In atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting particles result in different Doppler shifts, the cumulative effect of which is the line broadening.[1] This resulting line profile is known as a Doppler profile. A particular case is the thermal Doppler broadening due to the thermal motion of the particles. Then, the broadening depends only on the frequency of the spectral line, the mass of the emitting particles, and their temperature, and therefore can be used for inferring the temperature of an emitting body.

Saturated absorption spectroscopy, also known as Doppler-free spectroscopy, can be used to find the true frequency of an atomic transition without cooling a sample down to temperatures at which the Doppler broadening is minimal.

## Derivation

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:

${\displaystyle f=f_{0}\left(1+{\frac {v}{c}}\right),}$

where ${\displaystyle f}$ is the observed frequency, ${\displaystyle f_{0}}$ is the rest frequency, ${\displaystyle v}$ is the velocity of the emitter towards the observer, and ${\displaystyle c}$ is the speed of light.

Since there is a distribution of speeds both toward and away from the observer in any volume element of the radiating body, the net effect will be to broaden the observed line. If ${\displaystyle P_{v}(v)\,dv}$ is the fraction of particles with velocity component ${\displaystyle v}$ to ${\displaystyle v+dv}$ along a line of sight, then the corresponding distribution of the frequencies is

${\displaystyle P_{f}(f)\,df=P_{v}(v_{f}){\frac {dv}{df}}\,df,}$

where ${\displaystyle v_{f}=c\left({\frac {f}{f_{0}}}-1\right)}$ is the velocity towards the observer corresponding to the shift of the rest frequency ${\displaystyle f_{0}}$ to ${\displaystyle f}$. Therefore,

 ${\displaystyle P_{f}(f)\,df={\frac {c}{f_{0}}}P_{v}\left(c\left({\frac {f}{f_{0}}}-1\right)\right)\,df.}$

We can also express the broadening in terms of the wavelength ${\displaystyle \lambda }$. Recalling that in the non-relativistic limit ${\displaystyle {\frac {\lambda -\lambda _{0}}{\lambda _{0}}}\approx -{\frac {f-f_{0}}{f_{0}}}}$, we obtain

 ${\displaystyle P_{\lambda }(\lambda )\,d\lambda ={\frac {c}{\lambda _{0}}}P_{v}\left(c\left(1-{\frac {\lambda }{\lambda _{0}}}\right)\right)\,d\lambda .}$

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

${\displaystyle P_{v}(v)\,dv={\sqrt {\frac {m}{2\pi kT}}}\,\exp \left(-{\frac {mv^{2}}{2kT}}\right)\,dv,}$

where ${\displaystyle m}$ is the mass of the emitting particle, ${\displaystyle T}$ is the temperature, and ${\displaystyle k}$ is the Boltzmann constant.

Then

${\displaystyle P_{f}(f)\,df={\frac {c}{f_{0}}}{\sqrt {\frac {m}{2\pi kT}}}\,\exp \left(-{\frac {m\left[c\left({\frac {f}{f_{0}}}-1\right)\right]^{2}}{2kT}}\right)\,df.}$

We can simplify this expression as

${\displaystyle P_{f}(f)\,df={\sqrt {\frac {mc^{2}}{2\pi kTf_{0}^{2}}}}\,\exp \left(-{\frac {mc^{2}\left(f-f_{0}\right)^{2}}{2kTf_{0}^{2}}}\right)\,df,}$

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

${\displaystyle \sigma _{f}={\sqrt {\frac {kT}{mc^{2}}}}\,f_{0}}$

and full width at half maximum (FWHM)

 ${\displaystyle \Delta f_{\text{FWHM}}={\sqrt {\frac {8kT\ln 2}{mc^{2}}}}f_{0}.}$

## Applications and caveats

In astronomy and plasma physics, the thermal Doppler broadening is one of the explanations for the broadening of spectral lines, and as such gives an indication for the temperature of observed material. Other causes of velocity distributions may exist, though, for example, due to turbulent motion. For a fully developed turbulence, the resulting line profile is generally very difficult to distinguish from the thermal one.[2] Another cause could be a large range of macroscopic velocities resulting, e.g., from the receding and approaching portions of a rapidly spinning accretion disk. Finally, there are many other factors that can also broaden the lines. For example, a sufficiently high particle number density may lead to significant Stark broadening.

Doppler broadening can also be used to determine the velocity distribution of a gas given its absorption spectrum. In particular, this has been used to determine the velocity distribution of interstellar gas clouds.[3]

Doppler broadening has also been used as a design consideration in high-temperature nuclear reactors. In principle, as the reactor fuel heats up, the neutron absorption spectrum will broaden due to the relative thermal motion of the fuel nuclei with respect to the neutrons. Given the shape of the neutron absorption spectrum, this has the result of reducing neutron absorption cross section, reducing the likelihood of absorption and fission. The end result is that reactors designed to take advantage of Doppler broadening will decrease their reactivity as temperature increases, creating a passive safety measure. This tends to be more relevant to gas-cooled reactors, as other mechanisms are dominant in water cooled reactors.

## References

1. ^ Siegman, A. E. (1986). Lasers.
2. ^ Griem, Hans R. (1997). Principles of Plasmas Spectroscopy. Cambridge: University Press. ISBN 0-521-45504-9.
3. ^ Beals, C. S. "On the interpretation of interstellar lines". adsabs.harvard.edu.
Absorption band

According to quantum mechanics, atoms and molecules can only hold certain defined quantities of energy, or exist in specific states. When such quanta of electromagnetic radiation are emitted or absorbed by an atom or molecule, the energy of the radiation changes the state of the atom or molecule from an initial state to a final state. An absorption band is a range of wavelengths, frequencies or energies in the electromagnetic spectrum which are characteristic of a particular transition from initial to final state in a substance.

Annihilation radiation is a term used in Gamma spectroscopy for the gamma radiation produced when a particle and its antiparticle collide and annihilate. Most commonly, this refers to 511-keV gamma rays produced by a normal (negative) electron colliding with a positron.

Annihilation radiation is not monoenergetic, unlike gamma rays produced by radioactive decay. The production mechanism of annihilation radiation introduces Doppler broadening. The annihilation peak produced in a gamma spectrum by annihilation radiation therefore has a higher full width at half maximum (FWHM) than other gamma rays in spectrum. The difference is more apparent with high resolution detectors, such as Germanium detectors, than with low resolution detectors such as Sodium iodide detectors.

Because of their well-defined energy (511 keV) and characteristic, Doppler-broadened shape, annihilation radiation can often be useful in defining the energy calibration of a gamma ray spectrum.

Clover (detector)

A clover detector is a gamma-ray detector that consists of 4 coaxial N-type high purity germanium (Ge) crystals each machined to shape and mounted in a common cryostat to form a structure resembling a four-leaf clover.

Corona

A corona (Latin, 'crown') is an aura of plasma that surrounds the Sun and other stars. The Sun's corona extends millions of kilometres into outer space and is most easily seen during a total solar eclipse, but it is also observable with a coronagraph. The word corona is a Latin word meaning "crown", from the Ancient Greek κορώνη (korōnè, “garland, wreath”).

Spectroscopy measurements indicate strong ionization in the corona and a plasma temperature in excess of 1,000,000 kelvins, much hotter than the surface of the Sun.

Light from the corona comes from three primary sources, from the same volume of space. The K-corona (K for kontinuierlich, "continuous" in German) is created by sunlight scattering off free electrons; Doppler broadening of the reflected photospheric absorption lines spreads them so greatly as to completely obscure them, giving the spectral appearance of a continuum with no absorption lines. The F-corona (F for Fraunhofer) is created by sunlight bouncing off dust particles, and is observable because its light contains the Fraunhofer absorption lines that are seen in raw sunlight; the F-corona extends to very high elongation angles from the Sun, where it is called the zodiacal light. The E-corona (E for emission) is due to spectral emission lines produced by ions that are present in the coronal plasma; it may be observed in broad or forbidden or hot spectral emission lines and is the main source of information about the corona's composition.

Dicke effect

Dicke effect, also known as Dicke narrowing (or sometimes collisional narrowing) in spectroscopy, named after Robert H. Dicke, refers to narrowing of the Doppler broadening of a spectral line due to collisions the emitting species (usually an atom or a molecule) experiences with other particles.

Doppler

Doppler may refer to:

Doppler (surname), a surname and a list of people with the name

Christian Doppler (1803–1853), Austrian mathematician and physicist

Doppler effect

Doppler (building), a building in Amazon.com's corporate headquarters

Doppler (crater), a lunar impact crater

Doppler (novel), a novel by Erlend Loe

3905 Doppler, an asteroid

Doppler, the mascot of the WNBA's Seattle Storm

Equivalent width

The equivalent width of a spectral line is a measure of the area of the line on a plot of intensity versus wavelength. It is found by forming a rectangle with a height equal to that of continuum emission, and finding the width such that the area of the rectangle is equal to the area in the spectral line. It is a measure of the strength of spectral features that is primarily used in astronomy.

Fast-neutron reactor

A fast-neutron reactor (FNR) or simply a fast reactor is a category of nuclear reactor in which the fission chain reaction is sustained by fast neutrons (carrying energies of 5 MeV or greater), as opposed to thermal neutrons used in thermal-neutron reactors.

Such a reactor needs no neutron moderator, but requires fuel that is relatively rich in fissile material when compared to that required for a thermal-neutron reactor.

This effect is similar to Gaussian blur effect in image processing produced by convolution with the Gaussian function.

The term is named after Carl Friedrich Gauss.

Long-slit spectroscopy

In astronomy, long-slit spectroscopy involves observing an elongated celestial object (such as a nebula or along the major axis of a disc galaxy at high inclination) through an elongated slit aperture, and refracting this light with a prism or diffraction grating. This type of spectrograph causes the Doppler shift-induced frequency distribution of the collected light to manifest as a spatial distribution through differential refraction, revealing the amplitude of the rotation curve.

Neutron cross section

In nuclear and particle physics, the concept of a neutron cross section is used to express the likelihood of interaction between an incident neutron and a target nucleus. In conjunction with the neutron flux, it enables the calculation of the reaction rate, for example to derive the thermal power of a nuclear power plant. The standard unit for measuring the cross section is the barn, which is equal to 10−28 m2 or 10−24 cm2. The larger the neutron cross section, the more likely a neutron will react with the nucleus.

An isotope (or nuclide) can be classified according to its neutron cross section and how it reacts to an incident neutron. Nuclides that tend to absorb a neutron and either decay or keep the neutron in its nucleus are neutron absorbers and will have a capture cross section for that reaction. Isotopes that fission, are fissionable fuels and have a corresponding fission cross section. The remaining isotopes will simply scatter the neutron, and have a scatter cross section. Some isotopes, like uranium-238, have nonzero cross sections of all three.

Isotopes which have a large scatter cross section and a low mass are good neutron moderators (see chart below). Nuclides which have a large absorption cross section are neutron poisons if they are neither fissile nor undergo decay. A poison that is purposely inserted into a nuclear reactor for controlling its reactivity in the long term and improve its shutdown margin is called a burnable poison.

Neutron temperature

The neutron detection temperature, also called the neutron energy, indicates a free neutron's kinetic energy, usually given in electron volts. The term temperature is used, since hot, thermal and cold neutrons are moderated in a medium with a certain temperature. The neutron energy distribution is then adapted to the Maxwellian distribution known for thermal motion. Qualitatively, the higher the temperature, the higher the kinetic energy of the free neutrons. The momentum and wavelength of the neutron are related through the De Broglie relation. The large wavelength of slow neutrons allows for the large cross section.

Plasma diagnostics

Plasma diagnostics are a pool of methods, instruments, and experimental techniques used to measure properties of a plasma, such as plasma components' density, distribution function over energy (temperature), their spatial profiles and dynamics, which enable to derive plasma parameters.

Reverberation mapping

Reverberation mapping is an astrophysical technique for measuring the structure of the broad emission-line region (BLR) around a supermassive black hole at the center of an active galaxy, and thus estimating the hole's mass. It is considered a "primary" mass estimation technique, i.e., the mass is measured directly from the motion that its gravitational force induces in the nearby gas.

Newton's law of gravity defines a direct relation between the mass of a central object and the speed of a smaller object in orbit around the central mass. Thus, for matter orbiting a black hole, the black hole mass is related by the formula

${\displaystyle GM_{\bullet }=fR_{\mathrm {BLR} }(\Delta V)^{2}}$

to the RMS velocity ΔV of gas moving near the black hole in the broad emission-line region, measured from the Doppler broadening of the gaseous emission lines. In that formula, RBLR is the radius of the broad-line region; G is the constant of gravitation; and f is a poorly known "form factor" that depends on the shape of the BLR.

While ΔV can be measured directly using spectroscopy, the necessary determination of RBLR is much less straightforward. This is where reverberation mapping comes into play. It utilizes the fact that the emission-line fluxes vary strongly in response to changes in the continuum, i.e., the light from the accretion disk near the black hole. Put simply, if the brightness of the accretion disk varies, the emission lines, which are excited in response to the accretion disk's light, will "reverberate", that is, vary in response. But it will take some time for light from the accretion disk to reach the broad-line region. Thus, the emission-line response is delayed with respect to changes in the continuum. Assuming that this delay is solely due to light travel times, the distance traveled by the light, corresponding to the radius of the broad emission-line region, can be measured.

Only a small handful of AGN (less than 40) have been accurately "mapped" in this way. An alternative approach is to use an empirical correlation between RBLR and the continuum luminosity.

Another uncertainty is the value of f. In principle, the response of the BLR to variations in the continuum could be used to map out the three-dimensional structure of the BLR. In practice, the amount and quality of data required to carry out such a deconvolution is prohibitive. Until about 2004, f was estimated ab initio based on simple models for the structure of the BLR. More recently, the value of f has been determined so as to bring the M-sigma relation for active galaxies into the best possible agreement with the M–sigma relation for quiescent galaxies. When f is determined in this way, reverberation mapping becomes a "secondary", rather than "primary," mass estimation technique.

SN 2008ha

SN 2008ha was a type Ia supernova which was first observed around November 7, 2008 in the galaxy UGC 12682, which lies in the constellation Pegasus at a distance of about 21.3 megaparsecs (69 Mly) from Earth.SN 2008ha was unusual in several ways: with an absolute V band magnitude of −14.2 it is one of the faintest supernovae ever observed; its host galaxy type very rarely produces supernovae. Another unusual feature of SN 2008ha was its low expansion velocity of only ~2000 km/s at maximum brightness, which indicates a very small kinetic energy released in the explosion. For comparison, SN 2002cx expanded at a velocity of ~5000 km/s whereas typical

SN Ia expand at around ~10,000 km/s. The low expansion velocity of SN2008ha resulted in relatively small Doppler broadening of spectral emission lines and this led to higher quality data.

The supernova was studied with ultraviolet, optical, and near-infrared photometry as well as optical spectra, using the Magellan telescopes in Chile, the MMT telescope in Arizona, the Gemini and Keck telescopes in Hawaii, and NASA's Swift satellite. Spectroscopically, SN 2008ha was identified as a SN 2002cx-type, a peculiar sub-class of SN Ia. SN 2008ha had a brightness period of only 10 days, which is significantly shorter than that of other SN 2002cx-like objects (~15 days) or normal Ia supernovas (~20 days). From the peak luminosity and the brightness time it was estimated that SN 2008ha generated (3.0 ± 0.9) × 10−3 M⊙ of 56Ni, had a kinetic energy of 2 × 1048 ergs, and ejected 0.15 M⊙ of material.

Saturated absorption spectroscopy

In experimental atomic physics, saturated absorption spectroscopy or Doppler-free spectroscopy is a set-up that enables the precise determination of the transition frequency of an atom between its ground state and an optically excited state. The accuracy to which these frequencies can be determined is, ideally, limited only by the width of the excited state, which is the inverse of the lifetime of this state. However, the samples of atomic gas that are used for that purpose are generally at room temperature, where the measured frequency distribution is highly broadened due to the Doppler effect. Saturated absorption spectroscopy allows precise spectroscopy of the atomic levels without having to cool the sample down to temperatures at which the Doppler broadening is no longer relevant (which would be on the order of a few millikelvins). It is also used to lock the frequency of a laser to the precise wavelength of an atomic transmission in atomic physics experiments.

Saturated spectroscopy

Saturated spectroscopy is the method by which the exact energy of the hyperfine transitions within an atom can be found. When a monochromatic light is shone through an atom, the absorption cross-section is broadened due to Doppler broadening. Saturated spectroscopy allows the doppler broadened peak to be resolved so that the exact transitions can be found.

More than a decade after the first demonstration of spectral hole burning (or Lamb dip, a result of saturated absorption process) inside HeNe laser cavity at 1.1 μm in 1962, the greater majority of SA spectroscopy research was carried out with gas lasers and molecules in the mid-infrared.

But because SA requires high laser intensity, and the gas molecules usually have widely spread strong absorption spectra only in the mid-IR, while compact widely tunable mid-IR lasers were slow to develop, the SA technique has not been widely used for molecular chemical analysis besides precision metrology, which only been limited to the isolated wavelengths of HeNe and CO2 lasers and limited number of molecules.

Spectral line

A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to identify atoms and molecules. These "fingerprints" can be compared to the previously collected "fingerprints" of atoms and molecules, and are thus used to identify the atomic and molecular components of stars and planets, which would otherwise be impossible.

Uranium-238

Uranium-238 (238U or U-238) is the most common isotope of uranium found in nature, with a relative abundance of 99%. Unlike uranium-235, it is non-fissile, which means it cannot sustain a chain reaction in a thermal-neutron reactor. However, it is fissionable by fast neutrons, and is fertile, meaning it can be transmuted to fissile plutonium-239. 238U cannot support a chain reaction because inelastic scattering reduces neutron energy below the range where fast fission of one or more next-generation nuclei is probable. Doppler broadening of U-238's neutron absorption resonances, increasing absorption as fuel temperature increases, is also an essential negative feedback mechanism for reactor control.

Around 99.284% of natural uranium's mass is uranium-238, which has a half-life of 1.41×1017 seconds (4.468×109 years, or 4.468 billion years).

Due to its natural abundance and half-life relative to other radioactive elements, 238U produces ~40% of the radioactive heat produced within the Earth. 238U decay contributes 6 electron anti-neutrinos per decay (1 per beta decay), resulting in a large detectable geoneutrino signal when decays occur within the Earth. The decay of 238U to daughter isotopes is extensively used in radiometric dating, particularly for material older than ~ 1 million years.

Depleted uranium has an even higher concentration of the 238U isotope, and even low-enriched uranium (LEU), while having a higher proportion of the uranium-235 isotope (in comparison to depleted uranium), is still mostly 238U. Reprocessed uranium is also mainly 238U, with about as much uranium-235 as natural uranium, a comparable proportion of uranium-236, and much smaller amounts of other isotopes of uranium such as uranium-234, uranium-233, and uranium-232.

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