Excitation and Ionisation in High Energy Density Plasmas Steven Rose - PowerPoint PPT Presentation

Excitation and Ionisation in High Energy Density Plasmas Steven Rose Imperial College London University of Oxford, UK

Overview • Laser-produced plasmas can produce the highest laboratory thermal radiation fields (i.e. excluding laser fields). • We consider in these lectures the direct effect of those radiation fields on the excitation and ionisation of high energy density plasmas (not the indirect effect of heating by the radiation field). • We describe the theory describing excitation and ionisation in HED plasmas, particularly the contribution from the radiation field. • We discuss experiments which test models with little or no radiation field, with a narrow-band radiation field and with a broad-band radiation field. • We finish with some recent work that links laboratory HED plasmas with astrophysical plasma physics and spectroscopy. • Conclusions

(energy in a Planckian radiation field)/(total energy) 4 aT ε = ρ + energy 4 C T aT V

(pressure from a Planckian radiation field)/(total pressure) P ε = rad + pressure P P mat rad

Radiation interacts with bound electrons through free electrons e-i bound electrons free electrons e-i ff bb s bf ions radiation

Radiation interacts with bound electrons directly e-i bound electrons free electrons e-i ff bb s bf ions radiation

Collisional and radiative rates β ’ β ’ c c r r R R R R ′ ′ ′ ′ α → β β → α α → β β → α β β r r c c R R R R α → β β → α β → α α → β α α

Collisional rates c R Electron collisional excitation (~n e ) α → β c R Electron collisional de-excitation (~n e ) β → α c R Electron collisional ionisation (~n e ) + Autoionisation ′ α → β c Three-body recombination (~n e 2 ) + Dielectronic recombination (~n e ) R ′ β → α

Radiative rates Photoexcitation - stimulated absorption r R α → β r Photo de-excitation – spontaneous and stimulated emission R β → α r R Photoionisation – stimulated absorption ′ α → β r R Photorecombination (~n e ) – spontaneous and stimulated emission ′ β → α

Level of detail describing atomic states Detailed Configuration Detailed Term Accounting Accounting DCA DTA

Rate equations The rate equations governing the populations are for each level α dn ( t ) ∑ ∑ = − + + + α c r c r n ( t ) ( R R ) n ( t )( R R ) α α → β α → β β β → α β → α dt β β which have the constraint ∑ = e Z ( t ) n ( t ) n α α α and for the case of the system having come to a steady-state dn α ( t ) = 0 dt

Photo excitation / de-excitation rates Both rates involve an integral over the line shape and the radiation field ∞ ∫   → = π σ ν ν ν ν R r 1 4 ( ( ) I ( ) / h ) d   ν = ν α β α → β 2 I ( ) ( 2 h / c ) 3   ν −   exp( h / kT ) 1 0 r   ∞ Ω ∫   α − ν = π σ ν ν + ν ν ν r h / kT U 3 2 R 4 α → β ( ( ) / h )(( 2 / ) I ( )) e d e  e  h c β → α α → β Ω   β 0 β − α   E ( ) E ( )     = U π α → β e 2 ( ) ( ) kT σ ν = φ ν e f α → β α → β α → β mc

Photopumping If photon intensity is roughly constant over the line shape then we can approximate it as a delta function ( ) ( ) φ ν = δ ν − ν α → β 0 which leads to the radiative excitation and de-excitation rates being = r R A n α → β β → α 1 ph = n = + ν − ph r h / kT R A ( 1 n ) e 1 0 b β → α β → α ph π ε Ω 2 2 2 8 e α = 0 A f β → α α → β Ω 2 3 h mc β

Photoionisation / photorecombination rates Both rates involve an integral over the photoionisation cross-section and the radiation field   1   ν = ν 2 I ( ) ( 2 h / c ) 3   ∞ ν −   exp( h / kT ) 1 ∫ = π σ ν ν ν ν R r 4 ( ( ) I ( ) / h ) d r ′ ′ α → β α → β 0   3 / 2 ∞   Ω e 2   n h ∫ α − ν = π   σ ν ν ν + ν ν h / kT r 3 2 U R 4   α → β ′ ( ( ) / h )(( 2 h / c ) I ( )) e d e e ′   ′ β → α α → β π Ω   2 2 k m T   ′ β e e 0 ′ β − α   E ( ) E ( )   =   U ′ α → β kT e

Radiative rates - no radiation field If there is no ambient radiation field only spontaneous de-excitation and radiative recombination remain = A R r β → α β → α   ∞ 3 / 2   Ω e 2   n h ∫ α − ν = π   σ ν ν ν ν r h / kT U 3 2 ′ R 4   α → β ( ( ) / h )( 2 / ) e d e e h c   β ′ → α α → β ′ π Ω   2 2 k m  T  ′ β e e 0

No external radiation field – optical depth But even if no radiation is incident on the plasma from outside, internally generated radiation from transitions can influence the populations. τ This introduces the photon escape probability g ( 0 ) = r R 0 α → β 1 − φ ν ( ) = 2 x = τ e r R A g ( ) π ν ∆ β → α β → α 0 D For a Doppler lineshape this is given by = ν − ν ∆ ν x ( ) / 0 D ∞ ν 1 ) / 1 2 = ( 2 kT x )) τ = − − τ − 2 ∆ ν ∫ g ( ) ( 1 exp( e dx i 0 τ 0 0 D m c − ∞ 0 π Ω 2 e τ = φ ν − α mc f ( )( n n ) y Optical depth α → β α β Ω 0 0 β

Cauchy (Dirac-Fuchs) mean chord theorem volume V For any convex shape, the mean chord length is given by the Dirac-Fuchs theroem as y = 4V/A surface area A For a slab geometry, the mean chord length y = 4V/A = 2z 0 60 0 z 0 Dirac P A M, Technical Report MS-D-5, part I, Public Records Office, Kew (1943). Dirac PAM, Fuchs K, Peierls R and Preston P, Technical Report MS-D-5, part II, Public Records Office, Kew (1943).

Level of detail describing atomic states Detailed Configuration Detailed Term Average-atom Accounting Accounting DCA DTA

Average-atom model

Collisional and radiative rates c c c c r r R → R → R → R → n c c n n c c n r r c c r r R → R → R → R → n r r n n r r n n n c c r r R → R → R → R → m n n m m n n m m m

NIMP – equations and solutions sum of radiative and collisional rates from shell m to n  −  dP P = − − − +   + + − − − c r n n P  1  ( R R ) → → m m n m n ω   dt n screened-hydrogenic number of probability of a excitation and ionisation electrons hole in shell n energies, collisional in shell m and radiative rates used dP = 1 f ( P ,...., P , R ) 1 1 n max dt  dP = n max f ( P ,...., P , R ) n max 1 n max dt solved for P 1 ,...., n P max

Reconstruction of individual level populations from the average shell populations From the average populations (P 1 ,….,P n max ) the fraction of ions in a real configuration α (n 1 ,….,n n max ) can be calculated assuming that the orbitals are not statistically correlated ω − n n    −  ω i i i ! P P     α = ∏ i i i P ( ) 1     ω − ω ω     ( n )! n ! i i i i i i The fraction of ions in level a , degeneracy 2 J a +1, which is derived from the configuration α , is then    ω  ! ( )   = + α i   P ( a ) 2 J 1 / P ( )   a ω −     ( n )! n ! i i i

Photoionisation cross sections FeXXIII (1s 2 2s 2 ) FeXXIV (1s 2 2s)

Dielectronic recombination / autoionisation rates ---------------------- f ___________________________ limit _______________ n dP n dt dP _______________ m m - dt dP g _______________ g - dt ∗ → Z P Q Q W ( gf mn ) g m n dr ∗ = → with P Q R R Z Q W ( gf mn ) → → g m g m g m n dr ∗ → = → 0 0 0 0 0 0 0 detailed balance Z P Q Q W ( gf mn ) P P Q W ( mn gf ) g m n dr n m g a Albritton and Wilson, Phys Rev Letts, 83 , 1594 (1999); JQSRT, 65 , 1 (2000)

Comparison of NIMP and detailed (DCA) models with / without DR Se n e =5x10 20 cm -3 , T e =1000eV 1 fraction 0.1 DCA - no DR NIMP - no AI/DR DCA - DR NIMP - AI/DR 0.01 22 23 24 25 26 27 28 29 Mg B Na Ne F N C O * Z Lee, JQSRT, 38 , 131 (1987)

Comparison between theory and experiment No or small radiation field

Non-LTE ionisation distribution measurement Conditions in the gold plasma were characterised by Thompson scattering, radiography and X-ray spectroscopy. Foord, Glenzer, Thoe, Wong, Fournier, Wilson, Springer, Phys Rev Letts, 85 , 992 (2000). Foord, Glenzer, Thoe, Wong, Fournier, Albritton, Wilson, Springer, JQSRT, 65 , 231 (2000).