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State-to-State Kinetics ofMolecular and Atomic Hydrogen Plasmas
MARIO CAPITELLI
Department of Chemistry, University of Bari (Italy)CNR Institute of Inorganic Methodologies and Plasmas Bari (Italy)
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MOLECULARDYNAMICS
KINETICMODELS
FLUID DYNAMICS
state-resolvedelementary process probability
non-equilibrium distributionson internal degrees of freeedom
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MOLECULAR DYNAMICS
electron-impactinduced processes
atom-moleculecollision processes
gas-surfaceinteraction processes
KINETICMODELS
FLUID DYNAMICS
state-resolvedelementary process probability
non-equilibrium distributionson internal degrees of freeedom
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ELECTRON-IMPACT INDUCED PROCESSES
PROCESSES vibronic excitation direct dissociation
THEORETICAL APPROACHES semiclassical impact parameter method similarity function approach
INPUT DATA molecular potential of involved electronic states transition moment Born high-energy cross section value
DYNAMICAL INFORMATION energy and vibrational dependence of cross sections (complete set) isotopic effect investigation state-resolved and macroscopic rate coefficients radiative-decay vibrational excitation cross sections (EV)
FEATURES satisfactory accuracy (agreement with experimental data within the error of other methods) low computational load (except in the ab-initio step) suitable only for dipole-allowed transitions no treatment of resonances
Adamson et al., Chem. Phys. Lett., 436 (2007)
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GAMESS(General Atomic and Molecular
Electronic Structure System )
IMPACTCross Section
FAUSTBORN Cross Section
DYNAMIC CALCULATIONsolution of the radialSchroedinger equation forquantum vibrational levelsand estimation of structural anddynamical factors (evaluation ofintegrals)
AB-INITIO CALCULATIONpotentials, transition dipolemoments and GOS(Multi ReferenceConfiguration Interaction)
Radiative-DecayVibrational Excitation
maxwellianeedf distribution
Einstein coeffsof spontaneousemission
Rate Coefficients
INTERFACING SCHEME of DIFFERENT CODES
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GAMESS(General Atomic and Molecular
Electronic Structure System )
IMPACTCross Section
FAUSTBORN Cross Section
DYNAMIC CALCULATION
AB-INITIO CALCULATION
maxwellianeedf distribution
Rate Coefficients
SIMILARITYAPPROACHCross Section
rapid estimation ofcross sections fromoscillator strengthand transition energy
INTERFACING SCHEME of DIFFERENT CODES
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X 1Σg (νi=0) → B 1Σu X 1Σg (νi=0) → C 1Πu
Impact Parameter MethodCeliberto 2001
recommended valuesItikawa 2008
similarity approach
Impact Parameter MethodCeliberto 2001
recommended valuesItikawa 2008
similarity approach
D’Ammando, in preparationItikawa et al., J. Phys. Chem. Ref. Data, 37 (2008)Celiberto et al., ADNDT, 77 (2001)
electron-impact induced VIBRONIC EXCITATION CROSS SECTIONS for H2comparison of different approaches
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X 1Σg (νi) → B 1Σu
Impact Parameter MethodCeliberto 2001
similarity approach
VIBRATIONALLY-RESOLVED CROSS SECTIONS for H2
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X 1Σg (νi) → C 1Πu
Impact Parameter MethodCeliberto 2001
similarity approach
VIBRATIONALLY-RESOLVED CROSS SECTIONS for H2
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PROCESSES dissociative attachment resonant vibrational excitation (eV processes)
THEORETICAL APPROACH local resonance theory
INPUT DATA molecular potential of involved electronic states resonance characterization (width and energy dependence)
DYNAMICAL INFORMATION angular, energy and vibrational dependence of cross sections (complete set) isotopic effect investigation
FEATURES excellent agreement with experimental data resonable computational load
RESONANT PROCESSES in e- H2 collisions
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LIVX
Calculation of complete rovibrational ladders
relative to all the possible reactants/products,
including quasibound states by WKB method
DINX
Quasiclassical dynamics of atom-diatom collision
processes. Distribution of computations on a wide
computational grid with periodic check of results,
automatic re-calculations for uncompleted jobs and
collection of good ones
PES from literature
ATOM-MOLECULE COLLISIONS: from PES to complete sets of rovibrationalCROSS SECTIONS and RATES
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QCTX
Trajectory analysis and cross section results for
reactive and non-reactive processes as well as
dissociation. The quasibound states are here
considered as bound states. Various reactant and
product weight functions can be used to analyze the
trajectory results
XRATE
From translational energy dependent rovibrational cross
sections to rate coe fficient. This module includes the
possibility of selecting and mixing rate coe fficients on
the base of lifetime of initial/final states and of the level
of detail requested (e.g. total thermal rate, sum of rates
on final rotation, dissociation summed to state-to-state
rate coefficients with low lifetime, etc.)
Recombination can be obtained
in this module by two methods:
detailed balance applied to
dissociation rates and
orbiting resonance applied to
rovibrational state-to-state rates
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the two mechanisms can be used in a complementaryway, by choosing accurately the states involved in thecalculations
a detailed database of rate coefficients including initialand final rotation and state selected dissociation forH+H2 collision process is necessary
Two possible mechanisms: orbiting resonance theory and directthree body recombination (by detailed balance of dissociation data)
TERMOLECULAR HYDROGEN RECOMBINATION
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normalized recombination vsfinal vibrational quantum number
recombination rates vsfinal rotational quantum number
Esposito&Capitelli J. Phys. Chem. A 133 (2009)
TERMOLECULAR HYDROGEN RECOMBINATION
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PES used:L.P. Viegas, A. Alijah & A.J.C. Varandas, J. Chem. Phys. 126 (2007)
couplings calculated from the potential matrix, withparticular care for sign consistency of eigenvectorsEsposito, in preparation
a vibrationally detailed database of rate coefficientsshould be calculated, using trajectory surface hopping,including initial/final rotation and dissociationEsposito, in preparation
ION-MOLECULE COLLISION PROCESSES
comparison between cross sectionsobtained by DSM (Downhill SimplexMethod - full lines) and by QCT (QuasiClassical Trajectory - symbols)
This method is useful when, for a given chemical process, there are not cross sections data, but there are informationabout rate coefficients in a given range of temperatures (Minelli, to be published). The only data required are a goodfunctional form for cross section and a temperature dependent rate. The reference rate coefficients are input to astandard nonlinear optimization algorithm (Downhill Simplex Method, DSM (Nelder and Mead, Computer Journal 7(1965)) to give reconstructed cross sections that satisfy a convergence criterion.
Case study to test methoddissociation of the hydrogen molecule by atom impact
Experimental or TheoreticalRate Coefficients Cross Sections
deconvolution
DECONVOLUTION of temperature-dependent RATE COEFFICIENTSto energy-dependent CROSS SECTIONS
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MOLECULAR DYNAMICS for GAS/SURFACE INTERACTION PROCESSES
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ISOTOPIC EFFECT: H/D RECOMBINATION on GRAPHITE
ionsources
plasma-walltransition
electrostaticprobe
laser-inducedplasma
hypersonicplasma flow
Electric thrusters Negative ion sources Plasma shock tube
Divertor region Sheath physics: - see - instability - electronegativity
Ion collection Electron collection Dusty
LIBS Under-water cavitation bubble dynamics
MHD bow shock Space charge region
fluid models (CFD,MHD,SPH)kinetic models (Vlasov, PIC-MCC)
PLASMA NUMERICAL EXPERIMENTS
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0
10000
20000
30000
40000
50000
-3 -2 -1 0 1 2 3 4 5
Gas T
em
pera
ture
(K
)
spatial coordinate x (m)
x=0 shock front
upstream
T1 = 5000 K
P1 = 10
-2 atm
u1 = 5
downstream
T2(x=0) > T
1
P2 (x=0)> P
1
u2 (x=0)< u
1
x = 1.6 m x = 3.1 m
10-4
10-3
10-2
10-1
100
101
-3 -2 -1 0 1 2 3 4 5
Mola
r F
ractions
spatial coordinate x(m)
H
H+ - e
-
thick! = 0
thin! = 1
Euler Equations
Radiative Transfer Equation
CR Master &Electron Boltzmann Equations
H2 kinetics
ATOMIC HYDROGEN under SHOCK WAVE
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10-13
10-11
10-9
10-7
10-5
10-3
10-1
0 2 4 6 8 10 12 14
x(m) = 1.6
Ni /(
NH g
i)
level energy (eV)
! = 1
Ne = 1.20 10
13 cm
-3
NH = 5.44 10
16 cm
-3
Nn=2
= 3.32 1010
cm-3
!=0
Ne = 1.30 10
15 cm
-3
NH = 3.32 10
17 cm
-3
Nn=2
= 2.02 1011
cm-3
10-17
10-15
10-13
10-11
10-9
10-7
10-5
10-3
10-1
0 5 10 15 20 25 30
x(m) = 1.6
eedf (e
V-3
/2)
electron energy (eV)
!=0
Ne = 1.30 10
15 cm
-3
NH = 3.32 10
17 cm
-3
Nn=2
= 2.02 1011
cm-3
! = 1
Ne = 1.20 10
13 cm
-3
NH = 5.44 10
16 cm
-3
Nn=2
= 3.32 1010
cm-3
10-18
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
0 5 10 15 20 25 30
x(m) = 3.1
ee
df
(eV
-3/2
)
electron energy (eV)
! = 1
Ne = 1.88 10
15 cm
-3
NH = 2.54 10
17 cm
-3
Nn=2
= 7.52 1010
cm-3
! = 0
Ne = 7.64 10
14 cm
-3
NH
= 3.43 1017
cm-3
Nn=2
= 9.99 1010
cm-3
10-10
10-8
10-6
10-4
10-2
100
0 2 4 6 8 10 12 14
x(m) = 3.1
Ni /(
NH g
i)
level energy (eV)
! = 0
Ne = 7.64 10
14 cm
-3
NH
= 3.43 1017
cm-3
Nn=2
= 9.99 1010
cm-3
! = 1
Ne = 1.88 10
15 cm
-3
NH = 2.54 10
17 cm
-3
Nn=2
= 7.52 1010
cm-3
x = 1.6 m x = 1.6 m
x = 3.1 mx = 3.1 m
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1000
10000
100000
-2 -1 0 1 2 3 4 5
TGAS
(K)
TH[0-1]
Te-
Te
mp
era
ture
(K
)x [m]
!
S" = j" /#"
!
S" = B" (T )
Source function isthe ratio of emissivity
over absorption coefficient
at equilibriumequals the blackbody intensity
upstream condition
P1=10-2atmT1=1000Ku=20
non-equilibrium distributionsof spectral quantities
LOCAL SOURCE FUNCTIONoptically thin plasma (λ=1)
10-10
10-9
10-8
10-7
10-6
0 2 4 6 8 10 12 14 16
Sn=je
n/k
n
Bn(T)
E [ev]
B(T
) and je
!/k! [J
m-2
s-1sr
-1H
z-1]
(5)T=12775 K
10-9
10-8
10-7
10-6
10-5
0 2 4 6 8 10 12 14 16
S!=je
!/"
!
B!(T)
B(T
) and je
!/k! [J
m-2
s-1sr
-1H
z-1]
E [ev]
T=17817 K(4)
x = 3.29mTgas = 17817 K
10-10
10-9
10-8
10-7
10-6
0 2 4 6 8 10 12 14 16
Sn=je
n/k
n
Bn(T)
B(T
) and je
!/k! [J
m-2
s-1sr
-1H
z-1]
E [ev]
(6)T=11461 K
x = 3.28mTgas = 12775 K
x = 3.30mTgas = 11461 K
x=0 shock front
Tgas (K)
Telectrons (K)
TH[1-2] (K)
Capitelli et al., J. Phys. B, 43 (2010)
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MOLECULAR DYNAMICS
KINETICMODELS
FLUID DYNAMICS
state-resolvedelementary process probability
non-equilibrium distributionson internal degrees of freeedom
electron-impactinduced processes
atom-moleculecollision processes
gas-surfaceinteraction processes
TRANSPORTTHEORY
transportproperties
collisionintegrals
THERMODYNAMICS
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THERMODYNAMIC PROPERTIES of multicomponent plasmasincluding Debye-Hückel correction
SIMPLIFIED TWO and THREE-LEVEL PARTITION FUNCTION
ATOMIC HYDROGEN LEVELS in a BOX in the presence of Coulomband Debye-Hückel potentials
THERMODYNAMIC PROPERTIES
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• the method is based on the idea of combining a multitude of atomic energy levels into two or threegrouped levels. The partition function for atomic hydrogen can be approximated as
• the method has been applied to many atomic and ionic systems
D’Ammando et al., Spectrochim. Acta B, 65 (2010)
SIMPLIFIED TWO-LEVEL PARTITION FUNCTION
nmax =100nmax =10
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Capitelli&Giordano, Physical Review A, 80 (2009)
n
box radius = 103 ao
Bohr atomparticle in a spherical box
ENERGY LEVELS for ATOMIC HYDROGEN in a SPHERICAL BOX
Coulomb potential
screened Coulombpotential (Debye length = 102 ao)
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DATABASE OF TRANSPORT CROSS SECTION and COLLISIONINTEGRALS for He-H2 interactions in a wide temperature range [100-50000 K]
EFFECT OF MAGNETIC FIELD (tensorial reformulation)
EFFECT OF ELECTRONICALLY EXCITED STATES (EES)
Bruno et al., Physics of Plasmas (2010) in press
TRANSPORT PROPERTIES
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OMEGA
CHAPMAN-ESKOGCODE
TRANSPORT PROPERTYCALCULATION
solution of a system of linearequations algebraic equations oforder n*K, n being the numberof species and K theorder of approximation of theexpansion in Sonine polynomials
DYNAMICAL CALCULATION
classical collision integrals forelastic collision between speciesfrom (accurate, model orphenomenological) interactionpotentials
HIERARCHICAL CODE
calculation of the species-concentrationas a function of temperature
heat flux &diffusion
INTERFACING SCHEME of DIFFERENT CODES
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ground-state collisionintegrals
including EEScollision integrals
Bruno et al., Physics of Plasmas 14 (2007)
INTERNAL CONDUCTIVITY for HYDROGEN PLASMA including EES