 | This article may be too technical for most readers to understand.(September 2024) |
Aperiodic crystals are crystals that lack three-dimensional translational symmetry, but still exhibit three-dimensional long-range order. [1] In other words, they are periodic crystals in higher dimensions.[ clarification needed ] They are classified into three different categories: incommensurate modulated structures, incommensurate composite structures, and quasicrystals. [2]
The X-ray diffraction patterns of aperiodic crystals contain two sets of peaks, which include "main reflections" and "satellite reflections". [1] Main reflections are usually stronger in intensity and span a lattice defined by three-dimensional reciprocal lattice vectors. Satellite reflections are weaker in intensity and are known as "lattice ghosts". These reflections do not correspond to any lattice points in physical space and cannot be indexed with the original three vectors.
The history of aperiodic crystals can be traced back to the early 20th century, when the science of X-ray crystallography was in its infancy. At that time, it was generally accepted that the ground state of matter was always an ideal crystal with three-dimensional space group symmetry, or lattice periodicity. However, in the late 1900s, a number of developments in the field of crystallography challenged this belief. Researchers began to focus on the scattering of X-rays and other particles beyond just the Bragg peaks, which allowed them to better understand the effects of defects and finite size on the structure of crystals, as well as the presence of additional spots in diffraction patterns due to periodic variations in the crystal structure. These findings showed that the ground state of matter was not always an ideal crystal, and that other, more complex structures could also exist. These structures were later classified as aperiodic crystals, and their study has continued to be an active area of research in the field of crystallography. [1]
To understand aperiodic crystal structures, one must use the superspace approach. [3] In materials science, "superspace" or higher-dimensional space refers to the concept of describing the structures and properties of materials in terms of dimensions beyond the three dimensions of physical space. This may involve using mathematical models to describe the behavior of atoms or molecules in a materials in four, five, or even higher dimensions. [4]
Aperiodic crystals can be understood as a three-dimensional physical space wherein atoms are positioned, plus the additional dimensions of the second subspace. [4]
Dimensionalities of aperiodic crystals:
The "" represents the dimensions of the first subspace, which is also called the "external space" () or "parallel space" ().
The "" represents the additional dimension of the second subspace, which is also called "internal space" () or "perpendicular space" (). It is perpendicular to the first subspace.
In summary, superspace is the direct sum of two subspaces. With the superspace approach, we can now describe a three-dimensional aperiodic structure as a higher dimensional periodic structure. [4]
To index all Bragg peaks, both main and satellite reflections, additional lattice vectors must be introduced:
With respect to the three reciprocal lattice vectors spanned by the main reflection, the fourth vector can be expressed by
is modulation wave vector, which represents the direction and wavelength of the modulation wave through the crystal structure.
If at least one of the values is an irrational number, then the structure is considered to be "incommensurately modulated".
With the superspace approach, we can project the diffraction pattern from a higher-dimensional space to three-dimensional space. [4]
The biphenyl molecule is a simple organic molecular compound consisting of two phenyl rings bonded by a central C-C single bond, which exhibits a modulated molecular crystal structure. Two competing factors are important for the molecule's conformation. One is steric hindrance of ortho-hydrogen, which leads to the repulsion between electrons and causes torsion of the molecule. As a result, the conformation of the molecule is non-planar, which often occurs when biphenyl is in the gas phase. The other factor is the -electron effect which favors coplanarity of the two planes. This is often the case when biphenyl is at room temperature. [4]
In mathematics, a geometric algebra is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions.
In geometry and physics, spinors are elements of a complex number-based vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation, but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360°. It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors.
In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are quite popular in theoretical physics, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory. Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models includes the presence of solitons. Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang–Baxter equation and quantum groups. The solution of these models has given insights into the nature of phase transitions, magnetization and scaling behaviour, as well as insights into the nature of quantum field theory. Physical lattice models frequently occur as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory to prevent divergences or to perform numerical computations. An example of a continuum theory that is widely studied by lattice models is the QCD lattice model, a discretization of quantum chromodynamics. However, digital physics considers nature fundamentally discrete at the Planck scale, which imposes upper limit to the density of information, aka Holographic principle. More generally, lattice gauge theory and lattice field theory are areas of study. Lattice models are also used to simulate the structure and dynamics of polymers.
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions x, y, z, ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numbers rather than real numbers. The ordinary space dimensions correspond to bosonic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom.
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 mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.
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. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension which spans the vector space . For any basis of , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regular tiling of a space by a primitive cell.
In crystallography, the monoclinic crystal system is one of the seven crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a parallelogram prism. Hence two pairs of vectors are perpendicular, while the third pair makes an angle other than 90°.
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
The reciprocal lattice is a term associated with solids with translational symmetry, and plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier transform of the lattice associated with the arrangement of the atoms. 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, which is the dual of physical space considered as a vector space, and the reciprocal lattice is the sublattice of that space that is dual to the direct lattice.
In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra, possibly with extended supersymmetry.
Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.
In condensed matter physics and crystallography, the static structure factor is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation of scattering patterns obtained in X-ray, electron and neutron diffraction experiments.
In geometry, a point reflection is a transformation of affine space in which every point is reflected across a specific fixed point. When dealing with crystal structures and in the physical sciences the terms inversion symmetry, inversion center or centrosymmetric are more commonly used.
In physics, a phason is a form of collective excitation found in aperiodic crystal structures. Phasons are a type of quasiparticle: an emergent phenomenon of many-particle systems. The phason can also be seen as a degree of freedom unique to quasicrystals. Similar to phonons, phasons are quasiparticles associated with atomic motion. However, whereas phonons are related to the translation of atoms, phasons are associated with atomic rearrangement. As a result of this rearrangement, or modulation, the waves that describe the position of atoms in the crystal change phase -- hence the term "phason". In the language of the superspace picture commonly employed in the description of aperiodic crystals in which the aperiodic function is obtained via projection from a higher dimensional periodic function, the 'phason' displacement can be seen as displacement of the (higher-dimensional) lattice points in the perpendicular space.
In crystallography, a fractional coordinate system is a coordinate system in which basis vectors used to the describe the space are the lattice vectors of a crystal (periodic) pattern. The selection of an origin and a basis define a unit cell, a parallelotope defined by the lattice basis vectors where is the dimension of the space. These basis vectors are described by lattice parameters consisting of the lengths of the lattice basis vectors and the angles between them .
The empty lattice approximation is a theoretical electronic band structure model in which the potential is periodic and weak. One may also consider an empty irregular lattice, in which the potential is not even periodic. The empty lattice approximation describes a number of properties of energy dispersion relations of non-interacting free electrons that move through a crystal lattice. The energy of the electrons in the "empty lattice" is the same as the energy of free electrons. The model is useful because it clearly illustrates a number of the sometimes very complex features of energy dispersion relations in solids which are fundamental to all electronic band structures.
In solid state physics, the magnetic space groups, or Shubnikov groups, are the symmetry groups which classify the symmetries of a crystal both in space, and in a two-valued property such as electron spin. To represent such a property, each lattice point is colored black or white, and in addition to the usual three-dimensional symmetry operations, there is a so-called "antisymmetry" operation which turns all black lattice points white and all white lattice points black. Thus, the magnetic space groups serve as an extension to the crystallographic space groups which describe spatial symmetry alone.
Theo Willem Jan Marie Janssen, better known as Ted Janssen, was a Dutch physicist and Full Professor of Theoretical Physics at the Radboud University Nijmegen. Together with Pim de Wolff and Aloysio Janner, he was one of the founding fathers of N-dimensional superspace approach in crystal structure analysis for the description of quasi periodic crystals and modulated structures. For this work he received the Aminoff Prize of the Royal Swedish Academy of Sciences in 1988 and the Ewald Prize of the International Union of Crystallography in 2014. These achievements were merit of his unique talent, combining a deep knowledge of physics with a rigorous mathematical approach. Their theoretical description of the structure and symmetry of incommensurate crystals using higher dimensional superspace groups also included the quasicrystals that were discovered in 1982 by Dan Schechtman, who received the Nobel Prize in Chemistry in 2011. The Swedish Academy of Sciences explicitly mentioned their work at this occasion.
