Inorganic Chemistry
Table of Contents
1. Point Group
1.1. Schönflies Notation
1.1.1. Symmetry Operations
- Cₙ Rotation: n-fold rotation
- σ Reflection
- σₕ Horizontal Reflection: reflection with respect to the plane perpendicular to the principal axis.
- σᵥ Vertical Reflection: reflection with respect to the plane parallel to the principal axis.
- i Inversion
- S₂ₙ Improper Rotation: Rotation about an axis and reflection along the axis.
- From German Spiegel 'mirror'
1.1.2. Point Groups
- \( C_1 \) Only identity
- \( C_i \) Identity and inversion
- \( C_s \) Identity and improper rotation
- \( C_{\infty v} \) Linear along an axis and vertical reflections
- \( D_{\infty h} \) Linear along an axis and horizontal reflections
- \( C_{n} \) (Cyclic) Only rotations along a single axis are possible
- \( C_{nv} \) (Pyramidal) Rotations and vertical reflections
- \( C_{nh} \) (Reflection) Rotations and horizontal reflection
- \( D_n \) (Dihedral) Rotations and 2n C₂ rotations perpendicular to principal axis.
- \( D_{nh} \) (Prismatic) Dihedral and horizontal reflection
- \( D_{nd} \) (Antiprismatic) Dihedral and dihedral (vertical) reflections
- \( S_n \) Dihedral and improper rotations
- \( T, T_h, T_d \) (Tetrahedral)
- \( O, O_h \) (Octahedral)
- \( I, I_h \) (Icosahedral)
1.2. Hermann-Mauguin Notation
- International Notation
- It is the standard in International Tables For Crystallography.
1.2.1. Symmetry Operations
- n Rotation: n-fold rotation
- m Reflection
- n/m the mirror plane is perpendicular to the n-fold rotation axis,
- n̄ Rotoinversion: rotation by 2π/n and inversion
- 1̄ is the pure inversion
1.2.2. Point Groups
The symmetries in primary, secondary, tertiary directions are listed in order. The secondary directions are the all equivalent directions perpendicular to the primary direction, and tertiary directions are all equivalent directions between secondary directions. Note that the tertiary directions only occur when the rotational symmetry around the primary direction is even.
When the rotational symmetry is more than 10-fold, paranthesis is used to make it clear that it is not rotational symmetries along two directions.
Translation from Schönflies notation
- \( n \): \( C_n \)
- \( nm, nmm \): \( C_{nv} \). one \( m \) if \( n \) is odd, two \( m \) if \( n \) is even.
- \( \tfrac{n}{m} \): \( C_{nh} \)
- \( n2, n22 \): \( D_n \)
- \( \bar{n}\tfrac{2}{m} \): \( D_{nd} \)
- \( \tfrac{n}{m}\tfrac{2}{m}\tfrac{2}{m} \): \( D_{nh} \)
2. Miller Indices
Miller indices denotes each crystal faces. They are the macroscopic faces.
Tracht ('costume') is the set of kinds of faces. Habitus is the relative sizes of faces
- Isometric
- Needle-like
- Plate-like
- Cubically
- Column
Lattice plane is any plane that contains at least three nonlinear Bravais lattice points, and a family of Lattice planes are periodic and parallel lattice planes.
Miller indices is the notation system for each family of lattice planes, expressed by three integers (h k l). The integers are the coefficients of the reciprocal lattice vectors \( \mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3 \) that is parallel to the normal vector of the plane. A bar is placed if the coefficient is negative. They are also the numbers of planes poked by each lattice vector. Similar to covector, if you know.
The spacing between adjacent lattice planes are called the d spacing.
3. Lattice
The background lattice for describing crystal structure. The lattice points do not neccesarily coincide with atoms.
The set of points given by integer linear combination of three primitive translation vectors: \[ \Lambda := \{ n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3 \in \mathbb{R}^3\colon n_1, n_2, n_3 \in \mathbb{Z}\}. \]
3.1. Unit Cell
Unit cell can be defined on top of this lattice. It must tile the space by translation given by infinite subset of \( \Lambda \).
Primitive unit cell is the parallelepipel given by the primitive translation vectors.
- It is the parallelopiped with smallest possible volume that can be constructed from lattice points.
- It contains only one effective lattice point: one eighth of a point in each corner.
Conventioanl unit cell is parallelopiped given by three integer linear combinations of primitive translation vectors. They are used to show the symmetries clearly. This is what is meant when chemist refer to a "unit cell", and they are defined to be one of the crystal family.
Lattice vectors are the edges of a conventioanl unit cell. They are denoted with \( \mathbf{a},\mathbf{b},\mathbf{c} \). Depending on the centering types, they are defined to be specific integer linear combination of the primitive translation vectors.
In primitive system (P, one point per cell), they are equal to the primitive vectors:
\begin{align*} \mathbf{a} &= \mathbf{a}_1, \\ \mathbf{b} &= \mathbf{a}_2, \\ \mathbf{c} &= \mathbf{a}_3 . \end{align*}In body-centered system (I, two points per cell):
\begin{align*} \mathbf{a} &= \mathbf{a}_1 + \mathbf{a}_2, \\ \mathbf{b} &= \mathbf{a}_3 + \mathbf{a}_{1} , \\ \mathbf{c} &= \mathbf{a}_2 + \mathbf{a}_3. \end{align*}In base-centered system (S, two points per cell), the primitive vectors parallel to the base is mixed. For example in C system:
\begin{align*} \mathbf{a} &= \mathbf{a}_1 - \mathbf{a}_2, \\ \mathbf{b} &= \mathbf{a}_1 + \mathbf{a}_2 , \\ \mathbf{c} &= \mathbf{a}_3. \end{align*}In face-centered system (F, four points per cell):
\begin{align*} \mathbf{a} &= -\mathbf{a}_1 + \mathbf{a}_2 + \mathbf{a}_3, \\ \mathbf{b} &= \mathbf{a}_1 - \mathbf{a}_2 + \mathbf{a}_3, \\ \mathbf{c} &= \mathbf{a}_1 + \mathbf{a}_2 - \mathbf{a}_3. \end{align*}3.2. Bravais Flocks
- Bravais Lattices
- 14 in three dimensions
The lattice systems are classified by the constraints the lattice points have. The constraints imposed on the \( \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \) is being considered here.
| Lattice System | Primitive (P) | Base (S: A, B, C) | Body (I) | Face (F) |
|---|---|---|---|---|
| Triclinic | aP | |||
| Monoclinic | mP | mS | ||
| Orthorombic | oP | oS | oI | oF |
| Tetragonal | tP | tI | ||
| Rhombohedral | hR | |||
| Hexagonal | hP | |||
| Cubic | cP | cI | cF |
Due to the convention that c axis must have the highest rotational symmetry, rhombohedral lattice (R) is defined differently from the others:
\begin{align*} \mathbf{a} &= \mathbf{a}_1 - \mathbf{a}_2, \\ \mathbf{b} &= \mathbf{a}_2 - \mathbf{a}_3, \\ \mathbf{c} &= \mathbf{a}_1 + \mathbf{a}_2 + \mathbf{a}_3. \end{align*}The conventinal unit cell looks like rhombohdedral centered monoclinic system, in which two body centers are staggered in at one third and two third of the height.
This is the conjugacy class of lattice point groups. (Expansion required)
3.3. Fractional Coordinates
The coordinates system within a conventional unit cell. The lattice vectors are used as the basis vectors.
4. Crystal System
- 7 in three dimensions
- Types of unit cells based on their shape.
Unit cells can be chosen on top of lattice, and they are categorized based on symmetry. If a unit cell that looks like cubic system but does not have the required symmetry, it is not classified as the cubic system. We run down the list and match the system with the highest symmetry that matches.
The crystallographic axes (or lattice vectors) are notated with a, b, and c, where c is the axis with the largest rotational symmetry.
Other axis can be notated as a linear combination of the principal axes:
- [210] 2a + 1b direction
All the axes that are equivalent under symmetry is notated with angle brackets:
- ⟨210⟩ set of axis equivalent to [210] axis.
Unit cell should be chosen to be the smallest parallelepipel whose edges and faces best matches the symmetry axis and planes. Here, symmetries come from the actual atoms.
The cell constants (a, b, c) and cell angles (α, β, γ) are constrained differently based on crystal family.
| Crystal Family | Constants | Angles | Crystal System | Point Group Symmetry | Point Groups | Bravais Lattices | Lattice System | Space Groups |
|---|---|---|---|---|---|---|---|---|
| Triclinic (a) | none | none | Triclinic | none | 2 | 1 | Triclinic | 2 |
| Monoclinic (m) | none | α = γ = π/2 | Monoclinic | one 2, or one m | 3 | 2 | Monoclinic | 13 |
| Orthorhombic (o) | none | α = β = γ = π/2 | Orthorohmbic | three 2, or one 2 and two m | 3 | 4 | Orthorhombic | 59 |
| Tetragonal (t) | a = b | α = β = γ = π/2 | Tetragonal | one 4 | 7 | 2 | Tetragonal | 68 |
| Hexagonal (h) | a = b | α = β = π/2, γ = 2π/3 | Trigonal | one 3 | 5 | 1 | Rhombohedral | 7 |
| Hexagonal | 18 | |||||||
| Hexagonal | one 6 | 7 | 1 | 27 | ||||
| Cubic (c) | a = b = c | α = β = γ = π/2 | Cubic | four 3 | 5 | 3 | Cubic | 36 |
| 6 | 7 | 32 | 14 | 7 | 230 |
They are ultimately determined by the symmetry as well.
4.1. Crystal Classes
- 32 in three dimensions
The point group of unit cell.
The quotient group of space group by translations.
5. Space Group
- 230 in three dimensions
Crystals are categorized based on their space groups, and consequently their unit cell shapes.
5.1. Symmetry Operations
Symmetry operations in a space group is represented by affine transformations. The origin can be arbitrarily chosen since transformation matrix is only representation of the group.
- P, I, S, F the centering type tells the structure of translation operations
- m Reflection
- n Rotation: n-fold rotation
- n̄ Rotoinversion: rotation by 2π/n and inversion
- 1̄ is the pure inversion
- Glide reflection: Translation and reflection with respect to a glide plane
- a, b, c: translation by half unit along that axis, the reflection plane is often specified by its normal direction.
- n: reflection about one of the face and translation along the face diagonal by half unit in each axis parallel to the plane.
- d: quarter face-diagonal. Four atoms along an axis.
- e: doubly glided with respect to a single glide plane. Four atoms that forms a parallelogram.
- nₘ Screw: Translation by m/n unit, and rotation about that direction by 2π/n. If translate beyond the unit cell, you repeat back into the cell from the other side.
5.2. Space Groups
Space group is denoted similarly to Hermann-Mauguin notation. The first position is the translation symmetry, and the following three denote the symmetry along the three viewing directions that differ depending on crystal system.
The position with no symmetry is often omitted.
5.2.1. Viewing Directions
| Crystal System | 1st | 2nd | 3rd |
|---|---|---|---|
| Triclinic | |||
| Monoclinic | b | ||
| Orthoclinic | a | b | c |
| Tetragonal | c | a | [110] |
| Trigonal | c | a | [210] |
| Hexagonal | c | a | [210] |
| Cubic | a | [111] | [110] |
The empty spots are arbitrary and differs per crystal.
5.2.2. Examples
P2₁/n (P12₁/n1) \[ \mathrm{P2_1/n} = \mathbb{Z}^3\rtimes \{E, I, S, N\} \] where the representation of symmetry operations on three dimensional affine space are:
\begin{align*} E &\equiv \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix}, \\ S &\equiv \begin{bmatrix} -1 & 0 & 0 & \frac{1}{2}\\ 0 & 1 & 0 & \frac{1}{2}\\ 0 & 0 & -1 &\frac{1}{2} \\ 0 & 0 & 0 & 1\\ \end{bmatrix}, \\ E &\equiv \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix}, \\ N &\equiv \begin{bmatrix} 1 & 0 & 0 & \frac{1}{2}\\ 0 & -1 & 0 & \frac{1}{2}\\ 0 & 0 & 1 &\frac{1}{2} \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}. \\ \end{align*}The translation and rotation basis vectors are the set of lattice vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \). Note that the two glide planes and four screw axes are equivalent to each other under left composition with translation. Therefore, this is the exhaustive list of all the symmetry operations.
5.3. International Tables for Crystallography
A hollow circle means a general position, that is, arbitrary coordinates \( (x,y,z) \). Positive or negative sign next to the circle represent the z coordinate. Upon screw or glide operation 1/2 or 1/4 can be prepended. The comma in the circle means that it is inverted.
A small circle represent the inversion center.
Polygon represent the rotation axis. With hooklets, it becomes the screw axis. The direction of hooklets notates the screw direction.
6. Crystal Structure
This are how the actual atoms are arranged. Space group, Bravais lattice, crystal system are defined independently.
Sometimes the name of Bravais lattice is used for crystal structure if the atoms and lattice points match one to one.
6.1. Close Packing
Hexagonal close packing (hcp)
- hP Lattice
- ABA
- Two atoms per unit cell
Cubic close packing (ccp)
- cF Lattice
- ABC
- Four atoms per unit cell (one atom per lattice point)
- Since the lattice and atoms match, the cubic close packing is sometimes called face-cented cubic (FCC).
6.2. Metalic Crystal
They mostly form hexagonal close packing, face-centered cubic, or body-centered cubic.