Crystal structure

Incrystallography,crystal structureis a description of ordered arrangement ofatoms,ions, ormoleculesin acrystalline material.Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions ofthree-dimensional spacein matter.
The smallest group of particles in material that constitutes this repeating pattern is theunit cellof the structure. The unit cell completely reflects symmetry and structure of the entire crystal, which is built up by repetitivetranslationof unit cell along its principal axes. The translation vectors define the nodes ofBravais lattice.
The lengths of principal axes/edges, of unit cell and angles between them arelattice constants, also calledlattice parametersorcell parameters. Thesymmetryproperties of crystal are described by the concept ofspace groups.All possible symmetric arrangements of particles in three-dimensional space may be described by 230 space groups.
The crystal structure and symmetry play a critical role in determining many physical properties, such ascleavage,electronic band structure, andoptical transparency.
Crystal structure is described in terms of the geometry of arrangement of particles in the unit cells. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure.The geometry of the unit cell is defined as aparallelepiped, 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 thefractional coordinates(xi,yi,zi) along the cell edges, measured from a reference point. It is thus only necessary to report the coordinates of a smallest asymmetric subset of particles, called the crystallographic asymmetric unit. The asymmetric unit may be chosen so that it occupies the smallest physical space, which means that not all particles need to be physically located inside the boundaries given by the lattice parameters. All other particles of the unit cell are generated by the symmetry operations that characterize the symmetry of the unit cell. The collection of symmetry operations of the unit cell is expressed formally as thespace groupof the crystal structure.
Vectors and planes in a crystal lattice are described by the three-valueMiller indexnotation. This syntax uses the indicesh,k, andℓas directional parameters.
By definition, the syntax (hkℓ) denotes a plane that intercepts the three pointsa1/h,a2/k, anda3/ℓ, 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 areintegerswith 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 (hkℓ) planes intersecting one or more lattice points (thelattice planes), the distancedbetween adjacent lattice planes is related to the (shortest)reciprocal latticevector orthogonal to the planes by the formula
The crystallographic directions are geometriclineslinking nodes (atoms,ionsormolecules) of a crystal. Likewise, the crystallographicplanesare geometricplaneslinking 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:
Some directions and planes are defined by symmetry of the crystal system. In monoclinic, trigonal, tetragonal, and hexagonal systems there is one unique axis (sometimes called theprincipal axis) which has higherrotational symmetrythan the other two axes. Thebasal planeis 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 denoteda); similarly for the reciprocal lattice. So, in this common case, the Miller indices (ℓmn) and [ℓmn] both simply denote normals/directions inCartesian coordinates. For cubic crystals withlattice constanta, the spacingdbetween 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:
Forface-centered cubic(fcc) andbody-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 cubicsupercelland hence are again simply theCartesian directions.
The spacingdbetween adjacent (hkℓ) lattice planes is given by:
The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on the crystal lattice leaves it unchanged. All crystals havetranslational symmetryin 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 has twofold rotational symmetry about this axis. In addition to rotational symmetry, a crystal may have symmetry in the form of mirror planes, and also the so-called compound symmetries, which are a combination of translation and rotation or mirror symmetries. A full classification of a crystal is achieved when all inherent symmetries of the crystal are identified.
Lattice systems are a grouping of crystal structures according to the point groups of their lattice. All crystals fall into one of seven lattice systems. They are related to, but not the same as the sevencrystal systems.
The most symmetric, thecubicor isometric system, has the symmetry of acube, that is, it exhibits four threefold rotational axes oriented at 109.5° (thetetrahedral angle) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, arehexagonal,tetragonal,rhombohedral(often confused with thetrigonal crystal system),orthorhombic,monoclinicandtriclinic.
Bravais lattices, also referred to asspace lattices, describe the geometric arrangement of the lattice points,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 includingquasicrystals, 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, thebasis, 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 itscrystallographic 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.
In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.
Thecrystallographic point grouporcrystal classis 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
Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively calledsymmetry 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, thespace groupof the crystal structure contains translational symmetry operations. These include:
There are 230 distinct space groups.
By considering the arrangement of atoms relative to each other, their coordination numbers, interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing them.
The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stackingclose-packedatomic 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 were placed directly over plane A, this would give rise to the following series:
This arrangement of atoms in a crystal structure is known ashexagonal 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:
This type of structural arrangement is known ascubic 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.
One important characteristic of a crystalline structure is its atomic packing factor (APF). This is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres which can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows:
Another important characteristic of a crystalline structure is its coordination number (CN). This is the number of nearest neighbours of a central atom in the structure.
The APFs and CNs of the most common crystal structures are shown below:
The 74% packing efficiency of the FCC and HCP is the maximum density possible in unit cells constructed of spheres of only one size.
Interstitial sites refer to the empty spaces in between the atoms in the crystal lattice. These spaces can be filled by oppositely charged ions to form multi-element structures. They can also be filled by impurity atoms or self-interstitials to forminterstitial defects.
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.Impurities may also manifest aselectron spinimpurities 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 semiconductingferromagneticalloysmay lead to different properties as first predicted in the late 1960s.
Dislocations in a crystal lattice are line defects that are associated with local stress fields. Dislocations allowshearat lower stress than that needed for a perfect crystal structure.The local stress fields result in interactions between the dislocations which then result in strain hardening orcold working.
Grain boundaries are interfaces where crystals of different orientations meet.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 asingle crystalcut 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 theHall–Petchrelationship. Since grain boundaries are defects in the crystal structure they tend to decrease theelectricalandthermal conductivityof 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 theprecipitationof new phases from the solid. They are also important to many of the mechanisms ofcreep.
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 asdiffusionandplasticity. 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 anamorphous solid.
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 asevolutionary algorithms, random sampling, or metadynamics.
The crystal structures of simple ionic solids (e.g., NaCl or table salt) have long been rationalized in terms ofPauling's rules, first set out in 1929 byLinus Pauling, referred to by many since as the "father of the chemical bond".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. Pauling was therefore able to correlate the number of d-orbitals in bond formation with the bond length, as well as with 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.
In theresonating 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 theaxial 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.
After postulating a direct correlation between electron concentration and crystal structure in beta-phase alloys,Hume-Rotheryanalyzed 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.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.
In crystal structure predictions/simulations, the periodicity is usually applied, since the system is imagined as being unlimited 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(lattice parameters including angles), even if the system is subject to external stress.
Polymorphismis the occurrence of multiple crystalline forms of a material. It is found in many crystalline materials includingpolymers,minerals, andmetals.  According to Gibbs' rules of phase equilibria, these unique crystalline phases are dependent on intensive variables such as pressure and temperature.  Polymorphism is related toallotropy, which refers toelemental solids. The complete morphology of a material is described by polymorphism and other variables such ascrystal habit,amorphous fractionorcrystallographic defects. Polymorphs have different stabilities and may spontaneously and irreversibly transform from a metastable form (or thermodynamically unstable form) to thestableform at a particular temperature.They also exhibit differentmelting points, solubilities, andX-ray diffractionpatterns.
One good example of this is thequartzform ofsilicon dioxide, or SiO2. In the vast majority ofsilicates, the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral {SiO4} 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 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 {SiO4} tetrahedra are shared with others, yielding the net chemical formula for silica: SiO2.
Another example is elementaltin(Sn), which is malleable near ambient temperatures but isbrittlewhen cooled. This change in mechanical properties due to existence of its two majorallotropes, α- and β-tin. The twoallotropesthat are encountered at normal pressure and temperature, α-tin and β-tin, are more commonly known asgray tinandwhite tinrespectively. Two more allotropes, γ and σ, exist at temperatures above 161 °C and pressures above several GPa.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 adiamond cubiccrystal structure, similar todiamond,siliconorgermanium. Gray tin has no metallic properties at all, is a dull gray powdery material, and has few uses, other than a few specializedsemiconductorapplications.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.
Twenty of the 32 crystal classes arepiezoelectric, and crystals belonging to one of these classes (point groups) displaypiezoelectricity. All piezoelectric classes lackinversion symmetry. Any material develops adielectricpolarization 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 arepolar. All polar crystals arepyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.
There are a few crystal structures, notably theperovskite structure, which exhibitferroelectricbehavior. This is analogous toferromagnetism, 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).
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