The X-ray Crystallographer's Dilemma · AI Generator #GEN-1324D92A

Physics · modern-physics

Problem

A crystallographer is studying a new metal alloy using X-ray diffraction. She has two X-ray sources available: one produces characteristic X-rays from copper (Cu,

Z = 29

) and another from molybdenum (Mo,

Z = 42

). Both sources produce their respective

K_\alpha

X-rays through electron bombardment. The copper source requires electrons to be accelerated through a potential difference of

V_{Cu} = 9.0\;\text{kV}

to efficiently produce

K_\alpha

radiation, while the molybdenum source requires

V_{Mo} = 21.0\;\text{kV}

. The crystallographer wants to determine which source produces X-rays with a de Broglie wavelength closest to the spacing between crystal planes in her sample, which is

d = 2.5\;\text{Å}

. Using Moseley's law, where the frequency of

K_\alpha

X-rays follows

\sqrt{\nu} = R(Z - \sigma)

with

R = 2.48 \times 10^7\;\text{Hz}^{1/2}

and screening constant

\sigma = 1.0

for

K_\alpha

transitions, determine which X-ray source produces photons whose de Broglie wavelength is closer to the crystal plane spacing.

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