1.1 Bravais lattices

Bravais lattices are the building blocks of crystalline solids, defining how atoms arrange in 3D space. These infinite arrays of points form 14 distinct types, classified into 7 crystal systems based on symmetry and lattice parameters.

Understanding Bravais lattices is crucial for grasping crystal structures and their properties. They provide a framework for describing atomic arrangements, interpreting X-ray diffraction patterns, and analyzing electronic and vibrational properties of materials.

Definition of Bravais lattices

• Bravais lattices are infinite arrays of discrete points that appear exactly the same from every point in the array
• They form the basis for describing the atomic structure of crystalline solids in three dimensions
• The points in a Bravais lattice represent the locations of atoms, ions, or molecules in a crystal

14 types of Bravais lattices

7 crystal systems

Top images from around the web for 7 crystal systems

• 

  Introduction to crystals View original

  Is this image relevant?

• 

  Lattice Structures in Crystalline Solids | Chemistry View original

  Is this image relevant?

• 

  Introduction to crystals View original

  Is this image relevant?

• 

  Introduction to crystals View original

  Is this image relevant?

• 

  Lattice Structures in Crystalline Solids | Chemistry View original

  Is this image relevant?

1 of 3

Top images from around the web for 7 crystal systems

• 

  Introduction to crystals View original

  Is this image relevant?

• 

  Lattice Structures in Crystalline Solids | Chemistry View original

  Is this image relevant?

• 

  Introduction to crystals View original

  Is this image relevant?

• 

  Introduction to crystals View original

  Is this image relevant?

• 

  Lattice Structures in Crystalline Solids | Chemistry View original

  Is this image relevant?

1 of 3

• The 14 Bravais lattices are classified into 7 crystal systems based on their symmetry: triclinic, monoclinic, orthorhombic, tetragonal, cubic, trigonal, and hexagonal
• Each crystal system is defined by the relationships between the lattice parameters (lengths and angles) and the presence of certain symmetry elements (rotation axes, mirror planes, etc.)
• The crystal systems impose constraints on the possible Bravais lattices that can occur within them

Lattice centering

• Bravais lattices can have different types of lattice centering: primitive (P), body-centered (I), face-centered (F), and base-centered (A, B, or C)
• Primitive lattices have lattice points only at the corners of the unit cell, while non-primitive lattices have additional lattice points at the center of the unit cell or on the faces
• The type of lattice centering affects the number of atoms per unit cell and the atomic packing density

Properties of Bravais lattices

Lattice parameters

• Lattice parameters are the lengths (a, b, c) and angles (α, β, γ) that define the size and shape of the unit cell in a Bravais lattice
• The relationships between the lattice parameters determine the crystal system and the symmetry of the lattice
• Lattice parameters can be measured experimentally using techniques such as X-ray diffraction or neutron scattering

Symmetry operations

• Bravais lattices possess various symmetry operations that leave the lattice unchanged, such as translations, rotations, reflections, and inversions
• The set of symmetry operations that a Bravais lattice possesses determines its point group and space group
• Symmetry operations are important for understanding the physical properties of crystals, such as electronic band structure and phonon modes

Primitive vs non-primitive cells

Wigner-Seitz cell

• The Wigner-Seitz cell is a primitive unit cell that contains only one lattice point and is constructed by drawing perpendicular bisector planes between a chosen lattice point and its nearest neighbors
• It represents the smallest repeating unit of the Bravais lattice and has the same symmetry as the lattice
• The Wigner-Seitz cell is useful for visualizing the atomic arrangement and calculating properties such as the atomic packing factor

Bravais lattices in reciprocal space

Reciprocal lattice vectors

• The reciprocal lattice is a Fourier transform of the real-space Bravais lattice and represents the wavevectors of plane waves that have the same periodicity as the lattice
• Reciprocal lattice vectors (a*, b*, c*) are defined as the vectors perpendicular to the planes of the real-space lattice and have magnitudes inversely proportional to the interplanar spacings
• The reciprocal lattice is important for understanding diffraction patterns and electronic band structures in crystalline solids

Importance in crystallography

Relationship to crystal structure

• Bravais lattices provide a mathematical framework for describing the periodic arrangement of atoms in crystalline solids
• The crystal structure of a material can be described as a Bravais lattice with a basis, which is the arrangement of atoms associated with each lattice point
• Understanding the Bravais lattice and crystal structure is essential for predicting and explaining the physical properties of materials

X-ray diffraction patterns

• X-ray diffraction is a powerful technique for determining the Bravais lattice and crystal structure of a material
• The diffraction pattern is a Fourier transform of the electron density in the crystal and consists of a series of peaks that correspond to the reciprocal lattice points
• The positions and intensities of the diffraction peaks provide information about the lattice parameters, symmetry, and atomic positions in the crystal

Bravais lattices vs space groups

Additional symmetry elements

• Space groups are a more complete description of the symmetry of a crystal than Bravais lattices alone
• In addition to the translational symmetry of the Bravais lattice, space groups include point group symmetry elements such as rotations, reflections, and screw axes
• There are 230 unique space groups in three dimensions, which are determined by the combination of the Bravais lattice and the additional symmetry elements

Examples of Bravais lattices

Cubic lattices

• There are three types of cubic Bravais lattices: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC)
• SC lattices have atoms only at the corners of the cube, BCC lattices have an additional atom at the center of the cube, and FCC lattices have additional atoms at the center of each face
• Examples of materials with cubic Bravais lattices include copper (FCC), iron (BCC), and polonium (SC)

Hexagonal lattices

• The hexagonal Bravais lattice is characterized by two equal lattice parameters (a = b) and one unique lattice parameter (c), with angles of 90°, 90°, and 120°
• It has a six-fold rotational symmetry axis along the c-direction and additional mirror planes and two-fold rotation axes
• Examples of materials with hexagonal Bravais lattices include graphite, zinc, and magnesium

Brillouin zones of Bravais lattices

First Brillouin zone

• The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice and represents the smallest repeating unit in reciprocal space
• It is constructed by drawing perpendicular bisector planes between the origin and the nearest reciprocal lattice points
• The first Brillouin zone is important for understanding electronic band structures and phonon dispersion relations in crystalline solids

Higher-order Brillouin zones

• Higher-order Brillouin zones are constructed by extending the perpendicular bisector planes beyond the first Brillouin zone
• They represent regions in reciprocal space that are further away from the origin and have higher wavevectors
• Higher-order Brillouin zones are important for understanding complex band structures and phenomena such as Fermi surface nesting

Applications of Bravais lattices

Electronic band structure

• The electronic band structure of a crystalline solid describes the allowed energy states of electrons as a function of their wavevector in reciprocal space
• The symmetry and periodicity of the Bravais lattice determine the shape and connectivity of the electronic bands
• Understanding the electronic band structure is essential for predicting properties such as electrical conductivity, optical absorption, and magnetism

Phonon dispersion relations

• Phonons are quantized lattice vibrations in a crystalline solid and can be described as waves propagating through the Bravais lattice
• The phonon dispersion relations represent the frequencies of the phonon modes as a function of their wavevector in reciprocal space
• The symmetry and periodicity of the Bravais lattice determine the number and degeneracy of the phonon modes and their dispersion relations
• Phonon dispersion relations are important for understanding thermal properties such as heat capacity and thermal conductivity