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For 1-, 2- and 3-dimensional spaces potential wells do always scatter waves, no matter how small their potentials are, what their signs are or how limited their sizes are. For a particle in a one-dimensional lattice, like the Kronig–Penney model , it is possible to calculate the band structure analytically by substituting the values for the ...
When talking about solid materials, the discussion is mainly around crystals – periodic lattices. Here we will discuss a 1D lattice of positive ions. Assuming the spacing between two ions is a, the potential in the lattice will look something like this: The mathematical representation of the potential is a periodic function with a period a.
The vertex arrangement of the 16-cell honeycomb is called the D 4 lattice or F 4 lattice. [2] The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space; [3] its kissing number is 24, which is also the same as the kissing number in R 4, as proved by Oleg Musin in 2003.
The Ising model is given by the usual cubic lattice graph = (,) where is an infinite cubic lattice in or a period cubic lattice in , and is the edge set of nearest neighbours (the same letter is used for the energy functional but the different usages are distinguishable based on context).
A 1D optical lattice is formed by two counter-propagating laser beams of the same polarization. The beams will interfere, leading to a series of minima and maxima separated by λ / 2 {\displaystyle \lambda /2} , where λ {\displaystyle \lambda } is the wavelength of the light used to create the optical lattice.
In condensed matter physics, the Su–Schrieffer–Heeger (SSH) model or SSH chain is a one-dimensional lattice model that presents topological features. [1] It was devised by Wu-Pei Su, John Robert Schrieffer, and Alan J. Heeger in 1979, to describe the increase of electrical conductivity of polyacetylene polymer chain when doped, based on the existence of solitonic defects.
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) that has a partial order making it into vector lattice that possesses a neighborhood base at the origin consisting of solid sets. [1]