Solid State Chemistry
Table of Contents
1. Crystal Structure
고체는 기본 단위가 가장 밀집되어 있는 형태
Packing creates crystal which is solid with regular structure.
1.1. X-ray Crystallography
X-ray diffraction is the main method to investigate the crystal structure.
- Bruker, D8 Advance
Low d-value planes are detected at high angles. When there is low crystallinity, more peaks are seen at low angles.
1.2. Electron Diffraction
Electron diffraction gives 2D information.
Particle accelerator can be used to hit electron to crystal, and get 2D diffraction image. Shellex program developed by Sheldrich analysis gives atomic position, richer than simple X-ray diffraction.
- 포항 입자가속기
1.3. Lattice
The lattice describe the translation symmetry of a crystal structure, and the lattice points do not neccesarily coincide with atoms.
The set of points given by integer linear combination of three primitive translation vectors: \[ \Lambda := \{ n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3 \in \mathbb{R}^3\colon n_1, n_2, n_3 \in \mathbb{Z}\}. \]
Regularity, aka lattice, arises when atoms are packed most densely.
이상적으로는 고체 내부의 원자는 화학반응도 하지 않고 움직이지 않는다.
1.3.1. Lattice Types
Categorized based on the placement of lattice points (translational symmetries) in unit cell.
| Lattice Types | Symbol | Description |
|---|---|---|
| Primitive | P | One at origin |
| Body-centered | I | One at origin, one at body center |
| Face-centered | F | One at origin, three at each face |
| C-centered | C | one at origin, one at face normal to c |
| Rhombohedral | R |
1.3.2. Bravais Lattices
- Bravais Lattices
- These are the conjugacy classes of lattice point groups. (Expansion required)
There exists 14 Bravais lattices in 3 dimensions, each have different combination of lattice system and lattice type.
| 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 |
The empty combination is the ones that admits other Bravais lattice with the unit cell smaller than itself. Which violates our rule of choosing unit cell that are the smallest that fully represent the symmetry.
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.
1.4. Unit Cell
There is no unique way of defining unit cell. They can be translated arbitrarily. The unit cell is conventionally chosen to be the (1) smallest, (2) that fully represents the symmetry of the structure.
Unit cell that contains one lattice point is called the primitive unit cell \(\mathbf{P}\). Centered unit cell \(\mathbf{C}\) contains two lattice points, one at the center.
Primitive unit cell is the parallelepipel given by the primitive translation vectors of the lattice.
Conventioanl unit cell is parallelepiped given by three vectors which are each integer linear combinations of primitive translation vectors. They are used to show the symmetries clearly.
1.4.1. Lattice Parameters
The sidelengths of unit cell \(a,b,c\) each for \(x,y,z\) axis respectively, and the angles \(\alpha, \beta, \gamma\) that are the angle between two axis other than the respective axis \(a,b,c\).
They define the lattice vectors which are denoted with \( \mathbf{a},\mathbf{b},\mathbf{c} \). Depending on the lattice type (or centering type), they are defined to be specific integer linear combination of the primitive translation vectors of the lattice.
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*}1.4.2. Lattice Systems
- 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.
1.4.2.1. Crystal Classes
- 32 in three dimensions
The point group of unit cell.
The quotient group of space group by translations.
Categorized based on the unit cell axis, the crystallographic axes.
1.4.3. Fractional Coordinates
Start by placing an atom at the origin. The origin must correspond to some atom. The basis vectors are the basis vectors \[ \mathbf{a}, \mathbf{b}, \mathbf{c} \] of conventional unit cell. Naturally the coordinates within the unit cell are relative coordinates that span each axis from 0 to 1.
The lattice vectors are used as the basis vectors.
1.4.4. Centering Vector
The addtional lattice points other than the ones at the corner of a unit cell is indicated with vectors in fractional coordinates.
1.5. Space Group
- 230 in three dimensions
- Point group + translation + glide + screw
Crystals are categorized based on their space groups, and consequently their unit cell shapes.
1.5.1. 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.
1.5.1.1. Hermann-Mauguin Notation
The centering type (P, I, S, F) is followed by one to three symmetry elements, where the first element notates both rotation and mirror symmetry of a specified axis using fraction or forward slash. e.g. P6/mmm.
The directional slots may be an axis or a set of axis:
| Crystal System | 1st | 2nd | 3rd |
|---|---|---|---|
| Triclinic | |||
| Monoclinic | b = [010] | ||
| Orthoclinic | a = [100] | b = [010] | c = [001] |
| Tetragonal | c = [001] | a = 〈100〉 | 〈110〉 |
| Trigonal | c = [001] | a (basal principle) | [210] (basal diagonal) |
| Hexagonal | c = [001] | 〈100〉 (basal principle) | 〈210〉 (basal diagonal) |
| Cubic | a = 〈100〉 | 〈111〉 | 〈110〉 |
The direction is specified using crystal direction notation. The empty spots are arbitrary and differs per crystal. The b-axis is chosen for the first slot in monoclinic system, beause \(\beta \neq \pi/2\).
Possible symmetry elements are:
- 1 Identity
- m Mirror plane
- n Rotation: n-fold rotation
- n̄ Rotoinversion: rotation by 2π/n and inversion
- \(\bar{1}\) 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. Diagonal glide plane.
- d: quarter face-diagonal. Four atoms along an axis. Diamond glide plane
- 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.
1.5.2. List of 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 .
The position with no symmetry is often omitted.
1.5.2.1. P2₁/n
- P12₁/n1
It is a non-split extension: \[ \mathrm{P2_1/n} = \mathbb{Z}^3\bullet \{E, I, S, N\} \] where the representation of point-group operations on three dimensional affine space are
\begin{align*} E &:= \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix}, \\ S &:= \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 &:= \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix}, \\ N &:= \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*}in the basis 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.
1.5.3. 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.
1.6. Packing
Atom of single element can be arranged, or packed, in various ways, with translation symmetry (lattice) and point-group symmetry.
In a packing the ratio of atoms are given such that the charge is neutral in a unit cell.
1.6.1. Closed Packing
Coordination number 12. There are octahedral holes \(O_c\) and tetrahedral holes \(T_d\).
Tolerance factor
Hexagonal Close Packing
- HCP
- hP Lattice
- The layer repeats as ABAB.
It makes hexagonal unit cell, with two atoms per unit cell.
Cubic Close Packing
- CCP
- cF Lattice
- The layer repeats as ABCA.
It has face-centered cubic unit with four atoms per unit cell. Since the lattice and atoms match, the cubic close packing is sometimes called face-cented cubic (FCC).
1.6.2. Body-Centered Cubic
- BCC
1.6.3. Primitive Cubic
1.7. Interstitial Sites
Triangular
Square Planar
Octahedral hole
- In CCP: \(\ce{NaCl}\)
- 4 in CPP
Tetrahedral hole
- 8 in CPP
1.8. Crystal Planes
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.
Crystal planes are closely related to the peaks of X-ray crystallography
1.8.1. Miller Indices
Miller indices describes a family of crystal planes that go through the centers of atoms. It corresponds to a covector, that describes how many crystal planes has been poked through by the lattice vectors.
Miller indices is defined with respect to the crystallographic axes, that is the conventional unit cell and lattice vectors.
The notation is \((hkl)\) with no comma, and \(\bar{h}\) for negative component. \(\{hkl\}\) denotes all planes that are equivalent to \((hkl)\) by symmetry.
A 3D diagram of crystal planes is conventionally drawn with \(y\) axis going up.
Miller indices also denotes macroscopic crystal faces.
1.8.2. Interplanar Spacings
- \(d\)-Spacings, \(d\)-Value
We can calculate the lattice parameters from Miller indices and the \(d\) using X-ray crystallography.
1.8.3. Crystal Faces
Tracht ('costume') is the set of kinds of faces. Habitus is the relative sizes of faces
- Isometric
- Needle-like
- Plate-like
- Cubically
- Column
1.9. Crystal Directions
The direction with respect to the crystal is described similar to Miller indices, but with the basis of lattice vectors.
\(\left\langle hkl \right\rangle\) denotes all directions that are equivalent to \([hkl]\) by symmetry.
1.10. Quasicrystal
Quasicrystals do not have a repeating unit cell like ordinary crystals do. The Nobel Prize was awarded to Dan Shechtman in Chemistry in 2011 for disovering such crystal. In 1982, Dan Shechtman observed an alloy whose electron diffraction pattern showed sharp spots with 5-fold symmetry.
These build structures with forbidden symmetries, especially icosahedral symmetry.
- pentagonal prisms
- rhombohedra
- icosahedra
Penrose tilings with rhombus/parallelogram-like tiles are a classic model.
Quasicrystal have unusual properties such as:
- low friction
- hardness
- poor heat conduction in some cases
2. Crystalline Solids
Various forces can form crystal structure.
- Ionic bond
- Covalent bond
- Metallic bond
- van der Waals force
2.1. Polymorphs
- 다형체
There exists phase within the solid region, each with different crystalline structure.
2.2. Ionic Solids
2.2.1. Diagrams
Tetrahedrons and octahedrons are used to notate the tetrahedral holes and octahedral holes. Usually metal is at the center of hole, and anions are at the vertices.
2.2.2. MX
Zinc Blend
- ZnS
- Diamond Structure
- Normal at STP
Four of eight tetrahedral holes are filled with zinc in cubic closed packing of sulfur. The zincs are (1/4, 1/4, 1/4), (3/4, 3/4, 1/4), (1/4, 3/4, 3/4), (3/4, 3/4, 3/4).
Wurtzite
- ZnS
One of the bond in tetrahedral hole, in which direction the axis is defined as the c-axis, is slightly stretched.
2.2.3. MX₂
Fluorite
- CaF₂, Calcium Fluoride
Anatase
Rutile
- TiO₂, Titanium Dioxide
- Absorb ultraviolet
Brookite
2.2.4. Others
Spinel
- AB₂O₄
- \(\ce{A^{2+}}, \ce{B^{3+}}\)
Perovskite
- ABO₃
Titanate
Ilmenite
3. See Also
CCDC is the only official organization that manages crystallographic data. The data is free but the software for processing them is not.
4. References
- Elaine A. Moore, Lesley E. Smart. Solid State Chemistry. Fifth edition.
- Schoenflies notation - Wikipedia
- Hermann–Mauguin notation - Wikipedia
- Point groups in three dimensions - Wikipedia
- Lattice plane - Wikipedia
- Bravais lattice - Wikipedia
- Crystal system - Wikipedia
- Space group - Wikipedia
- Frank Hoffmann - YouTube. Fascination of Crystals and Symmetry.