Crystal systems are the foundation of solid-state physics, categorizing how atoms arrange in crystals based on symmetry. They're key to predicting material properties and understanding crystal structures at the atomic level.
Bravais lattices, crystal symmetry, and the seven crystal systems form the for describing crystal structures. These concepts help us analyze everything from structures to complex materials used in modern technology.
Crystal systems
Crystal systems categorize the geometric arrangement of atoms in a crystal based on their symmetry properties
Understanding crystal systems is crucial for predicting and analyzing the physical properties of solid materials
Bravais lattices
Bravais lattices are the 14 unique three-dimensional lattice types that describe the periodic arrangement of atoms in a crystal
Each Bravais lattice is characterized by its unit cell, which is the smallest repeating unit that can generate the entire crystal structure through translation
Primitive vs non-primitive cells
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Primitive cells contain only one lattice point and are the smallest possible unit cells for a given crystal structure
Non-primitive cells contain more than one lattice point and are often used for convenience in describing crystal structures
Examples of non-primitive cells include body-centered (2 lattice points) and face-centered (4 lattice points) unit cells
Crystal symmetry
Crystal symmetry describes the various ways in which a crystal structure can be transformed without changing its appearance
Symmetry plays a crucial role in determining the physical properties of crystals, such as optical, electrical, and mechanical properties
Translation vs point symmetry
Translation symmetry involves the repetition of a unit cell throughout the crystal lattice
Point symmetry refers to the symmetry operations that leave at least one point in the crystal structure unchanged
Examples of point symmetry include , , and
Symmetry elements
Symmetry elements are geometric entities (points, lines, or planes) about which symmetry operations are performed
Examples of symmetry elements include rotation axes, mirror planes, and inversion centers
Symmetry operations
Symmetry operations are the specific transformations that leave the crystal structure unchanged
Examples of symmetry operations include rotation, reflection, inversion, and translation
Seven crystal systems
The seven crystal systems are , , , , , , and
Each crystal system is characterized by its unique set of symmetry elements and
Triclinic system
The triclinic system has the lowest symmetry, with no constraints on the lattice parameters (a=b=c, α=β=γ=90°)
An example of a triclinic crystal is copper(II) sulfate pentahydrate (CuSO4·5H2O)
Monoclinic system
The monoclinic system has one twofold rotation axis or one mirror plane, with lattice parameters a=b=c, α=γ=90°, β=90°
An example of a monoclinic crystal is gypsum (CaSO4·2H2O)
Orthorhombic system
The orthorhombic system has three mutually perpendicular twofold rotation axes or mirror planes, with lattice parameters a=b=c, α=β=γ=90°
An example of an orthorhombic crystal is olivine ((Mg,Fe)2SiO4)
Tetragonal system
The tetragonal system has one fourfold rotation axis, with lattice parameters a=b=c, α=β=γ=90°
An example of a tetragonal crystal is rutile (TiO2)
Trigonal system
The trigonal system has one threefold rotation axis, with lattice parameters a=b=c, α=β=γ=90°
An example of a trigonal crystal is quartz (SiO2)
Hexagonal system
The hexagonal system has one sixfold rotation axis, with lattice parameters a=b=c, α=β=90°, γ=120°
An example of a hexagonal crystal is graphite (C)
Cubic system
The cubic system has the highest symmetry, with four threefold rotation axes, with lattice parameters a=b=c, α=β=γ=90°
Examples of cubic crystals include (NaCl) and diamond (C)
Lattice parameters
Lattice parameters describe the size and shape of the unit cell in a crystal structure
They include the (a, b, c) and (α, β, γ)
Axial lengths
Axial lengths (a, b, c) represent the lengths of the three edges of the unit cell
They are typically measured in angstroms (Å) or nanometers (nm)
Interaxial angles
Interaxial angles (α, β, γ) represent the angles between the three axes of the unit cell
They are measured in degrees (°)
Crystal planes
are imaginary planes that intersect the crystal lattice at specific points
They are described using and are important for understanding the growth, cleavage, and defects in crystals
Miller indices
Miller indices (hkl) are a set of three integers that describe the orientation of a crystal plane relative to the unit cell axes
They are determined by finding the reciprocals of the fractional intercepts of the plane with the unit cell axes and clearing fractions
Families of planes
are sets of crystal planes that are equivalent by symmetry
They are denoted by enclosing the Miller indices in curly brackets, such as hkl
Reciprocal lattice
The is a mathematical construct that represents the Fourier transform of the real space lattice
It is useful for describing diffraction patterns and electronic band structures in crystals
Relationship to real space lattice
The reciprocal lattice vectors (a∗, b∗, c∗) are related to the real space lattice vectors (a, b, c) by:
a∗=a⋅(b×c)2π(b×c)
b∗=a⋅(b×c)2π(c×a)
c∗=a⋅(b×c)2π(a×b)
Wigner-Seitz cell
The is a constructed by drawing perpendicular bisector planes between a lattice point and its nearest neighbors
It represents the volume of space that is closer to a given lattice point than any other lattice point
First Brillouin zone
The is the Wigner-Seitz cell of the reciprocal lattice
It is important for describing the electronic band structure and phonon dispersion in crystals
Space groups
Space groups are the complete set of symmetry operations that describe the symmetry of a crystal structure
There are 230 unique space groups in three dimensions
Screw axes
are symmetry elements that combine a rotation with a translation along the axis of rotation
They are denoted by a number n (representing the rotation angle 360°/n) followed by a subscript m (representing the translation distance m/n times the lattice parameter along the axis)
Glide planes
are symmetry elements that combine a reflection with a translation parallel to the plane of reflection
They are denoted by a letter (a, b, c, n, or d) indicating the direction of the translation, followed by a subscript indicating the fraction of the lattice parameter translated
Examples of crystal structures
Crystal structures are the specific arrangements of atoms in a crystal, described by the lattice type and the positions of atoms within the unit cell
Some common crystal structures include simple cubic, , , , and
Simple cubic
Simple cubic is a primitive cubic structure with atoms at each corner of the unit cell
An example of a simple cubic crystal is polonium (Po)
Body-centered cubic (BCC)
is a non-primitive cubic structure with atoms at each corner and one atom at the center of the unit cell
Examples of BCC crystals include iron (Fe) at room temperature and tungsten (W)
Face-centered cubic (FCC)
is a non-primitive cubic structure with atoms at each corner and one atom at the center of each face of the unit cell
Examples of FCC crystals include copper (Cu), aluminum (Al), and gold (Au)
Hexagonal close-packed (HCP)
Hexagonal close-packed is a primitive hexagonal structure with atoms arranged in a close-packed configuration
Examples of HCP crystals include magnesium (Mg) and zinc (Zn)
Diamond cubic
Diamond cubic is a face-centered cubic structure with additional atoms at (1/4,1/4,1/4) and (3/4,3/4,3/4) positions within the unit cell
Examples of diamond cubic crystals include diamond (C) and silicon (Si)
Zinc blende
is a face-centered cubic structure with two atom types, one at (0,0,0) and the other at (1/4,1/4,1/4)
An example of a zinc blende crystal is gallium arsenide (GaAs)
Sodium chloride
Sodium chloride is a face-centered cubic structure with two atom types, one at (0,0,0) and the other at (1/2,1/2,1/2)
An example of a sodium chloride crystal is sodium chloride (NaCl)
Cesium chloride
is a primitive cubic structure with two atom types, one at (0,0,0) and the other at (1/2,1/2,1/2)
An example of a cesium chloride crystal is cesium chloride (CsCl)