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In crystallography, **crystal structure** is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material.^{ [3] } 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.

**Crystallography** is the experimental science of determining the arrangement of atoms in crystalline solids. The word "crystallography" derives from the Greek words *crystallon* "cold drop, frozen drop", with its meaning extending to all solids with some degree of transparency, and *graphein* "to write". In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography. X-ray crystallography is used to determine the structure of large biomolecules such as proteins. Before the development of X-ray diffraction crystallography, the study of crystals was based on physical measurements of their geometry. This involved measuring the angles of crystal faces relative to each other and to theoretical reference axes, and establishing the symmetry of the crystal in question. This physical measurement is carried out using a goniometer. The position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to each face is plotted on the net. Each point is labelled with its Miller index. The final plot allows the symmetry of the crystal to be established.

An **atom** is the smallest constituent unit of ordinary matter that has the properties of a chemical element. Every solid, liquid, gas, and plasma is composed of neutral or ionized atoms. Atoms are extremely small; typical sizes are around 100 picometers.

An **ion** is an atom or molecule that has a non-zero net electrical charge. Since the charge of the electron is equal and opposite to that of the proton, the net charge of an ion is non-zero due to its total number of electrons being unequal to its total number of protons. A cation is a positively charged ion, with fewer electrons than protons, while an anion is negatively charged, with more electrons than protons. Because of their opposite electric currents, cations and anions attract each other and readily form ionic compounds.

- Unit cell
- Miller indices
- Planes and directions
- Interplanar spacing
- Classification by symmetry
- Lattice systems
- Crystal systems
- Space groups
- Atomic coordination
- Close packing
- Grain boundaries
- Defects and impurities
- Prediction of structure
- Polymorphism
- Physical properties
- See also
- References
- External links

The smallest group of particles in the material that constitutes this repeating pattern is the **unit cell** of the structure. The unit cell completely reflects the symmetry and structure of the entire crystal, which is built up by repetitive translation of the unit cell along its principal axes. The translation vectors define the nodes of the Bravais lattice.

In Euclidean geometry, a **translation** is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

In geometry and crystallography, a **Bravais lattice**, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by:

The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called *lattice parameters* or *cell parameters*. The symmetry properties of the crystal are described by the concept of space groups.^{ [3] } All possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups.

The **lattice constant**, or **lattice parameter**, refers to the physical dimension of unit cells in a crystal lattice. Lattices in three dimensions generally have three lattice constants, referred to as *a*, *b*, and *c*. However, in the special case of cubic crystal structures, all of the constants are equal and we only refer to *a*. Similarly, in hexagonal crystal structures, the *a* and *b* constants are equal, and we only refer to the *a* and *c* constants. A group of lattice constants could be referred to as **lattice parameters**. However, the full set of lattice parameters consist of the three lattice constants and the three angles between them.

**Symmetry** in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together.

In mathematics, physics and chemistry, a **space group** is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called **Bieberbach groups**, and are discrete cocompact groups of isometries of an oriented Euclidean space.

The crystal structure and symmetry play a critical role in determining many physical properties, such as cleavage, electronic band structure, and optical transparency.

**Cleavage**, in mineralogy, is the tendency of crystalline materials to split along definite crystallographic structural planes. These planes of relative weakness are a result of the regular locations of atoms and ions in the crystal, which create smooth repeating surfaces that are visible both in the microscope and to the naked eye.

In solid-state physics, the **electronic band structure** of a solid describes the range of energies that an electron within the solid may have and ranges of energy that it may not have.

**Crystal optics** is the branch of optics that describes the behaviour of light in *anisotropic media*, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone–Dale relation. Crystals are often naturally anisotropic, and in some media it is possible to induce anisotropy by applying an external electric field.

Crystal structure is described in terms of the geometry of arrangement of particles in the unit cell. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure.^{ [4] } The geometry of the unit cell is defined as a parallelepiped, providing six lattice parameters taken as the lengths of the cell edges (*a*, *b*, *c*) and the angles between them (α, β, γ). The positions of particles inside the unit cell are described by the fractional coordinates (*x _{i}*,

In geometry, a **parallelepiped** is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square or as a cuboid to a rectangle. In Euclidean geometry, its definition encompasses all four concepts. In this context of affine geometry, in which angles are not differentiated, its definition admits only *parallelograms* and *parallelepipeds*. Three equivalent definitions of *parallelepiped* are

In crystallography, a **fractional coordinate system** is a coordinate system in which the edges of the unit cell are used as the basic vectors to describe the positions of atomic nuclei. The unit cell is a parallelepiped defined by the lengths of its edges and angles between them .

- Simple cubic (P)
- Body-centered cubic (I)
- Face-centered cubic (F)

Vectors and planes in a crystal lattice are described by the three-value Miller index notation. This syntax uses the indices *ℓ*, *m*, and *n* as directional orthogonal parameters, which are separated by 90°.^{ [6] }

By definition, the syntax (*ℓmn*) denotes a plane that intercepts the three points *a*_{1}/*ℓ*, *a*_{2}/*m*, and *a*_{3}/*n*, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it means that the planes do not intersect that axis (i.e., the intercept is "at infinity"). A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined. The Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in (123). In an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane.

Considering only (*ℓmn*) planes intersecting one or more lattice points (the *lattice planes*), the distance *d* between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula

The crystallographic directions are geometric lines linking nodes (atoms, ions or molecules) of a crystal. Likewise, the crystallographic planes are geometric *planes* linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows:^{ [3] }

- Optical properties: Refractive index is directly related to density (or periodic density fluctuations).
- Adsorption and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes.
- Surface tension: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface.

- Microstructural defects: Pores and crystallites tend to have straight grain boundaries following higher density planes.
- Cleavage: This typically occurs preferentially parallel to higher density planes.
- Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (Burgers vector) is along a dense direction. The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice.

Some directions and planes are defined by symmetry of the crystal system. In monoclinic, rhombohedral, tetragonal, and trigonal/hexagonal systems there is one unique axis (sometimes called the **principal axis**) which has higher rotational symmetry than the other two axes. The **basal plane** is the plane perpendicular to the principal axis in these crystal systems. For triclinic, orthorhombic, and cubic crystal systems the axis designation is arbitrary and there is no principal axis.

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted *a*); similarly for the reciprocal lattice. So, in this common case, the Miller indices (*ℓmn*) and [*ℓmn*] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant *a*, the spacing *d* between adjacent (ℓmn) lattice planes is (from above):

Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

- Coordinates in
*angle brackets*such as ⟨100⟩ denote a*family*of directions that are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions. - Coordinates in
*curly brackets*or*braces*such as {100} denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.

The spacing * d* between adjacent (

- Cubic:
- Tetragonal:
- Hexagonal:
- Rhombohedral:
- Orthorhombic:
- Monoclinic:
- Triclinic:

The defining property of a crystal is its inherent symmetry, by which we mean that under certain 'operations' the crystal remains unchanged. All crystals have translational symmetry in three directions, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration. The crystal is then said to have a twofold rotational symmetry about this axis. In addition to rotational symmetries like this, a crystal may have symmetries in the form of mirror planes and translational symmetries, and also the so-called "compound symmetries," which are a combination of translation and rotation/mirror symmetries. A full classification of a crystal is achieved when all of these inherent symmetries of the crystal are identified.^{ [8] }

These lattice systems are a grouping of crystal structures according to the axial system used to describe their lattice. Each lattice system consists of a set of three axes in a particular geometric arrangement. There are seven lattice systems. They are similar to but not quite the same as the seven crystal systems.

Crystal family | Lattice system | Schönflies | 14 Bravais Lattices | |||
---|---|---|---|---|---|---|

Primitive | Base-centered | Body-centered | Face-centered | |||

triclinic | C_{i} | |||||

monoclinic | C_{2h} | |||||

orthorhombic | D_{2h} | |||||

tetragonal | D_{4h} | |||||

hexagonal | rhombohedral | D_{3d} | ||||

hexagonal | D_{6h} | |||||

cubic | O_{h} |

The simplest and most symmetric, the cubic (or isometric) system, has the symmetry of a cube, that is, it exhibits four threefold rotational axes oriented at 109.5° (the tetrahedral angle) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal, tetragonal, rhombohedral (often confused with the trigonal crystal system), orthorhombic, monoclinic and triclinic.

Bravais lattices, also referred to as *space lattices*, describe the geometric arrangement of the lattice points,^{ [6] } and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including quasicrystals, fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above.

The crystal structure consists of the same group of atoms, the *basis*, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its crystallographic point group.

A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system.

Crystal family | Crystal system | Point group / Crystal class | Schönflies | Point symmetry | Order | Abstract group |
---|---|---|---|---|---|---|

triclinic | pedial | C_{1} | enantiomorphic polar | 1 | trivial | |

pinacoidal | C_{i} (S_{2}) | centrosymmetric | 2 | cyclic | ||

monoclinic | sphenoidal | C_{2} | enantiomorphic polar | 2 | cyclic | |

domatic | C_{s} (C_{1h}) | polar | 2 | cyclic | ||

prismatic | C_{2h} | centrosymmetric | 4 | Klein four | ||

orthorhombic | rhombic-disphenoidal | D_{2} (V) | enantiomorphic | 4 | Klein four | |

rhombic-pyramidal | C_{2v} | polar | 4 | Klein four | ||

rhombic-dipyramidal | D_{2h} (V_{h}) | centrosymmetric | 8 | |||

tetragonal | tetragonal-pyramidal | C_{4} | enantiomorphic polar | 4 | cyclic | |

tetragonal-disphenoidal | S_{4} | non-centrosymmetric | 4 | cyclic | ||

tetragonal-dipyramidal | C_{4h} | centrosymmetric | 8 | |||

tetragonal-trapezohedral | D_{4} | enantiomorphic | 8 | dihedral | ||

ditetragonal-pyramidal | C_{4v} | polar | 8 | dihedral | ||

tetragonal-scalenohedral | D_{2d} (V_{d}) | non-centrosymmetric | 8 | dihedral | ||

ditetragonal-dipyramidal | D_{4h} | centrosymmetric | 16 | |||

hexagonal | trigonal | trigonal-pyramidal | C_{3} | enantiomorphic polar | 3 | cyclic |

rhombohedral | C_{3i} (S_{6}) | centrosymmetric | 6 | cyclic | ||

trigonal-trapezohedral | D_{3} | enantiomorphic | 6 | dihedral | ||

ditrigonal-pyramidal | C_{3v} | polar | 6 | dihedral | ||

ditrigonal-scalenohedral | D_{3d} | centrosymmetric | 12 | dihedral | ||

hexagonal | hexagonal-pyramidal | C_{6} | enantiomorphic polar | 6 | cyclic | |

trigonal-dipyramidal | C_{3h} | non-centrosymmetric | 6 | cyclic | ||

hexagonal-dipyramidal | C_{6h} | centrosymmetric | 12 | |||

hexagonal-trapezohedral | D_{6} | enantiomorphic | 12 | dihedral | ||

dihexagonal-pyramidal | C_{6v} | polar | 12 | dihedral | ||

ditrigonal-dipyramidal | D_{3h} | non-centrosymmetric | 12 | dihedral | ||

dihexagonal-dipyramidal | D_{6h} | centrosymmetric | 24 | |||

cubic | tetartoidal | T | enantiomorphic | 12 | alternating | |

diploidal | T_{h} | centrosymmetric | 24 | |||

gyroidal | O | enantiomorphic | 24 | symmetric | ||

hextetrahedral | T_{d} | non-centrosymmetric | 24 | symmetric | ||

hexoctahedral | O_{h} | centrosymmetric | 48 |

In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

The crystallographic point group or *crystal class* is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include

*Reflection*, which reflects the structure across a*reflection plane**Rotation*, which rotates the structure a specified portion of a circle about a*rotation axis**Inversion*, which changes the sign of the coordinate of each point with respect to a*center of symmetry*or*inversion point**Improper rotation*, which consists of a rotation about an axis followed by an inversion.

Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called *symmetry elements*. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.

In addition to the operations of the point group, the space group of the crystal structure contains translational symmetry operations. These include:

- Pure
*translations*, which move a point along a vector *Screw axes*, which rotate a point around an axis while translating parallel to the axis.^{ [9] }*Glide planes*, which reflect a point through a plane while translating it parallel to the plane.^{ [9] }

There are 230 distinct space groups.

By considering the arrangement of atoms relative to each other, their coordination numbers (or number of nearest neighbors), interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing them.^{ [10] }

The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer was placed directly over plane A, this would give rise to the following series:

- ...
**ABABABAB**...

This arrangement of atoms in a crystal structure is known as **hexagonal close packing (hcp)**.

If, however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises:

- ...
**ABCABCABC**...

This type of structural arrangement is known as **cubic close packing (ccp)**.

The unit cell of a ccp arrangement of atoms is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the {111} planes of the fcc unit cell. There are four different orientations of the close-packed layers.

The **packing efficiency** can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows:

The 74% packing efficiency is the maximum density possible in unit cells constructed of spheres of only one size. Most crystalline forms of metallic elements are hcp, fcc, or bcc (body-centered cubic). The coordination number of atoms in hcp and fcc structures is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74. This can be compared to the APF of a bcc structure, which is 0.68.

Grain boundaries are interfaces where crystals of different orientations meet.^{ [6] } A grain boundary is a single-phase interface, with crystals on each side of the boundary being identical except in orientation. The term "crystallite boundary" is sometimes, though rarely, used. Grain boundary areas contain those atoms that have been perturbed from their original lattice sites, dislocations, and impurities that have migrated to the lower energy grain boundary.

Treating a grain boundary geometrically as an interface of a single crystal cut into two parts, one of which is rotated, we see that there are five variables required to define a grain boundary. The first two numbers come from the unit vector that specifies a rotation axis. The third number designates the angle of rotation of the grain. The final two numbers specify the plane of the grain boundary (or a unit vector that is normal to this plane).^{ [10] }

Grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve strength, as described by the Hall–Petch relationship. Since grain boundaries are defects in the crystal structure they tend to decrease the electrical and thermal conductivity of the material. The high interfacial energy and relatively weak bonding in most grain boundaries often makes them preferred sites for the onset of corrosion and for the precipitation of new phases from the solid. They are also important to many of the mechanisms of creep.^{ [10] }

Grain boundaries are in general only a few nanometers wide. In common materials, crystallites are large enough that grain boundaries account for a small fraction of the material. However, very small grain sizes are achievable. In nanocrystalline solids, grain boundaries become a significant volume fraction of the material, with profound effects on such properties as diffusion and plasticity. In the limit of small crystallites, as the volume fraction of grain boundaries approaches 100%, the material ceases to have any crystalline character, and thus becomes an amorphous solid.^{ [10] }

Real crystals feature defects or irregularities in the ideal arrangements described above and it is these defects that critically determine many of the electrical and mechanical properties of real materials. When one atom substitutes for one of the principal atomic components within the crystal structure, alteration in the electrical and thermal properties of the material may ensue.^{ [11] } Impurities may also manifest as spin impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of certain properties such as specific heat may be affected by small concentrations of an impurity, as for example impurities in semiconducting ferromagnetic alloys may lead to different properties as first predicted in the late 1960s.^{ [12] }^{ [13] } Dislocations in the crystal lattice allow shear at lower stress than that needed for a perfect crystal structure.^{ [14] }

The difficulty of predicting stable crystal structures based on the knowledge of only the chemical composition has long been a stumbling block on the way to fully computational materials design. Now, with more powerful algorithms and high-performance computing, structures of medium complexity can be predicted using such approaches as evolutionary algorithms, random sampling, or metadynamics.

The crystal structures of simple ionic solids (e.g., NaCl or table salt) have long been rationalized in terms of Pauling's rules, first set out in 1929 by Linus Pauling, referred to by many since as the "father of the chemical bond".^{ [15] } Pauling also considered the nature of the interatomic forces in metals, and concluded that about half of the five d-orbitals in the transition metals are involved in bonding, with the remaining nonbonding d-orbitals being responsible for the magnetic properties. He, therefore, was able to correlate the number of d-orbitals in bond formation with the bond length as well as many of the physical properties of the substance. He subsequently introduced the metallic orbital, an extra orbital necessary to permit uninhibited resonance of valence bonds among various electronic structures.^{ [16] }

In the resonating valence bond theory, the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": ^{1}⁄_{2}, ^{1}⁄_{3}, ^{2}⁄_{3}, ^{1}⁄_{4}, ^{3}⁄_{4}, etc. The choice of structure and the value of the axial ratio (which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers.^{ [17] }^{ [18] }

After postulating a direct correlation between electron concentration and crystal structure in beta-phase alloys, Hume-Rothery analyzed the trends in melting points, compressibilities and bond lengths as a function of group number in the periodic table in order to establish a system of valencies of the transition elements in the metallic state. This treatment thus emphasized the increasing bond strength as a function of group number.^{ [19] } The operation of directional forces were emphasized in one article on the relation between bond hybrids and the metallic structures. The resulting correlation between electronic and crystalline structures is summarized by a single parameter, the weight of the d-electrons per hybridized metallic orbital. The "d-weight" calculates out to 0.5, 0.7 and 0.9 for the fcc, hcp and bcc structures respectively. The relationship between d-electrons and crystal structure thus becomes apparent.^{ [20] }

In crystal structure predictions/simulations, the periodicity is usually applied, since the system is imagined as unlimited big in all directions. Starting from a triclinic structure with no further symmetry property assumed, the system may be driven to show some additional symmetry properties by applying Newton's Second Law on particles in the unit cell and a recently developed dynamical equation for the system period vectors ^{ [21] } (lattice parameters including angles), even if the system is subject to external stress.

Polymorphism is the occurrence of multiple crystalline forms of a material. It is found in many crystalline materials including polymers, minerals, and metals. According to Gibbs' rules of phase equilibria, these unique crystalline phases are dependent on intensive variables such as pressure and temperature. Polymorphism is related to allotropy, which refers to elemental solids. The complete morphology of a material is described by polymorphism and other variables such as crystal habit, amorphous fraction or crystallographic defects. Polymorphs have different stabilities and may spontaneously convert from a metastable form (or thermodynamically unstable form) to the stable form at a particular temperature. They also exhibit different melting points, solubilities, and X-ray diffraction patterns.

One good example of this is the quartz form of silicon dioxide, or SiO_{2}. In the vast majority of silicates, the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral {SiO_{4}} units linked together by shared vertices in different arrangements. In different minerals the tetrahedra show different degrees of networking and polymerization. For example, they occur singly, joined together in pairs, in larger finite clusters including rings, in chains, double chains, sheets, and three-dimensional frameworks. The minerals are classified into groups based on these structures. In each of the 7 thermodynamically stable crystalline forms or polymorphs of crystalline quartz, only 2 out of 4 of each the edges of the {SiO_{4}} tetrahedra are shared with others, yielding the net chemical formula for silica: SiO_{2}.

Another example is elemental tin (Sn), which is malleable near ambient temperatures but is brittle when cooled. This change in mechanical properties due to existence of its two major allotropes, α- and β-tin. The two allotropes that are encountered at normal pressure and temperature, α-tin and β-tin, are more commonly known as *gray tin* and *white tin* respectively. Two more allotropes, γ and σ, exist at temperatures above 161 °C and pressures above several GPa.^{ [22] } White tin is metallic, and is the stable crystalline form at or above room temperature. Below 13.2 °C, tin exists in the gray form, which has a diamond cubic crystal structure, similar to diamond, silicon or germanium. Gray tin has no metallic properties at all, is a dull gray powdery material, and has few uses, other than a few specialized semiconductor applications.^{ [23] } Although the α–β transformation temperature of tin is nominally 13.2 °C, impurities (e.g. Al, Zn, etc.) lower the transition temperature well below 0 °C, and upon addition of Sb or Bi the transformation may not occur at all.^{ [24] }

Twenty of the 32 crystal classes are piezoelectric, and crystals belonging to one of these classes (point groups) display piezoelectricity. All piezoelectric classes lack a center of symmetry. Any material develops a dielectric polarization when an electric field is applied, but a substance that has such a natural charge separation even in the absence of a field is called a polar material. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar. All polar crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.

There are a few crystal structures, notably the perovskite structure, which exhibit ferroelectric behavior. This is analogous to ferromagnetism, in that, in the absence of an electric field during production, the ferroelectric crystal does not exhibit a polarization. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarized. This polarization can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnet can be reversed. However, although they are called ferroelectrics, the effect is due to the crystal structure (not the presence of a ferrous metal).

- Brillouin zone
- Crystal engineering
- Crystal growth
- Crystallographic database
- Fractional coordinates
- Frank–Kasper phases
- Hermann–Mauguin notation – Notation to represent symmetry in point groups, plane groups and space groups
- Laser-heated pedestal growth
- Liquid crystal – State of matter with properties of both conventional liquids and crystals
- Patterson function
- Periodic table (crystal structure)
- Primitive cell
- Schoenflies notation – Notation to represent symmetry in point groups
- Seed crystal
- Wigner–Seitz cell

In physics, a **phonon** is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, like solids and some liquids. Often designated a quasiparticle, it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles.

A **crystallite** is a small or even microscopic crystal which forms, for example, during the cooling of many materials. The orientation of crystallites can be random with no preferred direction, called random texture, or directed, possibly due to growth and processing conditions. Fiber texture is an example of the latter. Crystallites are also referred to as **grains**. The areas where crystallites meet are known as grain boundaries. Polycrystalline or multicrystalline materials, or polycrystals are solids that are composed of many crystallites of varying size and orientation.

In physics, **Bragg's law**, or **Wulff–Bragg's condition**, a special case of Laue diffraction, gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident on an atom, they make the electronic cloud move, as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency, blurred slightly due to a variety of effects; this phenomenon is known as Rayleigh scattering. The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible.

In crystallography, the terms **crystal system**, **crystal family**, and **lattice system** each refer to one of several classes of space groups, lattices, point groups, or crystals. Informally, two crystals are in the same crystal system if they have similar symmetries, although there are many exceptions to this.

In physics, a ferromagnetic material is said to have **magnetocrystalline anisotropy** if it takes more energy to magnetize it in certain directions than in others. These directions are usually related to the principal axes of its crystal lattice. It is a special case of magnetic anisotropy.

In crystallography, the **cubic****crystal system** is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.

In materials science, a **dislocation** or **Taylor's dislocation** is a crystallographic defect or irregularity within a crystal structure. The presence of dislocations strongly influences many of the properties of materials.

**Miller indices** form a notation system in crystallography for planes in crystal (Bravais) lattices.

A **grain boundary** is the interface between two grains, or crystallites, in a polycrystalline material. Grain boundaries are 2D defects in the crystal structure, and tend to decrease the electrical and thermal conductivity of the material. Most grain boundaries are preferred sites for the onset of corrosion and for the precipitation of new phases from the solid. They are also important to many of the mechanisms of creep. On the other hand, grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve mechanical strength, as described by the Hall–Petch relationship. The study of grain boundaries and their effects on the mechanical, electronic and other properties of materials forms an important topic in materials science.

**Crystal twinning** occurs when two separate crystals share some of the same crystal lattice points in a symmetrical manner. The result is an intergrowth of two separate crystals in a variety of specific configurations. The surface along which the lattice points are shared in twinned crystals is called a composition surface or twin plane.

The **crystallographic restriction theorem** in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.

**Powder diffraction** is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is called a **powder diffractometer**.

In geometry, a **rhombohedron** is a three-dimensional figure like a cube, except that its faces are not squares but rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells.

In materials science, a **slip system** describes the set of symmetrically identical **slip planes** and associated family of **slip directions** for which dislocation motion can easily occur and lead to plastic deformation. An external force makes parts of the crystal lattice glide along each other, changing the material's geometry. Depending on the type of lattice, different slip systems are present in the material. More specifically, slip occurs on close-packed planes, and in close-packed directions. The magnitude and direction of slip are represented by the Burgers vector. The picture on the right shows a schematic view of the slip mechanism. The slip planes and slip directions in a crystal have specific crystallographic forms. The slip planes are normally the planes with the highest density of atoms, i.e., those most closely spaced, and the direction of the slip is the direction in the slip plane that corresponds to one of the shortest lattice translation vectors. Often, this is the direction in which atoms are most closely spaced. A slip plane and a slip direction constitute a slip system. A critical resolved shear stress is required to initiate a slip. Slip is an important mode of deformation mechanism in crystals. For metals and technically used metallic alloys it is by far the most important deformation mechanism and subject to current research in materials science.

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 solid state physics, a **superstructure** is some additional structure that is superimposed on a given crystalline structure. A typical and important example is ferromagnetic ordering.

In crystallography, the **hexagonal crystal family** is one of the 6 crystal families, which includes 2 crystal systems and 2 lattice systems.

**Zone axis**, a term sometimes used to refer to "high-symmetry" orientations in a crystal, most generally refers to *any* direction referenced to the direct lattice of a crystal in three dimensions. It is therefore indexed with direct lattice-indices, instead of with Miller-indices.

**Geometrically necessary dislocations** are like-signed dislocations needed to accommodate for plastic bending in a crystalline material. They are present when a material's plastic deformation is accompanied by internal plastic strain gradients. They are in contrast to statistically stored dislocations, with statistics of equal positive and negative signs, which arise during plastic flow from multiplication processes like the Frank-Read source.

- ↑ Petrenko, V. F.; Whitworth, R. W. (1999).
*Physics of Ice*. Oxford University Press. ISBN 9780198518945. - ↑ Bernal, J. D.; Fowler, R. H. (1933). "A Theory of Water and Ionic Solution, with Particular Reference to Hydrogen and Hydroxyl Ions".
*The Journal of Chemical Physics*.**1**(8): 515. Bibcode:1933JChPh...1..515B. doi:10.1063/1.1749327. - 1 2 3 Hook, J.R.; Hall, H.E. (2010).
*Solid State Physics*. Manchester Physics Series (2nd ed.). John Wiley & Sons. ISBN 9780471928041. - ↑ West, Anthony R. (1999).
*Basic Solid State Chemistry*(2nd ed.). Wiley. p. 1. ISBN 978-0-471-98756-7. - ↑ International Tables for Crystallography (2006). Volume A, Space-group symmetry.
- 1 2 3 Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
- ↑ "4. Direct and reciprocal lattices".
*CSIC Dept de Cristalografia y Biologia Estructural*. 6 Apr 2017. Retrieved 18 May 2017. - ↑ Ashcroft, N.; Mermin, D. (1976). "Chapter 7".
*Solid State Physics*. Brooks/Cole (Thomson Learning, Inc.). ISBN 978-0030493461. - 1 2 Donald E. Sands (1994). "§4-2 Screw axes and §4-3 Glide planes".
*Introduction to Crystallography*(Reprint of WA Benjamin corrected 1975 ed.). Courier-Dover. pp. 70–71. ISBN 978-0486678399. - 1 2 3 4 Parker, C.B., ed. (1994).
*McGraw Hill Encyclopaedia of Physics*(2nd ed.). ISBN 978-0070514003. - ↑ Kallay, Nikola (2000).
*Interfacial Dynamics*. CRC Press. ISBN 978-0824700065. - ↑ Hogan, C. M. (1969). "Density of States of an Insulating Ferromagnetic Alloy".
*Physical Review*.**188**(2): 870–874. Bibcode:1969PhRv..188..870H. doi:10.1103/PhysRev.188.870. - ↑ Zhang, X. Y.; Suhl, H (1985). "Spin-wave-related period doublings and chaos under transverse pumping".
*Physical Review A*.**32**(4): 2530–2533. Bibcode:1985PhRvA..32.2530Z. doi:10.1103/PhysRevA.32.2530. PMID 9896377. - ↑ Courtney, Thomas (2000).
*Mechanical Behavior of Materials*. Long Grove, IL: Waveland Press. p. 85. ISBN 978-1-57766-425-3. - ↑ L. Pauling (1929). "The principles determining the structure of complex ionic crystals".
*J. Am. Chem. Soc.***51**(4): 1010–1026. doi:10.1021/ja01379a006. - ↑ Pauling, Linus (1938). "The Nature of the Interatomic Forces in Metals".
*Physical Review*.**54**(11): 899–904. Bibcode:1938PhRv...54..899P. doi:10.1103/PhysRev.54.899. - ↑ Pauling, Linus (1947). "Atomic Radii and Interatomic Distances in Metals".
*Journal of the American Chemical Society*.**69**(3): 542–553. doi:10.1021/ja01195a024. - ↑ Pauling, L. (1949). "A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds".
*Proceedings of the Royal Society A*.**196**(1046): 343–362. Bibcode:1949RSPSA.196..343P. doi:10.1098/rspa.1949.0032. - ↑ Hume-rothery, W.; Irving, H. M.; Williams, R. J. P. (1951). "The Valencies of the Transition Elements in the Metallic State".
*Proceedings of the Royal Society A*.**208**(1095): 431. Bibcode:1951RSPSA.208..431H. doi:10.1098/rspa.1951.0172. - ↑ Altmann, S. L.; Coulson, C. A.; Hume-Rothery, W. (1957). "On the Relation between Bond Hybrids and the Metallic Structures".
*Proceedings of the Royal Society A*.**240**(1221): 145. Bibcode:1957RSPSA.240..145A. doi:10.1098/rspa.1957.0073. - ↑ Liu, Gang (2015). "Dynamical equations for the period vectors in a periodic system under constant external stress".
*Can. J. Phys.***93**(9): 974–978. arXiv: cond-mat/0209372 . Bibcode:2015CaJPh..93..974L. doi:10.1139/cjp-2014-0518. - ↑ Molodets, A. M.; Nabatov, S. S. (2000). "Thermodynamic Potentials, Diagram of State, and Phase Transitions of Tin on Shock Compression".
*High Temperature*.**38**(5): 715–721. doi:10.1007/BF02755923. - ↑ Holleman, Arnold F.; Wiberg, Egon; Wiberg, Nils (1985). "Tin".
*Lehrbuch der Anorganischen Chemie*(in German) (91–100 ed.). Walter de Gruyter. pp. 793–800. ISBN 978-3-11-007511-3. - ↑ Schwartz, Mel (2002). "Tin and Alloys, Properties".
*Encyclopedia of Materials, Parts and Finishes*(2nd ed.). CRC Press. ISBN 978-1-56676-661-6.

- The internal structure of crystals... Crystallography for beginners
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- Intro to Minerals: Crystal Class and System
- Introduction to Crystallography and Mineral Crystal Systems
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- Interactive 3D Crystal models
- Specific Crystal 3D models
- Crystallography Open Database (with more than 140,000 crystal structures)

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