present a model which explains the formation of the lattice in terms of packing of spheres and surface tension forces and generalize these results to internal structures in other Lepidopteran scales.

Study of the paraelectric behavior of OH − ions in alkali halides with optical and caloric methods II. : Dynamics of the dipole alignment

The relaxation behavior of OH − dipole centers in ten alkali halides was studied with electro-optical and electro-caloric methods as a function of applied field, temperature and host lattice system. The reorientation kinetics of the 〈100〉 oriented OH − dipoles can be fitted successfully to a phonon-assisted (90°) orientational tunneling model. Below 5°K the observed time dependence of the optical dichroism and the T −1 dependence of the relaxation time gives clear evidence for the predominance of one-phonon processes, while above 5°K the T −4 dependence indicates multi-phonon processes. It could be demonstrated that in the one-phonon region of the dipole relaxation a population inversion among the dipole levels (with subsequent phonon emission) can be achieved after a fast reversal of the field polarity. An attempt is made to discuss the observed strong variation of the relaxation time (12 orders of magnitude) with the host lattice in terms of a tunneling matrix element renormalized due to the anisotropic lattice distortion around the defect.

Lattice Green's Function for the Orthorhombic Lattice in Terms of the Complete Elliptic Integral

The real and imaginary parts of the orthorhombic lattice Green's function at the origin are expressed as a sum of simple integrals of the complete elliptic integrals of the first kind. In order to give the expressions for all values of the variable from − ∞ to + ∞, use is made of the method of analytic continuation. The results of the numerical computations are shown by figures.

Lattice Theory of Second- and Third-Order Elastic Constants of Aluminum, Copper, and Nickel

Lattice theory has been used to obtain the expressions for the second- and third-order elastic constants for face-centered-cubic lattices in terms of the second- and third-order coupling parameters, considering the general interaction between the nearest-neighbor atoms. The number of constants have been reduced by expressing the third-order coupling parameters in terms of the coupling parameters of second order and second-order elastic constants. The general expressions of the elastic constants have been evaluated for Al, Cu, and Ni in a special case of central forces. The force constants involved in the expressions have been determined by representing the central interaction between the pair of atoms by the Morse potential function. It is found that the Cauchy relations for second-order elastic contants C12=C44 and for third-order elastic constants C112=C166 and C123=C456=C144 are satisfied in the case of central forces. It is also seen that the values of C111 for all the three metals are the largest and C112 is approximately half as that of C111 and almost all the values of C123 are negative and small compared with the other third-order elastic constants. The pressure derivatives of second-order elastic constants, the anisotropy factor, and the Debye temperatures have also been calculated for these metals. The values obtained are in good agreement with the experimental values available in the literature.

Lattice Green's Functions for the Cubic Lattices in Terms of the Complete Elliptic Integral

The real and imaginary parts of the lattice Green's functions for the simple cubic (actually the tetragonal), body-centered cubic, and face-centered cubic lattices, at the variable from −∞ to +∞, are expressed as a sum of simple integrals of the complete elliptic integral of the first kind. The results of the numerical calculations obtained with the aid of the formulas are shown by graphs.

Lattice Green's Function for the Simple Cubic Lattice in Terms of a Mellin-Barnes Type Integral

The lattice Green's function for the simple cubic lattice I(a)=1π3[triple intergral operator]0πdx dy dza+iε−cos x−cos y−cos zfor a > 3 is expressed as a Mellin-Barnes type integral. The analytic continuation gives simple and useful expressions in series for the numerical calculation of the real part IR(a) and the imaginary part II(a) of the integral for 0 < a < 1 and 1 < a < 3. The values at a = 1, a = 0, and a=√5 are obtained exactly: IR(1) = (π/2)[Γ(5/8)Γ(7/8)]−2, II(1)/(−i)=(√2)IR(1), and II(0) = 3·2−11/3π−4[(Γ(1/3)]6.

A Note on the Lattice Vibration with Glass-Like Disorder

The general formulation of the lattice vibration with glass-like disorder has not so far been given. The main difficulty is to choose a proper unperturbed system with which some iteration procedure may be started. For the one-dimensional lattice, however, useful models have been constructed and investigated. For instance Dean has proposed as a model a linear chain with force constants having a continuous probability distribution and has presented a· experiment by calculating its frequency spectrum through a direct numerical method.l) A Green's function method has been applied to Dean's model by Hidley,2l and it has been shown that much less computation based on the Green's function method can recover most of the results of computer experiment fairly accurately. Since the approximation adopted in Hindley's self-consistent treatment is nothing more than that used in other related problems (e. g., essentially the same as that of the Matsubara-Toyozawa method3l), one might hope that the Green's function method can provide a simple mathematical tool for discussing the glasslike disorder even in three dimensions, if, as Hindley suggested, the problem of disorder in structure can reduce to that of disorder in force constants by means of a suitable transformation. Unfortunately this is not the case, and even the mathematical realization of the three-dimensional glasslike disorder is not easy. Therefore as a first step let us define a weak structure disorder such that a one-to-one correspondence can be set up betweencfhe atoms in a regular lattice and those in a disordered lattice. In this note we shall point out that (i) a simple extension of the Hindley theory to more complicated systems with many internal degrees of freedom and of higher dimensions will give a fairly good result in calculating the phonon spectrum, and that (ii) in the course of the general formulation for weak structure disorder, argument can be much simplified by introducing effective complex force constants. These force constants are determined selfconsistently within the framework of the theory ~nd facilitate us to analy~e phonons iri a disordered lattice in terms of usual concepts used in a regular lattice. We shall illustrate our statement by using as an example a diatomic linear chain with glass-like disorder. The lattice dynamics of a regular diatomic linear chain can be described by a two-dimensional Green's function matrix D 0= {D~,} which satisfies

Formulas for the Lattice Green's Functions for the Cubic Lattices in Terms of the Complete Elliptic Integral

It is shown that the standard formulas of reducing elliptic integrals to the normal forms are useful in expressing the lattice Green's functions for the cubic lattices as a sum of simple integrals of the complete elliptic integrals.

Diffraction and channeling

A review of the scattering of fast charged particles by crystal lattices in terms of both classical and quantum mechanics is presented and a contrast and comparison of the two approaches is made. The anomalous transmission and absorption phenomena of the Borrmann effect of diffraction are analogs of channeling and blocking, which appear as classical mechanical effects in a mass independent limit. For conditions which correspond to localization of a wave packet in the crystal, accompanied by the elimination of observable diffraction effects, all features of the quantal model become analogous features of the classical model. The patterns seen in reflection, and in transmission with either thin or thick crystals, are then dominated by mass independent critical angles, commensurate with the disappearance of Planck's quantum of action and with the appearance of the laws of geometrical optics. An experimental study of the transmission of low energy protons through single crystals of gold is described and the different effects obtained for thin and thick specimens, analogous to those obtained in diffraction, are discussed. A consequence of the replacement of the extinction distance and the Bragg angle by a continuum of values for the orbital wavelength and critical angle respectively is that of a new phenomenon, referred to as quasi-channeling, which violates continuum potential concepts. It is proposed that quasi-channeling can give rise, in the best circumstances, to an oscillatory Rutherford scattering yield with depth in the `shadow' behind a crystal surface, analogous to the Pendellösung effects of diffraction, and to an additional yield due to scattering in the wake of point or clustered crystal defects.

Characterization of a De Morgan lattice in terms of implication and negation

Characterization of a De Morgan lattice in terms of implication and negation

Solution of the diffusion equation in periodic lattices in terms of trigonometric series

Solution of the diffusion equation in periodic lattices in terms of trigonometric series

Solution of Plane Ising Lattices by the Pfaffian Method

The partition function for the Ising problem for a general class of plane lattices is evaluated exactly using the Pfaffian method. A classification of plane lattices in terms of basic lattices with vertices decorated by sublattices is given, and the most general class which can be solved by the Pfaffian method described. The solutions given in this paper are for the special case when neighbouring lattice points of a basic rectangular lattice are connected by single bonds only. All of the exact solutions known so far are obtained as special cases. The critical points are defined as singularities of an analytic function and equations to determine their location are found. The derivatives of the partition function are evaluated in terms of elliptic integrals.