Step-by-Step KaTeX Derivation & Detailed Explanation
### Step-by-Step Derivation & Verification: 1. **FCC Basis Coordinates**: Basis atoms are at $(0, 0, 0)$, $(1/2, 1/2, 0)$, $(1/2, 0, 1/2)$, and $(0, 1/2, 1/2)$. 2. **Geometric Structure Factor Formulation**: $F_{hkl} = \sum_j f_j e^{-2\pi i (h x_j + k y_j + l z_j)} = f \left[1 + e^{-i\pi(h+k)} + e^{-i\pi(h+l)} + e^{-i\pi(k+l)}\right]$ 3. **Selection Rule Evaluation**: - **Case 1 (All Even)**: $h+k, h+l, k+l$ are all even $\implies e^{-i\pi(\text{even})} = +1 \implies F_{hkl} = 4f \neq 0$ (ALLOWED). - **Case 2 (All Odd)**: $h+k, h+l, k+l$ are all even $\implies F_{hkl} = 4f \neq 0$ (ALLOWED). - **Case 3 (Mixed Even/Odd)**: Two exponentials equal $-1$ and one equals $+1$, giving $1 + 1 - 1 - 1 = 0 \implies F_{hkl} = 0$ (FORBIDDEN). 4. **Inspection of Options**: - $(111)$: All odd (ALLOWED) - $(200)$: All even (ALLOWED) - $(220)$: All even (ALLOWED) - $(110)$: Mixed (two odd, one even) $\implies F_{110} = 0$ (FORBIDDEN). 5. **Conclusion**: Reflection $(110)$ is absent. Option (C) is correct.