🧶Inorganic Chemistry I Unit 13 Review

Crystal systems and Bravais lattices provide the framework for describing how atoms arrange themselves in crystalline solids. Every crystal, no matter how complex, can be described by one of just 14 unique lattice types built from 7 crystal systems. Getting comfortable with these classifications is essential for understanding solid-state structures, diffraction data, and ultimately a material's physical and chemical properties.

Lattice Fundamentals

Basic Building Blocks of Crystal Structures

A crystal is a solid with a periodic, repeating arrangement of atoms or ions extending in three dimensions. To describe that arrangement, you only need to define the smallest repeating chunk and the rule for how it repeats.

Unit cell: the smallest repeating structural unit that, when translated in all three directions, generates the entire crystal lattice.
Lattice points: mathematical points representing identical environments in the crystal. Each lattice point can represent a single atom, a molecule, or a group of ions (the basis).
Primitive cell: a unit cell containing exactly one lattice point (corners shared among adjacent cells sum to one point). Not all unit cells are primitive; sometimes a larger, non-primitive cell is chosen because it better reflects the symmetry.
Lattice parameters: the three edge lengths ( $a$ , $b$ , $c$ ) and three interaxial angles ( $\alpha$ , $\beta$ , $\gamma$ ) that define the shape of the unit cell.

Crystal System Classification

All crystals fall into one of seven crystal systems, classified by the relationships among their lattice parameters. Moving from highest to lowest symmetry:

Crystal System	Edge Lengths	Angles	Example
Cubic	$a = b = c$	$\alpha = \beta = \gamma = 90°$	NaCl
Hexagonal	$a = b \neq c$	$\alpha = \beta = 90°,\; \gamma = 120°$	Mg
Tetragonal	$a = b \neq c$	$\alpha = \beta = \gamma = 90°$	TiO₂ (rutile)
Trigonal (Rhombohedral)	$a = b = c$	$\alpha = \beta = \gamma \neq 90°$	Calcite
Orthorhombic	$a \neq b \neq c$	$\alpha = \beta = \gamma = 90°$	Aragonite
Monoclinic	$a \neq b \neq c$	$\alpha = \gamma = 90°,\; \beta \neq 90°$	Gypsum
Triclinic	$a \neq b \neq c$	$\alpha \neq \beta \neq \gamma \neq 90°$	Feldspar

Note that the trigonal system is sometimes listed as a subset of hexagonal (since trigonal crystals can be described with a hexagonal cell), but it is conventionally counted as the seventh crystal system.

Bravais Lattice Configurations

Within each crystal system, you can place lattice points in different positions within the unit cell. The four possible centering types are:

P (Primitive): lattice points only at the corners of the unit cell.
I (Body-centered): one additional lattice point at the center of the cell.
F (Face-centered): additional lattice points at the center of every face.
C (Base-centered): additional lattice points on just one pair of opposite faces (sometimes labeled A or B depending on which pair).

Not every combination of crystal system + centering type produces a unique lattice. Some combinations are equivalent to a simpler lattice after redefining the axes. When you eliminate these redundancies, exactly 14 distinct Bravais lattices remain. This is a mathematical result, not an arbitrary convention.

Basic Building Blocks of Crystal Structures, Introduction to crystals

Crystal System Types

Cubic System Characteristics

The cubic system has the highest symmetry: three equal axes at right angles ( $a = b = c$ , $\alpha = \beta = \gamma = 90°$ ). It supports three Bravais lattices:

Simple cubic (cP): lattice points at corners only. Rare in nature; polonium is the classic example. Coordination number = 6, packing efficiency ≈ 52%.
Body-centered cubic (cI): one additional point at the cell center. Found in many metals at room temperature, including Fe (α-iron), Cr, W, and the alkali metals. Coordination number = 8, packing efficiency ≈ 68%.
Face-centered cubic (cF): points at every face center. Very common for metals (Cu, Al, Ni, Au) and ionic compounds (NaCl adopts an FCC arrangement of each ion sublattice). Coordination number = 12, packing efficiency ≈ 74%.

The FCC lattice is identical to the cubic close-packed (CCP) structure, a point that comes up again in the close-packing section below.

Hexagonal and Tetragonal Systems

Hexagonal ( $a = b \neq c$ , $\alpha = \beta = 90°$ , $\gamma = 120°$ ): Only one Bravais lattice exists here: simple hexagonal (hP). The hexagonal close-packed (HCP) structure (found in Mg, Zn, Ti, Co) is built on this lattice with a two-atom basis, not a separate Bravais lattice type.

Tetragonal ( $a = b \neq c$ , $\alpha = \beta = \gamma = 90°$ ): Think of it as a cubic cell stretched or compressed along one axis. Two Bravais lattices: simple tetragonal (tP) and body-centered tetragonal (tI). A face-centered tetragonal cell can always be redescribed as a body-centered tetragonal cell with a smaller unit cell, so it's not counted separately. Tetragonal structures appear in rutile ( $\text{TiO}_2$ ) and several high-temperature superconductors.

Basic Building Blocks of Crystal Structures, Lattice Structures in Crystalline Solids | Chemistry for Majors

Lower Symmetry Crystal Systems

As symmetry decreases, more lattice parameters become independent, and crystals become harder to visualize.

Orthorhombic ( $a \neq b \neq c$ , $\alpha = \beta = \gamma = 90°$ ): All angles are still 90°, but all three edge lengths differ. This system supports the most Bravais lattices of any system: P, I, C, and F (four total). Examples include aragonite ( $\text{CaCO}_3$ ) and orthorhombic sulfur ( $\text{S}_8$ ).

Monoclinic ( $a \neq b \neq c$ , $\alpha = \gamma = 90°$ , $\beta \neq 90°$ ): One angle deviates from 90°. Two Bravais lattices: P and C. Gypsum ( $\text{CaSO}_4 \cdot 2\text{H}_2\text{O}$ ) and many molecular crystals (sucrose) crystallize in this system.

Triclinic ( $a \neq b \neq c$ , $\alpha \neq \beta \neq \gamma \neq 90°$ ): The lowest symmetry. No edges are equal and no angles are 90°. Only one Bravais lattice: P. Any centering in a triclinic cell can be reduced to a primitive cell. Feldspar minerals (e.g., albite, $\text{NaAlSi}_3\text{O}_8$ ) are common triclinic examples.

Advanced Lattice Concepts

Miller Indices and Crystallographic Planes

Miller indices are a compact notation for identifying planes and directions within a crystal.

To determine the Miller indices $(hkl)$ of a plane:

Find where the plane intercepts the $a$ , $b$ , and $c$ axes (in units of the lattice parameters).
Take the reciprocal of each intercept.
Clear fractions by multiplying through by the smallest common factor.
Enclose in parentheses: $(hkl)$ .

If a plane is parallel to an axis (intercept at infinity), the reciprocal is 0. Negative intercepts are written with a bar over the index: $(\bar{1}10)$ .

Some important planes to recognize:

$(100)$ : cuts the $a$ -axis at 1, parallel to $b$ and $c$ (a cube face in the cubic system).
$(110)$ : cuts $a$ and $b$ at 1, parallel to $c$ (a diagonal face).
$(111)$ : cuts all three axes at 1. In FCC, this is the close-packed plane with the highest atomic density.

For the hexagonal system, four-index Miller-Bravais indices $(hkil)$ are used, where $i = -(h+k)$ . The extra index makes symmetry-equivalent planes easier to identify.

Symmetry Operations in Crystals

Symmetry operations are transformations that map a crystal structure onto itself. They fall into two categories:

Point symmetry operations (leave at least one point fixed):

Rotation axes: $C_n$ where $n$ = 1, 2, 3, 4, or 6. Only these values are compatible with translational periodicity (this is the crystallographic restriction; 5-fold and >6-fold rotations cannot tile space).
Mirror planes ( $\sigma$ ): reflection across a plane.
Inversion center ( $i$ ): every point at $(x, y, z)$ maps to $(-x, -y, -z)$ .
Improper rotation axes ( $S_n$ ): rotation followed by reflection.

Space symmetry operations (involve translation):

Glide planes: mirror reflection combined with a translation parallel to the mirror plane.
Screw axes: rotation combined with a translation along the rotation axis.

Combining all compatible point symmetry elements gives 32 crystallographic point groups. Adding translational symmetry (glide planes, screw axes, and the 14 Bravais lattices) yields 230 space groups, which classify every possible 3D crystal symmetry.

Close-Packed Structures and Coordination

Close-packed structures maximize the fraction of space filled by identical spheres. Start with a single close-packed layer (call it A). The next layer sits in the hollows of layer A (call it B). For the third layer, there are two choices:

Place it directly above layer A → ABABAB... stacking → hexagonal close-packed (HCP).
Place it above the remaining set of hollows → ABCABC... stacking → cubic close-packed (CCP), which is the same as the FCC lattice.

Both HCP and CCP have:

Coordination number: 12 (each atom touches 12 neighbors).
Packing efficiency: ≈ 74% ( $\frac{\pi}{3\sqrt{2}} \approx 0.7405$ ), the theoretical maximum for identical spheres.

For comparison, BCC has a coordination number of 8 and packing efficiency of ≈ 68%, while simple cubic has a coordination number of 6 and packing efficiency of ≈ 52%.

The voids between close-packed atoms are also important in inorganic chemistry. Close-packed structures contain two types of interstitial holes:

Octahedral holes: surrounded by 6 atoms. There is 1 octahedral hole per atom in the close-packed structure.
Tetrahedral holes: surrounded by 4 atoms. There are 2 tetrahedral holes per atom.

Many ionic structures (NaCl, ZnS, $\text{Li}_2\text{O}$ ) can be described as a close-packed arrangement of one ion with the other ion occupying a specific fraction of octahedral or tetrahedral holes. This framework is extremely useful for rationalizing and predicting crystal structures of ionic compounds.

🧶Inorganic Chemistry I Unit 13 Review