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Collisional-Radiative Model
For Atomic Hydrogen Plasma
L. D. Pietanza, G. Colonna, M. Capitelli
Department of Chemistry, University of Bari, Italy
IMIP-CNR, Bari section, Italy
T, P or T, N
T uniquely defines:
internal level populations
(Boltzmann)
velocity distribution
(Maxwell)
ionization stage
(Saha)
LTE Plasma
Non-LTE Plasma
Microscopic Kinetic Approach
Collisional-Radiative Models
Level populations are obtained by solving a system of rate equations (Master Equations)
containg the rate coefficients of the main collisional-radiative elementary processes.
we need
Hydrogen plasma
H, H+, e-
1) set of energy levels for each species
2) cross sections and radiative transition probabilities
a) homogeneous, i.e. diffusion processes are neglected
b) quasi-neutral (ne = n H+ )
c) collision processes induced only by electrons
Collisional processes
10
-13
2
ex c itatio n cr o ss sectio n (cm )
10
-11
1) Excitation and de-excitation by electron impact

H (i)  e ( )
k ij




H ( j)  e ( )
10
10
-15
-17
1-2
2-3
10
-19
3-4
5-6
10
-21
10-11
k ji
15-16
10
dn
i
dt
  n i n e  k ij  n e  k ji n j
j i
20-21
-23
24-25
10
-25
0
j i
5
10
15
20
25
30
electron energy (eV)
10
k

H (i)  e ( )
ic




k ci
dn
dt
i
  n i n e k ic 


  e b ( b )
H  e ( )
2
n e n  k ci
10
-13
2
io n izatio n c ro ss sectio n s (cm )
 Ionization by electron impact and
2)
three body recombination
-12
10
10
10
10
10
10
10
-14
-15
-16
-17
i=1
i=2
i=3
i=5
i=10
i=15
i=20
i=25
-18
-19
-20
0

5
10
15
electron energy (eV)
20
25
30
Cross Sections and Detailed Balancing Principle
k

excitation-de-excitation by electron impact
H (i)  e ()
ij





H ( j)  e ( )
k ji
0 0
exc
de exc
0 0
n i f () v () ij ()  n j f (  E ij ) v (  E ij ) ji
(  E ij )

de exc
 ji
(  E ij ) 

gi
g j   E ij

generic reactive process by electron
impact

exc
e () 
 ij
()

 c i Pil  e (  *)
i
N e v () () f e ()  N e v (  *)  (  *) f e (  *)  ( N i f il )
ci

i
 (  *)  F()G (T ) ()
K eq 

i
Ni
ci

F() 

*
 gi
i
ci
 Qi
G (T ) 
i
ci
e
K eq
 h r ( KT )
Radiative Processes
1) Spontaneous emission and absorption
l A
ij ij
H (i)  
 H ( j)  h ij
with i
> j
Absorption is inserted by decreasing Aij (s-1) by a factor lij, i.e.

lij =1
Plasma optically thin:
dt

2) Radiative recombination

dn
lij <1
Plasma optically thick:


i
H  e () 
 H (i)  h
dn
dt
i
 ne n i
Aij* = lij Aij
i

*
*
 n jA ji  n i  A ij
j i
j i
Master Equations of CR Model
dn
i
 PD
i = 0, N liv

 dt

dn e  dn     dn i

dt
 dt
i dt
*
2
P    n j A ji  n e  n jk ji  n e n  k ci  n e n   i
j i
j i
*
D   n i  A ij  n i n e  k ij  n i n e k ic
j i
j i
Rate
coefficients


k 
f() electron energy distribution function
() cross section
v() electron velocity

 E t f ( ) ( ) v ( )d 

LTE
Maxwell distribution
Non-LTE
Boltzmann equation
f()
Coupling of CR Model with Boltzmann Equation for free electrons
rate coefficients
f()
Boltzmann equation
for free electrons
Master equations
level population
plasma composition
Boltzmann equation
df (, t )

dt
dJ E
d

dJ el

dJ e  e
d
d
 S an  S sup
J = flux in the energy space due to
1) JE electric field
2) Jel elastic collisions with atoms and ions


e (v i )  H(w i ) e (vf )  H(wf )




e (v i )  H (w i ) e (vf )  H (wf )
3) Je-e elastic electron-electron collisions






e (v i )  e (w i ) e (vf )  e (w f )
S = electron jumps in the energy space due to
1) San inelastic and ionization collisions



e ( E )  H ( k )  e ( E  E kn )  H ( n )



e ( E )  H ( k )  e ( E  E kn )  H
2) Ssup superelastic collisions




e ( E )  H ( k )  e ( E  E kn )  H ( n )
e

Case 1
only collisional processes
P=1 atm, Tg=5000 K, Texc=Te=TH=TH+ =20000 K
cH=ceeq(Texc)= 0.0356600, cH+=ce=ceeq(Texc)= 0.482170
H molar fraction vs time
1
1
0.1
H -e m ola r fra c tion
1.2
0.8
equilibrium
molar fraction
at T=5000 K
(0.999990)
0.001
equilibrium
molar fraction
at T=5000 K
-
0.6
0.01
+
H m ola r fra c tion
H+-e- molar fraction vs time
0.4
0.0001
0.2
10
10
10
-14
10
-12
10
-10
10
-8
10
-6
-4
10
time (s)
10
-2
10
0
10
2
10
4
10
6
(4.78032 10
-6
)
-5
-6
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
time (s)
10
-2
10
0
10
2
10
4
10
6
Case 1
electron distribution
hydrogen level distribution
1
1
0.01
n(i )/g(i )
t(s)=10
10
t(s)=10
-6
t(s)=10
t(s)=10
10
-8
t(s)=10
t(s)=1
t(s)=10
10
-8
10
t(s)=10
-4
-2
-1
6
10
10
10
10
t(s)=10
t(s)=10
-10
-12
t(s)=10
t(s)=1
-14
t(s)=10
= 4975 K
10
-12
-16
T
y = 0.50152 * e^(-2.3323x) R= 1
= 4978 K
fit
10
10
-8
t(s)=10
fit
10
t(s)=10
-6
-6
-10
T
t(s)=10
0.0001
-10
)
t(s)=10
-3/2
-4
t(s)=0
-12
e e df (e V
t(s)=10
10
0.01
t(s)=0
-18
y = 4.007 * e^(-2.3312x) R= 0.9997
-14
0
2
4
6
8
level energy (eV)
10
12
14
10
-20
0
5
10
15
(eV)
20
25
-12
-10
-8
-6
-4
-2
-1
6
Case 2
collisional processes + radiative:
spontaneous emission (l=1)
radiative recombination
P=1 atm, Tg=5000 K, Texc=Te=TH=TH+ =20000 K
cH=ceeq(Texc)=0.03566, cH+=ce=ceeq(Texc)=0.48217
electron distribution
hydrogen level distribution
1
0.01
0.01
n(i )/g(i )
10
t(s)=10
-8
t(s)=3 10
t(s)=5 10
-10
t(s)=10
10
-12
t(s)=10
t(s)=10
10
10
10
10
10
-14
t(s)=10
t(s)=10
t(s)=1
-16
t(s)=10
-18
-6
-5
-7
-7
10
0
2
4
6
8
level energy (eV)
10
12
14
t(s)=10
10
10
-8
-7
t(s)=3 10
-8
-10
t(s)=10
-12
t(s)=10
-3
-2
-9
t(s)=5 10
10
t(s)=10
-14
t(s)=10
-6
-5
-3
-2
-1
10
-16
t(s)=10
t(s)=1
6
10
10
-20
-6
-7
)
10
t(s)=10
t(s)=10
-8
-3/2
t(s)=10
-6
t(s)=10
0.0001
-9
e e df(e V
0.0001
10
t(s)=0
t(s)=0
-18
t(s)=10
-20
0
5
10
15
(eV)
20
25
-1
6
-7
-7
Case 2
H+- e- molar fractions
1
+
equilibrium
molar fraction
at T=5000 K
(4.78032 10
-6
)
0.0001
-
H -e m ola r fra c tion
0.01
10
10
10
-6
with radiative processes
without
-8
-10
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
time (s)
10
-2
10
0
10
2
10
4
10
6
Case 3
P=1 atm, Tg=20000 K, Texc=Te=TH=TH+ =5000 K
cH=ceeq(Texc), cH+=ce=ceeq(Texc)
collisional processes + radiative:
spontaneous emission (l=1)
radiative recombination
hydrogen level distribution
electron distribution
1
y = 0.22147 * e^(-0.62296x) R= 1
0.1
T
= 18628 K
y = 0.56268 * e^(-0.59823x) R= 0.99981
T
0.01
fit
= 19398 K
fit
0.001
0.0001
-5
10
-3/2
10
-7
e e df(e V
n(i)/g(i)
)
10
t(s)=0
10
t(s)=10
-9
t(s)=10
10
t(s)=10
-11
t(s)=10
t(s)=10
10
-13
-9
10
-5
-3
10
-12
t(s)=10
t(s)=10
-14
-16
t(s)=10
-9
-8
-7
-5
-3
-3
10
-3
t(s)=4 10
-18
t(s)=5 10
10
4
t(s)=10
t(s)=10
-15
2
-10
-7
t(s)=4 10
0
-8
t(s)=0
10
10
-8
t(s)=5 10
10
10
-6
6
8
level energy (eV)
10
12
14
-3
-3
-20
0
5
10
15
(eV)
20
25
Case 3
b(i)=n(i)/nSB(i)
molar fractions
10
100
1
i=1
i=2
1
0.48217
i=3
i=5
(i)
0.01
equilibrium
molar fractions
at T=20000 K
0.001
0.01
i=10
SB
0.03566
b(i)=n(i)/ n
m ola r fra c tions
0.1
i=25
0.0001
H
+
H -e
-
10
-6
0.0001
10
10
-8
-5
10
10
-3
10
-2
-10
10
-10
10
-9
10
-8
10
-7
10
-6
time (s)
time (s)
10
-5
10
-4
10
-3
10
-2
Conclusions
Collisional-radiative models are foundamental tools to characterize non-LTE plasma.
They calculate excited state populations, showing their possible deviation from LTE.
Departure from LTE occurs when the effect of radiative processes cannot be neglected
respect to electron collisions one, i.e. at lower temperature and electron density.
Works in progress
Implementation of a collisional-radiative model for N and O
k

ionization-three-body recombination
H ( i)  e ( )
ic



H



 e ( )  e b ( b )
k ci

Equilibrium constant vs T(K)
Equilibrium molar fractions vs T(K)
P=1 atm
10
10
10
10
18
14
12
e
10
10
10
10
0.8
6
4
1
0.01
0.0001
10
10
10
10
10
10
10
10
H
8
100
c
K (N /cm 3 )
1
16
fra z. m o lari
10
0.6
0.4
-6
-8
-10
0.2
-12
-14
-16
0
-18
-20
0
10000
20000
30000
T (K)
40000
50000
0
10000
20000
30000
T (K)
40000
50000
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