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A crystal or crystalline solid is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification.
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. A compact space is a space in which every open cover of the space contains a finite subcover. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice that satisfies at least one of these properties is known as a conditionally complete lattice. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete.
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.
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 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.
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets.
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .
In crystallography, a crystal system is a set of point groups. A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family.
In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
In physics, the reciprocal lattice emerges from the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system. The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, where refers to the wavevector.
In solid-state physics, the electronic band structure of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have.
In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, whose canonical class is big.
Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.
The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a primitive cell which has been constructed by applying Voronoi decomposition to a crystal lattice. It is used in the study of crystalline materials in crystallography.
In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a line, plane or rigid body – is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imaginary rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement, in which case it may be necessary to add an imaginary translation to change the object's position. The position and orientation together fully describe how the object is placed in space. The above-mentioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, and its position does not change when it rotates.
In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, , at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. Here, a lattice is an abstract structure with two binary operations, the "meet" and "join" operations, which must obey certain axioms; it is distributive if these two operations obey the distributive law. The union and intersection operations, in a family of sets that is closed under these operations, automatically form a distributive lattice, and Birkhoff's representation theorem states that every finite distributive lattice can be formed in this way. It is named after Garrett Birkhoff, who published a proof of it in 1937.
In crystallography, the hexagonal crystal family is one of the 6 crystal families, which includes two crystal systems and two lattice systems. While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent. In particular, there are crystals that have trigonal symmetry but belong to the hexagonal lattice.
Johann Friedrich Christian Hessel was a German physician and professor of mineralogy at the University of Marburg.
Moritz Ludwig Frankenheim was a German physicist, geographer, and crystallographer.
References
- ↑ Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: New York, 1976).
- ↑ H., Simon, Steven (2020). The Oxford Solid State Basics. Oxford University Press. ISBN 978-0-19-968077-1. OCLC 1267459045.
{{cite book}}: CS1 maint: multiple names: authors list (link) - ↑ J. B. Suck, M. Schreiber, and P. Häussler, eds., Quasicrystals: An Introduction to Structure, Physical Properties, and Applications (Springer: Berlin, 2004).
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