AMME1362 Chap.9 Unit Cells, Packing and Crystal Density
Unit Cells, Packing and Crystal Density
A unit cell represents the repeating geometry of a crystal. Counting fractional atoms, identifying where hard-sphere contact occurs and relating atomic radius to edge length allow coordination, packing factor and theoretical density to be calculated. The calculation is sensitive to powers of ten and volume conversion.
Differences between theoretical and measured density can reveal porosity, defects, chemistry or phase mixture rather than an algebra error.
What this chapter covers
- 01
Lattices and unit cells
- 02
Fractional atom sharing
- 03
Coordination numbers
- 04
Atomic packing factors
- 05
Radius-edge relations
- 06
Theoretical crystal density
Worked example · free
Calculate cubic crystal density
- 1Cell mass is 4 times 60 divided by 6.022 times 10²³, giving 3.985 times 10⁻²² g.
- 1Convert 0.400 nm to 4.00 times 10⁻⁸ cm.
- 1Cube the edge to obtain 6.40 times 10⁻²³ cm³.
- 1Divide cell mass by cell volume to obtain 6.23 g/cm³.
Key terms
- Unit cell
- A geometric repeat unit that generates a crystal lattice by translation.
- Coordination number
- The number of nearest neighbours around a selected atom or ion.
- Packing factor
- Hard-sphere atomic volume divided by total unit-cell volume.
- Crystal density
- Unit-cell mass divided by unit-cell volume for an ideal structure.
- Site occupancy
- The average fraction of a crystallographic site populated by a specified atom or ion. Partial occupancy changes the unit-cell mass contribution and must remain consistent with composition, charge balance and the structural model.
- Formula unit
- The smallest stoichiometric combination represented in an ionic or compound crystal. The number of formula units per cell multiplies the full formula molar mass when calculating ideal crystal density.
- Contact geometry
- The crystallographic line along which neighbouring hard-sphere atoms or ions touch in an ideal model. It determines the radius-edge relation and differs among simple cubic, body-centred cubic and face-centred cubic structures.
Unit Cells, Packing and Crystal Density FAQ
Why does a corner atom count as one eighth?
Eight neighbouring cubic cells meet at a corner, so each cell contains one eighth of that atom in the repeating bookkeeping model. Summing all corner contributions gives the effective count.
Why must the radius relation match the structure?
Atoms touch along different lines in simple cubic, body-centred cubic and face-centred cubic arrangements. Using the wrong contact diagonal produces an incorrect edge length, volume and density.
What can lower measured density below theory?
Porosity is a common cause, but vacancies, substitutional composition, multiple phases and measurement conditions can also matter. Compare the specimen state with the ideal unit-cell assumptions before interpreting the difference.
Exam move
Use a three-layer worksheet for every crystal calculation: visual inventory, geometric relation and physical audit. In the inventory layer, count contributions from corners, faces, edges, body positions and wholly internal sites. Write the sharing fraction beside each location and sum effective atoms or formula units before opening a density equation. The perspective sketch is evidence for the count, not a substitute for it.
In the geometry layer, mark where neighbouring hard spheres touch. Derive edge length from atomic or ionic radius along that line for the stated structure. Redraw simple cubic, body-centred cubic and face-centred cubic contact paths on separate cells. Do not transfer a face diagonal relation to a body diagonal arrangement. In the audit layer, check coordination, packing, stoichiometry and density magnitude.
Recalculate theoretical density with at least three combinations of atom count, molar mass and edge length. Write cell mass as effective atoms times molar mass divided by Avogadro's constant. Convert nanometres to centimetres before cubing and show the exponent. Reverse one problem to solve for edge length or molar mass so the relation is understood rather than recited.
Practise compounds by counting cations and anions separately. Reduce the cell inventory to the chemical formula and test charge neutrality where ionic charges are supplied. Only after that should the combined formula mass enter the numerator. A density result from a cell with the wrong stoichiometric ratio is not rescued by plausible units. Build a short comparison of coordination number and packing factor.
Coordination is a local nearest-neighbour count; packing factor is a cell-volume fraction in the hard-sphere model. Give examples of why related trends do not make the quantities interchangeable. Include voids as geometry created by packing rather than unlabelled empty space. Compare measured and theoretical density through hypotheses.
Lower bulk density may indicate porosity, vacancies, substituted atoms, phase mixture or temperature-dependent dimensions. Write which additional observation would distinguish these possibilities, such as microscopy for pores or composition analysis for substitution.
Finish each practice set by explaining periodicity: boundary atoms are shared because the lattice continues into adjacent cells, not because the atom itself is physically divided. Reproduce the complete derivation from cell inventory to grams per cubic centimetre on a blank page. Mark any step where the structure supplies information and any step where a unit conversion supplies information.
This separation makes most errors easy to locate and correct.
