Doppler broadening.The top shows the (random) motions of a group of particles.The purple vectors are the actual velocities; the green vectors are the radial components, which produce the Doppler shift. For each particle, identified by a number, the radial velocities are plotted below.The line profile is the sum of all the individual Doppler shifted signals, and with many more particles it would have a smooth appearance.

Line profile.We plot intensity as a function of wavelength.

containing the particles have some overall motion that just shifts the center wavelength of the line, the broadening would still be the same.

We can estimate the broadening as a function of temperature. If (v2) is the average of the square of the random velocities in a gas, and m is the mass per particle, the average kinetic energy per particle is (1/2)m(v2). If we have an ideal monatomic gas, this should equal (3/2)kT, giving

(1/2)m(v2) = (3/2)kT Solving for (v2) gives (v2) = 3kT/m

Taking the square root gives the root mean square (rms) speed

This gives us an estimate of the range of speeds we will encounter. (The range will be larger, since we can have atoms coming toward us or away from us, and since some atoms will be moving faster than this average, but the radial velocity is reduced since only the component of motion along the line of sight contributes to the Doppler broadening.) To find the actual wavelength range over which the line is spread out, we would use the Doppler shift expression in equation (5.4).

Example 6.1 Doppler broadening

Estimate the wavelength broadening in the Ha line in a gas composed of hydrogen atoms at T = 5500 K.

x v rms

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