A student draws an FCC unit cell showing 8 corner atoms and 6 face-center atoms. A classmate says the unit cell contains 14 atoms — one for each atom visible in the diagram. What is the correct count?
A14 — all atoms shown in the diagram belong to the unit cell
B6 — only the face-center atoms belong fully; corner atoms are shared with other cells
C4 — corner atoms contribute 1/8 each (total 1) and face-center atoms contribute 1/2 each (total 3)
D2 — only atoms in the strict interior of the unit cell are counted
Atoms are shared between adjacent unit cells: each corner atom sits at the intersection of 8 unit cells, contributing 1/8 per cell; each face-center atom borders 2 cells, contributing 1/2 per cell. FCC: 8×(1/8) + 6×(1/2) = 1 + 3 = 4 atoms. The classmate's error of 14 is the most common misconception — the drawing shows atoms positioned at cell boundaries, but most of their 'mass' belongs to neighboring cells. The correct count of 4 is what gives FCC its high atomic packing factor of 0.74.
Question 2 Multiple Choice
FCC has a higher atomic packing factor (0.74) than BCC (0.68). What is the primary reason?
AFCC has more atoms per unit cell (4 vs 2), and the face-centered arrangement allows atoms to touch along the face diagonal, achieving closer packing
BFCC atoms are smaller than BCC atoms, so they fit more tightly into the same lattice
CBCC has a body-center atom that creates unavoidable voids between corner atoms
DFCC has a larger unit cell volume than BCC, distributing atoms over more space and reducing overlap
In FCC, atoms touch along the face diagonal: 4r = a√2, giving the optimal relationship between atom radius and cell size. The face-centered arrangement gives each atom 12 nearest neighbors — the highest coordination number for equal-sphere packing — achieving the theoretical maximum packing of 74%. BCC atoms touch along the body diagonal (4r = a√3) with 8 nearest neighbors and 68% packing. The difference arises from the geometry of how spheres can nest together, not from differences in atom size.
Question 3 True / False
A corner atom in a unit cell contributes exactly 1/8 of an atom to that cell because it is simultaneously shared among 8 adjacent unit cells that meet at that corner.
TTrue
FFalse
Answer: True
In a three-dimensional crystal, each corner of a unit cell is where eight unit cells meet (imagine 8 cubes sharing a single corner point). The atom at that corner is shared equally among all 8 cells, contributing 1/8 to each. This convention ensures that when unit cells tile all of space, every atom is counted exactly once. The same logic gives face-center atoms a contribution of 1/2 (shared by 2 cells), edge atoms 1/4 (shared by 4 cells), and interior atoms 1 (belonging to one cell only).
Question 4 True / False
The lattice parameter 'a' of a pure metal is a fixed material constant that does not change with temperature or the addition of solute atoms.
TTrue
FFalse
Answer: False
Lattice parameters are not fixed constants. Thermal expansion increases average interatomic spacing as atomic vibration amplitudes grow with temperature, measurably increasing 'a'. Substitutional solute atoms larger than the host expand the lattice (positive misfit strain); smaller solutes contract it. These changes are not negligible — X-ray diffraction measurements of lattice parameter shifts are used in industrial quality control to measure residual stress, composition gradients, and the degree of solid solution formation.
Question 5 Short Answer
A materials engineer calculates the theoretical density of an aluminum sample from its FCC structure and lattice parameter (a = 4.05 Å), getting 2.70 g/cm³. The measured bulk density is 2.63 g/cm³. What does this discrepancy most likely indicate?
Think about your answer, then reveal below.
Model answer: The lower-than-theoretical density indicates structural defects that reduce average mass per unit volume. The most likely causes are: vacancies (missing atoms at lattice sites, reducing the actual atom count per unit volume below the perfect-crystal prediction), porosity (voids that contribute volume without mass), or substitutional impurities of lower atomic mass than aluminum. The theoretical density formula ρ = (n·A)/(V_c · N_A) assumes every lattice site is occupied by the correct atom.
Conversely, a higher measured density would suggest a denser impurity phase or measurement error. This diagnostic use of density — comparing measured to theoretical — illustrates how macroscopic, easily measured properties directly reflect atomic-scale crystal structure, making lattice parameter calculations practically useful for quality control.