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| {{More footnotes|date=September 2008}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
| [[File:Spectral lines en.PNG|thumb|Emission lines and absorption lines compared to a continuous spectrum.]]
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| In [[physics]], one thinks of '''atomic spectral line'''s from two viewpoints.
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| * An '''emission line''' is formed when an electron makes a transition from a particular discrete [[energy level]] {{math|''E''<sub>2</sub>}} of an atom, to a lower energy level {{math|''E''<sub>1</sub>}}, emitting a photon of a particular energy and wavelength. A spectrum of many such photons will show an emission spike at the wavelength associated with these photons.
| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
| * An '''absorption line''' is formed when an electron makes a transition from a lower, {{math|''E''<sub>1</sub>}}, to a higher discrete energy state, {{math|''E''<sub>2</sub>}}, with a photon being absorbed in the process. These absorbed photons generally come from background continuum radiation and a spectrum will show a drop in the continuum radiation at the wavelength associated with the absorbed photons. | | * Only registered users will be able to execute this rendering mode. |
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| The two states must be [[bound state]]s in which the electron is bound to the atom, so the transition is sometimes referred to as a "bound–bound" transition, as opposed to a transition in which the electron is ejected out of the atom completely ("bound–free" transition) into a [[continuous spectrum|continuum]] state, leaving an [[ionization|ionized]] atom, and generating continuum radiation.
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| A [[photon]] with an energy equal to the difference {{math|''E''<sub>2</sub> - ''E''<sub>1</sub>}} between the energy levels is released or absorbed in the process. The frequency {{math|''ν''}} at which the spectral line occurs is related to the photon energy by Bohr's frequency condition <math>E_2-E_1=h\nu</math> where {{math|''h''}} denotes [[Planck's constant]].<ref name="Bohr 1913">{{harvnb|Bohr|1913}}</ref><ref name="Einstein 1916">{{harvnb|Einstein|1916}}</ref><ref name="Sommerfeld 1923 43">{{harvnb|Sommerfeld|1923|p=43}}</ref><ref name ="Heisenberg 1925 108">{{harvnb|Heisenberg|1925|p=108}}</ref><ref name="Brillouin 1970 31">{{harvnb|Brillouin|1970|p=31}}</ref><ref name="Jammer 1989 113 115">{{harvnb|Jammer|1989|pages=113, 115}}</ref>
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| == Emission and absorption coefficients ==
| | <!--'''PNG''' (currently default in production) |
| An atomic spectral line refers to emission and absorption events in a gas in which <math>n_2</math> is the density of atoms in the upper energy state for the line, and <math>n_1</math> is the density of atoms in the lower energy state for the line.
| | :<math forcemathmode="png">E=mc^2</math> |
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| The emission of atomic line radiation at frequency {{math|''ν''}} may be described by an [[emission coefficient]] <math>\epsilon</math> with units of energy/time/volume/solid angle. ''ε dt dV dΩ'' is then the energy emitted by a volume element <math>dV</math> in time <math>dt</math> into solid angle <math>d\Omega</math>. For atomic line radiation:
| | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| :<math>\epsilon = \frac{h\nu}{4\pi}n_2 A_{21}\,</math> | | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| where <math>A_{21}</math> is the '''Einstein coefficient''' for spontaneous emission, which is fixed by the intrinsic properties of the relevant atom for the two relevant energy levels.
| | ==Demos== |
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| The absorption of atomic line radiation may be described by an [[absorption coefficient]] <math>\kappa</math> with units of 1/length. The expression ''κ' dx'' gives the fraction of intensity absorbed for a light beam at frequency ''ν'' while traveling distance ''dx''. The absorption coefficient is given by:
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| :<math>\kappa' = \frac{h\nu}{4\pi}~(n_1 B_{12}-n_2 B_{21}) \,</math>
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| where <math>B_{12}</math> and <math>B_{21}</math> are the Einstein coefficients for photo absorption and induced emission respectively. Like the coefficient <math>A_{21}</math>, these are also fixed by the intrinsic properties of the relevant atom for the two relevant energy levels. For thermodynamics and for the application of Kirchhoff's law, it is necessary that the total absorption be expressed as the algebraic sum of two components, described respectively by <math>B_{12}</math> and <math>B_{21}</math>, which may be regarded as positive and negative absorption, which are, respectively, the direct photon absorption, and what is commonly called stimulated or induced emission.<ref>Weinstein, M.A. (1960). On the validity of Kirchhoff's law for a freely radiating body, ''American Journal of Physics'', '''28''': 123-25.</ref><ref>Burkhard, D.G., Lochhead, J.V.S., Penchina, C.M. (1972). On the validity of Kirchhoff's law in a nonequilibrium environment, ''American Journal of Physics'', '''40''': 1794-1798.</ref><ref>Baltes, H.P. (1976). On the validity of Kirchhoff's law of heat radiation for a body in a nonequilibrium environment, Chapter 1, pages 1-25 of ''Progress in Optics XIII'', edited by E. Wolf, North-Holland, ISSN 00796638.</ref>
| | * accessibility: |
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| The above equations have ignored the influence of the spectral line shape. To be accurate, the above equations need to be multiplied by the (normalized) spectral line shape, in which case the units will change to include a 1/Hz term.
| | ==Test pages == |
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| For conditions of thermodynamic equilibrium, together the number densities <math>n_2</math> and <math>n_1</math>, the Einstein coefficients, and the spectral energy density provide sufficient information to determine the absorption and emission rates.
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
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| ===Equilibrium conditions===
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| The number densities <math>n_2</math> and <math>n_1</math> are set by the physical state of the gas in which the spectral line occurs, including the local [[spectral radiance]] (or, in some presentations, the local spectral radiant energy density). When that state is either one of strict [[thermodynamic equilibrium]], or one of so-called 'local thermodynamic equilibrium',<ref>Milne, E.A. (1928). The effect of collisions on monochromatic radiative equilibrium, ''Monthly Notices of the Royal Astronomical Society'', '''88''': 493-502. [http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1928MNRAS..88..493M&db_key=AST&link_type=ABSTRACT&high=4c3363690619220]</ref><ref>[[Subrahmanyan Chandrasekhar|Chandrasekhar, S.]] (1950), p. 7.</ref><ref name="Mihalas Mihalas 1984">[http://www.filestube.com/9c5b2744807c2c3d03e9/details.html Mihalas, D., Weibel-Mihalas, B. (1984). ''Foundations of Radiation Hydrodynamics'', Oxford University Press, New York, ISBN 0-19-503437-6.]</ref> then the distribution of atomic states of excitation (which includes <math>n_2</math> and <math>n_1</math>) determines the rates of atomic emissions and absorptions to be such that [[Kirchhoff's law of thermal radiation|Kirchhoff's law of equality of radiative absorptivity and emissivity]] holds. In strict thermodynamic equilibrium, the radiation field is said to be [[black body|black-body]] radiation, and is described by [[Planck's law]]. For local thermodynamic equilibrium, the radiation field does not have to be a black-body field, but the rate of interatomic collisions must vastly exceed the rates of absorption and emission of quanta of light, so that the interatomic collisions entirely dominate the distribution of states of atomic excitation. Circumstances occur in which local thermodynamic equilibrium does not prevail, because the strong radiative effects overwhelm the tendency to the Maxwell-Boltzmann distribution of molecular velocities. For example, in the atmosphere of the sun, the great strength of the radiation dominates. In the upper atmosphere of the earth, at altitudes over 100 km, the rarity of intermolecular collisions is decisive.
| | ==Bug reporting== |
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| In the cases of [[thermodynamic equilibrium]] and of [[local thermodynamic equilibrium]], the number densities of the atoms, both excited and unexcited, may be calculated from the [[Maxwell–Boltzmann distribution]], but for other cases, (e.g. [[laser]]s) the calculation is more complicated.
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| == Einstein coefficients ==<!-- This section is linked from [[Stimulated emission]] -->
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| In 1916, [[Albert Einstein]] proposed that there are three processes occurring in the formation of an atomic spectral line. The three processes are referred to as '''spontaneous emission''', '''stimulated emission''', and '''absorption'''. With each is associated an '''Einstein coefficient''' which is a measure of the probability of that particular process occurring. Einstein considered the case of isotropic radiation of frequency {{math|''ν''}}, and spectral energy density {{math|''ρ'' (''ν'')}}.<ref name="Einstein 1916"/><ref>Loudon, R. (2000), Section 1.5, pp. 16–19.</ref> Note that in some treatments, the black-body spectral radiance is used rather than the spectral radiation energy density.<ref>[[Subrahmanyan Chandrasekhar|Chandrasekhar, S.]] (1950), p. 354.</ref><ref name="Mihalas Mihalas 1984"/>
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| ===Spontaneous emission=== | |
| {{main|Spontaneous emission}}
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| [[File:AtomicLineSpEm.png|thumb|180px|Schematic diagram of atomic spontaneous emission]]
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| Spontaneous emission is the process by which an electron "spontaneously" (i.e. without any outside influence) decays from a higher energy level to a lower one. The process is described by the Einstein coefficient ''A''<sub>21</sub> (''s''<sup>−1</sup>) which gives the probability per unit time that an electron in state 2 with energy <math>E_2</math> will decay spontaneously to state 1 with energy <math>E_1</math>, emitting a photon with an energy {{math|''E''<sub>2</sub> − ''E''<sub>1</sub> {{=}} ''hν''}}. Due to the [[Uncertainty Principle#Energy-time uncertainty principle|energy-time uncertainty principle]], the transition actually produces photons within a narrow range of frequencies called the [[spectral linewidth]]. If <math>n_i</math> is the number density of atoms in state ''i'' then the change in the number density of atoms in state 2 per unit time due to spontaneous emission will be:
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| :<math>\left(\frac{dn_2}{dt}\right)_{\mathrm{spontaneous}}=-A_{21}n_2\,.</math> | |
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| The same process results in increasing of the population of the state 1:
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| :<math>\left(\frac{dn_1}{dt}\right)_{\mathrm{spontaneous}}=A_{21}n_2\,.</math>
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| === Stimulated emission ===
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| {{main|Stimulated emission}}
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| [[File:AtomicLineInEm.png|thumb|180px|Schematic diagram of atomic stimulated emission]]
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| [[Stimulated emission]] (also known as induced emission) is the process by which an electron is induced to jump from a higher energy level to a lower one by the presence of electromagnetic radiation at (or near) the frequency of the transition. From the thermodynamic viewpoint, this process must be regarded as negative absorption. The process is described by the Einstein coefficient <math>B_{21}</math> (J<sup>−1</sup> m<sup>3</sup> s<sup>−1</sup>), which gives the probability per unit time per unit spectral energy density of the radiation field that an electron in state 2 with energy <math>E_2</math> will decay to state 1 with energy <math>E_1</math>, emitting a photon with an energy {{math|''E''<sub>2</sub> − ''E''<sub>1</sub> {{=}} ''hν''}}. The change in the number density of atoms in state 1 per unit time due to induced emission will be:
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| :<math>\left(\frac{dn_1}{dt}\right)_{\mathrm{neg}\,\mathrm{absorb}}=B_{21}n_2 \rho(\nu)</math>
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| <br style="clear:both;" />
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| where <math>\rho(\nu)</math> denotes the spectral energy density of the isotropic radiation field at the frequency of the transition (see [[Planck's law]]).
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| Stimulated emission is one of the fundamental processes that led to the development of the [[laser]]. Laser radiation is, however, very far from the present case of isotropic radiation.
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| === Photo absorption === | |
| {{main|Absorption (optics)}}
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| [[File:AtomicLineAb.png|thumb|180px|Schematic diagram of atomic absorption]]
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| Absorption is the process by which a photon is absorbed by the atom, causing an electron to jump from a lower energy level to a higher one. The process is described by the Einstein coefficient <math>B_{12}</math> (J<sup>−1</sup> m<sup>3</sup> s<sup>−1</sup>), which gives the probability per unit time per unit spectral energy density of the radiation field that an electron in state 1 with energy <math>E_1</math> will absorb a photon with an energy {{math|''E''<sub>2</sub> − ''E''<sub>1</sub> {{=}} ''hν''}} and jump to state 2 with energy <math>E_2</math>. The change in the number density of atoms in state 1 per unit time due to absorption will be:
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| :<math>\left(\frac{dn_1}{dt}\right)_{\mathrm{pos}\,\mathrm{absorb}}=-B_{12}n_1 \rho(\nu)</math>
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| <br style="clear:both;" />
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| == Detailed balancing ==
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| The Einstein coefficients are fixed probabilities associated with each atom, and do not depend on the state of the gas of which the atoms are a part. Therefore, any relationship that we can derive between the coefficients at, say, thermodynamic equilibrium will be valid universally.
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| At thermodynamic equilibrium, we will have a simple balancing, in which the net change in the number of any excited atoms is zero, being balanced by loss and gain due to all processes. With respect to bound-bound transitions, we will have [[detailed balance|detailed balancing]] as well, which states that the net exchange between any two levels will be balanced. This is because the probabilities of transition cannot be affected by the presence or absence of other excited atoms. Detailed balance (valid only at equilibrium) requires that the change in time of the number of atoms in level 1 due to the above three processes be zero:
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| :<math>0=A_{21}n_2+B_{21}n_2\rho(\nu)-B_{12}n_1 \rho(\nu)\,</math>
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| Along with detailed balancing, at temperature {{math|''T''}} we may use our knowledge of the equilibrium energy distribution of the atoms, as stated in the [[Maxwell–Boltzmann distribution]], and the equilibrium distribution of the photons, as stated in [[Planck's law of black body radiation]] to derive universal relationships between the Einstein coefficients.
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| From the Maxwell–Boltzmann distribution we have for the number of excited atomic species ''i'':
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| :<math>\frac{n_i}{n}= \frac{g_i e^{-E_i/kT}}{Z}</math>
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| where ''n'' is the total number density of the atomic species, excited and unexcited, ''k'' is [[Boltzmann's constant]], ''T'' is the [[temperature]], <math>g_i</math> is the degeneracy (also called the multiplicity) of state ''i'', and ''Z'' is the [[partition function (statistical mechanics)|partition function]]. From Planck's law of black-body radiation at temperature {{math|''T''}} we have for the spectral energy density at frequency {{math|''ν''}}
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| :<math>\rho_\nu(\nu,T)=F(\nu)\frac{1}{e^{h\nu/kT}-1}</math>
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| where:
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| :<math>F(\nu)=\frac{8\pi h\nu^3}{c^3}</math>
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| where <math>c</math> is the [[speed of light]] and <math>h</math> is [[Planck's constant]].
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| Substituting these expressions into the equation of detailed balancing and remembering that {{math|''E''<sub>2</sub> − ''E''<sub>1</sub> {{=}} ''hν''}} yields:
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| :<math>A_{21}g_2e^{-h\nu/kT}+B_{21}g_2e^{-h\nu/kT}\frac{F(\nu)}{e^{h\nu/kT}-1}=
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| B_{12}g_1\frac{F(\nu)}{e^{h\nu/kT}-1}</math>
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| separating to:
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| :<math>A_{21}g_2(e^{h\nu/kT}-1)+ B_{21}g_2F(\nu)= B_{12}g_1e^{h\nu/kT}F(\nu)\,</math>
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| The above equation must hold at any temperature, so
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| :<math>A_{21}g_2 = B_{12}g_1F(\nu)\,</math>
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| :<math>- A_{21}g_2 + B_{21}g_2F(\nu) = 0\,</math>
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| Therefore the three Einstein coefficients are interrelated by:
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| :<math>\frac{A_{21}}{B_{21}}=F(\nu)</math>
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| and
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| :<math>\frac{B_{21}}{B_{12}}=\frac{g_1}{g_2}</math>
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| When this relation is inserted into the original equation, one can also find a relation between <math>A_{21}</math> and <math>B_{12}</math>, involving [[Planck's law]].
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| == Oscillator strengths ==
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| The oscillator strength <math>f_{12}</math> is defined by the following relation to the cross section <math>a_{12}</math> for absorption:
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| :<math>a_{12}=\frac{\pi e^2}{m_e c}\,f_{12}</math>
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| where <math>e</math> is the electron charge and <math>m_e</math> is the electron mass. This allows all three Einstein coefficients to be expressed in terms of the single oscillator strength associated with the particular atomic spectral line:
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| :<math>B_{12}=\frac{4\pi^2 e^2}{m_e h\nu c}\,f_{12}</math>
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| :<math>B_{21}=\frac{4\pi^2 e^2}{m_e h\nu c}~\frac{g_1}{g_2}~f_{12}</math>
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| :<math>A_{21}=\frac{8\nu^2 \pi^2 e^2}{m_e c}~\frac{g_1}{g_2}~f_{12}</math>
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| == See also ==
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| * [[Transition dipole moment]]
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| * [[Oscillator strength]]
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| * [[Breit–Wigner distribution]]
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| * [[Electronic configuration]]
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| * [[Fano resonance]]
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| * [[Siegbahn notation]]
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| * [[Atomic spectroscopy]]
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| * [[Molecular radiation]], continuous spectra emitted by molecules
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| == References ==
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| {{reflist}}
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| === Cited bibliography ===
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| {{refbegin}}
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| *{{cite journal
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| |last1=Bohr |first1=N.
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| |author1-link=Niels Bohr
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| |year=1913
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| |title=On the constitution of atoms and molecules
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| |url=http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Bohr_1913.pdf
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| | doi = 10.1080/14786441308634993
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| |journal=[[Philosophical Magazine]]
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| |volume=26 |pages=1–25
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| |ref=harv
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| }}
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| *{{cite book
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| |last1=Brillouin |first1=L.
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| |author1-link=Léon Brillouin
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| |year=1970
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| |title=Relativity Reexamined
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| |publisher=[[Academic Press]]
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| |isbn=978-0-12-134945-5
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| |ref=harv
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| }}
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| *[[Subrahmanyan Chandrasekhar|Chandrasekhar, S.]] (1950). ''Radiative Transfer'', Oxford University Press, Oxford.
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| *{{cite journal | author=Einstein, A. |author1-link=Albert Einstein| title=Strahlungs-Emission und -Absorption nach der Quantentheorie |journal=Verhandlungen der Deutschen Physikalischen Gesellschaft |volume= 18|pages= 318–323 | year=1916|bibcode = 1916DPhyG..18..318E }} Also {{cite journal
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| |last1=Einstein |first1=A.
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| |author1-link=Albert Einstein
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| |year=1916
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| |title=Zur Quantentheorie der Strahlung
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| |journal=[[Mitteilungen der Physikalischen Gessellschaft Zürich]]
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| |volume=18 |pages=47–62
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| |ref=harv
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| }} And a version nearly identical to the latter at {{cite journal
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| |last1=Einstein |first1=A.
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| |author1-link=Albert Einstein
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| |year=1917
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| |title=Zur Quantentheorie der Strahlung
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| |journal=[[Physikalische Zeitschrift]]
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| |volume=18 |pages=121–128
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| |bibcode=1917PhyZ...18..121E
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| |ref=harv
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| }} Translated in {{cite book
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| |last1=ter Haar |first1=D.
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| |author-link=Dirk ter Haar
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| |year=1967
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| |pages=167–183
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| |title=The Old Quantum Theory
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| |publisher=[[Pergamon]]
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| |lccn=66029628
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| |ref=harv
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| }} Also in [http://astro1.panet.utoledo.edu/~ljc/einstein_ab.pdf Boorse, H.A., Motz, L. (1966). ''The world of the atom'', edited with commentaries, Basic Books, Inc., New York, pp. 888–901.]
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| *{{cite journal
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| |last1=Heisenberg |first1=W.
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| |author1-link=Werner Heisenberg
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| |year=1925
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| |title=Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen
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| |journal=[[Zeitschrift für Physik]]
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| |volume=33
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| |pages=879–893
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| |ref=harv
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| |bibcode = 1925ZPhy...33..879H |doi = 10.1007/BF01328377 }} Translated as "Quantum-theoretical Re-interpretation of kinematic and mechanical relations" in {{cite book
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| |last1=van der Waerden |first1=B.L.
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| |author1-link=Bartel Leendert van der Waerden
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| |year=1967
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| |title=Sources of Quantum Mechanics
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| |pages=261–276
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| |publisher=[[North-Holland Publishing]]
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| |ref=harv
| |
| }}
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| *{{Cite book
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| |last1=Jammer |first1=M.
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| |author1-link=Max Jammer
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| |year=1989
| |
| |title=The Conceptual Development of Quantum Mechanics
| |
| |edition=second
| |
| |publisher=[[Tomash Publishers]] [[American Institute of Physics]]
| |
| |isbn=0-88318-617-9
| |
| |ref=harv
| |
| }}
| |
| *Loudon, R. (1973/2000). ''The Quantum Theory of Light'', (first edition 1973), third edition 2000, Oxford University Press, Oxford UK, ISBN 0-19-850177-3.
| |
| *{{cite book
| |
| |last1=Sommerfeld |first1=A.
| |
| |others=Brose, H. L. (transl.)
| |
| |author1-link=Arnold Sommerfeld
| |
| |year=1923
| |
| |title=Atomic Structure and Spectral Lines
| |
| |url=http://books.google.com/books/about/Atomic_structure_and_spectral_lines.html?id=u1UmAAAAMAAJ
| |
| |edition=from 3rd German
| |
| |publisher=[[Methuen Publishing|Methuen]]
| |
| |ref=harv
| |
| }}
| |
| {{refend}}
| |
| | |
| ==Other reading==
| |
| | |
| {{refbegin}}
| |
| * {{cite book | author=Condon, E.U. and Shortley, G.H. | title=The Theory of Atomic Spectra | publisher=Cambridge University Press | year=1964 | isbn =0-521-09209-4 }}
| |
| * {{cite book | author=Rybicki, G.B. and Lightman, A.P. | title=Radiative processes in Astrophysics | publisher=John Wiley & Sons, New York | year=1985 | isbn =0-471-82759-2 }}
| |
| * {{cite book | author=Shu, F.H. | title=The Physics of Astrophysics - Volume 1 - Radiation | publisher=University Science Books, Mill Valley, CA | year=1991 | isbn =0-935702-64-4 }}
| |
| * {{cite journal | author=Robert C. Hilborn | title=Einstein coefficients, cross sections, f values, dipole moments, and all that | journal=physics/0202029 | year=2002 | url = http://arxiv.org/abs/physics/0202029 }}
| |
| * {{cite journal | author=Taylor, M.A. and Vilchez, J.M. | title=Tutorial: Exact solutions for the populations of the n-level ion | journal=Pub. Astron. Soc. Pac. 121, 885 | pages=1257–1266 | year=2009}}
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| * [http://ioannis.virtualcomposer2000.com/spectroscope/amici.html#colorphotos Emission Spectra from various light sources]
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| | |
| [[Category:Emission spectroscopy]]
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| [[bg:Атомна спектрална линия]]
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| [[it:Linea spettrale atomica]]
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| [[pl:Widmo liniowe]]
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| [[zh:原子谱线]]
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