1.3 Crystal systems

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 basis for describing crystal structures. These concepts help us analyze everything from simple cubic 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 rotation, reflection, and inversion

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 triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic
• Each crystal system is characterized by its unique set of symmetry elements and lattice parameters

Triclinic system

• The triclinic system has the lowest symmetry, with no constraints on the lattice parameters ($a \neq b \neq c$, $\alpha \neq \beta \neq \gamma \neq 90°$)
• 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 $a \neq b \neq c$, $\alpha = \gamma = 90°$, $\beta \neq 90°$
• 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 $a \neq b \neq c$, $\alpha = \beta = \gamma = 90°$
• An example of an orthorhombic crystal is olivine ((Mg,Fe)2SiO4)

Tetragonal system

• The tetragonal system has one fourfold rotation axis, with lattice parameters $a = b \neq c$, $\alpha = \beta = \gamma = 90°$
• An example of a tetragonal crystal is rutile (TiO2)

Trigonal system

• The trigonal system has one threefold rotation axis, with lattice parameters $a = b = c$, $\alpha = \beta = \gamma \neq 90°$
• An example of a trigonal crystal is quartz (SiO2)

Hexagonal system

• The hexagonal system has one sixfold rotation axis, with lattice parameters $a = b \neq c$, $\alpha = \beta = 90°$, $\gamma = 120°$
• 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 $a = b = c$, $\alpha = \beta = \gamma = 90°$
• Examples of cubic crystals include sodium chloride (NaCl) and diamond (C)

Lattice parameters

• Lattice parameters describe the size and shape of the unit cell in a crystal structure
• They include the axial lengths ($a$, $b$, $c$) and interaxial angles ($\alpha$, $\beta$, $\gamma$)

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 ($\alpha$, $\beta$, $\gamma$) represent the angles between the three axes of the unit cell
• They are measured in degrees (°)

Crystal planes

• Crystal planes are imaginary planes that intersect the crystal lattice at specific points
• They are described using Miller indices 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

• 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 reciprocal lattice 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 ($\vec{a}^*$, $\vec{b}^*$, $\vec{c}^*$) are related to the real space lattice vectors ($\vec{a}$, $\vec{b}$, $\vec{c}$) by:
  • $\vec{a}^* = \frac{2\pi(\vec{b} \times \vec{c})}{\vec{a} \cdot (\vec{b} \times \vec{c})}$
  • $\vec{b}^* = \frac{2\pi(\vec{c} \times \vec{a})}{\vec{a} \cdot (\vec{b} \times \vec{c})}$
  • $\vec{c}^* = \frac{2\pi(\vec{a} \times \vec{b})}{\vec{a} \cdot (\vec{b} \times \vec{c})}$

Wigner-Seitz cell

• The Wigner-Seitz cell is a primitive cell 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 first Brillouin zone 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

• 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

• 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, body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HCP), and diamond cubic

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)

• Body-centered cubic 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)

• Face-centered cubic 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

• 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

• 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)