DAT General Chemistry: Crystal Lattices and Unit Cells

DAT High-Yield Concept

Crystalline vs. Amorphous Solids: Crystalline solids exhibit long-range 3D periodic order and sharp melting points; amorphous solids lack long-range order and soften over a temperature range.
Unit Cell Fractional Ownership: Corner atoms contribute $\frac{1}{8}$ , edge atoms contribute $\frac{1}{4}$ , face-center atoms contribute $\frac{1}{2}$ , and body-center atoms contribute $1$ full atom.
Net Atoms per Unit Cell ( $Z$ ): Simple Cubic (SC) = $1$ , Body-Centered Cubic (BCC) = $2$ , Face-Centered Cubic (FCC) = $4$ .
Cell Edge Length ( $a$ ) to Atomic Radius ( $r$ ): SC: $a = 2r$ ; BCC: $a = \frac{4r}{\sqrt{3}}$ ; FCC: $a = 2\sqrt{2}r$ .

Crystalline versus Amorphous Solids

Solids are classified into two major categories based on the structural arrangement of their constituent particles:

Crystalline Solids: Possess a highly ordered, repeating three-dimensional arrangement of atoms, ions, or molecules extending over long distances (long-range order). Because all intermolecular or ionic bonds in the crystal matrix are uniform in strength, crystalline solids melt sharply at a precise temperature. Examples include ionic salts like $\ce{NaCl}$ and metallic elements like $\ce{Fe}$ .
Amorphous Solids: Lack long-range periodic spatial organization (short-range order only). Their disordered internal structure yields varied bond strengths throughout the material, causing them to gradually soften over a range of temperatures rather than displaying a sharp melting point. Examples include glass, rubber, and synthetic polymers.

(Note: The physical behavior and bonding classifications of molecular solids are covered in depth in a separate lesson.)

Crystal Lattice and Unit Cell Terminology

A crystal lattice is a three-dimensional array of points illustrating the spatial distribution of particles in a crystalline solid. Every point in the lattice is termed a lattice point, which represents an atom, ion, or molecule.

The unit cell is the smallest repeating structural unit that generates the entire crystal lattice when repeatedly translated along its principal axes. A unit cell is characterized by six lattice parameters: three edge lengths ( $a$ , $b$ , and $c$ ) and three interaxial angles ( $\alpha$ , $\beta$ , and $\gamma$ ).

Image of a crystal unit cell definition. a, b, and c are unit cell lengths. alpha is the angle between b anc c, beta is the angle between a and c, and gamma is the angle between a and b. (Source: Mcpazzo · Public domain · Wikimedia Commons)

Key Concept: Fractional Atom Ownership Rules

Corner Position: Shared equally by 8 adjacent unit cells → contributes $\frac{1}{8}$ atom to each cell.
Edge Position: Shared equally by 4 adjacent unit cells → contributes $\frac{1}{4}$ atom to each cell.
Face Center Position: Shared equally by 2 adjacent unit cells → contributes $\frac{1}{2}$ atom to each cell.
Body Center Position: Completely contained within 1 unit cell → contributes $1$ whole atom.

Cubic Unit Cells and Atom Counting

On the DAT, you must know the three types of cubic unit cells where all edge lengths are equal ( $a = b = c$ ) and all axial angles equal $90^\circ$ ( $\alpha = \beta = \gamma = 90^\circ$ ):

1. Simple Cubic (SC) Unit Cell

Particles reside exclusively at the 8 corners of the cube. Corner atoms touch each other directly along the cube edges.

$\text{Net atoms per cell } (Z) = 8 \text{ corners} \times \frac{1}{8} = 1 \text{ atom}$

Because corner atoms touch along the edge, $a = 2r$ . The coordination number (number of nearest neighbor atoms) is $6$ , yielding a packing efficiency of $52.4\%$ . Polonium ( $\ce{Po}$ ) is the primary elemental example.

Simple cubic crystal structure. (Source: Vectorization: Stannered · CC BY-SA 3.0 · Wikimedia Commons)

2. Body-Centered Cubic (BCC) Unit Cell

Particles reside at the 8 corners plus one full atom located at the center of the unit cell cube. Corner atoms do not touch one another; instead, they touch the central atom along the body diagonal (length $d = 4r = \sqrt{3}a$ ).

$\text{Net atoms per cell } (Z) = \left(8 \text{ corners} \times \frac{1}{8}\right) + 1 \text{ central atom} = 2 \text{ atoms}$

Rearranging the body diagonal relationship yields $a = \frac{4r}{\sqrt{3}}$ . The coordination number is $8$ , with a packing efficiency of $68.0\%$ (e.g., $\ce{Fe}$ , $\ce{Na}$ ).

3. Face-Centered Cubic (FCC) Unit Cell

Particles reside at the 8 corners and at the center of each of the 6 square faces. Corner atoms touch the face-centered atoms along the face diagonal (length $d = 4r = \sqrt{2}a$ ).

$\text{Net atoms per cell } (Z) = \left(8 \text{ corners} \times \frac{1}{8}\right) + \left(6 \text{ face centers} \times \frac{1}{2}\right) = 1 + 3 = 4 \text{ atoms}$

Rearranging the face diagonal relationship yields $a = 2\sqrt{2}r$ . The coordination number is $12$ , attaining the maximum close-packing efficiency of $74.0\%$ (e.g., $\ce{Cu}$ , $\ce{Au}$ , $\ce{Al}$ ).

The three cubic unit cells, and how many atoms each one actually contains — A crystal is built by repeating one small block, the unit cell, in every direction, and the three cubic cells differ only in which extra positions are occupied. Counting the atoms in a cell means counting the fraction of each atom that lies inside it: an atom at a corner is shared among eight neighboring cells and counts as one eighth, an atom centered on a face is shared between two and counts as one half, and an atom at the center of the cell belongs to that cell alone. The simple cubic cell therefore contains one atom, the body-centered cubic cell two, and the face-centered cubic cell four.

Summary Comparison of Cubic Unit Cells

Unit Cell Type	Corner Contribution	Center/Face Contribution	Net Atoms ( $Z$ )	Coordination Number	Edge Length ( $a$ ) vs Radius ( $r$ )	Packing Efficiency
Simple Cubic (SC)	$8 \times \frac{1}{8}$	$0$	$1$	$6$	$a = 2r$	$52.4\%$
Body-Centered Cubic (BCC)	$8 \times \frac{1}{8}$	$1 \text{ (body center)}$	$2$	$8$	$a = \frac{4r}{\sqrt{3}}$	$68.0\%$
Face-Centered Cubic (FCC)	$8 \times \frac{1}{8}$	$6 \times \frac{1}{2} \text{ (face centers)}$	$4$	$12$	$a = 2\sqrt{2}r$	$74.0\%$

DAT Memory Trick: Atom Counts & Packing Efficiency
Remember the sequence S-B-F with 1 - 2 - 4 net atoms!
• Simple = 1 net atom (lowest packing density,

52\%

)
• Body-centered = 2 net atoms (medium packing density,

68\%

)
• Face-centered = 4 net atoms (highest packing density,

74\%

)

Worked Quantitative Example & Exam Trap

Question: Gold ( $\ce{Au}$ ) crystallizes in a face-centered cubic (FCC) unit cell with an edge length of $a = \pu{0.408 nm}$ . Calculate the atomic radius of gold in picometers ( $\pu{pm}$ ).

Step 1: Identify the cell geometry equation.
For an FCC lattice, atoms contact along the face diagonal, so $a = 2\sqrt{2}r$ .

Step 2: Rearrange for atomic radius $r$ .
$r = \frac{a}{2\sqrt{2}}$

Step 3: Convert unit cell edge length to picometers using dimensional analysis.
$a = \pu{0.408 nm} \times \frac{\pu{1000 pm}}{\pu{1 nm}} = \pu{408 pm}$

Step 4: Substitute into the equation and recalculate.
$r = \frac{\pu{408 pm}}{2\sqrt{2}} = \frac{\pu{408 pm}}{2.8284} = \pu{144.25 pm} \approx \pu{144 pm}$

Common Exam Trap: Students often incorrectly apply the simple cubic relationship (

a = 2r

) to all unit cells or miscalculate face contributions by counting face-centered atoms as whole atoms rather than halving them (

6 \times \frac{1}{2} = 3

). Always identify the specific unit cell type first! $

Critical Thinking Questions

If a metal transitions from a simple cubic lattice to a face-centered cubic lattice without changing its atomic radius, how will its macroscopic density change? Explain using packing efficiency.
A synthetic titanium oxide crystal has titanium atoms occupying all 8 corners of a cubic unit cell and oxygen atoms occupying the center of all 6 faces. What is the empirical formula of this oxide?
Why do crystalline solids like quartz ( $\ce{SiO2}$ ) melt at a sharp, single temperature, whereas amorphous silica glass ( $\ce{SiO2}$ ) gradually softens over a broad temperature range?

References

Sources returned by Google Search grounding during lesson verification.

12.1: Crystal Lattices and Unit Cells - Chemistry LibreTexts — chem.libretexts.org
10.6 Lattice Structures in Crystalline Solids - OpenStax Chemistry 2e — openstax.org

Crystal Lattices and Unit Cells

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