Collisional-Radiative Models and the Emission of Light

Transcription

1 Collisional-Radiative Models and the Emission of Light F.B. Rosmej Sorbonne Universités, Pierre et Marie Curie, Paris, France and Ecole Polytechnique, LULI-PAPD, Palaiseau, France Diagnostics Plasma, 1-3/12/2014, Orléans, France

2 Contents I. Introduction II. Collisional-Radiative models III. Quantum mechanical point of view IV. Conclusion and outlook

3 I. Introduction Then God said let there be light and there was light... Genesis

4 ...and since then, light is one of the most fascinating phenomena in nature...

5 We all know what light is, but it is not easy to say what it really is...

6 Historical overview 1180: Nekkam, magnetic needle : compass. 1289: Maricourt, reflection of the same type of magnetic poles. 1771, 1773: Cavendish, electrostatic experiments. 1785: Coulomb, electrostatic law of a point charge. 1831: Faraday, electromagnetic induction. 1864: Maxwell, dynamical theory of the electromagnetic field. 1900: Planck, beginning of the quantum theory of light, quantization of the energy of a harmonic oscillator (ref.: Verhandlungen der Deutschen Physikalischen Gesellschaft 2 (1900), 202 and 237). 1905: Einstein, photoelectric effect, corpuscularity of electromagnetic radiation (ref.: Annalen der Physik 17 (1905), 132). 1917: Einstein, interaction of electromagnetic radiation with atoms (ref.: Phys. Z. 18 (1917), 121). 1926: Lewis, the quantum of radiation was named photon (ref.: G.N. Lewis, Nature 118 (1926), 874). 1963: Glauber, coherent photon states (ref.: R.J. Glauber, Phys. Rev. 131 (1963), 2766).

7 Aurora light emission In 1924 McLennan & Shrum showed in a laboratory experiment (discharge with O + He), that the green color originates from the Oxygen.

8 Atomic population and light emission Consider a two-level atom: Line emission ε: ε = n 1 A 10 h ν 10 = energy time volume n = atomic population = number volume A = C = number of radiation decays time number of collisions time

9 Collisional-radiative equations d n 1 d t = n 0 C 01 n 1 A 10 n ( 1 C 10 + C ) d1 population by collisions depopulation by radiative decay depopulation by collisions Stationary case: d dt = 0 => n 1 = n 0 C 01 A 10 +C 10 +C d1 ε= n 0 C 01 h ν 10 A 10 A 10 + C 10 +C d1

10 The collisional rate C is proportional to the particle density: C n What happens for low particle densities? A 10 C 10 + C d1 ε= n 0 C 01 h ν 10 A 10 A 10 + C 10 +C d1 n 0 C 01 h ν 10 A 10 A 10 => ε n 0 C 01 h ν 10 At low particle densities, the intensity ε turns out to be independent of the transition probability A

11 The water-pool model I. water population flow excitation rate "C" h water pool atomic population "n" hole area A transition probability "A" water loss h A radiation intensity n A

12 The water-pool model II. small water population flow excitation rate "C" h=0 atomic population "n" hole area A transition probability "A" water loss radiation intensity water flow independent of A C

13 Spatial distribution of colors Why does the red color appear only at large altitudes?

14 ε= n 0 C 01 h ν 10 A 10 A 10 + C 10 +C d1 depopulation high density A n 0 C 01 h ν C 10 +C d1 all upper level population is quenched by collisions before radiative decay Quenching condition: A 10 << C 10 + C d1 depopulation A(red) 10-3 s -1, A(green) 10-1 s -1 A(red) << A(green) The red light is quenched for lower particle densities, so only at high altitudes where the density is low the red light can survive

15 II. Collisional-Radiative model Light emission Atomic population Theory: collisional-radiative model/atomic kinetics derive densities, temperatures,... = diagnostics non-perturbative

16 Distribution of ion densities n Z Z+1 d n Z dt = n ( Z n 2 e T Z,Z 1 + n e D Z,Z 1 + n e R Z,Z 1 + n e I ) Z,Z +1 + n ( Z +1 n 2 e T Z +1,Z + n e D Z +1,Z + n e R ) Z +1,Z + n ( Z 1 n e I ) Z 1,Z I I T D R T D R Z Z-1 T: 3-body recombination D: dielectronic recombination R: radiative recombination I: ionization X Z +1 + e + e X Z + e X Z +1 + e X Z** X Z** X Z* + ω X Z + ω ' X Z +1 + e X Z X Z + e X Z +1 + e + e

17 Charge stage distribution: stationary solution n Z +1 n Z = n e I Z,Z +1 n e R Z +1,Z + n e D Z +1,Z + n e 2 T Z +1,Z High density limit: n Z +1 n Z = 2 m ekt e 2π 2 3/2 g Z +1 g Z e EZ /kte n e Saha-Boltzmann Low density limit: n Z +1 n Z = I Z,Z +1 R Z +1,Z + D Z +1,Z Corona I Z,Z +1 R Z +1,Z + D Z +1,Z = F(T e ) Independent of density! n e 2 T << n e R + n e D

18 Charge stage distribution of Argon closed atomic shells create broad distribution for He-like and Ne-like ions MARIA-code: F.B. Rosmej, Europhysics Letters 76, 1081 (2006 ) F.B. Rosmej: Exotic states of high density matter driven by intense XUV/X-ray Free Electron Lasers, "Free Electron Laser", InTech 2012, editor S. Varró, p , ISBN (free download)

19 Spectral distribution I jz i Z ( ω ) = ω j Z i Z 4π n j Z A jz i Z ϕ ( jz i Z ω,ω ) jz i Z single line transition ( ) = I jz i Z ( ω ) I ω Z n Z =0 N N Z Z j Z =1 i Z =1 total transitions Atomic population of level "j" from charge state "Z": dn jz dt Z N n Z ' Z '=0 i Z ' =1 = n jz W jz i Z ' + Z N n Z ' Z '=0 k Z ' =1 n kz ' W kz ' j Z depopulation population Population matrix W: Atomic physics processes: W ij = W rad col ij + W ij W rad ij = A ij + Γ ij + P abs ij + P em ij + P rr iz ij + P ij W ij col = n e C ij + n e I ij + n 2 e T ij + n e R ij + n e D ij + Cx ij + n HP C HP ij + n HP I HP ij... A: radiative decay Γ: autoionization C: excitation/dexcitation I: ionization T: 3-body recombination R: radiative recombinaton D: dielectronic recombination P abs : photoabsorption P em : stimlated emission P rr : stimulated rad. recomb. P iz : photo ionization Cx: charge exchange C HP : heavy particle excitation I HP : heavy particle ionization

20 Spectral distribution & density variations I H-like Aluminum: Ly α : 1s 2p He-like satellites: 1s2l 2l2l' Collisional redistribution between atomic levels is density dependent F.B. Rosmej : X-ray emission spectroscopy and diagnostics of nonequilibrium fusion and laser produced plasmas, in Highly Charged Ion Spectroscopic Research, Taylor & Francis 2012, p. 267, ISBN:

21 Spectral distribution & temperature variations Resonance line intensity: res I k,j'i' = n e n k C kj' Dielectronic satellite line intensity: kt e sat I k,ji = n e n k D kj l A ji A jl + m Γ jm Dielectronic capture rate (Γ=autoinization rate): D kj = Γ jk g j g k ( ) exp E kj / kt e ( ) 3/2 cm 3 s 1 kt e sat I k, ji = n en(1s) D 1s j res I k ' j 'i ' n e n(1s) C 1s j ' = function( T e ) F.B. Rosmej : X-ray emission spectroscopy and diagnostics of non-equilibrium fusion and laser produced plasmas, in Highly Charged Ion Spectroscopic Research, Taylor & Francis 2012, p. 267, ISBN:

22 Line intensity ratios and diagnostics As atomic populations depend in general on collisional-radiative processes, their population is dependent on density, temperature, hot electrons,...: I jz i Z = γ ( T e,n e, f hot,...) I j 'Z ' i ' Z ' The ideal case of a temperature diagnostic is identified as : I jz i Z I j 'Z ' i ' Z ' = α ( T e ) Dielectronic satellites The ideal case of a density diagnostic is identified as : I jz i Z I j 'Z ' i ' Z ' = β ( n e )

23 Spectral distribution & density variations II Stark broadening of dielectronic satellites Experiment: High-spectral resolution Theory: spectral simulations atomic populations/kinetics line profiles F.B. Rosmej : X-ray emission spectroscopy and diagnostics of non-equilibrium fusion and laser produced plasmas, in Highly Charged Ion Spectroscopic Research, Taylor & Francis 2012, p. 267, ISBN:

24 Experiment: dense laser produced plasma Nd-Glass laser: 50 J, 12 ns, Mg-target I jz i Z I j 'Z ' i ' Z ' = γ T e,n e, f hot,... ( ) MARIA simulations : kt e = 210 ev, n e = cm 3, L eff = 500µm Multiple line ratios are requested Spectral simulations! F.B. Rosmej : X-ray emission spectroscopy and diagnostics of non-equilibrium fusion and laser produced plasmas, in Highly Charged Ion Spectroscopic Research, Taylor & Francis 2012, p. 267, ISBN: Adds line profile information to line ratios!

25 Suprathermal electrons I Suprathermal electrons are in strong "competition" with bulk electrons Hot to distinguish hot electrons from bulk electrons? F.B. Rosmej : X-ray emission spectroscopy and diagnostics of non-equilibrium fusion and laser produced plasmas, in Highly Charged Ion Spectroscopic Research, Taylor & Francis 2012, p. 267, ISBN:

26 Suprathermal electrons II Qualitative distortion of the distribution of ionic populations: Charge stages are mixed! F.B. Rosmej : X-ray emission spectroscopy and diagnostics of non-equilibrium fusion and laser produced plasmas, in Highly Charged Ion Spectroscopic Research, Taylor & Francis 2012, p. 267, ISBN:

27 III. Quantum mechanical point of view

28 Quantum mechanical two-level atom 2 1 ω 0 = E 2 E 1 Ĥ E ψ 1 = E 1 ψ 1 Ĥ E ψ 2 = E 2 ψ 2 ψ 1 ( r,t) = e ie 1 t / ψ 1 ( r ) ψ 2 ( r,t) = e ie 2 t / ψ 2 ( r ) Electromagnetic interaction: Ĥ = ĤE + Ĥ em( r,t) ψ ( r,t ) = C 1 ( t)ψ 1( r,t) + C2 ( t)ψ 2 ( r,t) C 1 C 1 ( t) H 11 + C 2 t ( t)e iω 0t H 21 + C 2 t ( ) ( )e iω 0t H 12 = i C t 1 t ( ) H 22 = i C ( t) 2 t H ij := ψ i * r ( ) Ĥ 1 ψ j r ( )dv

29 ( ) ρ 11 := C 1 ( t) 2 = N 1 t N ( ) ρ 22 := C 2 ( t) 2 = N 2 t N number of ground states = total number of atoms number of excited states = total number of atoms Atomic populations ρ 12 := C 1 ( t)c * 2 ( t) ρ 21 := C * 1 ( t)c 2 ( t) Quantum mechanical populations: Heuristic approach: dρ 22 dt dρ 12 dt = 1 ( 2 ivei ω 0 ω )t ρ ( 2 ive i ω 0 ω )t ρ 21 = 1 ( 2 ive i ω 0 ω )t ( ρ 11 ρ 22 ) Diagonal & non-diagonal elements are inherently coupled to each other! dρ jj dt k q = ρ jj W jk + ρ qq W qj Non-diagonal elements do not exist!

30 Quantum mechanical versus heuristic approach Straight forward expansion of the density matrix equations in terms of diagonal elements looks not possible... Photo-absorption and emission in heuristic approach: Electromagnetic interaction: A 21, B 12, B 21 : Einstein cofficients dn 2 dt = N 2 A BWN 2 + BWN 1 spontaneous emission stimulated emission stimulated absorption W : radiation field Time dependent solution: N 2 ( t) = NBW { ( } )t A + 2BW 1 e A+2 BW...to be compared with density matrix solution...

31 dρ 22 dt dρ 12 dt = 1 ( 2 ivei ω 0 ω )t ρ ( 2 ive i ω 0 ω )t ρ 21 2γ rad ρ 22 = 1 ( 2 ive i ω 0 ω )t ( ρ 11 ρ 22 ) ( γ rad + γ coll )ρ 12 V := ee 0X 12 Electromagnetic perturbation potential X 12 = ψ 1 * ( r ) x ψ 2 ( r ) dv Dipole matrix elements 2γ rad = A γ coll = elastic collisions No general solution...

32 ...except for the weak field limit... ρ 22 { ( )t ζ } 3 ( t) = 1 4 V 2 ζ 1 +ζ 2 2e γ rad +γ coll ζ 1 = ( γ rad + γ coll ) / γ rad ( ω ω 0 ) 2 + γ rad + γ coll ζ ( ) 2 2 = ( γ rad γ coll ) / γ rad e 2γ radt ( ω ω 0 ) 2 + ( γ rad γ coll ) 2 ζ 3 = ( ω 0 ω ) 2 + γ rad + γ coll ( ω 0 ω ) 2 + γ rad + γ coll ( )( γ rad γ coll )cos ( ω 0 ω )t ( ) 2 ( ω 0 ω ) 2 + ( γ rad γ coll ) 2 + 2γ coll ( ω 0 ω )sin ( ω 0 ω )t ( ω 0 ω ) 2 + ( γ rad + γ coll ) 2 ω 0 ω ( ) 2 + ( γ rad γ coll ) 2 Weak field limit in heuristic approach: N 2 ( t) = NBW { ( } )t A>>2 BW N 2 t A + 2BW 1 e A+2 BW ( ) = NBW A { 1 e At }

33 ...except for the weak field limit... ρ 22 { ( )t ζ } 3 ( t) = 1 4 V 2 ζ 1 +ζ 2 2e γ rad +γ coll ζ 1 = ( γ rad + γ coll ) / γ rad ( ω ω 0 ) 2 + γ rad + γ coll ζ ( ) 2 2 = ( γ rad γ coll ) / γ rad e 2γ radt ( ω ω 0 ) 2 + ( γ rad γ coll ) 2 ζ 3 = ( ω 0 ω ) 2 + γ rad + γ coll ( ω 0 ω ) 2 + γ rad + γ coll ( )( γ rad γ coll )cos ( ω 0 ω )t ( ) 2 ( ω 0 ω ) 2 + ( γ rad γ coll ) 2 + 2γ coll ( ω 0 ω )sin ( ω 0 ω )t ( ω 0 ω ) 2 + ( γ rad + γ coll ) 2 ω 0 ω ( ) 2 + ( γ rad γ coll ) 2 Weak field limit in heuristic approach: N 2 ( t) = NBW { ( } )t A>>2 BW N 2 t A + 2BW 1 e A+2 BW ( ) = NBW A { 1 e At } Both equations have still nothing in common...

34 ...the quantum mechanical phases are at the origin... I. "Broad band illumination": frequency spread of the radiation field exceeds considerably the line width 2(γ rad + γ coll ): + ρ 22 dω = πv 2 4γ rad { 1 e 2γ radt } II. Large collisional broadening: γ coll >> γ rad ρ 22 = ( γ rad + γ coll )V 2 / 4γ rad ( ω 0 ω ) 2 + γ rad + γ coll { } ( ) 2 1 e 2γ radt (provides also line profile)

35 ...the quantum mechanical phases are at the origin... I. "Broad band illumination": frequency spread of the radiation field exceeds considerably the line width 2(γ rad + γ coll ): + ρ 22 dω = πv 2 4γ rad { 1 e 2γ radt } Looks similar to II. Large collisional broadening: γ coll >> γ rad N 2 ( t) = NBW A { 1 e At } ρ 22 = ( γ rad + γ coll )V 2 / 4γ rad ( ω 0 ω ) 2 + γ rad + γ coll { } ( ) 2 1 e 2γ radt

36 IV. Conclusion and outlook Most of light emission is atomic in origin and therefore subject to quantum effects Atomic populations determine the spectral distribution of the light emission Collisional-radiative processes determine the atomic populations Simulation of the atomic populations allow to draw conclusions: temperature, density, hot electrons,... Quantum mechanical description: density matrix The relation between quantum and heuristic approach is not simple... and has not yet been solved for realistic multi-level systems Einstein theory invalid for high intensities: the Einstein coefficients are not constant, 1 st order perturbation theory, rotating wave approximation, randon orientation, non-interfering matter probe...