Close Packing, Unit Cells and Ionic Lattice Types
A unit cell describes a repeating crystal structure. Counting its shared particles and locating its interstitial sites connects the structure to density and coordination. Ionic size models provide useful estimates, but bonding and conditions can change the arrangement.
What a learner can do afterwards
- Counts atoms per unit cell for the cubic types and calculates a density from cell dimensions
- Distinguishes hexagonal from cubic close packing by their stacking sequence
- Locates tetrahedral and octahedral holes and links occupancy to a named structure type
- Uses a radius ratio to predict a coordination number and states where the prediction fails
1 · Read
A unit cell repeats by translation to describe a crystal. A primitive cell contains one lattice point; a conventional cell may be larger to show symmetry. In cubic cells, corners contribute 1/8, face centres 1/2 and interior atoms 1, according to how many cells share them.
For conventional monatomic cells, simple cubic has only corners. Body-centred cubic (BCC) adds an interior atom. Face-centred cubic (FCC) adds six face-centred atoms to eight corners. Count corner contributions of 1/8 and face contributions of 1/2. Ionic structures can have multi-atom bases.
Let a be the cell edge and r the sphere radius. For touching-sphere FCC, 4r = a sqrt(2). Cell mass is nM/N_A; volume is a cubed. Density is cell mass/volume = nM/(N_A a cubed), where n counts atoms, M is molar mass and N_A is Avogadro's constant. Use consistent units.
Hexagonal close packing (HCP) repeats ABAB; cubic close packing (CCP) repeats ABCABC and equals FCC. Letters label layer positions. Both ideal packings have 12 nearest neighbours per sphere. Other stacking sequences exist.
A tetrahedral hole has four surrounding spheres; an octahedral hole has six. For N close-packed spheres there are 2N tetrahedral and N octahedral holes. Rock salt (NaCl) has an FCC chloride array with all octahedral holes filled by sodium. Zinc blende has an FCC sulfide array with half the tetrahedral holes filled by zinc. FCC octahedral sites are at the body centre and edge midpoints; tetrahedral sites are one quarter or three quarters along each axis.
In an ideal hard-sphere ionic model, r+/r- from about 0.225 to 0.414 suggests coordination 4; 0.414 to 0.732 suggests 6; 0.732 to 1 suggests 8. Here r+ and r- are cation and anion radii. These ranges are approximate predictions, not laws. Covalent bonding, pressure and changes in effective ionic radii can defeat the model.
Count shared particles, calculate mass per cell volume, and distinguish stacking and hole occupancy. Radius ratios give model-based estimates.
Unit-cell sharing and cubic-structure explanations adapted from OpenStax Chemistry 2e, section 10.6. Additional teaching and questions were written for this project.
The rewritten readings were written for this dataset.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.