Everything about Unit Cell totally explained
In
mineralogy and
crystallography, a
crystal structure is a unique arrangement of
atoms in a
crystal. A crystal structure is composed of a motif, a set of
atoms arranged in a particular way, and a lattice. Motifs 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 properties, such as
cleavage, electronic
band structure, and
optical properties.
Unit cell
The crystal structure of a material or the arrangement of atoms in a crystal structure can be described in terms of its unit cell. The unit cell is a tiny box containing one or more motifs, a spatial arrangement of
atoms. The units cells
stacked in three-dimensional space describes the bulk arrangement of atoms of the crystal. The unit cell is given by its
lattice parameters, 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
measured from a lattice point.
Although there are an infinite number of ways to specify a unit cell, for each crystal structure there's a
conventional unit cell, which is chosen to display the
full symmetry of the crystal (see below). However, the conventional unit cell isn't always the smallest possible choice.
A
primitive unit cell of a particular crystal structure is the smallest possible volume one can construct with the arrangement of atoms in the crystal such that, when stacked, completely fills the space. This primitive unit cell doesn't always display all the symmetries inherent in the crystal. A
Wigner-Seitz cell is a particular kind of
primitive cell which has the same symmetry as the lattice.
In a unit cell each atom has an identical environment when stacked in 3 dimensional space. In a primitive cell, each atom may not have the same environment.
There are only
seven possible crystal systems that atoms can pack together to produce an infinite 3D space lattice in such a way that each lattice point has an identical environment to that around every other lattice point.
Classification of crystals by symmetry
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 degrees about a certain axis may result in an atomic configuration which 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.
Crystal system
The 7 Crystal systems
(Defining Symmetry) |
The 14 Bravais Lattices: |
triclinic
(none) |
|
monoclinic
(1 diad) |
simple |
base-centered |
|
|
orthorhombic
(3 perpendicular diads) |
simple |
base-centered |
body-centered |
face-centered |
|
|
|
|
hexagonal
(1 hexad) |
|
rhombohedral
(1 triad) |
|
tetragonal
(1 tetrad) |
simple |
body-centered |
|
|
cubic
(4 triads) |
simple |
body-centered |
face-centered |
|
|
|
The
crystal systems are a grouping of crystal structures according to the axial system used to describe their lattice. Each crystal system consists of a set of three axes in a particular geometrical arrangement. There are seven unique 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 109.5 degrees (the tetrahedral angle) with respect to each other. These threefold axes lie along the body diagonals of the cube. This definition of a cubic is correct, although many textbooks incorrectly state that a cube is defined by three mutually perpendicular axes of equal length – if this were true there would be far more than 14
Bravais lattices. The other six systems, in order of decreasing symmetry, are
hexagonal,
tetragonal,
rhombohedral (also known as trigonal),
orthorhombic,
monoclinic and
triclinic. Some crystallographers consider the hexagonal crystal system not to be its own crystal system, but instead a part of the trigonal crystal system. The crystal system and Bravais lattice of a crystal describe the (purely) translational symmetry of the crystal.
Angles of the 7 basic crystalline structures
triclinic = 100 and 80 degrees,
monoclinic = 100, 80 and 90 degrees,
orthorhombic = 90 degrees,
hexagonal = 90 and 120 degrees,
rhombohedral = 100 and 80 degrees,
tetragonal = 90 degrees,
cubic = 90 degrees
The 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 which 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 three-dimensional lattices, classified by crystal system, are shown to the right. 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 and space 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 can 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 and
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.
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 axis, which rotate a point around an axis while translating parallel to the axis, and
glide planes, which reflect a point through a plane while translating it parallel to the plane. There are 230 distinct space groups.
Physical properties
Defects or impurities in crystals
Real crystals feature
defects or irregularities in the ideal arrangements described above and it's 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 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.
Dislocations in the crystal lattice allow
shear at lower stress than that needed for a perfect crystal structure.
Crystal symmetry and physical properties
Twenty of the 32 crystal classes are so-called
piezoelectric, and crystals belonging to one of these classes (point groups) display
piezoelectricity. All 21 piezoelectric classes lack
a center of symmetry. Any material develops a
dielectric polarization
when an electric field is applied, but a substance which 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 behaviour. This is analogous to
ferromagnetism, in that, in the absence of an electric field during production, the ferroelectric crystal doesn't exhibit a polarisation. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarised. This polarisation can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnet can be reversed. However, it's important to note that, although they're called ferroelectrics, the effect is due to the crystal structure, not the presence of a ferrous metal.
Incommensurate crystals have period-varying translational symmetry. The period between nodes of symmetry is constant in most crystals. The distance between nodes in an incommensurate crystal is dependent on the number of nodes between it and the base node.
Further Information
Get more info on 'Unit Cell'.
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