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| [[File:Spectral lines en.PNG|thumb|Emission lines and absorption lines compared to a continuous spectrum.]]
| | == Xiaomengチー、市清懐シーン、警察 == |
| 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.
| | 非常に強い、と会衆に駆けつけランツィだけその車に停止を逃れることがすぐに散乱おびえ、突然起動停止しました [http://www.dmwai.com/webalizer/kate-spade-4.html ケイトスペード ショルダーバッグ]。<br><br>車は人に大きな安堵が、彼の人生は彼らの最後の足かもしれ作ることができる以上の犯罪の側に駐車し、それはその時点で、高速で、彼はそれがマウスの良心だと思った、それだけでされている車を見張ることができるものであるた女性はそれは、Lakaichemenは、ダイビングが行くようになった私は、私は考えず罪プロラッシュに行きたいしている勇敢な犯罪のように見えますが、彼女のスイング回路図をカット。<br><br>ブーム [http://www.dmwai.com/webalizer/kate-spade-14.html kate spade マザーズバッグ]...ブーム [http://www.dmwai.com/webalizer/kate-spade-12.html ケイトスペード 財布 値段]...ブーム [http://www.dmwai.com/webalizer/kate-spade-4.html ケイトスペード 時計 取り扱い店]...カープラススロットル、前進しようとし、マチェーテスティックチームのグループの前の方向は、その銃の男がちょうど車の女歯以来、彼の手を上げ、燃料ドア、うわー、彼は車の音に向かって男怖い情事ガンを急いで、ああ、母、顧客のラウンジに彼の人生を注いだ別の捨てられた客の中にそれを隠して、地面を走った。<br><br>は彼を急いでいなかったが、再び鋭い反転は、洗浄ラインのうち、スピードでレースすることは離れて引っ張って [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペードニューヨーク 財布]...<br>救急車が行く準備ができて<br>Xiaomengチー、市清懐シーン、警察 |
| * 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 (the full spectrum of electromagnetic radiation) and a spectrum will show a drop in the continuum radiation at the wavelength associated with the absorbed photons.
| | 相关的主题文章： |
| | <ul> |
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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.
| | == お祝いを取得 == |
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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>
| | 、単独で涙を拭った [http://www.dmwai.com/webalizer/kate-spade-8.html kate spade 財布 ゴールド]。<br>お祝いを取得<br>、喜びだけスキムの涙の歌を遠慮すること、知っていました [http://www.dmwai.com/webalizer/kate-spade-9.html バッグ ケイトスペード]。<br><br>私の子供たちは泣く！ [http://www.dmwai.com/webalizer/kate-spade-14.html ケイトスペード クラッチバッグ] ？これは希少性とすることができ、マウスは、彼が重かった、としてつぶやいた犯罪の数を知っている：「話すことで徐によると、あなたの出生記録を開始してから選択された意図的に省庁を消去され、続いているシークレットサービス捜査隊、旧地方練習を募集罪の上...... [http://www.dmwai.com/webalizer/kate-spade-11.html ケイトスペードのバッグ] 2が存在するよりも少しだけよりは、レコードがないので、今、あなたも復元したい、また、一時半のものです......许平秋前に来て、私が話を夜には、そのようなことは、私は同意通過することですが、この特定の状況が、彼はそれは一つのことにまでバックルを意図していた場合には言った、私は何よりもそれの多くを行うことができない場合があります怖いていたので、私たちは保たれていますドラムには......」<br>でも、彼はそれが少し残酷だと感じ、彼は少し行くことができなかったので、マウスの声がゆっくりと、ますます小さくなったときに、次に<br> [http://www.dmwai.com/webalizer/kate-spade-15.html kate spade マザーズバッグ]。あなたはすべて取り除か、親戚、友人、クラスメート、すべてはあなたを知って、あなたが知っている、元の家から生活環を生きてます |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://qzhzt.mdiscuz.com/forum.php?mod=viewthread&tid=47766 http://qzhzt.mdiscuz.com/forum.php?mod=viewthread&tid=47766]</li> |
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| | <li>[http://talee.net/blogs/post/9092 http://talee.net/blogs/post/9092]</li> |
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| | </ul> |
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| == Emission and absorption coefficients == | | == 彼は横に走った、私はもつれ罪明らかに恐れて == |
| 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.
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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:
| | 多くのHotShotsのはあまり、彼はまた、深刻な疑問ゆうErxiongの弟で、瞬間をHongchuan場合があります。<br><br>」ではないちゃトピックを行い、......これらのことを知っているかもしれませんねえガオ林姉妹兄弟が、あなたが言う、それを聞いて？ [http://www.dmwai.com/webalizer/kate-spade-10.html ケイトスペード バッグ ショルダー] '私は罪を尋ねた。<br><br>'。あなたの夢中酒に見て、私はあなたを伝えることができ、「高尚な悪い笑顔で道を冗談：'遼長官、あなたは高速アウト、彼は絶対に知っている自転車rruu彼を求めるだろう。私はあなたに言ったことはもちろんのこと、ああ行く。 [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード バッグ 激安] '<br>彼は横に走った、私はもつれ罪明らかに恐れて<br>、私はすぐに走った、バス罪、かまの外を見た [http://www.dmwai.com/webalizer/kate-spade-4.html ケイトスペード バッグ 新作]。しかし、予想外に速い車は、私はあまり罪の方向をかわす、私はドライバーの罪はねビープ泥は、裁判所の死刑ああ判決を批判した [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード バッグ 激安]。そして、罪の顔のタッチよりも、とは早くなくなってようときに車の彼の目を開いた。<br><br>「ママは力を持っていたので、私はディレクターに就任、最初にオープンしました。 [http://www.dmwai.com/webalizer/kate-spade-12.html ケイトスペード 財布 値段] '<br>怒っ<br>彼はホッピングが、批判振り返ってみると、多くのデューティ·部屋のドアでの同僚やポインティングが彼を見て笑ったのを見呪わ、彼はこの恥ずかしいな方法でカバーされた、Xiufenの下で、彼女をカバーし |
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| :<math>\epsilon = \frac{h\nu}{4\pi}n_2 A_{21}\,</math>
| | == '豆暁波は、本格的なヨセフだった == |
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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.
| | 広告ミックスの業界が、あまり知られて頭を善戦。 [http://www.dmwai.com/webalizer/kate-spade-1.html ケイトスペード 財布 セール] '<br><br>「投資の広告と同等、右、ちょっと、細部はあなたが知らない、とにかく、あなたを教えていないだろう。 [http://www.dmwai.com/webalizer/kate-spade-1.html ケイトスペード リボン バッグ] '豆暁波は、本格的なヨセフだった。<br>二人は多分小さな広告が肌本当に厚くすることができる掲示、ドンShaojunは言うまでもないバック熊剑飞を見て、なぜ、少し恥ずかしい知って、この大きな男は確かに、本質的にポーターであり、高尚と林ゆうジンポップ散布し笑い、話す<br>グッドHunfanは、彼が心配して尋ねた、と言うことではない罪の上に少し薄い小さな子供に見えることができます： '？私の子は、ここで手に入れた方法」<br><br>「私......それを唱え、飢餓の食事、十分な食事の上に、マウスのおかげで、それらを打つようにします [http://www.dmwai.com/webalizer/kate-spade-15.html kate spade マザーズバッグ]。「私は静かに本物の罪、この男は本物でそれを置くために持って生まれて製造嘘です、ドンShaojunこと本格的な文章は、マウスを見て、間違いなく、彼は奇妙で、驚いて、「うん、マウスは、あなたは良いテーブルの上に置かれ、それはまた、死ぬことができる、子どもたち以上に売春援助に戻らないのだろうか？。」 [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペード 人気バッグ]。<br><br>「私は [http://www.dmwai.com/webalizer/kate-spade-6.html ケイトスペード マザーズバッグ]......」、表では、見て、勝つために、マウスの瞬間である |
| | 相关的主题文章： |
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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:
| | == shut the door == |
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| :<math>\kappa' = \frac{h\nu}{4\pi}~(n_1 B_{12}-n_2 B_{21}) \,</math>
| | 'Time does not, and why would a wedding anniversary? Why can not every [http://www.dmwai.com/webalizer/kate-spade-14.html kate spade マザーズバッグ] day is Memorial Day and I know that you are the police, but the police can not do at home, [http://www.dmwai.com/webalizer/kate-spade-14.html kate spade マザーズバッグ] right? Outside is a world, shut the door, the couple may also [http://www.dmwai.com/webalizer/kate-spade-0.html ケイトスペード バッグ] be a world that world, it should be you say, every day. 'Mo experts said.<br><br>'how can I do?' [http://www.dmwai.com/webalizer/kate-spade-8.html ハンドバッグ ケイトスペード] Lin Yu Jing asked, the mood began to suddenly see the light.<br><br>'do all day every day like a wedding on the line. longer boring man, [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペード 人気バッグ] also like the kind of mood. And you did not have the charm of age, stature, body, face, no worse than anyone else ah.' Mo expert laughed.<br><br>Lin Yu Jing embarrassed touch cheeks, embarrassed authentic with: 'I arrived on the back office not long before all the field work, long run, but for those really did not notice.'<br><br>'It now began to pay attention to it, believe me, your charm enough to save a man's heart, even the stars polygamous, huh.' Mo experts laughed.<br><br>'This ...... he really a little flower.' |
| | | 相关的主题文章： |
| 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>
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| The above equations have ignored the influence of the [[spectroscopic 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.
| | <li>[http://decklink.cn/plus/view.php?aid=361802 http://decklink.cn/plus/view.php?aid=361802]</li> |
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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.
| | <li>[http://www.qbike.com/cgi-bin/classifieds.cgi http://www.qbike.com/cgi-bin/classifieds.cgi]</li> |
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| ===Equilibrium conditions===
| | <li>[http://jbhq.org/plus/feedback.php?aid=6 http://jbhq.org/plus/feedback.php?aid=6]</li> |
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| 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">Mihalas, D., Weibel-Mihalas, B. (1984), pp. 329–330.</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.
| | </ul> |
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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>
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| ===Various formulations===
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| Hilborn has compared various formulations for derivations for the Einstein coefficients, by various authors.<ref>Hilborn, R.C. (2002). [http://arxiv.org/ftp/physics/papers/0202/0202029.pdf Einstein coefficients, cross sections, ''f'' values, dipole moments, and all that.]</ref> For example, Hertzberg works with irradiance and wavenumber.<ref>Herzberg, G. (1950).</ref> Yariv works with energy per unit volume per unit frequency interval;<ref name="Yariv">[[Amnon Yariv|Yariv, A.]] (1967/1989), pp. 171–173.</ref> also;<ref>Garrison, J.C., Chiao, R.Y. (2008), pp. 15–19.</ref> this is how the present account is formulated. Mihalas & Weibel-Mihalas work with radiance and frequency;<ref name="Mihalas Mihalas 1984"/> also Chandrasekhar;<ref>[[Subrahmanyan Chandrasekhar|Chandrasekhar, S.]] (1950), p. 354</ref> also Goody & Yung;<ref>Goody, R.M., Yung, Y.L. (1989), pp. 33–35.</ref> Loudon uses angular frequency and radiance.<ref>Loudon,R. (1973/2000), pp. 16–19.</ref>
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| ===Spontaneous emission===
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| {{main|Spontaneous emission}}
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| [[File:AtomicLineSpEm.svg|thumb|220px|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.svg|thumb|220px|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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| {{-}}
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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 ===
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| {{main|Absorption (optics)}}
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| [[File:AtomicLineAb.svg|thumb|220px|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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| {{-}}
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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><ref name="Yariv"/>
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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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| and
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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^3}~\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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| * [[Relativistic Breit–Wigner distribution|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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| *Garrison, J.C., [[Raymond Chiao|Chiao, R.Y.]] (2008). ''Quantum Optics'', Oxford University Press, Oxford UK, ISBN 978-019-850-886-1.
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| *Goody, R.M., Yung, Y.L. (1989). ''Atmospheric Radiation: Theoretical Basis'', 2nd edition, Oxford University Press, Oxford, New York, 1989, ISBN 0-19-505134-3.
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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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| }}
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| *Herzberg, G. (1950). ''Molecular Spectroscopy and Molecular Structure'', vol. 1, ''Diatomic Molecules'', second edition, Van Nostrand, New York.
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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
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| |title=The Conceptual Development of Quantum Mechanics
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| |edition=second
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| |publisher=[[Tomash Publishers]] [[American Institute of Physics]]
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| |isbn=0-88318-617-9
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| |ref=harv
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| }}
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| *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.
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| *[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.]
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| *{{cite book
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| |last1=Sommerfeld |first1=A.
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| |others=Brose, H. L. (transl.)
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| |author1-link=Arnold Sommerfeld
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| |year=1923
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| |title=Atomic Structure and Spectral Lines
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| |url=http://books.google.com/books/about/Atomic_structure_and_spectral_lines.html?id=u1UmAAAAMAAJ
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| |edition=from 3rd German
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| |publisher=[[Methuen Publishing|Methuen]]
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| |ref=harv
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| }}
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| *[[Amnon Yariv|Yariv, A.]] (1967/1989). ''Quantum Electronics'', third edition, John Wiley & sons, New York, ISBN 0-471-60997-8.{{refend}}
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| ==Other reading==
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| {{refbegin}}
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| * {{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 }}
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| * {{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 }}
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| * {{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 }}
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| * {{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 }}
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| * {{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}}
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| {{refend}}
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| ==External links==
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| * [http://ioannis.virtualcomposer2000.com/spectroscope/amici.html#colorphotos Emission Spectra from various light sources] {{Dead link|date=November 2012}}
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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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