Shortest Lattice Vector Research Articles

Why is the slip direction in CuZn and FeAl different than in CoTi?

The slip system in FeAl at low temperatures is <111>{110}. However, while in B2 compounds the slip plane is {110}, the slip direction is often <001> which is the shortest lattice vector in this structure. Differences among various B2 systems are obviously rooted in the nature of chemical bonds and it has often been assumed that it is the antiphase boundary energy that dictates the choice of the slip system. However, it is shown in this paper by comparing the three compounds CuZn, FeAl and CoTi that factors governing the choice of the slip direction are more varied. They are, in particular the elastic anisotropy, types of planar faults on {110} planes with which dislocations can dissociate and the energy of these faults that is not simply related to the order–disorder transition temperature. Analysis taking into account all these factors explains then different slip directions in FeAl and CoTi that have similar order–disorder transition temperatures and shows why in FeAl the slip system is the same as in CuZn.

On Nearly Orthogonal Lattice Bases and Random Lattices

We study lattice bases where the angle between any basis vector and the linear subspace spanned by the other basis vectors is at least $\frac{\pi}{3}$ radians; we denote such bases as “nearly orthogonal.” We show that a nearly orthogonal lattice basis always contains a shortest lattice vector. Moreover, we prove that if the basis vector lengths are “nearly equal,” then the basis is the unique nearly orthogonal lattice basis up to multiplication of basis vectors by $\pm 1$. We also study random lattices generated by the columns of random matrices with $n$ rows and $m \leq n$ columns. We show that if $m \leq c\,n$, with $c \approx 0.071$, then the random matrix forms a nearly orthogonal basis for the random lattice with high probability for large $n$ and almost surely as $n$ tends to infinity. Consequently, the columns of such a random matrix contain the shortest vector in the random lattice. Finally, we discuss an interesting JPEG image compression application where nearly orthogonal lattice bases play an important role.

On Free Planes in Lattice Ball Packings

This note, by studying the relations between the length of the shortest lattice vectors and the covering minima of a lattice, proves that for every d‐dimensional packing lattice of balls one can find a four‐dimensional plane, parallel to a lattice plane, such that the plane meets none of the balls of the packing, provided that the dimension d is large enough. Nevertheless, for certain ball packing lattices, the highest dimension of such ‘free planes’ is far from d.

The Ajtai random class of lattices

Ajtai has recently given a reduction from the problem of approximating a short basis for a lattice in the worst case, to the problem of finding a short lattice vector for a uniformly chosen lattice in a certain random class of lattices. Here we give an explicit formula for the number of lattices of the type used by Ajtai. We also prove some results about the average volume of the fundamental cell of such a lattice.

In-situ STM investigation of specific anion adsorption on Cu(111)

The specific anion adsorption of chloride and sulfate on Cu(111) from acidic aqueous electrolytes has been studied using scanning tunneling microscopy (STM) and cyclic voltammetry. At positive potentials adsorbed chloride forms a well ordered (√3×√3)R30° superstructure; at negative potentials the bare copper surface can be imaged with atomic resolution. By use of a so-called potentiodynamic STM measurement it is possible to correlate directly the appearance and the disappearance of the chloride superstructure to the anodic and the cathodic peak in the cyclic voltammogram. The chloride adsorbate influences the surface topography strongly in such a way that copper steps preferentially run along close packed chloride rows after chloride adsorption. An enhanced surface mobility caused by the chloride adsorbate leads to an ‘electrochemical annealing’ effect. Surface defects such as pits heal rapidly and the decay of copper stacks is promoted. Although the cyclic voltammograms of Cu(111) in chloride acid electrolyte and in sulfuric acid electrolyte are similar, both showing characteristic adsorption and desorption peaks, the surface structures and the adsorption kinetics are extremely different. At positive potentials the sulfate adsorbate forms an anisotropic Moiré structure which occurs in three rotational domains. High resolution STM images reveal an additional species which is assigned to coadsorbed water molecules. Close packed sulfate rows are separated by zig-zag chains of these water molecules. The short range lattice vectors of the sulfate structure are similar to those found for sulfate adlayers on other fcc(111) surfaces (Au, Pt, Rh). Due to a strong kinetic hindrance it is possible to observe directly the mechanism of the sulfate adlayer formation and the decay process. The Moiré formation starts locally at upper step edges and spreads from the step edges over the upper terraces. During the adlayer formation process a mass transport out of the top copper layer takes place, resulting in a drastic change of the surface topography due to the formation of characteristically shaped islands and step edges. Different models are discussed explaining the long range periodicity of the Moiré pattern and the topographic changes during the adlayer formation and decay processes.

Lattice basis reduction, Jacobi sums and hyperelliptic cryptosystems

Using the LLL-algorithm for finding short vectors in lattices, we show how to compute a Jacobi sum for the prime field Fp in Q(e2πi/n) in time O(log3p), where n is small and fixed, p is large, and p = 1 (mod n). This result is useful in the construction of hyperelliptic cryptosystems.

Improved low-density subset sum algorithms

The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of sufficiently low density. Both methods rely on basis reduction algorithms to find short non-zero vectors in special lattices. The Lagarias-Odlyzko algorithm would solve almost all subset sum problems of density<0.6463 ... in polynomial time if it could invoke a polynomial-time algorithm for finding the shortest non-zero vector in a lattice. This paper presents two modifications of that algorithm, either one of which would solve almost all problems of density<0.9408 ... if it could find shortest non-zero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.

Dislocations and Plastic Deformation In Molybdenum Disilicide

A composite of MoSi2 (reinforced with SiC whiskers) has been deformed at high temperatures and the resulting dislocation substructure studied by means of transmission electron microscopy (TEM). Classical g.b analysis of the dislocations in the MoSi2 matrix has shown that the Burgers vectors are (100), (110), and 1/2(111) which are the shortest lattice vectors in the body‐centered tetragonal lattice. Trace analysis showed that there are numerous possible slip planes for each of the determined slip directions. This result suggests that various operative slip systems can contribute to the deformation, depending on the orientation of the particular grain with respect to the stress state.

Structural research on non transition metal double nitrites: Crystal structure, optical absorption and emission of TlBa2(NO2)5

The thallium-barium double nitrite, TlBa2(NO2)5, is pyroelectric in the 77-600 K range and crystallizes in thePca2 1 space group. The lattice constants at 293 K are: a = 17.868(12),b = 4.934(3),c = 13-426(11) A (MoKa, λ = 0.71069 A). There are four stoichiometric units in the unit cell of volume V= 1184(1) A3 (D o = 3.98,D x = 3.979 Mgm−3),F (000) = 1232, μ = 20.36mm−1. The crystal structure was solved by Patterson and Fourier methods and refined by least-squares to a final conventional agreement indexR = 0.053 for 1371 independent reflections collected in a θ range of 3–30°, using MoKα radiation. There are two independent barium atoms surrounded by NO 2 − groups, both with coordination number 10 and distances in the ranges Ba-O = 2.69(4)-3.18(4) A and Ba-N = 3.01(4)-3.18(4) A. The environment of thallium is clearly affected by the lone-pair stereoactivity and involves 12 Tl-O and Tl-N contacts less than 3.5A, but only four Tl-O distances are shorter than 3 A (min. 2.76(2), max. 2.85(3) A), with a pyramidal coordination and thallium at the apex of the pyramid. All these coordination polyhedra are joined in chains running along the shortest lattice vector [010].

On the shortest lattice vectors in a three-dimensional translational (Bravais) lattice

On the shortest lattice vectors in a three-dimensional translational (Bravais) lattice