In
atomic physicsAtomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...
,
Doppler broadening is the broadening of
spectral lineA spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies.- Types of line spectra :...
s due to the
Doppler effectThe Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer...
in which the
thermalIn physics and thermodynamics, heat is the process of energy transfer from one body or system due to thermal contact, which in turn is defined as an energy transfer to a body in any other way than due to work performed on the body....
movement of
atomThe atom is a basic unit of matter consisting of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s or
moleculeA molecule is defined as an electrically neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds. Molecules are distinguished from polyatomic ions in this strict sense...
s 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 frequency 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 frequency is a
normal distributionIn probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that cluster around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known...
with a standard deviation of
where is the central frequency of the profile, is the
speed of lightIn physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
, is the
temperatureIn physics, temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics...
, is the
Boltzmann constantThe Boltzmann constant is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant N
A:...
and the mass of the atom.
In
atomic physicsAtomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...
,
Doppler broadening is the broadening of
spectral lineA spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies.- Types of line spectra :...
s due to the
Doppler effectThe Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer...
in which the
thermalIn physics and thermodynamics, heat is the process of energy transfer from one body or system due to thermal contact, which in turn is defined as an energy transfer to a body in any other way than due to work performed on the body....
movement of
atomThe atom is a basic unit of matter consisting of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s or
moleculeA molecule is defined as an electrically neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds. Molecules are distinguished from polyatomic ions in this strict sense...
s 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 frequency 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 frequency is a
normal distributionIn probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that cluster around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known...
with a standard deviation of
where is the central frequency of the profile, is the
speed of lightIn physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
, is the
temperatureIn physics, temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics...
, is the
Boltzmann constantThe Boltzmann constant is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant N
A:...
and the mass of the atom. For widths that are much smaller than the central frequency, the Doppler profile in wavelength is also a normal distribution with standard deviation
where is the center wavelength. The widths are sometimes characterized by the full width at half maximum of the profile (FWHM) which is related to the standard deviation by:
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 shiftThe Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer...
in frequency will be:
where is the observed frequency, is the rest frequency, is the velocity of the emitter towards the observer, and
c is the
speed of lightIn physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
.
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 is the fraction of particles with velocity component to along a line of sight, then:
where is the mass of the emitting particle, is the temperature and is the
Boltzmann constantThe Boltzmann constant is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant N
A:...
.
In optics we measure frequency content (as opposed to velocity content), and it is convenient to re-express the distribution in terms of where is the probability of an observed photon having a frequency between and 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 , and rearranging terms of the Doppler shift equation above as such that , we find:
We can simplify this expression as:
,
which we immediately recognize as a
GaussianGAUSSIAN is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70...
peak with standard deviation
and full width at half maximum
We can also consider the above equation in terms of wavelength to express the probability of an observed photon having a wavelength between to according to the stationary observer. For widths that are small with respect to the central wavelength, we can make the approximation
.
and furthermore apply the change of variable . The Doppler profile in wavelength units is then also a Gaussian:
with standard deviation
and full width at half maximum
Nuclear technology
In a
nuclear reactorA nuclear reactor is a device in which nuclear chain reactions are initiated, controlled, and sustained at a steady rate.The most significant use of nuclear reactors is as an energy source for the generation of electrical power and for the power in some ships...
, 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
neutronThe neutron is a subatomic particle with no net electric charge and a mass slightly larger than that of a proton.Neutron are usually found in atomic nuclei. The nuclei of most atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of protons in a...
capture by
U-238Uranium-238 is the most common isotope of uranium found in nature. When hit by a neutron, it eventually becomes plutonium-239 ....
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-238Uranium-238 is the most common isotope of uranium found in nature. When hit by a neutron, it eventually becomes plutonium-239 ....
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-238Uranium-238 is the most common isotope of uranium found in nature. When hit by a neutron, it eventually becomes plutonium-239 ....
frame of reference, will be 6.67eV. This increases the number of neutrons absorbed by
U-238Uranium-238 is the most common isotope of uranium found in nature. When hit by a neutron, it eventually becomes plutonium-239 ....
atoms, reducing the number of neutrons available to cause the more useful
U-235Uranium-235 is an isotope of uranium making up about 0.72% of natural uranium. Unlike the predominant isotope uranium-238 it is fissile, i.e. it can sustain fission chain reaction...
to
fissionIn nuclear physics and nuclear chemistry, nuclear fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts, often producing free neutrons and lighter nuclei, which may eventually produce photons...
, reducing the reactivity by the reactor.
In some reactor designs, such as the
pebble bed reactorThe pebble bed reactor is a graphite-moderated, gas-cooled, nuclear reactor. It is a type of Very high temperature reactor [formally known as the high temperature gas reactor ], one of the six classes of nuclear reactors in the Generation IV initiative...
, this and other forms of natural
negative feedbackNegative feedback occurs when the output of a system acts to oppose changes to the input of the system; with the result that the changes are attenuated. If the overall feedback of the system is negative, then the system will tend to be stable.- Overview :...
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 heatDecay heat is the heat released as a result of radioactive decay. This is when the radiation interacts with materials and the energy of the alpha, beta or gamma radiation is converted into the thermal movement of atoms.-Natural occurrence:...
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 a loss of coolant accident.
Astronomy and plasma physics
In
astronomyAstronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere...
and plasma physics, Doppler broadening is one of the explanations for the broadening of
spectral lineA spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies.- Types of line spectra :...
s, and as such gives an indication for the temperature 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. Another cause is thought to be the large range of velocities resulting from the receding and approaching portions of a rapidly spinning accretion disk, presumed to be at the center of Seyfert galaxies.