5 Atomic radiation processes Institute for Astronomy

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A Line transitions,Einstein coefficients, probability that a photon in frequency interval in the solid angle. range is absorbed by an atom in the energy level El with a. resulting transition El Eu per second,dwabs l u B lu I d d. atomic property no of incident probability for probability for. photons absorption of with,photon with,probability for absorption. transition l u profile,Blu Einstein coefficient for absorption. Einstein coefficients,similarly for stimulated emission.
dwst l u Bul I d d,Bul Einstein coefficient for stimulated emission. and for spontaneous emission,dwsp l u A ul d d,Aul Einstein coefficient for spontaneous emission. dwst l u Bul I d d,4 dwabs l u B lu I d d, Einstein coefficients absorption and emission coefficients. Number of absorptions,stimulated emissions,I dF dV dF ds in dV per second. nl dw abs dV nu dwst dV,absorbed energy in dV per second.
stimulated emission counted, dE abs nl h dw abs dV nu h dwst dV as negative absorption. and also using definition of intensity, I ds d d dF Absorption and emission coefficients are a. function of Einstein coefficients occupation,numbers and line broadening. nl B lu nuBul,for the spontaneously emitted energy L. L nl B lu nuBul L,Relations between Einstein coefficients.
Einstein coefficients are atomic properties do not depend on thermodynamic. state of matter,We can assume TE S L B T,nuAul n Aul. nl B lu nu B ul nl Blu nnul B ul,From the Boltzmann formula e kT. nu gu h 2 2,for h kT 1 1 B T 2 kT,nl gl kT c,2 2 gu h A. c2 gl kT Blu gl,0 gl Blu guB ul 6,2 2 gu h A,gl Blu guB ul. c2 gl kT Blu gl,Relations between Einstein coefficients.
2 2 gu h A ul kT,c2 gl kT B lu h,2 h 3 A ul gu h,c2 B lu gl kT. 2 h 3 gl Note Einstein coefficients atomic,Aul B quantities That means any. c2 gu lu relationship that holds in a special,thermodynamic situation such as T. 2 h 3 very large must be generally valid,L B lu nl n. only one Einstein,h gl 2h 3 coefficient needed,Oscillator strength.
Quantum mechanics, The Einstein coefficients can be calculated by quantum. mechanics classical electrodynamics calculation,Eigenvalue problem using using wave function. Hatom l El l H atom Vnucleus Vshell, Consider a time dependent perturbation such as an external. electromagnetic field light wave E t E0 ei t, The potential of the time dependent perturbation on the atom is. V t e E ri E d d dipol operator,Hatom V t l El l,with transition probability l d u 2 8.
Oscillator strength,The result is B f,4 lu mec lu, flu oscillator strength dimensionless classical result from electrodynamics. 0 02654 cm2 s,Classical electrodynamics, electron quasi elastically bound to nucleus and oscillates within outer electric field as E. Equation of motion damped harmonic oscillator, x x 20 x E ma damping force restoring force EM force. me the electron oscillates preferentially at resonance. incoming radiation 0, constant resonant The damping is caused because the de and. accelerated electron radiates,2 20 e2 2 2,8 e 20 natural.
3 mec3 3 me c3 9,Classical cross section and oscillator strength. Calculating the power absorbed by the oscillator the integrated classical. absorption coefficient and cross section and the absorption line profile are. Z 2 nl number totcl classical,integrated over the line profile L cl. nl tot density of cross section,me c absorbers cm2 s. d Lorentz damping line profile, oscillator strength flu is quantum mechanical correction to classical result. effective number of classical oscillators 1 for strong resonance lines. From nl B lu neglecting stimulated emission,Oscillator strength.
h e 2 absorption cross section,B lu flu lu dimension is cm2. flu h B lu, Oscillator strength f value is different for each atomic transition. Values are determined empirically in the laboratory or by elaborate numerical atomic physics calculations. Semi analytical calculations possible in simplest cases e g hydrogen. flu 3 3 2 l u3,u2 g Gaunt factor,H f 0 6407,H f 0 1193. H f 0 0447 11,Line profiles, line profiles contain information on physical conditions of the gas and. chemical abundances, analysis of line profiles requires knowledge of distribution of opacity with.
several mechanisms lead to line broadening no infinitely sharp lines. natural damping finite lifetime of atomic levels, collisional pressure broadening impact vs quasi static approximation. Doppler broadening convolution of velocity distribution with atomic profiles. 1 Natural damping profile,finite lifetime of atomic levels line width. NATURAL LINE BROADENING OR RADIATION DAMPING, t 1 Aul 10 8 s in H atom 2 1 finite lifetime with respect to. spontaneous emission,E t h 2 uncertainty principle. 1 2 1 2 2 c,line broadening,e g Ly 1 2 1 2 10 4 A,1 4 H 1 2 4 6 10 4 A.
Lorentzian profile 13,Natural damping profile,resonance line excited line. natural line broadening is important for strong lines resonance. lines at low densities no additional broadening mechanisms. e g Ly in interstellar medium,but also in stellar atmospheres. 2 Collisional broadening, radiating atoms are perturbed by the electromagnetic field of neighbour atoms. ions electrons molecules, energy levels are temporarily modified through the Stark effect perturbation. is a function of separation absorber perturber, energy levels affected line shifts asymmetries broadening.
E t h C rn t r distance to perturbing atom, a impact approximation radiating atoms are perturbed by passing particles at distance r t. Duration of collision lifetime in level lifetime shortened line broader. in all cases a Lorentzian profile is obtained but with larger total than only natural damping. b quasi static approximation applied when duration of collisions life time in level. 5 Atomic radiation processes Einstein coefficients for absorption and emission oscillator strength line profiles damping profile collisional broadening Doppler broadening continuous absorption and scattering 2 3 A Line transitions Einstein coefficients probability that a photon in frequency interval in the solid angle range is absorbed by an atom in the energy level E l with a

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