In
mineralogy and
crystallography,
crystal structure is a unique arrangement of
atoms or
molecules in a
crystalline liquid or
solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting longrange order and symmetry. Patterns are located upon the points of a
lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the
lattice parameters. The
symmetry properties of the crystal are embodied in its
space group.
A crystal's structure and symmetry play a role in determining many of its physical properties, such as
cleavage,
electronic band structure, and
optical transparency.
Unit cell
The crystal structure of a material or the arrangement of atoms within a given type of crystal structure can be described in terms of its unit cell. The unit cell is a small box containing one or more atoms, a spatial arrangement of atoms. The unit cells
stacked in threedimensional space describe the bulk arrangement of atoms of the crystal. The crystal structure has a threedimensional shape. The unit cell is given by its
lattice parameters, which are the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (
x_{i} , y_{i} , z_{i}) measured from a lattice point.
Miller indices
Main article:
Miller index
Planes with different Miller indices in cubic crystals
Vectors and atomic planes in a crystal lattice can be described by a threevalue
Miller index notation (
ℓmn). The
ℓ,
m, and
n directional indices are separated by 90°, and are thus orthogonal. In fact, the
ℓ component is mutually perpendicular to the
m and
n indices.
By definition, (
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 simply means that the planes do not intersect that axis (i.e., the intercept is "at infinity").
Considering only (
ℓmn) planes intersecting one or more lattice points (the
lattice planes), the perpendicular distance
d between adjacent lattice planes is related to the (shortest)
reciprocal lattice vector orthogonal to the planes by the formula:
Planes and directions
The crystallographic directions are fictitious
lines linking nodes (
atoms,
ions or
molecules) of a crystal. Likewise, the crystallographic
planes are fictitious
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:
 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.

Dense crystallographic 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.
In the rhombohedral, hexagonal, and tetragonal systems, the
basal plane is the plane perpendicular to the principal axis.
Cubic structures
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
facecentered cubic (fcc) and
bodycentered 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.
Classification
The defining property of a crystal is its inherent symmetry, by which we mean that under certain 'operations' the crystal remains unchanged. 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 socalled "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.
^{[1]}
Lattice systems
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 geometrical arrangement. There are seven lattice systems. They are similar to but not quite the same as the seven
crystal systems and the six
crystal families.
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.
Atomic coordination
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 then.
HCP lattice (left) and the fcc lattice (right).
Close packing
The principles involved can be understood by considering the most efficient way of packing together equalsized spheres and stacking
closepacked 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 type of 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 crystal structure is known as
cubic close packing (ccp)
The unit cell of the ccp arrangement is the facecentered 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
closepacked layers.
The
packing efficiency could be worked out by calculating the total volume of the spheres and dividing that 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 (bodycentered cubic). The
coordination number of hcp and fcc is 12 and its
atomic packing factor (APF) is the number mentioned above, 0.74. The APF of bcc is 0.68 for comparison.
Bravais lattices
When the crystal systems are combined with the various possible lattice centerings, we arrive at the
Bravais lattices. They describe the geometric arrangement of the lattice points, and thereby the translational symmetry of the crystal. In three dimensions, there are 14 unique Bravais lattices that are distinct from one another in the translational symmetry they contain. All crystalline materials recognized until now (not including
quasicrystals) fit in one of these arrangements. The fourteen threedimensional lattices, classified by crystal system, are shown above. The Bravais lattices are sometimes referred to as
space lattices.
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 14 Bravais lattices. The characteristic rotation and mirror symmetries of the group of atoms, or
unit cell, is described by its
crystallographic point group.
Point groups
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.
Space groups
The
space group of the crystal structure is composed of the translational symmetry operations in addition to the operations of the point group. 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
 Glide planes, which reflect a point through a plane while translating it parallel to the plane.
There are 230 distinct space groups.
Grain boundaries
Grain boundaries are interfaces where crystals of different orientations meet. A
grain boundary is a singlephase 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).
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.
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.
Defects and impurities
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.
^{[2]} 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.
^{[3]}^{[4]} Dislocations in the crystal lattice allow
shear at lower stress than that needed for a perfect crystal structure.
^{[5]}
Prediction of structure
Crystal structure of sodium chloride (table salt)
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 highperformance 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".
^{[6]} Pauling also considered the nature of the interatomic forces in metals, and concluded that about half of the five dorbitals in the transition metals are involved in bonding, with the remaining nonbonding dorbitals being responsible for the magnetic properties. He, therefore, was able to correlate the number of dorbitals 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.
^{[7]}
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.
^{[8]}^{[9]}
After postulating a direct correlation between electron concentration and crystal structure in betaphase alloys,
HumeRothery 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.
^{[10]} 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 delectrons per hybridized metallic orbital. The “dweight” calculates out to 0.5, 0.7 and 0.9 for the fcc, hcp and bcc structures respectively. The relationship between delectrons and crystal structure thus becomes apparent.
^{[11]}
Polymorphism
Polymorphism refers to the ability of a solid to exist in more than one crystalline form or structure. According to Gibbs' rules of phase equilibria, these unique crystalline phases will be dependent on intensive variables such as pressure and temperature. Polymorphism can potentially be found in many crystalline materials including
polymers,
minerals, and
metals, and 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
Xray 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 threedimensional frameworks. The minerals are classified into groups based on these structures. In each of its 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 βtin. The two
allotropes that are encountered at normal pressure and temperature, αtin and βtin, 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.
^{[12]} 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 dullgray powdery material, and has few uses, other than a few specialized
semiconductor applications.
^{[13]} Although the αβ 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.
^{[14]}
Physical properties
Twenty of the 32 crystal classes are socalled
piezoelectric, and crystals belonging to one of these classes (point groups) display
piezoelectricity. All piezoelectric classes lack a centre 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 countercharge, in the same way that a ferromagnet can be reversed. However, it is important to note that, although they are called ferroelectrics, the effect is due to the crystal structure (not the presence of a ferrous metal).
See also
 For more detailed information in specific technology applications see Materials science, Ceramic engineering, or Metallurgy.
References
 ^ Ashcroft, N.; Mermin, D. (1976) Solid State Physics, Brooks/Cole (Thomson Learning, Inc.), Chapter 7, ISBN 0030493463
 ^ Nikola Kallay (2000) Interfacial Dynamics, CRC Press, ISBN 0824700066
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 ^ Courtney, Thomas (2000). Mechanical Behavior of Materials. Long Grove, IL: Waveland Press. pp. 85. ISBN 1577664256.
 ^ 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. Bibcode 1938PhRv...54..899P. doi:10.1103/PhysRev.54.899.
 ^ Pauling, Linus (1947). Journal of the American Chemical Society 69 (3): 542. doi:10.1021/ja01195a024.
 ^ Pauling, L. (1949). "A ResonatingValenceBond Theory of Metals and Intermetallic Compounds". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (19341990) 196 (1046): 343. Bibcode 1949RSPSA.196..343P. doi:10.1098/rspa.1949.0032.
 ^ Humerothery, W.; Irving, H. M.; Williams, R. J. P. (1951). "The Valencies of the Transition Elements in the Metallic State". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (19341990) 208 (1095): 431. Bibcode 1951RSPSA.208..431H. doi:10.1098/rspa.1951.0172.
 ^ Altmann, S. L.; Coulson, C. A.; HumeRothery, W. (1957). "On the Relation between Bond Hybrids and the Metallic Structures". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1934–1990) 240 (1221): 145. Bibcode 1957RSPSA.240..145A. doi:10.1098/rspa.1957.0073.
 ^ 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" (in German). Lehrbuch der Anorganischen Chemie (91–100 ed.). Walter de Gruyter. pp. 793–800. ISBN 3110075113.
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External links