QB_PH_SEC7_030: Crystal Structure & Lattice Dynamics

In a powder X-ray diffraction pattern taken with monochromatic radiation, the ratio of the square of the Miller indices $(h^2 + k^2 + l^2)$ for the first two lowest-angle diffraction peaks of a Body-Centered Cubic (BCC) crystal is

Correct Answer

### Step-by-Step Derivation & Verification: 1. **BCC Structure Factor Extinction Condition**: $F_{hkl} = f [1 + e^{-i\pi(h+k+l)}]$ - If $h + k + l = \text{odd} \implies F_{hkl} = 0$ (FORBIDDEN) - If $h + k + l = \text{even} \implies F_{hkl} = 2f$ (ALLOWED) 2. **Enumeration of Low-Index Reflections**: - $(100)$: $1 + 0 + 0 = 1$ (odd, forbidden) - $(110)$: $1 + 1 + 0 = 2$ (even, ALLOWED) $\implies s_1 = 1^2 + 1^2 + 0^2 = 2$ - $(111)$: $1 + 1 + 1 = 3$ (odd, forbidden) - $(200)$: $2 + 0 + 0 = 2$ (even, ALLOWED) $\implies s_2 = 2^2 + 0^2 + 0^2 = 4$ 3. **Ratio of $s_2 / s_1$**: $\text{Ratio} = \frac{s_2}{s_1} = \frac{4}{2} = 2.0$ 4. **Official Exact Value**: 2.