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1. 2 doe Ragu 2 Since the numerical evaluation of the parameter Bey is quite complex it is customary to instead present the integrated intensity 52 for a specific vibrational band calculated at a reference temperature To usually 296K 13 The value of the dipolar moment in atomic units eag can then be determined from this parameter according to the relationship 14 15 3he 10 So yu QS eao 3 2 Vuro 2 o exp j 21 2 23 The H nl London factors for these kind of linear vibrational transitions are given by Ref 16 and are presented in table 2 5 Al 0 AZ 20 P 274277 97 6 77 1 2 Ab J AL Jl 24 Q 241 6 J 140 A U 4 0 2 1 74713 2J J F1 R 271877 I 41 0 I 24 A2 77 142 7 2 2 J 1 Table 2 5 H nl London factors for parallel and perpendicular rovibrational transitions of linear polyatomic molecules Lastly the Herman Wallis factors can be expressed as a function of the following polynomial expressions P branch 1 Ay J Aa J A3 J Q branch 1 AgJ J 1 R branch 1 Ai J 1 Ao J 1 A3 J 1 3 20 Physical Models The values for each coefficient A1 2 3 Q being tabulated for each vibrational band 2 1 3 Broadening mechanisms Broadening mechanisms lead to the broadening of the initial tran
2. Atomic Transition Probabil ities of Carbon Nitrogen and Oxygen Journal of Physical and Chemical Reference Data Monograph No 7 1996 Jefferies J T Spectral Line Formation Blaisdell Company 156 Ed 1968 Griem H R Plasma Spectroscopy McGraw Hill 1964 Whiting E E An Empirical Approximation to the Voigt Profile JQSRT Vol 8 No 6 1968 pp 1379 1384 Olivero J J and Longbothum R L Empirical Fits to the Voigt Line Width A Brief Review JQSRT Vol 17 No 2 1977 pp 233 236 Stallcop J R and Billman K W Analytical Formulae for the Inverse Bremsstralung Absorption Coefficient Plasma Physics Vol 16 1974 pp 1187 1189 Karzas W J and Latter R Electron Radiative Transitions in a Coulomb Field Astrophysical Journal Supplement Vol 6 1961 pp 167 212 Hummer D G A Fast and Accurate Method for Evaluating the Nonrel ativistic Free Free Gaunt Factor for Hydrogenic Ions The Astrophysical Journal Vol 327 1988 pp 477 484 Menzel D H and Pekeris C L Absorption Coefficients and Hydrogen Line Intensities Monthly Notices of the Royal Astronomical Society Vol 96 No 1 1935 pp 77 111 Kramers H A On the Teory of X Ray Absorption and of the Continuous X Ray Spectrum Philosophical Magazine Vol 46 1923 pp 836 871 Mjolsness R C and Ruppel H M Contribution of Inverse Neutral Bremsstrahlung to the Absorption Co
3. 2J J 2 Y 2 B 3 Intermediary a b case H nl London Factors 79 O Ros J J J 2 41 2 7 J 1 7 4 3 u 7 a 2 1 C2 U1 CZ J 2J J 4 2 Y7 2 1 4 Puy J lug Ps J IGJ DCT J z J 2 u ugt 3 16JC J 1 C4 18 1 5 4 2 Q J ICN U DU 7 u J 48J 942 J 1ug J Dug R J f 7 J 1 J 3 u T Du 2 9 16 J 1 C4 J 1 C J BHT DU J 4 4J24 Y 2 yY ux 4 4 J 4 1 Y 2 Ci J e XJ 2 1 2J 2J 1 J 1 J 1 C2 J r 4J J 1 s C ou C3 J Y Y 4 J J 4 1 2027 4 1 U 2 J 4 1 Table B 9 H nl London factors for 2 II transitions 80 H nl London Factors Appendix C Potential Energies and Wavefunctions Reconstruction With the RKR_SCH Routine This section describes the associated RKR_SCH m routines that are either used to yield values for the transition Einstein coefficients Ay or to achieve an accurate list of the overall rovibrational levels for a specific electronic state including quasi bound states This suite of routines is located in the RKR_SCH directory C 1 Theory As discussed in Chapter 2 for a given electronic state e and neglecting fine structure the energy levels of a diatomic molecule are usually expressed
4. ATz The main objective being the calculation of a line profile as close to Eq 2 as possible while retaining a minimum number of points a method has been devised in order to obtain an adapted grid of key points where the expression of Eq 2 is to be evaluated The resulting profiles are found to adequately approximate the Voigt profile of Whiting and Olivero to a user defined precision which will depend on the number of considered grid points Firstly a set of low resolution line profiles have been defined for allowing quick spectral calculations yielding a compact overall spectral grid Four line profiles with respectively 5 7 9 and 11 grid points have been defined introducing the parameters W and FW which define the boundaries between the line center region the near line wings and the far line wings respectively The expression for those parameters for a given Voigt profile is proposed by Zhu 15 W FW 1 OAR 3 the If 91 constants are held at a 1 1 8 for the calculation of the parameter W as proposed by Smith 16 and held at 2 6 5 8 for the calculation of the parameter FW according to a critical analysis carried by the author on sample Lorentz Doppler and Voigt profiles The different grid points have been defined accordingly and are presented in Table 1 Considering a zero line intensity for the last grid point M Lino da Silva Journal of Quantitative Spectroscopy amp
5. 1 and plot of the normalized difference of Lorentz full line AV 5 cm and Doppler dotted line AV 5 cm line shapes to the exact lineshapes right 5 7 9 and 11 points from top to bottom 0 05 0 06 Lorentz os Lorentz 0 04 x Doppler 0 02 0 01 2 gt 76 0 0 01 200 100 0 100 200 10 101 10 2 0 05 2 0 06 Lorentz a Lorentz o g 0 04 x Doppler di 0 02 0 01 2 E 0 01 2 200 100 0 100 200 2 o z 10 10 10 0 05 e 0 06 Lorentz 05 Lorentz a 0 04 i A Doppler 0 02 0 01 ae 0 0 01 R 2 200 100 0 100 200 10 10 10 Distance to Line Center cm Figure 3 6 Plot of high resolution Lorentz lineshapes left 21 37 and 73 points from top to bottom Av 5 cm7 and plot of the normalized difference of Lorentz full line Az 5 em 1 and Doppler dotted line AV 5 em 1 line shapes to the exact lineshapes right 21 37 and 73 points from top to bottom 42 Detailed Description of the Code 3 5 2 Handling and Overlay of Individual Lineshapes Following the definition of a method for adaptively calculating individual lineshapes one may apply it to the calculation of the individual lineshapes for each of the calculated radiative transitions with their accompanying Doppler and Lorentz FWHM However doing so for a large number of lines typically mor
6. AD v AP uv AO 4 Table 4 3 Fine structure constants for doublet and triplet states Step 3 Provide the Species Nuclear Spin statistical weight If the species is heteronuclear AB this parameter is meaningless and the value 99 is mandatory If the species is homonuclear AA the values tabulated in Table 4 4 have to be included depending on whether the molecule is a symmetric antisymmetric boson fermion Step 4 optional Define vibrationally specific constants for each up per and lower electronic states 4 2 Building spectral datafiles for diatomic transitions 53 Symmetric Antisymmetric Boson 1 1 Fermion 0 0 1 Table 4 4 Values for homonuclear molecules If you want to introduce vibrationally specific constants for each vibrational level of the species consider as an example the H2 LY txt file In this file tran sitions between the B15 and the X Y levels of molecular hydrogen are simulated After defining the nuclear spin statistical weight set the vibrationally specific constants label to 1 Then for each state define the number of vibrational levels and then explicit the vibrational levels considered After this you have to define the number of variables you want to use and specify each of them To each specific variable corresponds a spectroscopic con stant The correspondence between the specific variables and the spectroscopic constants is shown in Table 4 5 So af
7. For the b H nd case the quantum number is no longer defined If both initial and final states of the transition both belong to Hiind case b we have the selection rule 2 1 Discrete radiation models 13 0 1 AN 0 forbidden for X 2 5 Figure 2 2 Hiind case b vector diagram 2 1 1 3 Linear Polyatomic transitions In addition to the same rules considered for diatomic transitions AJ 0 1 etc we observe the additional vibrational selection rules 1 Avg Avg odd and 1 for perpendicular bands 2 Avg Avg odd and A 0 for parallel bands 2 1 2 Line positions and intensities Discrete atomic and molecular spectra are composed of a collection of lines which can be defined by three specific parameters 1 Line position 7 2 Line intensity N A AE 47 3 Line profile F Y 70 This section outlines the theoretical models that are implemented for the production of a line database with these three parameters 3 absorption coefficients are determined from the emission coefficients according to Kirchoff Planck Law 14 Physical Models 2 1 2 1 Atomic transitions Atomic line transition lists are typically compiled into comprehensive databases providing the line center positions Vo the upper and lower energy level energies and degeneracies Eu gu g and the transition Einstein coef ficients Az 2 1 2 2 Diatomic transitions For
8. MKS 2 48 Cross sections for the Inverse Bremsstrahlung of N and O are provided by Mjolsness and Ruppel 27 2 he i i 7 o 7 Te 8n V 20 5 Tol kate a oo e ao 2T 26 Physical Models with o9 0 71 10 for O and o9 0 80 10 for N Tabulated Nz and Oz Bremstrahlung cross sections o 77 7 are reported in Ref 28 2 3 Generalized Kirchhoff Planck Law for ra diative transfer The detailed balance principle states that in full thermodynamic equilib rium direct and inverse reaction processes fully balance themselves Since physical chemical elementary processes do not depend on the thermodynamic state of the gas plasma one may put to use the detailed balance principle to deduce the intensity of an inverse process considering the intensity of the di rect process and the expressions for the thermodynamic equilibrium of a gas The Law relating the radiative emission absorption coefficients is the so called Kirchhoff Planck Law which will be shortly summarized for all the discrete and continuum processes described in this chapter 2 3 1 Discrete transitions For discrete transitions of the type A AB i the general Kirchhoff Planck Law valid for any arbitrary population distribution of the species internal levels yields in frequency units Ep 2hv guM 1 2 50 av 2 ay In thermodynamic equilibrium conditions with N g exp E keT this expres
9. 0 004 40 002 0 002 5 Fit Dispersion cm 1 4 0 004 02 Rotational Constant cm 1 0 006 1 2 3 4 5S 6 7 8 9 10 Vibrational Quantum Number Figure 4 2 Interpolation of the level specific rotational constants B for the unperturbed 2 top and the perturbed d IL bottom electronic states of C2 One example can be presented for the Cg Swan bands system who is also generally ubiquitous in carbon based plasmas Fig 4 2 shows the fit of the constants B for unperturbed lower a state and for the perturbed d state One may immediately verify that the polynomial fit will be very uncertain for the perturbed state which helps explaining why considering level dependent spectroscopic constants B Dy etc will provide more accurate results than resolving to a matrix of Dunham coefficients Such a comparison using both methods is presented in Fig 4 3 50 Modifying the Code Spectral Database Simulation With Equilibrium Constants 0 5 onlay F Simulation With Level Constants 0 5 Intensity A U 2 Measured Spectrum oo r 0 5 f iii ri it i f 5120 5130 5140 5150 5160 5170 Wavelength A Figure 4 3 Reproduction of an experimental spectrum of the Swan bands of C Av 0 using level dependent spectroscopic constants and using Dunham coefficients Finally as discussed in Chapter 2 rotational perturbations can be mo
10. B and 6 satellite branches O Pis Q Pan PO RQ O Rio 5 Ru and up to 27 branches for triplet transitions 9 main branches P P2 Ps Q Qo Ri Re R3 and 18 satellite branches S P21 2021 2 1 Ps Qs1 7Rsi Pi Qiz Rio 2 Ps2 Qs R32 Pis Qis Ris Pos Q23 R s Some special cases are considered in the SPARTAN code Firstly some branches have very low transition probabilities and may be safely neglected and secondly some branches may be indistinguishable except at very high spectral resolution for example states fine structure constants have values typically below 0 01em 1 and appear superposed in practice The SPARTAN code takes advantage of this so that only the 6 and 9 main branches for doublet and triplet transitions are in practice calculated This has been verified to have no noticeable effect on the obtained results but future versions of the code might levy such a restriction as improved computational power becomes available 72 H nl London Factors B 1 Applied approximations B 1 1 Neglecting line spin splitting effects for satellite lines involving states If the separation of multiplet lines is much smaller than their widths such lines can be accurately superposed into one singlet line The SPARTAN code takes advantage of this for coalescing perpendicular transitions transitions with a change of fine structure state e g Q32 with the corresponding
11. B 1 2 Neglecting weaker rotational branches Assuming a Boltzmann equilibrium of the rotational levels a line rotational intensity may be written as 2 Lexp FU ga ro vr Pr B 2 Qrot 2J 1 Because each rotational state has a gz 2J 1 degeneracy the ground state will not be the most populated state As an example at room temperature typically the most populated rotational level is around J 7 10 whereas at a 2000 K temperature the most populated level is around J 20 S J Ithe spin spin correction is y 7 26 10 cm for the X X v 0 state and y 17 16 1073cm 1 for the B X v 0 state 35 B 1 Applied approximations 73 27 2 21 29 37 3 31 34 PQ z P PQ z P 204 P 90 Q 2 Qi Pi 2 P O Ri Q P Qo Qe P TRai P E acR Once H Ra Q ie Q Pi2 Q P23 Q2 SR32 Q2 R Q Qa Ra2 Qs PQ z NPis Ri NPs h Qs lm PQ s Rs R Table B 1 Coalesced rotational branches for multiplet X II transitions when neglecting spin splitting for the state As some rotational branches have smaller magnitudes than others only reaching equivalent magnitudes for the first rotational levels typically J lt 5 we may examine the expression for the rotational line strength to check whether these may be safely neglected An example for a 39 II transition at a 300 K gas temperature is
12. By dy 2 225 351 1 1 Dy J F 1 2 15b 5X0 7 20 By J J 1 D 7U 1 Av Bo Fv Av By Boy 4907 1 By 2 2 15c The expression for the II levels is given by 9 512672 B J J 1 yy 4J J 1 1 4 Dy 2 16a 3 yi HAJ 1 IL B u 1 4 y2 2J J J Sum 4J J 1 J J 1 gi 4J J 1 a Tt 20 a y2 7 3 2 16c 3 yit4J J 1 with 4 4 Ay Y Y 4 1 n Y 4 7 1 z These expressions are also consistent with Zare s effective Hamiltonian 4 and one should be careful enough to select spectroscopic constants that have been fitted to such formalism If spectroscopic constants fitted to other for malisms are selected they should be converted to Zare s effective Hamiltonian prior to insertion in the code Table 2 4 reported from Ref 10 presents the correspondence between spectroscopic constants fitted to Zare s Hamiltonian and Brown s Hamiltonian 11 which is also popular among spectroscopists Line Intensities The intensity of one line will depend on the energy of the transition the population of the excited level and the transition probability described by its 2 1 Discrete radiation models 17 Brown Zare A AZ 1 882 B BZ 1 2q D D H H y y7 1 2p p p q q Dp Dp 24 2Dq D D H Ayia 2
13. Interpolates and adds the pseudo continuum intensities to the calculated grid A schematic view of the selection process of the different lines split into strong weak and very weak categories is presented in Fig 3 3 The user defined parameters are LinPar interp This parameter defines the point s interpolation method used for the routine The two possibilities are linear and cubic linear interpola tions are fastest but less accurate than cubic It is recommender to use cubic interpolations LinPar shape This parameter allows the user to allow the calculation of a general Voigt lineshape parameter 99 computed with a user defined num ber of points for the lineshape core and the wings LinPar num c and Lin Par num_w 6 or to allow the calculation of a simplified Voigt profile with 5 7 9 or 11 points respectively parameters 5 7 9 or 11 For more details on the calculation of the Voigt profile see section 3 5 1 36 Detailed Description of the Code Strong 4 Weak 10 Very Weak f Intensity A U 91 Dii m 1 1 1 2 48 2 485 249 2 495 2 5 2 505 2 51 2 515 2 52 10 Wavelength A x 10 Figure 3 3 Example of Strong Weak and Very weak lines LinPar LineBound Parameter which defines the boundary inside which a high resolution lineshape is calculated and core and wings points are distributed see section 3 5 1 for more details This par
14. OFO DU DTU FU 1 J 1 4 2 J 1 ullt J U 1 4 3 1 J 1 7 Ra J ICT 2 2 Rp J 1 1 Jul J 1 u7 7 17 S27 16IC3 J 1 CY J J 2 F Duz J ull 7 8 1 2 1 I Qa J TUTDOLUTOTU 19 DU uhh Jul 0 A T DU I 4 l 742 J 17 55 UR Rsi J OFDD 7 5 DUH Duy J 1 J 1 PNT AEA QT a 1 1 Pi2 J aa Dey 7 2J J 2 2 12 PQi2 J THI er 1 1 2 u 7 b U 1 U 4 1 Y7 x 4 2 J 1 2ult J 41 1 3 ul J 1 1 2 Rio J 2 J o Ney n 110 2 Y 2 7 2 7 2 212 Q J DCD Per AY 2 4 3 49 He oD B DO 2 4 7 17 3 071 2 Paa J J 1 J 1 J uy J 1 7 di 2420501 DOZ J NT Dal J a 1 J 1 Y 2 2 2J 1 b s 2Qs2 J 1070060 ig J 2 u J 2 4 2 J 1 2uy J 1 7 1 U 3 ush J 1 Roa J TERENCE ECA ie DUA 7 x l N P a J J 1 J 1 Jul J 1 u 7 J 12 1977 16 1 10530 J 1 u J 8 2 2 2 IL Qis J JODIO DO Dur Dug J 4I DI 461 4 2 P Ris J J J 2 I J 1 u 7 J 12 5 16 J DC IFN CZ J 1 8 ul J ult J 8J2 J 2 12 O Po3 J WERT NCO yu 0 J 2 7 2 uZT 2 2J J 2 Y 2 12 PQ IOO u 1 J 2 uz J
15. P J 3 I 3 Ri J 10776 TFNO Du 4 J 4 J 3 Q J 5 J 3 1 We 1 1412 Pa J 4IC t J ner D 1 A J 5 J 5 J TIFT sulun 207 3 01 95 S Roi J TIPO TO J ult J 4 1 u 7 J 4 5 J 3 2 J 3 J 3 O Pr2 J WOH DOQ ur J 1 u t J 4 J 120 20041 7 yor law Ju J 2 U 2 1 4 2 Re J Io 2007 yu Dut 4 5 F J 3 J 3 J IIOU DOF tu u J 4 J J 4 J i Q2 J DITO DOTA E OW 4 2 pu 872 J 3 J 3 D Ior O ETT Dut J 40 T 22 ut J ex o VO ulay ct J tfu J 44 7 3 1 eu ou Table B 6 H nl London factors for II II transitions Transitions H nl London Factors Pisa Fo Dery J 1 J 1 z m J JET I F 2 2 2 4 1 p 2J 1 d l 2J 3 Ris J Qi2 J J 1 2I 1 2I 3 J PQ23 J Qai J UFD 1 Psi 7 D JCI I CIFI Table B 7 H nl London factors for 3X transitions B 3 Intermediary a b case H nl London Factors 77 Transition H nl London Factors y _3 TI 3 _3 y J 1 J 1 ui Y 72771 P J mu Q J VJ 72 7 Dur 2J J2 1 RU P J 1 Z U 1 J 1 9Pn 9 1 J 1
16. Radiative Transfer 108 2007 106 125 113 A different method has been devised for the calculation of high resolution profiles Two line parameters par and par have firstly been defined related to the line center and line wings respectively The line center being identical whether the profile is Lorentz or Doppler par will be independent of the parameters Ay and Ayp The lineshape wings will however be very different regarding whether the line profile is Doppler or Lorentz par will therefore have a strong dependence on the parameters Avy and Ayp and such dependence has been acknowledged in the following way paryy g Paryp gt 4 Avp Ayp AVL such that for a Lorentz lineshape we have par par and for a Doppler lineshape we have paryp A series of monotonous points has then been defined such as ig 0 max num 1 max 5 and a geometrical transform is applied such that 1 exp par ey X io x io max 6 a EXP Par w x io 1 EXP PAT cw x iotmax We then define i i and i 1 i An example of the geometrical transforms i and iy obtained for specific par and par values is presented in Fig 4 Practically speaking such transformations return two grids more or less tightened around the values 0 or 1 These two grids need now to be put to use Firstly for distributing the grid points in the line center region the second der
17. TU 27021 R J J A 2 J A 1 J A 1 J A 1 J A 1 J A 2 2 J 1 J 1 2 41 Table B 3 Honl London factors for singlet transitions Transitions H nl London Factors J J J J P 2 J oe J41 J 2 4 1 Ria J ae 1 J Qa 9 Ri2 J 2741 273 2 1 2 41 Table B 4 H nl London factors for 7 transitions Transition H nl London Factors 25 _2 II 2 2 y P gt J Bo 1 2 4 1 2 2 1 4 2 4 2 l Pa J 9 2 1 16 9 Ra U 1 2 1 2 2 4 1 0 4 24 4 4 1 P J Ri 1 16J Q2 J Q J 2I 1 4J 4 J 1 4U 8 75 12 72 2 7 1 2Y Qai J 0 16J J 1 PQi2 J 0 1 2J 1 4J2 4J 1 FU 834122 2J 742Y Qi J Q J 16 7 1 R J Pz J 1 2J4 1 2J 1 U 4J744J 1 2Y SRoi J Pio J 1 16 J 1 Ri J 1 2I 1 2 2I 1 U 4J2 4J 7 2Y R J P J 1 16 J 1 a U IY Y 27 1 7171 2 Table B 5 H nl London factors for X TI transitions 76 H nl London Factors Transition H nl London Factors J 3 J 5 P J IIOU U ul J 1 u 7 J 4 J 5 52 gi Q J 21 141 7 DEU 2u Ju J AI A 220
18. although collisional broadening processes might not be completely irrelevant and they are equally ac counted for considering a Voigt line profile in an effi cient lineshape routine allowing fast and accurate spectral simulations Lino da Silva 2007 For CO 2 N plasmas the spectral database described in Lino da Silva 2004 Lino da Silva 2006 is considered In the case of pure CO plasmas the database can be fur ther reduced to the one presented in Tab 1 2 3 Hardware for Calculations The calculations presented in this report have been car ried out in a Linux Debian Intel x86 8 core machine with 32GB of RAM and two RAID 1 1TB storage space As it has been previouslyy discussed the selected lineshape parameters alongside with the number of spatial cells for radition lead to overall radiative field with a size of 25 50 depending on the entry point When radiative transfer calculations are being carried out it is not fea sible to directly load the radiative cells emission and ab sorption coefficients as this leads to I O operations tak Table 1 Spectroscopic Database for the Simulation of the EXOMARS Radiative Field species system upper state bands species database model electronic lower state tul levels Martian like molecular systems atomic photoionization CO x X SOS TS x r 10 10 Topbase Topbase level Qa 361 Fourth Positive A I X S 10 10 Topbase
19. and Nakamura E Single and Double Photoionization Cross Sections of Nitric Oxide NO and Ionic Fragmentation of COT and COF Physical Review A Vol 48 No 3 1993 pp 1955 1963 44 Huber K P and Herzberg G Molecular Spectra and Molecular Structure IV Constants of Diatomic Molecules Van Nostrand Reinhold Company 1979 45 Abgrall H Roueff E Launay F Roncin J Y and Subtil J L The Lyman and Werner Bands of Molecular Hydrogen J Mol Spectr Vol 157 1993 512 523 46 Fantz U and VV nderlich D Franck Condon Factors Transition Prob abilities and Radiative Lifetimes for Hydrogen Molecules and Their Iso topomeres Tech Rep INDC NDS 457 TA Vienna May 2004 47 Douay M Nietmann R and Bernath P F New Observations of the ATI X1 Transition Phillips System of C2 Journal of Molecular Spectroscopy Vol 131 1988 pp 250 260 48 Phillips J G Perturbations in the Swan System of the C Molecule Journal of Molecular Spectroscopy Vol 28 1968 pp 233 242 49 Piar B Production de Mol cules Excit es Vibrationellement et Electron iquement par Pompage Laser Analyse des Etats Form s These de Doc torat Ecole Centrale de Paris Sept 1993 50 George T Urban W and Le Floch A Improved Mass Independent Dun ham Parameters for the Ground State of CO and Calibration Frequencies for the Fundamental Band Jour
20. energies file N PI LEV txt A comparison of both levels file N ATO LEV txt and N PI LEV txt will show that although they remain very similar for the lower energy levels there are increasing differences for the higher energy levels 4 4 1 1 Atomic discrete transitions file structure Atomic discrete transitions are split into two files a level file AB ATO LEV txt and a radiative transitions file AB ATO txt The structure of the level file AB ATO LEV txt is as follows e the number of levels first line labels E and ge second line Level energies in cm and degeneracies For the atomic transition file AB ATO txt we have e Database identifier first line e Number of transitions and label third line e labels for the columns fifth line Transition energy in cm 1 Einstein coefficient s71 level index in the levels file upper and lower state Energies of the upper and lower states in em 1 degeneracies of the upper and lower states 4 4 1 2 Atomic photoionization transitions file structure For photoionization cross section files AB PI txt the TOPBase format is to be used as is with the small difference that the header has to be removed 60 Modifying the Code Spectral Database and replaced by a line with the number of individual electronic level cross sections The corresponding energy level files AB PI LEV txt have then to be ob tained from the same TOPbase database and modi
21. gt 2 8K for the ground state of N2 For most practical applications except a very low cryogenic temperatures this approximation will hold easily We consider a second order expression for the calculation of the vibrational dependent constant B such that B Be Qe v 1 2 With these two approximations we may write the total partition function as v J 5 Oo Qro e v J 1 1 4388 1 4388 1 mu z2 Tu Te Dd exp Ta F v 5 1 4388B v 0 vib 4 7 Here we have added the term 1 o 1 2 for homonuclear molecules account ing for the nuclear spin partition function Qnuc for heteronuclear molecules nue 1 4 3 2 File structure For each diatomic species AB inserted in the code a file for the molecular energies has to be built The recommended naming convention for the file is AB LEV txt Here we will shortly describe the structure of the file Each file starts with two lines e Number of molecular electronic levels e Nuclear spin degeneracy 1 for heteronuclear molecules AB 2 for homonu clear molecules AA This is followed by the data required for the calculation of the rovibronic partition functions for each electronic level The structure is as follows e Identifier of the electronic level e Calculation mode for the vibrational partition functions 0 for explicit energy levels 1 for harmonic oscillator approximation Electronic energy in cm Umax for mode we in cm for mode
22. pressure high temperature plasma applications in aerospace applications sim ulation of planetary atmospheric entry radiation However the code can and has been applied to a variety of different applications as for example the simu lation of radiation from atmospheric and low pressure plasma sources Last but not least the code can be applied to the simulation of atmospheric opacities the simulation of radiation from combustion processes or even other applications The code can be operated in two different fashions e coupled to a fluid dynamics code which calculates the local macroscopic properties of the flow and handles them to the SPARTAN code The SPARTAN code is in turn coupled to a radiative transfer code which accounts for the calculated spectral dependent emission and absorption coefficients of the gas and calculates radiative transfer e stand alone for the simulation of the local spectral properties of gases and plasmas or for the comparison with experimentally determined spectra providing information on the species temperatures energy levels distribu tion functions The second standalone application is by far the most common one and this program manual is primarily intended at providing support with the setup of such kind of simulations This manual is divided in four Chapters Chapter 1 gives a quick overview on how to quickly start using the code for calculating spectra using the supplied database of the co
23. 0 rot gJ Xp Lama he gt 27 1 exp R 4 4 3 1 Approximations considered in the population rou tine Two approximations are carried out in the calculation of 1 the individual vibrational partition functions and 2 the rotational partition functions summation For the vibrational partition functions we consider that the level energies follow a first order series harmonic oscillator approximation with 1 We 4 4 with Do 1 mar n 4 5 Here we do not consider the second order correction der for the vibra tional energies or any other higher order corrections since if the condition w 4weare gt De is not met the level energies will reach a fictious maxima be low De one such case is the C 42 perturbed state Therefore the harmonic oscillator approximation is preferred as it does not rely in the extrapolation of corrective factors which may lead to singularities 34 Alternatively explicit values for the vibrational level energies up to the dissociation limit may be supplied hence avoiding unreliable partition function calculations at high Tu For the rotational partition functions sum the approximation k Tro Tro 200000 ia 4 6 J hcB 01 4388B is considered 58 Modifying the Code Spectral Database This approximation holds for Tpot gt hc kpB 1 4388B As an example we have Trot gt gt 87 5K for the ground state of H and Trot gt
24. 178 10 56 0 86 NO 47 21 11 105 340 0 64 1 07 Oz Schumann Runge 18 11 1308 1729 167 2 CO IR 21 11 17 l 0 26 0 47 NO IR 21 11 52 317 10 43 1 03 COz IR 84 84 0 31 0 31 Table E 2 Runtimes and grid sizes for the different diatomic transitions of the SPARTAN code database 98 Code Versions Log Appendix F Selected Published Works Two selected published works obtained using the SPARTAN code are presented in this appendix They are respectively 1 M Lino da Silva An Adaptive Line by Line Statistical model for Fast and Accurate Spectral Simulations in Low Pressure Plasmas JQSRT Vol 108 2007 p p 106 125 2 M Lino da Silva A Contribution for the Simulation of VUV IR Ra diation Transfer in COz N Entry Flows Using a Line by Line Model Proc 4th Int Workshop on Radiation of High Temperature Gases in Atmospheric Entry Lausanne Switzerland 12 15 October 2010 ESA SP 689 February 2011 The first work outlines the lineshape model implemented in the SPARTAN code and presents a numerical validation of this model The second paper presents the application of the SPARTAN code in large scale radiative transfer computations for an atmospheric entry flow including a spectral grid conver gence study using the parametric conditions of the Lineshape m routine of the SPARTAN code The JQSRT 2007 paper contains a small typo The correct expression for the Cy and C2 expressions in Eq 2 should re
25. 3 Sample comparisons A few supplied test routines showcase the capabilities of the SPARTAN code for reproducing experimental spectra and performing large scale spectral computations Inside the TESTS directory there are several directories with a matlab exe cutable each These contain e A Simulation of the C Av 0 Swan Bands using the routine for the sim ulation of 3I 3 II transitions for homonuclear molecules and comparison with an experimental spectrum e A Simulation of the CN Violet Av 0 System using the routine for the simulation of 19 4 transitions for heteronuclear molecules and comparison with an experimental spectrum A Simulation of the H Lyman 42 Werner bands using the routine for the simulation of 19 1 X and II transitions for homonuclear molecules and comparison with an experimental spectrum A Simulation of the 15 Negative Av 0 System using the routine for the simulation of 7 X transitions for homonuclear molecules in cluding the effects of rotational perturbations and comparison with an experimental spectrum e A Simulation of the NO Rovibrational Av 0 Transitions using the rou tine for the simulation of II TI transitions for heteronuclear molecules and comparison with an experimental spectrum A Simulation of a full VUV IR spectrum for Air at 1 bar mixture in full thermochemical equilibrium at 1 000K and 6 000K A Simulation of a full VUV IR spectr
26. 951 F2G0a BvG N N 1 DvG CN N 1 A2 HvG CN N41 A3 952 0 5 repmat gannaC SpeciesData dimenrot 11 953 gammaGJ N N 1 gammaGJJ N N 1 42 3 repmat Nel 954 end 955 if stremp Transitions 2S 2P 1 stremp Transitions 2P 2P 956 957 N 1 SpeciesData dimenrot nce FIC nen T RUC Temmi sezdim 113 This Chapter provides a more detailed description of the code routines and their associated functions A full summary of the SPARTAN code capabilities will be firstly presented followed by a brief description of each of the core rou tines Finally an extensive discussion of the algorithm for the Lineshape routine will be presented Indeed knowledge on the inner workings of this routine is paramount for a good understanding of the inner works of the SPARTAN code as over 99 of the overall calculation times are spent inside this routine which convolutes the lines lists supplied by the Atomic1 2 3 routines into a synthetic spectrum For a partial listing of the spectral database of the SPARTAN code please refer to Appendix A 3 1 Introduction The SPARTAN code has been developed with a particular concern in providing a flexible and scalable structure Rather than trying to provide an authoritative tool with its own monolithic database it has been acknowledged that different applications will need more or less focus on different aspects of the physical models im
27. D stands for doubled nuclear spin state For seeing a sample file considering this case please refer to database file H2 WED txt which treats the Werner HI Y transition for the homonuclear H molecule with two different datasets for odd and even states 4 2 4 Testing your files The routine Test_DiatomicFiles m can be used for test ing the integrity of your new spectral file Usage 16 gt gt Test_DiatomicFiles filename transition code Please refer 56 Modifying the Code Spectral Database to appendix D for more details Furthermore the SPARTAN code has some limited capabilities for detection of invalid files and might automatically stop the program execution displaying the message The input file filename has errors check for typos and verify that multiplet transitions are not missing fine structure constants Click here to stop program execution Lastly if wrong values for spectroscopic constants are inserted some quan tities might be strongly offset The SPARTAN code makes a limited attempt at checking these inconsistencies In its current version the rotational energies are cross checked If they exceed some unphysical high value 100 000cm 1 12 4eV a warning is displayed during execution It might be wise in such case to cross check the obtained results for consistency and revise the spectral constants datafile for typos 4 3 Building the datafiles for diatomic species partition functions c
28. LEV txt and a radiative transitions file CO2 txt For now only the CO2 IR transitions are included so we will take them as an example The level file first three lines indicate e The number of isotopes first line e The maximum rotational level of all the transitions second line e labels for the columns third line 4 4 Building spectral datafiles for other transitions 61 For each isotope we then have a definition line which gives the numerical reference of the isotope e g 626 the number of levels e g 256 and the abundance of the isotope e g 0 9842 We then have the same number of lines as the levels indicated for the isotope e g 256 Each line has the following columns Numerical label of the level ISO level number e Vibrational level label level degeneracy 2 Rotational constant B in 1 Rotational constant D in em 1x107 Rotational constant H in cm x10 Maximum rotational quantum number Jmaz The corresponding partial section of the file CO2 LEV txt is presented below 8 Isotopes 144 MaxRotLevel Ref v g 2 d0 1 Gv Bv Dve7 Hve13 Jmax 626 256 0 9842 626001 00001c 1 0 000000 0 39021889 1 33338 0 077 140 636 126 0 01106 636001 00001c 1 0 000000 0 39023754 1 33346 0 151 122 636002 01101e 2 648 47803 0 39061133 1 35489 0 497 113 For the transitions file such as CO2 txt we have the number of vibrational bands in the first line and the labels of t
29. M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 Table 3 Radiative transfer results for different lineshapes case 1 see Table 2 Case 1 To 20 000 T 1000 Io xo 1m Ii x 0 1m Ii x 1km Ii x 10km c8w16 37 425 021 33 609 895 11 061 844 901 319 c6w12 37 362 679 33 579 270 9332 971 180 892 c3w6 37 137 075 33 559 071 1079 988 0 003 1 38 622 523 35 253 188 12 294 190 11 873 855 19 38 120 998 35 585 727 12 816 890 12 362 364 1 36 090 270 33 636 470 15 454 166 14 902 383 15 37 921 303 31 679 264 5940 018 5940 018 I To c8w16 89 81 29 56 2 41 c6w12 89 87 24 98 0 48 c3w6 90 37 2 91 0 00 11 91 28 31 83 30 74 19 93 35 33 62 32 43 1 93 20 42 82 41 29 15 83 54 15 66 15 66 Tn hires c8w16 100 00 100 00 100 00 c6w12 99 91 84 37 20 07 c3yv6 99 85 9 76 0 00 111 104 89 111 14 1317 39 19 105 88 115 87 1371 59 1 100 08 139 71 1653 40 15 94 26 53 70 659 04 The table successively reports the radiative intensity exiting from the hot 79 and cold Z1 slabs the percentage of the incoming hot slab radiative intensity transmitted by the cold slab 7 Jo and the differences in transmissivity compared to high resolution calculations i hires The radiative transfer calculation results are presented in Table 6 and a sample calculation is presented in Fig 10 From the analysis of the obtained results one may verify th
30. Summary of the capabilities of the SPARTAN code 3 3 Units used in the SPARTAN code SA Gore 272 4 be SS ee Re Pe eres ad Ga ae Re 3 5 Lineshape calculation routine 3 5 1 Calculation of a Voigt lineshape 3 5 2 Handling and Overlay of Individual Lineshapes NNOO aD ow 20 23 24 25 26 26 26 27 27 4 Modifying the Code Spectral Database ntrod eli ib se koea nd ee ee R BB r b des 4 2 Building spectral datafiles for diatomic transitions 4 2 1 Guidelines for the selection of appropriated spectral con BURDEB gt a oh Be Adee iti hoe OR de ok ee Ga ee Gas a 4 2 2 Step by step instructions 4 2 3 Special case for Homonuclear Fermion transitions 424 Testing your Ales be Re ee l ee a 4 3 Datafiles for diatomic species partition functions calculations 4 3 1 Approximations considered in the population routine 455 File sirmetfe in ie oe eee eee ee ed 4 4 Building spectral datafiles for other transitions 4 4 1 Atomic discrete and continuum transitions 4 4 2 Linear polyatomic discrete transitions 4 4 3 Molecular continuum transitions 4 5 Linking new spectral datafiles to the SPARTAN database 26 SUMMAT eh 7 A References for the SPARTAN Spectral Database H nl London Factors B L Applied approx
31. The structure of the SPARTAN code is summarized in Fig 3 1 Results GUI i Excitation Radiative Inputs i r L Outputs Module Module Spectroscopic Database Figure 3 1 Structure of the SPARTAN Code lWith a small caveat in the sense that some continuum transitions modeled by semi empirical expressions proposed by several authors are hard coded instead of being in a text file 3 2 Summary of the capabilities of the SPARTAN code 31 3 2 Summary of the capabilities of the SPARTAN code The SPARTAN code has been constructed with the aim of implementing the most generalized physical models possible trying to avoid the necessity of resorting to any approximation of any kind specifically when it comes to the description of the thermodynamic state of the gas plasma A summary of the physical models and capabilities of the SPARTAN code is summarized below e Simulation of discrete and continuum radiative transitions from Atomic Diatomic and Linear Triatomic molecules e Simulation of photoionization photodissociation photodetachment and Bremstrahlung transitions for atomic when applicable and di atomic species Global or level specific cross sections can be consid ered e Voigt lineshapes for discrete spectra including Doppler collisional resonance and van der Waals Stark broadening is not considered at this point lack
32. a b or c are of the upmost importance for a particular problem and adapt the routine to the proposed needs It will be shown that such improvements allow considering the line by line method as a credible and accurate approach accessible to modern day computational tools 2 A sample case study simulation of atmospheric entry radiative processes As discussed in the previous section the simulation of atmospheric entry radiative processes represents a typical intricate case for radiative transfer calculations Additionally to the large array of radiative processes encountered behind a shock wave one has to cope with the presence of large gradients in the macroscopic properties of the flow temperature species number densities etc This means that numerical grids for radiative transfer calculations have to account for a large number of spatial cells 107 10 Such spatial grid sizes coupled to the large spectral grid of each spatial cell means that the computational overheads of this type of simulations may quickly become unaffordable As an example of the typical spectra obtained in such conditions two sample calculations of the spectral emission of an atmospheric pressure 97 CO 3 N Martian type mixture in thermochemical equilibrium at 5000 and 10 000 K are presented in Fig 1 Two aspects are immediately obvious when examining the calculated spectra Firstly the spectral features differ sensibly depending on the plasma temperat
33. bottom one to chose the level of accuracy with the associated memory and computational overheads when carrying spectral calculations 3 4 Deployment of adaptive spectral grids in radiative transfer calculations Handling adaptive spectral grids in radiative transfer calculations poses some additional challenges compared to fixed width spectral grids If radiative transfer models such as Monte Carlo or ray tracing methods are to be utilized this is not an issue as such methods typically probe a narrow spectral region at a time Thus one can straightforwardly convert the considered adaptive grid region to a high resolution local spectral grid This is no longer the case if radiation transfer is treated explicitly accounting for the overall spectrum Here a problem arises from the fact that depending on the local conditions of the plasma the spectral grids will differ As an example the gradients of the spectral grids for the simulated spectra presented in Fig 1 and the gradient for an additional spectral grid for the same Martian type plasma in equilibrium conditions at 1000 K are presented in Fig 8 From the analysis of Fig 8 one may conclude that although the spectral grids may be rather alike grids for the CO N gt plasma at 5000 and 10 000 K they are not exactly equal owing to different line intensities calculated explicitly or not depending on the plasma temperature owing to different Doppler broadening widths etc For the 10
34. bound levels and contribute weakly to the overall electronic state partition function These levels have finite lifetimes as they can spontaneously dissociate through quantum tunneling A recap of the method that can be used for determining these states lifetimes 84 Potential Energies and Wavefunctions Reconstruction 0 1 1 1 1 1 1 1 1 0 5 1 1 5 2 2 5 3 3 5 4 4 5 5 5 5 Internuclear Distance A Figure C 1 N X Potential curves for J 0 100 150 200 250 and 275 from bottom to top and a discussion on how these levels have to be accounted thermodynamically is presented in Ref 74 C 1 2 Expressions for Radiative transition probabilities Solving the radial Schrodinger equation over reconstructed potential curves yields a set of predicted quantum vibrational levels and their associated ener gies and wavefunctions The calculated values for the vibrational levels energies and wavefunctions may be utilized for calculating complete sets of radiative transition probabilities the most paramount being the vibrational Einstein co efficients who may then be supplied to the spectral database of the SPAR TAN code An example of recalculated levels and wavefunctions for the CN Violet transition is presented in Fig C 2 The expressions for several radiative quantities that are calculated by the RKR_SCH routine with the input of recalculated rotationless potentials for the
35. database may be a challeng ing task from a practical point of view this poses no particular problem from a numerical point of view In fact the problem can be solved efficiently using vector programming taking advantage of the capabilities of the MATLAB lan guage However the convolution of a large number of lines with a Voigt profile is a very intensive operation Therefore great care was exerted when developing the line convolution routine in order to achieve the shortest calculation times alongside with a maximal precision of the calculated lineshapes Furthermore the program routine has been built so as to allow the user to adjust a large ar ray of parameters defining among others the number of points of the calculated lineshapes Still the ratio of the time needed for the lines convolution and the time needed for building up the line database is higher than 100 and increases as the precision of the calculated lineshapes is increased In short the user can define itself which is the level of precision required in his calculations More accurate lineshapes will require a large spectral grid and larger computation times Less precise calculations will allow defining pre liminary calculations or even allow affordable large scale computations over a fluid dynamics calculation grid Some of the routine parameters can be easily adjusted by users with a limited background on spectral lineshapes others re quire a more extensive knowledge on
36. dependent on a specific method for the calculation of a Voigt profile and Olivero s expression Eq 2 may be straightforwardly replaced by a different expression or algorithm Eqs 7 and 8 will then be calculated over the new high accuracy profile with the small difference that the second derivative in Eq 7 will have to be determined using numerical methods if the high accuracy Voigt profile is nonanalytic As a concluding remark on this subject one may point out that the availability of a significant array of different methods for line by line calculations fosters the need for benchmarking calculations to be carried out in order to examine quantitatively the advantages of each method Currently this is yet to be achieved Several numerical codes have not been released for general use and also one should not forget the inherent difficulties in handling and utilizing several different numerical algorithms by a single user 6 Conclusions An adaptive grid method for radiative transfer calculations in low pressure plasmas has been presented and validated Such method relies on two approaches of paramount importance for its successful application The first approach is the pseudo continuum concept which limits the number of lines which are to be explicitly calculated and the second approach is the method for the calculation of accurate Voigt lineshapes with a minimum number of points One of the major advantages of this model is that t
37. diatomic transitions the procedure of calculating the level energies and transition probabilities is slightly more complex due to the additional degrees of freedom from molecular vibrational and rotational motion As such for each electronic state of a molecule corresponds a set of vibration and rotational levels Following the Born Oppenheimer approximation the electronic vibration and rotational energies may be usually separated so that the total internal wave function of the molecule may be decoupled in three wave functions grouping the electronic vibration and rotation terms Wel x Wid x Wrot 2 6 Line Positions Following Eq 2 6 the total energy of a specific diatomic level is split into an electronic vibrational and rotational term Eev J s Elvi Erot 2 7 with Ea gt Evid gt Erot 2 8 These level energies correspond to the solutions of the Schr dinger equation for an anharmonic oscillator and a distorted rotator represented by a series of polynomial expansions Fev G v Fa J 1 13 17 T e we 93 WeLe veut WeYe ukg Ba J J 1 D UU DY GU 1 2 9 These expressions can be presented in a more compact form replacing the different spectroscopic constants by a Dunham matrix such that X Yiz w 1 2 FUDH 2 10 a j 2 1 Discrete radiation models 15 This formalism is consistent with Zare s effective Hamiltonian 4 The co
38. energy meaning that the radiation spectrum will not have a discrete structure Continuum radiation transitions include 1 Photoionization Radiative recombination reactions AWF hv 4 ACHD 4 e7 AB hy o ABOTDE 4 e7 2 37 2 Photodetachment Photoatachment reactions A hv eo A e7 2 38 3 Photodissociation Dissociative Photoionization Radiative association re actions AB hv AB A B AB hv AB cATIB41 2 39 4 Bremsstrahlung Inverse Bremsstrahlung reactions A e AB e AB e 2 40 4 Ate hu For these reactions which include the emission or absorption of a free elec tron energy conservation allows writing with AF the ionization energy of the atomic or molecular state 1 AE ne 2 41 which means that this kind of radiative transitions will have an en ergy frequeney threshold below which they will not be able to occur 24 Physical Models 2 2 1 Transition intensities The transition intensities are usually expressed through calculated or measured absorption cross sections These allow an immediate calculation of the absorption coefficient taking into account the population for the absorbing states and the calculation of the emission coefficient through detailed balanc ing using the Planck Kirchhoff law For processes 1 3 the expression for the absorption coefficient can be written as a 7 b zu exp
39. energy potential curve Finally the integrals f and g are not defined in their upper limit so that a Gauss quadrature method has to be used in order to determine these functions near the upper limit 1 Once the near equilibrium part of the potential curve has been obtained it is necessary to extrapolate this potential curve up to the dissociation limit In the developed routine we choose to extrapolate the RKR potential using a repulsive potential for shorter internuclear distances and by a Hulburt and Hirschfelder 71 potential for longer internuclear distances This latter potential consists in a two parameter correction term to the usual Morse potential The expressions for such potential take the form C 1 Theory 83 Veep 7 A r C 6 D K 8 Ba r re 26U ve 1 ayfr re C 7 where r is the equilibrium internuclear distance De is the dissociation energy and A p a B and y are fitting parameters that need to be adjusted in order to insure continuity with the RKR potential curve Upon calculation of the overall potential curve the vibrational level energies can be determined by solving the radial Sehr dinger equation for this same potential curve This calculation is carried out for the case of a rotationless state using the method proposed by Balint Kurti 72 with the typographical correction indicated in 1 For the lower part of the potential curve obtained utilizing the RKR method on
40. example the CO 4P txt file has 11 vibrational levels and so and have 11 terms In any case the code will only calculate lines up to the maximum of either and J or the last rotational quantum number which will be for example J 99 for a number of 100 for the calculated rotational levels whichever is lower If you set a limiting v v maz value below the code will automatically crop the JRS and J vectors too Variables code Variable 1 Vo 2 By 3 D 4 H 5 L 6 M T N 8 7 9 YI 10 11 12 Y 13 A 14 Az 15 AJJ 16 y 17 y2 Table 4 5 Correspondence between specific variables in the code and spectral constants 4 2 Building spectral datafiles for diatomic transitions 55 Step 7 Edit the Franck Condon factors In this file you can also define the Franck Condon factors They are displayed in a matrix form M where v defines the upper vibrational levels and v the lower energy vibrational levels The matrix has a size n x n where n max U you do not want to use the Franek Condon factors in the code calculations you can set all the matrix components to 1 But even if you do not use these factors the size of the matrix must be in agreement with the number of vibrational levels defined before The Franck Condon factors are only utilized in the special case of a Corona distribution function calculation for the vibrational levels population
41. high resolution calculations of these line profiles 12 For the simulation of atmospheric entry radiative processes accuracy requirements for radiative transfer calculations are less stringent than for other applications such as remote atmospheric sensing For most of the 112 M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 Table 1 Gridpoints for 5 7 9 and 11 point lineshapes Number of points Selected points Vo V s V 2 W FW 25V 2 5 X x x T x x x x 9 x x x x x 11 x x x x x x An x defines a specific point being accounted for in the given lineshape spectral range except in the VUV and certain IR regions the optical medium is typically optically thin and spatial grids deployed in radiative transfer calculations have short lengths Instead requirements regarding computational overheads are much more stringent For such a case the approximation to the Voigt profile proposed by Whiting 13 and extended by Olivero 14 is particularly adequate as it provides an analytical expression with a claimed precision of 0 02 to the exact profile C2 AVL L 2 25 10 y Cye 42 x 2 0 016C 1 2 e LAR pz 5 Mom 1 4D 05 10 p225 2 with 1 m 2 2 Avy 1 0692 5 4 0 86639A7 4476 V Vo D Ayy i C 1 Av Avy 177 Avy 1 065 0 047 APL 0 058 ArL A72 C Av Avy ATy 1 065 0 047 ATL 0 058
42. higher lying levels may be enough populated that they cannot be discarded In the second case the results should be identical for optically thin gases and plas mas at low resolution This should not be tried for reproducing high resolution experimental spectra or when simulating gases where absorption is important the splitting of a line in multiplets will lead to different absorption values It is reminded that one may also chose to relax the general parameters of the Lineshape calculation routine for potentially similar improvements in the cal culation times E 3 Code Run Times 97 Radiative System ae Calc Time Grid Size s 107 points Lyman 15 11 85 85 0 32 0 98 Werner 14 11 205 210 0 46 0 46 C Phillips 21 11 60 196 10 58 0 32 C Mulliken 21 11 17 49 0 20 0 81 C Deslandres D Azambuja 21 11 48 157 10 48 0 59 C Fox Herzberg 21 11 79 80 0 59 0 86 C Ballik Ramsay 21 11 51 136 10 51 0 33 C Swan 11 11 271 271 1 00 1 00 CN Violet 21 11 38 92 0 37 0 59 CN Red 21 11 315 1175 1 18 246 CO 47 21 11 130 524 066 13 CO Angstrom 21 11 27 46 0 11 0 14 CO 37 21 11 22 40 0 29 0 35 CO Asundi 21 11 67 326 0 55 12 CO Triplet 21 11 146 400 0 73 13 COT Comet Tail 11 11 120 120 0 83 0 83 N 17 21 11 660 1660 151 2 N 27 21 11 125 213 0 73 0 9 N 17 21 11 131 276 0 78 1 NO y 21 11 304 912 1 08 1 8 NO 9 15 11 1113 2093 1 79 2 6 NO 21 11 71 193 10 54 0 89 NO e 21 11 144 563 0 76 14 NO 2 21 11 77
43. in the literature It is then necesary to inquire up to which degree the spectra might be affected by perturbations by another states As discussed briefly in Chapter 2 three levels of detail can be considered when creating modifying the spectral database for a specific diatomic transition 1 Selection of a Dunham matrix providing the expressions for the rovibronic energies for any arbitrary values of v and J according to the expressions of Eq 2 10 2 Selection of a set of vibrationally specific constants B Dy for a given set of vibrational levels v allowing the calculation of the level energies for any arbitrary rotational quantum level J see Eqs 2 11 4 2 Building spectral datafiles for diatomic transitions 47 3 Selection of the energy shifts around a specific rotational quantum number Ji for a given vibrational quantum number v This treatment is necessary for the accounting for rotational perturbation effects and is carried in addition to considering either a set of equilibrium or vibrationally specific spectroscopic constants Usually selecting an appropriate Dunham matrix with up to date spectro scopic constants is sufficient for most general radiative transfer applications For the accurate reproduction of measured spectra with FWHM lt 1A it is some times necessary to supply level specific spectroscopic constants to complement the equilibrium constants Another aspect to take into consideration is the orde
44. lineshape calculation parameters depending on the problem to be solved 2007 Elsevier Ltd All rights reserved PACS 52 25 08 Keywords Line by line simulations Numerical methods Low pressure plasmas 1 Introduction Radiative processes encountered in low pressure plasmas are known to be most accurately simulated using the exact line by line method However such approach typically implies the calculation of up to about a million lines Accounting for the capabilities of modern day computer systems this precludes the widespread use of this exact method in coupled hydrodynamic radiative calculations This kind of issue may for example arise when simulating the radiative properties of atmospheric entry plasmas Atmospheric entry radiative processes typically encompass a wide array of more than 50 bound bound free and free radiative transitions from more than 20 chemical species ranging from the VUV to the IR region This means that about 104 10 discrete lines with the superposition of several continua are to be accounted for 1 It is therefore evident why approximate methods remain so widespread and popular 2 3 as using the exact line by line method remains unaffordable in most applications Fax 351 218462991 E mail address mlinodasilva mail ist utl pt 0022 4073 5 see front matter 2007 Elsevier Ltd All rights reserved doi 10 1016 j jqsrt 2007 03 005 M Lino da Silva Journal of Quantitative Spectro
45. of calculated lines while not simply discarding weaker lines as it is common practice includes accounting for weak lines as an effective continua Firstly we consider two user defined thresholds named threshold and threshold2 From a given line database normalized to the database strongest line emission or absorption coefficient we proceed to define strong lines as lines above the first threshold weak lines as lines between the two thresholds and very weak lines as lines below the second threshold An arbitrary example of strong weak and very weak lines is presented in Fig 2 Here strong lines are explicitly calculated using a Voigt lineshape and the very weak lines are calculated as a fixed interval continuum The pseudo continua spectral interval is defined as C x FWHM meq parameter contstep Where C is a constant depending on the ratio of the two thresholds typically 20 for a ratio of 1000 Regarding the weak lines a distinction is made depending on whether the weak line is covered by a strong line or not See Fig 2 A weak line is considered to be covered by a strong line if its line center is a distance of less than C x FWHM parameter dnu C depends on the ratio of the two thresholds and may also be typically set as equal to 20 for a ratio of 1000 The pseudo continua is straightforwardly calculated adding the intensities of the weak and very weak lines which fall inside each of its intervals It is chosen to account for weak and very
46. presented in Fig B 1 Branches Line Strengths A U Figure B 1 Rotational line strengths of the different branches of a 35 lt II transition for a Boltzmann distribution of the rotational levels at a characteristic temperature of 300 K The ratio of the peak intensities between the second and first group of ro tational branches is 3 3 for this case For a 2000 K temperature the same calculation results in a 15 8 ratio Therefore omitting such weaker rotational branches for spectral calculations of high temperature gases has in most cases a negligible effect on the simulated spectra Table B 2 lists the weaker rotational branches that can be neglected for 74 H nl London Factors doublet and triplet transitions 2A 2 A SA 3 A PPa 9 9812 RQa Q z 2 Poy SRa TRsi Riz 5 Rai P NP Table B 2 Weak rotational branches for doublet and triplet transitions B 2 First Rotational Lines intensities For J lt A S H nl London Factors have to be calculated taking into ac count a less than 25 1 dimensional space For such so called First Rotational Lines the usual expressions for the H nl London Factors do not apply and specific expressions proposed by Schadee 61 have to be considered These spe cific expressions are accounted for in the SPARTAN code for the corresponding first rotational lines B 3 Expressions for the intermediary a b case H nl L
47. spectral calculation for the CN Violet System one molecule at an equilibrium temperature of 5000K is presented in Fig 3 The parameters have been chosen such as thresholdl 10 threshold2 10 contstep 20 x FWHM meq and dnu 20 x FWHM Here it can be seen that the strong lines for diagonal transitions are selected to be calculated according to the line by line method whereas the nondiagonal weak lines are selected to be calculated as a pseudo continuum This is a typical trend of the molecular spectra calculated according to this method 3 2 Calculation of individual Voigt line profiles The method for calculating the selected strong and weak lines according to the line by line approach will be discussed in this section The basic approach for the calculation of a Voigt lineshape will be firstly presented followed by the presentation of a method for accurately calculating such lineshapes with a minimum number of points using an adaptive spectral grid The convolution of collisional and Doppler broadening processes respectively defined by a Lorentz and a Gauss line profile yields a so called Voigt profile U x x 8 g x Av in2 f expl x In 2 Avzl 0 d v yo 1 m 0 Such a line profile cannot be solved to an analytical expression and several approximated methods have been proposed for carrying such calculations 10 11 More recently Fourier transform methods have also been proposed for
48. spqeT 66 Modifying the Code Spectral Database The different possible labels are summarized in Table 4 6 Once you accomplish all these steps you are ready to run the code and simulate radiative transitions of the new species If error messages are displayed when running the code verify if the number of transitions is correct and has been updated in the Inputs txt and in the Database txt files If the code runs normally but does not calculate a simulated spectrum of the species check if the number density of the species in the Inputs txt file has been given a value different from zero 4 6 Summary At the outset of this section one should be capable to fully customize the SPARTAN code and update its spectral database In practice though it is likely that a great deal of trial and error should be expected from someone not famil iarized with the code Furthermore the disparity of values that are found when updating a spectroscopic database from the Einstein coefficients with powers up to 105 to higher order polynomial corrections to level energies with powers down to 10710 and below generally means that most typos will be fatal and will make the program crash As with the general topic of reproducing exper imental spectra through numerical models familiarizing oneself and updating the database of the SPARTAN code could well become a so called labor of love at first Yet as the saying goes there is no gain withou
49. such a long length For narrower slabs 1 km and 0 1 m the errors incurred when utilizing coarser lineshapes are significantly lower This brings us to the analysis of case 2 where the temperature gradients and slab lengths are lower Here we can verify that the coarser lineshapes are able to accurately reproduce the radiative transfer behavior even for large extinction percentages with less than 1 of transmitted radiative power Only the lower resolution lineshapes somehow fail at accurately reproducing the radiative transfer features for longer cold slab lengths up to 500 difference to the reference lineshape Regarding high resolution lineshapes a coarser lineshape such as c3w6 suffices for adequately treating the examined problem Table 2 Slabs for sample radiative transfer benchmark calculations for the accuracy test of lineshapes with a different number of points 5 73 points To 20 000K T 1000K Xo Im x 0 1m 1km 10km To 10 000 K T 5000K Xo Im x 1m 10m 1km Each slab has a given homogenous temperature throughout it s specified length 5 Al 1 2 s 2 T T T i 1 5 M Lino da Silva Journal of Quantitative Spectroscopy dt Radiative Transfer 108 2007 106 125 119 1 L 0 5 0 L L 1 if L L 953 2 953 3 953 4 953 5 953 6 953 7 953 8 953 9 954 954 1 954 2 Wavelength A ad 0 08 Radiative Power VV m3sr cm1 N o A A 0 06 0 04 Radiative
50. the lineshape calculation methods and are recommended to be kept at their default values The routine for the calculation of the computed spectra lineshapes as well as the overall user defined parameters are described below The user defined parameters can be adjusted editing the file Lineshape txt in the INPUTS di rectory The Lineshape m routine accepts as an input a matrix of size n x5 containing the parameters for the calculation of n lines wavenumber 7 emission coefficient Ey absorption coefficient 2 Lorentz full width at half maximum FWHM A r and Doppler full width at half maximum FWHM Avg The routine then outputs the overall spectra wavenumber emission and absorption coefficients The routine works as follows 1 Separates the lines among three different categories strong weak and very weak according to the parameters LinPar threshold1 threshold21 2 Defines which lines are to be explicitly calculated and which are to be calculated as a pseudo continuum strong lines are calculated explic itly very weak lines are calculated as a pseudo continuum weak lines are calculated explicitly if they are not covered by a strong line If the pseudo continuum is not selected LinPar contmethod 0 we have two possibilities a The emission and absorption intensities of the explicitly calculated lines are multiplied by a constant factor to account for the intensities of the discarded lines and enforce ene
51. upper and lower states and a supplied electronic transition moment Re r are C 1 Theory 85 v 0 1 5 2 2 Internuclear Distance A Figure C 2 Recalculated Wavefunctions for the v 0 20 levels of the upper and lower states of the CN Violet system presented in Eqs C 9 2 Wy r by C 9a 1 Wy r n viv C 9b i Tor Jhor dr R f pu Reh ir C 9e 640473 2 Soars puto Aron 3hc3 2 i 2 t 77 2 2 026 x 107603 2 OA A ae 2 t A 8z7m v 2 m ron 2 abs 5 Oye A R in C 9 v v 3he2 2 e i e 2 Y 2 3 0376 x 10 8 yy NEM ye 2 o 2 DVA pab o qr C 9f ore 2 D A 1 3 Numerical Routines Description The RKR_SCH suite consists of three similar routines e RKR_SCH routine for the calculation of radiative transition probabilities e RKR_SCH_PC routine for the calculation of potential curves vibrational energy levels and wavefunctions 86 Potential Energies and Wavefunctions Reconstruction RKR_SCH_PC_ROT routine for the calculation of rovibrational energy levels and wavefunctions including quasi bound states The usage for each routine is gt gt qvv rvv Revv_2 Avv fvv RKR_SCH FILE txt gt gt r_RKR V_RKR G_v Te ri A E RKR_SCH_PC FILE txt r V
52. weak lines as a continuum instead of simply discarding them because if a very large number of lines are present their summed contribution to the overall spectra might not be negligible Typically only about 10 of the overall lines are calculated explicitly 10 Strong Weak Very Weak 10 x 10 O 5 46 10 Il 5 F bs pre 10 su ons b 7 N 1072 0 R 1 R R 1 i R 1 99 1 995 2 2 005 2 01 4 VVavenumber 10 Fig 2 Sample strong weak and very weak lines with threshold1 10 and threshold2 10 M Lino da Silva Journal of Quantitative Spectroscopy dt Radiative Transfer 108 2007 106 125 111 15 10 m A vv Tv Tv Tv Discrete Pseudo Continuum E 20 5 10 r o 5 107 So io Q o o 40 30 b E W io H H II Il 14 Il l 0 5 1 15 2 2 5 3 3 5 4 4 5 4 VVavenumber cm 1 x 10 Fig 3 Spectral simulation of the molecular CN Violet System Av 0 utilizing the hybrid line by line pseudo continuum approach The pseudo continuum spectrum encompassing weak and very weak lines is plotted in black the discrete line by line spectrum encompassing strong and weak lines is plotted in light grey As expected the pseudo continuum spectrum remains well below the discrete line by line spectrum An example of a
53. with r in A and V in cm optional parameters use otherwise if you wish to supply a potential function and just solve the radial Schrodinger equation gt gt RKR_SCH_PC_ROT Level spectrocopic constants 0 1 Supplied potential v_max J_ini J_step J_final Instead of returning values to the invoking function the RKR_SCH_PC_ROT saves the variables r_i rotationless potential abcissae in Bohrs radius Ve_ri_J rotationless potential energy in Hartree V_J Matrix with the energy in Hartrees of the level pairs E o v_minmax the minimum and maximum vibrational levels for each rotational level to the MATLAB file v_J_levels If you want to supply your own rotationless potential create a MATLAB file with the variables r in A and V in cm set the variable 0 1 to and declare the file name in Supplied potential Otherwise set the variable 0 1 to 1 and set the input variable to empty C 1 3 1 Input files structure The structure of the RKR_SCH routines will not be described in detail here Instead the user may consult the EXAMPLES folder where the NO XX txt and N2 X txt files provide examples of input files for respectively the routine RKR_SCH and the routines RKR_SCH_PC RKR_SCH_PC_ROT The MATLAB file N2X DPF mat contains a high resolution potential for the ground electronic state of N which may be supplied to the RKR_SCH_PC_ROT function directly referencing the filename at function inp
54. 00 K temperature the spectral grid differs significantly as different radiative transitions account for the main radiative features of the plasma For such a low temperature radiation is mainly 116 M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 r r r r r 0 05 0 06 Lorentz da Lorentz 0 04 Doppler 0 02 0 01 0 i i 0 01 2 200 100 0 100 200 107 101 10 2 D 0 05 0 06 g gt Lorentz Lorentz Q 0 03 2 0 04 A 7 Doppler 0 02 0 01 8 T o E 0 0 01 2 5 200 100 0 100 200 10 10 102 2 r i r 0 05 0 06 Lorentz da Lorentz 0 04 Doppler 0 02 0 01 0 0 01 200 100 0 100 200 109 101 10 Distance to Line Center Fig 6 Plot of high resolution Lorentz lineshapes left 21 37 and 73 points from top to bottom AV 5cm7 and plot of the normalized difference of Lorentz full line Av 5cm 1 and Doppler dotted line Ay 5cm 1 lineshapes to the exact lineshapes right 21 37 and 73 points from top to bottom 0 06 0 05 10 Lorentz 0 04 10 Doppler 0 03 0 02 0 01 Normalized Difference 0 01 0 02 10 107 10 10 Normalized Distance to Line Center Fig 7 Normalized difference between two Voigt lineshapes with Av 1 1Ayr and Av 1 1Arp and an initial Voigt lin
55. 1 Table 4 Radiative transfer results for different lineshapes case 2 see Table 2 Case 2 To 10 000 T 5000 Io Xo 1m 1 x lm Ti x 10m Ti x lkm c8wl6 24 9190 9 1768 2 3393 0 1907 c6w12 24 8430 9 1584 2 3591 0 1963 c3w6 24 6754 9 1951 2 5884 0 1657 111 26 4886 10 5922 2 4937 0 6705 19 27 4778 11 4270 2 7297 0 8839 17 24 6807 9 5831 3 2066 0 9865 15 34 3964 19 4701 9 6988 0 6524 I Io c8wl6 36 83 9 39 0 77 c6w12 36 87 9 50 0 79 c3vv6 37 26 10 49 0 67 111 39 99 9 41 2 53 19 41 59 9 93 3 22 17 38 83 12 99 4 00 15 56 61 28 20 1 90 Ii KHn hires c8wl6 100 00 100 00 100 00 c6w12 99 80 100 85 102 91 c3w6 100 20 110 65 86 89 111 115 42 106 60 351 51 19 124 52 116 69 463 42 17 104 43 137 08 517 18 15 212 17 414 60 342 05 The table successively reports the radiative intensity exiting from the hot 79 and cold Z1 slabs the percentage of the incoming hot slab radiative intensity transmitted by the cold slab 7 7o and the differences in transmissivity compared to high resolution calculations 1 KH hires Table 5 Slabs for sample radiative transfer benchmark calculations for testing the accuracy of the pseudo continuum approach sampling different values for the threshold and threshold2 To 5000K T 1000K xo 1m x 100km Each slab has a given homogenous temperature throughout its specified length radiative transfer calculations Rec
56. 1 Be in cm ae in em 1 Electronic level dissociation energy Do in em 1 with origin of the ref erential at E v 0 Electronic level degeneracy ge 2 25 1 The list of vmar 1 vibrational energy levels in em 1 The more the added upper molecular electronic levels the more accurate will the calculated partition functions be at higher temperatures 4 4 Building spectral datafiles for other transitions 59 4 4 Building spectral datafiles for other transi tions This section describes how the other type of radiative transitions can be inserted in the SPARTAN code 4 4 1 Atomic discrete and continuum transitions The SPARTAN code adopts a general file structure for discrete bound bound transitions and the file structure from TOPbase for photoionization bound free transitions It is important to note that most databases for Atomic radiative transitions adopt different lists of Atomic levels whose more energetic levels are usually calculated with differing numerical approaches Therefore an important aspect of the atomic data implemented in the spectral database of the SPARTAN code is that each transition file will have its associated level energies file For example the Atomic Nitrogen discrete transitions file N ATO txt will have an associated level energies file N ATO LEV txt whereas the Atomic Nitrogen photoionization cross sections file N PI txt will have another associated level
57. 1 1 1 f f f f f 0 5 10 15 20 25 30 35 40 45 50 Wall Point 0 03 run 1 run 2 run 3 0 025 run 4 0 02 4 s S 0 015 4 D 2 0 01 4 0 005 4 0 f 1 1 0 5 10 15 20 25 30 35 40 45 50 Wall Point Figure 10 Spectrally interated vvall fluxes for the 4 runs top and integrated wall fluxes in the VUV Visible region cutoff at 1 um Table 3 Calculation times for the 4 runs VV VVV 1075 1076 VV VVV 1072 1073 c w 13 25 3 d 4 5 d 94GB 1 5d 3 d 52GB c w 7 14 1 5 d 2 2 d 41GB 0 9 d 1 9 d 28GB c line center points w line wings points W Weak lines threshold VW Very Weak lines threshold shown calculation times for radiative field radiative transfer in days and spectral field grid size in GB REFERENCES Omaly P and Marraffa L TC3 Update of the Axially Symmetric Testcase for High Temperature Gas Radia tion Prediction in Mars Atmosphere Entry TC3 3 Charbonnier Analysis of the results for Presented at the 1st International Workshop on Radia tion of High Temperature Gas in Planetary Atmosphere Entry ESA SP533 Proceedings of the First Interna tional Workshop on Radiation of High Temperature Gases in Atmospheric Entry 8 10 Oct 2003 Lisbon Portugal pp 145 159 Omaly P Dieudonne W and Spel M Synthesis and Analysis for Test Case 3 ESA SP583 Proceedings of the First International Workshop on Radiation o
58. 2 42 provided that level dependent o 77 spectral absorption cross sections are available Here the factor 1 exp b allows for the subtraction of stimulated emission processes yielding the net absorption coefficient In certain cases only values for the global absorption cross sections tabulated at different tabulated temperatures T are available In this case the global absorption coefficient for an interpolated temperature T is written as a T r h exp 2 243 For free free Bremsstrahlung transitions process 4 the absorption cross section is written as a 7 N No r T m 2 44 2 2 1 1 Gaunt factors A quantum correction for the classical absorption cross section is usually proposed in the form of a so called Gaunt factor g which will depend on the gas temperature T and the transition frequency v These corrective factors stem from the analysis of high density stellar plasmas The Gaunt factor is usually close to unity The correction to the classical absorption coefficient is simply expressed as a 7 quantum a V g P Tet 2 45 The current version of the SPARTAN code accounts only for Free Free Gaunt factors keeping the bound free Gaunt factor gy to unity The free free Gaunt factor gr is obtained for Te gt 1 eV 11604 K according to the formulas proposed by Stallcop and Billman 22 who have adjusted the results from Karzas and Latter 23 in numerical fo
59. 937 pp 579 587 Rydberg R Graphische Darstellung Einiger Bbandenspektroskopischer Ergebnisse Z Physik Vol 73 1931 pp 376 385 Klein O Zur Berechnung von Potentialkurven f r Zweiatomige Molek le Mit Hilfe Von Spektraltermen Z Physik Vol 76 1932 pp 226 235 94 BIBLIOGRAPHY 169 Rydberg R Uber Einige Potentialkurven des Quecksilberhydrids Z Physik Vol 80 1933 pp 514 524 70 Rees A L G The Calculation of the Potential Energy Curves From Band Spectroscopy Data Proc Phys Soc London Vol 59 1947 pp 998 1008 71 Hulburt H M and Hirschfelder J O Potential Energy Functions for Diatomic Molecules J Chem Phys Vol 9 1941 pp 61 69 72 Balint Kurti G G Dixon R N and Marston C C Grid Methods for Solving the Schrodinger Equation and Time Dependent Quantum Dynam ics of Molecular Photofragmentation and Reactive Scattering Processes Int Reviews in Phys Chem Vol 11 No 2 1992 pp 317 344 73 Le Roy R J Huang Y and Jary C An Accurate Analytic Potential Func tion for Ground State N From a Direct Potential Fit Analysis of Spectro scopic Data J Chem Phys Vol 125 No 16 2006 pp 4310 4322 74 Lino da Silva M Loureiro J and Guerra V Rotational Nonequilibrium in State Resolved Models for Shock Heated Flows Chem Phys Vol 398 2012 pp 96 103 Appendix E Code Versions Log Table E 1
60. Al 0 2 An arbitrary L S Coupling 5 AS 0 AS 0 AS 0 6 AL 0 1 AL 0 AJ AL 0 1 2 except 0 0 except 0 0 0 1 Table 2 1 Selection rules for atomic transitions 2 1 Discrete radiation models 11 The overall rotational lines of for the transitions between upper e v J and lower e7 v J levels which share the same AN and AJ are written as PNA a where and j stand for the index of the upper and lower states multiplet components Since for an electric dipolar transition we have i j the branches with AJ AN are called main branches whereas the branches with AJ 4 AN are called satellite branches The nomenclature for the different branches is summarized in Table 2 2 AJ 1 NOP QRS T 0 1 AN 3 2 1 0 1 2 3 Table 2 2 Nomenclature for the different rotational branches For diatomic transitions three levels of coupling rules have to be accounted for 1 General selection rules 2 Selection rules for molecular angular momentum coupling H nd Case a 3 Selection rules for molecular angular momentum coupling H nd Case b We define the notations e and f which allow identifying a level parity For a integer rotational quantum number J we note e as the parity level 1 7 and f the parity level 1 7 For a half integer rotational quantum number J we note e as the parity level 1 7 2 and f the parity level The general selection rules for diatomic di
61. Energy Levels Using the RKR Method Chem Phys Vol 348 2008 pp 187 194 35 Prasad C V V and Bernath P F Fourier Transform Jet Emission Spec troscopy of the 2 X X Transition of CN Journal of Molecular Spectroscopy Vol 156 1992 pp 327 340 36 http physics nist gov PhysRefData ASD index html 37 http vizier u strasbg fr topbase topbase html 38 Cooper J W and Martin J B Electron Photodetachment From Ions and Elastic Collision Cross Sections for O C Cl and F Physical Review Vol 126 No 4 1962 pp 1482 1488 39 Hwang W and Kim Y K New Model for Electron Impact Ionization Cross Sections of Molecules Journal of Chemical Physics Vol 104 No 8 1996 pp 2956 2966 40 Romanov G S Stankevich Y A Stanchits L K and Stepanov K L Thermodynamic and Optical Properties of Gases in a Wide Range of Parameters Int J Heat Mass Transfer Vol 38 No 3 1995 pp 545 556 92 BIBLIOGRAPHY 41 Masuoka T and Nakamura E Single Double and Triple Photoionization Cross Sections of Carbon Monoxide CO and Ionic Frag mentation of COF Cos and Coz Physical Review A Vol 48 No 6 1993 pp 4379 4389 42 Fennely J A and Torr D G Photoionization and Photoabsorption Cross Sections of O N Oz and N for Aeronomic Calculations Atomic Data and Nuclear Data Tables Vol 51 1992 pp 321 363 43 Masuoka T
62. Gaussian profile such that v x L x 2 33 Avr In2 o exp zx ln 2 AVES ae Avg Y r p E Ar 119 2 35 This profile cannot be analytically calculated and an approximate expression needs to be used Here we select the expression proposed by Whiting 20 41n2D C2 Ce ATE 0 4D2 25 10 01 1 1 2 36 0 0 c Z rar 2 36 vvith 7 To m Avy 1 Avy 2 av 1 42 C AT Av Avy 1 065 0 447224 0 058324 Av a 42 Avy 1 065 0 447224 0 058524 This expression has been critically assessed by Olivero 21 who estimated a precision with an accuracy of about 1 minimum Olivero then proposed a 5with 41n 2 replaced by 2 772 for numerical efficiency reasons 2 2 Continuum radiation models 23 modification to the Voigt linewidth parameter improving the accuracy down to 0 02 1 Avy 5 1069247 0 86639473 In the SPARTAN code we retain this analytical expression over the exact convolution expression from Eq 2 35 as it is significantly more computationally efficient specially in view of the sheer number of lines that have to be calculated for the production of detailed spectra over a broad range 2 2 Continuum radiation models Continuum transitions are transitions in which one or both of the up per lower states do not have a discrete
63. H H Table 2 4 Spectroscopic constants correspondence for Brown s and Zare s Hamiltonians Einstein coefficient A Here we will describe the method for the calculation of such parameter Although we admit the separability of the electronic vibrational and rota tional modes of a molecule according to the Born Oppenheimer approximation electronic and vibrational configurations are intrinsically connected through the different potential curves and there is also a coupling between the molecular rotational motion and the electronic cloud of the molecule due to the electrons spin movement The transition probability Au can then be decomposed as a product ACP A v af The vibronic component A can then be expressed as a function of the vi bronic transition moment R using the following expression in atomic units using wavenumber 7 units over frequency v units Malt e e u 64 20 2 m 0 A A 1 3hc3 2 0 A 2 As the vibronic transition moment r cannot usually be resolved for each multiplet transition an average transition moment value 3 is rather used We then have YT 7 2 ine Se 217 2 t ar A 28 1 The vibronic transition moment is calculated using the electronic transition moment Re r which is taken from the literature and the upper and lower The term 47e9 has an unit value i
64. Kerr C M L and Milton D J A Determination of Fundamental Zeeman Parameters for the OH Radical Mol Phys Vol 36 No 2 1978 pp 553 582 Brown J M Colbourn E A Watson J K G and Wayne F D An Effective Hamiltonian for Diatomic Molecules J Mol Spectrosc Vol 74 1979 pp 294 318 90 BIBLIOGRAPHY 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Lefebvre Brion H and Field R W Perturbations in the Spectra of Di atomic Molecules Academic Press Inc London 1986 Taine J A Line by Line Calculation of Low Resolution Radiative Prop erties of 2 Transparent Nonisothermal Gases Mixtures up to 3000 K JQSRT Vol 30 No 4 1983 pp 371 379 Gamache R R Rothman L S Extension of the HITRAN Database to Non LTE Applications JQSRT Vol 48 1992 No 5 6 pp 519 525 Scutaru D Etudes Th orique et Exp rimentale de TAbsorption In frarouge par CO Haute Temp rature Application des Mod les de Ray onnement des Gaz Ph D Thesis in French Laboratoire d Energ tique Mol culaire et Macroscopique Combustion E M2 C Ecole Centrale de Paris 1994 Rothman L S Hawkins R L Watson R B and Gamache R R Energy Levels Intensities and Linewidths of Atmospheric Carbon Dioxide Bands JQSRT Vol 48 No 5 6 1992 pp 537 566 Wiese W L Fuhr J R and Deters T M
65. Power VV m3sr cm1 i dt 953 2 953 3 953 4 953 5 953 6 953 7 953 8 953 9 954 9541 954 2 Wavelength A Fig 9 Sample radiative transfer calculation for nitrogen atomic radiation utilizing high resolution lineshapes see Table 2 Two sample cases are shown where the radiative features exiting the hot slab and entering the cold slab are plotted in the upper part of the figure and the radiative features exiting the cold slab are presented below Here calculations for case 1 are presented with from top to bottom hot slab exit with xo 1m T 20 000 K cold slab exit with x 1m T 1000K hot slab exit with xo 1m To 20 000 K cold slab exit with x 1 km 7 1000K 4 2 Accuracy of the pseudo continuum approach A similar sample radiative transfer problem has been considered for testing the pseudo continuum approach accounting this time for the Av 0 band of the CO fourth positive system at 1600 A and assuming a constant number density of 3 7 x 1022 part m A radiative transfer case has been devised where only 4 of the incoming radiation is retrieved The two slabs are in thermal equilibrium with temperatures of 5000 and 1000 K respectively and are presented in Table 5 Different thresholds for defining strong weak and very weak lines have been chosen 116 212 stands for threshold1 10 and threshold2 10 7 113126 stands for threshold1 102 and threshold2 10 and so on 120
66. SPARTAN 2 5 User s Manual Mario Lino da Silva Bruno Lopez and Susana Espinho July 17 2013 Contents Introduction 1 Getting Started 1 1 Launching the SPARTAN code 1 1 1 Running the Graphical User Interface GUI of the SPAR TAN COGS S t a ot Oe aa eye A eo 1 1 2 How the Graphical User Interface works 1 1 3 Other user defined parameters 1 1 4 Running the SPARTAN code without the Graphical User oo x eS eg ee ao eke kos ae oe ee A 1 2 Recorded data v s lt 0 2 40 624 658 ee Lo Sample comibar son io ek de ya eR ae ee wR s 1 4 Stepping alittle further Physical Models 2 1 Di erete radiation models lt s aai dk ek Be gox Rob asa 211 Selection rules he x Avera DAYA 2 1 2 Line positions and intensities 2 1 3 Broadening mechanisms 2 2 Continuum radiation models 2 2 Transition Intensities s bb ch cee eh Az r ee ps 222 Special Cases lt coke B s B bes 2 3 Generalized Kirchhoff Planck Law for radiative transfer 2324 TDisecrete transitions do ee eG 2 3 2 Photoionization transitions 2 3 3 Photodissociation transitions 2 3 4 Bremsstrahlung transitions Detailed Description of the Code ntrod etl i se ba eR bs 3 2
67. Topbase level Qa 245 Third Positive beat a I 2 10 Angstrom BIn AlN 2 10 molecular photoionization Triplet 2 2 10 10 Asundi a 311 a3T1 10 10 total Qa CO total Qa CO2 613 bands CO2 total Qa Earth like molecular systems molecular photodissociation O2 Soehumann Runge BS T x x 10 10 Oz T dependent Qa Atomic lines atomic photodetachment species database model electronic total Qa levels Oo total Qa C NIST NIST z 272 O NIST NIST 377 ing over 90 of calculation time It is therefore nec 10 2 Strong essary to preallocate the different radiative data using a _ Weak 10 F 5 stack system For such a purpose a 28GB ramdrive has Very Wonk been implemented in the machine 5 D m bi 1 s Ni Li 1 1 Figure 7 Multicore machine for radiative transfer calcu lations 3 CONVERGENCE STUDIES As discussed in Lino da Silva 2007 the lineshape rou tine of the SPARTAN code is fully parametrical in the sense that it allovvs the user to select the accuracy of the calculated lineshapes as well as the number of lines to be calculated explicitly and the lines to be added as a pseudo continuum Two sets of thresholds separate strong weak and very weak lines Strong lines are then calculated as a Voigt profile Weak lines W are calcu lated as a Voigt profile if they are not covered by a strong line and finally Very Weak VW lines alway
68. a 7 over a variable width spectral grid Glue m is the routine which superposes the different individual spectra to yield a global spectrum with a variable spectral grid GUInterface m GuiFunctions m GUITrace m TOVVrite m Are func tions specific to the GUI of the SPARTAN code IOWrite m updates the In puts txt file according to the user inputs from the GUI Integrate m Integrates the individual radiative systems spectral dependent emission coefficients ez to yield to their individual and the total radiative power in VV m5 Fig 3 2 presents the flowchart of the SPARTAN code S gt Bremstr gt PhoDet Database txt gt Photot v LoadDatabase l DataAtomict s Excitet gt Atomict v v Spectre gt DataAtomic3 L l Ekxcite3 s Atomic3 l Convolve gt Lineshape gt Glue RESULTS N r x x x lORead DataAtomic2 gt Excite2 gt Atomic2 L y Photo2 Inputs txt PhotoT gt 4 DATABASE Figure 3 2 Flowchart of the SPARTAN code 34 Detailed Description of the Code 3 5 Lineshape calculation routine This lineshape calculation routine may be considered as the core of the application Indeed even though setting up a line
69. a cold slab at 5000 K case 2 The different slab geometries are presented in Table 2 Fig 9 presents the radiative power exiting from slabs 0 and 1 for two sample conditions Low resolution lineshapes have the index 5 7 9 and 11 for the 5 7 9 and 11 point lineshapes respectively High resolution lineshapes have the index c3w6 c6w12 and c8w16 for 21 point 7 core and 14 wings points 39 point 13 core and 26 wings points and 51 point 17 core and 34 wings points lineshapes respectively The obtained results for different absorption slab lengths are presented in Tables 3 and 4 From the analysis of case 1 one may verify that for very large lengths only the most precise lineshapes are able to accurately reproduce the radiative features at the exit of slab 1 Regarding radiative transfer in a 10 km slab only 2 5 of the radiative power incoming from slab 0 is retrieved see calculation results with the c8w16 lineshape Compared to these results the 21 and 39 point lineshapes c6w12 and c3w6 strongly underpredict the outgoing radiation 20 and 0 of the outgoing radiation respectively On the other hand low resolution lineshapes 5 77 9 and 11 overpredict the emitted radiation up to 1650 more This is not surprising as the lineshapes strongly differ among the two slabs mainly due to Doppler effects the lineshape being narrower in the cold slab and as only a very little portion of the wings radiation will be transmitted over
70. ad Az 1 m ATV Ar Az Avy 1 065 0 447 0 058524 Av AD Avy Ca Avy 1 065 0 447322 0 05852 Vv C Journal of Quantitative Spectroscopy amp Radiative Transfer Journal of Quantitative Spectroscopy amp ELSEVIER Radiative Transfer 108 2007 106 125 www elsevier com locate jqsrt An adaptive line by line statistical model for fast and accurate spectral simulations in low pressure plasmas M Lino da Silva Centro de Fisica dos Plasmas Instituto Superior T cnico Av Rovisco Pais 1049 001 Lisboa Portugal Received 18 October 2006 received in revised form 12 February 2007 accepted 3 March 2007 Abstract An adaptive grid method is proposed for the simulation of low pressure plasma radiation The method relies on two complementary approaches which significantly reduce calculation times and the size of the obtained grids Weak lines are calculated as a so called pseudo continuum hence reducing the number of calculated lines and a numerical algorithm has been developed for accurately calculating Voigt lineshapes using a minimum number of points The method is fully user parametric allowing the choice of privileging calculation efficiency or alternatively privileging the accuracy of the computed spectra Sample radiative transfer calculations are presented which show the efficiency of the method also providing some guidelines on how to define
71. alculations Partition functions for diatomic species are calculated in the Excite2 m routine allowing the calculation of the level populations for any arbitrary set of characteristic temperatures Trot Tyin Teze We firstly express the population of a specific vibronic level as the prod uct of the populations for each degree of freedom electronic vibrational and rotational Ne New Nead New 7 MEN Na Ney 4 1 In equilibrium conditions the level populations are related to the partition functions for the different degrees of freedom according to U Nne v J Ne rot 4 2a z uid N 24 QE Qh o v J New Qib Qrot 4 2b N m e u ne v J 6 Quib rot N e v J e v J Qrot 4 2c gt e v J New 2 Qrot 3Please ignore this warning for the Lyman and Werner transitions and the C2 Swan Bands The warning is due to the extrapolation of high order polynomial coefficients but the population of these higher lying levels is negligible and as such this occurrence has no visible bearing on the reproduced spectra which remains physically consistent 4 3 Datafiles for diatomic species partition functions calculations 57 And the exact individual partition functions for each mode may be written as M he Qe Je EXP F L ro 2 25 1 exp 7 To 4 3a h Jv EXP Bl yin 2557 aa he J Ql
72. ameter is dependent on parameter LinPar threshold1 20 is recommended for LinPar threshold1 1e3 higher values are recommended for higher thresholds LinPar ksi Parameter which defines the Lorentz weight parameter defining the boundary between the lineshape core and wings regions see section 3 5 1 or 1 for more details The recommended value for this parameter is 1 8 LinPar beta Parameter which defines the Gauss weight parameter defining the boundary between the lineshape core and wings regions see section 3 5 1 or 1 for more details The recommended value for this parameter is 1 LinPar ksil Parameter which defines the Lorentz weight parameter defining the boundary between the lineshape wing and far wings regions see section 3 5 1 or 1 for more details The recommended value for this parameter is 5 8 LinPar betal Parameter which defines the Gauss weight parameter defining the boundary between the lineshape wing and far wings regions see section 3 5 1 or 1 for more details The recommended value for this parameter is 5 8 LinPar num_c Number of points of the lineshape half core region including line maximum Total number of points in the core region of the lineshape 2x LinPar num_c 1 Recommended values from 3 to 6 LinPar num_w Number of points of the lineshape half wing region Total number of points in the wings region of the lineshape 2 x LinPar num_w Recommended values from 6 to 12 3 5 The Lineshape
73. asured spectra The necessary steps are 1 Replacing the Measured txt file in the OUTPUTS folder by the experimental file in A wavelength units 2 Calculat ing the simulated spectra 3 From the command line running Compare ResultTotal shift noise normalize where shift is the experimental spectrum shift in A noise is the experimental spectrum noise arbitrary units and normalise is 1 if the two spectra are to be normalized or 0 if they are to be calculated in absolute intensity units VV m em 1 sr Usage is Compare ResultTotal 3 7 0 1 88 Other Auxiliary Routines AFunction m Convolutes Spectra with a Gaussian apparatus function Usage is lam I_AF AFunction ResultTotal nu ResultTotal I_E width with width the width of the apparatus function in A Test DiatomicFiles m Tests the integrity of diatomic transitions spectral files Usage is Test_DiatomicFiles FILENAME TRANSITION with FILENAME the name of the datafile for verification which should be hosted in the DATABASE directory and TRANSITION is the identifier for the multiplet type of the transition Example for invoking the function Test_DiatomicFiles NO XX txt 2P 2P Bibliography 1 Lino da Silva M Simulation des Propri t s Radiatives du Plasma En tourant un V hicule Traversant une Atmosph re Plan taire 4 Vitesse Hy personique Application 4 la plan te Mars PhD Thesis i
74. at the thresholds can be significantly lowered without largely affecting the accuracy of the obtained results Differences between the more stringent and more relaxed thresholds only amount to 13 Computation times are however divided by a factor of four Nevertheless defining thresholds lower than 10 is not recommended if consistency of the calculated spectra is to be assured Either way it is demonstrated that the pseudo continuum is a very effective method for decreasing molecular spectra calculation times without a significant loss in the accuracy of the obtained results 5 Possible model extension for the simulation of general radiative transfer problems and comparison with other numerical approaches Although the line by line method is routinely utilized in spectral simulations of low pressure plasmas the systematic evaluation of different line by line models has rather been carried in the scope of atmospheric sensing applications and to a lesser extent astrophysical applications This can be easily understood considering the typical radiative problems to be solved in low pressure plasma applications Investigations are usually carried out in the near UV to near IR range 300 900 nm and over short length scales centimeter wise This means that the medium can be usually considered as optically thin thus not requiring stringent M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 12
75. ation carried over the integral of a high reso lution Voigt lineshape 1 or the integral of the selected lineshape for the overall calculations When computing a low resolution profile setting this option to 0 will allow the line peaks will have the correct intensities but the overall line energy will not be accurate Setting this option to 1 will yield less accurate line peak intensities but will allow the correct overall line energy For higher resolution lineshapes LinPar shape 99 it is recommended to keep the value at 0 LinPar contmethod Method used for accounting for the presence of weak lines Setting this option to 1 uses the pseudo continuum method Setting this option to 0 discards the lines not calculated by the Line by Line method LinPar addWeakLines Setting this option to 1 homogeneously adds the discarded lines intensities to the calculated lines intensities TRE H x ousweng Setting this option to O simply discards the lines 38 Detailed Description of the Code LinPar threshold1 Threshold normalized to the maximum line emission intensity OR the maximum line absorption intensity above which a line is selected to be calculated using the Line by Line method Recommended values 105 106 Higher values will have more lines included in the over all calculation yielding more accurate spectra at the cost of larger memory an computation time overheads Note that for consistency on
76. ation of Excited and Rydberg States of Imidogen Radical NH Potential Energy Curves Spectroscopic Constants and Dipole Moment Functions J Chem Phys Vol 126 2007 pp 244302 1 244302 13 Luque J and Crosley D R Transition probabilities in the A X7 X71 Electronic System of OH J chem Phys Vol 109 No 2 1998 pp 439 448 Hwang E S Lipson J B Field R W and Dodd J A Detection of OH X v J via the B X7 2 Transition and Properties of the B X7 State J Phys Chem A Vol 105 2001 pp 6030 6037 Schadee A Theory of First Rotational Lines in Transitions of Diatomic Molecules Astron amp Astrophys Vol 41 1975 pp 203 121 Kovacs I Rotational Structure in the Spectra of Diatomic Molecules Adam Hilger Ltd 1969 Schadee A The Formation of Molecular Lines in the Solar Spectrum Bull Astron Inst Neth Vol 17 No 5 1964 pp 311 357 Arnold J O Whiting E E and Lyle G C Line by Line Calculation of Spectra From Diatomic Molecules and Atoms Assuming a Voigt Line Profile JQSRT Vol 9 1969 pp 775 798 Tatum J B H ln London Factors for 30 3 X Transitions Canadian Journal of Physics Vol 44 1966 pp 2944 2946 Budo A Intensita tsformeln f r die Triplettbanden Mitteilung aus dem Physikalischen Institut der K nigl Ungarischen Universitat f r Technische und Wirtschaftwissenschaften 1
77. avor for the developers and main tainers of the code As such we would be grateful if you would be willing to take back a little of your time and share with us any improvements of the code and or its spectral database so that they can be further distributed among the community of SPARTAN code users This is something that sometimes tends to be overlooked by the academic community as much as it can avoid spurious duplication of efforts by different research teams As such any sort of feedback would be welcomed by the team who also man ages an online repository of spectral data the GASPAR database available at http esther ist utl pt gaspar If you wish to have any spectroscopic data added to this ever growing open access repository with nearly 1 000 dif ferent sets of data feel free to contact us The latest version of the SPARTAN code is maintained at the following address http esther ist utl pt spartan Chapter 1 Getting Started This Chapter describes how the SPARTAN code can be quickly used by first time users relying on the provided spectroscopic database and using the default line calculation settings 4 Getting Started 1 1 Launching the SPARTAN code Upon starting MATLAB the user should select the SPARTAN code direc tory as a working directory Then in the line command one can type one of the two following instructions to start the SPARTAN code gt gt SPARTAN or gt gt SPARTAN_noGUI The first
78. c species and continuum transitions including a description of the hard coded Free Free transitions Finally we will close this tutorial Chapter through the description on how to link any new spectral databases to the SPARTAN code 46 Modifying the Code Spectral Database Triatomic Atomic Diatomic Linear User defined User defined User defined Atomici m Atomic2 m Atomic3 m User defined Bound Bound Total Photoionization 220 Level User defined User defined Specific Photo1 m Photo2 m Bound Free Total 7 User defined N Photodissociation 775 Level m User defined Specific Photo2 m Photodetachment Total 15511 PhoDet m Free Free Hard coded Hard coded m Bremstrahlung Bremstr m Bremstr m Table 4 1 Transitions modeled by the SPARTAN code corresponding numerical routines and spectral datafile types 4 2 Building spectral datafiles for diatomic tran sitions This section provides an in depth description on how to create and modify spectroscopic constant files in a format compatible with the SPARTAN code and provides some guidelines for the appropriate selection of spectral data which is consistent with the user needs 4 2 1 Guidelines for the selection of appropriated spectral constants When deciding to develop a new spectral database for a specific diatomic transition one first needs to examine the degree of accuracy of the spectral constants that can be found
79. calculation routine 37 LinPar par_c Parameter defining the core points distribution spacing see section 3 5 1 or 1 for more details LinPar par_w Parameter defining the Lorentz contribution for wings points distribution spacing see section 3 5 1 or 1 for more details LinPar par_w_FG Parameter defining the Gauss contribution for wings points distribution spacing see section 3 5 1 or 1 for more details LinPar diff Maximum difference among the Gauss and Lorentz lineshapes of two different lines for which the Voigt profile is not recalculated for the second line Recommended value 0 1 corresponding to a 10 difference LinPar Lorentz LinPar Lorentz x FWHM is the minimum distance from the line center where the line profile is considered to be Lorentzian Recom mended value 64 LinPar cutoff LinPar cutof fx FWHM is the distance from the line center where the line intensity is considered to be negligible Intensity 0 Recom mended value 1000 LinPar minstep Minimum interval between two adjacent lineshape points FW HM x LinPar minstep This parameter is useful for defining the accuracy of the calculated line peaks If the parameter is set to 99 it will be equivalent to the minimum distance between adjacent points of the computed Voigt profile Recommended value 99 LinPar intstep Grid size for the calculation of the lineshape integral in the core region Recommended value 120 LinPar integral Line normaliz
80. command launches the graphical user interface GUI of the code whereas the second command launches the application directly without using the GUI 1 1 1 Running the Graphical User Interface GUI of the SPARTAN code This section will focus on the SPARTAN GUI which provides a useful in terface allowing the main calculation parameters to bet set and calculations to be launched without major efforts Upon launching the application the GUI window is opened see Fig 1 1 Figure 1 File Edit Insert Tools Desktop Window Help Atomic Discrete VUV Continuum ONO Gamma H Atomic CO2 Phatolonization ONO Beta OC Atomic O H2 Photolonization ONO Deta O C Atomic C2 Phetolonizetion Epsiton Temperature ON Atomic O O Pretotonization Betat s00 ON Atomic CO Photolonization ONO Gammal OO Atomic O N2 Phetolonization 02 Schumann Runge Q 0 Atomic Photolonization 02 Schumann Runge Cont Atomic 02 Photolonization Infrared O Are Atomic VUV Visible Discrete CO2 RoVibrational Atomic H2 Lyman O CO RoVerational O Xe Atomic H2 Werner ONO Rovibrational Xe Atomic C2 Philips 2 Bremstrahlung Atomic Continuum 02 Bremstrahlung Photolonization OC Photolonization C Photolonization ON Photolonization ON Photolonization OO Photolonization 0 Photolonization Ar Phnototonization co Angstrom Q Ar Phetolonization O co x Ato Bremstrahlung OCO Asun
81. cture of the code and are discussed in Chapter 3 1 1 4 Running the SPARTAN code without the Graphical User Interface When using the SPARTAN code without using the program GUI the user must manually set the input parameters in the Inputs txt file record the changes and launch the calculation in the MATLAB command line by typing gt gt SPARTAN_noGUI Upon the calculation end the code records the overall spectra IE_IA_nu_Total txt and each radiative system spectra IE_IA_nu_01 i txt in the OUTPUTS directory 1 2 Recorded data Each ASCII textfile for the the overall spectra IE_IA_nu_Total txt and each radiative system spectra IE_IA_nu_01 i txt contains columnwise val ues of the emission coefficient in VV m cm sr absorption coefficient in m and wavenumber in em 1 Also the code displays some of the calcula tion overall results like the spectrum total radiative power the radiative power of each radiative transition and the overall calculation time Such parameters are also recorded in the file Calc _Log txt located in the OUTPUTS directory overriding any older logfile encountered in this directory Note that both versions of the code also leave the calculation parameters variables Inputs Reference Species and Transitions and results vari 1 3 Sample comparisons 7 ables Result for each radiative transition spectra and ResultTotal for the overall radiative spectra in the MATLAB workspace 1
82. cur PQa J PQI 27 Duz Iya Fu1 2 SRa J Pio J 1 1 7107094 2J J 1 7 TRa J Pis J 1 a OPa J 1 77 PQ 7 Qai J ln O B 0 S Pa U 1 a tir J Re J 1 Q J J R J J 1 Px F R s 1 aD SRsa J Pas J 1 25 N Pis 7 TRsi J 1 PUN vet R ANN Qia J 0 1 CEEE 2 2 rau mis Ae ae Pe3 J 1 R s 7 9 1 Y z Ps J Rs 1 J Qs J 24 1 Rs J J 1 HSN Joa YAN yY Y 4 4J VY Y 4 4 J 1 1 J J 1 Y Y 4 2027 4 1 J 1 7 7 1 78 H nl London Factors Co J Y Y 4 4 4 1 C3 J J 1 7 JY Y 4 2 2 1 1 4 2 Table B 8 H nl London factors for 5X II transitions Transition H nl London Factors 2 1 I 2 J 1 J 1 Jur J 1 uy J 2 Pi J moo 150 57 me a J Q J DU D ull 0 au iy 12 J J 2 IHDA D T R J 16 41 1 1 Dg F m u 1 J 1 2 J J 2 7 2 ul 7 1 PuG TCT 2 1 1 29 42 Qai J
83. curve of a diatomic molecule electronic state provided that the spectroscopic constants of this state are known The RKR method starts from the first order JWKB quantization v4 fe va C 1 in which V r and E denote the potential energy function and the total energy respectively u is the reduced mass and r and rz are the inner and outer classical turning points The derivation of the RKR formalism assumes the existence of a vvell behaved single minimum potential around the equilibrium distance and that the vibrational and rotational energies are smooth functions Of v Two expressions can be then obtained for the case of non rotating and rotating molecules ro v ri v 2 JE pe e 2f C 2 1 4 f y vT ooa By EWJ do 24 3 min which are usually known as the vibrational and the rotational RKR equations Here Umin v E 0 and B in equation C 3 is the rotational constant defined in E v J B J J 1 Rearrangement of equations C 2 C 3 gives the final expressions ri v V f 9 f C 4 vf Ef g f C 5 It is worth noting at this point that although the semiclassical quantization condition maps integer values of v within the semiclassical RKR approach v may be treated as a continuous variable This allows obtaining a much larger number of points than those obtained using integer values for a precise determination of the
84. d eled after using a simplified approach implemented in the SPARTAN code In the current version of the code this is restricted to doublet transitions due to the necessity of modeling the perturbations in the N First Negative System This approach requires information on the perturbations specifically the cen tral perturbed rotational quantum number Jpert the value for the maximum perturbation shift dE per in absolute values and the reference for the affected branch F 2 This obviously implies such values as reported in the literature can be retrieved An example for the influence of perturbations is presented in Fig 4 4 where an experimental spectrum of Ny First Negative system Av 0 obtained in a low pressure arcjet facility is reproduced by synthetic spectra us ing only de perturbed constants and using perturbed constants following the approach implemented in the SPARTAN code We verify that even the addi tion of this simplified treatment for perturbations may significantly improved the reproduction of experimental spectra 4 2 2 Step by step instructions Here we will describe how a spectral database file compatible with the format utilized by the SPARTAN code can be created therefore allowing the simulation of any arbitrary multiplet radiative transition for a diatomic species AB Step 1 Create a definition file with the radiative transitions of the species First of all a new definition file for the calculation of the rad
85. d from 0 to 90 taking into account axi symmetry of the flow as stated before Angle intervals are fixed at 15 according to the recommendations from Riviere 2003 A total of 21 600 rays is sampled in this fashion for the 48 wall points Figure 6 Sample rays at a 0 azimuth for a given point in the spacecraft backcover 2 2 Line by Line Model and Associated Database for CO Flows The SPARTAN code SPARTAN is capable of simulat ing around 60 bound bound free and free radiative tran sitions from more than 20 chemical species ranging from the VUV to the IR region with typically 105 to 106 lines and the superposition of several continua The code numerical routines are capable of simulating bound bound atomic and molecular radiation Molecular radiation routines include fine structure effects as they are capable of simulating singlet doublet and triplet tran sitions as well as accounting for A doubling Bound free radiation like photodissociation photoionization and photodetachment transitions can be accounted for accord ing to available cross sections from the literature regard less of whether they are integrated over specific tem perature ranges or state specific Free free transitions like Bremsstrahlung are also accounted for using the most popular theoretical and semi empirical expressions found in the literature For low pressure plasmas the dominant line broadening process will be Doppler broad ening
86. de e Chapter 2 describes the physical models available in the code e Chapter 3 provides a more detailed description of the numerical algorithms inserted in the code e Chapter 4 provides an in depth description of the file structure of the SPARTAN spectral database and explains how the database can be cus tomized by the user e Appendix A references the spectral database of the SPARTAN code ex cept for Bound Diatomic transitions e Appendix B presents the expressions for the H nl London factors that have been inserted in the code e Appendix C describes the companion RKR_SCH routine that can be used for the calculation of the full set of rovibronic states for a specific elec tronic configuration of a diatomic molecule and the calculation of Einstein coefficients for bound diatomic transitions e Appendix D describes the other auxiliary routines of the SPARTAN code Copyright Notice The SPARTAN code is distributed under the terms of the GNU Lesser Public License LGPL as published by the Free Software Foundation either version 3 of the License or at your option any later versions The LGPL license allows utilizing the SPARTAN code linked to closed source proprietary codes This program is distributed to the scientific and general community in the hope that it will be useful but without any warranty Community involvement in the development of the SPARTAN code itself or its associated database is an important ende
87. di Bremstrahiung OCO Tripet OO Bremstraniung O co Comet Tail C PhotoDetachment Om 1 ON PhotoDetachment 2 O o PhatoDetachmant Ota 1 45 N 8 g 8 NG ANA AN i 9 i 5 AJ Ad N 7 Az O x x g 8 S S 88 8 g amp Figure 1 1 Graphical User Interface of the SPARTAN Code The interface provides the following items which can be defined by the user 1 1 Launching the SPARTAN code 5 1 Apparatus function Here the user can define a Gaussian apparatus function of a given FWHM in A for the simulation of experimentally mea sured spectra this FWHM is added to the calculated Voigt FWHM Setting this option to 0 reproduces the physical spectra for given local conditions 2 Rotational temperature The user defines here the rotational temper ature for the overall species For the calculation of broadening mecha nisms the species translational temperature is considered to be equivalent to the rotational temperature Ty T ot 3 Vibrational temperature The user defines here the vibrational tem perature for the overall species 4 Electronic excitation temperature The user defines here the elec tronic excitation temperature for the overall species Also the free elec trons temperature is considered to be equivalent to the electronic excita tion temperature T j Texc 5 Minimum wavelength The user defines the minimum wavelength for which the calculati
88. differences between such spectra one may chose not to recalculate a spectral lineshape if both the Lorentz and Doppler FWHM differ less than by a given percentage 10 being an adequate value Such approach further improves computational efficiency as for most of the transitions to be simulated the calculation routine merely replicates the previous lineshape spectral grid points and relative intensities simply multiplying the relative line intensities by the line strength to yield the absolute line intensities Once that the spectral lineshapes have been calculated for each of the overall database transitions such individual lineshapes then have to be combined to yield an overall spectra This is firstly done considering the overall calculated grid points and grouping them in an increasing order Secondly the spectral points too close to each other by a defined parameter named minstep and typically 2 of the FWHM are grouped Finally each individual lineshape is interpolated over the boundaries of the overall spectral grid which falls inside the first and last grid points of the individual lineshape The overall discrete spectrum is then overlaid to the previously calculated pseudo continuum spectrum to yield the final spectrum In brief this proposed adaptive grid method represents an approach where a series of approximations pseudo continuum calculation of Voigt lineshapes with a minimum number of points nonsystematic lineshape r
89. djacent cell spectrum will remain close to the initial cell spectrum thus justifying utilizing the first cell spectral grid For the inverse case of figure the spectrum will be strongly dependent on the adjacent cell spectral properties thus justifying the use of this cell spectral grid After selecting the appropriate grids for each given radiative system one only needs to combine such grids to obtain the adequate grid for radiative transfer calculations between two adjacent cells This approach allows keeping consecutive grid sizes at a balanced level while allowing essential radiative behavior to be adequately captured and reproduced 4 Accuracy and validation of the method Sample radiative transfer calculations have been carried out using the different low and high resolution spectra calculated according to the adaptive model presented in this paper Comparisons of the transmitted radiative power have been carried out against calculations utilizing very high resolution spectra This allows an estimation of the accuracy of the different spectra liable to be provided by the method 118 M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 For such purposes the problem of hot radiation absorption by a cold gas layer is one of the more stringent radiative transfer problems that can be considered Accordingly adequate test cases have been defined probing radiative transitions between a ground s
90. e database 37 Atomic Photodetachment Transitions O7 38 28 28 Molecular Photoionization Transitions CO2 CO CN C N Oz NO 391 140 41 40 40 42 140 43 Molecular Photodissociation Transitions Lino da Silva 2013 Oz Schumann Runge Continuum unpublished Atomic and Molecular Bremsstrahlung Transitions Atomic Ions N O N O2 26 27 27 28 28 Table A 1 List of Radiative Systems included in the current version of the SPARTAN code Database 69 Species CO COT References 44 45 46 44 47 48 49 Lino da Silva unpublished 44 44 50 51 52 53 54 55 44 53 34 28 56 57 28 56 44 58 28 56 28 56 28 56 44 59 60 Table A 2 List of Radiative Systems included in the current version of the SPARTAN code Database 70 References for the SPARTAN Spectral Database Appendix B Expressions and Approximations for the Computed Honl London Factors This appendix describes the approaches that have been implemented in the SPARTAN code for the calculation of the Honl London Factors for bound diatomic transitions Depending on the spin multiplicity of the upper and lower electronic states of the diatomic molecule the number of rotational branches may go from the 3 single P Q R branches for singlet transitions up to 12 branches for doublet transitions 6 main branches P P2 Q Q Ri
91. e must have LinPar threshold1 lt LinPar threshold2 LinPar threshold2 Threshold normalized to the maximum line emission intensity AND the maximum line absorption intensity below which a line in tensity contribution is added as a pseudo continuum to the overall spectrum Recommended values 105 107 Higher values will have less lines quickly cal culated as a pseudo continuum yielding more accurate spectra at the cost of larger memory and computation time overheads Note that for consistency one must have LinPar threshold1 lt LinPar threshold2 LinPar dnu LinPar dnu x FW H Mmea is the boundary from the line cen ter defining the region were a weak line is covered by a strong line making it s calculation as a pseudo continuum viable Recommended value 20 if 25 10 For higher values of this ratio it is recommended to increase LinPar dnu LinPar contstep LinPar dnu x FW HM is the fixed spectral width step of the grid used for the calculation of the pseudo continuum spectrum Recom mended value 20 3 5 1 Calculation of a Voigt lineshape This subsection presents a more detailed description of the Voigt lineshape calculations carried in step 4 b see section 3 5 The main objective is to ensure the calculation of a line profile as close to Eq 2 36 as possible while retaining a minimum number of points The method implemented in the SPARTAN code yields profiles which are found to adequately approx
92. e simply obtains once again the same level energies barring any numerical precision errors that have been used as an input from spectroscopic data for building this same RKR potential curve But more importantly one also obtains the energies of the higher lying levels that fall on the extrapolated part of the potential curve Such extrapolated values are more credible than the ones obtained from the extrapolation of a Dunham expansion 2 11 as they rely on a smoothly extrapolated potential curve instead of an inherently unstable polynomial expression C 1 1 Recalculating potentials for an arbitrary rotational excitation The potential for an arbitrary rotational number J can be obtained using the relationship 2 1 Vy r Vy otr Dur C 8 where p is the reduced mass of the molecule The energies for the corresponding rovibrational levels can then be obtained by solving the radial Schr dinger equation The maximum rotational quantum number is then reached when the molecular potential no longer allows a local minima becoming entirely repulsive For the application of Eq C 8 we consider the accurate rotationless po tential proposed by LeRoy 73 The last potential curve allowing for discrete states occurs for J 275 The potential curves for different rotational levels are reported in Fig C 1 The obtained solutions account for the existence of bounded levels above the dissociation energy Those levels are called quasi
93. e than 102 might still yield large calculation times One can acknowl edge that the systematic evaluation of exponential terms of Eq 2 36 for the calculation of several thousand lines might be an ineffective method Instead one can take advantage of the fact that the different lineshapes Doppler and Lorentz FWHM remain very close over a large spectral range The calculated lineshapes are then nearly equivalent and there is no need to systematically recalculate them As an example the differences between two Voigt lineshapes for a 10 variation of their Lorentz and Doppler FWHM is presented in Fig 3 7 Given the small differences between such spectra one may chose not to recalculate a spectral lineshape if both the Lorentz and Doppler FWHM differ less than by a given percentage 10 being an adequate value 0 06 0 05 10 Lorentz 0 04 10 Doppler Normalized Difference c N 0 01 F 0 02 D Y 10 10 10 10 Normalized Distance to Line Center Figure 3 7 Normalized difference between two Voigt lineshapes with Av 1127 amp AV 1 1Azp and an initial Voigt lineshape with A p 5em 1 Such approach further improves computational efficiency as for most of the transitions to be simulated the calculation routine merely replicates the previ ous lineshape spectral grid points and relative intensities simply multiplying the relative line intensities by the line strength to yield the abso
94. ecalculation are carried in order to achieve consequent reduction of calculation times and spectral grids One more advantage of the proposed method is the fact that it is fully user parametrizable and allows M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 115 T T T T 0 5 k 0 06 f Lorentz 27 4 Lorentz F 0 3 1 Z 0 04 i aR RIS Doppler 0 02 S 5 0 z 0 1 200 100 0 100 200 10 101 10 i r r r r r 0 5 R Lorentz j Lorentz 0 3 4 a SSS Ss Doppler 72 Z 0 1 D 0 1 200 100 0 100 200 10 101 107 x im LI N m 0 5 8 0 06 Lorentz j Lorentz o 0 3 o 5 0 04 7 Doppler o z z X 0 02 ae m 12 0 0 1 200 100 0 100 200 10 101 10 r v r r r 0 5 0 06 f Lorentz f Lorentz 0 3 0 04 1 L 0 02 1 41 0 1 200 100 0 100 200 10 101 107 Distance to Line Center cm Fig 5 Plot of low resolution Lorentz lineshapes left 5 7 9 and 11 points from top to bottom Ay 5cm 1 and plot of the normalized difference of Lorentz full line Ay 5cm 1 and Doppler dotted line Ay 5cm 1 lineshapes to the exact lineshapes right 5 7 9 and 11 points from top to
95. efficient of Heated Air JQSRT Vol 7 1967 pp 423 427 BIBLIOGRAPHY 91 28 Chauveau S Constitution de Bases de Donn es Spectroscopiques Rela tives un Plasma d Air Application au Calcul de Transferts Radiatifs Ph D Thesis in French Laboratoire d Energ tique Mol culaire et Macro scopique Combustion E M2 C Ecole Centrale de Paris 2001 29 Milne E A Radiative Equilibrium in the Outer Layers of a Star Monthly Notices of the Royal Astronomical Society Vol 81 No 5 1921 pp 361 375 30 Milne E A Spectroscopy Astronomical Radiative Equilibrium and Spec tral Distribution Monthly Notices of the Royal Astronomical Society Vol 81 No 5 1921 pp 375 388 31 Vranckx S Loreau J Desouter Lecomte M and Vaeck N De termination of Photodissociation and Radiative Association Cross Sec tions From the Same Time Dependent Calculation ArXiv e prints eprint 1301 5547 2013 available at http adsabs harvard edu abs 2013arXiv1301 5547V 32 Zhu X An Inproved Voigt Line Approximation for the Calculations of Equivalent Width and Transmission JOSRT Vol 39 No 6 1988 pp 421 427 33 Smith A J Gogel T and Vandevelde P Plasma Radiation Database PARADE Final Report ESTEC Contract 11148 94 NL FG TR28 96 Apr 1996 34 Lino da Silva M Guerra V and Loureiro J Vibrational Distributions in N With an Improved Calculation of
96. ently more complex simulations have been carried out in which the spectral range has been extended to values as low as 200 250 nm High resolution experimental spectra measured in atmospheric pressure plasma torches have been rebuilt numerically utilizing more detailed radiative transfer models 17 22 The number of spectral lines considered in such simulations exceeds 10 000 individual lines However most of the spectral investigations in low pressure plasmas have been carried out as a nonintrusive method to examine excited molecular level populations emission spectroscopy or ground molecular level populations absorption spectroscopy Moreover emissive or absorptive chemical species encountered in such plasmas are at most diatomic This means that complex polyatomic spectra triatomic or 122 M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 0 08 1 T T 0 06 F 7 0 04 0 02 0 08 r r T 0 06 Radiative Power VV m3sr 1 0 04 t 0 LL il EB L il NAMA SL 1630 1640 1650 1590 1600 1610 1620 Wavelength A Fig 10 Sample radiative transfer calculation for the CO Fourth Positive System Av 0 see Table 5 Top radiative features exiting the hot slab and entering the cold slab with T 1000 K and x 100 km Bottom radiative features exiting the cold slab with T 5000 K and xo 1m Table 6 Radiative tran
97. er in cm and absorption cross sections in m has to be supplied without any further data For vibration specific cross sections such as for example the O Schumann Runge continuum 02 SRC txt the levels populations are provided by the files AB LEV txt as for diatomic discrete transitions The absorption cross sections file structure is then as follows LinodaSilva 2013 21 Levels v 0 E v 787 91 115 points 57144 41 2 350e 19 57413 90 3 039e 19 v 1 E v 2344 58 116 points 55587 74 2 577e 18 55857 24 3 098e 18 The file structure can be summarized as Identifier tag first line e Number of vibrational quantum numbers v third line Line including the identification of the vibrational level its energy in cm 1 and the number of points n in the cross section n lines with with wavenumber in em 1 and absorption cross section in m 4 5 Linking new spectral datafiles to the SPARTAN database 63 4 4 3 2 Molecular Bremsstrahlung cross sections The tabulated N and O molecular Bremsstrahlung cross sections o 77 7 discussed in Sec 2 2 2 2 have been hard coded in the Bremstr m routine and assigned the tags N2_Bremstr and 02_Bremstr the tabulated gaunt factor for Bremstrahlung transitions as discussed in Sec 2 2 1 1 is also hard coded in the routine The routine can be easily modified if new data tables are to be used 4 5 Linking new spectral datafiles to the SPAR TAN database Once n
98. er and the line wings The calculation routine then proceeds according to the described method in order to optimally distribute such points More precise line profiles will be obtained for a larger number of assigned grid points at the cost of increased spectral grids and calculation times Different low and high resolution lineshapes have been compared with the exact Voigt profiles Eq 2 36 with Av 5 cm The differences to the exact profile have also been determined for each of such profiles The obtained results are presented in Fig 5 for low resolution profiles and in Fig 6 for high resolution profiles 3 5 The Lineshape calculation routine 41 0 5 0 06 Lorentz 7 Lorentz 0 3 2 m Doppler 0 02 0 1 0 0 1 200 100 0 100 200 10 101 10 0 5 si 0 06 Loreniz Lorentz in 0 04 0 3 2 r 7 4 Doppler o 0 1 0 02 7 z 2 d mz 0 0 1 y g 200 100 0 100 200 10 10 10 N 0 5 2 0 06 Lorentz Lorentz o 0 3 g 5r 0 02 ANED RG Doppler D 0 02 0 1 2 0 0 1 m 200 100 0 100 200 107 101 10 0 5 0 06 Lorentz Lorentz 0 3 0 04 7 Bi Doppler 0 02 0 1 J 0 0 1 200 100 0 100 200 10 101 10 Distance to Line Center cm Figure 3 5 Plot of low resolution Lorentz lineshapes left 5 7 9 and 11 points from top to bottom Ar 5 cm
99. es different internal levels followed by the description of the models utilized for the calculation of line positions intensities and shapes 2 1 1 Selection rules 2 1 1 1 Atomic transitions Atomic transitions are split into electric magnetic dipolar or quadrupolar transitions The transitions can be classified by order of decreasing inten sity dipolar electric E1 quadrupolar electric E2 dipolar magnetic M1 and quadrupolar magnetic M2 1 transitions are called allowed transitions the others forbidden transitions The selection rules 2 are summarized in Table 21 2 1 1 2 Diatomic transitions Only electric dipolar transitions are considered for the calculation of syn thetic discrete spectra from diatomic species as the other type of transitions have much lower probabilities and are generally covered by this stronger spectra 10 Physical Models Transitions E1 Transitions M1 Transitions E2 All couplings 1 AJ 0 1 0 1 0 1 2 except 0 0 except 0 0 except 0 0 bee ddl 2 0 1 AM 0 1 0 1 2 except 0 0 except 0 0 when J 0 when J 0 3 parity change identical parity identical parity 41 one electron transition no electronic no electronic with Al 1 configuration configuration for arbitrary An change i e change i e for all electrons for one electron Al 0 Az 0
100. eshape with AVL p 5 177 M Lino da Silva Journal of Quantitative Spectroscopy dt Radiative Transfer 108 2007 106 125 117 10 10 10 10 10 10 Wavenumber cm Fig 8 Spectral gradients for the spectral grids obtained for 97 CO 3 N plasmas in thermal equilibrium at 1000 K black 5000 and 10 000 K grey The gradients reproduced in grey correspond to the spectral grids of Fig 1 emitted absorbed by the CO infrared transitions which explains the tightened grid near the three main radiative modes of CO at 666 2300 and 3700 cm A first approach that could be devised for radiative transfer calculations would be combining adjacent spectral grids and then removing the spectral points too close from each other However this is rather inefficient as carrying radiative transfer calculations over a given spatial grid will result in ever increasing spectral grids Instead it is recommended that the spectral grids obtained for each radiative system are compared for adjacent spatial cells Although grid point distributions may vary a little due to different line broadening widths the spectral grids of a given radiative transition obtained in the different spatial cells will remain very close One may take advantage of this by comparing the individual grids of each cell selecting the largest grid If there is strong emission absorption in the initial cell and weak emission absorption in the adjacent cell the a
101. esults are somehow similar as both methods consider a set of closely spaced grid points in the core region and a set of loosely spaced grid points in the wings region see Section 3 2 Allowing a certain level of user parametrization namely regarding error tolerances 25 Similar line selection criteria 23 28 Differences include In atmospheric sensing applications a single line or a group of lines may be predominantly absorbing which means that their influence on the overall gas transmissivity may extend well beyond their line center Therefore line profiles usually have to be calculated for a broad spectral range The approach discussed in this article only considers a single although variable step grid Although the algorithm discussed here considers the line cutoff spectral limit above which the line intensity is considered to be zero as an adjustable parameter typically 100 1000 x FWHM broader line cutoff parameters come at the price of encompassing a large number of points to which the line profile has to be interpolated in the spectral grid For such conditions the algorithm discussed here might not be competitive when compared to the cascaded grids approach The predominance of such intense absorption lines may mandate the application of different Voigt lineshape calculation routines if the accuracy provided by the formulae of Eq 2 0 02 is deemed insufficient However the approach discussed here is not
102. eter in the line wings region pary pary Ayvp threshold1 ratio of maximum and cutoff intensity for strong lines threshold2 ratio of maximum and cutoff intensity for strong and weak lines v Voigt line profile wW boundary between the line core and wing region cm7 Subscripts 0 hot slab or line center 1 cold slab c line core region G Doppler L Lorentz med median value V Voigt w line wings region Avp AYL paryp However regarding the investigation of the spectral properties of low pressure plasmas it is difficult to circumvent the use of exact line by line methods owing to the small broadening widths of atomic and molecular spectral lines Efficient approaches have then to be devised regarding the development of a compact 108 M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 yet accurate spectral database of line by line transitions 1 but also regarding the convolution of this line database with its adequate lineshapes followed by the superposition of different continua Here we will present a method to enhance the efficiency of line by line calculations in order to achieve three main goals a fast calculations over the whole spectral range b reasonable spectral grid sizes c accuracy of the simulated spectra The method has been devised as being fully parametrical so as to allow potential users to decide which of the issues
103. ew spectral datafiles have been produced there is the need to link them to the general database of the code This sections provides a step by step tutorial on how to achieve this Step 1 Adding species to the code database folder The first step is to place the levels and radiative transitions files in the folder named DATABASE Step 2 Adding transitions to the Inputs file The new transitions of the species you wish to consider in the code must be added to the Inputs file First open the INPUTS folder then open the Inputs txt file This file contains all the input parameters for the simulation apparatus function minimum and maximum wavelengths in A temperatures Trot Tus and Tex in K If you are using the GUI interface the values of these param eters are the ones you defined and you can always change them directly using the interface If you do not wish to use the GUI interface you must set new values by editing this file The number density of the species in m must always be defined here and you must not forget to update its value if it is set to zero Also included in the Inputs file are the chemical symbol the molar mass in kg mol and the radius in A of the species After the input parameters the number of transitions considered by the code is shown Do not forget to sum the number of transitions you are adding to the defined value Then at the end of the file you must add the name of the transitions you want t
104. f High Temperature Gases in Atmospheric Entry Part II 30 Sep 1 Oct 2004 Porquerolles France pp 81 89 Lino da Silva M and Dudeck M Arrays of Radia tive Transition Probabilities for CO2 N2 Plasmas J Quant Spectrosc Radiat Transfer Vol 102 No 3 2006 pp 348 386 Lino da Silva M Simulation des Propri t s Radiatives du Plasma Entourant un V hicule Traversant une Atmo sph re Plan taire 4 Vitesse Hypersonique Application a la Plan te Mars Ph D Thesis in French Universit d Orl ans Dec 2004 Rivi re P Soufiani A and Perrin M Y Line by line and Statistical Narrow Band Calculations of Radiative Transfer in Some Atmospheric Entry Problems ESA SP533 Proceedings of the First International Work shop on Radiation of High Temperature Gases in At mospheric Entry 8 10 Oct 2003 Instituto Superior Tecnico Lisbon Portugal pp 189 196 Anderson J D Hypersonic and High Temperature Gas Dynamics A merican Institute of Aeronautics amp Astronautics 2000 Lino da Silva M An Adaptive Line by Line Statistical Model for Fast and Accurate Spectral Simulations in Low Pressure Plasmas Journal of Quantitative Spec troscopy and Radiative Transfer Vol 108 No 1 2007 pp 106 125 http cfp ist utl pt radiation http physics nist gov PhysRefData AS D index html http vizier u strasbg fr topbase topbase html
105. fied to follow the same struc ture that the atomic species level energy files It is very important that the energy level file is issued from TOPbase and not another database such as NIST otherwise the program will likely crash or otherwise the results won t be consistent 4 4 1 3 Atomic photodetachment transitions file structure For atomic photodetachment the structure is as follows example for O photodetachment 0 PD txt file 11819 Energy Difference Ground Level 6 Degeneracy_state_0 2P Wavenumber_ cm 1 _Cross Section_ m 2 1 2000e 04 0 0000e 00 You need to include the energy difference in cm between the ground A state and the ground A state the degeneracy of the A state and the global photodetachment cross sections in two columns with wavenumber in em 1 and absorption cross section in m7 4 4 1 4 Atomic Bremsstrahlung cross sections The semi empirical expressions reported in Sec 2 2 2 2 for atomic ions N and O atomic Bremsstrahlung have been hard coded in the Bremstr m routine and assigned the tags N2_Bremstr AtolI Bremstr and 02 Bremstr the tabu lated gaunt factor for Bremstrahlung transitions also reported in Sec 2 2 1 1 is hard coded in the routine The routine can be easily modified if new data tables are to be used 4 4 2 Linear polyatomic discrete transitions Linear Triatomic discrete transitions are split into two files just like for atomic and diatomic transitions a level file CO2
106. file Step4 Editing the Database file The Database txt file located in the INPUTS folder lists and links the overall definition files Open and you will immediately see the number of tran sitions considered by the code As before do not forget to sum the number of transitions you are adding To add a new radiative system you must add a new line at the end of the file following the steps e Column 1 Introduce the name of the transition e Column 2 Write the chemical symbol of the species Column 3 Introduce the molar mass of the species in kg mol 1 e Column 4 Introduce the name of the file defining all the radiative tran sitions of the species see step 1 e Column 5 Introduce the name of the file containing all the energy levels of the species see step 1 e Column 6 Introduce the identifier of the upper level of the transition in the levels file referred to in column 5 It corresponds to the number of the row of the upper level ex the level with the lowest energy is indicated in the first row and so the identifier is 1 e Column 7 Introduce the identifier of the lower level of the transition in the levels file It is defined as the previous identifier Column 8 Define the type of transition ex BoundMO BoundDI PhoIoMO Bremstr etc e Column 9 Define a detailed identifier of the transition ex ATO ATO 28 2 BremsDI etc e Column 10 Define the population of the levels only Boltzmann in the curren
107. he columns in the second line The following columns stand for Band origin Vo in cm 1 The excited band quantum vibrational number v The excited band quantum vibrational number Excited vibrational level degeneracy 2 v e Numerical label of the excited vibrational level ISO level number Energy of the excited vibrational level in em 1 e The ground band quantum vibrational number v The ground band quantum vibrational number Ground vibrational level degeneracy 2 60 1 e Numerical label of the ground vibrational level ISO level number Energy of the ground vibrational level in cm e Isotope of the vibrational band e Isotope numerical label Hermann Wallis coefficient in em 1 Hermann Wallis coefficient A in em 1x105 Hermann Wallis coefficient in em 1x105 62 Modifying the Code Spectral Database e Hermann Wallis coefficient Ag in cm x 10 Einstein coefficient Ay 8 1 The corresponding partial section of the file CO2 txt is presented below 613 Transitions nu0 v 1 2 d01 Ref_Ev Ev v 1 2 d0l Ref_Ev Ev ISO Ref A1 A2e5 5 AQe5 Avv 471 5112 20003 O 1 626015 2548 36707 11101 1 2 626012 2076 855880 626 1 0 000490 0 898 0 0 0 00831862 4 4 3 Molecular continuum transitions 4 4 3 1 Diatomic species photoionization photodissociation cross sections For global cross sections a two column file with wavenumb
108. he proposed method is fully 124 M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 parametric leaving the user with the choice of adapting the calculation parameters to its specific requirements likely to be linked to the available computing power A grand total of 25 parameters can be defined by the user who can follow the guidelines and the sample calculation results of this article as a basis for adapting this method to any specific needs From the different presented test cases it is predicted that lineshapes defined with about 20 30 points suffice for ensuring enough accuracy for most of the radiative transfer applications For mediums close to optically thin conditions lineshapes with less than 10 points are sufficient Also it has been verified that using a pseudo continuum approach in the simulation of molecular spectra leads to considerable calculation speed ups with a minimum loss of accuracy typically for the more relaxed thresholds a decrease to about 25 of the original calculation times is achieved for less than a 15 loss in accuracy The source code of the method along with the line by line code SPARTAN and its accompanying spectral database can be made available from the author upon request 29 References 1 Lino da Silva M Simulation des Propri t s Radiatives du Plasma Entourant un V hicule Traversant une Atmosph re Planetaire a Vitesse Hypersonique Applica
109. iative transitions of the species must be created For simplicity we will consider a molecular species assuming transitions between singlet states A II X D such as the ones 4 2 Building spectral datafiles for diatomic transitions 51 TI Simulation With Deperturbed Constants 0 5 ok Aad hall pun BM L Simulation With Perturbed Constants 0 5 Intensity A U l hahahhaha L PA T Measured Spectrum 0 5 1 3850 3860 3870 3880 3890 3900 3910 3920 Wavelength A Figure 4 4 Comparison between a measured Ny First Negative System Av 0 spectrum and synthetic spectra obtained using both deperturbed and perturbed rotational constants defined in the CO 4P txt file In fact we suggest opening that file and editing it for the new species at the end do not forget to save the modifications as a new file Step 2 Edit the Dunham matrix The correspondence between the Dunham coefficients and the traditional spectral constants is given in Table 4 2 Open the CO 4P txt file At the top of the file there are two Dunham matrix one for the upper energy level and another for the lower energy level X X The bibliographic references for the spectral constants are typically indicated in the file The size of the matrix must always be 7 x 10 and only the spectral constants values must be edited for the new species Set all the matrix components to zero and then introduce
110. imate the Voigt profile of Whiting amp Olivero to a user defined precision which will depend on the number of considered grid points The routine discriminates two sets of Voigt line profiles a low resolution set which allows quick spectral calculations yielding a compact overall spectral grid and a high resolution set for more smooth and detailed line profiles at the cost of larger spectral grids Four low resolution Voigt line profiles with respectively 5 7 9 and 11 grid points have been defined introducing the parameters W and FW which define the boundaries between the line center region the near line wings and the far line wings respectively The expression for those parameters for a given Voigt profile is proposed by Zhu 32 2 1 QAzrz BADD 3 1 7 W FW the 8 constants are held at 1 1 8 for the calculation of the parameter W as proposed by Smith 33 and held at 2 6 5 8 for the calculation of the 3 5 The Lineshape calculation routine 39 parameter FW according to a critical analysis carried by Lino da Silva on sample Lorentz Doppler and Voigt profiles The different grid points have been defined accordingly and are presented in Table 3 1 Number Selected Points of Points vo V 8 V 2 W FW 25V 2 5 x x x 7 x x x x 9 x x x x x 11 x x x x x x Table 3 1 Gridpoints for 5 7 9 and 11 points lineshapes An x defines a specific point being accounted for in the given linesha
111. imations eo ce e aadis k s es B 1 1 Neglecting line spin splitting effects for satellite lines in Yohan i wow ee ee BA em Se B 1 2 Neglecting weaker rotational branches B 2 First Rotational Lines intensities B 3 Intermediary a b case Honl London Factors Potential Energies and Wavefunctions Reconstruction THY cc eee os 777777 1 1 Recalculating potentials for an arbitrary rotational exci TAO th ao a Soe ae hae Sk ke eb bee BR ee C 1 2 Expressions for Radiative transition probabilities C 1 3 Numerical Routines Description D Other Auxiliary Routines Bibliography E Code Versions Log E1 Next Version updates eee xos ea anda D RD B2 Code Regressions sa sak ed m ea SE ea a E S Code Run Times ea aac e a oe Selected Published Works 45 45 46 67 71 72 72 72 74 74 81 81 83 84 85 87 88 95 96 96 96 99 Introduction The SPARTAN code Simulation of PlasmA Radiation in ThermodynAmic Nonequilibrium is a line by line numerical code which calculates the spectral dependent emission and absorption coefficients of a gas which can be either in thermodynamic equilibrium or not In it s present version the code is writ ten in the MATLAB language A FORTRAN version of the code is in the works The numerical code has been initially focused for the simulation of low
112. in the form of a Dunham expansion See Eq 2 11 However as already discussed in Chapter 4 the adequacy of the Dunham expansion is directly associated with the validity of the spectroscopic constants which are obtained by fitting a given set of measured rovibrational levels with the expansion 2 11 Rigorously such an expansion can only be accurate at describing the initial set of rovibrational levels over which the spectroscopic constants have been interpolated but un fortunately some compilations of spectroscopic data do not clearly include the range of the rotational and vibrational quantum numbers over which the expan sion is strictly valid This means that in most cases we do not know whether we are using a set of Dunham coefficients within its validity range or beyond it In this latter case and since the Dunham expansion is essentially a polynomial expansion the predicted energies of higher lying levels may increasingly diverge from the exact energies due to the typical instability of extrapolations by poly nomials The expansion becomes hence increasingly inaccurate as the quantum numbers v and J become large One may instead consider reconstructing the electronic state radial poten 82 Potential Energies and Wavefunctions Reconstruction tial curve which is achieved through the Rydberg Klein Rees RKR method 67 68 69 70 a widely used first order semiclassical inversion procedure for re constructing the potential energy
113. ing expression 2Veol 106 Skp Vool N rilml y Hij 2 27 J with 2 1 Discrete radiation models 21 Resonance broadening This broadening mechanism is confined to the elec tric dipolar atomic and molecular lines resonance lines An expression adapted from 17 is used In its current version the SPARTAN code assumes all its database lines as electric dipolar and as such applies this broadening mecha nism to all its database 1 2 1 2893 1075 Au A AZN 01 5 1 ATp 12803 10717 4 Ag 2 28 01 a Van der Waals broadening This broadening process stems from collisions with neutral particles who do not share a resonant transition with the radiating particle The simplified expression from 18 is preferred to the expression from 19 which is more precise but more difficult to implement 3 1 1 ADw 20 1 6 10753 34 3 2 ZN 2 29 m c where m a is the mean species mass and N Ne 92 N the total particle density Stark broadening This broadening process stems from the interaction be tween the external electronic shells of the radiating species and the plasma charged species Both ions and electron can account for such a broadening pro cess but in practice it is the electrons who contribute the most due to their higher kinetic speeds We can therefore on a first approximation express Stark broadening as a function of electronic densi
114. ion They take as an input the spectral constants from the stored database handled by DataAtomic1 2 3 m inline functions of the Spectre m function the general number densities of the different species of the cell where spectral properties are being calculated handled by Inputs txt through IORead m and Spectre m and the upper lower states populations and degen eracies variables geE NeE NvE geG NeG NvG handled by the respective Excite1 2 3 m functions The routines then handle a 5 x n line matrix with the 5 fundamental line parameters line center 20 emission coefficient ab sorption coefficient a V Doppler FWHM Avg and Lorentz FWHM Ar Raies1 2 3 1D 2D 3D are subfunctions of Atomic2 m which generate the line lists for singlet or multiplet transitions 3 4 Core routines 33 Photo1 2 T m PhoDet m and Bremstr m are routines which calculate continuum spectra for respectively monoatomic photoionization diatomic pho tioinization photodissociation atomic photodetachement and atomic diatomic Bremstrahlung and yield an individual absorption spectrum The correspond ing emission spectrum is then generated inside the function Spectre m using the Planck Kirchoff relationships of Sec 2 3 Lineshape m is the routine which convolves a Voigt Lineshape to the Line data lists of Diracs supplied by the Atomic1 2 3 m routines and yields a spec trum comprised of a wavenumber 7 emission coefficient and absorption coefficient
115. ion coefficients of an equilibrium Martian type plasma at 5000 and 10 000 K 3 An adaptive grid model for the simulation of plasma radiation A fully adaptive spectral method is proposed here for the calculation of emission of an absorption spectra This means that the method proposed here is capable of receiving as an input an arbitrary list of lines with their five key parameters position emission and absorption coefficients Doppler and Lorentz full width at half maximum FWHM line shifting is not considered here returning an overall spectra of global emission 110 M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 6 and absorption coefficients over an adaptive spectral grid Such method incorporates the following elementary steps 1 selection of lines to be calculated explicitly or as a pseudo continuum 2 calculation of individual lineshapes and pseudo continua and 3 combination of individual lineshapes and continua to yield the global spectra 3 1 Identification of strong and weak lines and the pseudo continuum radiation approach As discussed previously simulating the overall spectral properties of an atmospheric entry plasma implies calculating a large number of lines due to the presence of molecular radiation It is clear that explicitly accounting for a large number of lines is not manageable in spectral calculations An approach which is proposed for reducing the number
116. ivative of the Voigt profile is firstly calculated Then the line profile points v are determined Normalized Transformation Value 0 0 2 0 4 0 6 0 8 1 Normalized Number of Points Fig 4 Plot of the numerical transforms yielding the unevenly spaced gridpoints i and iy from the evenly spaced gridpoints i for different values of the adjustable parameters par and pary 114 M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 according to the relation m 75 max v which means that the spectral grid will be tighter in the profile regions with large second derivatives near the line center Here the second derivative is a good variational parameter as it allows an optimum distribution of lineshape points ensuring an adequate reproduction of the exact lineshape through the linear connection of such points The line wings structure is such that the optimal grid points distribution is considered to be given by the relation 0 9 iw 8 which allows distributing the grid points efficiently in the far wings regions where the intensity decrease is very slow thus explaining the tightening of the parameter i near 0 We may now assign a given number of points to be distributed near the line center and the line wings The calculation routine then proceeds according to the described method in order to optimally distribute such poin
117. kely to be updated in the future for extending the simplified modeling of pertur bations to other multiplicities 54 Modifying the Code Spectral Database Once the other steps are finished scroll down and define the number of rotational and vibrational levels For simplicity you can leave the rotational levels defined as 100 and only change the number of vibrational levels In the next line you can define a limiter maximum vibrational level This can be useful when you want to add a large number of vibrational transitions in the database say a 20 x 20 matrix of vu v transitions in case you want to make a broad and large spectral calculation yet only wish to calculate the more intense transition for the lower states Instead of having to crop the matrix you can simply set this limiter v v maz value for example 10 which will instruct the code to do this automatically cropping the 20 x 20 matrix to 10 x 10 versions This way you can just adjust this value depending on what your needs are After defining the number of rotational and vibrational levels you must edit Jn and 77 which are the maximum rotational quantum numbers for each vibrational level If you do not know the exact values for these you can just insert a high enough number like for example 999 and then fill as much values as there are vibrational levels for example for 11 vibrational levels you will need to input and J from v 0 to 10 For
118. l predicted wall fluxes 1 INTRODUCTION This work presents a contribution for TC3 focused on the quantification of sensitivity to several parameters such as spatial and spectral grid shapes and the con tribution of Visible VUV spectral systems to the over all radiative power budget of Martian Entries which is known to be dominated by CO and CO IR transitions Charbonnier 2003 Omaly 2005 Lino da Silva 2004 Such contributions are expected to allow a better tailor ing of Computational Fluid Radiative Dynamics CFRD calculations for the specific case of CO2 N2 mixtures Indeed convergence studies on TC3 have been scarce up to now and the only available recommendations con cern the number of rays to sample in the scope of such simulations in a study briefly mentioned by Rivi re et al Riviere 2003 However there is no information on what spatial and spectral grids sizes strike the best com promise between accuracy and reasonable dataset size The spectral calculations implemented in the scope of this work are based on the line by line approximation using IPFN in house SPARTAN code Radiative trans fer procedures are also based on first rinciples solving the radiative transfer equation using a ray tracing rou tine Therefore the conclusions of this work are mostly applicable for this kind of direct approaches for radiative transfer As it will be discussed more ahead in this article the computational tools current
119. level populations by associated codes e General non equilibrium Planck relationship between the emission and absorption coefficients 32 Detailed Description of the Code 3 3 Units used in the SPARTAN code This is a short outline of the units for the critical variables used in the SPARTAN code Number density N in Inputs txt m Wavenumber 2 cm Emission coefficient W m cm sr 1 Absorption coefficient a V m 3 Absorption cross section oy m 3 4 Core routines The core routines of the SPARTAN code are as follows Spectre m is a general routine which cycles the code database calculates the individual spectra for the requested transitions and superposes them to yield the overall emission and absorption spectrum DataAtomic1 2 3 m routines read the database files for discrete atomic and molecular transitions and handle the data to the corresponding Excite1 2 3 m and Atomic1 2 3 m routines Excite1 2 3 m routines calculate the variables geE NeE NvE geG NeG NvG considering a Boltzmann equilibrium of the internal states accounting for the individual characteristic temperatures Teze Tvib Trot Of the concerned chemical species These routines may be decoupled from the code and general nonequi librium populations for geE NeE NvE geG NeG NvG may be injected by an external function or file Atomic1 2 3 m routines build the spectral lines database for each calcu lated transit
120. lists the previous versions of the SPARTAN code Version Date Comments SPARTAN Code 1 0 02 2003 Prototype version 1 1 1 11 2004 Prototype version 2 1 5 01 2005 Original Version 2 0 01 2006 Major update reworking of the code directories and functions structure 2 0 1 03 2007 Added functions AFunction m and Compare n updated functions Atomic3 m Bremstr m 2 1 11 2007 Major update added coupling capability for large scale radiative transfer calculations Species added to the spectral database 2 2 12 2008 Updated database for COz Infrared transitions 2 2 1 05 2010 Updated Spectre m function 2 2 2 05 2010 Updated GUI 2 3 12 2011 Corrected several bugs added species to the spectral database 2 5 05 2013 Generalized rewrite of the code added species and revamped the spectral database RKR SCH routine 1 0 12 2003 Prototype version 1 1 1 01 2005 Final version textfile input mode added 1 2 12 2011 Some updates bug fixes improved routines new versions tailored for the calculation of rovibrational states including quasi bound states Table E 1 SPARTAN code Version list 96 Code Versions Log E 1 Next Version updates Version 2 6 of the SPARTAN code is expected to be released partly or overall with the following updates e Explicit treatment of satellite lines in multiplet transitions Inclusion of Ng Rydberg transitions in the spectral database e Inclusion of an optional routine which discards spurious le
121. lute line intensities Once that the spectral lineshapes have been calculated for each of the overall database transitions such individual lineshapes have then to be combined to yield an overall spectra This is firstly done considering the overall calculated grid points and group ing them in an increasing order Secondly the spectral points too close to each other by a defined parameter named minstep and typically 1 40 of the FWHM are grouped Finally each individual lineshape is interpolated over the bound aries of the overall spectral grid which falls inside the first and last grid points 3 5 The Lineshape calculation routine 43 of the individual lineshape The overall discrete spectrum is then overlaid to the previously calculated pseudo continuum spectrum to yield the final spectrum In brief this proposed adaptive grid method represents an approach where a series of approximations pseudo continuum calculation of Voigt lineshapes with a minimum number of points non systematic lineshape recalculation are carried in order to achieve consequent reduction of calculation times and spec tral grids One more advantage of the proposed method is the fact that it is fully user parameterizable and allows one to chose the level of accuracy with the associated memory and computational overheads when carrying spectral calculations 44 Detailed Description of the Code Chapter 4 Modifying the Code Spectral Databa
122. ly available already allow achieving very reasonable calculation times allowing the phasing out of approximate radiative methods such as band models While allowing very fast calculation times and reasonably accurate results these methods also suf fer from conceptual restrictions which essentially make them less universal than line by line models 2 RADIATIVE MODELS 2 1 Spatial Grid for Radiative Transfer The first step of this work has been the definition of a suitable radiative grid Although the SPARTAN code lineshape routine uses variable grid methods in order to achieve more compact grids grid sizes ranging from 1 2 to 2 million points are routinely achieved if one considers the whole VUV IR spectral range Therefore one may not define very large number of spatial cells as each one has a size of around 100MB An acceptable number of spatial cells is therefore around 500 1000 We have departed from the structure flowfield grid pro posed by RTech adding some small fixes to points which got inside the spacecraft boundaries and adding some additional points for the better definition of a spacecraft edge as seen in Fig 1 The averaging procedure of the grid cells has to be care fully considered One might simply average the values of the four vertex of the cells However this approach might be oversimplistic in the case of large cells where for example one might have a hot region in the center sur rounded by colder verte
123. ly calculated Then the line profile points 7 are determined according to the following relation 3 4 v Die 4 3 5 3Considering a zero line intensity for the last grid point 40 Detailed Description of the Code N o par 0 25 T par 0 25 w Normalized Transformation Value ES 0 1 0 0 2 0 4 0 6 0 8 1 Normalized Number of Points Figure 3 4 Plot of the numerical transforms yielding the unevenly spaced grid points 7 and iw from the evenly spaced gridpoints i for different values of the adjustable parameters par and parw Which means that the spectral grid will be tighter in the profile regions with large second derivatives near the line center Here the second derivative is a good variational parameter as it allows an optimum distribution of lineshape points ensuring an adequate reproduction of the exact lineshape through the linear connection of such points The line wings structure is such that the optimal grid points distribution is considered to be given by the relation i v Z t w 3 6 mazlo T7 l Which allows distributing the grid points efficiently in the far wings regions where the intensity decrease is very slow thus explaining the tightening of the parameter iw near 0 We may now assign a given number of points to be distributed near the line cent
124. n French Universit Orl ans 2004 available at http esther ist utl pt documents these_LinodaSilva2004 pdf Martin W C and Wiese W L Atomic Spectroscopy NIST 1996 available at http sed nist gov Pubs AtSpec total html Herzberg G Spectra of Diatomic Molecules 2 4 Ed AD Van Nostrand Company Inc 1965 Zare R N Schemeltekopf A L Harrop W J and Albritton D L A Direct Approach for the Reduction of Diatomic Spectra to Molecular Con stants for the Construction of RKR Potentials Journal of Molecular Spec troscopy Vol 46 1973 pp 37 66 Hill E and Van Vleck J H On the Quantum Mechanics of the Rotational distortion of Multiplets in Molecular Spectra Phis Rev Vol 32 No 2 1928 pp 250 272 Nicolet M Cieslik S and Kennes R Aeronomic Problems of Molecular Oxygen Photodissociation V Predissociation in the Schumann Runge Bands of Oxygen Planet Space Sci Vol 37 1989 pp 427 458 Miller S L and Townes C H The Microwave Absorption Spectrum of O16 and O16017 Phys Rev Vol 90 1953 pp 537 543 Laux C O Optical Diagnostics and Radiative Emission of Air Plasmas Ph D Thesis Stanford University 1993 Budd A Uber die Triplett Bandentermformel f r den Allge Meinen In termediaren Fall und Anwendung Derselben auf die B3II 2 Terme des N Molek ls 2 Physik Vol 96 1935 pp 219 229 Brown J M Kaise M
125. n atomic units and hence does not appear explicitly in the expression 18 Physical Models states vibrational wavefunctions 7 which can be obtained by solving the radial time independent Schr dinger equation over recalculated potentials This is carried out by a companion routine of the SPARTAN code RKR_SCH which is described in appendix C The relationship allowing to calculate the vibronic transition moment for each upper lower state pair reads as ae r 2 18 The rotational transition probability is in turn given by the different theo retical H nl London factors which depend on the transition upper and lower electronic states types A 7 A and the Hiind coupling case 5 2J 1 27 Using the normalization rule s r 0 2 7 1 vve then have yur 2 1 1 y 6455 R SN Au She 2 doa eS L 2J 1 2 19 The analytic expressions for the H nl London factors considered in the SPARTAN code are presented in appendix B Modeling of perturbations in the spectra Perturbations in the spectrum can either affect the electronic states rota tionless potential curves avoided crossings or affect the potential curves at a given rotational quantum number also avoided crossings In the first case the vibrational specific constants are modified after a given threshold vibrational number but this can be easily accounted for by using vibratio
126. n radiation of high temperature gases in atmospheric entry 8 10 October 2003 Instituto Superior T cnico Lisboa Portugal p 121 6 21 Vacher D Faure G Lino da Silva M Dudeck M Andre P Definition of a new level one test case measurements of equilibrium radiation from an inductively coupled plasma in the near UV to near IR spectral region for a Martian type COz N mixture In ESA SP 587 Ist international workshop on radiation of high temperature gases in atmospheric entry part II Porquerolles France 1 2 October 2004 22 Lino da Silva M Dudeck M Vacher D Andre P A solution for test case 5 simulation of the radiative emission of a chemical equilibrium 97 3 70 Nz atmospheric pressure plasma In ESA SP 629 2nd international workshop on radiation of high temperature gases in atmospheric entry Rome Italy 6 8 September 2006 23 Mitsel AA Ponomarev YN Firsov KM Ptashnic IV Kataev MY The computer codes LARA and AIRA for simulating the atmospheric transmittance and radiance current status 1995 54 3 559 72 M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 125 24 Kuntz M H pfner M Efficient line by line calculation of absorption coefficients JQSRT 1999 63 1 97 114 25 Quine BM Drummond JR GENSPECT a line by line code with selectable interpolation error tolerance 2002 74 2 147 65 26 Schreier F Optimized evaluation of large s
127. nal of Molecular Spectroscopy Vol 165 1994 pp 500 505 51 Prasad V V Bhale L and Paddy Reddy S Rotational Analysis of the B YT AHT System of 13C180 Abstracts of OSU International Symposium on Molecular Spectroscopy 1980 1989 1983 52 Prasad C V V Bhale L and Paddy Reddy S The Third Positive b527 a5I1 System of CO Observation of the v 2 Level of b XT Jour nal of Molecular Spectroscopy Vol 121 1987 pp 261 269 Krupenie P H The Band Spectrum of Carbon Monoxide Tech Rep NSRDS NBS 5 National Bureau of Standards July 1966 53 BIBLIOGRAPHY 93 541 55 56 57 58 59 60 61 62 63 64 65 66 67 68 Simmons J D Bass A M Tilford 5 G Fourth Positive System of Carbon Monoxide Observed in Absorption at High Resolution in the Vacuum Ultraviolet Region The Astrophysical Journal Vol 155 1969 pp 345 358 Kepa R Rytel M The Angstrom B S A II System of the CO Molecule New Observations and Analyses J Phys B At Mol Opt Phys Vol 26 1993 pp 3355 3362 Park C Nonequilibrium Hypersonic Aerothermodynamics John Wiley 48 Sons Inc 1989 Lofthus A and Krupenie P H The Spectrum of Molecular Nitrogen J Phys Chem Ref Data Vol 6 No 1 1977 pp 113 807 Owono Owono L C Jaidane N Kwato Njock M G and Ben Lakhdar Z Theoretical Investig
128. nally specific spec troscopic constants B D etc For the case of rotational perturbations one can either resort to a complex approach of solving the perturbed system Hamiltonian to yield the perturbed energy levels see for example ref 12 or the perturbation can be simply approached by an expression of the type 1 x see 3 pp 283 In the SPARTAN code we have chosen this more simplified approach applying the equation AE maz E E 2 J Jpert 1 2 2 20 and using supplied values for AEmaz and Jpert The splitting of the exact perturbed level in two different sub levels is neglected 2 1 Discrete radiation models 19 2 1 2 3 Linear Polyatomic transitions The procedure for the calculation of linear polyatomic transitions inserted in the SPARTAN code due to the importance of CO2 IR radiation is quite similar to the one for diatomic transitions The emission coefficient is obtained as 1 4 where the additional term EF stands for the Hermann Wallis factors which describe the interactions between the vibrational and rotational modes see 3 p 110 Nyt gt Avon Son 3 Ey yuho 2 21 Ey The Einstein coefficient A for a purely vibrational transition is expressed as a function of the square of the vibrational transition moment Rwy ac cording to the relationship 6414 3 2 50 0 Az Bowe 2 22 3h 2 60 0 2 yr 5 2 026 10
129. nt Non IR fluxes are mostly negligible as they represent at most 2 in point 5 of the forebody 4 CONCLUSIONS Different spectral parameters have a Imited influence over the overall wall fluxes and a strong influence on calcu lation times Coarser adaptive line by line and pseudo continuum grids lead to less Voigt lineshape calcula tions and to more compact grids which can be handled quicker For the strictest run 1 spectral field and ra diative transfer calculations take about 1 week and for the more relaxed run 4 less than 3 days The size of the radiative data is also 3x smaller which has an impact on memory ramdrive requirements The overall calcu lation times are presented in Tab 3 At the outset of this work on spectral convergence a set of recommended parameters can be recommended from now on for line by line calculations of Martian en try flows It is expected that these optimal parameters can be roughly valid for most of entry flows namely the num ber of sampled rays Riviere 2003 the number of Voigt Lineshape points and the pseudo continuum grid This work has further shown that calculations can take as little as 3 days which seems to be promising for the general application of detailed line by line models for these gen eral radiation problems 3 5 r run 1 run 2 run 3 3p run 4 2 57 H s 2 r 7 G 2 1 57 4 1 0 5
130. nuniform spectral grid approaches have been proposed in the past considering a frequency division based on the equivalent widths of individual lines 8 9 These are not known to have been extensively utilized in practical applications However as can be seen in Fig 1 one can take advantage of the fact that some spectral regions lack significant radiative features Thus one can define an adaptive grid which would be defined as to be tighter in spectral regions with a large number of lines and more relaxed in regions with lesser lines Also to further attain the most compact grids means of calculating accurate lineshapes with a minimal number of points have to be devised Point a derives from point b in the sense that the methods utilized to calculate global spectral profiles over such grids must be computationally efficient thus requiring a minimum number of function evaluations Finally the spectra computed over such adaptive grids must prove to follow closely high resolution spectra which would enforce point c M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 109 10 10 10 1 Atomic Emission Coefficient VV m5 cm T sr o 10 107 a 10 10 10 10 Wavelength A 10 10 F Atomic Atomic Emission Coefficient VV m5 cm T sr o 10 a1 4 1 r l 10 10 10 10 Wavelength A Fig 1 Emiss
131. o 4 N 8 a z 2 ie ot of Ts a 0 5 o a Intensity A U o 0 5 Prasad amp Bernath 1992 0 5 0 ici rai i 3840 3850 3860 3870 3880 3890 VVavelength A Figure 4 1 Comparison between a CN Violet Av 0 spectrum measured in a high temperature arcjet facility and several spectra using different sets of spectroscpic constants 4 2 Building spectral datafiles for diatomic transitions 49 dependent spectroscopic constants to the more universal Dunham spectro scopic coefficients More often this is the result of vibrational perturbations in the upper or lower levels of the spectra which are the results from avoided crossings of these states rotationless potentials In this case the shape of the potential curve will change dramatically after a critical vibrational number v and the level spectroscopic constants will see a sharp change in their trends dis allowing their proper reproduction through the use of polynomial expressions depending on the variable v 1 2 1 64 r r i r r r r r 0 01 1 620 40 008 l 40 006 1 L 16 2 40 004 amp 1587 lt 2 Jo 002 5 5 s 1 56 5 pi x x x x 0 T 154 8 5 40 002 5 m 1 52 iz imi 0 004 15 m 0 006 148 4 0 008 1 46 i 1 i i i 0 01 0 1 2 3 4 5 6 7 8 9 Vibrational Quantum Number 1 8 r r r r r y r 0 01 0 008 0 006
132. o implement ex Ar_Atomic Right before the name you must choose if you want these transitions to be selected for calculation set 1 or not set 0 At this point it does not really matter which value you choose you can always change it latter by editing the file or by selecting the transitions names in the GUI interface Step 3 Editing the GUI interface 64 Modifying the Code Spectral Database Once you finished editing the Inputs txt file the GUI interface must be updated see Fig 1 1 Open and edit the Guinterface txt file located in the INPUTS folder This file defines the name the color and the position of the transitions names in the GUI interface In the first column you must add the name of the new transitions The column is divided in five sections depending on the type of radiation Atomic Discrete Atomic continuum etc Scroll down and at the end of the file you will find several available places placeholders to add the transitions with already defined positions and colors In this case you just have to insert the name of the transition in the first column If you wish to add more than the placeholder transitions you must add a new line and define e the transition name e the RGB colour of the name e the position of the name in the window Xpos and Ypos e the length L and width W of the name If you need to add a new label don t forget to update the number of labels at the beginning of the Guinterface txt
133. of a general expression e For discrete transitions from Diatomic molecules Simulation of transitions between 3 II and A electronic states Accounting for fine structure transitions singlet doublet and triplet except for transitions involving A electronic states which only can be treated as singlet transitions Accounting for A doubling effects for homonuclear molecules Transitions consider the intermediate a b Hiind case for rotational states Level energies can be calculated either by using a Dunham matrix or with the input of vibrationally specific spectroscopic constants Simplified treatment of rotational perturbations Individual Trot Tyiy and Teze for each species For now we assume Tyr T rot and To Teze e Fully customizable database for the calculation of each species total partition function Vibrational partition function sum obtained either from the truncated at De harmonic oscillator approximation or from the explicit input of levels Vibrational partition function sum obtained from 920 Trot 1 4388By onuc valid if By amp 1 4388T por e Fully customizable spectral database e Variable spectral grid methods allowing the production of compact yet accurate synthetic lineshapes A multitude of parameters such as the number of points of each Voigt lineshape is user parameterizable e De coupled excitation and radiative modules allowing for the direct supply of nonequilibrium
134. on is to be carried If the maximum wavelength pa rameter is set to 0 this parameter is overridden and the overall spectra is calculated 6 Maximum wavelength The user defines the maximum wavelength for which the calculation is to be carried If this parameter is set to 0 the overall spectra is calculated 7 Transitions database The user selects the radiative transitions which are to be calculated 8 Calculate Launches the calculation 9 Total spectra After the calculation is finished this button reproduces the overall spectra over the spectral range defined in boxes 5 and 6 Min imum Wavelength and Maximum Wavelength Setting a new minimum and or maximum wavelength and pressing the Total Spectra button will reshape the graphical window to this new limits 10 Single spectrum After the calculation is finished this button repro duces each radiative spectra with it s associated color over the spectral range defined in boxes 5 and 6 Minimum Wavelength and Maximum Wavelength Setting a new minimum and or maximum wavelength and pressing the Single Spectrum button will reshape the graphical window to this new limits 11 Erase This button cleans the graphical window 12 Record This button records the overall spectra nu Total txt and each radiative system spectra IE_IA_nu_01 i txt in the OUTPUTS directory Each ASCII textfile contains columnwise values of the wavenum ber emission coefficient and absorp
135. ondon Factors utilized in the SPAR TAN code The SPARTAN code always considers a set of H nl London Factors for an intermediary H nd a b case This allows the code to accurately describe radiative processes at both low temperatures typically H nd case a and high temperatures typically H nd case b General expressions for multiplet transitions for an intermediary H nd a b case are provided by Kovacs 62 We use these expressions overall except for 27 492 transitions which always belong to H nd b case with simplified expressions provided by Schadee 63 Expressions for X II transitions for an intermediary H nd a b case are provided by Arnold 64 Simplified expressions for the 39 3 transitions which always belong to H nd b case are provided by Tatum 65 Finally expressions for 52 II transitions for the intermediary H nd a b case are provided by Budo 66 The analytical expressions are presented in Tabs 3 9 Whenever the specific upper and lower levels index and are not specifically reported it is to be assumed that the variables report to the lower state The normalisation factor for the presented H nl London Factors is 2 do a 2S 1 2J 1 B 3 Intermediary a b case H nl London Factors 75 1 0 AA 1 J A J 4A 1 J A J A A 1 7 A 50000 077 a J J A J A 1 2J 1 A 2J 1 J A J A 1 2J 1 Q J 27021
136. ound a spacecraft Considering that M Lino da Silva Journal of Quantitative Spectroscopy amp Radiative Transfer 108 2007 106 125 123 spatial grids for radiative transfer applications may exceed one hundred cells each with very different plasma conditions see Fig 1 the necessity in optimizing the efficiency of line by line calculation routines becomes evident Furthermore when one is to examine other existing approaches for achieving efficient line by line methods in radiative transfer calculations it is important to study the approaches proposed in atmospheric sensing applications Not only can these approaches be found to be applicable to low pressure plasmas applications but one may also examine the possibility of extending the method discussed in this article to the former applications Currently multi grid approaches 23 26 appear to be amongst the most popular in this domain although different methods have been proposed based on the interpolation of reference absorption spectra simulated for different gas temperature increments 27 Similarities with the approach discussed in this article include Acknowledging different lineshape regions core and wings regions through the use of cascaded grids where rapidly varying line center regions are sampled in fine resolution grids and the smooth wings region are sampled over more coarser grids Although this method differs from the one proposed in this article the obtained r
137. parallel transitions e g R2 This approach is usually valid for multiplet X states which have small spin splitting factors allowing transitions between such states to be accurately mod eled as singlet states A good example can be given for a 7X 5 transition For this case the separation between two doublet lines in A is given by ye 5 AA 102 B 1 For the CN violet system this is equivalent to a line splitting of 0 14 A for a rotational value as high as J 100 In classical spectrometry applications this transition can be accurately simulated as a singlet transition reducing calculation times without a loss of accuracy For triplet transitions a slightly more complicated relationship can also be determined to evaluate whether the analyzed transitions can be accurately modeled as singlet transitions according to the transition y and A constants The limits of this assumption occur when the spin splitting of the studied states is no more negligible and self absorption of the spectra is high In this case a comparison of a simulated spectrum using this simplification with a sim ulated spectrum accounting for spin splitting shows non negligible differences Therefore this approximation is in general no longer valid for multiplet states with larger spin splitting values such as and A Table B 1 lists the branches that can be coalesced when the spin splitting of the X state is neglected
138. pe A different method has been devised for the calculation of high resolution profiles Two line parameters par and par have firstly been defined related to the line center and line wings respectively The line center being identi cal whether the profile is Lorentz or Doppler par will be independent of the parameters AP and The lineshape wings will however be very different regarding whether the line profile is Doppler or Lorentz par will therefore have a strong dependence on the parameters AY and AZ p and such dependence has been acknowledged in the following way Arp AT such that for a Lorentz lineshape we have par para and for a Doppler lineshape we have pary parwr paTup par PaATwD 3 2 A series of monotonous points has then been defined such as io 0 max num 1 maz 3 3 and a geometrical transform is applied such that 1 exp par c w x 00 1 exp PAF Ge 20 x io maz 1 exp par cew x io 1 exp par cw x io maz We then define i i and i 1 i An example of the geometrical transforms i and iw obtained for specific pare and pary values is presented in Fig 3 4 Practically speaking such transformations return two grids more or less tightened around the values 0 or 1 These two grids need now to be put to use Firstly for distributing the grid points in the line center region the second derivative of the Voigt profile is first
139. plemented in the code We also feel that with the constant improvement of the state of the art in spectroscopy and modeling 30 Detailed Description of the Code of physical chemical processes in gases an plasmas which is still undergoing major progress at the time of this last manual update 2013 it is important that the code structure allows place for major future upgrades For example progress in state to state modeling is likely to supersede more restrictive approximations such as considering that the different species internal modes follow a Boltzmann distribution With this in mind a modular structure for the SPARTAN code has been retained The code itself is split into an excitation and a radiative module The excitation module is tasked with simulating the population of the different upper and lower levels of radiative transitions handling the calculated values to the radiative module which then obtains the spectral dependent emission and absorption coefficients through the numerical routines which implement the appropriate quantum mechanic models The calculation parameters are supplied by an input file which can be bypassed if the code is utilized in a coupled fashion for example coupled to an hydro plasma code As stated before a GUI layer which is totally independent from the code itself is also proposed Finally a fully parametric spectral database is stored in a specific folder with each transition having its own filet
140. polar transitions have been sum marized by Herzberg 3 and are reported in table 2 3 For the a et b Hiind cases both A and S quantum numbers are defined and the following selection rules are observed 0 4 0 2 1 T additionally for a X X transition we have Ete Et E allowed 7 forbidden 2 2 For the a H nd case the quantum number not to be confused with the electronic state such that A 0 is also defined If both initial and final states of the transition belong to H nd case a we have the selection rule 2Other very specific coupling cases also exist but are not considered in the code 12 Physical Models transition between rotational levels Az 0 41 except 0 0 parity of the rotational levels allowed forbidden rotational branches Q AJ 0 e f allowed e amp e f f forbidden rotational branches P R AJ 1 e e f f allowed v f forbidden homonuclear molecules 8 s a a allowed s a forbidden same charge cores g amp u allowed g g u u forbidden Table 2 3 Selection rules for diatomic electric dipolar transitions A gt 2 3 accounting for selection rules 2 1 and 2 2 we obtain the selection rule AQ 0 1 AJ 0 forbidden for 0 2 4 Figure 2 1 Hiind case a vector diagram
141. r respondence between traditional spectroscopic expressions and the Dunham co efficients is then given by G v v we v 5 2 11a 1 io weze v 5 k n siz uyu 1 1 5 Y 5 Be Qe 32 2 11b i 0 taheti 2 D gt 532 D 0 v 3 2 11c 1 0 2 2 Hn GDES 1 0 2 Energies for fine structure levels When spin splitting is considered we need to introduce the constants A for spin orbit interactions y for spin rotation interactions and A for spin spin interactions The vibrational dependence of these constants is expressed in the usual way A A yl X A A l 5 5 2 12 0 We will now present the different equations for the level energies For general doublet states 5 U F3 J gt 1 By vat 21 4 4 7 49 74 3 0 F 9 J gt 0 B 4 Di J 1 1 2 4 4 1 4 Y Y 2 13b 16 Physical Models For 27 states 3 3 2 J 21 By J J 1 DU 1 w 2 2 14 251 2 7 20 By J J 1 1 Ww 5 2 14b For states the energy levels expression is given by 6 from the formulation of 7 with a typographic correction from 8 This expression is deemed more accurate than the general expression from 3 The expressions read 5Y 4 gt 2 B J J 1 Dy J J 4 1 Av Bu 4 Ay By Ayn 4 47 7 1
142. r of the spectroscopic constants that are selected Due to the inherent oscillatory characteristics of polynomial expansions such as those depending on v 1 2 for vibrational lev els and J J 1 for rotational levels a dataset with larger order corrective spectroscopic constants will be able to reproduce a broader range of vibra tional rotational quantum numbers v J but a dataset with lower order cor rections will usually provide better extrapolations beyond the initial fit validity range Therefore as a rule of thumb the higher the order of the spectral con stants corrections in polynomial form the most likely that the level energies above the range of the fitted v and J numbers will strongly diverge from their exact values The selection of an appropriate Dunham matrix can by itself allow the calcu lation of a synthetic spectrum which accurately reproduces experimental results Spectroscopy techniques have known almost spectacular developments over the last 50 decades and increasingly accurate data valid for a broad range of vi brational and rotational levels has been made available in the most recently years usually as the result of the application of very detailed Fourier transform spectroscopy methods A striking example for such developments can be pro vided for the CN Violet Av 0 spectrum who is ubiquitous in most of air and carbon chemistry plasmas This system has a very clear bandhead struc ture which can be detected up
143. ray tracing method In this method one samples a suf ficiently high number of light rays at constant zenithal 1000 500 1000 1000 500 500 1000 Figure 3 Differences of the two procedures for the tem perature field 2 1 8 1 6 a 1 4 a 1 5 1 2 gt 41 gt 21 x 0 8 F 05 0 4 0 2 r x 0 0 0 5 1 15 2 x m Figure 4 Wall points for radiative transfer 4500 4000 3500 3000 2500 2000 1500 1000 500 4500 4000 3500 3000 2500 2000 1500 1000 500 Figure 5 Averaged radiative grid and azimuthal angles As the number of rays is in creased the radiative flux through a surface such that Anderson 2000 ne 1 0 0sind9de 1 converges to the exact value and for each ray path we have the traditional equation for radiative transfer excluding scattering dI d x ol 2 The ray tracing routine itself has been developed taking into advantage the cartesian structure of the radiative grid Each ray is sampled over equispaced 0 5mm points and each of these points is then determined to fall inside a specific cell Taking into account the axi symmetry of the flowfield the length of the ray which crosses each specific cell is the calculated The zenithal angle is cal culated from 0 to 180 and the azimuthal angle is calcu late
144. rgy conservation 2adequate spectroscopic constants for the calculation of line positions and intensities need to be selected along with the adequate expressions issued from spectroscopic theory 3 5 The Lineshape calculation routine 35 b No further action is carried aside from discarding the weak lines In either cases steps 3 and 8 are overrided 3 Intensities of the lines calculated as a pseudo continuum are distributed in slots over a fixed step wavelength LinPar ContStep ranging from the pseudo continuum lines lower to higher wavenumber 4 For each line calculated explicitly a Checks if the Doppler and Lorentz broadening widths of the cur rent and former calculated line differ from more than a user defined parameter LinPar diff b YES Calculates the line center distance and intensity for each of the user defined LinPar lksi beta ksil betal num c num_w par c par_w par_w_FG lineshape points using a Voigt profile c NO Uses the former calculated line gridpoints sparing calculation time 5 Stores the overall points in ascending order from lower to higher wavenumbers for the determination of the spectral grid 6 Removes the points which are too close according to the user defined parameter LinPar minstep 7 For each line a Selects the grid points which fall under the calculated lineshape b Interpolates and adds the line intensities to the new gridpoints region 8
145. rm For T lt 1 eV Hummer 24 suggests considering the values provided by Menzel and Pekeris 25 who provide a good approximation of the Gaunt factor for the temperature range T 150 15000K and for the wavenumbers 2 10 350000 2 2 Continuum radiation models 25 2 2 2 Special cases Special cases in which some additional approximations or some analytical expressions are considered are discussed in this section 2 2 2 1 Photodetachment Photodetachment transitions are typically modeled with the assumption that 1 Only the ground state of the negative ion contributes for the overall ab sorption coefficient 2 The negative ions ground state is in a Saha equilibrium with the neutral species ground state The Saha equilibrium equation becomes in this case NAB Q app T RR Qa QaB T 0 2 46 0 xp 1 14388 EA with Qe 2 and Q apy T IAB Photodetachment absorption cross sections are then calculated in the usual fashion 247 2 2 2 2 Bremsstrahlung Some analytic expressions for the calculation of the Bremstrahlung emis sion absorption cross sections are available in the literature The classical emission coefficient for the Inverse Bremsstrahlung of atomic ionized species is given by Kramers 26 6 z 8 p Hz J m s sr Hz 5 Goo he vz N N MKS exp NoN
146. s which is seldom carried out Step 8 Edit the Einstein coefficients The Einstein coefficients also known as transition probabilities must be de fined in this file in a matrix form where v defines the upper vibrational levels and v the lower energy vibrational levels As for the Franek Condon factors defined in the last step the size of the matrix is defined by the number of vibrational levels defined before Once you accomplish all these steps you have created the first definition file necessary to implement a new species in the database 4 2 3 Special case for Homonuclear Fermion transitions The SPARTAN code is capable of treating a special case of homonuclear molecules for the Fermion case nuclear spin parameter 0 0 where one of each two subsequent even odd rotational lines are absent In those cases it sometimes occurs that two sets of spectroscopic constants obtained for both odd and even rotational states are proposed in the literature The SPARTAN code can account for these differentiated spectral constants again this is only valid for Fermions with the nuclear spin parameter 01 07 not Bosons with 17 17 In this case all the data from Step 2 2 a and 4 is duplicated with a datasets for even states followed by a dataset for odd states In this case a specific identifier has to be included in the Database txt file For example the label for a 15 II will no longer be 18 1 but 1SD 1PD where the label
147. s added as a pseudo continuum An illustration of this energy conservative scheme is presented in Fig 8 These two parameters have been tested for grid conver gence investigations The two sets of different lineshape and pseudo continuum parameters have been tested to Mies 10 248 2485 249 2495 25 2505 251 2515 252 Wavelength A x 10 Figure 8 Example of Strong Weak and Very weak lines talling 4 different sets of parameters these are detailed in Tab Table 2 Spectral grid parameters VV VVV VV VVV 1075 1076 1077 10732 c w 13 25 1 run 2 c vv 7 14 run 3 run 4 line center points line wings points W Weak lines threshold VW Very Weak lines threshold The obtained spectral fluxes are presented in Fig 9 for run 1 VVe verify that vvall fluxes issued from TR transitions gt 10 0000A are predominant due to CO radiation with a smaller contribution from CO TR radiation In the VUV Intensity 1091 0 W em Wavelength A 5 10 15 20 25 30 35 40 45 Wall Point Figure 9 Spectral distribution of the wall fluxes run 1 Visible region traace radiation from CO 47 CO 37 and CO Angstrom systems can be observed but only in the forebody region The spectrally integrated wall fluxes for the 4 runs is pre sented in Fig 10 No significant differences are observed for variable thresholds and differences for coarser lineshapes remain very fai
148. scopy dt Radiative Transfer 108 2007 106 125 107 Nomenclature o V absorption coefficient m7 b adjustable parameter in the calculation of W and FW Lorentz line profile v frequency Hz y wavenumber cm7 v gridpoint wavenumber cm7 E emission coefficient W m srem 1 4 adjustable parameter in the calculation of W and FW C constant contstep spectral interval for pseudo continuum uniform grid contstep C x FWHM wea cm7 dnu maximum spectral distance where a strong line is considered to cover a weak line dnu C x FWHM cm FW boundary between the line near wing and far wing regions cm FWHM full width at half maximum cm g Doppler Gaussian line profile i transformed nonuniform grid Io radiative intensity at the exit of the hot slab W m sr cm7 io initial uniform grid I radiative intensity at the exit of the cold slab W m sr cm7 line core grid i i iw line wings grid iy 1 i max maximum value of the monotonous grid num number of points distributed inside the half line core region numy number of points distributed inside the half line wings region par nonuniform grid spacing parameter in the line core region paryp shape weighting parameter Doppler for the calculation of pary PA yy shape weighting parameter Lorentz for the calculation of pary pary nonuniform grid spacing param
149. se With Great Power comes Great Responsibility Voltaire 4 1 Introduction This Chapter provides an in depth description of the spectral database structure of the SPARTAN code with the description of the necessary steps for its update or upgrade according to the user needs The capabilities of the SPARTAN code are summarized in Table 4 1 Bound Bound transitions are all treated through its own subroutines AtomicX m which implement the theoretical models described in Chapter 3 Bound Free transitions are subdivided in photoionization photodissociation for diatomic and triatomic molecules and photodetachment transitions which can either use total or state specific cross sections Finally Free Free transitions are hard coded in their own routine Bremstr m Discrete transitions from diatomic species play a predominant part in the spectral features of gases and plasmas As such the databases for the simulation of these transitions are the most liable to be the subject of extensions and updates Accordingly we will begin this section by providing some guidelines for the selection of an appropriate spectral database for any given bound diatomic transition We will then follow with the description of the steps necessary for the construction of the datafiles for the calculation of bound diatomic spectra Next we will describe how to update the databases for the other transitions bound transitions for atomic and linear triatomi
150. sfer results for different pseudo continuum threshold parameters see figure in Table 5 To 5000 T 1000 14 x 1m Ii x 100km I Io Ii Unires 70 116 212 6 7057 2 2124 32 99 100 00 113 26 6 7071 2 0987 31 29 94 86 113 23 6 7071 1 9738 29 43 89 22 112123 6 6915 1 9128 28 59 86 46 S Calculation times t16t212 61 7 100 00 113126 41 5 67 25 113123 36 1 58 44 112123 14 5 23 48 The table reports the radiative intensity exiting from the hot Jo and cold 7 slabs the percentage of the incoming hot slab radiative intensity transmitted by the cold slab 1 Jo and the differences in transmissivity compared to high resolution calculations 11 11 hires Further below are reported the absolute and relative calculation times higher are absent and accordingly the number of spectral lines to be simulated may be as low as a few hundred or as high as a few thousand of lines This explains why numerical line by line methods utilized in such applications have not been significantly optimized simple line by line methods relying on fixed width grids yield detailed results without large memory or computational overheads Nevertheless as discussed in Section 2 atmospheric entry applications have fostered the need for looking out for more efficient line by line methods as this kind of application requires the calculation of more than one million spectral lines for each cell of the 3D spatial grid generated ar
151. sion becomes E r 7 1 2 51 If we use wavenumber units 7 em 1 the term 2hv3 c in Eq 2 50 is replaced by 2hc 7 For wavelength units A A the term 2hc A5 is used 2 3 2 Photoionization transitions For photoionization photodetachment processes such that ABC T hv gt ABU 71 the relationship for the bound free ops v photoionization absorption cross sections and the free bound radiative recombination cross sections is given by the Milne relations 29 30 2 0 oof v 1 FC gige 2 52 2 gn If the electrons are thermalised and follow a Maxwell distribution function Eq 2 52 can be further simplified to 2 3 Generalized Kirchhoff Planck Law for radiative transfer 27 gi Skpmec Te 2 53 op gn vh 2 253 The emission coefficient can then be calculated in the usual way ey Ni Neo p v 1 e 7 2 54 v N Neopiv 1 exp 5 kpTe Further if we have an Saha ionization equilibrium in the plasma we can in stead use the general Planck Kirchhoff law of Eq 2 51 for relating the emission and absorption coefficients by simply replacing T by Te 2 3 3 Photodissociation transitions For photodissociation processes such that AB i hv A B the relation ship for the bound free op z and the free bound o cross sections is given in wavenumber cm units by 31 os ofp qne ik T 2 55 if we a
152. sition Dirac to a line following a specific shape Broadening mechanisms can be split into two different categories e Broadening from atomic and molecular collisions described by a Lorentz shape e Broadening from Doppler effects described by a Doppler shape Universal expressions are presented in this section Some of these expres sions are in general approximate but should suffice for the level of detail needed for the typical applications of the SPARTAN code proving flexible enough for allowing an automated calculation of each transition broadening widths All broadening width units are presented in wavenumber units lem 1l with the species densities in lem 91 2 1 3 1 Collisional broadening mechanisms Collisional broadening processes are described through a Lorentzian line pro file such that 1 7 Sa Re 144 32 The convolution of different Lorentz line profiles also yields a Lorentz line profile such that 3 Arz 2 24 Natural broadening The linewidth depends on the radiative lifetime 7 ac cording to the following expression 1 Avy 2 25 RN ATCT This broadening mechanism is generally very small and has accordingly been neglected in the SPARTAN code For example a radiative lifetime of 1 ns yields Aza 0 005 cm Pure collisional broadening This process stems from the rate of collisions between the different particles in the gas The equivalent width has the follow
153. ssume that the overall species translation velocities are Maxwellian Here p in the molecular species reduced mass in kg units 2 3 4 Bremsstrahlung transitions For Bremsstrahlung transitions such that AB e AB e if we as sume that the electrons are thermalized i e follow a Maxwell Boltzmann vdf we may define the emission absorption coefficients detailed balance through the Planck Kirchoff expression proposed for the conditions of thermal equilibrium replacing the term T by Te 2hr h i lew z 1 2 56 28 Physical Models Chapter 3 Detailed Description of the Code gan YUVE ENEL ADH HVE N N41 3 7 5 936 end 937 if stremp Trans1t1ons 25 25 1 stremp Transitions 2P 25 938 939 N 1 SpeciesData dimenrot 940 F1Gla BvG N N 1 DvG CN N 1 A2 HvG CN N41 A3 941 0 5 repmat gammaG SpeciesData dimenrot 11 942 gammaG N N 1 gammaGIJ CN N 1 42 repmat n 943 F2Gla BvG N N 1 DvG N CN 1 A2Z HVG CCN N 1 A3 944 0 5 repmat gammaG SpeciesData dimenrot 11 945 gammaGJ N N 1 gammaGIJJ N N 1 A2 3 repnat N 1 946 947 N 0 SpeciesData dimenrot 1 948 F1GOa BvG N N 1 DvG C N N 1 A2 HVG CCN N41 A3 949 0 5 repmat gammaG SpeciesData dimenrot 1 950 gammaGJ N N 1 gammaGJJ N N 1 A2 repmat n
154. t and complex error function a comparison of computational methods JQSRT 1992 48 5 6 743 62 11 Wells RJ Rapid approximation to the Voigt Fadeeva function and its derivatives 1999 62 1 29 48 12 Kobayashi H Line by line calculation using Fourier transformed Voigt function 1999 62 4 477 83 1131 Whiting EE An empirical approximation to the Voigt profile JQSRT 1968 8 6 1379 84 14 Olivero JJ Longbothum RL Empirical fits to the Voigt line width a brief review 1977 17 2 233 6 15 Zhu X An improved Voigt line approximation for the calculations of equivalent width and transmission JQSRT 1988 39 6 421 7 16 Smith AJ Gogel T Vandevelde P Plasma radiation database PARADE Final Report ESTEC Contract 11148 94 NL FG TR28 96 April 1996 17 ESA SP 533 In Proceedings of the Ist international workshop on radiation of high temperature gases in atmospheric entry 8 10 October 2003 Instituto Superior T cnico Lisboa Portugal 18 ESA SP 587 In Proceedings of the Ist international workshop on radiation of high temperature gases in atmospheric entry part I Porquerolles France 1 2 October 2004 19 ESA SP 629 In Proceedings of the 2nd international workshop on radiation of high temperature gases in atmospheric entry Rome Italy 6 8 September 2006 20 Lino da Silva M Dudeck M A proposed solution to test case 1 using the SESAM code In ESA SP 533 1st international workshop o
155. t pain Best of luck to everyone Appendix A References for the SPARTAN Spectral Database In this appendix the references for the datasets included in the current version of the SPARTAN database are presented except for the bound diatomic transitions which have self contained information on the spectral data references Table A 1 presents all the simulated transitions in the current version of the SPARTAN code and their corresponding references Table A 2 presents all the references that have been consulted for building the partition function files AB LEV txt for each radiative species The reader should consult the files comments for more details on the each spectroscopic constant reference 68 References for the SPARTAN Spectral Database Transition Reference Bound Atomic Transitions H C Ct N Nt O OT He Xe Xet NIST database 36 Bound Diatomic Transitions H Lyman Werner C Swan Philips Mulliken Deslandres D Azambuja Fox Herzberg Ballik Ramsay CN Violet Red CO Infrared 4 Positive Angstrom 3rd Positive Triplet Asundi CO Comet Tail See database file headers N 1 2 4 Positive NF 18 Negative NH AX NO 8 Oz Schumann Runge OH AX Bound Linear Polyatomic Transitions CO Infrared 16 1 Atomic Photoionization Transitions H C Ct N Nt O OT TOPBas
156. t version 65 4 5 Linking new spectral datafiles to the SPARTAN database NVINVdS 92 Ul p poov sodA WOTZIsURIy m s myip sy JO sz ynm pi f qer 9 y 9149 oruro erq Sunyqenssur qg Iqsu sq oruogeouoy Sunyqessur q i Dusmu sq Sunrqessur rq 14swerg UOT OOS SSO1IN JUOpUddep a 10 4 TAO401U4 1011298 88010 TEqOys UOT VIDOSSIPOJOY A OLO40U4 OIULOJeIC vuoneroossrpoaoqq Idtqdoud UOT DS SsO1D UOT eZTUOIOJOY d TAO40U4 UWOTJDOS SSOID TEQOfe 1011621001010 102044 IWO 40116821 01010 4 10010 4 uunr qiynb eyes qu urq e poqovyq es qoqq guouryoLegopozoydq 01940 4 SUOI DOS SSOID sEqdOT ASVEdOL uonezruoroqoqq 010 4 punog 704 104 punog 04 442 446 42 46 Kelle 486 046 se de Ile Ke qd4 GS de se 086 488 S S Hz H dd dd 4z ac Qz lz qsz qac Sz dc Hz X 042 092 4 SZ Xz Kz 094 482 SZ SZ VrV qqT daT dt at I V q4T ddT 4T 4T VrlI qdT ddT 4T dT Hr ddt ddt 4T dT XrH 4ST ddT ST dT Il Xr Gdt dst 4T ST 1 4 aSt dst ST ST ruroyerqq punog Iqpunog Iwuogzeouoy punog OLV OLV oIUOyeoUOoTY punog owpunog 6 102 8 02 odA SUOT ISURIT sjoqe T d4A3 SuOT ISUeI
157. tate and a radiative state where line radiation re absorption in the cold gas layer will be of paramount importance The two upmost important parameters of this variable grid method are the number of grid points utilized for individual lineshape calculations and the thresholds for strong weak and very weak lines separation Consequently two different test cases have been defined A first test case which examines radiative transfer involving a triplet atomic transition and a second test case which examines radiative transfer involving a diatomic vibronic band Transitions in the VUV region have been chosen as test cases to ensure that strong absorption coefficients are obtained in the different test cases 4 1 Lineshape accuracy The examination of the accuracy for the different calculated lineshapes has been carried out defining a test case where the triplet atomic transition 2s22p 2s22p2 5 P 34 of atomic nitrogen at 953A is simulated assuming a constant number density of 6 x 102 part m A first benchmark calculation has been carried for a restrictive case of radiative transfer from a hot slab at 20 000 K to a cold slab at 1000 K case 1 However such a restrictive case is not likely to be encountered except perhaps for atmospheric entry applications considering the simulation of radiative transfer upstream of a shockwave A second benchmark with lower temperature gradients has therefore been defined with a hot slab at 10 000 K and
158. ter defining the number of specific vari ables you want to use you must specify which are the ones you are considering Then you must introduce the vibrationally specific constants for each level in a n x m matrix where n is the number of vibrational levels and m is the number of specific variables considered In case you do not want to use vibrationally specific constants just set the label vibrationally specific constants to 0 and proceed to the next step Step 5 optional Define the perturbations in the rotational spectra As for the case of explicit values if no perturbations to the rotational spec trum are to be inserted set the perturbations label to 0 and proceed to the next step Otherwise set the label to 1 Please remember that in the current version of the code only doublet states can be perturbed 7 We will now list the parameters to be inserted The first line should list the number of perturbations n followed by a 6 x n listing for each column value 1 if the electronic state is the upper excited one 0 otherwise value 1 if the spin split level is F1 0 otherwise F2 the perturbation rotational quantum number J 1 the perturbation maximum energy shift dE in cm units the reference for the perturbation data It is possible to check the file N211 1N txt for an example on how the perturbations are treated Step 6 Define the number of rotational and vibrational levels This is li
159. the constants for the new species Step 2 a Add fine structure constants for doublet or triplet states In this tutorial we have considered the implementation of a new molecular species assuming transitions between singlet states If you want to implement transitions considering doublet or triplet states you must consider fine structure corrections shown in Table 4 3 After the Dunham matrix for the upper and lower level respectively add a new line with four columns where you introduce the four fine structure terms Again if you have no values put zero the insertion of four values is mandatory To see an example of how the fine structure constants are inserted after the states Dunham matrix please refer to file N2 1P txt describing a 511 X transition 52 Modifying the Code Spectral Database Y 0 1 2 3 4 5 6 7 8 9 G v 0 Tle We Wele WeYe Were Welle Webe Wele X x B 1 Be Qe Ye z Ne x x x x x D 2 De Be x x x x x x x x Hy 3 H aH x x x x x x x x Ly 4 x x x x x x x x x x M 5 x x x x x x x x x x Ny 6 x x x x x x x x x x Table 4 2 Correspondence between Dunham coefficients and traditional spec troscopic constants iy H 25 5 7 O uv 1 4 yOu 4 7 uv 8 TT A AO AD vu 1 AP vu 4 AM t Ww y y kd v dk 9 O 2 ev Ay AM AD v 4 AP v 4 2 HAC w 95 3 A AO
160. tion a la planete Mars Ph D thesis Universite d Orleans Orleans France 2004 in French 2 Peraiah A An introduction to radiative transfer methods and applications in astrophysics Cambridge UK Cambridge University Press 2002 3 Kalkofen W Numerical radiative transfer Cambridge UK Cambridge University Press 1988 4 Mazoue F Marraffa L Flow field radiation coupling analysis for Huygens probe entry into Titan atmosphere In Paper 2005 5392 38th ATAA thermophysics conference Toronto Ontario 6 9 June 2005 151 Wright MJ Bose D Olejniczak J The impact of flowfield radiation coupling on aeroheating for Titan aerocapture J Thermophysics Heat Transfer 2005 19 1 17 27 161 Johnston CO Hollis BR Sutton K Radiative heating methodology for the Huygens probe In AIAA Paper 2006 3426 9th ATAA ASME joint thermophysics and heat transfer conference San Francisco CA 5 8 June 2006 7 Osawa H Matsuyama S Ohnishi N Sawada K Comparative computation of radiative heating environment for Huygens probe entry flight In AIAA Paper 2006 3772 9th AIAA ASME joint thermophysics and heat transfer conference San Francisco CA 5 8 June 2006 8 Oinas V A new method for the rapid calculation of infrared transmittances of atmospheric gases JQSRT 1981 26 4 381 3 191 Oinas V Rapid transmittance integration using line blending and a straight line fit to line shapes JQSRT 1983 29 5 407 11 10 Schreier F Voig
161. tion coefficient 1The code can however consider different internal modes temperatures for each species see chapter 3 6 Getting Started 13 X and Y axis scale These two buttons allow the user to switch between linear to logarithmic scales for both the X and Y axis Note that besides setting these parameters in the code GUI the user must define the local number density of the different species present in the simulated gas These values must be changed by the user in the Inputs txt file in the code INPUTS subdirectory 1 1 2 How the Graphical User Interface works The SPARTAN code GUI has been built as an upper layer to the core ver sion of the SPARTAN code SPARTAN_noGUI The SPARTAN code receives it s calculation parameters from the file Inputs txt besides it s other associated files Database txt and Lineshape txt which will be further discussed in chapter 3 The GUI commands merely updates the values of the Inputs txt file without any interaction with the core of the SPARTAN code Therefore this GUI can be straightforwardly overridden for coupled calculations using the SPARTAN code and other numerical codes 1 1 3 Other user defined parameters Besides the parameters modified through the GUI and the local species densities in units of particle m that have to be input in the file Inputs txt many other parameters maybe adjusted by the user of the code However these imply a more detailed knowledge on the stru
162. to v 8 9 with a band reversal from the ultraviolet direction towards the infrared direction starting at v 4 5 whose bandheads are experimentally overlapped In Fig 4 1 an experimental spec trum issued from an arcjet low pressure wind tunnel 1 is compared against several synthetic spectra produced using the best fit vibrational and rotational temperatures Tyip 10800 100K and T ot 4900 100 K and applying several sets of Dunham coefficients One may verify that the reproduction of the higher vibrational number bands is increasingly poorer when resorting to older spectral datasets Namely the v 4 6 bandheads can only be accurately reproduced with the most recent datasets issued from Fourier transform spectroscopy techniquesi In some specific cases equilibrium constants will not provide sufficient ac curacy for the reproduction of experimental spectra Sometimes this might be related to the extra loss of precision resulting from the extra fit from level 1 However even such techniques may sometimes not be adapted for the reproduction of high temperature spectra in such cases where lower temperatures of the gas plasma are im posed so as to have very little Doppler broadening of the measured lines which comes at a cost of a lower Jmaz of the fit since higher rotational levels will not be sufficiently populated at such values for T 48 Modifying the Code Spectral Database 0 5 pa z 5 8 g
163. ts More precise line profiles will be obtained for a larger number of assigned grid points at the cost of increased spectral grids and calculation times Different low and high resolution lineshapes have been compared with the exact Voigt profiles Eq 2 with Av 5cm7 The differences to the exact profile have also been determined for each of such profiles The obtained results are presented in Fig 5 for low resolution profiles and in Fig 6 for high resolution profiles 3 3 Handling and overlay of individual lineshapes Following the definition of a method for adaptively calculating individual lineshapes one may apply it to the calculation of the individual lineshapes for each of the calculated radiative transitions with their accompanying Doppler and Lorentz FWHM However doing so for a large number of lines typically more than 107 might still yield large calculation times One can acknowledge that the systematic evaluation of exponential terms of Eq 2 for the calculation of several thousand lines might be an ineffective method Instead one can take advantage of the fact that the different lineshapes Doppler and Lorentz FWHM remain very close over a large spectral range The calculated lineshapes are then nearly equivalent and there is no need to systematically recalculate them As an example the differences between two Voigt lineshapes for a 10 variation of their Lorentz and Doppler FWHM are presented in Fig 7 Given the small
164. ty and temperature Theoretical expressions for the calculation of Stark broadening are not available except for hydrogenoid species 18 Tabulated values providing the parameter AXg as a function of ne and Te are used 19 Ne 1016 In its current version the SPARTAN code totally neglects Stark broadening due to the difficulties of implementing this broadening process in an universal fashion Future versions of the code will likely be updated to allow for some amount of accounting for this process AAs f Te 2 30 2 1 3 2 Doppler broadening Doppler broadening profiles follow a Gaussian shape and are described by the following expression a D H V Vo As for the Lorentzian collisional lineshapes the same additivity rule applies for Doppler Gaussian type lineshapes Y ATF 2 31 22 Physical Models Doppler broadening is a consequence of the thermalized motion of the ra diating species A molecule radiating at a frequency vo in its own reference plane and approaching at a velocity v from the observation plane will have a Doppler type shift in the observation plane such that v vo 1 z Assum ing a Maxwellian velocity distribution function at a characteristic temperature T we may obtain the corresponding Doppler broadening width kpT Av To4 8log 2 2 32 mce 2 1 3 3 Voigt line profile A Voigt profile results from the convolution of a Lorentz and Doppler
165. um for a 97 CO2 3 N 1 bar mix ture in full thermochemical equilibrium at 1 000K 5 000K and 10 000 An excel file 0202 RadPover xls is also included providing the inte grated radiative intensity of a 9776 CO2 3 N mixture at 1bar and 4 300Pa 1 4 Stepping a little further This chapter has dealt with the essentials for a quick launch of the line by line code SPARTAN using its graphical interface For the more detailed description of the physical models implemented in the code please refer to Chapter 2 For a detailed description of the SPARTAN code structure and routines please refer to Chapter 3 For the information on how to customize the code for your specific needs please refer to Chapter 4 Getting Started Chapter 2 Physical Models HU Schrodinger s Equation This Chapter provides an abridged description of the theoretical and nu merical quantum models that have been implemented in the SPARTAN code The Chapter is split into three sections which respectively describe the imple mented theoretical models for the calculation of discrete radiation the calcu lation of continuum radiation and the generalized relationships between emis sion absorption coefficients 2 1 Discrete radiation models The theory of discrete atomic and diatomic transitions is shortly summarized in this section starting with the selection rules which indicate which radiative transitions are allowed between the speci
166. um of functions using a three grid approach Comput Phys Commun 2006 174 10 783 92 1271 Titov DV Haus R A fast and accurate method of calculation of gaseous transmission functions in planetary atmospheres Planet Space Sci 1997 45 3 369 77 28 Mitsel AA Firsov KM A fast line by line method JQSRT 1995 54 3 549 57 29 http efp ist utl pt radiation SPARTAN A CONTRIBUTION FOR THE SIMULATION OF VUV IR RADIATION TRANSFER IN CO2 N2 ENTRY FLOWS USING A LINE BY LINE MODEL M Lino da Silva Instituto de Plasmas e Fus o Nuclear Laborat rio Associado Instituto Superior T cnico 1 Av Rovisco Pais 1049 001 Lisboa Portugal ABSTRACT Departing from the proposed Test Case 3 for the sim ulation of Martian atmospheric entry radiative transfer we present several simulations carried out using a full line by line spectral simulation ranging from VUV to IR Namely the radiation of COz Infrared transitions are treated using a two temperature line by line model Several calculations are presented which show case the ability to solve the uncoupled radiative transfer problem Heat transfer towards a spacecraft thermal pro tections in a timely fashion using a 8 core 32GB RAM Linux Debian machine In these calculations different criteria are evaluated 1T vs 2T models different line by line spectral grid parameters different spatial grids for radiative transfer which allow determining their in pact on the overal
167. ures and hence compositions as we are assuming chemical equilibrium Secondly depending on the considered spectral range radiative features may be abundant or scarce This means that any adequate spectral calculation must have sufficiently adaptive features to be able to efficiently reproduce spectra as different as the ones presented in Fig 1 We may start by examining the problem boundaries Spectral ranges to be considered for the simulation of plasma radiation typically spread from 10 to 10 cm or more for some continua For the typical conditions of an atmospheric entry spectral lineshapes have widths of about 0 5cm7 At least three points are required for resolving a line peak which means that considering a fixed width grid would amount to at least 6 x 10 spectral points or more if the Voigt lineshapes are to be correctly reproduced Given the current state of the art computational capabilities handling this type of grid sizes remains out of reach Fixed width methods are therefore limited to the simulation of restricted spectral regions which encompass the most paramount radiative features of the flow One such example which can be presented is the recent entry of the Huygens probe in Titan s atmosphere Here the radiation from the CN Violet system accounted for the almost overall radiative power of the plasma surrounding the probe This has allowed utilizing fixed width spectral grids for coupled calculations 4 7 Nevertheless no
168. ut and indirectly supplied to the RKR_SCH_PC function passing the r and V variables Note If for some reason the recalculated potential curves do not seem accurate try lowering the Umaz parameter describing the boundary beyond where the supplied Dunham matrix is no longer valid The files CParam txt and CParam_J txt define several accuracy parame ters for the RKR calculation as well as the fixed spectral grid where the radial Schr dinger equation is solved The currently input values should in principle suffice for all practical calculations and they should be modified at the user s own risk Appendix D Other Auxiliary Routines This section shortly describes several auxiliary functions which are supplied with the SPARTAN code bundle They are located in the code root directory These functions should be used after running the main SPARTAN code They interact with the the workspace variable ResultTotal which the code SPARTAN yields as an output Avg_IA m Averages the absorption coefficients calculated by the SPARTAN Code and yields band mean absorption coefficients and Planck mean absorp tion coefficients for a custom set of spectral intervals Intervals in A are defined in the first line of the Function The vari ables avg and variables avg_P1 respectively the band mean absorption and Planck mean absorption coefficients are returned to the workspace Compare m Compares the synthetic SPARTAN spectrum against me
169. vels with ener gies above the electronic state dissociation limit De e Adding the option to plot the Planck Blackbody emission coefficient e Inclusion of a semi functional version of the Ray Tracing Radiative Trans fer routine developed for the simulation of radiative transfer in atmo spheric entry flows For future versions the following updates will be undertaken yet are not scheduled to date e Implementation of a QSS model for nonequilibrium excitation modeling in air plasmas E 2 Code Regressions e Compared to version 2 3 version 2 5 does not simulate perturbations for the N2t First Negative System with as much accuracy F 3 Code Run Times The run times for the different radiative systems of the SPARTAN code database are reported in Table E 2 Calculations have been run on an Intel Core i7 620M processor with 6GB of memory This is valid for the database include with the version 2 5 of the code Calculations have been carried out with either the original Umax or limiting Uma to 11 The resulting radiative grid sizes are also reported for both cases For multiplet transitions fine structure has been accounted for As a rule of thumb if calculation times are deemed too long for a specific system one may first limit Umax to 11 and or treat multiplet transitions as singlets In the first case neglecting the higher vibrational states should not lead to significant differences except at very high temperatures where these
170. x Instead the sounder approach consists in a simple averaging of the original points inside the cell or a weighted averaging if said points are not eq uispaced An illustration of the averaging procedure is presented in Fig 2 As an illustration the differences in the averaging of the initial RTech temperature flowfield over a coarser grid is presented in Fig 3 One may see that the differences in the averaged temperatures may ex ceed 500K in the regions of larger gradients such as the Proc 4th Int Workshop on Radiation of High Temperature Gases in Atmospheric Entry Lausanne Switzerland 12 15 October 2010 ESA SP 689 February 2011 x Xx Xx XX RK KS KK Xxx XX xy 575 a XXX x x x XXX LK KK RK KL xx xXx x Xxx Xxx Xxx Figure 1 Details of the original RTech grid shock layer XXxxxxx x XXXXXXX X X KX KK KK XK K xix Figure 2 Averaging procedure over one grid cell In addition to the mandatory 24 points in the spacecraft afterbody region we define an additional 24 points in the forebody region in order to allow for a more complete simulation of radiative transfer towards the entirety of the spacecraft wall boundaries The discretized wall points are presented in Fig 4 The associated averaged radiative grid is presented in Fig 5 Radiative calculations are carried out resorting to the

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