The Significance of Associations in a Square Point Lattice

1947 ◽  
Vol 9 (1) ◽  
pp. 99 ◽  
Author(s):  
D. J. Finney
Keyword(s):  
Point Lattice

scholarly journals Vector Arithmetic in the Triangular Grid

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 373
Author(s):  
Khaled Abuhmaidan ◽  
Monther Aldwairi ◽  
Benedek Nagy
Keyword(s):  
Coordinate System ◽  
Triangular Grid ◽  
Vector Addition ◽  
Plane Space ◽  
Physical Simulations ◽  
Rectangular Grids ◽  
Cartesian Coordinate ◽  
Zero Sum ◽  
Coordinate Geometry ◽  
Point Lattice

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

Continuum QCD2 from a fixed-point lattice action

1981 ◽  
Vol 132 (2) ◽  
pp. 463-481 ◽  
Author(s):  
Paolo Rossi
Keyword(s):  
Fixed Point ◽  
Lattice Action ◽  
Point Lattice

THEORY OF THE INTERACTION BETWEEN THE LATTICE VIBRATIONS AND THE ROTATIONAL MOTION IN SOLID HYDROGEN

1966 ◽  
Vol 44 (2) ◽  
pp. 313-335 ◽  
Author(s):  
J. Van Kranendonk ◽  
V. F. Sears
Keyword(s):  
Rotational Motion ◽  
Zero Point ◽  
Solid Hydrogen ◽  
The Self ◽  
Intermolecular Potential ◽  
Lattice Vibrations ◽  
Energy Effect ◽  
Self Energy ◽  
Blowing Up ◽  
Point Lattice

The effects of the interaction between the rotational motion of the molecules in solid hydrogen and the lattice vibrations, resulting from the anisotropic van der Waals forces, have been investigated theoretically. For the radial part of the anisotropic intermolecular potential an exp–6 model has been adopted. First, the effect of the lattice vibrations, and of the anistropic blowing up of the crystal by the zero-point lattice vibrations, is discussed. The effective anisotropic interaction resulting from averaging the instantaneous interaction over the lattice vibrations is calculated by assuming a Gaussian distribution for the modulation of the relative intermolecular separations by the lattice vibrations. Secondly, the displacement of the rotational levels due to the self-energy of the molecules in the lattice is calculated both classically and quantum mechanically, and the resulting shifts in the frequencies of the rotational transitions in solid hydrogen are given. Finally, the splitting of the rotational levels due to the anisotropy of the self-energy effect is calculated. The theory is applied to the calculation of the asymmetry of the S0(0) triplet in the rotational Raman spectrum of solid parahydrogen, and of the specific heat anomaly in solid hydrogen at low ortho-concentrations.

Molecular Dynamics simulation of organic materials: Structure, potentials and the MiCMoS computer platform

CrystEngComm ◽  
2022 ◽  
Author(s):  
Angelo Gavezzotti ◽  
Leonardo Lo Presti ◽  
Silvia Rizzato
Keyword(s):  
Molecular Dynamics ◽  
Molecular Dynamics Simulation ◽  
Structure Determination ◽  
Organic Materials ◽  
Single Point ◽  
Lattice Energy ◽  
Dynamics Simulation ◽  
Organic Crystals ◽  
X Ray ◽  
Point Lattice

The science of organic crystals and materials has seen in a few decades a spectacular improvement from months to minutes for an X-ray structure determination and from single-point lattice energy...

scholarly journals A Semimodular Imbedding of Lattices

1960 ◽  
Vol 12 ◽  
pp. 582-591 ◽  
Author(s):  
D. T. Finkbeiner
Keyword(s):  
Modular Lattice ◽  
Dimensional Lattice ◽  
General Lattice ◽  
Basic Properties ◽  
Arithmetic Properties ◽  
Finite Dimensional ◽  
Unpublished Work ◽  
Restricted Type ◽  
Point Lattice

The study of structural or arithmetic properties of a general lattice often can be facilitated by imbedding as a sublattice of a lattice of a more restricted type whose properties are known. However, if is too restricted, a general imbedding is impossible; for example, cannot be modular because , as a sublattice of , would then have to be modular. One of the best results of this nature has been given by Dilworth in an unpublished work in which he shows that any finite dimensional lattice is isomorphic to a sublattice of a semi-modular point lattice (1, pp. 105 and 110). In the present paper Dilworth's imbedding process is modified to obtain a sharper result: Any finite dimensional lattice is isometrically isomorphic to a sublattice of a semi-modular lattice which has the same number of points as and which preserves basic properties of the join-irreducible arithmetic of .

HIGH Tc SUPERCONDUCTIVITY IN DIBORIDES BY MICRO-STRAIN AND FERMI LEVEL TUNING

2002 ◽  
Vol 16 (11n12) ◽  
pp. 1591-1598 ◽  
Author(s):  
A. BIANCONI
Keyword(s):  
Chemical Potential ◽  
Resonance Energy ◽  
Zero Point ◽  
Shape Resonance ◽  
Modulation Amplitude ◽  
Lattice Vibrations ◽  
High Tc ◽  
Critical Range ◽  
Ε Phase ◽  
Point Lattice

It is shown that the process of T c amplification in diborides occurs in a particular region of the (ρ,ε) phase diagram, where ρ is the charge density and ε is the micro-strain in the metallic boron plane. The T c (ρ,ε) shows that the superconducting phase occurs while the chemical potential is tuned near the "shape resonance" of the diboride superlattice and the micro-strain is in a critical range. The range of the high T c phase is determined by the modulation amplitude Δshape of the shape resonance energy due to the zero-point lattice vibrations Δu rms (ε), pointing towards an electronic or vibronic pairing mechanism. It has been discussed that the McMillan's formula breaks down for the diborides.

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