Q.Define unit cell.

A unit cell is the smallest three-dimensional portion of a crystal lattice that, when repeated in all directions, reproduces the whole crystal structure.

A crystal lattice is a regular three-dimensional arrangement of points (representing constituent particles - atoms, ions, or molecules) in space. The unit cell is the fundamental building block of this lattice: it is characterised by its edge lengths (a, b, c) and the angles between them (alpha, beta, gamma). Translating (repeating) the unit cell along all three crystallographic axes reconstructs the complete crystal lattice.

Note: this concept belongs to the Solid State chapter, removed from the current rationalised NCERT Class-12 syllabus, so it is flagged out of syllabus even though the answer is given here.

AgBr is the textbook example that shows both a Schottky and a Frenkel defect together.

A Schottky defect occurs when an equal number of cations and anions leave their lattice sites completely, creating vacancies while keeping the crystal electrically neutral; it lowers the density of the solid and is common in ionic solids where the cation and anion are similar in size (e.g. NaCl, KCl, CsCl).

A Frenkel defect occurs when a smaller ion — usually the cation, because it is small enough to fit — is displaced from its normal lattice site into an interstitial site instead of leaving the crystal. Density is unaffected because no ion is actually lost.

Most ionic solids show only one of the two, depending on the relative sizes of their ions. Silver bromide (AgBr) is unusual: Ag⁺ is small enough to occupy interstitial sites (favouring a Frenkel defect), while the Ag⁺/Br⁻ size ratio is also close enough to allow paired vacancies (a Schottky defect). As a result, AgBr genuinely exhibits both defects at once.

Radius ratio predicts ionic-crystal geometry; unit cell is the basic repeating building block of any crystal lattice.

1. Radius Ratio The radius ratio of an ionic compound is defined as: $\text{Radius ratio} = \frac{r_{cation}}{r_{anion}}$ Since cations are usually smaller than the anions surrounding them, this ratio is normally less than 1. Its value determines how many anions can pack around a central cation without touching each other — i.e., it determines the coordination number and geometry of the ionic crystal. For example, radius ratio 0.225–0.414 favours tetrahedral coordination (CN=4), 0.414–0.732 favours octahedral coordination (CN=6), and 0.732–1.0 favours cubic coordination (CN=8).
2. Unit Cell A unit cell is the smallest repeating structural unit of a crystal lattice which, when repeated (translated) indefinitely in all three dimensions, generates the entire crystal structure. It is characterised by:

• three edge lengths $a, b, c$
• three angles between the edges $\alpha, \beta, \gamma$

Different combinations of these parameters define the seven crystal systems (cubic, tetragonal, orthorhombic, etc.).

In bcc, the 8 corner atoms together contribute 1 atom (each shared 8 ways) and the single body-centre atom contributes fully, giving 2 atoms per unit cell.

Contribution counting in a bcc unit cell:

• Corner atoms: 8 corners, each atom shared among 8 adjacent unit cells, so each contributes $\dfrac{1}{8}$ atom. Total from corners $= 8 \times \dfrac{1}{8} = 1$.
• Body-centre atom: 1 atom lies entirely within the unit cell (not shared with any neighbouring cell), contributing the full atom $= 1$.

Total atoms per bcc unit cell $= 1 + 1 = 2$.

(For comparison: a simple cubic cell has 1 atom/cell; an fcc cell has 4 atoms/cell — $8 \times \dfrac{1}{8} + 6 \times \dfrac{1}{2} = 1 + 3 = 4$.)

An FCC unit cell contains 4 effective atoms, and the number of tetrahedral voids equals twice the number of atoms (2Z), giving 8.

In a face-centred cubic (FCC) lattice, the effective number of atoms per unit cell is:

$Z = 8 \times \tfrac{1}{8} \text{(corners)} + 6 \times \tfrac{1}{2} \text{(face centres)} = 1 + 3 = 4$

For any close-packed arrangement (FCC or HCP), the number of octahedral voids equals Z, and the number of tetrahedral voids equals 2Z. This is because each atom in a close-packed structure is surrounded by 2 tetrahedral voids per atom on average.

So, tetrahedral voids = 2 × 4 = 8.