Q.Calculate the number of atoms in an FCC 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 …

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.

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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 …

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 $= 8 \times \dfrac{1}{8} = 1$. …

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:

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

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