UPSC Combined Geo-Scientist, Mains · 2023 · Mains Paper II
Q1(d). Planes that diffract: simple cubic, a = 5 Å
Question
A simple cubic crystal diffracts an X-ray beam (λ = 3 Å) at an angle of 17.5°. Identify the set of planes that cause this diffraction. Given that a = 5 Å.
Answer in brief
d = 4.99 Å ≈ a, so h² + k² + l² = 1: the (100) planes (the {100} set), first-order reflection.
Key points
- With θ = 17.5° and n = 1: d = nλ/2 sin θ = 3/(2 × 0.30071) = 4.988 Å.
- h² + k² + l² = (a/d)² = (5/4.988)² = 1.005 ≈ 1.
- Only (hkl) = (100) gives 1 ⇒ the (100) planes, first order.
- Check: θ(100) = 17.46°, θ(110) = 25.10°, θ(111) = 31.31°; second order would need d = 9.98 Å > a.
- A simple cubic lattice reflects from every (hkl); bcc and fcc show systematic absences, so (100) itself would be missing there.
The full model answer
As written for the answer booklet: the diagram carries the working, the lines stay short. Read here only; it is not offered as a download.
CoversBragg's lawX-ray diffractionMiller indicessimple cubicinterplanar spacing
All parts of Question 1
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- 1(b) Exceptional behaviour of hydrogen and helium5 marks
- 1(c) Unit cell of solid C₆₀ and its carbon atoms5 marks
- 1(d) Planes that diffract: simple cubic, a = 5 Å5 marks
- 1(e) Chemical potential and its link with Gibbs energy5 marks
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- 1(g) Activation energy of 2HI → H₂ + I₂5 marks
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- 1(j) pH from a calomel–quinhydrone cell5 marks
- 1(k) Degeneracy of the level 6h²/8mL² in a cubic box5 marks
- 1(l) Is Peⁱᵏˣ + Qe⁻ⁱᵏˣ an eigenfunction?5 marks
- 1(m) Zero-point energy in joule and in cm⁻¹5 marks
- 1(n) Microwave activity of SF₆, CO, H₂, H₂O and CO₂5 marks
- 1(o) Stark–Einstein law and the quantum yield5 marks
- 1(p) Decay routes of a photoexcited molecule5 marks