Introduction to Crystallography
A crystal is formed by the periodic repetition of a group of atoms in all directions. That group of atoms is called the basis.
- The basis of a crystal can be one or more atoms.
- An ideal crystal consists of infinite repetitions of the basis.
- The lattice looks identical from whichever points you view the array.
- Note that lattice is not a crystal.
- In three dimensions, the lattice can be identified by three independent vectors \(\mathbf{a}_1\), \(\mathbf{a}_2\) and \(\mathbf{a}_3\).
- The position of each point \(\mathbf{R}\) can be written as a linear combination of these vectors, \[\mathbf{R}=n_1 \mathbf{a}_1+n_2 \mathbf{a}_2+n_3 \mathbf{a}_3 \tag{1}\] where \(n_1\), \(n_2\), and \(n_3\) are integers.
- The vectors that can generate or span the lattice are not unique. For example, see the following figure.
- Recall that the volume of a parallelepiped with axes \(\mathbf{a}_1\) , \(\mathbf{a}_2\) and \(\mathbf{a}_3\) is given by \(|\mathbf{a}_1\cdot(\mathbf{a}_2\times \mathbf{a}_3)|\) .
Primitive Cells vs. Unit Cells
- The unit cell may or may not be identical to the primitive cell.
Unit cell of a face-centered cubic structure
Lattice Parameters and Packing Fraction
Lattice Parameters
Packing Fraction
- We try to pack \(N\) hard spheres (cannot deform)
- The total volume of the spheres is \[V_s=N \frac{4}{3}\pi R^3\]
- The volume these spheres occupy \(V>V_S\) (there are spacing)\[{\rm Packing\ Fraction}=\frac{N \frac{4}{3}\pi R^3}{V}\]
Bravais Lattices
- Auguste Bravais, a French physicist, identified 14 distinct lattices in three dimensions.
- There are 5 Bravais lattices in two dimensions (shown below)
- In crystallography, all lattices are traditionally called Bravais lattices or translation lattices.
Five Bravais Lattices in Two Dimensions
 |  |  |
| Oblique | Rectangular | Centerd Rectangular |
 |  | |
| Square | Hexgonal (Rhombic) |
Seven Crystal Systems
Isometric (or Cubic) $a=b=c$ $\alpha=\beta=\gamma=90^\circ$ | Tetragonal $a=b\ne c$ $\alpha = \beta = \gamma = 90^\circ$ | Orthorhombic $a\ne b\ne c$ $\alpha = \beta = \gamma = 90^\circ$ | Hexagonal $a=b\ne c$ $\alpha = \beta = 90^\circ, \gamma = 120^\circ$ |
Triclinic $a\ne b\ne c$ $\alpha \ne \beta \ne \gamma$ | Monoclinic $a\ne b\ne c$ $\alpha = \beta = 90^\circ \ne \gamma$ | Rhombohedral (or Trigonal) $a=b=c$ $\alpha = \beta = \gamma < 120^\circ, \ne 90^\circ$ |
Cubic Lattices
| Primitive | Body-Centered | Face-Centered |
Simple Cubic (SC) Crystal
- Location of all lattice points \[\mathbf{R}=a(u\, \hat{\mathbf{x}}+v\, \hat{\mathbf{y}}+w\, \hat{\mathbf{z}})=a[u\ v\ w]\]
- Polonium (Po) is the only metal that forms a simple cubic unit cell.
- Each lattice point is shared by 8 neighboring units
- average volume occupied by each lattice point\(=\)
average volume occupied by each atom in SC\(=\Omega_{SC}=\frac{a^3}{8\times\frac{1}{8}}=a^3\)
Body-Centered Cubic (BCC) Crystal
- Location of all lattice points \[\begin{aligned} \mathbf{R}&=a(u\, \hat{\mathbf{x}}+v\, \hat{\mathbf{y}}+w\, \hat{\mathbf{z}})=a[u\ v\ w]\\ \mathbf{R}&=a((u+0.5)\, \hat{\mathbf{x}}+(v+0.5)\, \hat{\mathbf{y}}+(w+0.5)\, \hat{\mathbf{z}})=a[u+0.5\ v+0.5\ w+0.5]\end{aligned}\]
- average volume occupied by each lattice point\(=\)
average volume occupied by each atom in BCC\(=\Omega_{BCC}=\frac{a^3}{8\times\frac{1}{8}+1}=\frac{a^3}{2}\) - Nearest neighbor distance = \(d=2r=a\frac{\sqrt{3}}{2}\)
- \[\text{Packing fraction}=\frac{2\times\frac{4\pi}{3}\times(\frac{a\sqrt{3}}{4})^3}{a^3}=\frac{\pi\sqrt{3}}{8}\approx 0.68\]
The primitive translation vectors are: \[\mathbf{a}_1=\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{y}}-\hat{\mathbf{z}})\] \[\mathbf{a}_2=\frac{a}{2}(-\hat{\mathbf{x}}+\hat{\mathbf{y}}+\hat{\mathbf{z}})\] \[\mathbf{a}_3=\frac{a}{2}(\hat{\mathbf{x}}-\hat{\mathbf{y}}+\hat{\mathbf{z}})\]
- The primitive unit cell is a rhombohedron of edge \(a\frac{\sqrt{3}}{2}\)
- The angle between adjacent edges is \(109^\circ 28'\)
- Number of nearest neighbors = 8
- Number of second nearest neighbors = 6
- Second nearest neighbor distance = \(a\)
Some Elements with BCC structure
| Barium | Ba | Chromium | Cr |
| Cesium | Cs | \(\alpha-\)Iron | \(\alpha-\)Fe |
| Potassium | K | Lithium | Li |
| Molybdenum | Mo | Sodium | Na |
| Niobium | Nb | Rubidium | Rb |
| Tantalum | Ta | Titanium | Ti |
| Vanadium | V | Tungsten | W |
Face-Centered Cubic (FCC) Crystal
- Location of all lattice points \[\begin{aligned} \mathbf{R}&=a[u\ v\ w]\\ \mathbf{R}&=a[u+0.5\ v\ w]\\ \mathbf{R}&=a[u\ v+0.5\ w]\\ \mathbf{R}&=a[u\ v\ w+0.5]\end{aligned}\]
- average volume occupied by each lattice point\(=\)
average volume occupied by each atom in FCC\(=\Omega_{FCC}=\frac{a^3}{8\times\frac{1}{8}+6\times\frac{1}{2}}=\frac{a^3}{4}\) - Nearest neighbor distance = \(d=2r=\frac{a}{\sqrt{2}}\)
- \[\text{Packing fraction}=\frac{4\times\frac{4\pi}{3}\times(\frac{a\sqrt{2}}{4})^3}{a^3}=\frac{\pi\sqrt{2}}{6}\approx 0.74\]
The primitive cell vs the unit cell
The primitive translation vectors are: \[\mathbf{a}_1=\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{y}})\] \[\mathbf{a}_2=\frac{a}{2}(\hat{\mathbf{y}}+\hat{\mathbf{z}})\] \[\mathbf{a}_3=\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{z}})\]
- The angle between adjacent edges is \(60^\circ\)
The reference atom is red. Blue, green, and orange points are the 12 nearest neighbors of the reference atoms, and the purple points are its 6 next nearest neighbors atoms.
- Packing fraction \(\frac{1}{6}\pi\sqrt{2}\approx 0.740\)
- Some elements with fcc structure: Ar, Ag, Al, Au, Ca, Ce, \(\beta-\)Co, Cu, Ir, Kr, La, Ne, Ni, Pb, Pd, Pr, Pt, \(\delta-\)Pu, Rh, Sc, Sr, Th, Xe, Yb
Characteristics of Cubic structure
| simple cubic | b.c.c. | f.c.c. | |
|---|---|---|---|
| volume of conventional cell | \(a^3\) | \(a^3\) | \(a^3\) |
| no. of lattice points per cell | 1 | 2 | 4 |
| no. of nearest neighbors (coordination number) | 6 | 8 | 12 |
| no. of 2nd nearest neighbors | 12 | 6 | 6 |
| nearest neighbor distance | \(a\) | \(\frac{\sqrt{3}}{2}a\approx 0.866 a\) | \(\frac{a}{\sqrt{2}}\approx 0.707a\) |
| 2nd nearest neighbor distance | \(a\sqrt{2}\) | \(a\) | \(a\) |
| packing fraction | \(\pi/6\approx 0.52\) | \(\pi\sqrt{3}/8\approx 0.68\) | \(\pi\sqrt{2}/6\approx 0.74\) |
[111], [101], and [110] describe directions \(m\), \(t\), and \(n\), respectively.
Crystal Structure May Change with Temperature
- If the temperature changes, some materials might undergo a phase change.
- For example, at atmospheric pressure (105 Pa) and below 912 °C, pure iron (Fe) has a BCC structure, known as ⍺-iron. If we heat iron above 912 °C, its structure changes to an FCC structure, known as 𝛾-iron. Above 1394 °C, the structure changes back to BCC, known as 𝛿-iron. You may see the phase diagram of pure iron in the following figure.
Hexagonal Close-Packed (HCP) Structure
- 30 elements crystallize in hcp form.
- HCP crystal has a hexagonal lattice and a multi-atom basis.
- It can be viewed as two nested simple hexagonal Bravais lattice shifted by \(\mathbf{a}_1/3+ \mathbf{a}_2/3+\mathbf{a}_3/2\).
where
\[\mathbf{a}_1=a\hat{\mathbf{x}},\ \mathbf{a}_2=\frac{a}{2}\hat{\mathbf{x}}+\frac{a\sqrt{3}}{2}\hat{\mathbf{y}},\quad \mathbf{a}_3=c\,\hat{\mathbf{z}}\]
-
- average vol. occupied by each lattice point \[\Omega_{hex\ lattice}=\frac{\sqrt{3}}{2}a^2 c\]
- average vol. occupied by each atom \[\Omega_{HCP}=\frac{\Omega_{hex\ lattice}}{2}\]
Comparing Close-Packed Structures
| hcp | fcc |
- In this figure, the left structure is hcp, and the right is fcc
- volume fraction = 0.74
- number of nearest neighbors (coordination number) is 12 for both hcp and fcc structures
- Although a hexagonal close-packing of equal atoms is only obtained if \(c/a=\sqrt{8/3}\approx 1.63\), the term hcp is used for any structure described previously.
Elements with hcp structures
| Element | Element | ||
|---|---|---|---|
| Ideal | 1.63 | ||
| Be | 1.56 | Cd | 1.89 |
| Ce | 1.63 | \(\alpha\)-Co | 1.62 |
| Dy | 1.57 | Er | 1.57 |
| Gd | 1.59 | He (2K) | 1.63 |
| Hf | 1.58 | Ho | 1.57 |
| La | 1.62 | Lu | 1.59 |
| Mg | 1.62 | Nd | 1.61 |
| Os | 1.58 | Pr | 1.61 |
| Re | 1.62 | Ru | 1.59 |
| Tb | 1.58 | Ti | 1.59 |
| Tl | 1.60 | Tm | 1.57 |
| Y | 1.57 | Zn | 1.59 |
[adapted from Ashcroft, Mermin, Solid State Physics]
Diamond Crystal
- It is the structure of carbon in a diamond crystal
- It can be viewed as two interpenetrating fcc lattices displaced by \(\frac{a}{4}(\hat{\mathbf{x}}+\hat{\mathbf{y}}+\hat{\mathbf{z}})\)
- Or it can be imagined as the fcc lattice with two point basis \(\mathbf{0}\) and \(\frac{a}{4}(\hat{\mathbf{x}}+\hat{\mathbf{y}}+\hat{\mathbf{z}})\).
- Coordination number is 4
- Packing fraction is \(\frac{\sqrt{3}}{16}\pi\approx 0.34\)
 | |
| (a) Tetrahedral bond in a diamond structure | (b) Diamond structure projected on a cube face. Fractions denote the height above the base in units of \(a\) |
3D Diamond Structure
- Elements with diamond structure: C (diamond), Si, Ge, \(\alpha\)-Sn (gray)
- average volume occupied by each atom: \(\Omega_{DC}=\frac{\Omega_{FCC}}{2}=\frac{a^3}{8}\)
Sodium Chloride Structure
- Na\(^{+}\) and Cl\(^-\) ions are placed on alternate points of a simple cubic structure
- The lattice is fcc; the basis consists of Na\(^{+}\) and Cl\(^-\)
NaCl structure, Blue atoms represent Na atoms and Green ones represent Cl atoms.
Some compounds with sodium chloride structure
| LiF | LiCl | LiBr | LiI | ||
| NaF | NaCl | NaBr | NaI | ||
| RbF | RbCl | RbBr | RbI | ||
| CsF | |||||
| AgF | AgCl | AgBr | |||
| MgO | MgS | MgSe | |||
| CaO | CaS | CaSe | CaTe | ||
| SrO | SrS | SrSe | SrTe | ||
| BaO | BaS | BaSe | BaTe |
[Adapted from Ashcroft, Mermin, Solid State Physics]
Cesium Chloride Structure
- Cs\(^{+}\) and Cl\(^-\) ions are placed at \(\mathbf{0}\) and body center position \(\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{y}}+\hat{\mathbf{y}})\), respectively.
- The lattice is simple cubic; the basis consists of Cs\(^{+}\) and Cl\(^-\)
CsCl structure. Blue spheres represent Cl atoms and the big red sphere represents the Cs atom which is much larger than the Cl atoms.
- Some compounds with the cesium chloride structure:
| CsCl | CsBr | CsI |
| TlCl | TlBr | TlI |
Miller Indices
Miller Indices for Directions in Cubic Structure
- It is often required to specify certain directions and planes in crystals. To this end, we use Miller indices.
- \([hkl]\) represents the direction vector \(h\hat{\mathbf{x}}+k\hat{\mathbf{y}}+l\hat{\mathbf{z}}\) of a line passing through the origin.
- These integers \(h\), \(k\) and \(l\) must be the smallest numbers that will give the desired direction. i.e. we write \([111]\) not \([222]\).
- If a component is negative, it is conventionally specified by placing a bar over the corresponding index. For example, we write \([1\bar{1}1]\) instead of \([1 -1 1]\).
- Coordinates in angle brackets such as \(\langle123\rangle\) denote a family of directions that are equivalent due to symmetry operations, such as [123], [132], [321], [\(\bar{1}23\)], [\(\bar{1}\bar{2}3\)], etc.
[111], [101], and [110] describe directions \(m\), \(t\), and \(n\), respectively.
Miller Indices for Planes in Cubic Structure
- A crystallographic plane is denoted by the Miller indices of the direction normal to the plane, but instead of brackets we use parenthesis, i.e. (\(hkl\))
| $(100)$ | $(110)$ | $(111)$ |
- “Coordinates in curly brackets or braces such as \(\{100\}\) denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.”
- For a cubic system, a family \(\{hkl\}\) consists of all the planes given by the permutations of the number \(h\), \(k\), \(l\) and their negatives.
- If the symmetry of the system is lower than that of cubic, not all the planes given by the permutations necessarily belong to a family. For example, in a rhombohedral system, we have \(\{100\}=\{(100),(\bar{1}00),(010),(0\bar{1}0),(001),(00\bar{1})\}\), but in an orthorhombic system \(\{100\}\) family has only two members \((100)\) and \((\bar{1}00)\).
Miller Indices for hcp
- Miller indices contain 4 digits instead of 3 digits
- \([hkil]\) means: \[\{\alpha (h \mathbf{a}_1+k\mathbf{a}_2+i\mathbf{a}_3+l \mathbf{c}): \alpha\in\mathbb{R}\}\] and \[h+k+i=0\]
- Directions along axes \(\mathbf{a}_1\), \(\mathbf{a}_2\) and \(\mathbf{a}_3\) are of type \(\langle \bar{1} 2\bar{1} 0\rangle\).
- \((hkil)\) is a plane the normal direction of which is \([hkil]\).
 |  |
| Determination of indices for a digonal axis if Type I - $[2\overline{11}0]$ | Determination of indices for a digonal axis if Type II - $[10\overline{1}0]$ |
Further Reading
- Ashcroft, N.W., Mermin, N.D., Solid State Physics, Harcourt College Publishers, 1976.
- De Graef, M., McHenry, M.E., Structure of Materials, Cambridge University Press, 2007.
- Kittel, C., Introduction to Solid State Physics, 8th ed., Wiley, 2004.