the use of atomic data in collisional-radiative...
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THE USE OF ATOMIC DATA IN COLLISIONAL-RADIATIVE
MODELINGJOE ABDALLAH
LOS ALAMOS NATIONAL LABORATORY
Workshop on Atomic and Molecular Data for Fusion Energy Research
28 August – 8 September, 2006 ICTP, Trieste, Italy
THE USE OF ATOMIC DATA IN COLLISIONAL-RADIATIVE MODELING
• Introduction and motivation• Atomic Physics Data• Collisional-Radiative Modeling-Methods• Radiation Properties of Plasmas• Opacity
What are plasmas?
The Most Successful Magnetic Confinement Configuration is the Tokamak
CurrentCurrent driven through coils distributed
around torus creates primary magnetic fieldmagnetic field
Stabilizing and shaping
magnetic fields are generated by currents in
other coils
vacuumvessel
plasma
An external transformer induces a current in plasmacurrent in plasmaaround torus that creates a smaller magnetic fieldmagnetic field
The Largest Tokamak Is the Joint European Torus in England
cutaway drawing
view of inside during maintenance
Working Tokamaks Are Usually Hidden by Support Systems & Diagnostics
LAPD/UCLA
He Gas jet experiments, TRIDENT LASER
Why Collisional-Radiative (CR) Modeling?
• Plasmas emit and absorb electromagnetic radiation.
• CR models provide atom/ ion state populations that are required to simulate radiation properties of plasmas.
• CR calculations provide diagnostic information about plasma conditions, e.g. , density and temperature by comparison with experiment.
• CR include electron collisions which are generally responsible for populating excited states.
He-like and H-like Al lines from a laser produced plasma
#366638#366684
Notation for Electron Configurations
322 221:nitrogen ofion configurat ground Example,
24 of valuemaximum orbital)in electrons ofber number(num occupation
notation picspectroscoby designated,...,3,2,1,0
number quantum momentumangular ,...3,2,1
number quantum principle)( orbitals ofset
pss
lw
spdfnl
ln
n nl w
+=
====
Block Diagram NeI
Energy level structure of pd configuration
Rate Equations and Definitions
• Nil = Population per unit volume(cm-3) of level l of ion species i• Ailjm = Rate matrix(sec-1)• Rate coefficients for processes which alter level populations,
integrated atomic cross sections.• These include electron collisions, photon collisions and
spontaneous processes • Depends on Ne, the number of free electrons per unit volume
(cm-3), for processes involving electron collisions, cross sections for the various processes, electron energy distribution, and radiation field.
jmjm
iljmil NA
dtdN ∑=
CATS:Cowan Code
RATS:relativistic
GIPPER:ionization
ACE:e- excitation
Atomic Models
CFG-based
fine structure
ATOMIC
LTE or NLTE
low or high-Z
spectral modeling
power loss
Opacity Tables
Another Theoretical Opacity Modeling Integrated Code
gf-valuese- excitatione- ionizationhυ-ionization
auto-ionization
energy levelsemissionabsorption
http://aphysics2.lanl.gov/tempweb/
Atomic Physics Codes
UTA’s
Electron Energy Distribution Function(EEDF)
• Describes the distribution of free electrons as a function of electron energy. The fraction of electrons per unit energy interval. Unit is (energy)-1, usually eV.
• Described by the Maxwellian function for a given electron temperature kTe( see below) for thermal plasmas.
• kTe expressed in energy units, usually eV.• kTe = 11605 °K
∫∞−
==02/3
/
1)( ,)(
2)( dEEFkTeEEF
e
kTE e
π
∫∞
=0
1)( dEEF
Radiation field• The number of photons per unit energy interval per unit volume
between hν and hν +d hν , units of (volume)-1(energy)-1, usually cm-3eV-1.
• kTr in eV.• For thermal plasmas the radiation field is given by the Planck
function.
)1()(8)( /33
2
−=
rkThechhhG ν
νπν
Electron and photon distributions at various temperatures
Maxwellian Planckian
1eV
10
100
1000 1eV
10
100
1000
Processes• Electron impact excitation/de-excitation:
e + Ail ⇔ e +Aim
• Radiative excitation/spontaneous decay: hν + Ail ⇔ Aim
• Electron impact ionization / 3 body recombination: e + Ail ⇔ Ai+1m+e+e
• Photo-ionization / radiative recombination: hν + Ail ⇔ Ai+1m+e
• Auto-ionization / di-electronic capture: Ail
*⇔ Ai+1m+e
Example: integrated rate coefficient
tcoefficien rate0
)()()(
,
thenis secondper per volume excitation ofnumber totalthe
)()()(
between secondper per volume sexcitation ofnumber
0')('
)'()(-
=∫∞
=
=
→=
+
+=−+=
+−
↔+
EdEEEvEFs
where
seNilN
dEEmiEvEFeNilN
dEE and E
EEilEimEEE
EimAeilAEe
σ
σ
LTE - Saha Equations
i
kTEil
i
il
ee
i
kTEili
iliie
i
kTie
i
ei
Zeg
NN
hkTmZ
egZ
NqNZeZZ
NNN
il
il
i
/
2/3
2
/
/11
22
−
−
−++
=
⎟⎠⎞
⎜⎝⎛=
=
=
=
∑
∑
π
φ ϕi = ionization energy of ion i
qi = charge of ion i
Eil = energy of level l
gil = statisical weight of level l
Ni=population of ion i
Ne=electron density
Nil=population of level l ion I
Zi,Ze=partition functions
Principle of Detailed Balance
• Under LTE conditions the forward rate of each process is balanced by the inverse rate
• Therefore, the cross section for the inverse process may be derived from the cross section for the forward process
• This is required for the solution to approach the LTE limit
.section cross excitation theof terms
in 'section cross excitation-de for the expressionan Derive
/2),(,,for ipsrelationsh
properties LTE using from calculated becan ' Therefore
')'(')'()'()()()(
between secondper per volume sexcitation ofnumber
LTEat rate excitation-de rate excitation
)('
)'()(
σ
σ
σσ
σσ
emEvEFeNilN
dEEimEvEFeNimNdEEmiEvEFeNilN
dEE and E
ilEimEEE
EimAeilAEe
=
→=→
+
=
−+=
+−
↔+−
Detailed balance for excitation / de-excitation
Coronal Equilibrium
• Ii=net ionization rate coefficient from from ground state of ion i including inner shell excitation/autoinization
• Ri=net recombination rate from ground state of ion i including recombination into excited states of ion i-1
• Low density limit• Solar corona model• Only ground states significantly populated• Radiative decay rates >> e- impact excitation rates
0)( 1111 =++−= ++−− iiiiiiii NRNIRNI
dtdN
Solution of Rate Equations
Numerical Integration
Steady Stategeneral matrix solution
Local Thermodynamic Equilibrium
Coronal Equilibrium
∞→t
0=dt
dNil
High Ne, or Tr=Te
Low Ne, Tr=0
e- Impact Excitation ↔ De-Excitation
sec/in and
,
)()(),(
')'()(
3
/
0
0
0
0
cmts
seggt
dEEEvkTEFs
NtNNsNdt
dNEEE
EEEAEeAEe
ekTE
im
il
E e
imeileil
ilim
imil
=
=
+−=
−=−=
+↔+
∫∞
−−
σ psesetsTNN
EE' E
e
e
il
211 :Ext coefficien rate excitation-de
tcoefficien rate excitation emperatureelectron t densityelectron
population state
energy threshold0
energy final energyimpact
2 +↔+
=====
===
−−
Photo-Excitation ↔ Decay
in eV,hνu,y,z in
ug
ilmgfz
hg
ilmgfy
hg
ilmgfmchey
kThGg
gfm
heu
zNyNuNdt
dNEEhAAh
`
im
im
ime
ril
ilm
e
imimilil
ilim
imil
01
20
7
2032
22
0
2
0
sec
)(
)()(10338.4
)()(8
),()(
−
=
×=
=
=
++−=
−=↔+
ν
νπ
νπ
νν
112 211 :Exfieldradiation no with 0
tcoefficien ratedecay stimulated t)coefficienA (Einstein
tcoefficien ratedecay sspontaneou tcoefficien rate excitation u
etemperaturradiation population state
energy transition0
energyphoton
psshzu
z
y
kTN
hvhv
r
il
↔+
===
====
==
ν
Fe+5: 3d3-3d24f
Photo-Ionization ↔ Radiative Recombination
.in 0,, ,sec/3
in ,,,1
sec is
2/3
/022
1066.1
)0(2/1
2)0(
)0 ,(22/32/1
2
1
)()0
,(
0
)(
eVhEkTcmqrp
pkT
kThe
jmg
ilgqr
dEEhiljmE
Eh
ekTEFcemjmg
ilgr
cdhhiljmh rkThGp
jmNeqNjmNerNilpNdt
ildN
EhEilEjmEh
jmAEeilAh
ν
ν
νσν
ννσν ν
νν
ν
−
−×=+
++
∫∞=
∫∞=
++−=
+=+−=
+−
↔+
−+↔+
==
=+==
==
esshq
kTkTqr
r
hvhv
re
12 11:Exfieldradiation no with 0
emission, stimulated with tcoefficien rateion recombinat radiative
tcoefficien rateion recombinat radiative tcoefficien rateation photoioniz u
energy ionization0
energyphoton
ν
e- Impact Ionization ↔ Three-Body Recombination
eVEe
kT
c
c
ekT
ekTEe
jmg
ilg
b
dEEQEvE e
kTEFc
jmN
ebN
ilN
ecN
dt
ildN
EEEEEil
Ejm
EE
jmAEeEe
ilAEe
in 0
and
sec/6
cmin bsec,/3
cmin
2/3)(
/022
1066.1
)()()0
,(
2
'''0
'''
)''()'()(
−×=
∫∞=
+−=
++=++−=
+−
+−
↔+−
−−−
−
++↔+
=
=
=
==
eese
e
b
e
EeE
-
12 11s :Exsection. cross
ionizationimpact Qonly. Maxwellian
t,coefficien rate ion recombinatbody - three
t.coefficien rate ionizationimpact c
energy. ionization0
energy.impact
Auto-Ionization ↔ Di-Electronic Capture
eVekTE
da
a
ekT
ekTEe
jmg
ilgd
ailjmAa
jmNedNilaNdt
ildN
ilEjmEE
jmAEeilA
in and 0
sec/3
cmin and 1-
secin
2/3)(
/022
1066.1
0
)0(
−−
×=
=
+−=
−=
+−
↔
−+↔
==
=
ess
d
E
12 22like-He :Exonly. Maxwellian
t,coefficien rate capture electronic-di
rate. ionization-auto a
energy. ionization0
Fe+5:3d14f2-3d2
Emission spectra
• Line or bound-bound contribution:• The number of emitting photons per second per volume per unit
energy interval per transition:
HWHMwew
whhw
dhh
hEEh
hhg
gfmcheN
whh
ilm
ilm
ilim
ilmim
ilm
eim
=
=Φ
+−=Φ
=Φ
=Φ
−=
Φ
−−
∞
∫
220 /)(2ln2
220
0
0
2032
22
)/(2ln
or )(
/ e.g.
1)(
shape line)(
)()()(8
ννπ
ννπ
νν
νν
ννπ
Emission spectra, lines, cont.
• The total amount radiated energy per second per volume per unit energy interval, sum over all transitions:
)()()(8 3032
22
ννπ hhg
ilmgfmcheN ilm
imeim Φ∑
Emission spectra
• Recombination or free-bound continuous contribution• The total amount radiated energy per second per volume
per unit energy interval, sum over all transitions:
EhEEEhv
NN
iljm
ejm hiljmE
h
ekTEFcemjmg
ilg
+=+−=
∑
0
)(2/1
3)(
),(22/32/1
2
1
ν
νσν
Emission spectra• free-free or Bremsstralung continuous contribution• free electron decelerates in plasma and emits a photon• simple hydrogenic formula• The total amount radiated energy per second per volume per unit
energy interval, sum over all transitions:
iion of chargeiion of population
)(10585552.9 2
2/1
/14
==
× ∑−
−
i
i
iie
kThe
qN
qNkTeN eν
Power Loss Is Obtained by Integrating Emission Over ALL Photon Energies
ii
ieeff
eiljmiljmiljmim
ejmbf
ilimilmilmim
imbb
ffbfbbtot
NqNkTP
kTEEqrNNP
EEzyNP
PPPP
∑
∑
∑
−×=
+−+=
−+=
++=
22/114 )(1055.9
))2/3()((
))((
P in eV/(sec cm3), 1J=6.24x1018eV, 1W=1J/sec
What is opacity?
∂Iω x( )∂x
=−κωρIω x( )
Iω x( )=Ioexp−κωρx( )
Applications of Opacities
γ
eγ e
bound-bound bound-free free-free
eγ
γ
scattering e
Physical processes contributing to the opacity:
These photon absorption and scattering processes Combine to give the frequency dependent opacity:
( ) ( )( )1 expbb bf ff skTωκ κ κ κ ω κ= + + − − +
)e(temperatur)(density mass
)/(1101.4
)(1
)()(1
0
3
2
32/70
23
2
KelvinTcmg
NqkThT
N
hN
hg
gfcm
heN
iiie
eff
iljmil
ilbf
ilmil
ilm
eililbb
=
=
×=
=
Φ=
−
− ∑
∑
∑
ρ
νρκ
νσρ
κ
νπρ
κ
Components of the frequency dependent opacity(bound-bound, bound-free, free-free and scattering)
10 -5
10 -4
10 -3
10 -2
10 -1
100
10 1
10 2
10 3
10 4
10 5
106
10 7K
(cm
2 /gm
)
2000150010005000hnu (eV)
kbbl97fee kbfl97fee kffl97fee kscl97fee
WorkOp-IV:97 KtotFeE: T=100 eV ρ=1e-4g/cc
10-5
10-4
10-3
10 -2
10 -1
10 0
10 1
102
103
104
105
106
107
K (c
m2 /g
m)
2000150010005000hnu (eV)
ktotl97fee WorkOp-IV:97 KtotFeE: T=100 eV ρ=1e-4g/cc
The resultant total frequency dependent opacity
Exercises
(I) Consider the 2 state system. Let N1and N2 be the population density of the 2 states respectively. Let N = the constant total number density. Let c be the rate coefficient for excitation from state 1 to state 2, d be the de-excitation rate coefficient from state 2 to state 1, and r be the spontaneous decay rate from state 2 to state 1. Let Ne denote a given electron density.
(1)Write down the rate equations for N1 and N2 and derive an expression for the steady-populations of both in terms of N.
(2) Derive a formula for N1 and N2 as a function of time(t) by directly solving the differential rate equations.
(3) Check the formula to if the steady-result is obtained as t gets large, also check the formula for Ne small ( coronal limit), and for large Ne (LTE).
(4) Derive an expression to first order for the relaxation time, the time it takes for N1 to reach its steady-state value.
Exercises
(II) Assume the 2-level system of the previous example to be the 1s and 2p configurations of H-like Lithium. This example is actually of interest in Lithography because of its wavelength. Calculate the power (Watts/cm3) emitted by the 2p-1s emission line at N=?, Ne=? and kTe=?. Use the LANL atomic physics website to evaluate the required atomic data.
(III) Use the LANL atomic physics website to identify the lines observed in the Al spectra shown above. Match the observed line position with calculated values.