Summary of the previous lecture

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Summary of the previous lecture
Particles Which species do we have how
much of each
Momentum How do they move?
Energy What about the thermal motion internal
energy
What do we want/need to know in detail?
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Particles Plasma Chemistry
Energy Plasma Light
Momentum Plasma Propulsion
3
Transport Modes
Fluid mean free paths small mfp ltlt L
There are many conditions for which some plasma
components behave fluid-like whereas others are
more particle-like
Hybride models have large application fields
4
Discretizing a Fluid Control Volumes
Plasma
Particles
Particles
Energy
Energy
Momentum
Momentum
For any transportable quantity ?
Transport via boundaries
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Examples of transportables
Densities Momenta in three
directions 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
Transport Fluxes Linking CV (or NPs)
? ? ? -??
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Simularities
Thus The Fluid Eqns Balance of Particles
Momentum Energy
The Momenta of the Boltzmann Transport Eqn.
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 Creation Molecules atoms i
ons/electrons etc.
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MathNumerics a FlavorSourceless-Diffusion
?T Cst
?T - k?T
-?T /k ?T
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Discretized
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Matrix Representation
1 2 3 4
-Tin 0 0 -Tout
T1 T2 T3 T4
- 2 1 1 -2 1 1 -2 1
1 -2
1 2 3 4

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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
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More general S-less Diffusion/Convection
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Laplace and Poisson
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Examples
Simularities !!
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A Capacitor space-charge zero
-?. ?V ?/?o 0
0
V
Basically a 1-D problem
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A resistor Ohms law
-?.??V 0
1-D problem
Provided ? is Cst
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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
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Concept disturbed Bilateral Relations
?
A proper channel
?
N? ?f
N? ?b
Equilibrium Condition ?t/?b ltlt 1 or ?t ?b ltlt
1 The escape per balance time must be
small
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?
Mixed Channel
?
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
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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
Distribution function Known in shape
Continuum
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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 Start outside
the plasma with I?(?) 0. Entering plasma
I?(?) grows afterwards absorption.
dI?(?)/ds j? - k(?)I?(?)
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Ray Tracing
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General Procedure ??
Fluid Swarm Collection
h i e ?
h i es ef ?
E
h i esef ?
E
h i esef ?
E
h i esef ?
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.
Examples e A, A, A, A, etc, N, N2, etc. NH, H2
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.
The BTE states that
?tfi ?.fiv ?v.fi a (?tfi)CR
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Simularities of the Boltzmann Transport equation
?tn ?. nv S
Leads to
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Generalization to 6-D phase space
Normal space
?tn ?. nv S
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
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The momenta of the BTE Specific balances
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
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,
In many cases ?t ?i ui ?. ?i iuiui , niqiui?B
and ?i g negligible ?p, ni qi E and
?Fij, dominant
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Drift Diffusion continued
Mostly Ffric gtgt Fthermo
Fij fric -(pipdom /pDij ) (ui - u )
Fij fric -(pi/Di ) (ui - u )
Fij fric -(kTi /Di ) ni (ui - u )
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Drift - Diffusion II
Normally
ni (ui u ) - (Di / kTi ) ?pi (ni qi Di /
kTi ) E,
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Ambipolar Diffussion
ni (ui u ) - (Di / kTi ) ?pi (ni ?i) E
with ?i qiDi/kTi mobility and ? ?i ni?i qi
conductivity
j ?i niqi (ui u ) - ?i ?i?pi ? E
In most cases the current density j is closely
related to the external control parameter I and
E the result
Eamb ?-1 ?i qi ?i?pi
Eamb kTe /qe?pe/pe
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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
More general
?all mi Spart, i 0 or ?all Smass, i 0
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Reaction Conservative II charge
AB ? A B- qAB qA qB
Reactions
Each creation of couple A and B associated with
destruction AB
SAB - SA -SB
Thus
qAB SAB qASA qBSB 0
More general
?all qi Si 0 or ?all Scharge, i 0
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The Composition
Bulk in Mass
?t ?m ?. ?mu 0
gives
with ?m ?all nimi and u ? ?nimiui /?m
barycentric or bulk velocity
Bulk in Charge
?t ?q ?. j 0
?t ni qi ?.ni qi ui Scharge,i
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 RN(NH3) 1 and
R of H in NH3 3 or RH(NH3) 3 or
Ri? 3 with i NH3 and ? H
Now consider NH3 ? N 3H
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
?t H ?. ?H 0
gives
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
?j ni uj
Number of X nuclei in j
Efflux of j
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Removing 1 H atom
Equivalent removing 3 H atoms
Removing 1 H3 molecule
?H 3 ?H3
In general ?H ? RHi ?j
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Simularities
Total mass transport
?t ?m ?. ?mu 0
Total charge transport
?t ?q ?. j 0
?t ni qj ?.ni qi uj Scharge,i
Total X-nuclei transport
?t X ?. ?X 0
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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
Navier Stokes
?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
Continuum Free electron states
Bound electron state
In principle ? bound states Should we treat them
all?
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CRM Black Box the ground state as entry
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
Response on influx at
Typical recombing Syst
Again dependent on ne and Te
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More than 2 entries?
If atoms in stat p are transportables
Transport sensitive
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Radiative pumped
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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 CRMs are used for Diagnostics We need info of
the ASDF non-BB
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What info is in the ASDF?? Example
The lower part of the ASDF gives insight in
trapsport the pLSE contains info of the
electron gas.
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Action Plan
Which levels are TS Transport Sensitive
or LC Local Chemistry
Cooperation TS ?? LC
Competition of Radiation versus Collisions
More general Determine structure and tasks of a
CRM
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Local Chemistry levels
np np Pp /Dp - ?? np up/ Dp
?? np up n(u/L)
np Pp /Dp
In many cases for excited states
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Competition e-collisions versus radiation
ne dependent independent
Influence Te In many cases we see 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
The corresponding Restoring Proper Boltzmann
Balance
?b(2) ?b(1) exp -E12/kTe
Escape of Radiation
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General Impact Radiation Leak
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
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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 pcr9 ne 9 1023 Z7
For pltpcr levels radiative For pgtpcr 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
64
However ne K gtgt 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.
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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
pLSE settles for Ip ? 0 since ?t/?b ? 0
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The Excitation Saturation Balance an improper
balance
y(1) y()(1?t?b) ?b1 b1 1 ?t?b
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Shape of the ASDF in ESB
?
1
? n?t ?.nw
y(p) y()(1?t?b) ?b(p) b(p) 1 ?t?b
?b ns(p)K(p,p1) ns(p) scales with
p2 K(p,p1) with p4
Restoring to P large close to cont
?b(p)boDa(ne?)-2p-6
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The Saha density mnemonic
? s(p) (ne/2) (n/g) h3/(2?mekTe)3/2 exp
(Ip/kTe)
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
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General Structure II
? n(p)/?t ?.(n(p) w(p)) P(p) n(p) D(p)
Transport Production Destruction
P(p) ?q n(q) D(q,p) nD(,p) Production
Term D(p) ? ?q D(p,q) neK(p) A(p)
Destruction factor
Transition Frequencies D(p,q) ne K(p,q)
A(p,q)
Total Destruction A(p)?l A(p,l) and
neK(p)?q K(p,q)
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Global Structure III
S(p) T(p) Source 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 Usually ground
levels LC 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
Config. space Excitation space
Config. space Transport Chemistry Transport
- n? D(?,?) n? D(?,?) T(?) n? D(?,?)
- n? D(?,?) T (?)
Transport Configuration space ruled by T(?) and
T(?) Chemistry 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
- 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 (?)

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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) 0 Gives
n(i) D(i,i)-1 D(?,i) n(?) D(i,i)-1D(?,
i)n(?)

In Matrix
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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?
T E? n? D(?,i) Ei - ni D(i,i) Ei n?
D(?,i) Ei 0 n? D(?,?)E? ni D(i,?) E?
- n? D(?,?)E? T (?) E?
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Energy Sources II
?kltj ?j nk D (k,j)- nj D (j,k)Ekj E?? T(?)

D De D? CR-Decomposition Gives ?kltj ?j
nk De (k,j)- nj De (j,k)Ekj ?kltj ?j nk D?
(k,j)- nj D? (j,k)Ekj E?? T(?)
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Matrix Representation
85
Preparing for Non-eek
Different Agents (energy investing) in the
System e, ? and h
Kirchhoff junction rule The Algebraic sum of the
energy current into the junction ( system) is
zero.
86
Coefficients for Radiation and Transport
R-Matrix known from ASDF
J-Matrix is known from task 2
The R- and J-matrix provide means for the
e-Energy Equation
87
General Matrix Representation
Or
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
M rules the normal EEK
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Radiation Sensitive Levels
n? D(?,i) - ni D(i,i) n? D(?,i) 0
Rhs 0 due to spatial transport D ?
10-3s Thus S(p) must be zero. The level depends
on the LC solely.
Now Suppose resonant A(i, ?) range 107s-1 And
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 0lt??1 Better normalized net
emission factor since ?gt1 stim.
emission ?lt0 absorption
n(?)D? (?,i) - n(i) D? (i,i) n(?)D(?,i)
n(i)A(i,?) -?n(?)B(?,i) n(i)B(i, ?)?? d?
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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 outside the plasma with
I?(?) 0. Entering 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