Doppler Broadening

It is the broadening of the spectral line due to Doppler Effect of the random thermal motion of the molecules. Doppler effect is the change in the frequency of a wave for an observer moving relative to the source of the wave. Spectral line is basically an isolated bright or dark line in the spectrum (portion of the electromagnetic spectrum that is visible by the human eye) produced by emission or absorption of light of a single wavelength.

There is a lack of sharpness in the spectrum line which causes the broadening and is related to the distribution of velocities of the atoms or molecules. The spontaneous emission of the different particles which is the transition from a higher state to lower state, have different velocities which undergo Doppler shifts (wavelength and frequency of light emitted by the moving object is shifted). The cumulative effect of the different shifts of the atoms or molecules in a medium causes the broadening of the spectrum line. The derived line profile is called as Doppler profile.

Thermal Doppler broadening

In a gas, the individual atoms or molecules are always moving in haphazard directions, with an average velocity directly proportional to the temperature of the gas which can be understood from the formula given below:

(1/2mv2 )= 3/2kT

1/2mv2 denotes the kinetic energy of the particles

m stands for mass

v is the average velocity of the particles

T is the temperature of the gas

K is Stefan-Boltzmann constant

Any specific atom or molecule could either be moving along or perpendicular to the electromagnetic radiation propagation or both. Every spectral line emitted is Doppler shifted relative to the observer, resulting in a slight change in the observed wavelength. The spreading of the spectral line due to the temperature of the emitting medium is called as thermal Doppler broadening. The spreading of the spectral line depends on the frequency, mass of the emitting particles and the temperature. This finds huge application in determining the temperature of the emitting body.

Doppler Broadening Derivation

The random motion of a particle which is called as thermal motion causes it to move towards the observer, whereas the emitted radiation is shifted to a higher frequency. The frequency is lowered if the emitter moves away. For non-relativistic thermal velocities (does not follow the law of relativity), the Doppler shift in terms of frequency will be:

Doppler Broadening equation 1 Picture


Where ƒ is the observed frequency, ƒ0is the rest frequency, v is the velocity of the emitter moving towards the observer and c is the speed of light.

There is a distribution of speeds both toward and away from the observer in any volume element of the radiating body which is a means for integrating a function with respect to volume in various coordinate systems and denoted as dV. Due to this phenomenon, the net effect will broaden the observed line.

If Pv(v)dv is the fraction of particles with velocity component v to v+dv along a line of sight (electromagnetic radiation propagation), then the resulting distribution of the frequencies is

Doppler Broadening equation 2 Image


Where Doppler Broadening equation 3 Photo

The above equation gives the velocity towards the observer corresponding to the shift of the rest frequency ƒ0to ƒ . Therefore,

Doppler Broadening equation 4 Picture


Frequency is always inversely proportional to wavelength which is denoted as λ.Therefore, the broadening terms can be expressed in terms of the wavelength. In terms of non-relativistic limit

Doppler Broadening equation 5 Image



Doppler Broadening equation 6 Photo


In the case of thermal Doppler broadening, the velocity distribution is given by the Maxwell distribution

Doppler Broadening equation 7 Picture


Where m is the mass of the emitting particle, T is the temperature and k is the Boltzmann constant.


Doppler Broadening equation 8 Picture


The expression is further simplified as

Doppler Broadening equation 9 Photo


Which we immediately recognize as a Gaussian profile with the standard deviation

Doppler Broadening equation 10 Image


and full width at half maximum (FWHM)

Doppler Broadening equation 11 Picture

Doppler Broadening equation 12 Image


with the standard deviation

Doppler Broadening equation 13 Photo


and FWHM

Doppler Broadening equation 14 Picture


Doppler broadening on nuclear reactors

When a reactor gets hotter, the accelerated motion of the atoms in the fuel increases the probability of neutron capture by U-238 (Uranium-238 atoms. The Doppler effect arises when the temperature of the fuel changes. When the uranium is heated, its nuclei move more rapidly in random directions, and therefore generate a wider range of relative neutron speeds. U-238, which forms the bulk of the uranium in the reactor, is much more likely to absorb fast neutrons. This reduces the number of neutrons available to cause U-238 fission, reducing the power output by the reactor.

In pebble bed nuclear reactor, this effect ensures safety through negative feedback. Here a small amount of uranium, inside each small sphere increases the temperature and it absorbs more and more neutrons which reduces the number for subsequent fission. The Doppler broadening occurs at around 900 degrees centigrade, which prevents the potential temperature in each sphere from rising.

Doppler Broadening Applications and other causes

Thermal Doppler broadening is a consequence of the distribution of velocities of the emitting particles. The line width which forms, gives the information on temperature of the emitting particle. The reason behind the broadening of spectral lines is usually explained using this theory.

There might be other causes of velocity distributions, one of them being turbulent motion. In the presence of turbulence, the resulting line profile is quite similar to that of the thermal one. Another cause could be macroscopic velocities as seen in an accretion disc. Here, the disc is basically a structure which is formed by diffuse material in orbital motion around a central body like a star in the universe. Gravity causes the material in the disc to move inwards in a spiral direction towards the central body, due to which emission of electromagnetic radiation takes place. Stark broadening is another significant factor behind the spectral line broadening where the presence of an external static electric field causes shifting and splitting of the spectral lines.