1.2 Primitive cells and basis

Primitive cells and crystal basis are fundamental concepts in solid state physics. They describe how atoms arrange themselves in crystalline materials, forming the building blocks of complex structures.

Understanding these concepts is crucial for analyzing material properties. By combining primitive cells with crystal basis, we can describe a wide range of crystal structures, from simple metals to complex semiconductors and exotic materials.

Primitive cells

• Fundamental building blocks of crystal structures that contain one lattice point each
• Primitive cells are the smallest repeating unit that can reproduce the entire crystal lattice through translation operations
• Understanding primitive cells is crucial for analyzing the symmetry and properties of crystalline materials in solid state physics

Definition of primitive cells

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• Smallest volume of space that contains exactly one lattice point
• Can be translated through primitive translation vectors to reproduce the entire crystal lattice
• Choice of primitive cell for a given crystal structure is not unique (can be chosen in different ways as long as it contains one lattice point)

Wigner-Seitz primitive cell

• Constructed by drawing perpendicular bisector planes between a chosen lattice point and its nearest neighbors
• Region enclosed by these planes defines the Wigner-Seitz primitive cell
• Advantageous because it preserves the full symmetry of the lattice (useful for studying electronic properties)

Primitive translation vectors

• Set of linearly independent vectors $\vec{a}_1$, $\vec{a}_2$, $\vec{a}_3$ that can translate the primitive cell to reproduce the entire lattice
• Primitive translation vectors are not unique (different choices possible as long as they span the lattice)
• Examples:
  • Simple cubic lattice: $\vec{a}_1 = a\hat{x}$, $\vec{a}_2 = a\hat{y}$, $\vec{a}_3 = a\hat{z}$ (where $a$ is lattice constant)
  • Face-centered cubic lattice: $\vec{a}_1 = \frac{a}{2}(\hat{y}+\hat{z})$, $\vec{a}_2 = \frac{a}{2}(\hat{x}+\hat{z})$, $\vec{a}_3 = \frac{a}{2}(\hat{x}+\hat{y})$

Crystal basis

• Arrangement of atoms, ions, or molecules associated with each lattice point in a crystal structure
• Basis describes the physical content of the crystal, while the lattice describes the underlying geometric structure
• Understanding the crystal basis is essential for determining the physical properties and symmetry of crystalline materials

Definition of crystal basis

• Set of atoms, ions, or molecules that are attached to each lattice point
• Basis is repeated at every lattice point to generate the complete crystal structure
• Can consist of a single atom (simple basis) or multiple atoms (complex basis)

Relationship between basis and lattice

• Lattice provides the structural framework, while basis fills in the physical content
• Combining the lattice and basis gives rise to the full crystal structure
• Different basis choices on the same lattice can result in different crystal structures with distinct properties

Examples of crystal basis

• Diamond structure: Two identical carbon atoms per lattice point, arranged in a tetrahedral geometry
• Sodium chloride (NaCl) structure: Alternating Na+ and Cl- ions arranged on a face-centered cubic lattice
• Perovskite structure (ABX3): Complex basis with three different types of atoms (A, B, and X) arranged in a specific geometry

Lattice with basis

• Complete description of a crystal structure obtained by combining the primitive cell (lattice) and the basis
• Essential for understanding the physical properties, symmetry, and behavior of crystalline materials in solid state physics
• Allows for the classification and analysis of a wide range of crystal structures

Combining primitive cell and basis

• Place the basis (atoms, ions, or molecules) at each lattice point defined by the primitive cell
• Repeat this arrangement throughout space using the primitive translation vectors
• Results in the complete crystal structure that describes the periodic arrangement of atoms in the material

Unit cells vs primitive cells

• Unit cells are a choice of repeating unit that can reproduce the entire crystal structure through translation
• Primitive cells are a special case of unit cells that contain exactly one lattice point
• Non-primitive unit cells (e.g., conventional unit cells) may contain multiple lattice points and are often chosen for convenience or to highlight symmetry

Conventional unit cells

• Commonly used unit cells that may be larger than the primitive cell and contain multiple lattice points
• Examples:
  • Face-centered cubic (FCC) conventional unit cell contains 4 lattice points
  • Body-centered cubic (BCC) conventional unit cell contains 2 lattice points
• Chosen to emphasize the symmetry of the crystal structure or for ease of visualization and computation

Primitive reciprocal lattice vectors

• Reciprocal lattice is a Fourier transform of the real-space lattice, representing the wave vectors of plane waves that have the same periodicity as the crystal lattice
• Primitive reciprocal lattice vectors are essential for describing the electronic structure, diffraction patterns, and phonon dispersion in crystalline materials

Definition of reciprocal lattice

• Set of all wave vectors $\vec{K}$ that yield plane waves with the same periodicity as the crystal lattice
• Defined by the condition: $e^{i\vec{K} \cdot \vec{R}} = 1$ for all lattice vectors $\vec{R}$
• Each point in the reciprocal lattice corresponds to a set of lattice planes in the real-space lattice

Reciprocal lattice primitive cell

• Primitive cell of the reciprocal lattice, known as the first Brillouin zone
• Constructed using the Wigner-Seitz method in reciprocal space
• Contains all unique wave vectors that describe the electronic and vibrational properties of the crystal

Reciprocal lattice translation vectors

• Primitive translation vectors of the reciprocal lattice, denoted as $\vec{b}_1$, $\vec{b}_2$, $\vec{b}_3$
• Defined by the relation: $\vec{a}_i \cdot \vec{b}_j = 2\pi\delta_{ij}$ (where $\vec{a}_i$ are primitive translation vectors of real-space lattice)
• Examples:
  • Simple cubic lattice: $\vec{b}_1 = \frac{2\pi}{a}\hat{x}$, $\vec{b}_2 = \frac{2\pi}{a}\hat{y}$, $\vec{b}_3 = \frac{2\pi}{a}\hat{z}$
  • Face-centered cubic lattice: $\vec{b}_1 = \frac{2\pi}{a}(\hat{x}-\hat{y})$, $\vec{b}_2 = \frac{2\pi}{a}(\hat{y}-\hat{z})$, $\vec{b}_3 = \frac{2\pi}{a}(\hat{z}-\hat{x})$

Brillouin zones

• Primitive cells of the reciprocal lattice that contain all unique wave vectors relevant to the electronic and vibrational properties of the crystal
• Understanding Brillouin zones is crucial for analyzing band structures, Fermi surfaces, and phonon dispersion relations in solid state physics

First Brillouin zone

• Primitive cell of the reciprocal lattice, constructed using the Wigner-Seitz method
• Contains all unique wave vectors that are closest to the origin of the reciprocal lattice
• Boundaries of the first Brillouin zone are defined by planes that bisect the lines connecting the origin to the nearest reciprocal lattice points

Higher order Brillouin zones

• Regions in reciprocal space that are further away from the origin than the first Brillouin zone
• Constructed by extending the Wigner-Seitz method to include more distant reciprocal lattice points
• Higher order Brillouin zones are less commonly used in solid state physics, as most essential information is contained within the first Brillouin zone

Brillouin zone vs Wigner-Seitz cell

• Both are primitive cells constructed using the Wigner-Seitz method, but in different spaces
• Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice, while the Wigner-Seitz cell refers to the primitive cell of the real-space lattice
• Brillouin zone contains information about the electronic and vibrational properties, while the Wigner-Seitz cell describes the real-space structure of the crystal

Crystal structures

• Periodic arrangement of atoms, ions, or molecules in a crystalline solid
• Determined by the combination of the underlying lattice and the basis associated with each lattice point
• Understanding crystal structures is essential for predicting and explaining the physical, chemical, and electronic properties of materials in solid state physics

Simple crystal structures

• Crystal structures with a single atom or a simple basis associated with each lattice point
• Examples:
  • Simple cubic (SC): One atom at each corner of a cubic unit cell
  • Body-centered cubic (BCC): One atom at each corner and one at the center of a cubic unit cell
  • Face-centered cubic (FCC): One atom at each corner and one at the center of each face of a cubic unit cell
  • Hexagonal close-packed (HCP): Two-atom basis arranged in a hexagonal lattice

Complex crystal structures

• Crystal structures with a more intricate basis, often involving multiple types of atoms or molecules
• Examples:
  • Diamond structure: Two identical atoms per lattice point, arranged in a tetrahedral geometry
  • Zinc blende structure: Two different types of atoms arranged in a tetrahedral geometry
  • Perovskite structure (ABX3): Three different types of atoms arranged in a specific geometry, with A at the corners, B at the center, and X at the face centers of a cubic unit cell

Crystal structure notation

• Shorthand notation used to describe the arrangement of atoms in a crystal structure
• Examples:
  • Pearson symbols: Indicate the crystal system, lattice type, and number of atoms per unit cell (e.g., cF4 for FCC, hP2 for HCP)
  • Strukturbericht notation: Uses letters to denote specific crystal structures (e.g., A1 for FCC, B1 for NaCl structure)
  • Space group notation: Describes the symmetry of the crystal structure using a combination of letters and numbers (e.g., Fm$\bar{3}$m for FCC)

Symmetry in crystal lattices

• Fundamental property of crystalline materials that describes the invariance of the lattice under certain transformations
• Symmetry plays a crucial role in determining the physical properties, electronic structure, and behavior of materials in solid state physics
• Three main types of symmetry: translational, point group, and space group symmetry

Translational symmetry

• Invariance of the lattice under translation by a lattice vector
• Characterized by the primitive translation vectors of the lattice
• Gives rise to periodic boundary conditions and Bloch's theorem, which are essential for understanding electronic band structures and phonon dispersion relations

Point group symmetry

• Symmetry operations that leave at least one point of the lattice fixed (e.g., rotations, reflections, inversions)
• Describes the symmetry of the unit cell or the basis
• Determines the anisotropy of physical properties, such as electrical conductivity, thermal expansion, and optical response

Space group symmetry

• Combines translational and point group symmetry to describe the full symmetry of the crystal structure
• There are 230 distinct space groups in three dimensions, each characterized by a unique set of symmetry operations
• Space group symmetry dictates the allowed electronic and vibrational states, as well as the presence of certain physical phenomena (e.g., piezoelectricity, ferroelectricity)
• Examples:
  • FCC lattice: Space group Fm$\bar{3}$m (cubic symmetry with face-centering and inversion)
  • Diamond structure: Space group Fd$\bar{3}$m (cubic symmetry with diamond basis)
  • Graphene: Space group P6/mmm (hexagonal symmetry with inversion)