Radiative and non-radiative transitions

Transcription

1 Radiative and non-radiative transitions Radiative bound bound absorption spontaneous emission stimulated emission bound free: photoionization free bound: radiative recombination fluorescence / Auger effect Collisional excitation deexcitation collisional ionization/recombination Piotr Życki; PhD Course Astrophysical spectroscopy, CAMK, 2007,

2 Reminder energy levels Hydrogen

3 Radiative transitions Bound bound transitions: Einstein coefficients 1. Spontaneous emission A ul = transition probability per unit time for spontaneous emission (sec -1 ) 2. Absorption B lu J = transition probability per unit time for absorption J J - line profile d ; d = Stimulated emission B ul J = transition probability per unit time for stimulated emission Relations between Einstein coefficients: g l B lu =g u B ul (transitions between levels 1 and 2 have to balance) g, l g u statistical weights of the levels involved, A ul = 2 h 3 B c 2 ul e.g., g=2 J 1

4 Einstein coefficients Quantum mechanical calculations of transition probabilities w if = 4 2 e 2 m 2 c J if 2 f exp i k r l j if i 2 probability of transition exp i k r =1 i k r 1 2 i k r 2 exp i k r 1 dipole transition; higher orders: electric quadrupole, magnetic dipole, etc... in the dipole approximation: w if = 1 2 B if J if = c ħ 2 d 2 if J if d 2 if = d x if 2 d y if 2 d z if 2 d e hence B ul = 8 2 d ul 2 A 3 c ħ 2 ul = d 3 h c 3 ul 2 For degenerate states: A ul = ul 1 d 3 hc 3 g ul 2 u -- dipole operator where the sum is over all substates of the lower and upper levels j r j d x if 2 = f e x i 2

5 Oscillator strength Absorption classically: tot = e2 mc =B h classic lu lu 4 B classic lu = 4 2 e 2 h lu mc B lu = 4 2 e 2 h ul mc f lu f lu = 2m 3ħ 2 g l e E u E 2 l d lu 2 f oscillator strength quantum correction to the classical value of B Emission B ul = 4 2 e 2 h lu mc f ul g l f lu =g u f ul Oscillator strength for emission are negative g u A ul = 8 2 e 2 2 ul m c 3 g u f ul = 8 2 e 2 2 ul mc 3 g l f lu

6 Natural line width Energy levels are somewhat broadened as a result of of Heisenberg's uncertainty principle. Finite life-time of a level means some spread in energy, Level lifetime, t~1/ A 21 1/ E t~ħ transition, E t ~sin 2 0 t 0 e t/2 hence profile in frequency space (Fourier transform) /4 2 = 0 2 /4 2 Long-lived levels give very narrow lines (e.g. H 21 cm line)

7 Radiative transitions Permitted (allowed) transitions: dipole matrix element does not vanish Semi-forbidden transitions: dipole transitions but with a change of spin Forbidden transitions: dipole matrix element vanishes (may be possible at higher orders)

8 Selection rules exp i k r =1 i k r

9 Transitions with spin flip Intercombination (semi-forbidden) lines Grotrian diagram CIII] 1909 Å, 6.50 ev (2s2p 2s 2 ) 3 S 3 P 3 P o 3 D 1 S 1 P 1 P o 1 D

10 The He-like ions line triplet Ground state: 1s 2 1 S 0 1s2p 1 P 1 permitted (resonance) line 1s2p 3 P 1 intercombination 1s2s 3 S 1 forbidden (magnetic dipole) 1s2p 3 P 2 magnetic quadrupole (ΔL=1, ΔS=0, ΔJ=1) (ΔL=1, ΔS=1, ΔJ=1) (ΔL=0, ΔS=1, ΔJ=1) (ΔL=1, ΔS=1, ΔJ=2) 3 S 3 P o 3 D 1 S 1 P o 1 D

11 Forbidden lines Example: [OIII] 5007 Å line configuration: 2s 2 2p 2 2s 2 2p 2 line components: 4931 Å 3 P 0 1 D 2, E Å 5007 Å, 3 P 1 1 D 2, M1, E2 3 P 2 1 D 2, M1, E2 L=1, S=1, J =2 L=1, S=1, J =1 L=1, S=1, J =0 3 S 3 P 3 P o 3 D 3 D o 1 S 1 P 1 P o 1 D 1 D o

12 Strictly forbidden transitions Matrix element vanishes to all perturbational orders. Example: Lyα radiative decay of hydrogen: 2s to 1s two spherically symmetric wave functions. Possibilities: collisional shifting 2s 2p, then 2p 1s. A = s -1 two-photon process: '= A = 8.2 s photon dominates if n<10 4 cm -3.

13 Transition rates: bound bound, hydrogen A simple case: Hydrogen h =Ry n 2 n ' 2 Ry e2 =13.6 ev 2 a 0 Bound-bound transitions Lyman-α transition: other Lyman series: Absorption: g f = =0.83 A 21= s 1 g 1 f 1n = 29 n 5 n 1 2n 4 3 n 1 2n 4 ; n 1 g 1 f 1n ~ 1 n 3 B 12 =8.3 A 21 = s 1 Absorption cross section (at the line center, assuming the natural line width only): =B 12 h cm 2

14 Transition rates: bound bound Number of absorptions: n i R ij =n i B ij J d n i B ij J ij =n i 4 ij J ij h =n i 4 ij J h d ij Number of stimulated emissions: n j B ji J d =n j B ji J ij =n j g i g j B ij J ij =n j 4 g i g j ij J ij h Number of spontaneous emissions: 3 2h n j A ji ij d =n j c 2 3 2h ij B ji =n j 4 g i c 2 g j ij 4

15 Transition rates: bound free Bound-free transitions (photoionization) Final energy of the electron: Cross section: bf 2 9/2 Z 5 c 7 /2 3a 0 3/2 7/2 E f =ħ ħ more accurate form: Gaunt factor 1 bf = me 10 Z ch 6 n 5 g,n,l, Z 3 for 1 2 S level of H-like ions (Osterbrock 1989): bf = A 0 Z 2 1 4arctan 4 1 exp 2 / 4 exp A 0 = e a 2 4 0= cm 2, 1 = 1 1 h 1 =Z 2 h 0 =13.6 Z 2 ev Number of photoionizations: n i R ik =n i 4 ik J 0 h d

16 Transition rates Radiative recombination the inverse process to photoionization Number of recombinations (spontaneous + stimulated), obtained from the principle of detailed balance: n k R' ki, spon R' ki, stim n k n k i n R ki=n k n k i n 4 sometimes used: recombination coefficient: n k R' ki, spon R' ki, stim n k n e RR T ik Collisional rates: excitation/deexcitation, ionization/three body recombination Upward rate i->j, where j is a bound or free state: n i C ij =n i n e ij v f v vdv n i n e q ij T v 0 Downward rates: in equilibrium: n i C ij =n j C ji hence: 0 h 2 h 3 c 2 J e h /kt d k ion i atom J J d m e v 0 2 =E 0 - threshold for the ionization n j C ji =n j n i /n j C ij =n j n i /n j n e q ij T

17 Collisional transitions Autoionization Doubly excited states - two electrons in excited levels may have energy higher than the ionization potential of the ion in ground state. Then autoionization energetically favorable. One electron leaves the ion, the other returns to the ground state. Dielectronic recombination An ion collides with with an energetic electron: doubly excited state may form. This may autoionize, or a radiative downward transition can take place, leaving a bound atom with a single excited electron. He 0 (1s 2 ) + e He - (1s 2p 2 ) He - (1s 2p 2 ) He 0 (1s 2 2p) + hυ 1 (satellite line) He - (1s 2 2p) He 0 (1s 2 2s) + hυ 2

18 Fluorescence/Auger effect An electron is removed from an inner (e.g., K) shell and a highly excited ion is formed. This may decay with the emission of a fluorescent line (e.g. Kα line). Or, the excitation energy may be used to eject a number of electrons (Auger process).