Lattice (group) in the context of "Diamond cubic"

Spatial statistics is a field of applied statistics dealing with spatial data.It involves stochastic processes (random fields, point processes), sampling, smoothing and interpolation, regional (areal unit) and lattice (gridded) data, point patterns, as well as image analysis and stereology.

In crystallography, the orthorhombic crystal system is one of the seven crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal.

In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space ⁠${\displaystyle \mathbb {R} ^{n}}$⁠, forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense.

Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid".

A clathrate is a chemical substance consisting of a lattice that traps or contains molecules. The word clathrate is derived from the Latin clathratus (clatratus), meaning 'with bars, latticed'. Most clathrate compounds are polymeric and completely envelop the guest molecule, but in modern usage clathrates also include host–guest complexes and inclusion compounds. According to IUPAC, clathrates are inclusion compounds "in which the guest molecule is in a cage formed by the host molecule or by a lattice of host molecules." The term refers to many molecular hosts, including calixarenes and cyclodextrins and even some inorganic polymers such as zeolites.

Clathrates can be divided into two categories: clathrate hydrates and inorganic clathrates. Each clathrate is made up of a framework and guests that reside the framework. Most common clathrate crystal structures can be composed of cavities such as dodecahedral, tetrakaidecahedral, and hexakaidecahedral cavities.Unlike hydrates, inorganic clathrates have a covalently bonded framework of inorganic atoms with guests typically consisting of alkali or alkaline earth metals. Due to the stronger covalent bonding, the cages are often smaller than hydrates. Guest atoms interact with the host by ionic or covalent bonds. Therefore, partial substitution of guest atoms follow Zintl rules so that the charge of the overall compound is conserved. Most inorganic clathrates have full occupancy of its framework cages by a guest atom to be in stable phase. Inorganic clathrates can be synthesized by direct reaction using ball milling at high temperatures or high pressures. Crystallization from melt is another common synthesis route. Due to the wide variety of composition of host and guest species, inorganic clathrates are much more chemically diverse and possess a wide range of properties. Most notably, inorganic clathrates can be found to be both an insulator and a superconductor (Ba₈Si₄₆). A common property of inorganic clathrates that has attracted researchers is low thermal conductivity. Low thermal conductivity is attributed to the ability of the guest atom to "rattle" within the host framework. The freedom of movement of the guest atoms scatters phonons that transport heat.

Low-energy ion scattering spectroscopy (LEIS), sometimes referred to simply as ion scattering spectroscopy (ISS), is a surface-sensitive analytical technique used to characterize the chemical and structural makeup of materials. LEIS involves directing a stream of charged particles known as ions at a surface and making observations of the positions, velocities, and energies of the ions that have interacted with the surface. Data that is thus collected can be used to deduce information about the material such as the relative positions of atoms in a surface lattice and the elemental identity of those atoms. LEIS is closely related to both medium-energy ion scattering (MEIS) and high-energy ion scattering (HEIS, known in practice as Rutherford backscattering spectroscopy, or RBS), differing primarily in the energy range of the ion beam used to probe the surface. While much of the information collected using LEIS can be obtained using other surface science techniques, LEIS is unique in its sensitivity to both structure and composition of surfaces. Additionally, LEIS is one of a very few surface-sensitive techniques capable of directly observing hydrogen atoms, an aspect that may make it an increasingly more important technique as the hydrogen economy is being explored.

Geometry of numbers, also known as geometric number theory, is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in ${\displaystyle \mathbb {R} ^{n},}$ and the study of these lattices provides fundamental information on algebraic numbers. Hermann Minkowski (1896) initiated this line of research at the age of 26 in his work The Geometry of Numbers.

The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.

In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessarily have unit size, or even a particular size at all. Rather, the primitive cell is the closest analogy to a unit vector, since it has a determined size for a given lattice and is the basic building block from which larger cells are constructed.

The concept is used particularly in describing crystal structure in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its unit cell, which is a section of the tiling (a parallelogram or parallelepiped) that generates the whole tiling using only translations.