A certain atom emits light of frequency $f_0$ when at rest. A monatomic gas composed of these atoms is at temperature $T$. Some of the gas atoms move toward, and others away from, an observer due to their random thermal motion. Using the rms speed of thermal motion, show that the fractional difference between the Doppler-shifted frequencies for atoms moving directly toward the observer and directly away from the observer is $\Delta f / f_0 \approx 2 \sqrt{3 k T / m c^2}$. Assume $m c^2 \gg 3 k T$.

Solution

Verified**Problem Definition and Givens:**

A certain atom emits monochromatic light with frequency $f_0$ when at rest.

A gas composed of the this atom is heated to a temperature $T$.

This result in some gas particles moving toward the observer and some are moving away from the observer due to the random thermal motion.

We are required to find the relation between the difference between red shifted and blue shifted frequency, against the original frequency produced by the atoms.