X-ray Crystallography Detective Work · AI Generator #GEN-EEA05BDB

Physics · modern-physics

Problem

An X-ray crystallography lab discovers an unknown metallic element in a meteorite sample. They bombard the sample with high-energy electrons to generate characteristic X-rays. The unknown element produces a prominent Kα_\alpha X-ray line at wavelength λ=0.154  nm\lambda = 0.154\;\text{nm}. From previous analysis of the meteorite, they know the element has atomic mass A=64  uA = 64\;\text{u} and forms a simple cubic crystal structure with lattice parameter a=3.61×10−10  ma = 3.61 \times 10^{-10}\;\text{m}. Using Moseley's law, the Kα_\alpha X-ray frequency is given by: ν=R∞c⋅34⋅(Z−σ)2\nu = R_\infty c \cdot \frac{3}{4} \cdot (Z - \sigma)^2 where R∞=1.097×107  m−1R_\infty = 1.097 \times 10^7\;\text{m}^{-1} is the Rydberg constant, c=3.00×108  m/sc = 3.00 \times 10^8\;\text{m/s}, and σ=1.0\sigma = 1.0 is the screening constant for Kα_\alpha transitions. (a) Determine the atomic number ZZ of the unknown element. (b) Calculate the number density of atoms in the crystal (atoms per unit volume). (c) Estimate the de Broglie wavelength of the thermal motion of atoms at room temperature (T=300  KT = 300\;\text{K}), assuming the atoms behave as a classical ideal gas. Use kB=1.38×10−23  J/Kk_B = 1.38 \times 10^{-23}\;\text{J/K} and atomic mass unit u=1.66×10−27  kgu = 1.66 \times 10^{-27}\;\text{kg}.

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