Transcript Presentatie Lunteren 98

Summary of the previous lecture
Particles:
Which species do we have
how “much” of each
Plasma chemistry
Momentum: How do they move?
Plasma propulsion
Energy:
Plasma light
What about the thermal motion
internal energy
What do we want/need to know in detail?
TU/e
Particles
Particles
Energy
Energy
Momentum
Momentum
Particles:
Plasma Chemistry
Energy:
Plasma Light
Momentum: Plasma Propulsion
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Transport Modes
Fluid
mean free paths small
mfp << L
mean free paths large
mfp > L
Hybride
Quasi Free Flight
Sampling and tracking
There are many conditions for which
some plasma components behave “fluid-like”
whereas others are more “particle-like”
Hybride models have large application fields
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Discretizing a Fluid: Control Volumes
Particles
Particles
Energy
Energy
Momentum
Momentum
For any transportable
quantity

Source
Transport
via boundaries
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Examples of transportables
Densities
Momenta in three directions
How many species?
Mean energy (temperature)
Depends on Equilibrium
departure
As we will see:
In many fluid/hybrid cases
Energy: 2T: {e} and {h}
Momentum: for the bulk Navier Stokes
for the species: Drift Diffusion
Species:
the transport sensitive
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Nodal Point communicating via Boundaries
Mean properties  Nodal Points
Transport at boundaries
Transport Fluxes: Linking CV (or NP’s)
 = Source,
t  + = S
Steady State
General structure:
Transient
  = u -D 
Convection

Diffusion
 -
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Simularities
Thus: The Fluid Eqns: Balance of
Particles
Momentum
Energy
The Momenta of the Boltzmann Transport Eqn.
Other Example: Poisson: .E = /o
E = - V
 = S
  = u -D
Thus no “convection”
Can all be Treated as  -equations
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The


Variety
D
S
Temperature
Heat cond
Heat gen
Momentum
Viscosity
Force
Density
Diffusion
Molecules
atoms
ions/electrons
etc.
Creation
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MathNumerics: a Flavor
Sourceless-Diffusion
Continuum
Tin
Rod
Tout
t  + = S
0 + T = 0
T = Cst
T = - kT
T
-T /k =T
x
Take k = Cst
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Discretized
Continuum
Tin
Rod
Tout
Discretized
1
2T2 = T1 + T3
2
3
Intuition;
4
T = Cst
T2 = (T1 + T3)/2
Tin -2T1 + T2
T1 - 2T2 + T3
T2 - 2T3 + T4
T3 - 2T4 + Tout
=0
=0
=0
= 0TU/e
Matrix Representation
1 2
1
2
3
4
-2 1
1 -2 1
T1
-Tin
T2
0
=
3
1 -2 1
T3
0
4
1 -2
T4
-Tout
In matrix:
M T = b
A Sparce Matrix
Many zeros
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Sourceless-Diffusion in two dimensions
1
1–4 1
1
N
W P E
S
T5 = (T2 + T4 + T6 + T8 ) /4
Provided k = Cst !!
In general:
TP 
T
NB
NB
4
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More general S-less Diffusion/Convection
 c T

 c
NB
If k Cst
TP
NB
TP
NB
NB
 c* T

 c*
NB
Convection
Diffusion
NB
NB
NB
NB
NB
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Laplace and Poisson
  = u -D
 = S
If no “convection”
- D = S
  = -D
Poisson
Laplace
S 0
S=0
- D = S
- D = 0
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Examples
Ener balance
Ion balance
Space charge
.E = /o
Ohms’ law
.j = 0
j = E
. E = 0
E = -V
-.kT = S
-.Dn = Ion
- Rec
-. V = /o
Simularities !!
-.V = 0
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A Capacitor space-charge zero
0
-. V = /o = 0
N
+V
Basically a 1-D
problem
Each V the
Average of
the two adjacent
V
d
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A resistor: Ohms law
-.V = 0
0
+V
1-D problem
N
Each V the
Average of
the two adjacent
Provided  is Cst
V
d
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Ordering the Sources
 = S
S = P - L
L ~ D
Source combination Production and Loss
Large local - value in general leads
to large Loss
Source of ions
Example ions:  n+u+ = P+ - n+D+
RecombinationTU/e
Concept disturbed Bilateral Relations

A proper channel

N f
t = Nt
N b
Equilibrium Condition: t/b << 1 or t b << 1
The escape per balance time
must be small
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
Mixed Channel

N D
t = Nt
N D
P - nD = n u
The larger D
The less important transport for 
The more local chemistry determined
Note D is more general than b in dBR: collective chemistry
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Ingredients for non-fluid codes
The more equilibrium is abandoned the more info we need
Tracking the particles: Integrating Eqn of Motion F = ma
Interaction by chance: Monte Carlo
Field contructions: a) positions giving charge density  E
b) motion giving current density  B
Cells are needed
To organize the
field contributions
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Fluid versus particles (swarms)
Particle codes
Directly binary
interacting
individual particles
Bookkeeping
Position/velocity
Each indivual part
Particle in cell
interaction via
self-made field
Sampling
Distribution
Over r and v
Hybride
particles in a fluid
environment
Continuum
Distribution function
Known in shape
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A quasi free flight example
Radiative Transfer
Ray-Trace Discretization spectrum.
Network of lines (rays)
Compute I (W/(m2 .sr.Hz) along the lines
dI()/ds = j - k()I()
Start
Entering
outside the plasma with I() = 0.
plasma I() grows afterwards absorption.
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Ray Tracing
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General Procedure ??
Fluid
Swarm Collection
{h} {i} {e}
{}
{h} {i} {es} E
{h} {i}
{h}
{ef} {}
{es}{ef} {}
E
E
{i} {es}{ef} {}
{h} {i} {es}{ef} {}
Pressure
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The BTE: basic form
The BTE deals with fi(r,v,t) defined such that
fi(r,v,t)d3rd3v  the number of particles
of kind “i” in a volume d3r of the configuration space centered
around r with a velocity in the velocity space element d3v around v.
Note that “i” may refer to
an atom in a ground state
the same atom in an excited state
An ion or molecule, etc. etc.
Examples
e
A, A+, A*, A**, etc,
N, N2, etc.
NH, H2
The BTE states that:
tfi + .fiv + v.fi a = (tfi)CR
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Simularities of the Boltzmann Transport equation
tn + . nv = S
Source S
Leads to
tn accumulation
. nv and/or Efflux
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Generalization to 6-D phase space
tn + . nv = S
Normal space
Accumulation
Transport
Source
tfi + .fi v = (tfi)CR
Phase space 
The Boltzmann Transport Eqn
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The BTE: general form
Use  the divergence in the 6 dim  space (r x v).
tfi + .fi v = (tfi)CR
BTE: Shorthand notation
This is a  equation in  space
Representing
tfi + .fiv + v.fi a = (tfi)CR
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From Micro to Macro ordering using BTE
Fluid approach: assume shape of f is known
Procedure
I
Multiply BTE for “i”with a function g(v) and integrate over v.
For each species:
specific balances of mass; charge; current; momentum; energy
II Higher structures units:
Elements; Bulk: mass; charge; current; momentum; energy
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The momenta of the BTE: Specific balances
BTE
g(v)
t fi + .fi v = (tfi/dt)CR
Particle
1
t ni + .ni ui = Spart, i
mi
t ni mi + .ni mi ui = Smass,i
qi
t ni qi + .ni qi ui = Scharge,i,
0-mom Mass
Charge
1-mom Momentum
2-mom Energy
miv
 =1/2miv2
t ni mi ui+ .ni mi uiui = Smom, i,
t ni  i + .ni i ui = Senergy,i
Note:  is in configaration space solely
u is systematic velocity in configuration space
Smom contains p and .: This approach is questionable
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The zero order momenta
Particle
1
t ni + .ni ui = Spart, i
Mass
mi
t ni mi + .ni mi ui = Smass,i
Charge
qi
t ni qi + .ni qi ui = Scharge,i
Note that
Smass,i = mi Spart,i
Scharge,i = qi Spart,i
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Simplifications for specific mom balance
ui is omnipresent: simplifications of the origen: mom bal
t ni miui + .ni miuiui = Smom, i,
or
t i ui+ . i iuiui = -pi+ .i + i g + ni qi E + niqiuiB + Fij,
In many cases:
t i ui + . i iuiui , niqiuiB and i g negligible
p, ni qi E and Fij, dominant
Drift/Diffusion equation:
0 = -p + ni qi E + Fij,
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Drift Diffusion continued
0 = -pi + ni qi E + Fij,
F = Ffric + Fthermo
Fij fric = (pipj/pDij ) (uj – ui )
Mostly: Ffric >> Fthermo
Fij
fric
= -(pipdom /pDij ) (ui - u )
Case one dominant
species with udom = u
Fij fric = -(pi/Di ) (ui - u )
p = pdom
Fij fric = -(kTi /Di ) {ni (ui - u )}
pi = ni kTi
ni (ui –u) = - (Di / kTi ) pi + (ni qi Di / kTi ) E,
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Drift - Diffusion II
Normally:
ni (ui – u ) = - (Di / kTi ) pi + (ni qi Di / kTi ) E,
Thus
ni (ui – u ) = - (Di / kTi ) pi + (ni i) E
or
With i = qiDi/kTi
Einstein relation
ni (ui – u ) = - Dini + (ni ) E
If T  0
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Ambipolar Diffussion
ni (ui – u ) = - (Di / kTi ) pi + (ni i) E
with i = qiDi/kTi mobility
j = i niqi (ui – u ) = - i ipi +  E
and  = i nii qi conductivity
In most cases the current density j is closely related
to the external control parameter I; and E the result
E = -1 (j + i qi ipi)
or
E = Ej + Eamb
Eamb = -1 i qi ipi
Eamb = {kTe /qe}pe/pe
  ne eqe
qe e >> qii
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For the ions
ni (ui – u ) = - (Di / kTi ) pi + ni Di {Te /Ti } {qi /qe}pe/pe,
For the electrons
neqeue = j
Beware of the signs!!
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Reaction Conservatives I : mass
AB  A + B
mAB = mA + mB
Reactions
Each creation of couple A and B
associated with disappearence AB
SAB = - SA = -SB
Thus
SAB mAB + mASA + mBSB =0
Since Smass = m Spart
More general
all mi Spart, i = 0
or
all Smass, i = 0
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Reaction Conservative II: charge
AB  A+ + BqAB = qA + qB
Reactions
SAB = - SA = -SB
Each creation of couple A and B
associated with destruction AB
Thus
qAB SAB + qASA + qBSB =0
Since Scharge = q S
More general
all qi Si = 0
or
all Scharge, i = 0
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The Composition
all Smass,i = 0
Bulk in Mass

t ni mi + .ni mi ui = Smass,i
gives
t m + . mu = 0
all
with m = all nimi
and
u  nimiui /m
all Scharge,i = 0
Bulk in Charge

barycentric or bulk
velocity
t ni qi + .ni qi ui = Scharge,i
t q + . j = 0
all
With j = ni q1 ui
Current density
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Reaction Conservatives III: Nuclei
In general: species can be composed
e.g. NH3 is composed out of one N nucleus and three H
We say R of N in NH3 = 1 or
and
R of H in NH3 = 3 or
or
Now consider
NH3  N + 3H
RN(NH3) = 1
RH(NH3) = 3
Ri = 3 with “i” = NH3 and “” = H
RH(NH3) Spart(NH3) = RH (N) Spart(N) + RH(H) 3 Spart(H)
In general:
all RH(i) Spart, i = 0
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Elemental transport of H

all RHj Spart, j= 0
t ni RHi + .ni RHi uj = Smass,i
gives
t {H} + . {H} = 0
all
With  ni RHi = {H} and {H} =  ni RHi uj
In steady state:
. {H} = 0
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In general
t {X} + . {X} = 0
The change in time of the number density of nuclei of type X
Equals minus the efflux of these nuclei;
The efflux {X} of X is the weigthed sum:
{X} =  RXi j
Number of X nuclei in j
j = ni uj
Efflux of “j”
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Removing 1 H atom
Removing 1 H3 molecule
{H} = 3 H3
Equivalent: removing 3 H atoms
In general:
{H} =  RHi j
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Simularities
Total mass transport

all mj Spart,j= 0
t ni mi + .ni mi uj = Smass,i
t m + . mu = 0
all
all qj Spart,j = 0
Total charge transport

t q + . j = 0
t ni qj + .ni qi uj = Scharge,i
all
Total X-nuclei transport

all RHj Spart,j = 0
t ni RXi + .ni RXi uj = Smass,i
t {X} + . {X} = 0
all
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The momentum balance
on higher structure levels
The elements: simple addition of the DD equation.
The bulk:
i { ti ui+ . iuiui = -pi+ .i + i g + ni qi E + ni qi uiB + Fij }
=0
=0
Charge neutrality
Navier Stokes
Action = -Reaction
t u+ . uu = -p+ . + j  B + g
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Metal Halide Lamp
LTE or LSE is present (??)
Still not uniform
LTE: at each location the composition
prescribed by the Temperature
and elemental concentration
Convection and diffusion results in non-uniformity
10 mBar NaI and CeI
in 10 bar Hg
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If LSE is not established
CRM needed
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Collisional radiative models
1
p
Bound electron state
+
Continuum
Free electron states
In principle  bound states
Should we treat them all?
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CRM Black Box: the ground state as entry
1
+
CRM as a Black Box
With two entries
Typical Ionizing system
Generation of efflux of photons and radicals
As a result of input at entry 1
Response on influx largerly depends on ne and Te
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The ion state as entry
1
+
Response on influx at “+”
Typical recombing Syst
Again dependent on ne and Te
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More than 2 entries?
1
p
+
If atoms in stat “p” are transportables:
Transport sensitive
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Radiative pumped
1
p
Radiative input
+
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Superposition
Black Box
With several entries
Superposition theorema
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Is the Black Box description sufficient?
For modelling the BB approach is sufficient
If CRM’s are used for Diagnostics
We need info of the ASDF: non-BB
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What info is in the ASDF?? Example
Ion
Ground
state
state
Ionization flow
log( elementary occupation )
Influx
Efflux
real
distribution
b
p
The lower part of the ASDF
gives insight in trapsport
the pLSE contains info of
the electron gas.
pLSE
Saha distribution
I
1
I
0
p
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Action Plan
Which levels are
or
Cooperation
TS: Transport Sensitive
LC: Local Chemistry
{TS}  {LC}
Competition of Radiation versus Collisions
More general: Determine structure and tasks of a CRM
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Local Chemistry levels
Transport
Local Chemistry
 np up = Pp – np Dp
np = np Pp /Dp -  np up/ Dp
For excited states D = 107 s-1
[ np up] = n(u/L)
u/L << uth/L
103/10-3 = 106s-1
In many cases for excited states:
[ np up] /Dp << 10-1
np = Pp /Dp
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Competition e-collisions versus radiation
The competition electron collisions
ne dependent
ne k
Influence Te
In many cases we see:
radiation
independent
A
low ne ~ high Te
high ne ~ low Te
We start with low ne ; the corona balance
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The Corona Balance: an improper balance
 1
2
The corresponding Restoring:
Proper Boltzmann Balance
b(2) = b(1) exp -E12/kTe
Escape of Radiation
y() = y()[1+ tb]
with tb = A*/ne K(2,1) and y = /b
Thus: the larger ne the smaller departure
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General: Impact Radiation Leak
 1
2
p
y(p) = y(1)[1+ tb]-1
with
tb = A*(p)/ne K(p,1)
Define: N = A(p)/neK(p)
A(p)  p-4.5
K(p)  p4
N (p)p-9
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The CR Boundary
N (p) p-9  radiation impact decreases rapidly
Define Level pcr such that N (pcr) =1
Found that
For p<pcr
For p>pcr
pcr9 ne = 9 1023 Z7
levels radiative
collisional
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Radiation (Leak) Versus E-collisions (Restoring)
Essential is: t b : the escape per balance time
t determined by radiative transition probability
b determined by the e-collision cross section
Both depend on the oscillator strength
If e-transitions are optically allowed (dipole)
Dependent on values of the electron density
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However:
ne K >> A for all levels does not mean
That the system is in equilibrium
The leak of electron-ion pairs will
Modify the ASDF .
Take Saha as a standard.
TU/e
Last presented slide
Note: the slides hereafter were not discussed
during the lecture of 12-12-2003.
They will be presented, probably, in a future lecture
Joost van der Mullen
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Ion Efflux Effecting the ASDF
Ion
Ground
state
state
Ionization flow
log( elementary occupation )
Influx
Outflux
real
distribution
b
p
pLSE
pLSE settles
for Ip  0
Saha distribution
since t/b  0
I
1
I
0
p
TU/e
The Excitation Saturation Balance: an improper balance
Influx
=+
p
 =1
Ionization flow
restoring
n1 f
n+b
Outflux
+ = n+t =.n+w+
y(1) = y(+)(1+tb)
b1 = b1 –1 = tb
b =ns(1)K(1,2)
t =.n+w+/n+= Dn-2
b1=B1Da(ne n)-2
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Shape of the ASDF in ESB
=p
1
Influx
=+
Ionization flow
Outflux
npneK(p,p+1)
+ = n+t =.n+w+
n+b
nspK(p,p+1)
y(p) = y(+)(1+tb)
b(p) = b(p) –1 = tb
Restoring to P
large: close to cont
b =ns(p)K(p,p+1)
ns(p) scales with p2
K(p,p+1)
with p4
b(p)=boDa(ne)-2p-6
TU/e
log( elementary occupation )
ESB
real
distribution
b
p
pLSE
Saha distribution
I
1
I
0
p
TU/e
The Saha density: mnemonic
 s(p) = (ne/2) (n+/g+) [h3/(2mekTe)3/2] exp (Ip/kTe)
That is
 s(p) = e  + [V(Te) ] exp (Ip/kTe)
Look at balance Ap  A+ + e
A+ + e
bound  free pair
Number density of bound {e +} pairs in state p:  s(p)
Equals the density of pairs within V(Te) e  + [V(Te) ]
Weighted with the Boltzmann factor exp (Ip/kTe)
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General CRM Structure
 n(p)/t +.(n(p) w(p))
n1 neK(1,p)
e-ground st. exc.
n(q) neK(q,p)
e-induced prod.
- n(p) neK(p,1)
B1 e- de-excitation
- n(p) ne  K(p,q)
B e-induced destruction
nen+neK+(p)
- n(p) S(p)
three prt. recomb. S e-induced ionization
n(l)B (l,p)d
absorption
- n(p)  [A(p,l)+ B (p,l) d]
P emission (spont. + sti
n(l)B (l,p)d - n(p)  [A(p,l)+ B (p,l) d]
absorption
P emission (spont. + stim.)
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General Structure II
 n(p)/t +.(n(p) w(p)) =
Transport
P(p) –
Production
n(p) D(p)
Destruction
P(p) = q n(q) D(q,p) + n+D(+,p)
Production Term
D(p)  q D(p,q) = neK(p) + A*(p)
Destruction factor
Transition Frequencies:
Total Destruction:
D(p,q) = ne K(p,q) + A*(p,q)
A(p)=l A*(p,l) and neK(p)=q K(p,q)
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Global Structure III
S(p) =
Source
T(p)
Transport
Note: S(p) addition (remnants) Proper Balances B, S, P
Non-equilibrium
TS:
Of one PB leads to transport: S = T
LC:
Of one PB leads to imbalance other
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Simplifying Assumptions
QSSS:
Transport Dt(p) 103s-1 ; excited states D(p)  107s-1
Thus:
P(p)/D(p) - n(p) = Dt(p)/D(p,p)0
n(p) = P(p)/D(p).
Different Levels
TS levels
LC levels
Usually ground levels
Usually excited states
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Electron Excitation Kinetics
D (p) = ne K(p) + A*(p)
neK(p)=q K(p,q) and A(p)=l A*(p,l)
Atomic:
Hierarchy in rates
K(p)  p4
A(p)  p-4.5
Cut-off procedure: Reduces number of levels
Bottom:
Numerical
Top:
Analytical
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Exploration {2 Entries/2 Levels}


T ( )
Config. space
Transport
D( , )
Excitation space
Chemistry
- n D(,) + n D(,)
n D(,) - n D(,)
Transport
Chemistry
D(,  )
T ()
Config. space
Transport
= T()
= T ()
Configuration space ruled by T() and T()
Traffic excitation space: Sources
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Task Allocation
S()  n()D(,) - n()D(,)
Note: S()= - S()
Thus: T() = - T()
Needed:
nsys = n() + n()
n()/n() = [D(,)+ Dt ()] / D(,)
Task allocation
Fluid
ne , Te, nsys and T; i.e. Dt (),
Chemical:
CRM: D(,) and D(,).
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Exploring: {2 Entries/ 3 Levels}

T ( )

i
D(,  )
D( , )
- n D(,) + ni D(i,) + n D(,) = T ()
n D(,i) - ni D(i,i) + n D(,i) = 0
n D(,) +ni D(i,) - n D(,) = T ()
With: Mpq = D(q,p)
M  M M
e

for pq
T ()
S  M n and S  T
and Mpp = - D(p,p)   q p D(p, q)
Mepq = neK(q,p) and Mpq = A(q,p)
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The CRM Tasks
1)
the atomic state distribution function ASDF
2)
the effective conversion rates
3)
source terms of the energy equations
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Task 1: The ASDF
- n D(,) + ni D(i,) + n D(,) = T ()
n D(,i) - ni D(i,i) + n D(,i) = 0
n D(,) +ni D(i,) - n D(,) = T ()
LC levels expressed in TS levels
Using
n D(,i) - ni D(i,i) + n D(,i)
Gives
n(i) = D(i,i)-1 D(,i) n() + D(i,i)-1D(, i)n()
In Matrix:
n  Rn
=0
t
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Task 2: Effective Conversion
Substitute
n(i) = D(i,i)-1 D(,i) n() + D(i,i)-1D(, i)n() into
n D(,) +ni D(i,) - n D(,) = T ()
Give J(, ) = D (, ) + D(i,) D(i,i)-1 D(, i)
Presence internal level enhances traffic 
The effective frequency of the   conversion equals
that of the direct process plus that of the excitation
[D(i,)] of that part [D(i,i)-1 D(, i)] of the n(i) population
which originates from the  level.
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Task 3 Energy Sources
- n D(,) + ni D(i,) + n D(,) = T () x E
n D(,i) - ni D(i,i) + n D(,i) = 0
x Ei
n D(,) +ni D(i,) - n D(,) = T () x E
- n D(,) E + ni D(i,) E + n D(,)E
n D(,i) Ei - ni D(i,i) Ei
+ n D(,i) Ei
n D(,)E + ni D(i,) E - n D(,)E
= T E
=0
= T () E
Addition:
 n D(,k) Ek +  ni D(i,k) Eik +  nD(,k) Ek = E 
k<j j [nk D (k,j)- nj D (j,k)]Ekj = E ()
We used
D(p,p)   q p D(p, q);
Ejk = Ek-Ej and
T () = -T () = 
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Energy Sources II
k<j j [nk D (k,j)- nj D (j,k)]Ekj = E T()
D = De + D
CR-Decomposition
Gives
k<j j [nk De (k,j)- nj De (j,k)]Ekj
+ k<j j [nk D (k,j)- nj D (j,k)]Ekj = E T()
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Matrix Representation
= T E
=0
= T()E
- n D(,) E + ni D(i,) E + n D(,)E
n D(,i) Ei - ni D(i,i) Ei
+ n D(,i) Ei
n D(,)E + ni D(i,) E - n D(,)E
E Mn E T
*
+ gives
Use
CR-Decomposition
This Gives:
Note that for EEK
is negative
M M M
e
*

E M n  E M n E T
*
e
*

*
E M n
*

TU/e
Preparing for Non-eek
Different Agents (energy investing) in the System
{e}, { } and {h}
Then
with summation
over {e,, h, etc.}
a E M n  E T
*
a
*
Kirchhoff junction rule:
The Algebraic sum of the energy current into
the junction (= system) is zero.
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Coefficients for Radiation and Transport
n  Rn
Using
t
E M n  E M R nt
*


*
Or
E M n  L nt
*

*
E T E J n
*
*
t
L E M R
*
*

R-Matrix known
from ASDF
J-Matrix is known
from task 2
The R- and J-matrix provide means for the e-Energy Equation
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General Matrix Representation
tt
S t  M
    tl
 0   M
t
lt
M  nt 
ll   l 
M  n 
Or
 S t   M tt n t  M lt n l 
    tl t
ll l 
M
n

M
n 
0
  
t
n and n densitiesof the TC and LC levels
Division S into St and Sl with Sl =0
Sub-matrices for different traffic
routes
Mtl for the traffic tl
Mll for the internal ll traffic
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Extra Sources
Molecular example
Ar2+ + e  Ar* + Ar
Provides an extra source in the Ar excitation space
Radiation example Extra population due to irradiation
Extra source in excitation space
S  M  n  S ex
M rules the normal
EEK
TU/e
Radiation Sensitive Levels

T ( )

i
D( , )
D(,  )
n D(,i) - ni D(i,i) + n D(,i)
T ()
=0
Rhs = 0 due to spatial transport
D  10-3s
Thus S(p) must be zero. The level depends on the LC solely.
Now
And
Suppose resonant A(i, ) range 107s-1
re-absorption
A(i, )  into A*(i, )= (i,) A(i, )
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The Escape Factor
n(i)(i,) A(i, ) = n(i)A(i,) - [n()B(,i) – n(i)B(i, )]  d
Usually  named escape factor: misnomer
Better
normalized net emission factor since
0<1
>1 stim. emission
<0 absorption
n()D (,i) - n(i) D (i,i)+ n()D(,i) =
n(i)A(i,) -[n()B(,i) – n(i)B(i, )] d
S  M  n  S ex
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Ray-trace Control-volume
Determination Sex
Step 1 start with plasma composition   using RT
gives  new values of , thus .
Step 2 use CV for new plasma
Step 1 returned
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Radiative Transfer
Ray-Trace Discretization spectrum.
Network of lines (rays)
Compute I (W/(m2 .sr.Hz) along the lines
dI()/ds = j - k()I()
Start
Entering
outside the plasma with I() = 0.
plasma I() grows afterwards absorption.
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Concluding
CRM essential for non-equilibrium chemistry in plasmas.
Two schemes 1) RadTrans not important
2) RadTrans essential
1)
CRM module separated from fluid
CRM for transport coefficients and source terms in
fluid Reactive plasma + lab plasmas He + Ar.
2)
Iteration procedure essential
Light generation plasmas
Matrix representation for tedious algebra
And generalization
TU/e