Crystal Packing Efficiency
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Direct answer
Packing efficiency measures the fraction of a unit cell's volume actually occupied by spheres: 52.4 per cent for simple cubic (π/6), 68 per cent for body-centred cubic (π√3/8) and 74 per cent for face-centred cubic, cubic close-packed and hexagonal close-packed alike (π/(3√2), the mathematical maximum for equal spheres). Coordination numbers run 6, 8 and 12 in the same order. Each percentage is derivable in two lines from Z, the sphere-touching geometry and the volume of the spheres — and those derivations, not the memorised numbers, are what JEE Advanced actually tests.
What you must remember
- The three fractions in exact form: sc = π/6 ≈ 52.4%; bcc = π√3/8 ≈ 68%; fcc, ccp and hcp = π/(3√2) ≈ 74% — hcp and ccp are equally efficient, differing only in stacking sequence (ABAB versus ABCABC).
- Touching geometry that generates them: sc spheres touch along the edge (2r = a); bcc along the body diagonal (4r = √3 a); fcc along the face diagonal (4r = √2 a).
- Coordination numbers: 6 for sc, 8 for bcc, 12 for ccp and hcp.
- Void census: per sphere, one octahedral and two tetrahedral voids; per fcc cell, 4 octahedral and 8 tetrahedral — cations select voids by size.
- Radius-ratio limits: 0.225-0.414 tetrahedral; 0.414-0.732 octahedral; above 0.732 cubic coordination — the predictor for ionic structures.
- Named structures: CsCl 8:8, NaCl 6:6 (chloride fcc with sodium in every octahedral void), zinc blende ZnS 4:4 (zinc in alternate tetrahedral voids), CaF2 4:8 (fluoride in all tetrahedral voids of the calcium fcc lattice).
- Structure habits: bcc for Li, Na, K at room temperature and for chromium, tungsten and alpha-iron; ccp for Cu, Ag, Au, Al, Ni; hcp for Mg, Zn, Cd, Ti; iron converts bcc to fcc near 1180 K.
- Density caution: packing sets volume use, but density also tracks atomic mass — hcp magnesium at 74% packing is still lighter than bcc iron at 68%.
Deriving the 68 per cent of bcc
Take a bcc cell: Z = 2 atoms, and spheres touch along the body diagonal so 4r = √3 a, giving r = √3 a/4. One sphere's volume is (4/3)π r^3 = (4/3)π × 3√3 a^3/64 = π√3 a^3/16; two spheres give π√3 a^3/8; dividing by the cell volume a^3 leaves π√3/8 = 0.680 — 68 per cent. Run the same film for fcc: Z = 4, r = a/(2√2) from 4r = √2 a, so each sphere contributes (4/3)π a^3/(16√2) and four give π a^3/(3√2), a fraction of 0.7406. Then apply the void language to rock salt: chloride ions form the fcc lattice, sodium occupies every octahedral void — four per cell, matching the four chlorides — so the formula and Z = 4 drop out of geometry alone; and with r(Na+)/r(Cl−) = 95/181 ≈ 0.52 sitting inside the octahedral window of 0.414-0.732, the radius-ratio rule independently votes for six-coordination. Derivation, verification, prediction — one geometry serving all three.
Packing questions the Advanced paper likes
Alongside solid state's removal from the JEE Main syllabus in the 2023-24 rationalisation, expect JEE Advanced — which retains the chapter — to probe derivations and ionic geometry rather than recall: derive the bcc fraction, locate the tetrahedral voids of fcc at the quarter-points along body diagonals, or run a radius-ratio prediction such as Cs+ (169 pm) against Cl− (181 pm) equal to 0.93, pointing to the cubic site and the CsCl 8:8 structure. Two confusions account for most lost marks. First, voids per cell versus per atom: an fcc cell offers four octahedral voids to four spheres (1:1) but eight tetrahedral voids (2:1), and questions deliberately switch the denominator. Second, "hcp packs better than fcc" — false; both sit at 74 per cent, and any density difference between an hcp and an fcc metal is a mass effect, not a packing one.
Frequently asked questions
Why do fcc and hcp share the same packing efficiency?
Both are closest packings of equal spheres with coordination 12; only the stacking sequence differs (ABCABC versus ABAB), and that does not change the occupied fraction π/(3√2).
What radius ratio predicts octahedral coordination?
Between 0.414 and 0.732 — at the lower limit the cation exactly touches six anions arranged octahedrally; sodium chloride's ratio near 0.52 sits comfortably inside.
How many tetrahedral voids accompany each sphere in close packing?
Two per sphere, eight per fcc cell of four spheres — which is why zinc blende fills only alternate tetrahedral voids to keep ZnS at 4:4.
Which common metals adopt bcc structure?
The alkali metals lithium, sodium and potassium at room temperature, plus chromium, tungsten and alpha-iron — 68 per cent packing with eight nearest neighbours.
How is the packing fraction of simple cubic derived?
One sphere of radius a/2 per cell gives volume (4/3)π(a/2)^3 = π a^3/6; dividing by a^3 yields π/6, about 52.4 per cent.