The unit cell completely defines the symmetry and structure of the entire crystal lattice, which is built up by repetitive translation of the unit cell along its principal axes. The repeating patterns are said to be located at the points of the Bravais lattice. 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.
The symmetry properties of the crystal are described by the concept of space groups. The crystal structure and symmetry play a critical role in determining many physical properties, such as cleavage , electronic band structure , and optical transparency. 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.
The positions of particles inside the unit cell are described by the fractional coordinates x i , y i , z i along the cell edges, measured from a reference point. It is only necessary to report the coordinates of a smallest asymmetric subset of particles. This group of particles 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 the space group of the crystal structure. Vectors and planes in a crystal lattice are described by the three-value Miller index notation.
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. 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 1 2 3. 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. 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.
The minerals are classified into groups based on these structures. The characteristic rotation and mirror symmetries of the unit cell is described by its crystallographic point group. 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. Grain boundaries are interfaces where crystals of different orientations meet. In nanocrystalline solids, grain boundaries become a significant volume fraction of the material, with profound effects on such properties as diffusion and plasticity.
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. 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.
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:. 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 hkl lattice planes is given by: 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. The crystal is then said to have a twofold rotational symmetry about this axis.
A full classification of a crystal is achieved when all of these inherent symmetries of the crystal are identified. 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. 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 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,  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. In total there are seven crystal systems: 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.
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.
By considering the arrangement of atoms relative to each other, their coordination numbers or number of nearest neighbors , interatomic distances, types of bonding, etc. 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:.
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:. The unit cell of a ccp arrangement of atoms is the face-centered cubic 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:.
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. This can be compared to the APF of a bcc structure, which is 0. Grain boundaries are interfaces where crystals of different orientations meet. 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 7 crystal systems consist of 32 crystal classes corresponding to the 32 crystallographic point groups as shown in the following table:. Point symmetry can be thought of in the following fashion: This is the 'inverted structure'.
If the original structure and inverted structure are identical, then the structure is centrosymmetric. Otherwise it is non-centrosymmetric. Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure.
This is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral enantiomorphic and its symmetry group is enantiomorphic. A direction meaning a line without an arrow is called polar if its two directional senses are geometrically or physically different.
A polar symmetry [ clarification needed ] direction of a crystal is called a polar axis. A polar crystal possess a "unique" axis found in no other directions such that some geometrical or physical property is different at the two ends of this axis. It may develop a dielectric polarization , e.
A polar axis can occur only in non-centrosymmetric structures. There should also not be a mirror plane or twofold axis perpendicular to the polar axis, because they will make both directions of the axis equivalent. The crystal structures of chiral biological molecules such as protein structures can only occur in the 65 enantiomorphic space groups biological molecules are usually chiral. The distribution of the 14 Bravais lattices into lattice systems and crystal families is given in the following table.
In geometry and crystallography , a Bravais lattice is a category of symmetry groups for translational symmetry in three directions, or correspondingly, a category of translation lattices. These lattices are classified by space group of the translation lattice itself; there are 14 Bravais lattices in three dimensions; each can apply in one lattice system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.
All crystalline materials must, by definition fit in one of these arrangements not including quasicrystals. For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3 or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor The Bravais lattices were studied by Moritz Ludwig Frankenheim in , who found that there were 15 Bravais lattices.
This was corrected to 14 by A. The following conditions for the lattice parameters define 23 crystal families.
The names here are given according to Whittaker. The names for these three families according to Brown et al are given in parenthesis. The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table. The number of enantiomorphic pairs are given in parentheses. Here the term "enantiomorphic" has a different meaning than in the table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral enantiomorphic structures. In the current table, "enantiomorphic" means that a group itself considered as a geometric object is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P3 1 and P3 2 , P4 1 22 and P4 3