X Rays and the Ordering of the Elements — Question 7

Question 7

A Bragg angle converts a characteristic X-ray line into an atomic-number estimate

A crystal with plane spacing $d=.200\,\mathrm{nm}$ gives a first-order reflection at angle $\theta=30.0^\circ$ measured from the planes. Use $2d\sin\theta=\lambda$, $hc=1.240\,\mathrm{keV\,nm}$, and an approximate calibration $E_\alpha=.0100(Z-1)^2\,\mathrm{keV}$.

1. Find the wavelength and photon energy.

2. Infer the nearest integer atomic number within this approximate model.

Question 7 - Solution

A Bragg angle converts a characteristic X-ray line into an atomic-number estimate

(a) Method and calculation

$$\lambda=2(.200)\sin30^\circ=\boxed{.200\,\mathrm{nm}}.$$$$\boxed{E_\alpha=1.240/.200=6.20\,\mathrm{keV}}.$$

(b) Interpretation and further calculation

$$Z=1+\sqrt{E_\alpha/.0100}=1+\sqrt{620}=25.90.$$$$\boxed{Z\simeq26}.$$

Physical check

The Bragg angle is measured relative to the crystal planes, not their normal. The integer assignment is approximate because the empirical calibration is rounded.