MATS1192 Chap.2 Crystal Structures and Atomic Packing
Crystal Structures and Atomic Packing
A crystalline material has long-range periodic order. The unit cell is the smallest repeating geometric description, but it should not be confused with a physical grain. A grain contains many unit cells with one orientation, and a grain boundary separates regions whose crystal orientations differ.
An amorphous solid retains local neighbours without a repeating long-range pattern.
The hard-sphere model supports comparisons among simple cubic, body-centred cubic, face-centred cubic and hexagonal close-packed structures. Coordination number counts nearest neighbours, while atomic packing fraction compares occupied sphere volume with unit-cell volume.
Simple cubic has coordination six and packing fraction about 0.52. Body-centred cubic has coordination eight and packing fraction about 0.68. Face-centred cubic and hexagonal close-packed both have coordination twelve and packing fraction about 0.74. Their shared efficiency does not make them identical because their stacking sequences differ.
Close-packed planes contain repeating A and B layer positions.
A third layer placed over the first gives the ABAB sequence of hexagonal close packing. A third distinct position produces the ABCABC sequence associated with face-centred cubic packing. This stacking test is more discriminating than memorising only the shared coordination number.
Ceramics cannot be packed by size alone.
Charge neutrality constrains the ratio of cations and anions, and radius ratio affects how many anions can surround a cation stably. Covalent directionality can impose further geometrical limits. When comparing real densities, packing fraction is only one input; atomic masses and lattice dimensions also matter.
What this chapter covers
- 01
Short-range and long-range atomic order
- 02
Unit cells, lattice points, grains and grain boundaries
- 03
Coordination number as a local neighbour count
- 04
Packing fraction as a hard-sphere volume ratio
- 05
Simple cubic, body-centred cubic and close-packed structures
- 06
ABAB and ABCABC stacking sequences
- 07
Charge neutrality and radius constraints in ceramics
Worked example · free
Distinguish two close-packed structures
- +1The first two close-packed layers occupy A and B positions.
- +1A third distinct set of hollows gives C, so the repeating sequence is ABCABC.
- +1ABCABC close packing corresponds to the face-centred cubic structure; ABAB would identify hexagonal close packing.
Key terms
- Crystal lattice
- A periodic array describing repeated positions in a crystalline solid.
- Unit cell
- The smallest repeating geometric unit used to describe a crystal.
- Coordination number
- The count of nearest neighbours around a selected site.
- Packing fraction
- Occupied hard-sphere volume divided by total unit-cell volume.
- Close packing
- An arrangement of equal spheres with maximum packing efficiency.
- Grain boundary
- An interface separating crystals with different orientations.
Crystal Structures and Atomic Packing FAQ
Are face-centred cubic and hexagonal close-packed identical?
No. They share coordination twelve and packing fraction about zero point seven four, but their layer sequences differ clearly. Face-centred cubic uses ABCABC stacking, while hexagonal close packing uses ABAB.
Does amorphous mean atoms have no arrangement?
No. Amorphous materials still have local neighbour relationships and short-range order. What they lack is the periodic long-range repetition that defines a crystalline solid structure.
Can packing fraction alone predict material density?
No. It compares geometric occupation for a sphere model. Real density also depends on the mass of the atoms associated with the cell and the actual cell dimensions.
Why are ceramic structures harder to predict?
Ionic ceramics must satisfy charge neutrality and stable cation coordination, while covalent character can add directionality. These constraints mean equal-sphere packing efficiency is not the only selection rule.
Is a grain the same thing as a unit cell?
No. A unit cell is a repeating geometric description. A grain is a much larger crystalline region containing many unit cells in one orientation and bounded by differently oriented neighbours.
Exam move
Build a comparison table from nearest-neighbour count, packing fraction and layer sequence. Derive at least one packing result from atom counts and geometry rather than memorising every value. Sketch A and B layers, then decide whether the next layer returns to A or occupies C.
Keep the scale hierarchy visible: atoms form unit cells, repeated cells form crystals, and differently oriented crystals form grains in a polycrystalline sample. For ceramics, add charge neutrality and radius compatibility before making a coordination claim. Recreate the close-packed layer sketch from a blank page and label which hollows remain available after the second sheet is placed.
Use the third sheet to distinguish the two sequences, then state the shared coordination and packing values as checks rather than identifiers. For a packing calculation, write occupied fraction and unoccupied fraction on separate lines and confirm that they sum to unity. When comparing real materials, list atomic mass and cell dimensions beside packing efficiency so a geometric result is not mistaken for a density prediction.
Finish ceramic questions by testing charge ratio, cation size and bond directionality together; a candidate arrangement that fails any one constraint needs revision.
