Crystal Structure (20 marks): 21-Mat-A4 Deformation Behaviour and Properties of Materials, May 2013, Q2

Question II — Crystal Structure (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

II.1 — Planes and directions

The (111) plane in a cubic cell intercepts all three axes at one lattice parameter and is the close-packed plane family of the FCC structure; the (210) plane intercepts the $a$-axis at $a/2$, the $b$-axis at $a$, and never crosses the $c$-axis (it runs parallel to $c$). In the hexagonal system, $(10\bar{1}0)$ is one of the six equivalent prismatic side faces of the hexagonal prism (it contains the $c$-axis and is perpendicular to the basal plane), and $[11\bar{2}0]$ is a close-packed basal-plane direction connecting the cell centre to a prism corner, perpendicular to the $c$-axis.

(111): equal intercepts on all three axes — the close-packed triangular plane of an FCC cell.

(210): intercepts at a/2, b, and parallel to c — a plane tilted only about the c-axis.

Hexagonal prism: the shaded face is (10-10); the blue arrow is [11-20] in the basal plane.

II.2 — MgO ionic packing factor and density

Given. NaCl-type (rock-salt) structure: cation and anion touch along the cube edge, $r(\text{Mg}^{2+}) = 0.078\ \text{nm}$, $r(\text{O}^{2-}) = 0.132\ \text{nm}$; molar masses $M_{\text{Mg}} = 24.31\ \text{g/mol}$, $M_{\text{O}} = 16\ \text{g/mol}$; $N_A = 6.023\times10^{23}\ \text{mol}^{-1}$.

Find. The ionic packing factor and the mass density of MgO.

Approach. In the rock-salt structure the cell edge is set by cation–anion contact along the edge, $a = 2(r_{Mg}+r_O)$; the conventional cubic cell holds 4 Mg²⁺ and 4 O²⁻ ions (an FCC lattice of anions with cations filling every octahedral hole).

Rock-salt MgO: O2- on an FCC sublattice, Mg2+ filling every octahedral interstice — each ion is 6-coordinate.

Cell edge. $$a = 2(r_{Mg}+r_O) = 2(0.078+0.132) = \boxed{0.420\ \text{nm}}$$
Ionic packing factor. Four Mg²⁺ and four O²⁻ ions per conventional cell: $$\text{IPF} = \frac{4\cdot\tfrac{4}{3}\pi r_{Mg}^3 + 4\cdot\tfrac{4}{3}\pi r_O^3}{a^3} = \frac{4\cdot\tfrac{4}{3}\pi(0.078)^3 + 4\cdot\tfrac{4}{3}\pi(0.132)^3}{(0.420)^3} = \boxed{0.627}$$ About 62.7% of the cell volume is occupied by ions — higher than the 0.52 of simple cubic packing because the small cation nests efficiently in the octahedral hole between six anions.
Mass density. Cell mass $= 4(M_{Mg}+M_O)/N_A$, cell volume $=a^3$ (convert $a$ to cm: $0.420\ \text{nm} = 4.20\times10^{-8}\ \text{cm}$): $$\rho = \frac{4(24.31+16.00)}{(6.023\times10^{23})(4.20\times10^{-8})^3} = \boxed{3.61\ \text{g/cm}^3}$$ This is close to the handbook value for MgO periclase (≈3.58 g/cm³), confirming the rock-salt model.

Question II — final results
Quantity	Value
Cell edge, $a$	0.420 nm
Ionic packing factor	0.627 (62.7%)
Mass density	3.61 g/cm³

Topic. Miller/Miller–Bravais indexing and the rock-salt (NaCl-type) ionic structure.

Key relations. Edge-contact condition $a=2(r_c+r_a)$; $\text{IPF}=\sum V_{\text{ions}}/V_{\text{cell}}$; $\rho = nM/(N_A V_{\text{cell}})$.

Why this works. MgO's rock-salt structure is two interpenetrating FCC sublattices (O²⁻ and Mg²⁺), each ion 6-fold coordinated by the other, so the geometry reduces to the same edge-contact and lever-of-four-ions bookkeeping used for NaCl itself.

Common pitfall. Using the face-diagonal contact condition (correct for same-size touching spheres in FCC) instead of the edge condition that actually applies to rock-salt's cation–anion contact.

Source. Callister & Rethwisch, Materials Science and Engineering, 10th ed., Ch. 12 (Ceramic structures) and Ch. 3 (Miller indices).

Question 2 of 8: Crystal Structure (20 marks)

Question II — Crystal Structure (20 marks)

II.1 — Planes and directions

II.2 — MgO ionic packing factor and density