IChO 2003 · Theory problem 13

X-ray diffraction angles

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The smallest diffraction angle of a monochromatic beam of X-rays in a certain experiment is 11.5°. Based on this we must expect a beam of X-rays diffracted at:

(a) 22.0 degrees　( )

(b) 22.5 degrees　( )

(c) 23.0 degrees　( )

(d) 23.5 degrees　( )

(e) 24.0 degrees　( )

(f) 24.5 degrees　( )

Official English text of the problem.

Answers

  • (a) Incorrect.
  • (b) Incorrect.
  • (c) Incorrect.
  • (d) Correct: \(23.5^\circ\).
  • (e) Incorrect.
  • (f) Incorrect.

Solution

Constructive interference of X-rays reflected from successive crystal planes obeys Bragg’s law:

\[ 2d\sin\theta_n=n\lambda, \]

where \(d\) is the spacing between the planes, \(\lambda\) is the X-ray wavelength, and \(n=1,2,\ldots\) is the diffraction order. Here \(\theta_n\) is the Bragg angle, measured relative to the planes.

The smallest diffraction angle corresponds to the first order, \(n=1\):

\[ 2d\sin 11.5^\circ=\lambda. \]

For the second order,

\[ 2d\sin\theta_2=2\lambda. \]

Dividing these equations eliminates both \(d\) and \(\lambda\):

\[ \sin\theta_2=2\sin11.5^\circ=0.3987359. \]

Therefore,

\[ \theta_2=\arcsin(0.3987359) =23.4992^\circ \approx\boxed{23.5^\circ}. \]

Thus (d) is the correct choice. The other listed angles do not correspond to an integer diffraction order. On the answer sheet, mark (d) and leave (a), (b), (c), (e), and (f) unmarked.

The answers were checked against the official answers published by the organisers.