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UPSC Civil Services (Mains)

Chemistry optional

  • 2 papers
  • 250 marks each
  • 3 h per paper
  • 5 of 8 questions, Q1 and Q5 compulsory

Paper I covers physical and inorganic chemistry.

UPSC Civil Services (Mains), Chemistry optional · 2016 · Paper I

Q1(c)(i). Derivation of the Bragg equation

  • 5 marks
  • Answered
Question

Derive the relation nλ=2dθ in crystallography.

Answer in brief

X-rays reflected by successive lattice planes, d apart, travel paths differing by 2dθ. They reinforce only when this difference is a whole number of wavelengths: nλ=2dθ.

Key points

  • Treat the crystal as a stack of parallel lattice planes a distance d apart. X-rays of wavelength λ meet them at the glancing angle θ, and each plane reflects a small part of the beam (angle of reflection = angle of incidence).
  • Ray 1 is reflected at O in plane 1; ray 2 at B in plane 2, directly below O. Draw OA and OC perpendicular to the incident and reflected parts of ray 2. Ray 2 travels the extra path AB + BC.
  • The angle AOB equals θ (sides mutually perpendicular to those of the glancing angle). In triangle OAB, OB =d, so AB =dθ; in the same way BC =dθ: path difference =2dθ
  • The reflected rays are in phase only when the path difference is a whole number of wavelengths: nλ=2dθ n=1,2,3,…

The full model answer

Written out in full, as it should go in the answer booklet. The solution page with its diagrams follows it.

  1. Treat the crystal as a stack of parallel lattice planes a distance d apart. X-rays of wavelength λ meet them at the glancing angle θ, and each plane reflects a small part of the beam (angle of reflection = angle of incidence).
  2. Ray 1 is reflected at O in plane 1; ray 2 at B in plane 2, directly below O. Draw OA and OC perpendicular to the incident and reflected parts of ray 2. Ray 2 travels the extra path AB + BC.
  3. The angle AOB equals θ (sides mutually perpendicular to those of the glancing angle). In triangle OAB, OB =d, so AB =dθ; in the same way BC =dθ: path difference =2dθ
  4. The reflected rays are in phase only when the path difference is a whole number of wavelengths: nλ=2dθ n=1,2,3,…
  5. n is the order of reflection. At other angles the waves from the many planes cancel, so sharp reflections appear only at the Bragg angles; since θ≤1, λ≤2d is needed.

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Model answer to Q1(c)(i), 2016 Paper I: solution page 5
Solution page 5 of 46
Read the whole 2016 solution2016 question paper

All parts of Question 1