San José State University

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Thayer Watkins
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 The Line Shape Produced by Doppler Broadening

A molecule has a variety of vibration modes. When radiation of the same frequency as one of these modes hits the molecule, the molecule vibrates and then reradiates that energy but in a random direction. Thus some of the impinging radiation will be radiated back in the direction it came from. This is the mechanism of the greenhouse effect. Radiation which is not of a frequency equal to one of the vibration modes of the molecule does not interact with the molecules.

## The Broadening of the Spectral Lines

The discrete set of vibration frequencies of a molecule is called its spectrum; this is both a la Kirchhoff's Law its absorption spectrum and its emissions spectrum. If the impinging radiation had to have exactly the wavelength of the discrete spectral lines there would not be much interaction between the radiation and the molecules because of the small probability of the radiation having exactly that wavelength.

The spectrum is modified by the motion of the molecules. The Doppler effect is the modification of the perceived frequency of radiation due to the motion of the molecule. If the molecule is traveling in opposite direction from the incoming radiation the perceived frequency of the radiation is greater. Thus if radiation were slightly lower frequency than a vibration frequency of a molecule the Doppler effect could bring about a coincidence with the vibration frequency of the molecule. If a molecule were traveling in the same direction as incoming radiation the Doppler effect lowers its perceived frequency and thus could result in the absorption of radiation of a slightly higher frequency.

The relationship that prevails is that if V is the velocity of the molecule relative to the direction of propagation of the radiation then the perceived frequency νd is given by

#### νd = ν0(1 − V/c)-1

where ν0 is the true frequency and c is the speed of light.

## The Doppler Line Shape Function

The distribution of the velocities of the molecules in a gas is Gaussian; i.e., the normal bell-shaped curve. This leads to a distribution of the frequencies of the absorbed radiation which is also Gaussian; i.e.,

#### (1/√2πσ)exp(−(ν-ν0)²/(2σ²))

where σ is the line width.

In effect the lines of the absorption spectrum are broadened by the Doppler effect. The width of the broadened curve σ about the line at frequency ν0 is given by

#### σ = (ν0/c)(2kTln(2)/m)½

where m is the mass of the molecule and T is the absolute temperature of the gas.

For information on other line shape see Line Shapes