Chemical Crystallography I: Lattices, Unit Cells & Close-Packed Topologies

The fundamental mathematical description of crystalline matter rests upon the distinction between the abstract geometrical space lattice and the physical crystal structure.

The Mathematical Definition of a Crystal

A crystal structure is formed when an identical group of atoms, ions, or molecules—termed the basis or motif—is attached identically to every mathematical point of a three-dimensional space lattice:

A mathematical space lattice is an infinite, periodic array of discrete points in space defined by translation vectors:

where $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar primitive translation vectors. The atomic environment viewed from any lattice point $\mathbf{r}$ is identical to that viewed from $\mathbf{r}' = \mathbf{r} + \mathbf{T}$.

Primitive vs Non-Primitive (Conventional) Unit Cells

A unit cell is a parallelepiped defined by three basis vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ with lengths $a, b, c$ and interaxial angles $\alpha = \angle(\mathbf{b}, \mathbf{c})$, $\beta = \angle(\mathbf{a}, \mathbf{c})$, and $\gamma = \angle(\mathbf{a}, \mathbf{b})$.

1. Primitive Cell ($P$): Contains exactly one lattice point per cell ($Z_{\text{lattice}} = 1$). A primitive cell represents the minimal volume unit that tiles all space under translational operations without gaps or overlaps.

2. Wigner-Seitz Primitive Cell: The unique, geometrically symmetric primitive cell constructed by drawing lines from a chosen lattice point to all adjacent lattice points and constructing normal bisecting planes. The region enclosed by these planes forms the Wigner-Seitz cell, which exhibits the full point group symmetry of the lattice.

3. Conventional (Non-Primitive) Unit Cells: Cells chosen with larger volumes ($Z_{\text{lattice}} > 1$) in order to display the full rotational and mirror symmetry of the crystal system:

• Body-Centered ($I$, Innenzentriert): Additional lattice point at $(1/2, 1/2, 1/2) \implies Z_{\text{lattice}} = 8(1/8) + 1 = 2$.
• Face-Centered ($F$, Flächenzentriert): Additional lattice points at $(1/2, 1/2, 0)$, $(1/2, 0, 1/2)$, $(0, 1/2, 1/2) \implies Z_{\text{lattice}} = 8(1/8) + 6(1/2) = 4$.
• Base-Centered ($C, A, \text{ or } B$): Additional lattice points on one pair of opposite faces \implies Z_{\text{lattice}} = 8(1/8) + 2(1/2) = 2$.

By combining spatial translational periodicity with rotational and inversion symmetry, Auguste Bravais demonstrated in 1848 that there exist precisely 14 distinct space lattices in three dimensions, grouped into 7 crystal systems.

The 7 Crystal Systems

The 7 crystal systems are classified by their essential characteristic symmetry elements, which impose geometric constraints upon unit cell edge lengths ($a, b, c$) and interaxial angles ($\alpha, \beta, \gamma$):

1. Cubic (Isometric): Essential symmetry: $4$ threefold rotation axes along body diagonals ($4C_3$).

• Constraints: $a = b = c$, $\alpha = \beta = \gamma = 90^\circ$.
• Bravais Lattices ($3$): Primitive ($P$), Body-Centered ($I$), Face-Centered ($F$).

2. Tetragonal: Essential symmetry: $1$ fourfold rotation or rotary-inversion axis ($1C_4$ or $1\bar{4}$).

• Constraints: $a = b \neq c$, $\alpha = \beta = \gamma = 90^\circ$.
• Bravais Lattices ($2$): Primitive ($P$), Body-Centered ($I$).

3. Orthorhombic: Essential symmetry: $3$ mutually perpendicular twofold axes or mirror planes ($3C_2$ or $3\sigma_v$).

• Constraints: $a \neq b \neq c$, $\alpha = \beta = \gamma = 90^\circ$.
• Bravais Lattices ($4$): Primitive ($P$), Body-Centered ($I$), Face-Centered ($F$), Base-Centered ($C$).

4. Hexagonal: Essential symmetry: $1$ sixfold rotation or rotary-inversion axis ($1C_6$ or $1\bar{6}$).

• Constraints: $a = b \neq c$, $\alpha = \beta = 90^\circ, \gamma = 120^\circ$.
• Bravais Lattice ($1$): Primitive ($P$).

5. Trigonal (Rhombohedral): Essential symmetry: $1$ threefold rotation axis ($1C_3$ or $1\bar{3}$).

• Constraints: $a = b = c$, $\alpha = \beta = \gamma \neq 90^\circ < 120^\circ$ (in rhombohedral axes) or $a = b \neq c, \gamma = 120^\circ$ (in hexagonal axes).
• Bravais Lattice ($1$): Rhombohedral ($R$).

6. Monoclinic: Essential symmetry: $1$ twofold rotation axis or mirror plane ($1C_2$ or $1m$).

• Constraints: $a \neq b \neq c$, $\alpha = \gamma = 90^\circ, \beta \neq 90^\circ$ (unique $b$-axis setting).
• Bravais Lattices ($2$): Primitive ($P$), Base-Centered ($C$).

7. Triclinic: Essential symmetry: Only identity ($1$) or inversion center ($\bar{1}$).

• Constraints: $a \neq b \neq c$, $\alpha \neq \beta \neq \gamma \neq 90^\circ$.
• Bravais Lattice ($1$): Primitive ($P$).

Many elemental metals, noble gases, and ionic anion sublattices crystallize in dense structures derived from the closest packing of identical hard spheres in three dimensions.

Close-Packed Two-Dimensional Layers (Layer A)

In a single two-dimensional close-packed plane, each sphere is in contact with $6$ coplanar neighbors in a hexagonal arrangement. The surface area of the hexagonal unit rhombus containing $1$ sphere is:

The area occupied by the circular cross-section is $A_{\text{circle}} = \pi R^2$. The 2D packing fraction is:

Stacking in the Third Dimension: Voids B and C

Within a close-packed layer $A$, there exist two sets of triangular interstitial dips pointing in opposite directions:

• Set $B$: Triangular depressions pointing "upward".
• Set $C$: Triangular depressions pointing "downward".

A second close-packed layer can be placed in either set $B$ or set $C$. Choosing set $B$ establishes layer sequence $AB$. When adding the third layer, two distinct stacking choices emerge:

1. Hexagonal Close-Packing (hcp): Stacking sequence $ABABAB\dots$

• Spheres of the third layer sit directly above the spheres of the first layer $A$.
• Symmetry: Hexagonal system, space group $P6_3/mmc$.
• Ideal Axial Ratio: For touching hard spheres:

• Examples: $\text{Mg}, \text{Ti}, \text{Zr}, \text{Zn}, \text{Cd}, \text{Co}$.

2. Cubic Close-Packing (ccp / Face-Centered Cubic fcc): Stacking sequence $ABCABC\dots$

• Spheres of the third layer occupy void set $C$, resting above neither $A$ nor $B$.
• Symmetry: Cubic system, face-centered Bravais lattice ($Fm\bar{3}m$).
• Close-packed $\{111\}$ planes intersect at $70.53^\circ$, giving four equivalent close-packed directions $\langle 110 \rangle$.
• Examples: $\text{Cu}, \text{Ag}, \text{Au}, \text{Al}, \text{Ni}, \text{Pb}, \text{Pt}$.

Both hcp and ccp achieve the identical maximum atomic packing factor:

and each sphere has a coordination number of $\text{C.N.} = 12$ ($6$ coplanar, $3$ above, $3$ below).

The interstitial cavities remaining between close-packed spheres host cations in ionic lattices (such as $\text{NaCl}, \text{ZnS}, \text{CaF}_2$) and interstitial atoms in alloys.

1. Tetrahedral Interstitial Voids ($T_d$)

A tetrahedral void is bounded by four mutually touching host spheres arranged at the vertices of a regular tetrahedron.

• In the fcc unit cell (lattice parameter $a = 2\sqrt{2}R$), the cell can be subdivided into 8 smaller octants of edge length $a/2$.
• The center of each octant is surrounded by 4 spheres (one corner atom and three adjacent face-centered atoms) forming a regular tetrahedron.
• Fractional Coordinates: The 8 tetrahedral voids reside at:

• Radius Ratio Derivation:

Distance from corner $(0,0,0)$ to octant center $(a/4, a/4, a/4)$ is:

Substitute $a = 2\sqrt{2} R$:

2. Octahedral Interstitial Voids ($O_h$)

An octahedral void is bounded by six host spheres arranged at the vertices of a regular octahedron.

• In the fcc unit cell:

1. Body-Center Void: Located at $(1/2, 1/2, 1/2)$, coordinated by 6 face-centered spheres at distance $a/2$.

2. Edge-Center Voids: Located at the center of all 12 unit cell edges: $(1/2, 0, 0), (0, 1/2, 0), \dots$ Each edge-center is shared by 4 adjacent unit cells.

• Total Count per Cell:

• Radius Ratio Derivation:

Along any unit cell edge, two host spheres of radius $R$ at $(0,0,0)$ and $(1,0,0)$ sandwich an interstitial void of radius $r_{\text{oct}}$:

A universal stoichiometric relationship governs interstitial sites in all close-packed lattices, regardless of whether the stacking is cubic ($ABC$) or hexagonal ($AB$).

The Universal Stoichiometric Rule

For any close-packed assembly containing $N$ host spheres:

Rigorous Proof for the FCC / CCP Lattice

In a conventional face-centered cubic unit cell:

1. Host Atoms ($N$):

2. Octahedral Voids:

• $1$ at cell body-center $(1/2, 1/2, 1/2)$ $\implies 1 \times 1 = 1$.
• $12$ at edge-centers $\implies 12 \times (1/4) = 3$.
• Total $N_{\text{oct}} = 1 + 3 = 4 = N$.

3. Tetrahedral Voids:

• $1$ inside each of the 8 constituent octants $\implies 8 \times 1 = 8 = 2N$.

Structural Archetypes Derived from Interstitial Filling

• Rock Salt (NaCl): $100\%$ of octahedral voids occupied in ccp anion array ($6:6$ coordination).
• Zinc Blende (ZnS): $50\%$ of tetrahedral voids occupied alternately in ccp anion array ($4:4$ coordination).
• Fluorite (CaF$_2$): $100\%$ of tetrahedral voids occupied in ccp cation array ($8:4$ coordination).
• Nickel Arsenide (NiAs): $100\%$ of octahedral voids occupied in hcp anion array ($6:6$ coordination).
• Wurtzite (ZnS): $50\%$ of tetrahedral voids occupied alternately in hcp anion array ($4:4$ coordination).
• Cadmium Iodide (CdI$_2$): $50\%$ of octahedral voids occupied in alternate layers of hcp iodide ($6:3$ coordination).

In 1929, Linus Pauling formulated geometric principles governing ionic crystal coordination polyhedra. Pauling's First Rule states that a coordinated polyhedron of anions is formed about each cation, with the coordination number dictated by the radius ratio $\rho = r_+ / r_-$.

Derivation of Critical Radius Ratios

• Linear Coordination (C.N. = 2):

• Trigonal Planar Coordination (C.N. = 3):

• Tetrahedral Coordination (C.N. = 4):

• Octahedral Coordination (C.N. = 6):

• Cubic Coordination (C.N. = 8):

• Cuboctahedral / Close-Packed (C.N. = 12):

The efficiency with which atoms fill space in a crystal lattice dictates mechanical density, vacancy migration barriers, and compressibility.

Mathematical Definition of the Atomic Packing Fraction

The Atomic Packing Fraction (APF) is:

• Simple Cubic (sc): $N = 1, a = 2R \implies \text{APF} = \frac{\pi}{6} \approx 52.36\%$.
• Body-Centered Cubic (bcc): $N = 2, a\sqrt{3} = 4R \implies \text{APF} = \frac{\pi\sqrt{3}}{8} \approx 68.02\%$.
• Face-Centered Cubic (fcc): $N = 4, a\sqrt{2} = 4R \implies \text{APF} = \frac{\pi}{3\sqrt{2}} \approx 74.05\%$.
• Diamond Cubic: $N = 8, a\sqrt{3} = 8R \implies \text{APF} = \frac{\pi\sqrt{3}}{16} \approx 34.01\%$.

Theoretical Mass Density Derivation

where $Z$ is the number of formula units per unit cell, $M$ is the molar mass ($\text{g/mol}$), $N_A = 6.02214 \times 10^{23}\text{ mol}^{-1}$, and $V_{\text{cell}}$ is in $\text{cm}^3$.

When host metal lattices incorporate solute elements, solid solutions form via interstitial or substitutional mechanisms.

Interstitial Solid Solutions & Hägg's Rule

Stable interstitial phases form when:

• Carbon in Iron (Steel Metallurgy):
• Austenite ($\gamma$-Fe, fcc): Large symmetric octahedral voids ($r_{\text{oct}} = 0.53\text{ \AA}$) allow up to $2.14\text{ wt}\%$ carbon solubility at $1147^\circ\text{C}$.
• Ferrite ($\alpha$-Fe, bcc): Tetragonally distorted octahedral voids with a compressed axis opening of only $0.19\text{ \AA}$ severely restrict carbon solubility to $< 0.022\text{ wt}\%$.

Substitutional Solid Solutions & Hume-Rothery Rules

Extensive solid solubility requires:

1. Size Factor: Atomic radius mismatch $\le 15\%$.

2. Crystal Structure: Identical space lattice and crystal symmetry.

3. Electronegativity: Minimal electronegativity difference to avoid intermetallic precipitation.

4. Valence Electron Concentration ($e/a$): Characteristic ratios govern phase stability:

• $\alpha$-phase (fcc solid solution): $e/a < 1.36$
• $\beta$-phase (bcc): $e/a = 21/14 = 1.50$
• $\gamma$-phase (complex cubic): $e/a = 21/13 \approx 1.62$
• $\epsilon$-phase (hcp): $e/a = 21/12 = 1.75$