Definitions

broadening

Doppler broadening

In atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect in which the thermal movement of atoms or molecules shifts the apparent frequency of each emitter. The many different velocities of the emitting gas result in many small shifts, the cumulative effect of which is to broaden the line. The resulting line profile is known as a Doppler profile. The broadening is dependent only on the wavelength of the line, the mass of the emitting particle and the temperature, and can therefore be a very useful method for measuring the temperature of an emitting gas.

The Doppler profile in wavelength is a normal distribution with a standard deviation of

Deltalambda=lambda_0sqrt{frac{kT}{mc^2}}

where lambda_0 is the center wavelength of the profile, c is the speed of light, T is the temperature, k is the Boltzmann constant and m the mass of the atom. For widths that are much smaller than the central wavelength, the Doppler profile in frequency is also a normal distribution with standard deviation

Delta f=f_0sqrt{frac{kT}{mc^2}}

where f_0=c/lambda_0 is the central frequency. The widths are sometimes characterized by the full width at half maximum of the profile (FWHM) which is related to the standard deviation sigma by:

FWHM=2sigmasqrt{2ln(2)}

Derivation of Doppler profile

When thermal motion causes a particle to move towards the observer, the emitted radiation will be shifted to a higher frequency. Likewise, when the emitter moves away, the frequency will be lowered. For non-relativistic thermal velocities, the Doppler shift in frequency will be:

f = f_0left(1+frac{v}{c}right)

where f is the observed frequency, f_0 is the rest frequency, v is the velocity of the emitter towards the observer, and c is the speed of light.

Since there is a distribution of speeds both toward and away from the observer in any volume element of gas, the net effect will be to broaden the observed line. The distribution of speeds towards and away from an observer is given by the Maxwell distribution. If P(v)dv is the fraction of particles with velocity component v to v+dv along a line of sight, then:

P(v)dv = sqrt{frac{m}{2pi kT}},expleft(-frac{mv^2}{2kT}right)dv

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

In optics we measure frequency content (as opposed to velocity content), and it is convenient to re-express the distribution in terms of P(f) where P(f)df is the probability of an observed photon having a frequency between f and f+df relative to that stationary observer.

The Doppler shift equation can be used to express velocity in terms of the frequency. Using the relationship from probability that P(v)dv=P(f)df, and rearranging terms of the Doppler shift equation above as v=c(f/f_0-1) such that dv/df=c/f_0, we find:

P(f)df=left(frac{c}{f_0}right)sqrt{frac{m}{2pi kT}},expleft(-frac{mleft[cleft(frac{f}{f_0}-1right)right]^2}{2kT}right)df

We can simplify this expression as:

P(f)df=sqrt{frac{mc^2}{2pi kT {f_0}^2}},
expleft(-frac{mc^2left(f-f_0right)^2}{2kT {f_0}^2}right)df,

which we immediately recognize as a Gaussian peak with standard deviation

sigma_{f} = sqrt{frac{kT}{mc^2}}f_0

and full width at half maximum

Delta f_{text{FWHM}} = sqrt{frac{8kTln 2}{mc^2}}f_{0}.

We can also consider the above equation in terms of wavelength P(lambda) to express the probability of an observed photon having a wavelength between lambda to lambda+dlambda according to the stationary observer. For widths that are small with respect to the central wavelength, we can make the approximation

frac{lambda-lambda_{0}}{lambda_{0}} approx -frac{f-f_0}{f_0}.

and furthermore apply the change of variable df = (-c/lambda^{2})dlambda. The Doppler profile in wavelength units is then also a Gaussian:

P(lambda)dlambda = sqrt{frac{mc^2}
{2pi kTlambda_0^2}},expleft(-frac{mc^2(lambda-lambda_0)^2}{2kTlambda_0^2}right)dlambda

with standard deviation

sigma_{lambda} = sqrt{frac{kT}{mc^2}}lambda_{0}

and full width at half maximum

Delta lambda_{text{FWHM}} = sqrt{frac{8kTln 2}{mc^2}}lambda_{0}.

Nuclear technology

In a nuclear reactor, this effect reduces the reactivity of a reactor as the reactor temperature increases.

When a reactor gets hotter, the accelerated motion of the atoms in the fuel increases the probability of neutron capture by U-238 atoms. When the uranium is heated, its nuclei move more rapidly in random directions, and therefore see and generate a wider range of relative neutron speeds. U-238, which forms the bulk of the uranium in the reactor, has very distinct energies at which it absorbs neutrons, so that it will be thousands of times more likely to absorb a 6.67eV neutron than a 8eV neutron. As the random motion of the U-238 atoms increases though, the more likely it is that atom will be moving away from the 8eV neutron at the right speed that the neutron speed, in the U-238 frame of reference, will be 6.67eV. This increases the number of neutrons absorbed by U-238 atoms, reducing the number of neutrons available to cause the more useful U-235 to fission, reducing the reactivity by the reactor.

In some reactor designs, such as the pebble bed reactor, this and other forms of natural negative feedback places an inherent upper limit on the temperature at which the chain reaction can proceed. Such reactors are said to be "inherently safe" because a reactor failure cannot generate a criticality excursion. It is worth noting, however, that because of decay heat emitted from the decay of fission products, a meltdown is still theoretically possible if the ability to cool the reactor is lost, and thus the reactor design must be designed to prevent loss of coolant accident.

Astronomy

In Astronomy, Doppler broadening is one of the explanations for the broadening of spectral lines, and as such gives an indication for the relative temperatures of observed material. There are, however many other factors which can broaden the lines as well. For example high surface gravity (a sign of small stars) leads to high pressure, which in turn leads to Stark broadening (see Spectral line).

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