Crystal Structures

1.1 Lattices and Basis

A crystal is defined by a lattice (infinite array of points with translational symmetry) and a basis (the arrangement of atoms associated with each lattice point).

The lattice is specified by primitive lattice vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$ Such that every lattice point is at:

$\mathbf{R} = n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3, \quad n_i \in \mathbb{Z}$

1.2 Bravais Lattices

There are 14 Bravais lattices in three dimensions, classified into 7 crystal systems:

System	Lattices	Constraints
Cubic	SC, BCC, FCC	$a = b = c$ , $\alpha = \beta = \gamma = 90^\circ$
Tetragonal	P, I	$a = b \neq c$ , $\alpha = \beta = \gamma = 90^\circ$
Orthorhombic	P, C, I, F	$a \neq b \neq c$ , $\alpha = \beta = \gamma = 90^\circ$
Monoclinic	P, C	$a \neq b \neq c$ , $\alpha = \gamma = 90^\circ \neq \beta$
Triclinic	P	$a \neq b \neq c$ , $\alpha \neq \beta \neq \gamma$
Hexagonal	P	$a = b \neq c$ , $\alpha = \beta = 90^\circ$ , $\gamma = 120^\circ$
Trigonal	R	$a = b = c$ , $\alpha = \beta = \gamma \neq 90^\circ$

Notation: P = primitive, I = body-centred, F = face-centred, C = base-centred, R = rhombohedral.

1.3 Common Crystal Structures

Simple Cubic (SC): One atom per unit cell. Coordination number $= 6$ . Packing fraction $= \pi/6 \approx 0.52$ .

Body-Centred Cubic (BCC): Atoms at corners and body centre. Two atoms per cell. Coordination Number $= 8$ . Packing fraction $= \sqrt{3}\pi/8 \approx 0.68$ . Examples: Fe ( $\alpha$ -iron), W, Cr.

Face-Centred Cubic (FCC): Atoms at corners and face centres. Four atoms per cell. Coordination Number $= 12$ . Packing fraction $= \sqrt{2}\pi/6 \approx 0.74$ . Examples: Cu, Al, Au, Ag.

Hexagonal Close-Packed (HCP): Two atoms per primitive cell. Coordination number $= 12$ . Same Packing fraction as FCC. Examples: Zn, Mg, Ti.

Diamond structure: Two interpenetrating FCC lattices offset by $(a/4, a/4, a/4)$ . Eight atoms per Conventional cell. Examples: C (diamond), Si, Ge.

1.4 Miller Indices

A plane with Miller indices $(hkl)$ intersects the crystallographic axes at $a/h$ , $b/k$ , $c/l$ .

Procedure: (1) Find intercepts with axes in units of lattice constants. (2) Take reciprocals. (3) Reduce to smallest integers.

Negative indices are written with a bar: $(\bar{h}kl)$ . Families of equivalent planes are denoted $\{hkl\}$ .

Directions are written as $[hkl]$ ; families of equivalent directions as $\langle hkl \rangle$ .

1.5 Wigner-Seitz Cell

The Wigner-Seitz cell is the primitive cell constructed by drawing perpendicular bisector planes Between a lattice point and all its neighbours. It is the region of space closer to the given lattice Point than to any other.

1.6 Packing Fractions and Density

The packing fraction (also called atomic packing factor) is the fraction of volume in a unit cell Occupied by atoms:

$\mathrm{APF} = \frac{N \cdot V_{\mathrm{atom}}{V_{\mathrm{cell}} = \frac{N \cdot \frac{4}{3}\pi R^3}{V_{\mathrm{cell}}}}}$

Where $N$ is the number of atoms per cell, $R$ is the atomic radius, and $V_{\mathrm{cell}}$ is the Cell volume.

The theoretical density of a crystal:

$\rho = \frac{nM}{N_A V_{\mathrm{cell}}}$

Where $n$ is the number of formula units per cell, $M$ is the molar mass, $N_A$ is Avogadro”s Number, and $V_{\mathrm{cell}}$ is the cell volume.

Worked Example: FCC Packing Fraction

In FCC, nearest neighbours touch along the face diagonal. For lattice constant $a$ and atomic radius $R$ :

$4R = \sqrt{2}\,a \implies R = \frac{a\sqrt{2}}{4}$

Four atoms per conventional cell:

$\mathrm{APF} = \frac{4 \times \frac{4}{3}\pi R^3}{a^3} = \frac{4 \times \frac{4}{3}\pi \left(\frac{a\sqrt{2}}{4}\right)^3}{a^3} = \frac{4 \times \frac{4}{3}\pi \cdot \frac{2\sqrt{2}\,a^3}{64}}{a^3} = \frac{\pi\sqrt{2}}{6} \approx 0.7405$

Worked Example: Density of BCC Iron

$\alpha$ -iron is BCC with lattice constant $a = 0.2866$ nm, molar mass $M = 55.845$ g/mol, and 2 atoms Per conventional cell.

$\rho = \frac{2 \times 55.845}{6.022 \times 10^{23} \times (2.866 \times 10^{-8})^3}$

$(2.866 \times 10^{-8})^3 = 23.55 \times 10^{-24}\ \mathrm{cm}^3 = 2.355 \times 10^{-23}\ \mathrm{cm}^3$

$\rho = \frac{111.69}{6.022 \times 10^{23} \times 2.355 \times 10^{-23}} = \frac{111.69}{14.18} = 7.88\ \mathrm{g}/cm^3$

This matches the accepted experimental density of iron ( $7.87\ \mathrm{g}/cm^3$ ).

Worked Example: HCP Packing Fraction

For HCP with ideal $c/a = \sqrt{8/3}$ Lattice constant $a$ And atomic radius $R = a/2$ :

Two atoms per primitive cell. The cell volume is $V_{\mathrm{cell} = \frac{\sqrt{3}}{2}a^2 \cdot c = \frac{\sqrt{3}}{2}a^2 \cdot a\sqrt{8/3} = \sqrt{2}\,a^3}$ .

$\mathrm{APF} = \frac{2 \times \frac{4}{3}\pi (a/2)^3}{\sqrt{2}\,a^3} = \frac{\frac{\pi a^3}{3}}{\sqrt{2}\,a^3} = \frac{\pi}{3\sqrt{2}} = \frac{\pi\sqrt{2}}{6} \approx 0.7405$

This confirms that HCP and FCC have the same packing fraction, as both are close-packed structures.

1.7 Crystal Defects

Real crystals are never perfect. Defects are classified by their dimensionality.

Point defects (0D):

Vacancy: Missing atom at a lattice site.
Interstitial: Extra atom between lattice sites.
Substitutional: Foreign atom replacing a host atom.
Frenkel defect: Vacancy-interstitial pair (atom moves to an interstitial site).
Schottky defect: Pair of vacancies in ionic crystals (cation + anion vacancies to maintain charge neutrality).

The equilibrium concentration of vacancies follows from minimising the free energy $F = n_v E_v - k_B T \ln\binom{N}{n_v}$ :

$n_v = N\,e^{-E_v/(k_B T)}$

Where $E_v$ is the vacancy formation energy ( $\sim 1$ eV).

Line defects (1D):

Edge dislocation: An extra half-plane of atoms inserted into the lattice. The Burgers vector $\mathbf{b}$ is perpendicular to the dislocation line.
Screw dislocation: The lattice is helically distorted. $\mathbf{b}$ is parallel to the dislocation line.
Mixed dislocation: Combines edge and screw character.

Dislocations enable plastic deformation at stresses far below the theoretical shear strength. The Peach-Koehler force per unit length on a dislocation:

$\mathbf{F} = (\boldsymbol{\sigma}\cdot\mathbf{b}) \times \hat{\mathbf{t}}$

Planar defects (2D):

Grain boundary: Interface between two crystallites of different orientation.
Stacking fault: Error in the stacking sequence (e.g., ABCABC $\to$ ABCABABC in FCC).
Twin boundary: Mirror plane across which the lattice is reflected.

1.8 X-ray Diffraction Analysis

X-ray diffraction is the primary experimental tool for determining crystal structures. X-rays are Produced by bombarding a metal target with electrons, generating both Bremsstrahlung (continuous Spectrum) and characteristic radiation (sharp lines at element-specific energies, e.g., Cu $K_\alpha$ at $\lambda = 1.5406$ Å).

Three principal experimental methods:

Laue method: A stationary single crystal is exposed to a polychromatic X-ray beam. Each set of planes selects the wavelength satisfying Bragg’s law, producing a spot pattern revealing symmetry.
Rotating crystal method: A single crystal is rotated in a monochromatic beam. As each set of planes sweeps through the Bragg angle, a diffraction spot is recorded.
Powder (Debye—Scherrer) method: A polycrystalline sample is exposed to monochromatic X-rays. Randomly oriented crystallites produce diffraction cones, recorded as concentric rings on a detector.

:::caution Common Pitfall The coordination number is the number of nearest neighbours, not the number of atoms in the unit Cell. For FCC, the coordination number is 12 even though there are only 4 atoms per conventional Cell. Do not confuse the basis size with the coordination number.

Worked Example: Interplanar Spacing for SC, BCC, FCC

For a cubic crystal with lattice constant $a$ The interplanar spacing for $(hkl)$ is:

$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$

For SC, all $(hkl)$ reflections are allowed. For BCC, only $(hkl)$ with $h+k+l$ even are present. For FCC, only all-even or all-odd $(hkl)$ are present.

The first few allowed reflections and their $d$ -spacings for FCC ( $a = 0.405$ nm, aluminium):

$(hkl)$	$h^2+k^2+l^2$	$d$ (nm)	$2\theta$ (Cu $K_\alpha$ )
(111)	3	0.2338	38.5°
(200)	4	0.2025	44.7°
(220)	8	0.1432	65.1°
(311)	11	0.1221	78.2°
(222)	12	0.1169	82.4°

Note that (100) is absent in FCC (since 1+0+0 = 1 is odd but not all odd). The sequence of Allowed $h^2+k^2+l^2$ values (3, 4, 8, 11, 12, …) is characteristic of FCC.

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