This cheat sheet covers the three most common metallic crystal structure types: body-centered cubic, face-centered cubic, and hexagonal close-packed. Students need these structures to connect atomic arrangement with density, packing efficiency, coordination number, and material properties. A clear reference helps compare the diagrams, formulas, and key values that often appear in chemistry and materials science problems.
The core ideas are unit cells, atoms per unit cell, coordination number, atomic radius relationships, and atomic packing factor. For cubic structures, the edge length connects to atomic radius through geometry, while HCP uses a hexagonal cell with an ideal ratio . Density problems use , where is atoms per unit cell and is the unit cell volume.
Comparing BCC, FCC, and HCP shows why FCC and HCP are more closely packed than BCC.
Key Facts
- In a body-centered cubic unit cell, the number of atoms per unit cell is because corner atoms contribute atom total and the body-center atom contributes atom.
- In a face-centered cubic unit cell, the number of atoms per unit cell is because corners contribute atom total and face atoms contribute atoms total.
- For BCC, atoms touch along the body diagonal, so and .
- For FCC, atoms touch along the face diagonal, so and .
- The atomic packing factor is .
- BCC has coordination number and atomic packing factor .
- FCC and HCP both have coordination number and atomic packing factor .
- Crystal density is calculated with , where is molar mass and .
Vocabulary
- Unit cell
- A unit cell is the smallest repeating three-dimensional block that shows the symmetry and arrangement of atoms in a crystal.
- Body-centered cubic
- Body-centered cubic, or BCC, is a cubic structure with atoms at the corners and one atom at the center of the cube.
- Face-centered cubic
- Face-centered cubic, or FCC, is a cubic structure with atoms at the corners and at the centers of all six faces.
- Hexagonal close-packed
- Hexagonal close-packed, or HCP, is a close-packed structure with layers arranged in an repeating pattern.
- Coordination number
- Coordination number is the number of nearest neighboring atoms touching a given atom in a crystal structure.
- Atomic packing factor
- Atomic packing factor is the fraction of a unit cell's volume occupied by atoms, given by .
Common Mistakes to Avoid
- Counting corner atoms as whole atoms is wrong because each corner atom is shared by unit cells, so each corner contributes only atom.
- Using the BCC radius formula for FCC is wrong because BCC atoms touch along the body diagonal, while FCC atoms touch along the face diagonal.
- Assuming BCC is close-packed is wrong because BCC has , while close-packed FCC and HCP have .
- Forgetting to convert radius units is wrong because density calculations require consistent units, such as converting to before finding in .
- Using for all unit cells is wrong because BCC has , FCC has , and HCP depends on the specific unit cell chosen.
Practice Questions
- 1 A BCC metal has atomic radius . Calculate the unit cell edge length using .
- 2 An FCC metal has molar mass and edge length . Calculate its density using with .
- 3 For an FCC unit cell, show how the atoms from corners and faces add to atoms per unit cell.
- 4 Explain why FCC and HCP have the same coordination number and packing efficiency even though their layer stacking patterns are different.
Understanding Crystal Structure Types (BCC, FCC, HCP)
A crystal is a repeating arrangement, not a collection of isolated unit cells. Each cell joins seamlessly to cells on every side. This is why atoms drawn at corners or faces must be shared with neighboring cells when counting them.
The drawing is a bookkeeping model. It helps describe an enormous solid using one small repeating piece.
Students should separate the visible spheres in a diagram from the effective number of whole atoms assigned to one cell. This prevents one of the most common errors in crystal structure questions.
The directions where atoms touch determine the geometry of a structure. In a cubic cell, an edge, a face diagonal, and a body diagonal have different lengths. Only one of these lines passes through touching atoms for a given structure.
For body-centered cubic, the important line runs from one corner through the center atom to the opposite corner. For face-centered cubic, the important line lies across a face. Sketching this line before using a radius relationship is safer than memorizing formulas.
It shows why the square root of three appears in one case and the square root of two appears in the other. It also helps students notice that atoms are treated as hard spheres in this model, which is a useful approximation rather than a perfect picture of electron clouds.
Packing affects how metals behave when forces act on them. Metals with face-centered cubic structures include aluminum, copper, silver, and gold. Their atoms can slide along several closely packed layers.
This contributes to high ductility, meaning that many of these metals can be drawn into wires or shaped into sheets. Magnesium, zinc, and some forms of titanium have hexagonal close-packed structures. Their packed layers have a different stacking pattern, and they have fewer easy directions for atomic layers to slide at room temperature.
This can make forming them harder. Iron shows another important idea.
It has a body-centered cubic structure at ordinary temperatures, yet its structure changes when heated. A change in crystal structure can alter density, strength, and the way a metal responds during manufacturing.
Density calculations connect the atomic model to measurements made in a lab. The mass of one unit cell comes from the effective number of atoms and the mass of each atom. The cell volume comes from its dimensions.
Units matter throughout this process. If molar mass is in grams per mole, the final volume should usually be in cubic centimeters when density is wanted in grams per cubic centimeter. A length given in nanometers or picometers must be converted before cubing it.
Small conversion mistakes become much larger after the volume is calculated. Real crystals can contain vacancies, impurities, and grain boundaries, so measured density may differ slightly from the ideal value. The ideal calculation still gives a strong reference for identifying materials and checking experimental results.