Crystal Detector Calibration · AI Generator #GEN-66FD1319

Physics · modern-physics

Problem

A vintage crystal radio detector uses a galena (lead sulfide) crystal with cubic structure. The crystal has a lattice parameter

a = 5.94 \times 10^{-10}\;\text{m}

. To calibrate the detector, an engineer shines X-rays of wavelength

\lambda = 1.54 \times 10^{-10}\;\text{m}

onto the crystal surface at various angles. She observes that the first-order diffraction peak occurs when the X-ray beam makes an angle of

\theta = 15.0°

with the crystal surface. The engineer needs to determine which set of crystal planes is responsible for this diffraction peak. In a cubic crystal, the spacing between parallel planes with Miller indices

(h, k, l)

is given by:

d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}

Find the Miller indices

(h, k, l)

of the crystal planes causing this diffraction peak, assuming the simplest possible combination of small integer values.

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