The X-ray Crystallographer's Puzzle · AI Generator #GEN-7355566E

Physics · modern-physics

Problem

A crystallographer is analyzing an unknown metal using X-ray spectroscopy. She observes that when electrons with kinetic energy

35.0\;\text{keV}

strike the metal target, the shortest wavelength (minimum wavelength) X-ray produced is

\lambda_{\text{min}} = 35.4\;\text{pm}

. However, she also notices a characteristic X-ray line at wavelength

\lambda = 154\;\text{pm}

, which corresponds to a

K_\alpha

transition where an electron drops from the

L

shell to fill a vacancy in the

K

shell. The energy difference between the

L

and

K

shells in this metal is

8.05\;\text{keV}

. The crystallographer wants to determine whether her electron beam has sufficient energy to knock out

K

-shell electrons and produce this characteristic line. (a) What is the binding energy of the

K

-shell electrons in this metal? (b) Can her electron beam produce the observed

K_\alpha

line? Justify your answer quantitatively. Constants:

h = 6.63 \times 10^{-34}\;\text{J·s}

c = 3.00 \times 10^8\;\text{m/s}

1\;\text{eV} = 1.60 \times 10^{-19}\;\text{J}