A "I'IIEDRETICAL AND DiPERIMENTAL INVESTIGATION OF THE MECHANISMS OF THE HYDROGEN -FLUORIDE PULSED CHEMICAL LASER Cissertation for the Degree of Ph. D. MICHIGAN SEATE UNIVERSITY EOSEPH 3. T. HOUGH 1975 LIBRARY MchiganStaoc Univasity This is to certify that the thesis entitled 1 A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF THE MECHANISMS OF THE HYDROGEN-FLUORIDE PULSED CHEMICAL LASER presented by Joseph J. T. Hough has been accepted towards fulfillment of the requirements for Doctor of Philosophy degreein Mechanical Engineering fauna/OZ @4. Major professor Date March 6, 1975 0-7639 3‘. “#951171- ~ - L alumna av . , " ‘IIOAB & SONS' I ~BOOK BINDERY' LIBRARY amoens "3 v, . ,n I ‘ 1d“ t11’.|'> "I .‘-"- U ' .I‘E‘l‘wu rare. .‘ u .(2 . > “213'“ r' I .1 _. "L's: I, ‘1’ ‘ibnbcv. t“:\w.I.‘n;:‘-.i hier- ‘. ‘ " . an my". . ‘ . ‘ . l .241. s ‘3 r \ r . .5”). ‘.. u" ..,i = .s ,. _ - v 0‘. \ €- ' ' 1- . .> ' I , s - n- ~ ' file {1| TA “ 7"“ f ‘1‘" " ‘3 .RPNLD'» I. TIME?!) to a we is, :’h:«,~:_mv.-i:u. .w “i-‘I'T' I) myrf'é-a‘i‘thre .3!" V >. . I , I 2' :7 v more mun: 31w; (0 run-NW? ' -.'(~ . ‘ ‘~‘:1‘S"H*IMT 7n3mle-26‘f: 13/33.;{5' Th‘ ._ ,L. ‘ _x_‘ ,1 ' .- “W10; NRWME gig: ~ (J‘IOT, Sand *“' s p.41»... 53$“ ”% gig. -_ A 3'4WWW KI A“ (/(A ABSTRACT A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF THE MECHANISMS OF THE HYDROGEN-FLUORIDE PULSED CHEMICAL LASER by Joseph J. T. Hough Kinetic mechanisms in the pulsed HF chemical laser are examined by comparing predictions from improved theoretical models with experimental observations. Additional insight into these mecha- nisms is obtained by examining the predictions of the models in regions outside the domain of the present experiments. Two rate-equation models are developed for the F + H2 and F2 + H2 lasers. Both models simulate cavity transients such as the interaction of the chemical kinetic and radiative phenomena within the active medium in the pulsed HF chemical laser. Calculations show that the growth in intensity is sufficiently slow that the gain may attain levels far above threshold before lasing begins. Intensity increases sharply after the gain is far above threshold, which causes the gain to drop rapidly and oscillate near the threshold value. This fluctuation in gain, which is a result of fluctuation in HF(v,J) populations, makes the calculations more sensitive to relaxation processes than has been shown by earlier constant-gain models. The model of the F + H2 laser is further expanded to study the effects of preferential pumping into rotational levels, rotational relaxation, and rotational nonequilib- rium. Rotational relaxation is modeled by associating with each (v, J) Joseph J. T. Hough state a relaxation time constant (7) consistent with available experimental data and theoretical calculations. The results show that, as the rotational relaxation rate decreases, the laser output decreases and pulse energy is more uniformly distributed among the lasing transitions within a given band. The predicted time-resolved spectrum exhibits strong simultaneous lasing on many vibrational- rotational transitions. The model is capable of predicting the perfor- mance of lasers operating in the line-selected mode. A comparison is made of the output from a three-transition cascade with the same three transitions operated in the single-line mode. The sum of the output from the three single-line calculations is 65% of the output from the cascade. Concurrent with the theoretical work, output from an electrically initiated, transverse pin discharge SF6-H2 pulsed laser was character- ized with model predictions, and the results are in good agreement. Experiments were conducted to investigate the effect of cavity losses and cavity threshold on the performance of the laser operating in the single line as well as the multiline mode. Losses within the optical cavity as small as 5%, typical of many lasers, can easily result in a 30% reduction of output power. This is especially significant for high-Q cavities. Model predictions graphically illustrate the effect of cavity threshold on laser output. These results should be helpful in the design of line-selected extraction techniques for high—power, pulsed (and continuous) HF lasers . i ', (W'EDHEORETICAL AND EXPERIMENTAL INVESTIGATION OF THE ‘ MECHANISMS OF THE HYDROGEN-FLUORIDE PULSED CHEMICAL LASER BY Joseph J. T. Hough A DISSERTATION Submitted to. Michigan State University in partial fulfillment of the requirements for the degree of poems. or PHILOSOPHY Department of Mechanical Engineering 1975 ACKNOWLEDGMENTS I express my appreciation to my thesis adviser, Dr. Ronald Kerber, for his guidance throughout the preparation of this thesis; his friendship and generosity of his time has been a real source of encouragement. I thank the faculty members who served on my doctoral guidance committee: Dr. Jes Asmussen, Dr. Jack Kinsinger, Dr. George Leroi, Dr. Mahlon Smith, and Dr. Richard Zeren. Their reviews and critiques have been most valuable. The Department of Mechanical Engineering and the Division of Engineering Research, Michigan State University, have provided me with teaching and research assistantships during my tenure as a graduate student. For this, I am sincerely grateful. I am also indebted to Mr. Paul C. Fisher, President of the Fisher Pen Company, whose financial assistance gave me the peace of mind to carry out my studies. I sincerely thank The Aerospace Corporation and Mrs. Polly Hicks of the very able editorial staff for assistance in the final prepara— tion of this thesis for publication. Finally, I would like to express my appreciation to my wife and parents whose constant optimism, faith, and patience have been the mainstay of my inspiration. TABLE OF CONTENTS 1. INTRODUCTION ........................... . 1. 1 History ............................ 1. 2 Present Work .............. . ......... 2.. COMPUTER SIMULATION OF PULSED HF CHEMICAL LASER .............. . ....... . . . 2. 1 Model Formulation .................. . . . 2. 2 Effect of Relaxation Oscillations ...... . . . . . 2. 2.1 Comparison With Existing Models ...... 2. 2. 2 Other Model Features ........... . . . 2. 3 Effect of Rotational Nonequilibriurn ...... . . . . 2. 4 Summary . . ....................... . . 3. EFFECT OF CAVITY LOSSES ON PERFORMANCE OF THE SF -H PULSED CHEMICAL LASER: THEORYA D XPERIMENT . . . . . . . ..... . . . . 3.1 Introduction.. ...... ..... 3. 2 Experimental Details . . . . . . . . . 3. 3 Computer Simulation . . . . . ....... . 3. 4 Results and Discussion . . . . . . . . . 3.5 SumInary.......... . . ..... 4. SUMMARYAND CONCLUSIONS ......... APPENDIX A: APPENDIX B: APPENDIX C: APPENDIX D: APPENDIX E: APPENDIX F: REFERENCES RATE COEFFICIENTS FOR H2 + F2 CHEMICALLASER.................... DERIVATION OF THE RADIATIVE TRANSFER EQUATION. . . . . . . ...... . . . DERIVATIONOFX rad DERIVATION OF THE ENERGY EQUATION . . CALCULATION OF PULSE ENERGY . . . . . . DESCRIPTION OF COMPUTER SIMULATIONS FOR THE PULSED HF CHEMICAL LASER . . . iii 70 72 76 78 81 84 123 1‘11"?!” 5" Fe 9. F.10. F.11. F. 12. LIST OF TABLES Page Effect of HF VV rate coefficient change on model predictions ................... 22 Model predictions of effect of rotational relaxation rate on laser output ........... 34 Model predictions for a representative rotational relaxation rate .............. 40 Rate coefficient for H + F chemical 2 2 laser ............................ 71 Identification of variables Y(1, N) in H + F model ..................... 88 Z 2 Identification of variables Y(1, N) in SF -H model ..................... 89 6 2 Nomenclature ....................... 90 Sample computer output for H2 + F.2 model ........................... 92 Sample computer output for SF 6-H2 model ........................... 94 Program MODELC ............... . . . . . 95 Subroutine GAINC ..................... 100 Subroutine DIFFUNC ................... 101 Program MODELG .................... 105 Subroutine GAING ..................... 110 Subroutine DIFFUNC ..... . ............. 111 Subroutine FLASH . . . . . ................ 115 Subroutine DIFSUB . . . . . . . . . . . . ........ 116 iv Figure 1.1. LIST OF FIGURES Comparison of time-resolved spectral output 3.. Experimental data reported in Reference 29 for a 1 H :1 F2:60 He mixture at 50 Torr .................. b. Calculated spectrum for the same mixture from Reference 23 where rotational equilibrium was assumed ....................... Comparison of power histories as determined by present model and by constant-gain model ............. . .............. Model calculations of power histories of 2-1 band a. Comparison of the predictions of the present model with that of the constant- gain model ....................... b. Individual transitions in the 2-1 band as determined from the present model ...... . Comparison of time history of gain for selected transitions of standard case as modeled in present model and in constant-gain model of Reference 23. (a) P2(9), (b) Pz(10), (c) P2(11). Exact time histories for the constant-gain model for the times when the gain is less than threshold were not available; therefore, these portions of the curves are shown schematically ..... . ................ Time evolution of gain and output power for Pz(10) transition for standard case ........ Page 20 24 Figure Page 2. 5. Effect of initial F2 dissociation on pulse energy and initiation efficiency as computed with recent rate coefficient compilation of Cohen ........................... 26 2. 6. Effect of initial F2 dissociation on pulse length as computed with recent rate coefficient compilation of Cohen ................. 27 2. 7. Comparison of power histories for three different rotational relaxation rates ....... 36 2. 8. Power histories for all transitions in 2-1 band for three different rotational relaxation rates a. Rotational equilibrium .............. 37 b. A,1_=2X10'8T'-1/2 ........... 37 c. AT=2X10'7T'1/z ................ 37 2. 9. Comparison of power histories as determined by present model and by Lyman' 3 model with experimental result of Beatie et a1. 3 The present model result has been smoothed for this comparison .................. 38 3. 1. Michigan State University SF6-H2 pulsed chemical laser facility ................ 44 3. 2. Circuit used to trigger spark gap for appli- cation of pulsed high-voltage across discharge electrodes ................. 45 3. 3. Gas- handling system that provided calibrated gas mixtures for SF6- Zlaser . . . . . . 46 3. 4. Experimental setup for introduction of variable loss into laser cavity ................ 49 3. 5 a. Typical oscilloscope trace of discharge current during laser initiation ......... 54 b. Oscilloscope trace of laser output for 1 H2:1 SF6:10 He mixture at 50 Torr. Attenuator angle 0 set at 0 deg ......... 54 vi Figure 3. 5. c. Laser output predicted by present model at same gas mixture and cavity condition. The predicted time to threshold, pulse width, and pulse shape are similar to that in (b) ....................... Effect of cavity losses on total laser output for three gas mixtures a. 0.2 H2:1SF6:10 He at 33 Torr ......... b. 1 H2:1SF6:10 He at 33 Torr ......... c. 1H :1SF :10 He at 50 Torr ......... 2 6 Effect of cavity losses on output of laser operating in single-line mode for two selected transitions. (a) P1(4), (b) P2(3). Note significant loss in useful laser output that resulted from parasitic oscillations, as shown in (a) .................... Model calculations of the effect of cavity threshold on pulse energy for a 1 H :5 SF6 mixture at 120 Torr, with laser operating in three different modes a. Multiline mode .................. b. Single-line mode on P2(3) transition ..... c. Cascade mode of three transitions, P (6), P (7), andP (8) ............. 3 2 1 Optical cavity .............. . ....... Computer simulation of pulsed HF chemical laser .......................... Page 54 56 56 56 58 62 62 62 83 86 1. INTRODUCTION 1. 1 History The basic principles of laser (or optical maser) action have been well understood since the early days of quantum theory and were clearly enunciated by Einstein in his paper on the quantum theory of radiation. 1 It was not until much later, however, that its practical significance became clear. The first successful maser was devel- oped by Townes2 at Columbia University in 1954. It was a gaseous ammonia maser that operated in the microwave range. In 1958, Schawlow and Townes3 proposed an extension of the microwave technique to the infrared and optical range, and the laser was born. In 1960, Maiman4’5 published experimental results for the first succesful laser, a ruby laser. By 1972, laser action had been achieved with atoms, ions, and molecules in gases, liquids, solids, glasses, flames, plastics, and semiconductors.6 The characteristics of the laser, with its unique capability of delivering intense, coherent electromagnetic radiation, make it a valuable tool in the laboratory. Its temporal and Spatial coherence properties have been exploited in such applications as inteferometry and holography, and made it possible to focus the laser output into an extremely small spot size, attaining power densities not possible with any other source of light. The laser quickly became a valuable tool in industry as well. In 1965, Western Electric announced the first use of laser light in a mass production application: They had developed a laser system that could pierce holes in diamond dies for drawing wire. Industry is now using the laser to measure process parameters, scribe, drill, evaporate, and weld, in a variety of applications . 7 Recognition of the advantages and potential of the conversion of chemical energy to laser power has brought about increased interest in the development of chemical lasers. The chemical laser is unique because it is capable of yielding very-high specific power densities with reaction initiation energies much less than those emitted in the laser beam. In other laser systems, the population inversion necessary for laser action must be achieved through the initiation energy, but in the chemical laser, the initiation energy only serves to prepare chemical species that react exothermically to produce the laser active medium. Solid-state lasers have achieved output energies of §5% of the initiation energy, while the CO2 gas laser has demonstrated an initiation efficiency of as high as 25%. Chemical lasers, on the other hand, have been operated with 317070 initiation efficiency. 8’ 9 The first chemical laser was demonstrated in 1965 by Kasper and Pimentel. 10 it was a flashlamp—initiated, pulsed HCl laser pumped by the reaction H + C1 9 HC1(v) + C1 (AH = -45. 2 kcal/mol) (1.1) 2 where AH is the net change in molar enthalpy of the chemical sys- tem resulting from this reaction. Thereafter, a great many other A compounds were made to lase, although most of the effort has been directed toward diatomics of the hydrogen halide type. These mole- cules are favored because they are generally produced by highly exothermic reactions and are, thus, capable of achieving the popula- tion inversion of the vibrational levels necessary for lasing to occur. Furthermore, they have large electric dipole moments, which results in large cross sections for stimulated emission. 11’ 12 Laser action from the HF molecule initiated by flash photoly- sis of UF6-H2 mixtures was reported by Kompa and Pirnente113 shortly after the initial operation of the HCl chemical laser. At about the same time, Deutsch14 reported similar laser action result- ing from the initiation by pulsed electrical discharge of SFé-H2 mixtures. Both lasers were pumped by the reaction F + H2 -> HF(v) + H (AH = -31.7 kcal/mol) (1. 2) which produced a population inversion ratio15 of N1:N2:N = 0. 31: 3 1. 00:0.47, where Nv represents the population of HF molecules in the vibrational level (v). With the discovery of the chemical laser, it was recognized that one essential advantage of this laser was its potential for high efficiency, and that this efficiency could be realized through the use of a chain reaction to achieve the population inversion. The first chemical laser operating on a chain reaction was, again, the HCl laser. In 1969, Batovskii16 and Basov17 separately reported the first HF chain- reaction lasers, which were initiated by electrical discharge and flash photolysis, respectively, in mixtures of H2 4 and F2 gases. The chain consists of a cold reaction, Reaction (1. 2), and a hot reaction, H+FZ-)HF(V) +H (AH: -97.9kca1/mol) (1.3) where the adjectives cold and hot refer to the relative exothermicity of the reactions. The population inversion ratio of Reaction (1 . 3) 19 has been measured by Jonathan18 and Polanyi and has a maximum at v = 6. Many other researchers subsequently investigated the H2 + F2 chain-reaction laser. Both theoretical and experimental studies were carried out. Many initiation techniques, e. g. , electron-beam- irradiated discharge, electrical discharge, flash photolysis, and laser photolysis, with a large assortment of experimental apparatus and a variety of gas mixing procedures were employed to obtain a wide range of results. The highest energy output (2360 J) and initia- tion efficiency (> 170%) attained to date were reported by Greiner et al. , 9 who utilized relativistic electron beams for initiation. Kerber et al. 8 also obtained 170% initiation efficiency with a laser initiated by electrical discharge. Current efforts in high-energy pulsed sys- tems are directed toward achieving successful large-volume initia- tion of the laser medium. The electron beam is considered a prime candidate for accomplishing this objective because of its ability to deposit large amounts of energy uniformly over a large volume of the gas mixture. 20 Although both the HF and HCl laser Systems have been the subject of much research activity, the H2 + F2 laser is generally preferred to the H2 + Cl2 laser because of the significantly higher exothermicity of the H2 + I“2 chain (one reaction of the H2 + Cl2 chain is actually slightly endothermic). In addition, the HF deuterated analogue, DF, has its transitions in a transmission window of the atmosphere. Knowledge gained from the study of the HF system will facilitate the understanding of the DF system at a fraction of the cost that would otherwise be incurred in direct investigations of that system. 1. 2 Present Work Current efforts to improve the performance of the HF laser requires an understanding of the detailed mechanisms within this device. It is the purpose of this thesis to make a systematic and detailed investigation of the competing mechanisms in the pulsed HF chemical laser utilizing both the theoretical predictions of a rate- equation model and laboratory observations. The specific objective of this study is to examine the kinetic mechanisms and the effect of the following: 1. The transient behavior of the laser parameters, which includes the interaction of gain and intensity before and during lasing. 2. Nonlinear deactivation mechanisms, specifically, the vibration-vibration (VV) energy transfer process. 3. Laser performance as a function of level of initiation. 4. Rotational nonequilibrium resulting from lasing and prefer- ential pumping into rotational levels. 5. The effect of cavity losses and cavity threshold on the output of the SFé-H2 laser. Several theoretical models for the analysis of the H2 11,21-26 27 2 developed. These models follow the assumption that lasing begins + F2 and the SF6-H chemical lasers have already been when gain reaches threshold and that gain equals loss during the lasing period. It has been shown, however, that rate-equation solu- tions of laser performance with the constant-gain assumption will minimize the effect of the very fast VV exchange reactions. 23’ 25 The model presented here is similar to that of Rockwood et a1. , 28 which was developed for pulsed C0 laser simulation. Model features include the determination of the time evolution of the gain, intensi- ties, and species concentrations. This formulation permits the observation of the time history of the interaction of gain and inten- sity. Since the gain levels are permitted to fluctuate in a more realistic manner than in constant-gain solutions, the modeling of nonlinear deactivation mechanisms is more accurate. This model, therefore, permits a more careful evaluation of the effect of the VV energy transfer process . The time-resolved spectral output of the experimental data from Suchard et a1. 29 is compared with the calculations of the com- puter model from Kerber et al. 23 in Figure 1. 1. The V and J on the ordinate designate the lower level quantum numbers of the transitions, and the horizontal bars represent the periods of lasing on these transi— tions. The calculations predict a rigid J-shifting pattern and no multiline lasing, whereas the experimental data indicate significant deviation from the J shifting and definite multiline lasing. * Multiline lasing was also seen in the work reported in References 30-32. The model of Reference 23 assumes a Boltzmann distribution of the rota— tional states at the translational temperature; the effects of preferential * The discrepancy in pulse duration has been removed by a later study,'26 in which revised kinetic rate coefficients were used. V J ‘3 _ T I I I _ 3 g _.«P4IaI 5 o_.—_— .: 10‘: —_ 2 E __—_/P3(5) g 0:: _: 10‘:— 1' l I —“— : o-:_ —_ .5 l0‘—_——_ _: :—__/P 5 : 0 E l( I (a) : 9 — I I I I ‘0 T I I : 5 : __.—_ P6I10)§ o_:_ P (10) 3 '0-3- ——5A:: 4 ; .——— é o-:— 94“”. _= 10‘;— _ .—_ a E ——_ a 05— Pan” _: lO-z— _— __ 2 §_——— 5 : P (H) : O—_—- 2 A ; lo-:— _—_ -: I E _—— i :- P (10) : --'_ l _'. IO‘E— _—\'; 1 0 §_—-— I» a o "5' I I I I 5 0 20 40 60 so 1% TIME AFTER INITIATION. usoc Figure 1. 1. Comparison of time-resolved spectral output. (a) Experi- mental data reported in Ref- erence 29 for a 1 112:1 F2:60 He mixture at 50 Torr; (b) cal- culated spectrum for the same mixture from Reference 23 where rotational equilibrium was assumed. pumping and rotational hole-burning resulting from lasing are ignored. This is also true of the other existing models. The conjecture here is that rotafional nonequilibrium does, in fact, exist and is the result of lasing and preferential pumping. By taking into account the effects of these two mechanisms, the present model will attempt to resolve these discrepancies. In the following sections, the model is presented in its sequential phases of development. In the initial phase, an H2 + F2 model is for- mulated in which a Boltzmann distribution for the rotational states is assumed. It is compared with earlier model calculations of Kerber et al. 23 in order to illustrate the effects of the present model assump- tions. Other model features are also discussed, including the cavity transients and predictions of laser performance as a function of level of initiation, using the recent chemical kinetic rate coefficients Sug- gested by Cohen. 33 In the subsequent phases of development, the model includes the effects of rotational nonequilibrium. Unfortunately, this endeavor also increases the complexity of the model considerably, and makes the numerical computations longer than is desirable. The model is, therefore, limited to the simpler case of the SFé-H2 laser. For this model, the individual rotational level populations are determined from the solution of rate equations. Relaxation of the rotational popula- tion is accomplished by the assumption of a characteristic rotational- relaxation time similar to the model of Schappert34 for C02. With the rotational populations of the first four vibrational levels identified, an accurate assessment of the effect of rotational nonequilibrium on laser performance can be made. Calculations were made to assess the effect of several rotational-[relaxation rates on the spectral output of the laser. 9 The effect of preferential pumping into rotational levels is also considered. This formulation permits, for the first time, the study of a laser operating in the line-selected mode; the effect of cascading is evident from the results of these calculations. An SF6-H2 laser was constructed in order to make an experi— mental check of the model predictions. Reaction of the SF6-H2-He (diluent) gas mixture is initiated by a helical array of electrical dis- charges; pulsed high voltage is supplied through the use of a triggered spark gap and a capacitor. The results are presented of an experi- mental and theoretical investigation conducted to assess the effect of cavity losses and cavity threshold on the performance of this device. Model predictions of laser performance are compared with experi— mental data, and the validity of the model assumptions is examined. 2. COMPUTER SIMULATION OF PULSED HF CHEMICAL LASER 2. 1 Model Formulation The formulation of the chemical laser computer simulation is described. A general model was developed and then tailored to the two particular cases of concern. The reactions used to represent the chemical processes are: 1. The H2 + F2 chain F+H :HF(V)+H 2 H+F =HF(V)+F 2 2. Vibrational-translational (VT) deactivation HF(v) + M :3 HF(v') + M H2(v) + M%H2(v - 1) + M 3. VV quantum exchange HF(v) + HF(v')==HF(v + 1) + HF(v’ - 1) HF(v) + H2(v’)%HF(v + 1) + H2(v’ - 1) 10 11 4 . Dis so ciation— recombination F2+M¢M+F+F H2+M==M+H+H HF+M¢M+H+F The major provisions in the models are: 1. The dominant kinetic processesfire represented by the reac- tion system suggested by Cohen (Table A. 1). 2. The reacting mixture is homogeneous and is contained in a Fabry-Perot laser c'avity. 3. All possible transitions within a band are assumed to have low initial intensities that grow if the gain rises above threshold. Lasing is always assumed to be in the P branch. Initial inten- sity levels can be selected individually or set proportional to the spontaneous emission rate. 4. Initiation is modeled by the introduction of a finite concentra- tion of F atoms into the gas mixture. The chemical reactions are written I; (2 1) 2 “r1 N1;- 2 I3r1 Ni ' i -r i where Ni is the molar concentration of species i, ari and firi are stoichiometric coefficients, and k1‘ and k_r are forward and backward rate coefficients. The rate of change of concentration for nonlasing molecules is _1. = x, (2.2.a) and, for HF molecules, one has dNH dt F(v,J)_ — Xi +Xrad(v,J) - Xrad(v - 1, JL) + A(v,J) (2.2.b) where the xrad terms are rates of change in concentration as a result of lasing into and out of level (v, J). The lower-level rotational quan- tum numbers are J and JL for the transitions v + 1 —~ v and v —.v - 1, respectively. The net rate of spontaneous emission into level (v, J) is given by A(v,J). The chemical reactions yield a concentration change a . fl . Xi =2 (firi - ari) (kr HNj rJ - k-ran U) (2.3) r J J and Xrad(v’ J) = g(v,J) f(v,J) (2-4) where g(v, J) is the gain on the v + 1 -> v transition with lower-level J and f(v, J) is the lasing flux on the same transition. The rate equa— tion for the lasing flux is (Appendix B) m = c[g(v,J) - athr] f(v,J) (2.5) dt where c is the speed of light and _ 1 athr — - Z—I: ln(RORL) (2.6) 13 where L is the length of the active medium and R0 and RL are the mirror reflectivities. The gain is hNA 81": J) = 4." 90‘2“]! J) ¢(V9J) B(V, J) ZJ + 1 x [(5.1 71) NHF(v+1,J-1) ' NHF(v,J):| (2'7) where wc(v, J) is the wave number of the transition, B(v, J) is the Einstein isotropic absorption coefficient based on the intensity, and ¢(v,J) is the Voigt profile at line center as given in Reference 22. The firSt term of Equation (2. 5) determines the rate of increase in the intensity of the radiation field within the laser cavity; the second term gives the rate energy is lost from the cavity. The lost energy includes that extracted through the output coupler as laser output as well as real losses resulting from such mechanisms as absorption, scattering, and extraneous reflections. The energy equation for a constant density gas is ENC ir—P d—NiH (28) ivit"L'Zdt i ' 1 where Cv is the molar specific heat at constant volume, Hi is the 1 molar enthalpy of species i, and PL is the output lasing power per unit volume. The output power in the v + 1 —- v band is PLVIt) =2 hc NAatherIV.-T) f(v,J) (2.9.a) J 14 and PLIt) =;PLVIt) (2.9.b) where the only cavity loss is as3umed to be the laser output. In making comparisons with experimental measurements, however, real losses must be accounted for, as discussed in Chapter 3 and Appendix E. Numerical integration of Equations (2. 2), (2. 5), and (2. 8) by the modified Adams-Moulten method of Gear35 determines the time evolution of the species concentrations, temperature, pressure, the gain on all transitions, and the intensities on all lasing transitions. The laser energy extracted in each band is then determined by inte- grating the power E tcP d o v-jo. Lv t (2.1.a) where tc is the length of laser pulse and the total pulse energy is E :2 Ev (2.10.b) V 2. 2 Effect of ReLELation Oscillation—s The term "relaxation oscillation" frequently has been used to describe the phenomenon that results from gain fluctuations during pulsed-laser operation. In this section, the results of computations of the performance of a laser pumped by the H2 + F2 chain reaction are given. These results are compared with those of the model of Reference 23, and the unique features of the present model are ‘ 15 graphically illustrated. In this formulation, a Boltzmann distribution for the rotational populations at the translational temperature is assumed; hence, NHF(V, J) = NHF(v)|: " ' -thv/kT 2‘1" 1 e J (2.11) Qr(T) where the values of the rotational partition function Q:(T) and the rotational energy E}, are from the data of Mann et al. 36 2. 2. 1 Comparison with Existing Models The most extensive calculations of pulsed H2 + F2 chain- reaction chemical laser performance have been made by Kerber et al. 23 by means of the constant—gain model.22 The present model predictions are compared with those of Reference 23. Comparisons are made by the use of the same chemical equations and rate coeffi- cients given in Table II of Reference 23. In Figure 2.1, a comparison is shown of power histories as computed from both models for the case F:F2:HZ:Ar = 0. 1:1:1:50 at an initial pressure (Pi) of 1. 207 atm and an initial temperature (Ti) of 300°K. The cavity conditions were set at RC = 0. 8, RL = 1.0, and L = 100 cm. In general, the predicted pulse shape and pulse dura— tion of the two models are comparable. The small fluctuations resulting from J- shifting in the model of Reference 23 were deleted in their figure. However, it is clear that the characteristics of the present model facilitate simulation of power fluctuations and that these fluctuations are significant during at least half of the pulse. This is even more graphically illustrated when the results of PULSE POWER, x104 W/cm3 Figure 2. 1. [ I INITIAL GAS MIXTURE 0.1 F:1 F2:1H2:50 Ar Ti = 300°K, Pi = 1.207 atm CAVITY CONDITIONS ‘\R0 = 0.8, RL = 1.0 L = 100 crri TIME, usec Comparison of power histories as determined by present model and by constant-gain model. 17 Reference 23 for the power on the 2-1 band are compared with those of the present model in Figure 2. 2. a. In this figure, the details of the fluctuations from the constant- gain solution are plotted without smoothing. In Figure 2. 2.b, the time evolution of the laser intensity on the various transitions in the P2”1 band is shown. The present model permits simultaneous lasing on all transitions that reach thres- hold; however, the present assumption of rotational equilibrium causes J-shifting in the manner shown in the earlier models. Previous mod- els show lasing only on the transition with maximum gain. Lasing output spectra are found to be on lower J levels than the J levels of Reference 23; this result is more consistent with experiment. 30 Actual laser performance often exhibits the more erratic oscillations predicted by the present model; this is confirmed by the pilse profiles given in References 8 and 26. The small-signal gain of the standard case from Reference 23 was compared with that calculated by use of the present model to check the thermodynamic, spectroscopic, and kinetic data. These results compare to within 1%. The effect of including spontaneous emission on the small-signal gain is less than 3%. Since the rate of relaxation of HF(v) by VV processes is pro- portional to [HF(v)] [HF(v’)], these rates are nonlinear with respect to excited HF populations. With this in mind, the time evolution of the gain of selected transitions of the present model is compared in Figure 2. 3 with that of the constant-gain model of Reference 23. The significance of the constant-gain assumption is the prediction of lower inversions, and thereby the effect of deactivation mechanisms is minimized. «1 ° I I'NITIAL GAS MIXTURE is O.1F:1F2:1H2:50Ar .2 2% T' = 300°K, Pl=1.207atm_‘ x . \ CAVITY commons PRESEN _ §1_ MODELT \ L"°°°"‘ _ \ consrmnoA a (— MODEL m 3 CL n 1 I I v 2 3 1 TIME, 113°C In) 6 10 PZI3I P214) 2(5) 92(7) P (8) 2 / 92(9) PZHO) 104 p (11) \ 3 92(12) E 3 g 10 l l l 1020 1 2 3 4 TIME, use: (b) Figure 2. 2. Model calculations of power histories of 2-1 band. (a) comparison of the predictions of the present model with that of the constant-gain model; (b) individual transitions in the 2-1 band as determined from the present model. ”“35",8 TIC-I .l-' '5 iJ '..',"._ ' .A' 'J ‘1' i" ’ "o‘!h'{~ " on; n} va-Iiiom -. ». Intangi- '. In" ‘J: .‘t‘ sonnets}! To I‘mun! mug-Jazz’s .,_) .II .t‘ I1... 'a-IM omi: 1:12le {I :1. -> 335M me o.“ o.m coed “wood moood Nouod Nfimod memod «20.0 hm >> “~me 0.“ N.m owed «wood moood Noood veuod oumod vwfiod whmvndum Munoz «Gomoum o; OJ» rod .0 $50.0 N306 mmuod mmmod oovod «muod hm >> Swag o.« «.v 2: .o wvood aged m:o.o mouod unmod womod pndvndum HOCOE GMdOIuamumcoO N\fiu a» m mom vmm mvm mmm «NH cum 3mm 00?; 68MB out. Swuocm mnoflogoum HOOOE no OMGNAO «Gumofiuoou 3.2 >> hm mo «ovuflm .« .N OBNB 23 used, indicates the presence of relaxation oscillations similar to that shown in Figure 2. 1. However, as the characteristic times of the chemical kinetic and radiation processes decrease relative to the period of one round trip in the cavity, the effect of these oscillations decreases. Therefore, as the level of initiation in the chemical laser decreases, which implies F/F2 initial decreases, the effect of the additional features of the present model diminish. Even in this range, the additional information contained in the prediction of the time evolution of the gain and intensity on each transition makes these calculations instructive. 2. 2. 2 Other Model Features For all the calculations presented, the initial photon flux was 14 mol/cmz. For reductions in this flux of as much as set at 10- 10-3, the pulse length changed by less than 1% and the pulse energy changed by less than 3%. Therefore, the calculations are relatively insensitive to the initial flux levels. If desired, these flux values may be set proportional to the rate of spontaneous emission. The gain and intensity profile of the P2(10) line are shown in Figure 2.4. When the gain reaches threshold, the intensity begins to grow, slowly at first; but, since its growth is nonlinear, the rise becomes sharp as gain and intensity become larger. The resulting high flux causes the gain to drop below threshold and to oscillate near that value while lasing continues. The gain profile shows several other "dips" before the P2(10) line begins lasing. These dips are caused by lasing on the P2(7), P2(8), and Pz(9) lines. This is to be expected since the depopulation of a J-level as a result of lasing must be "shared” by other rotational ”"1 24 10‘1 I I I 105 5‘. P (10) If 2 I 4 10‘2- '1' ‘. "0 ,_ 912.10) I I I 5 ' I I i I {—5 i E THRESHOLD 1 : / 3 10'3— N : ‘10 II I . I I I I I I I I I I l I ‘4 I I l l 102 10 CI 1 2 3 4 TIME, usec Figure 2. 4. Time evolution of gain and output power for P2(10) transition for standard case. POWER, W/cm3 25 levels to maintain the Boltzmann distribution. This loss of population in the upper level causes a drop in the gain profile of all the J—levels. Those nearest the lasing J are most significantly affected, even though the Boltzmann distribution is maintained. This is also the reason for the termination of the P2(10) line. As the P2(11) line begins to lase, there is a drop in the gain profile of all J-levels. The P2(10) line is already at threshold gain, and the drop causes gain to fall far below threshold and lasing terminates. Hence, if a Boltzmann distribution for the rotational levels is assumed, there is not an extensive amount of simultaneous lasing among the J-levels of a given band. The intensity profiles of the P2(J) lines are shown in Figure 2. 2.b. Although the gain reaches values well above threshold, a total population inversion has never been observed; therefore, lasing on the R-branch is not possible. The kinetic model recently reviewed by Cohen33 has been used to study the effect of the initial F/F2 ratio on pulse energy and pulse duration as shown in Figures 2. 5 and 2. 6, respectively. The initial F—atom concentration is varied while the stoichiometric balance is maintained between H and F. Other initial conditions are held constant. The initial gas mixture is YF:1H2:XF2:50He with 2X + Y = 2. The initial temperature and pressure are 300°K and 50 Torr, respec- tively. The cavity parameters are set at RC = 0. 8, RL = 1.0, and L = 100 cm. For the specific mixture and cavity conditions considered, the present calculations indicate that the maximum pulse energy will be Obtained at initial F/F2 concentrations of approximately 0. 15. For the more preliminary kinetic model of Reference 23, this optimum oc- curred at a ratio of F/F2 of approximately 0. 4. v. 26 PULSE ENERGY, x 10'3 J/cm3 0 I INITIAL GAS MIXTURE ‘ FzszzF:He \ X:1:Y:50 \ 2X + Y = 2 \ T1 = 300°K, Pi = 50 Torr CAVITY CONDITIONS R0 = 0.8, RL =1.0 L = 100 cm 120 4 10'3 Figure 2. 5. 10‘2 [Fl/(ZEFZJ + [PD 1 10'1 10° Effect of initial F2 dissociation on pulse energy and initiation efficiency as computed with recent rate coefficient compilation of Cohen. 2 PULSE ENERGY/DISSOCIATION ENERGY 27 3 10 l I INITIAL GAS MIXTURE F2:H2:F:He le :Y:50 2X + Y = 2 TI = 300°K, PI = so Torr CAVITY CONDITIONS ‘02— R0 = 0.8, RL = '00 _‘ L = 100 cm 8 n a m“ 2 .— Io‘ — — mo I I [FJ/Iztrzl + [F]) Figure 2. 6. Effect of initial F2 dissociation on pulse length as compute$ with recent rate coefficient compilation of Cohen. 9 28 In Figure 2. 5, the ratio of pulse energy to the energy required to produce the initial F-atom concentration by dissociation of F2 is also shown. Model calculations become very long and expensive as the ratio F/F2 is decreased; therefore, an optimum point was not determined. Note, however, that this ratio is still increasing sharply with decreasing F/IE‘2 at a ratio equal to 0.016. The par- ticular optimum for this curve is, of course, very dependent upon mixture composition and cavity parameters. The variation of the pulse time parameters shown in Figure 2.6 is similar to that presented in Reference 23. 2. 3 Effect of Rotational Nonequilibrium The significance of rotational relaxation on the characteristics of laser performance was investigated. Because of the complexity of the calculations, the study was restricted to the simpler case of the SF6-H2 laser. Since the model was developed to compare with subsequent experiments, some features of the model are peculiar to that purpose. The laser cavity is assumed initially to contain a homogeneous mixture of H2 and SF6' Since less than 5% of the SF6 is typically consumed during the discharge, 38 it is treated strictly as a diluent, and the net result of the F-atom producing plasma kinetics is incor- 27 * porated separately. Lyman found the F-atom production rate to This model, therefore, does not incorporate deactivation by ions, at least one of which, F‘, is considered significant. (J. S. Whittier, The Aerospace Corporation, private communication, March 1974). 29 be roughly preportional to the input power. This proportionality is assumed in this study. The contribution of the input power to the translational temperature rise is also included. The kinetic processes are the same as in the preceding H2 + F2 model, except that only the F + H2 reaction is considered and the diluents are dif- ferent. Since the model formulation is the same as that previously used, with the exception of the initiation simulation and the rotational relaxation feature, the details will not be repeated. Rotational relaxation is incorporated into the model by asso- ciating a relaxation time constant, T(v,J), with each HF(v, J) state. These constants are formulated such that their characteristics are consistent with available experimental and theoretical data concern- ing the relaxation mechanism. This formulation permits a straight- forward analysis of the dependence of rotational relaxation without resorting to the detailed molecular collision dynamics, and it pro- vides a simple approach to evaluating the broad effects of rotational nonequilibrim'n in the lasing process. Available experimental data on rotational~relaxation time con- stants are generally the result of analysis of acoustic absorption or * thermal conductivity measurements. A partial compilation of *Several laboratories are presently conducting experiments to measure HF rotational- relaxation rates. Techniques such as chemiluminescence (T. L. Cool, Cornell University, Ithaca, New York), singles- lse pump-probe with a Pocket's cell used for optical switching,3 ' and double resonance 40 are used. Because of frequency coincidence problems, the double resonance results are of dubious value. No results are yet available from the chemi- luminescence studies. The pIimp-probe technique ave k F-HF = (7. 8 :I: 0. 2) X 10'7 sec'1 Torr" and (4. 9 :t 0.4) X 10 sec- Torr"1 for the self—relaxation rates of the P1(5) and P1(6) transitions, respectively. 30 existing data are given in References 41 and 42. Although all measurements are not in agreement, most concur with the following observations: 1. The rotational relaxation time constant (T ) increases for . . . 41 42 R decrea51ng moment of 1nert1a. : . . . 41, 42 2. TR increases for decrea51ng dipole moment. . . . 43-47 3. TR increases for increasmg temperature. TR is defined by dER : ER(T) - ER(t) dt T (2.12) R where ER(T) is the rotational energy of the system in equilibrium at temperature (T) and ER(t) is that energy at time (t). Strictly speak- ing, this equation is valid only for small departures from equilibrium. Raff and Winter,45 and more recently Polanyi and Woodall, 48 have developed rate-equation models that describe rotational relaxation in which rate coefficients depend upon the rotational energy separation. Their results indicate that it is insufficient to associate a relaxation time with each vibrational level; instead, one such parameter, r(v, J), is necessary for each vibrational-rotational state. If -r(v, J) were assmned to be an increasing function of the rota- tional energy separation (AEV, J)’ the first and third observations would be satisfied. Such an assumption is realistic since, for the increasing energy separation between adjacent levels, the probability of a collision being energetic enough for energy transfer to occur is decreased. A discussion of the consistency with these observations follows . 31 For a rigid rotor, AE =E -E (2.13.a) and AEJ = hz ”+1? (2.13.b) 41v I which shows that AE increases linearly with the rotational quantum J number J and is inversely proportional to the moment of inertia of the molecule. Hence, as I is decreased, AE and T(v,J) increase. J Similarly, as temperature is increased, the higher J-states become more populated, and the larger T(v,J) associated with these higher J- states become important in the overall equilibrium process. Thus, the time scale for relaxation increases. Although this argu- ment is specifically for a rigid rotor, it is clear that a similar argument may be used for the general case. When it is considered that a smaller dipole moment implies a decrease in the interaction cross section, it is not surprising that TR increases for a decreasing dipole moment. Polanyi and Woodall48 ignored this in their compu- tations of rotational relaxation of HCl, and still obtained excellent results. This effect is als.o neglected in this study. Under extreme conditions of rotational nonequilibrium, J -J lasing may contribute to the relaxation process. This effect has also been neglected. There are two processes that contribute to the perturbation of the equilibrium among the rotational states: lasing and preferential pumping. It is assumed that the vibrational-relaxation process will 32 not disturb the rotational equilibrium since it is a relatively slow process. The rate equation for the HF(v, J) populations, i. e. , Equation (2. 2.b), is modified to dNHF(v, J) T = P(v, J) + B(v,J) R(v) + g(v,J) f(v,J) B NHF(v, J) ' NHF(v, J) T _g(v-1,J+1)f(v-1,J+1)+ RIV’J (2.14) where P(v, J) is the pumping rate into level (v, J), R(v) is the net rate of vibrational relaxation into level v, B(v,J) is the Boltzmann rota- tional distribution at (v, J) for unit concentration, N§F(V,J) is the instantaneous rotationally equilibrated concentration of the level (v, J), and TR(v,J) is the time constant of rotational relaxation for level (v, J). Although some experimental data concerning P(v, J) are avail- 15,49 able, they are far from complete. Monte Carlo trajectory calculations have been made by Muckerman, 50 Blais and Truhlar, 51 and Wilkins . 52 For this study, the computations of Wilkins are used; a linear fit of his results at 300 and 500°K is assumed, and a check is made of the effect of this assumption. The relaxation time,rR(v, J), is assumed to be of the form BAE /kT TR(v.J) = ATe v,J (2.15) where A;1 is selected as some fraction of the binary collision fre- quency and B is a parametric constant in the exponential that further approximnes the transition probability of the collisions. For this study, B is set at 1.0x 10'3. 33 The initiation energy is added to the energy equation to yield dT dNi , NicviET; = P1 ' PL ‘2? H1 (2'16) 1 1 where PI is the input power. This model evaluates the time history of concentration corresponding to the first sixteen rotational states of the lower four vibrational states. The resulting set of nonlinear first-order ordinary differential equations are solved by the same modified Adams-Moulten technique used earlier. Model calculations were made at various rotational relaxation rates. The initial gas mixture used was 1 HZ:5 SF6 at 120 Torr and 300°K. Mirror reflectivities of the optical cavity were taken to be 1. 0 and 0. 8, and the gain length was 50. 8 cm. The total atomic fluorine introduced into the mixture was 5.6% of the SF6 concentra- tion. The results are given in Table 2.2 of four selected cases: ( 1) rotational equilibrium maintained; (2) the rotational relaxation time constant approximately the inverse of the gas kinetic collision frequency; (3) rotational relaxation modeled at 10% of the rate in case (2); and (4) rotational relaxation modeled at 1% of the second case. The first and second cases have very similar results. The relaxation time for the second case is comparable to the time required for light to traverse the length of the active medium. At slower rotational relaxation times, the populations in the various J-states do not contribute instantaneously to the lasing in any given line. This has a two-fold consequence: (1) Energy is extracted less rapidly, which allows more time for the deactivation mechanisms 34 m2 .o owns 385 8:56 «386 $256 Nth. 72 x N 93 .o won .o 3.8 .o $23 .o $38 .o Shoo .e Nth. w-2 x N can .0 can: ommé 88 .o 2:85 3805 25.8.0 Q79 m13 x N 8:29:35 o2 .o o~m.o 8.8 .o 3:86 3:88 oomoo .o ”accustom N39. 3 m NMM «NH Sm r. .4. oomi .653. 00:. Qwhonm «.9350 Honda :0 3.2 nemusxgou 3:03.30.“ mo uoommo mo $339.6on ~35: .N .N 03MB 35 to act. This results in a more uniform distribution of output energy among the lasing transitions of a given band and a net decrease of the total pulse energy (Figure 2. 7). (2) Since the higher J-state popula- tions are not instantaneously coupled to the populations of the lasing transitions, these higher J-transitions reach threshold earlier and multiline lasing becomes more prevalent. Lasing on higher J levels also increases as TR increases (Figure 2. 8). The decrease in the predicted total laser output as a result of rotational nonequilibrium effects was expected. Since previous model predictions have been significantly higher than actual experimental values, inclusion of this effect is helpful in achieving agreement between theory and experi- ment. This, however, is not necessarily the only source of dis- crepancy between theory and experiment. * In Figure 2. 9, the third case is compared with Lyman's model and the experimental result from Beattie et al. 38 This result indicates that even with rotational relaxation, model predictions of pulse output are still higher than the experimental values. ** Lyman used for his calculations essentially the initial rate coefficients compiled by Cohen in 1971. For the calculations of this study, rate coefficients reflecting more recent experimental data were used. 33 One of the most significant results of the incorporation of rota- tional relaxation into the model is the appearance of multiline lasing (compare Figures 2.8.a. 2.8.b, and 2.8.c). The overlapping of _*,____ See Reference 26, where the discrepancy was ascribed mostly to parasitics. ** The data in Figure 2. 9 were taken from Figure 7 of Reference 27. Note that the calculations in Reference 27 were made with RL = R = 1. o 36 Io , I 1 I ROTATIONAL EQUILIBRIUM 8 _ , AT=2 x10'8T'I/2 _ I m -7 -1 2 g If, A,- = 2 x 10 T / E ,' ' v0 6 _ '0 \. ._ '1! \ INITIAL GAS MIXTURE “ X 0 II I H :5 SF 5 ,'I' II 2 06 g 4 — Ii ‘,\ Ti = 300 K, Pi = 120 Torr a II- II CAVITY CONDITIONS ‘ u, I I Q i \I L = 50.8 cm 2 _ f -| __ \Q o l NHL I J 0 0.2 0.4 0.6 0.8 TIME, usec Figure 2. 7. Comparison of power histories for three different rotational relaxation rates. 37 10 23f .. I/ ._ ._ I) w 3 m I 1,2“IIIIIIII|\\\ .0 UI POWER, x 10" W/cm3 ‘0 & I I a 3 I POWER, x104 W/Cm3 0 0J ~ \ ~ ‘- \ i POWER, x107 III/cm3 I2 0 0.2 0.4 0.6 o .4 0.6 TIME, uuc TIME, usoc In) (b) 1.0 I I 7 0.0 — — INITIAL oAs MIXTURE 0-6 — I 142:5 sq _ 1'I = 300°K. P, = 120 Torr . Io CAVITY CONDITIONS 0.4 ~— ‘ .- /“\ R0 = 0.0. RL = 1.0 , H L = 50.8Cm I 1m“ 0'2 _ " 4I\ 13 _ 2 IN I) ///‘ ‘§ 14 . ,I 4 x 0 rli‘qutaék 0 0.2 0.4 0.6 TIME, mac (c) Figure 2. 8. Power histories for all transitions in 2-1 band for three different rotational relaxa- tion rates. a) roiational equilibrium (b) A,r =2x10- '1"-1?-,(c).A.,r =2x10- T-i/Z. 1.0 0.8 m E fl 3 mo — 0.6 X a. ll] 5 a 0.4 l.|.l V) _| D D. 0.2 0 Figure 2.9. 38 I I [PRESENT MODEL LYMAN's MODEL— INITIAL GAS MIXTURE _‘ I I Ti = 3000 K, PI = 120 TOI‘I’ I CAVITY CONDITIONS I I IR—0.08,RL-10 I, L = 50.8 cm I V IL/ I I XI , 0.2 0.4 0.6 TIME, psec Comparison Of power histories as determined by present model and by Lyman's model with experi- mental result Of Beatie et al. The present model result has been smoothed for this comparison. 39 the transitions in a given band has always been seen 30, 32 Time-reSOIVCd spectra that follow a J-shifting experimentally. sequence similar to that shown in Figure 2. 8 are reported in References 31 and 32. However, the spectrum found by Suchard et al. 29’ 30 shows significant departure from the sequential J-shift pattern. One possible explanation of their result was thought to be the non-Boltzmann product distribution Of the pumping reactions. For these calculations, a Boltzmann distribution was not assumed, but the results still show a consistent J- shifting pattern. The results, however, are expected to be sensitive to cavity losses. A second calculation was made for the third case, with an assumed Boltzmann distribution among the rotational states Of the product molecule HF(v), in order to check the significance Of the role Of preferential pumping. The results in Table 2. 3 show that, for the gas mixture and the cavity condition considered, the effect is small. Preferential pumping resulted mainly in a slight increase in output energy for the higher J-transitions. One added advantage of modeling rotational relaxation is that the model has the capability Of estimating performance in line- selected modes. In Table 2. 3, a comparison is made Of the laser performance for various modes of Operation. It is apparent that maximum output energy is obtained from a laser where all tran- sitions are allowed to reach threshold and lase. The effect of cascading is also evident. The sum Of the pulse output from the three single-line calculations, i.e. , P1(8), Pz(7), and P3(6), is only 65% of the output for the case with these three transitions operating in a cascading manner. The "Off-J" cases, i.e. , P1(9), 40 .r N\«I.H. wlcu X N M 4m wflfignm nwflcou 309nm oi .o mmm .o 8.3 .o 828 .o e58 .o ~38 .o oz fits «£532 o3 .o de $85 .2585 82.85 8.255 5mm .ENC .Aasm e: .o SN .o 33 .o 2:5 .0 ~88 .o 3:55 Sena .ENO .8?” m8 .0 one .o :58 .o Booed o o 33 oawfim Sena Q: .0 SN .o 325 .o o 338 .o o 83 Emma E~m o2 .o mmm .o >38 .0 o o :38 .o 83 236 €an 2: .o 8». .o 3.3 .o $25 .o $25 .o 325 .o 35:32 NE“ 3 m 2m Em 2m 069.. .083. 00:. Smuoam 0602 mcflduomo 3m.“ Gofimxflou 3:03.30.“ o>flflnom0990h s new unoflowvonm dove; .m .N BANE d 41 P2(7), and P3(5), show a decrease in output energy when compared with the cascading case. 2. 4 Summary Rate-equation models that graphically illustrate the effect Of gain fluctuation and rotational nonequilibrium on pulsed HF laser performance are presented. In the present investigation, it was determined that incorporation of these effects into chemical-laser models is essential for the accurate investigation Of the spectral content in the laser pulse. Examination of the effect Of line-selected Operation on pulse energy indicates that considerable care in line selection must be made to avoid large energy losses and that a model similar to the one developed in this investigation will be helpful in selecting transitions for beam propagation. 3. EFFECT OF CAVITY LOSSES ON PERFORMANCE OF THE SF6'H2 PULSED CHEMICAL LASER: THEORY AND EXPERIMENT 3. 1 Introduction Because of the potential for high performance, the reaction Of hydrogen and the halogens have been used widely in the production of 13'38'53'55 Of these, the HF active species for chemical lasers. and DF systems have been studied most Often because of the potential for developing lasers with high specific power densities through the the use of a chemical chain reaction. In an electrical discharge, SF6 will provide the F atoms for the reaction F+H2-rHF(v)+H (3.1) The vibrationally excited HF can then be used as the laser medium. The SFé-H2 laser has proved to be a valuable tool in the examination of mechanisms in HF and DF lasers. With this laser, the extra expense and complications Of handling a potentially dangerous gas are avoided, while sacrificing little in the study and interpretation of the mechanisms. 56-59 have investigated the effect Of the Several researchers use Of different initiation techniques to increase the efficiency of the SF‘6-H2 chemical laser. However, with the exception of the measure- ments of the variation Of laser output with output compling made by 42 43 Jones,60 no formal attempt has been made to systematically identify the role Of cavity losses in the performance Of this device. The results Of such an investigation are presented here, where the experimental data were Obtained using a transverse pin-discharge laser, and the theoretical predictions were made from the computer model given in Chapter 2, which includes in its computations the effects Of rotational nonequilibrium. 3. 2 Experimental Details Figure 3.1 is a schematic diagram Of the pulsed SF6-H2 laser facility used in this investigation. The gas mixture is contained in two 1-in. i.d. , 49-1/2 in. long, lucite tubes placed end to end. Both ends Of the tubes are sealed by CaF flats mounted at the Brewster 2 angle for 2. 8-p.m transmission. The electrode assembly is made up Of 338 resistor (4709) pin-pairs set transverse to the laser axis and mounted on the lucite tubes in a double helical configuration. The interelectrode gap is set at 2. 08 cm. The optical cavity is formed external to the tubes and consists of a 7. O-m radius, concave spherical, dielectric-coated (silicon substrate) for a nominal 99% reflectivity at 2. 5 to 3. 5 pm and a flat-output coupler, dielectric- coated (silicon substrate) for 95 + 1-5% reflectivity at 2. 5 to 3. 5 pm. Pulsed high voltage is obtained by means of a 5400 pF energy storage capacitor, the voltage Of which is applied across the electrodes by triggering a spark gap (EG&G, Model 14B). The trigger circuit used is shown in Figure 3. 2. Figure 3.3 is a schematic diagram of the gas-handling sys- tem. The gases (SF6, H2, and He) were research grade (99.99% pure) supplied by Mathe son and were used without further purifica- tion. The flow rates were controlled and measured with Matheson 44 SCREEN BOX—\ I- ——————— "1 l I : ESCILLOSCOPQ : I 9 I METEREO FLOWS l DETECTOR I SF6, H2, ”6 I 7 , , I A | [MONOCHROMATOIq I A L. I satzftsrems I L“- Qjfi LASER g ENERGY II I, , _—l [I , METER W ----------- EJ POWER SUPPLY-i ’ 4709 I- “I I I——IVW—oHIGI-I I 50 M52 (VOLTAGEI ‘I—I——1 I I 1 I CAPACITOR~1/ ——0 l 50 M9 I I L — — _ _ — _ J —_— TO VACUUM Figure 3. 1. Michigan State University SFé-H2 pulsed chemical laser facility. 45 E NERGY STOR AGE CAPACITOR I uF J 1 $29 \ESPARK If E686 TR 149 Figure 3. 2. Circuit used to trigger spark gap for application of pulsed high-voltage across discharge electrodes. 46 V - VERNIER VALVES R - LOW-PRESSURE REGULATORS HEGE GAUGE Vb : S U CHECK VALVE L, LASER _ II L [I CONTROL COMPONENT ms ‘1 fitmvs aiD—QDI~3IfIféis E XI-IAUST HASTINGS THERMOCOUPLE VACUUM GAUGE Figure 3. 3. Gas-handling system that provided calibrated gas mixtures for/SFé-HZ laser. 47 needle valves and flownmeters. Pressures were monitored on Heise Bourdon tube gauges. A mechanical pump (Kinney) is used to maintain a continuous gas flow through the laser and provide for Operation over a range of pressures. Pulse output is focused by a 2-in. -diameter gold-coated concave Spherical mirror of 45. 2-cm radius of curvature into a 0. 5-m monochro- meter (McKee-Pederson) equipped with a 295 line lmm grating blazed at 2.8 pm. Radiation emerging from the monochromator is detected by a AuzGe photodiode detector (Raytheon) Operated at 77°K and dis- played on a fast-rise (1. 2 nsec) oscillOSCOpe (Tektronix 485). Screens were placed before the detector to avoid saturation. Because Of the potential high gain of the HF system, under some Operating conditions, relatively strong output has been mea- sured from laser cavities with the mirrors removed. 60-62 This phenomenon is known as superradiance. Except for performance studies, typical runs in the present experiments were made at 33 Torr, in a 1 SF6:1 H2:10 He gas composition, and at 30-kV dis- charge voltage. The power output decreased by two orders Of magnitude with the mirrors removed; therefore, superradiance did not affect the measurements. At the beginning Of each run, the laser chamber is pumped- down to check for possible leaks and to re-zero the pressure gauges. The pulse repetition rate is limited by the time required to recharge the capacitors and the exchange rate Of the gases within the cavity. For best repeatability, approximately 15 sec was allowed between shots. A comparison of the output for a single shot and ten 48 superposed shots showed that, except for some erratic fringes, the pulses were repeatable . The experimental setup is shown schematically in Figure 3.4. A scheme is desired whereby losses may be introduced into the cavity without otherwise altering the existing Optical configuration (as may happen, for example, by changing the output coupler and realigning). An effective variation Of the output coupling is accomplished by inserting a CaF2 attenuator into the Optical cavity, as is shown in the Figure 3. 4. The threshold gain (Orthr) is then given by the equation (20 L) RR tze thr =1 (3.2) Where R0 and RL are the reflectivities of the output coupler and total reflector, respectively, t is the transmissivity Of the attenuator, and L is the gain length in the medium. Since t is a function of the angu- la. 1- position Of the CaF2 attenuator, a continuous range of Othr may be obtained by varying that angle. In addition to the attenuator, a Can compensator is placed outside the Optical cavity in the path of 11143 output radiation to compensate for the vertical diSplacement Of the la. 8 er output resulting from the attenuator. An iris is inserted in the oPtical cavity to increase the threshold for parasitics. It also £1111 ctions as a mode-control device. The iris Opening is decreased until further decreases result only in a monotonic decrease in output 130 Wer, indicating dominance by the TEMOO mode. To ensure a more ac Q“urate determination of the angular position of the attenuator, this aangle is measured by the calibrated position of a He-Ne beam 49 He-Ne ATTENUATOR MIRROR] CALIBRATED SCALE Ge: Au ________________ R DETECTOR Ro ”“5 éi H L [1%I f3 _________________ 3L3 COMPENSATOR 338 PIN PAIRS ATTENUATOR Figure 3.4. Experimental setup for introduction Of variable loss into laser cavity. 50 reflected from a mirror affixed to the attenuator (Figure 3. 4). A similar arrangement is used for the compensator. 3. 3 Computer Simulation The computer code is described in Chapter 2. The predicted output lasing power per unit volume is, from Equation (2. 9), PL(t) =2 hc NA arthr wC(V,J) f(v,J) (3. 3) V! where wc(v, J) is the wave number of the v + 1 —> v transition with lower level J, and f(v, J) is the lasing flux on the same transition, is the threshold gain defined by athr a =-—1—1n(R'R) (3 4) thr 2L 0 L ' are the Where L is the length of the active medium and R; and RL effective mirror reflectivities. The laser power PL(t), in Equation (3. 3), is the sum from both mirrors of the Cavity. The output intensity extracted from mirror ’ 0 R0 13 , (1 - R9 P - PL(t) (3.5) O - I 1/2 I 1].? [1 + (RO/RL) ][1 - (RORL) ] Initial conditions and model parameters are determined to reflect conditions Of the experiments. The electrical-discharge- lnitiated laser used in the present investigation produced columns of he a>1‘ly uniform plasma orthogonal to the laser axis. Since diffusion 51 is negligible during the pulse, the active region Of the laser may be approximated as homogeneous. The active medium length is taken to be the sum Of the diameters Of these columns and is estimated to be 169 cm. Various types of losses within the cavity are lumped with the 99% reflecting spherical mirror for an estimated value RL 2 O. 95. The output coupler is taken as R2) = R0 X t2(9), where t(9) is the transmittance of the Can attenuator set at angle 0 to the laser axis and R0 = 0. 95 is the true reflectivity of the output coupler. The power extracted from R2), as given by Equation (3. 5), includes losses resulting from reflection from the attenuator out of the optical cavity, as well as actual power passing through R0. This loss can be corrected for by means Of Equation (E. 6), Appendix E, to Obtain finally t(i - R8) (1 - R0) PL(t) Po(t) = [1 +(R2)/RL)1/2][1 _ (RgRL)1T2] [1 _ R23] (3.6) Electric discharge initiation, i. e. , abstraction Of F atoms fr Om SF6, is approximated by the relation dNF —-d—£- (3.7) = C PI(t) NSF6 Where PI(t) is the normalized discharge power profile and C serves as a parameter to couple the experimental results to theory. The pa. rameter C is determined by matching the computed pulse duration and the time that it takes for the laser to reach threshold with the 113 asured times for a selected case. Since the laser output was 52 attenuated, comparisons Of theory and experiment are accomplished by matching the outputs at a selected reference point. 3.4 Results and Discussion Typical oscilloscope traces for the discharge current and the total laser output along with the corresponding predicted laser pulse are shown in Figure 3. 5. The discharge current behaved like a damped oscillator with a one-half period (FWHM) of O. 26 usec. The resulting laser pulse was of similar duration (0. 24 nsec, FWHM), while the ”tail” of the pulse was considerably lengthened by the second half-period Of the discharge current. Calculations from the model showed that, near this level Of initiation, i. e. , F/SF6 = 0.005, the laser pulse shape is quite sensitive to that Of the discharge current. Figure 3. 5 shows that the predicted pulse shape compares very well with that Observed on the oscilloscope. The "knee" at the beginning of the pulse is due to lasing on the 2-1 band, which has the highest pumping rate. This lasing populated the upper level of the 1 — 0 band and depleted the lower level Of the 3-2 band, complementing flie pumping on those levels. Thus, gain on the 3-2 and 1-0 bands re se rapidly and lasing commenced on all bands, which caused the ab rupt rise in the Observed output pulse. Thereafter, lasing was 81.1 stained by the fast-pumping reaction. Pulse termination resulted when pumping subsided and lasing and collisional deactivation have d eInleted the pOpulation inversion. Model predictions of total laser output and the corresponding observed output are plotted in Figure 3. 6 as a function of cavity 1° 8 ses. The theoretical values were computed from Equation (3. 6). Figure 3. 5. a. 53 Typical oscilloscope trace Of discharge current during laser initiation. Oscilloscope trace Of laser output for 1H :1 SF :10 He mixture at 50 Torr. Attenuator angle 9 set at 0 deg. Laser output predicted by present model at same gas mixture and cavity condition. The predicted time to threshold, pulse width, and pulse shape are similar to that in (b). DISCHARGE CURRENT PULSE POWER PULSE POWER, RELATIVE UNITS 54 ‘llllllllll LI_I_II_I§OOI_. I I I I 0 0.4 I I I 0.8 1.2 1.6 2.0 TIME,IIsec (C) Figure 3. 5 55 In conjunction with Equation (3. 4), RCRL is a measure Of the sum Of the cavity losses, which are in the form Of laser output as well as transmission through the spherical mirror, reflection from the attenuator, and other extraneous losses. Three cases are shown: (1) 0.2 H2:1SF6:10 He at 33 Torr, (2)1H2:1SF6:10 He at 33 Torr, and (3) 1 HZ:1 SF6:10 He at 50 Torr. The lower gain cases, (1) and (2), were selected to avoid possible complications resulting from parasitic oscillations or superradiance. When agreement between theory and experiment was good, a higher gain case, (3), was run. In Figure 3. 6, the curves represent model predictions Of laser out- put as a function of Cavity threshold. The vertical bars represent the scatter in the measured data. In general, the calculated and observed quantities agree rather well. Both indicate a decrease in laser output with increasing cavity loss, i. e. , decreasing R'ORL. The experimental data, however, indicate a somewhat smaller slope fo r the curve than that predicted, especially in the region Of higher 10 s s. The higher loss conditions correspond to smaller attenuator angles (approximately 0 to 20 deg). It was demonstrated in Chapter 2 that rotational relaxation and rOtational nonequilibrium play a significant role in the Character and De rformance of the HF Chemical laser. Since the rotational relaxa- tion rates are generally unknown, it is appropriate that an examina- tion be made Of the sensitivity Of the present calculations to these 1.a-i:es. The rotational relaxation rate was increased by an order Of magnitude from the gas kinetic frequency to ten times the gas kinetic c ollision frequency between HF molecules and all collision partners. The results indicate that there is no extensive change for this case. 56 U, E 60 INITIAL GAS MIXTURE I I a I EXPERIMENTAL g 0.2I-Izzl SF6:10 H0 DATA E 40 _TI = 300°K, PI = 33 Torr __ a: CAVITY CONDITIONS a: THEORY / . Ro=0.9SXt2, RL =0.95 >. 0 L = 169 2 III III 3 2 o I I 0.70 0.00 0.90 RJRL Id) In I— 60 i INITIAL GAS MIXTURE I I 3 I H2:I SF6:10 Ha I EXPE§;¥§"TAL E TI = 300°K, Pi = 33 Torr j 40 —CAVITY CONDITIONS — III a: R0 = 0.95 x :2, RL = 0.95 «>5. L = 169 cm 5. 20 — —— 2 THEORY III III 3 E 0 I I 0.70 0.00 0.90 RgaL (b) V) I: 50 I I § INITIAL GAS MIXTURE IEXPERIMENTAL m I szl SF6:IO H0 DATA .2- TI = 300°K, Pl = m TOI’I’ / j 40 ”CAVITY CONDITIONS 1 — III 0: R°=0.95Xt2, RL=0.95 ,: L = 169 cm I/I 2 I/ THEORY _‘ Ill 20 '— 2 III III 3 3 I I a. 0 0.70 0.0 0.90 RO’RL (C) Figure 3. 6. Effect of cavity losses on total laser output for three gas mixtures. (a) 0.2 H :1 SF6:10 He at 33 Torr, (b) 1 H2:1 SF :10 He a 33 Torr, (c) 1H2 :15F6:10 He at 5 Torr 57 Because of the rotational relaxation provision in the model, it is well suited for making calculations Of laser Operation in the line- selected mode. The experiment was repeated for the laser operating on a single line. The transitions P1(4) and P2(3) were selected, since these were Observed to be among the strongest lines in the pulse output spectra. The results for the P1(4) and PZ(3) compari- sons are presented in Figures 3. 7.a and 3. 7.b, respectively. The discrepancy between theory and experiment is more severe in the P2(3) case. This behavior is not expected from considerations of the kinetic mechanisms involved, and may be partially due to the fact that mode structure and diffraction losses were not incorporated into the model. Further study is necessary to Clarify this point. Note that the value assigned to RL is somewhat arbitrary. Fortunately, calculations indicate that the shapes Of these theoretical curves are not sensitive tO that value. If, for example, R were L assigned the value 0. 90 instead of the 0. 95 presently used, the maxi- mum change in predicted pulse energy in Figures 3. 7. a and 3. 7. b for any given R’ORL, is less than 2.5%, which is significantly smaller than the experimental data scatter. The rotational-relaxation rate is an important factor in deter- mining the rate at which the population inversion of a given transition will be restored or maintained following laser action on that tran- sition. It is expected that, with the laser Operating in the single-line mode, as the rotational relaxation rate is increased, more energy would be extracted. This was verified by calculations. As in the multiline case, the rotational-relaxation rate was increased from gas kinetic to ten times gas kinetic with the laser Operating in the PULSE ENERGY. RELATIVE UNITS PULSE ENERGY, RELATIVE UNITS 58 72 I INITIAL GAS MIXTURE I EXPERIMENTAL 1H2:1 SF6:10 He DATA 0 __Ti = 300°K, Pi = 50 Torr CAVITY CONDITIONS R0 = 0.95 x t2, RL = 0.95 ”so” 4 P- Lfl'}; / I T "' / \ / \\_,’“7~_._/ 0 I OFF-AXIS PARASITICS 0.70 0.00 0.90 RoI RL (0) 16 I I INITIAL GAS MIXTURE IEXPERNENTAL DATA 12 I szl SP6:10 H0 _Tl = 3000K, Pi = 50 TOI’I’ —" CAVITY CONDITIONS R0 = 0.95 x t2, RL = 0.95 -— L=169Cm — I II I L/fi '— THEORY I I 0 0.70 0.80 0.90 “6% ‘ (b) Figure 3. 7. Effect of cavity losses on output of laser Operating in single-line mode for two selected transitions. (a) 131(4), (b) P2(3). Note significant loss in useful laser output that resulted from parasitic oscillations, as shown in (a) 59 single-line mode on the P2(3) transition. The laser output increased by approximately 6% for various values Of R'OR but had very little L effect on the overall shape Of the theoretical curve presented in Figure 3. 7.b. The U- shaped curve (Figure 3. 7.a) represents an interesting example Of Off-axis parasitics that resulted in a reduction Of laser output. Here, in the changing of the effective reflectivity of the out- put coupling, the attenuator was turned in the counterclockwise direction. When the normal to the attenuator was approximately 20 to 40 deg from the laser axis, the detector showed an abrupt drop in laser power. The CaF2 attenuator had apparently coupled itself to the walls of the laser chamber and produced Off-axis oscillations. Similar results were Observed for the P2(3) and multiline cases at these angles, although the power reductions were not as large. In the P1(4) case, lasing on the 1—0 band, which depleted the v = 1 level, and the fact that the pumping tO the v = 2 level is at least twice as fast as that tO any other vibrational level, lead to large population inversions in the 2-1 band transitions. Since lasing on those transitions was sup- pressed in this mode, parasitics were expected. Parasitic oscillations were not evident when the attenuator was turned in the clockwise direction, with all possible extraneous reflecting surfaces outside the laser chamber covered by a thin layer of sponge. The results shown in Figures 3. 6 and 3. 7 illustrate the impor- tance of "small" cavity losses that are Often neglected in the design Of Optical cavities. At the angles of maximum attenuation the CaF2 attenuator used in these experiments has typical transmission coefficients greater than 93%, and the power reduction,in some 60 cases, is more than 50%. When the extraneous losses in the cavity become comparable tO the output coupling, large power reductions will result. Thus, as the Q of the cavity is increased, the problem progressively worsens. One possible source of such losses in the gas laser is the Brewster window. While this window theoretically has a transmissivity of 100% for radiation with polarization parallel to the plane Of incidence, it may still be an important source Of losses for some laser systems. (In high power lasers, there may be window or mirror, or both, damage after a few shots.) In addition to losses re- sulting from scattering and absorption, the windows could serve as couplers for Off-axis parasitic oscillations. Given the reasonably good agreement between theory and experi- ment, one is encouraged to proceed with some confidence in the model. Model calculations Of the effect Of threshold condition on the laser pulse energy are given in Figure 3. 8. A more typical, higher gain gas mixture was selected, and a wider range for a was used. The gas thr composition was 100 Torr Of SF6 and 20 Torr Of H . Three repre- 2 sentative cases are presented: (1) total pulse energy; (2) P2(3) in the single line mode; and (3) P3(6), P2(7), and P1(8), in the cascade mode. For the DF laser, these three lines in the cascade are known to have good transmission coefficients through the atmosphere. * The re- sultant curves show the eXpected profile: At high threshold, lasing is delayed and sporadic and a large portion Of the pumping energy is lost through deactivation. At low threshold, energy is not extracted * D. J. Spencer, The Aerospace Corporation, private communication, September 1974. Figure 3. 8. 61 Model calculations of the effect of cavity threshold on pulse energy for a 1 H2:5 SF6 mixture at 120 Torr, with laser Operating in three different modes. (a) mulfi-line mode; (b)' single-line mode on P2(3), transition; (c) cascade mode Of three transitions, P3(6), P2(7), and P1(8) PULSE EKRGY, J/Iitar PULSE ENERGY. J/llter PULSE ENERGY. sylltor I I T I I I T I T T I I I I I I6 _ ROTATIONAL EOUILIsRIUM 14 I- 12 — III GAS KINETIC I0 . INITIAL GAS MIXTURE I 142:5 SF6 0 _ T, = 300°K. I:I . 120 Torr L 01 1 1 1 1 I 1 1 1 1 I 1 1 1 II I0" 10" 10‘2 10" “thr' cm" IeI T 1 T I T 1 T 1 I I T I I I I 3 .— GAS KINETIC 2 _ III AS KINETIC / G ' INITIAL GAS MIXTURE ’- ‘ H215 SF6 ‘ T' ' ”OK. Pl . 1mTOl’f O 1 1 1 11 1 1 l 1 I l 1 1 1 I I0" 10'3 I0‘2 I0" “thr“ cm" IbI T I 7 II T l I I I I I I I] ,2 __ GAS KINETIC Io — . I—- 6 III. GAS KINETIC ‘ — INITIAL GWRS 2 —- T. . m0" pl - 1m 'forr o l l 1 1 I 1 l l l l 1 l l l l 10“ 10'3 10‘2 10" “thr" ""4 IcI Figure 3 . 8 63 efficiently from the optical cavity, although intensities within the cavity reach relatively high levels. This condition is also the most susceptible to power reduction resulting from ”minor" cavity losses. The optimum performance lies between these two extremes. 3. 5 Summary A theoretical and experimental investigation was conducted to assess the role of cavity losses in the performance of the pulsed chemical laser in the single-line and the multiline modes. Although the investigation was carried out with the SFé-H2 laser, the results are believed to be of more general validity. It was found that losses within the optical cavity, of as little as 5%, typical of many lasers, can easily result in a 30% or more reduction in’power output. This problem becomes especially acute for high-Q cavities. In addition, model predictions are presented that graphically illustrate the effect of cavity threshold on laser output. This includes, for the first time, predictions of performance in the single-line and cascading modes . 4. SUMMARY AND CONCLUSIONS Computer models are presented that simulate the performance of the H + F chain reaction and the SF6-H pulsed chemical 2 2 2 lasers. Rate equations are used to represent the chemical kinetic and stimulated emission processes occurring in a representative unit volume within a Fabry-Perot cavity. All processes are assumed uniform throughout the cavity. Lasing is permitted on all lines in the vibrational-rotational bands that reach threshold and may respond to gain fluctuations during the lasing period. Rotational relaxation is incorporated into the model of the SFé-H2 laser to study the effect of rotational nonequilibrium on the laser perfor- mance. The individual rotational level populations are determined from the solution of rate equations . Relaxation of the rotational population is accomplished by the assumption of a characteristic rotational-relaxation time similar to the model of Schappert34 for C02. With the rotational populations of the first four vibrational levels identified, the effects of single-line operation can be examined. Comparisons of the predictions of this model with that of the constant-gain model illustrated the unique features of the present model. The present formulation facilitated the determination of the time evolution of gain, intensity, and species concentration. Inter- action of gain and intensity before and during lasing was observed. 64 65 Contrary to the constant-gain model, lasing did not commence upon gain reaching threshold. The growth of intensity was sufficiently slow such that lasing effectively began only after the gain was well above threshold. This usually resulted in a large initial spike in the intensity, which could be a potential source of damage to windows and mirrors in high power lasers. In addition to the prediction of transients and relaxation oscillations seen experimentally, this model demonstrated greater sensitivity to nonlinear mechanisms such as vibrational-vibrational (VV) deactivation. This resulted in the prediction of shorter pulses in the laser output. Calculations from the model showed that the assumption of a Boltzmann distribution among the rotational-state populations pre- cluded extensive multiline-lasing and that such multiline lasing became prevalent only when rotational-relaxation effects were con- sidered. Rotational relaxation also accounted for a decrease in predicted laser output; this prediction is more consistent with experiment. The modeling of rotational relaxation permitted, for the first time, an evaluation of the performance of a chemical laser when operating in the line-selected mode. Calculations were made of the SFé-H2 laser oscillating on a single line for each of the three transitions P1(8), P2(7), and P3(6). The sum of the pulse output from the three single-line calculations is only 65% of the output for the case with these three transitions operating in a cascading manner. For the gas mixture and cavity conditions considered, the effect of preferential pumping on rotational nonequilibrium and laser 66 performance was small. It resulted mainly in a slight increase in output energy for the higher J transitions. This behavior, however, may be sensitive to cavity threshold. This model was used in a theoretical and experimental investi- gation of the effect of cavity losses on the performance of a SF6-H2 laser. The experimental data are expected to provide a test of the validity of the model's predictions and assumptions. At low levels of initiation, as with the present case, the laser pulse shape is sensi- tive to that of the discharge current. The predicted pulse shape and pulse width are in good agreement with experimental observations. Good agreement between theory and experiment is also found in the prediction of pulse output variation with cavity losses, for the laser operating on the single transitions P1(4) and Pz(3), as well as in the multiline mode. Although not modeled, parasitic oscillations was shown to be an important consideration in the investigation of laser performance. Under some operating conditions, it could lead to significant losses in the useful laser output. Laser performance as a function of cavity threshold is also of interest. Calculations of the effect of output coupling are presented for the cases of multi- line operation, single-line operation on the P2(3) transition, and the cascade of the three transitions P3(6), P2(7), and P1(8). The model has demonstrated remarkably good agreement with experimental observations. One can conclude that there are no gross inaccuracies in the chemical kinetic or radiative formulation of the model. As shown in Chapter 3, however, there are factors such as cavity losses and parasitic oscillations which exist in real 67 lasers that are unaccounted for, but which could have appreciable effect on the laser character. These factors obviously are not easily modeled. There are other phenomena that are not modeled which exist within most lasers: Mode competition and mode beating, intensity distribution transverse to the laser axis, medium inhomogeneity, temperature gradients, acoustic effects, interaction with the walls, and plasma kinetics in the case of electrical discharge initiation. Attempts were made in the present experiments to eliminate or minimize the effects of some of these unknown inputs. An iris was used to maintain dominance of the TEMoo mode in the radiation field, thus mode competition is minimized. The particular configuration selected for this laser resulted in a high-gain path along the. axis of the laser away from the walls. Thus, in the time scale of the laser pulse, interaction with the wall is highly unlikely. The effect of medium inhomogeneity or temperature gradients have not been studied. These could result in irregularities along the gain path and distortions in the radiation field, but the net effect may average out along the laser axis and result in only minor alterations of the laser output. The lack of plasma chemistry in the model appeared to be a serious drawback in these experiments. Initiation through electrical discharge proceeds through the creation of a partially ionized plasma that includes in its composition excited and metastable Species as well as ion-electron pairs and neutral particles. Production of F atoms necessary for the pumping reaction is the result of complex processes within the plasma such as charge transfer, Penning 68 ionization, recombination, detachment, and attachment. The model presented here, however, incorporates no plasma kinetics. Nevertheless, since only a small fraction of the gas is ionized (typi- cally less than 5%; in this case, less then 2%), the effect of these charged particles on the neutral chemistry and the associated model predictions is expected to be minimal. The most important contribution of the plasma kinetics is F-atom production, which was found by Lyman to be roughly proportional to the input power. This fact was incorporated into the model empirically. The rotational-relaxation process modeled in this investigation represents only an initial step in a rapidly developing area of active research. Recognition of the importance of the role of rotational nonequilibrium and rotational relaxation in the behavior of chemical 63-65 and lasers has led to several recent studies, both theoretical experimental. * While the formulation adopted in this work has been valuable in the prediction of broad effects of rotational relaxation and in providing physical insight into the problem, the rotational relaxation rates now being measured will permit a more accurate study to be made of these phenomena through a careful model of the detailed collisional dynamics involved in the processes . To be complete, this model should include the effects of R-branch and J -J lasing. However, the increased computational complexities result- ing from these additions make this proposition impractical for the present day generation of computers. * See footnote, page 29. 69 Further experimental verifications of the model predictions are desirable. Experiments such as small- signal—gain probing and Raman scattering would provide valuable data on the species concentrations within the laser medium, and side-arm chemilumi- nescence could be used to verify the predicted high- gain overshoots APPENDICES APPENDIX A RATE COEFFICIENTS FOR H2 + F2 CHEMICAL LASER APPENDIX A RATE COEFFICIENTS FOR H2 + F2 CHEMICAL LASER The chemical kinetic model used for the rate -equation solution has been suggested by Cohen33 and is shown in Table A. 1. Rate coefficients k and k- designate forward and backward rates, respec- tively. For each reaction, the missing rate coefficient is determined from the equilibrium constant. '70 71 Table A. 1. Rate coefficients for H2 + F2 chemical lasera ReNaztion Reaction Rate Coefficient, cc/mol-Iec M, v lab F+HZ(0)=HF(1)+H kuez.6xio‘3“-°’° lb F+Hz(0):HF(Z)+H klbes.axio”"'6/° lc F+HZ(O)=HF(3)+H klc=4.4xio‘3"-6/° ld F+H2(0) :HF(4)+ H k_m=7.4x Io‘z‘°°5°’° le r + HZIO) = HF(S) + H k-” = Li '10‘3‘0-5l/9 Ir F+HZ(0)=HF(6)+H k_“:l.9x1013'0'56/e 2a H+FZ:HF(0)+F k2a=1,1x1012'2-4/° 2b H+FZ=HF(l)+F k2b=z.5xio‘z‘Z-‘/° Zc H+F2=HF(Z)+ r kzc=3.5xio‘z'Z-"° 2d H+F2=HF(3)+F k2d=3.6xio‘2'2-4’° 2e H+FZ=HF(4)+F kze:1.6\<1013'z'4/6 zr H+FZ=HF(S)+F sz:3.6x1013-Z.4/9 23 H+ rzeunenr k28=4.8xio‘3'2'“9 2h H+FZ=HF(7)+ F kZh: 5,5,. Io‘Z’Z-‘l/e 21 H+F2=HF(8)+ F kZI =2.5x10‘2'2"”e 3av° HF(v)+ M1 = HF(v - I)+ M‘ I.“ =v(Io”'°T‘°‘3 + I00" 73's) M1 = HF, v = I - - - 8 3b! HF(1)+MZ =HF(O)+MZ k3b:=l.5x10w'1‘”61‘ MZ=F 3bz HF(Z) + M2 = Hp(1)+ M2 ka2 :1-5 x 1010-0.5/9 T 3bv HF(V)+MZ:HF(v-l)+Mz k3b =I.5on‘°T v=3---8 3cv HF(v) 4» M4 = HF(v - I) + M4. kac: = (8 x10" T‘)v M4 = Ar, F2, 5176; v = I - - - 3dv HF(v)+MS:HF(v- I)+MS k3dv=v(8.7x10'7T5) MS=He v=l-- . a 3eV HF(v)+M6:HF(v')+M6 1(3'3V:l.8x1013.o‘7/0 szfl v:l- - . s, v' 1, 86 l. Initialize Variables [Y(1,J) , t] 2. Initialize PrOgram Control Parameters [H, HMIN, HMAX, EPS, MAXDER. TIMEL, TSKIP, NCT. JSTART, NRVAR, etC.] 3. Obtain a. Spectroscopic Data E,WC.B,O,F b. Thermodynamic Data CV, H C. Pumping Reaction Product Energy Distribution [$LOP, TRCEP] 4. Define Cavity Conditions [R0, RL. L] 41 1. Obtain Rate Coefficients [KFR, KBR] 2. Compute Normalized Boltzmann Distribution from Equation (2 ll) 3. Compute Rotational Relaxation Rate Constants from Equation (2.15) l I. Call Subroutine GAIN; 2. Compute (v, J) from Equation f2. 7) 1 Monitor all transitions with gain near or move threshold 1 [ Call integration subroutine DIFSUB ( Print This Set of YES Answers? ”2"" NDL N0 Guise Terminated? ) ' _J _J _J YES Figure F. 1. Computer simulation of pulsed HF chemical laser. 87 are related to the (I-1)th derivatives of Y(1,J). Tables F. 1.a and F. 1.b define these variables as used in the H2 - F2 and the SF6 - H2 models, respec- tively. The other symbols found in the computer programs are defined in Table F. 2. Typical computer outputs from the model calculations at a selected time in the duration of the laser pulse are given in Tables F. 3 and F. 4. Table F. 3 is from the H2 + F2 model and Table F.4 is from the SF‘6-H2 model. Complete listings of the programs and subroutines are given in Tables F. 5 through F. 12. 88 Table F. 1.a. Identification of variables Y(1,N) in H2 + F2 model N 1-9 10-16 17 18-20 21 22 23 24-79 Y(1, N) HF(v) concentration, mol-cm-3, v = N - 1 Undefined, reserved for additional variables H-atom concentration, mol-cm.3 H2(v) concentration, mol-cm-3, v = N - 18 F—atom concentration, mol-cm- F2 concentration, mol-cm"3 Translational temperature, °K Photon flux, f(v, J), mol-cm-Z-sec-1,vv = [(N - 17)/7i , J = Jifnax - 4, ' ° °, Jynax + 2, where Jmax indicates the transition of maximum gain. 89 Table F. 1.b. Identification of variables Y(1,N) in SF -H model N 1-7 8-23 24—39 40-55 56-71 72' 73-75 76 77 78 79-125 6 2 Y(1,N) II 2 HF(v) concentration, mol-cm-3, v HF(O, J) concentration, mol-cm-3, J = O, ' ° ', 15 HF(1,J) concentration, mol—cm_3, J = O, H °, 15 HF(Z, J) concentration, mol-cm-3, J = 0, ° ° ', 15 HF(3, J) concentration, mol-cm-3, J = O, ' ° °, 15 H-atom concentration, mol-cm- H2(v) concentration, mol-cm-3, v = N - 73 F-atom concentration, mol-cm- F2 concentration, mol-cm.3 Translational Temperature, °K Photon flux f(v, J); specific values of v and J selected by the program Table F. 2. Symbol in Text A(v, J) B(V, J) dt Nomenclature Symbol in Computer Program A(V. J) B(V, J) C CVI E(V, J) F(T, J) FLUX(V, J) ALPHA(V, J) EHPYI CRRNT K KFR, KBR LN TH Y(1,I) DERV1Y(I) 90 Definition Einstein isotropic coefficient for spontaneous emission, 1/molecu1e- sec Einstein isotro ic intensity absorption coefficient, cm /molecu1e-J-sec Speed of light, 2.997925 x 1010 cm/sec Molar specific heat at constant volume of species i, cal/mol-°K Rotational energy of state V, J, cm.1 Resonance broadening function, Reference 22 - 2 -1 Photon flux, mol-cm -sec Gain of transition (v + 1, J - 1) - (v, J), cm“ Plank's constant, 6.6256 X 10‘34 J-sec Specific enthalpy, kcal/ g Molar enthalpy of species i, kcal/mol Discharge current, amp Boltzmann's constant, 1. 38054 X 10.23 J-°K-1 Forward and backward rate constants, in terms of moles, centimeters, and sec- onds Length of active medium, cm Concentration of species i, mol-cm e e e -3 - Time-derivative of Ni’ mol-cm -sec Table F. 2. Symbol in Text ri ri thr ¢(V. J) (UC(V,J) ( Continued) Symbol in Computer Program NA POWER PIN Q(V, T) RO, RL R T Y(1, 23) THGAIN pHuvpn TAU(V,J) WC(V, J) 91 Definition Avogadro's number, 6.02252 X 1023 molecules-mol' Power density of laser output, W-cm'3 Power input from initiation, W—cm-3 Rotational partition function for level v Mir ro r reflectivitie 3 Universal as constant, 1.98725 cal- mol'1-°K‘ Time, sec Temperature, °K Stoichiometric coefficients of reaction r -1 Threshold gain, cm Normalized line profile of transition (v+ 1, J - 1)-0 (\,J), cm Rotational relaxation time constant, sec Wave number of transition (v + 1, J - 1) -0 (v,J),cm' Table F. 3. Sample computer output for the H2 + F2 model. .31252E‘07 SEC STEP SIZE: “529 TIMES, DIFSUB CALLED .67065E-02 JOULES/CC = .50“59E+03 Kc TEMPERATURE .76017E-0N SEC. TIHE= PULSE ENERGY .62237E+02 HBTTS/CC LflSlNG pOHER= CONCENTRATIONS IN HOLES/CC H2121 .38308t-D9 07 «u cm '1‘ am 1: F2 .59677E-09 .27312E-57 .2 H .1h620E-08 IN HOLES/SEC/CH-SOUAQEO OF THE LflSlNG LINES, PHOTON FLUX 92 a. annnnnnnn ¢dddduddd XI I I I l t I l DONGMMGO voooctonca meanc-IN—I—od O O O O O O I O A AMMMMMMQM ldddddddd XI 0 o O 0 I 0 0 «aluminum 11'ou 1300060006 313096606: OOOOODDOO >D°OGOOQO ‘dd—dddd-d—l moo-00000 . mnnnnnfinn tdddddddd XIOIOOIII (wwmmwwww Zoocaoooa ’DOOOOOOG OOGDOGOOO >QGOQDQOQ udddddddd u. e o o e e e o a - ”MMMMHHMM tOfiDGOOOO 300666009 .ODOOODQO >DOQGO°OO vddHflU-‘Hv-‘d M. o e o o e e O o x er Immo‘mmmoa‘ Ha >DdNMJmON IN WATTS/CC POHER OF THE LASING TRANSITIONS. NNNNNNNN «Adda-44d IOIIIIII wwummwmm «Nowcmme thchdm @HdQONJO N fidd‘holfid .1 :JMMFOMM NNNNNNNN dddddddfl IDIOOIII MUMUWWJM am:mmoo« :HNNODAM omnmow:: mwaocmew 3’34MMMW NNJchv-ION «coca—Iced I‘ltllfil uwuummuu MQQNNv-un"! osamnaam MHmMONJO :NQNMaNN ¢deNMdM O O O I 0 O O O NNNNN—INN dHQdOOdv-d I 0‘ I oo 0 l (umuwmwmiu Nfime‘INNnM MQ3L9NJJ ddONJNOm mWOHOdNH :J—d:dm~)n NNNNNNNN Hddddddd 10.0.0.0 umuuwumu DQOONKNd Q30NOQm0 OGNMHOWD mmnNoes: 31¢33MMM eoeeeeoe NNNNNNNN dddddddd I I I I I I I I muutumuutusu mowooao: smaoowmc m3¢00m00 ~00 :NC'JO‘Q: 3:3:TMMM NNNNNNNN dded—tdd I I I I i I I I inumwiuww OOMJJO‘O‘: hkmn-Iocom .3 3:331!!!” NOOQQNOO u OdNHJmON (Continued) Table F. 3. CONTRIBUTioN To PULSE ENERGV FRCH EACH TRANSITION IN JOULES/CC ¢o~m NOONNNQ -«uda dedddd 3000001000001 u «00 oweookq : arddionnsca dedddeQMNm 000000000000000 ace: ooamssmoscoaauo dddOOOOdddddddd 010000001100110 uuwummwwuw MNQQQNO‘OO—I HNNNdde-I OJ", '0") 000000000000000 N'QKU‘ 3MMOQN ommmwmwmwmm meCWNNSNN-d VUMDdNJ-fdog—i ILOU‘NNQQv-‘O‘ 1-1 INQMMmdNOdd own~o««mdn 0000000000000.0 0000c: addacaooaddd ”IIIIIIIIII. .WUUUUUUJLLJLUUJU :QMNJONQONQ.’ ~QU‘30‘NNJMJJJ deNONmadNM: IJAMQON©3NOO mflNQQH—dddm 000000000000000 GOO: QNNU‘ QM‘OM JIDNQQQQ Addd000000W—4d-‘dd 5001100010100110 owmmwwmwwuwmmuwm noomnswomcmsomos quoMOoomQKmU‘JOo ummmmomamsumommn IOQdWNJNOO¢dMONm mQNHMv-INNQ—IQJMMN 000000000000000 00WWQMMMHGON -«HGQQOOOQHV‘H 5111101001111 'UMUWWNWNWUUU NN¢dO4NdeONO VQNOJO‘3HNU‘NOH “NomQNNNNNOdM IWWOQNd3°QmH0 NOMMNHNMNNdN 00000000000000. no: Num333MMMMQN -d000000000v4d 9000001101110 0 mini-u wuwwuww 04 04:0‘ QNNNQQO -cmswcnmkmmno ilm Mchch NDO QMNN:NNm emN MWU‘HNO‘N:U\N«N 000000000000000 000 No:m::nmeooo~~c «00000000ddddfid 001100001000001 w mm wwwmuwmwwm an cmomODonao 0 HF(OvJI LU om «IN co: oo 0 NN hwmuwwwwaaeo O O 00000000000 newncmmsoaeamfism "0"“de IN 1/CH THE GAIN FOR ALL POSSIBLE TRANSITIONSI 93 -NNNNNNMM¢33Q83“ “00000009000000 0000100011101111 hwwuwmwuuwwwuuw ”ROADWONOMOQNNQ (dc-MONH'O-IQQQ ammo: IN:OO#deNNM~NNO QQOWOHNm¢M@mNQdM JflnnNNddeQNJNHJ ‘000000000000000 I I I I I I I I 1 ~NNNNMMN~MMMM¢JUI fiacoaaaooaoocaad 0101111101100111 cummwumummmmwmw amatomimo :QNN©M:‘ dommmuonmn chic-ca: INLNOMGmNNNleN” Qd0NLfimNd—1 OLA-{meals Q JNMNdNNfld‘DmHv-MDNO‘ (00000000000000. «NNNNN3NNNnMM383 ficcoooooooooacoo '100000101110011 mmwuwwmmuwmwmwu ~00m~anemccdnano (OdQNONOO‘m‘OQNOOO‘ IOQMQdeNKDNmMHS anooc:dmoonmmoa .JJNNJde—tdhfldh'fld (00000000000000. 0 I I 00 adv-INVWNQNNMHMJJ.’ ”00000000000000 .IIIIIIIIIIIIIII JMUJWNUHJWUJWHJNUJUUUJ ~3—u0dtorsaomonmsomh: (JdNQNMO‘~ONONNU‘O‘N IQNNNQC‘QHKH0QMNCO Quonmmo~a«m~wmma AddflsflhNNV‘HQJNonc-O (000000000000000 I 1 I I I I I addddddNJNMMMMQJ 5000000000000000 '101100111000111 MlIJLII mwmm'uwwluwwww 'dm @ODOdQNGNmOM CQO‘NNQNQOSLBOH300‘ Imhohdoflfidkhd‘d‘co QONO‘QJHQQdQ3‘0—1MW 4n::n~«n~«0m~d:q (000000000000000 I I 0 I 1 I I ndQQddddNNNnnn33 fiaooooaqaooooooo 0100000010000100 wwwwwmuwwuwmmmw UmMMNO~7Nmmr~omm¢o do:«MoaNmmmo=oco INOO‘J’O‘doxD-flbécwmd Q:dd¢@0d0dMQNOQN Audmnddddflefdmd «00000000000000. «aaamddwNNnMMJQ ”00000000000000 O¢+¢§+IIIIIIIIII «mmwwmumumwwwuu vommmmcon-«om4mo :NCNQMHMNNNLfikNv-‘N M J's-4 MM Q0 cfikgoodgmdga AdNMNHQMHdHO‘mNNN (000000000000000 0 1 0: 0 0 0 I 0 0 noaooooad«NM:M:d Soaccceooooacooo .000000011111111 owmwwwmwwmmwuuwm vosmxooaomsoaan 4003mwahm350ao«m Immo:¢nu::@ommov asam¢4dmaomaooom amehmndono4H36 (000-00000000000 IIIIIIIIIII fidNM-TUMDHOU‘OOONFO 3th dddddd Table F. 4. Sample Computer output for SF6'H2 model. .27103E-10 SEC SYEP SIZE: 1360 YIHESv DIFSUB CALLED .75h07E-03 JOULES/CC .«1a«9£¢03 K. YEPFERAYLFE .h69375005 HATIS/cc SEC, .1OGJJE-06 VINE: OULSE EhEFCY: LASING POHEQ S/CC ,- :— CONCENTRAYICNS IN HOL F2 C. ~J U0 “.0 .hlGL H E-C? .33679 OU‘U‘O‘O‘U‘U‘O‘U‘DODWV-(IUF) .1‘ ODOOOOOUOfifififlv-INHN 00010001100000000 Amwmwwumlmwwwwmmmm MNNtUU‘LDv-CNU‘U\OJ'J\D‘JJUN VP‘HVCJU‘UUHVQJQJJHHU‘WJNW mooammmmc~1~06r~mnsm 1‘JQWIUWLJ‘VJLJUIVFJ‘IJJ")'\(\M\J M‘Vu\wwJOU!VHU\MHCI'I\UH|V OOOOOOOOIOOOOOOOO QO‘QQoma‘G‘O‘O‘uoao-«ud 00c)0000000dflHH.-404d IIIIIIIIIIIIIIIII aluwwwmmmutuluuuwuImmwm NHJ‘T‘U\QMNH@U‘N'DMF$0HL0K VJCIDwNv )U‘Kd’flDO‘HLIM‘OJM ummaommaa‘muommmkmo‘a‘: IOJQNOU‘NMOMNJOWQLRLJO NJHHHHJNfiJflO‘mP‘flwPDv-d OOOOOOIOOOOOOOIOO NU‘UQQDWO‘U‘U'O')Q—IUQU 0000000000Hdflflfiflfi IIIIIIIIIIIIIIIII Awwlunnuuuutulu'uulunuulunutu HUHU(VWWKVU\U‘U"WN(VNWIU ~u\t_‘h'—1®\UHU‘U\NNMJJQU‘U‘ n.1smaarxmr10m30'r‘oomo ImfifikmmHJHnNrsrxum‘fiu‘ flQNNNflHOV’fiOMNJ'NHH NO‘QQOQU‘H‘O‘G‘fiv-Qdmm.’ .1 00051.3(3LWACJOOflv-INHHV-dvd 00000000000000000 Awmmwmmmmmmwwmu.unuu) UUNNNv-IQNOMWJCOO‘JNN vommlanHN )u\u«(rF’)V~N0\ LLNUU\U\.7MU\O‘MJNN :Nouors IHU‘C“ Nvmmo‘mmmu~~u:mmm dkHNNdO‘ IOMNLOHMQO‘H fl QHNfiJmONQOUHNM3m v-Iv-(Hv-‘Hv-I 94 GAIN OF EACH POSSIBLE TFANSIYION (1/CH) (JOULES/CCD 0F(1,J) ENERGY CF EACH LIht (ROTlS/CC) LINES POWER OF LASING ALPPA(C.J) ALFHA(1,J) ALPHA(ZyJ) HF(29J) HF(1,J) PF(2,J) hF(O,J) PF(0,J) 0000 I 00 0000000 00000 0000000 000000 0000000 0000 00 000000 J 00000 0 0000000 000000 :1“ 0000000 OOOOOOOOOOOOOOO ”(DUO UL) 00‘1TJOUQ OC‘OC‘I“) (‘0 L 1000000 000000 0000000 HNMmeDNQO‘0v-INMJID v-IHFIH'IH TIP‘(SEC0 ROTATICNAL RELAXATION HF‘i'J’ I‘F‘Z’J’ HF(qu'I HF(0,J) O‘U‘O‘U‘O’O‘O'O‘O‘U‘O‘O‘U'O‘O‘O‘ .30‘30000000000000 I111001010I00000 unmwwwwwwwmwwmwww mmuwcwmowmu‘whuo‘o' 00‘"|V0'IJU\KWU‘CJHIVI'IJUV nsooomoowowo‘mu‘o‘o‘ NNNNNNNNNNNNNNNN O‘O‘O‘U‘O‘U‘O‘O‘U‘O‘O‘O‘O‘U‘O‘O‘ wwmomwmwo‘a‘a‘mmwo‘o‘ 0000U00UUJ00LNJ040 IIIIIIIIIIIIIIII mummmwmwuuwwmmmu \DQJfiJNO‘NJOCONmeN QO‘v-«VWJJuJNQJu'v-HV")JUHU FNQQQGJQIJIJQJO‘U‘O‘O‘U‘U‘ NNNNNNNNNNNNNNN" U‘U‘U‘U‘U‘U‘O‘O‘U'U‘U‘U‘U‘U‘U‘O‘ 0............... 0|u-u-u'u'u'u-u‘u'u'u'u'u'u'u'u‘ 0000 100.100.1000ng IIIIIIIIIIIIIIII “lulu".InnuluuuuhNLJIJulmtnm \DtanwO‘NU-QHFlwlDON-Tm Inummuwovsd‘rn-iNJunDK hmccocmommmo‘mo‘a‘o‘ NNNKNBLNNNNLBNNN UGO‘U‘U‘O‘O‘O‘O‘O‘O‘O‘O‘O‘U‘O‘ U‘O‘U‘O‘U‘O‘O‘O‘O‘O‘O‘O‘U‘O‘O‘O‘ IJU'J' )IJQVJ' )0 ’I ’Qooutvcv 1010001100010000 mummwwmmmmmmmwwm sdacmmdmmma:km«m QUHNJU‘QK‘O‘WOJM VU‘KO soomoooonmmmmwmm NNNNNKKNNNNBKNNN U‘C‘U‘O‘U‘O‘U‘G‘O‘U‘O‘O‘O‘O‘O‘O‘ OHNMJWQDNOO‘OflNn-im fiHde-Ir. 95 Table F. 5. Program MODELC iaecx HO)ELC in 20 30 50 60 po)59nn ~09ELP¢I~°UT,0UTPHT,Tnp€1,ran?,rassx,rnoeu,7an7,Tnpea, 1 TADCQ,TA°F10,TA’E11,TAPF12) JIGENSION V(8.79).DCQv1Y(79),STORE(7) CO1HON YHAX(79),SAVF(1?,7°),ER?OR(TQ),°H(6u00),E(10,31),HC(9,18), 3(9,1«),o(o,s1),r(so,3o),xrn«1u2),K3a(1«2),ALPHA(a,1S), JMhX(%),CVF(17i,CVF?(17),EHPYH(17).E4PYH2(17),EHPYF(17), FHpvc7(17),FH°YHF(27),THGAIN,DL,7LUX,CVN?(17),AR,HE,N2, ‘ RCTF(197),PRTQ(1Q?),PF(1Q?),SIGNL(8),PLINE(8,1SI,ELINE(5,15) nrxL KFP,KB°,N? 95x0(1.su)r ,RF\0(2,10)HC RF\D(3,10)B QE\0(h,In)O 2Fx0(1o,g0)f orxo«a,so)cvr.cvr7,nv~2.rvsrs.cvuz,cvu= RF\O(9,?n)FHOYH,FHDYH9,FHPYF,EHPYF2,EH°YHF FOQHAT(SE16.S) F0!HhT(5¢16.59/GF16.9/8F1F.q/2516.9.QAX) F3!HAT( F?0.8) =OQHAI(q¢16.6) FOIHAT(6F1Q.S) PH! = 1. R‘ti-qfi Q=L=.Gh puz=so. PM! = 0. th‘hrfl ‘6ii_é o. KFI=23 FCIR ? .51925-7 FLJY=1.F-1h TIQEL = ?On.F-6 D‘,HIN = 1.F-19 ILIH = DELHIN/loc. H = .10 F-10 szn = DELMIN aux! = ?.F-f 5?; = 1. F-i NA(DEQ = R R0 =‘o.n RL 3 1.. LNTH = 100. T§(I° 3 1.5-6 YS°E° = T§KI° NOV : fl JC‘T375 J3\I" = JCNT PO'ER = 0. FV3Y = 0. PL : n. T : 0. JQTADT = 0 Ht 3 0 03 990 I 3 1,8 J’4\X(I) = ‘0 30 990 J = tqu pLINF(I,J) '3. ELENF(I,J) fl. 96 Table F. 5. (Continued) 990 IL’HA(I,J) = 00 1000 I 1000 YW\X(II 00 1009 nan(1) PCrBSI) 100‘ Qctt) = . HE QHF ' PCT? NZ BN2 ‘ FCTP A? 900 0 F070 T4;AIN = —AL00(:0 ' °L)/(?. ‘ LNTH) _ 00 1010 I = 1,17 1010 V(L,I) = 0. GHIIHH v11,101 = OH? 1 rc*° _ v11,19) = 0. 111.20) = 0. v11,’11 = 9:10 FCT° v11,22) = or? ' F019 v11,731 = 300. 00 1020 1 = 20,70 ___1020 Y(t,I) = FLUX nmr = 1v¢1.231 - 100.1/90. + 1.5 V7=IFIX(&KYD 00 1022 T=1,K* FF\D(1?930)KF° 1022 9:1017,301 000 102% 10.0 = T 01;? = (v11,?1) - 100.1/25. + 1.5 IV' = IFIX(AIKT) 1030 I=1KT.F0.IKT) cc *0 1000 MKT)IF = IKT,- K11 IFKKTDIF.GT.0) 00 TO 1026 9?:07 1033 L033h30lHATtll'TEPDEQATUPE was DROPEU‘II) 00 TO 1190 1039 KT 2 IKT 1u09,10“9_1“?,11KT01‘ RF\D(12.30)KFR 1000 9:1017,30) KBD _1050 IFKI:§0.0.) 60 To 1109 J31IN = Jantn + 1 IFIJGAIN - JCNT) 1000.10c0.1050 -———-—— 7 .1}.....;ICOEPUTE GAIN to: THE LINES NEAR J-HAX O Gin -10§2_001105§ IF11§.-_ 1021-1 JL)H = 000x111 - w ,J"LQH.?-J00XKI) f.‘ 00 1056 J=JLOH,JHIGH 1090 AL’HA(I.J) = 00100(IV,J,70,Y) _1_”“1§Q {9-11‘0 ““7 C C.........C0HDUTF CAIN =00 FDCH POSSI§LE TRANSITION T0 LOHER LEVEL (IV,J) C. ogo‘g’g’gglg CHFCK FO°_ J’SHIFT C 1058 StIN = 0 __.._-. 90 1920.121» Iv = I - 1 no 1060 J = 1.19 _10§0 00{0511,01_= 0010011v.J.79,v) 97 Table F. 5. (Continued) 0 C.........OETERHINF THF J (LCHEQ LEVEL) 0? «0! GAIN 0 00 1120 I = 1.0 qux0 = JPAY(I) JH\{(I) :00 J“ = u 00 1070 J = 0.13 It!QLPH0111J).LF.ALDHA¢I,JH)) GO to 1070 JH1X(I) =‘J J“ = JHAX(I) 107" EDITINUF IFKJ“§X(I) ,E0. 0) GO T? 1075 JN\X(I) = JNAY(I) - 1 1075 COITINUE IFIJQAX(I).F0.JMIXO’ 50 TC 1121 6“ Coo-cocoooIF A J'SHIFT HQ: “CCUCQFDQ QEDE‘INE THE APPROPPIATE F(V’J, 5112000909951NITIALI7F C JSFADT = 0 LJ 5 JHAXSI) - JHAxn L = 7 ' I + 16 I‘ILJ.LT.01 00 To 1192 IFKLJ.GE.7) Go To 1100 K° = 7 - LJ 00 1080 M = tykg _1080 Y(L:L+NPV:11111L!LJ*“’ DO 1090 N = 19LJ 1090 Y(l,L§N1KS) = FLUX ‘ “0, 59 TO 1120 _ 119? ML! = -LJ I‘KNLJ.GE.7) 00 TO 1100 ANK;_F ZthNLJ v DO 1193 M=1,NKS 1193 ST)RE(M) : V119L1M) -_.__13“ 1191_"=11NKS, 119$ Y(1,L+NLJ+H) = §T0°F(‘1 00 1195 M=1.NLJ “1195 111119H11=HEEQX 60 To 1120 1100 DO 1110 H = 1,7 .111QMY1L1L1H)_?WFL”X 1120 EOJTINUF c ggglogggooINTEGQATE C 1130 CI.L DIFSU9170,T,Y,°0VF,H,HHIN,HNAX,E°S,HF,YHAX,ERFOR,KFLAG, 1 . _J9TAPT,MAX0PC,PH,KFY) NC? 2 NOT 1 1 GO TO 1136 113! H : HHTN . flfiLN ?-H"IN’39- JS'ART = ‘1 50 TO 113" _113§-§01TINUF, . . . IFIKFLAG.LE. 0) GO TO 11:0 00 1610 I=Zh979 _IF1Y(1,I’0LTqFLUX, V11,I’:FLUX AND 98 Table F. 5. (Continued) 1610 coaTTNUP H1 = T - TOLO CQIST=2.05912 . THGAIN P0159 =0. 00 1137 121,0 (=1NAXII) - a L="I+16 0? 1137 J=1,7 PLSN‘(I,K¢J1 = Y(1,L+J) ' HC(I,K6J) ' CONST ' 0.180 P3069 = PnHPP + PLINE1I,K*J) 1137 EL!NE(I,K1J) = FLINPII.K+J) + PLINEII,<+J) 1 H1 EN;Y = FNCY + POHFD . 41 IPIT.GT.TINFL1 GO TP 1139 IFIT.LE.T§TEF) :3 TC 11:.0 "m H rsrgP = TPTSP y T§KTP 1139 PPINT 1100.T.v(1,P3),NCT.H1 1100 FOIMATI‘iTIME=‘,F11.S,' src,¥,10x,'TENPERATUFE=P.F11.5,P K,‘, H1 1)X,'DIFSUq CRLLFF‘,IE,' TIMES,‘.10X,‘STEP SIZF='.F11.5.' SEC‘) PPINT 1010.000FP,FNCY ‘ 1010 F0!MAT(/‘ LASIN: DOWFP=‘,611.S.‘ HAYTS/SC‘,10X,‘ PULSE ENERGY=', 1”_ U- F11.5,' JOULFSICC'I PPINT 1010 ‘ 1510 FOQHAT1/Il‘ CONCFNTPATIONP IN HULFS/CC‘) Lm_wme”°[NT”11K1110-,u_, 1151 razNATI/ 11X,‘H‘,15Y,'F‘.lhx,'FR’,12X,‘H2(O)‘,1?X,‘H2(1)'y12X, 1 ‘H7(’)‘) PPINT 110?,VI1,17I,Y(1.?1),YI1,72),YI1.10),YI1,191,Y(1.20) 11b? F0!NATIGF16.5) PPINT 1103 _;}93_rnzNAT(//7x,4ur¢0)0,10x,'HF(1)P, 10x,¥NF12)P.10x,PHF(3)’.10X, 1 ‘HF(h)‘.10X,‘HF(‘)‘,111,'HF(6)'glOX,‘HF(7)‘gioxg'HF(8)‘) PqtNT 111-0‘0, ‘V1191’0 I = 199) .1190 F02N0710516.5) PPINT 1570 1020 P01N0T(IIIP PHOTON FL”! or THE LASINS LINES, IN HOLFS/SEC/CH-SOUAR 1E0‘) I PQENT 1105 1110'; FGQNQT1/ ‘0‘,’V.,AY9‘JHGY‘,QX,‘:(V’JHAX'3).,SXQ‘F(V9JHAX‘2"95XQ 1 'F1V, “MY-1)'.SY.'V(V,J‘MX)‘,6X,‘F(V,JHAXOUH‘SM 1 "r1V1JMAXf7"Q‘SXg'F-(VyJNQX*3") DD 11‘07 I = 195 IV = I -1 L : 7 ‘ I 0 16 P3: NT 11Q6yIV9JMnX111,(Y‘19L1J)9 J = 197) 11.06 F02"AT(I§,IIU,7F1‘HC) ”1.157 CO‘TINUE PRINT 1530 1530 FOIHQT(’//‘ PDHFQ 01’ T41— LASINS TRANSITIONS, IN WATTS/CC") 00 15100» I31,“ IV=I-1 K=IMQY111 ' .0 pr: NT. 1155,1V,J-MAV(I,0101INE(I’K*J),J=1,7) 15H! COITINUE pR[ NT 1“" 1550 F0: HAIL'ICONTDIBUTIDN TO pULSE ENEPSY FRO” FACH TRANSITION IN JOUL 1FF'CC') 9°! NT 1560 1660 F0!MAY(/1X,'J‘,0¥,'“F10,J)‘yqx,'HF(1,J)‘,9X,‘HF(2,J)‘,QX,'HF(3,J)’ 1,91,.HF1109J’ '0979'Hc159J,',9X,‘HF(6,J)',9X,‘HF(7,J," DO 157" J=1015 PDIQT1.210,J,(FLINEII.J)91:1,8) 99 Table F. 5. (Continued) 1570 .1539 1200 1210 1220 11%? 1310 1150 .1160 -1165- 1170 1190 CCJTINUE P?[NT 1900 F"21‘101‘(///‘ THF 351” F09 ALL POSSIBLE TQANSITIONS, TN 1/CP‘) POINT 1’00 F0!MAT(//1Y,'J‘,6Y,‘ALDHA(0,J)'.8X,‘ALP4A(1,J)‘,6X,’ALPHI(ZyJ)'g 610'5LOHA(‘9J1'9579'ALPHQ(“,J)‘,6XL‘ALPHA(5,J)‘,6X,‘ALDHA(6 1,; p,6x,'nLPHA(7,JD‘) on 12?0 J=1915 D°[ NT 1,10,J’(ALPHA(TQJ, 9I31,8) FOIHQT(TQ,BV15.§) COJTINUE IFIY.E0.0.) GO TO 11‘0 IFIT.GT.19".F-6.HHD. FOHFF.LT.1.E’1) GO TO 1170 JSTAQT = 1 IF1TQLFQYTMFL) 50 Tn 10°C 50 TO 1170 Ir1HHTN .GF. ALT“, CO TO 113“ PPINT 1160,V¢LAG,T F01HAT(1X,‘STFP UNSUCCFSFFUL‘IIIS,E16.S) CA-L DIFFUN(T.Y,DFPv1Y) PPINT 1166,9F9V1Y F?}HOT(/1¥,SE16.5) 50 T0 1190 PQENT 1100 CDITINUE FN) 100 Table F. 6. Subroutine GAINC '36 g GAINC FUICTION BAINC1IV,J.N,Y) c c 7 THIS FUNCTION CALCULATES THE GAIN or A P- BRANCH TPANSITION HITH c 10150 LEVPL HPIv,JI c T4IS SUWPOUTINF Tn Pr USFP HITH PROGPAP 003E-c. 9 OIIENSION 910.70) c04H0N YHAxI79).SAvrI12,7P) .EPPOPI79),PH160001.EI10.311, HC(9, 1a), 1 .1 819111119019 61191.:(‘30'30’9KFR11Q?),K3Q(1I.2)’ALpHA(;’ 15,: 1 JHA!(0).CVF117).CVF?(17).EHPYH117),E4PYH?II7), FHDYF117), 1 EHDYF71171, FROYHF117)’THF’AINIOLycLUX,CVN?(17),AQ,HF, N2, 1 RPTPIIPP),PPTPI1«?1,PPI107).SIGNL(%).PLINE10.15).FLINEIB.151 RF|L NUP, NON, N2, LONTZ TF1 = Y(1,?!) __ ”an inn/100. f .5 IT = IFIXIAIT) 01 = 0. 0705230700 ‘ n2 2 0_ 0022020123 "A3 = 0. 009270627? Ah = 0. 0001820103 --- 105 P 0. 0002750677 E“ = 0. 0000030638 H4- = 20. 01 00PP ; 3.90113386-7 PNCIIV+1,J) P 50PTIIEH/04P1 H711=0. “r1230. I DO 10 1:105 HT11=HT“1 + Y(1gI1 10 HT12 3 HT"? 1 Y(1,!116) .HTQZ = HTH? + Hr + H? 1 AP H71 3 1 75 ' (HTV1 * Y(1g7) + Y(1, 8) P Y(109)) 1 .859 ‘ H'HZ LOITZ = 87.097 ‘ SO°T(TFN) ' (HTM P Y(1,IV P 1) ‘ F1IT9J, ‘ (1 1 -0 292./30°TITFH1) P 1.78)) .Yk:NEMEMQg§3Z§5§61 ‘ LQNT7 I 03°” IFIVLINP.0T.I?.01) 00 T0 20 PHI = (0.05071000100PP) P FXPIYLINEPVLINE) __”_VWE1LH;_PH1M3”11. P A1 P-YLINE P 02 P YLINEPPz + A3 P YLINE"3 + 00 IP ILINEPPu P As P VLINFPPP + A6 P YLINPPP61PPI-16) so To 30 _____2_0 $9051.13 -2. a! .YLINE‘EF?‘ - PHI = (1. - anSTPPI-1I P 3. P CONSTPPI-Z) - 15. P CONSTPPI-3) 11 105. P CONSTPPI-AITII3.1015976535 P LONTZ) __110_Nu!-%21111111123' (2. P FLOAT(J)-1.)/Q(IVP2,ITP1))‘CXP(-1.b387856 1 PEIIVP2.J)/TEP) N01 = IYII,IVP1)P (2 P FLOATIJIP1.)IOIIV+1.IT+1))PEXPI-1.u307006 ~.1. 1. PFIIVPIJJPIII’E“) GAINC = 3.17530ueP-11 P Hc11v+1.J) P PHI P SIIVP1,J) P ((12. P 1 PLOATIJ) +1.1712. P FLOAT(J) - 1. )1 P NUP - NON) ---EEEQSN,I.HMH EN) 101 Table F.7. Subroutine DIFFUNC ’JECK 6.- OI’FUNC ,§U!32QII?§.”IFFUN172V20F‘V1V) 0 THIS 91900UTINF T0 “F U°FU HITH HOUFLC 011éN§10N 770,79).0PPv1vI79),VVHP101,erPIa),VTH2(0),SEHIa) CJIHON VHAxI7o),SAvrI12.7PI.ERPOPI79),PHI$000),EI10.311.Hc19,101, “_m__1-_.W-IB!911011019161).Ft‘fl.30!;KFR(102).KPRILPZ).AL°HA(0115)1 1 JNAXIa).CVFI17),cvr?I17).FHPvH117),E4PYH2117),FHPYPI17), 1 gHovr9117).PHPVHFI17).THGAIN,PL.PLUP.CVNPI17I,AP,HE,N2, 1 PCTr1192),PCT01197),PFI142),SIGNL18),PLINEI8,15),FLINEI$,151 10 SW C THC r QCTII .q I n THE '0 n a; 60 -QQ,ZQ 1,: 1 PPIL N1,H?,N3.H«,H6,H6,H7,H0,H0.H10.KFP,K8P,N2 H5 3 Y(191) .01, 10 L, 3-? 19 NS = NS . €11.11 "h = Y(1.171 "9,=,Y(ljlfl) . Y(1,1Q) 1 Y(1,2]) PSIH = Y(1,?11 P Y(1.2?) HEIRP = HF . 0Q . rcuH "1.: H5 P 2.0 P FSUM P HE P 0R P H9 + Nb H2 = M5 P HFAPF P 2.5 P "0 P 20. P «a P N? H3 = #8 P HFAPP P HP P “h P N2 H6 = Y(1.21) N7 = Y(1.22) P 09 l! : HE _.P1) = HEARE P h. P “9 P_h. P Mu P N? P"04.001010 APF THF r‘0L0 PUPPING PEACTTONS a V11g18) ' Y(1,?1) ‘ KFQ110PI) QCFF11OPI) H "h 0019(10P1) 711,171 P Y(1,1PI) P K9PI10P1) FO.LOHING APE 14F HOT PUPPING PFACTIONS , DO 30 I P 1 9 PCIPI16P11' 711.17) P v11,22) P KFQ116PI) u “*0 RCIBI16PI) V119711 . V1191) ‘ K3R116PI’ F0.LOHING “OF THE V.T UFDCTIVATION REACTIONS 03 “Q 1-: 13" Rch125+11 = “5 ‘ Y1191111 ' KFQ125‘I1 QCr8125+T1 = N5 ' Y(1,!) ‘ K89125011 {2f5133311.? 95 ‘ V1111P1)_f KF°133111 QCr3133PI1 = “5 ' Y(1,!) ‘ K89133111 QCf‘1h1‘I) = ”6 ' Y119191) ‘ K:Q(Q1PI) _8Q[3(§1Pjgw: us P v11.11 P KsPIu1o1) PCFF1‘9OI) = M7 ‘ Y(19101) ‘ KFQ109 PI) RC181Q9111 = “7 ' Y(1,!) ‘ KBR1QQ§I1 “_£01F133111_P Hg P_Y(1.IP1) P KPPIP3PI) PCr919‘PI1 = "3 ‘ Y(1,!) ’ “89193111 pch1101P1) = N9 P Y(1.IP1) P K’R(101PI) PCIPIIpIPII = 09 P Y(1,!) P K391101PI) P0.L0H1NG APP HULTI-OUANTA v-T OEACTIVATIONS ‘PCIPIP7PYT'£"00 P Y(1,P) P KFPIP7P11 Pnr9157P1) = H0 P v11.0-1) P K3?(57PI) 00 50 1 = 1.7 102 Table F.7. (Continued) Nu P Y(Ioa’ P KFQ(65PI) nu P Y(1,n-I) P K3R(69PI) PCfF(6$PI) so 9078(65PI) on 60 I. ? 1.6 QCFF(720I) so RCF8(72PI) on 70-1 = 1 Rnrr(7n+r) ro QCYB(7API) on an I = 1.4 «a P Y(1,?) P KF°(72PI) Mu P Y(1.7-I) P K3Q(72PI) S «a P Y(1.6) P Kr9(raPI) Mu . y(1,fi—I) P KQQ(78PI) , RCTF(83PI) 81‘ QCFBUVSPI) 3” 90 I = 1 P Y(1,5) P KF9(%3PI) Mu P v¢1,<-I) P KqQ(83+I) II II I a ' RCFF(%7PI) = M“ P Y(1,“) P KFR1R7PI) 9" QC'B(87PI) = ”h P Y(1,h-I! P K9Q187PI) DO 100 I = 1” QCfF190PI) "h P Y(193) P KFR(QOPI) 100 QC'9(QOPI) Mk P Y(193'I, P K3?(90PI) QCFF(9‘) = ”h P Y(1,?)PKFC(93) QC'B(93) = “h P Y(1.1)PKPF(93) THF co.LouINr AQE Hc-Hr v-v evcunwce REACTIONS(NEGLECT 117,17u,13J) 'Oi1fin DO 11" I: 197 RC.F(IOQPI) 11F 9079(1090I) 00 120 I = 116 RCrF(117PI) = Y(IPIPi, P YtigIP?) P KF?(117PI) 12o ncrgg11zP1) = Y(1,!) P Y(1,IP3) P K?R(117PI) 0" 130 1:1’5 pC'F(12“PI’ 130 QCIB1IZQPI) 00 150 I319“ RCIF(I30PI’ = YIloIPI’ P V(1,IPH) P KF9113UPI) .QC.8(139 *I. = VI1,T, . Y(I’IPQ) P (BQ(13U§I) Y(1,IP1) P Y(1,IP1) P KF21109PI) Y(1,I) P V(1,IP?) P K3R(109PI) Y(19IP11 P Y(1,IP?) P K‘Q(1ZHPI) Y(1,I’ P Y(1,I+h) P K3P(1ZQPI) TH' FO.LOHINS ARF HC-H? V'V FXCHANGE QCACTIONS "3 n £7| QCFF(13hPI) 1&0 RC'911PHPI’ 90-150 1:1,? PC'FCIXBPT) RCYQI138PI) Y(qu) P Y(1,19) P KFQ(13QPI) Y(1gIP1) P Y(lylfli P (8R(13HPI) Y(qu) P Y(1,70) P KFQ(138PID Y(1,IP1) P Vt1,1qr P <9Q(13a+1) Pu: FOILOHING an; v-1 DFACTIVATIONS F02 n2(vr 0:1Nl ?3'F(1§OPI) ‘10 P Y(1,18PI) P KF?(1QO *1) 150 Qfir9(1hflPI) M10 P Y(1917PI) P KBQ(1h0 PI) 90 160 I = 11.1%? 169 RFII) = °CTV(I) - QPTfi(T) VTHFIIP‘IS THF V-T “FACIIVflTION RATc 0‘: HF FQOH HF(I) T0 HF(I‘l) '00-}, _,__"WDQ_1§§.1312§, 165 VTPFII) = RF(?€PI) P °F(31PI) P RF(u1PI) P RF(h9PI) P RF(93PI) P 1 R‘(101PI) £L__.m._-.wm*¢p.um_ “v-1- .. H .- H C VTHZIIPIIS THF V-T DEICTIVATION RATE OF H? FR01 42(1) T0 H21I-1) C “h-.. . wytigt1) = RFJJ§§)_P RF(13E) + art1zr) P RF(136) o RF11h1) Table F.7. (Continued) VT12(2) C 103 = R‘1139’ P 9‘11h0) P ?:(192) E_Ifl§1991913927535§05g§yfn C co_o = 9:111) P 9:11?) P PFtl!) P RF(1%) P 9F(16) P RFCI6) -1 + 33') .MYFF'P!,EE§, CII Der1Y(T) 891.; RF117) P 9:118) P 9:119) PF(2u) P RF129) P RF(20) P_2F(21) P °F(22’ P RF(23) son: v-v gxcyaqg: TFRMS :09 9:-9:. ADD TO 9:11-1) 1628 9:116) P QF¢23) P vv1:11) : RF(118) P PF1125) P 2:1131) - RF(136) - 9:1139) __ _ ..!Vt£!ll,?_BEILLEIMP_FF'1?6!.P 9‘1132’ - RF§136’ ' ”‘1150’ vv1:13) = -9:1118) P 9:1120) P 9:112?) P 9:1133) - 9:1137) vv1:1u) = -o:1119) P 9:1121) - ?F(1?5) P 8:1128) P 9:113u)-9:1138) _M _ yvj:16) 3m:3:1121) P 0:1173) - 9:1122) - 9:113?) VVP:16) = -9:1120) P D:1122) - ?F(126) P 9:1129) - o:1131) vv1:17) = -=:1122) - PF1128) - 8:1133) M,__«”!M!E£§1-EI:PE¢JZ§) - °F(129) - PF11391 c c TIP: “ERIVATIVF o: 78: CONCFNTRATIJNS on: to CiEHISTRY ONLY , 0:2v1 1) = PF(17) P 9:166) P 9:172) P 2:178) P 9:183) P ::187) P 1 8:190) P cr192) P 9:193) P RF(1IG) P v18:11) P vvn:11) - _,p:gv1112)_:_::111) P 9:118) P 1:169) P 9:171) P RF177) P RF182) P 1 9:186) P 9F(89) P a:191)'-‘§:1§3Y';’21 P 8:111b)‘P 1 9:1111) — er:11) P v19:12) - vvu:11) P vvu:12) _____ DF§V1Y131,319?11?3 P 9:119) P ?F(63) P 9:170) P 9:176) P RF(81) P 1 9:186) P 9F(8R) - 2:191) - 2:192) P 9:1110) - 1 2. P RF(111) P RF1112) - vrn:12) P VTH:13) - vvw:12) P 1 __ _NH,VVH:13) , 0:1V1V1u) = 9:113) P c:120) P RF(6?) P 9:169) P 9:175) P ::180) P 1 9:181) - 9:188) - 9:189) -9:190) P 9:1111) - n _1 2. P 9:111?) P RF1113) P v18:11) - vru:13) P vvu:1u) - 1 VVH:13) 0:1V1v16) = 9:116) P 2:171) P 2:161) P 2:168) P 9:171) P ::179) - fl - V1 “REPBP[_7 3:186) - 9:186) - 8:187) P 9:1112) - 1 2. P 8:1113) P 9:1118) P v18:16) - vru:1u) P vvu:15) - 1 vv8:1u) 0:1v1v16) = ::115) P 9:122) P 9:160) P 9:167) P RF(73) - 2:17;) - 1 9:180) - 9:181) - 9:182) - QF(83) P 9:1113) h 1 - 2. P ::111u) P 9:1115) P v19:16) - V’HFCS) P VV4F16) 1 - vvu:16) ' Q‘(99) P QF(66) - RF173) - PF(7“) - ?F(77) - RF(78) P RF(11E) - 2. P 9F11153 P 991116) P VTHF(7) - VTHF(6) P VVHF(7) - 1 PF(75) — 0F176) - 1 1 VVHF(5) 0’1V1Y18) = RF12h) P RFISBD - 1 99170) - RF(71) - j VTHF(fi) - VTHF(7) DFlV1Y(9) = °F(25) - ?F(68) - 1 RF1631 - RF16B) - On 170 I = 10,16 17“ DFIV1YII) = 0. 0‘2V1Y117’ = COL” - HOT DFIV1Y115’ : VTH211, - CFLD DFQV1Y(19) = VTH2C2) - VTH211) D‘!V1Y(20) 9 -VTH2(’) OEQV1Y12 1) cDERV1YC171 2F166) - ?F(67) - QF(68) - RF(69) - RF172) P QF1115) - 2. P 9F1116) P P VVHF(B) - VVHFCT) Q‘(GQ) - RF(60) - RF161) - RF(6?) - 9:165) P {F(lifi) - VTHF(B) ' VVHF16) 104 Table F.7. (Continued) 0:1v1v122) = -901 c 0 EFFECT OF 9ADIATIQN 9N HF(V) PONCENTRATION c sr;NL1I) 19 TornL 901: 0: EMISSION :901 9:11) To 9:11-11 r 1 03 1ROI_=1,H L = 7 P I P 16 K = 100x11) - L 180 $11; 8L1!) ; P v11,LP1) P ALDHA1I,KP2) P v11.LP2) P 1 819801!,wP3) P v11.LP3) P nL9801I,KPu) P v11,LP6) P 1 ALPHA(IPKPSI P v11.LP5) P AL:8A1I.KP6) ‘ Y(IPLPfi’ P 1 ALPHA(I,KP7) P v11,LP7) 0:1v1v11) 0:9v1v11) P FIGNL(1) OEQV1Y19, = 0F9V1Y19) P FIGNL13) 190 DF1V1Y111 = nFQVlVCI) P SIGNL111 P SIGVL11P1) CALCULPTE OUTPUT "OHF9 CO'C? $1 HC‘AVAGIDRO 80. = 2.05912 CAL-99/80L: 0015? : 2.05912 P THGGI” PL 3 0. DO 200 I = 1’3 K 3 JHQX111 ' h L 3 7 ' I P16 03 200 J = 197 20!! PL 3 PL 0 Y(19LPJ, . “C(I,K*J’ PL 3 PL ‘ CONST C C TE”PEQ\TURE DFRIVATIVF C I17 2 Y(19231/100. P .5 IT 3 IFIX(AIT) CISUH = 2.981 ‘ (HF P AR P Y(1117J) P CVN2(IT) :_fl§ P 1 5. 076 P 1v11,10) P 111,19) P 111,20)) P 5. 020 P 1v11. 1) Pw 1 v11,2) P v11,3) P v11,9) P 111,5) P Y(1,6) P 111, 7) P _~___1_____ v11l0) P v11,9)) P CVF(IT) P 111.21) P cv:21IT) P v11, 22) , ET! 9v = ((EHDYHllT) P 52.102) - (EHPYF(IT) P 10. 900)) P059v1v117) 1 P1supv:21IT)) P 069v1v122) P 1682182111)) P 0:9v1v110) __,__1 _ -_ P16Hpvn21IT)_ P 11.009) P 0:9v1v119) _,~.1_1 1 P(EHPYH?(IT) P 23.112) P 069v1v120) 1 P1EHovu:1IT) - 6a. 000) P 059v1v11) 1 P1689V8:1IT) - 93-923) P 0:9v1v12) 1P15H9vu:111) - 82.630) P DERV1Y(3) P (28:14:11?) - 32.202) P “‘ 1061v1v19) P 1:8:v8:111) - 22.392) P DERV1Y(5) P (EHPYHFCIT) - 1 12.959) P 9:8v1v16) P (FHPYHF_(IT) - 3.973) P 069v1v17) P 1 1: HDYHF(IT) P 6.572) P 0:9v1v10)P1:9:vu:111)P12.680)P0£9v1v19) ETPLov = :19L9v P 1. EP3 0:1v1v123) = -(°L P ETHLFY) / cvsun 00 210 I = 1.8 L P 7 P I P 16 K : JHAX(I) - P 03 210 J = 1,7 210 0:1v1v1LPJ) = (ALPHA(IPKPJ)-THGAIN) P 2.997925EP10 P v11,LPJ) 30 220 I 3 ?“,79 _ ___‘__ 220 I:1~11.I).L:.:Lux.Ano.059v1v1I).LT.0.) 0:9v1v11) = 0. RErURN EN) 105 Table F.8. Program MO DELG . DROGPAH HO"FL5(INPUT9 OUTPUT 9TAPE19TAPEZ9TADE39 TAPEQ9 TAPE79 TAPE89 1 TAPE99TA°F109 TAPE119TADE129TADE13’ Coo-9.9.99.9..“OCEL3 USES THE SAAE QATES AND WAT°IX ELFHENTS AS ”ODELC 919535100 Y(81135),DEQV1Y(125),XXX(39),YHAY(1?S),SAVECIS,125),ERRJ ' P112519PH116000’ COVHON €110,311, 81991819019161)9‘150930)9KF311QZ)9KBR113219ALPHA(391519 ELVL1719CVF117)9CVF2117)9EHPYH11719FHPYHZC17)9EHPYF(17)9 FHDYp7117,9EHPYHF(17)9THGAIN9PL9FLUX9CVN2117’9AR9HE9NZ9 9CTF11R219ECTB(1A2)9Qf(1§2}9SIGNL(h)PPLINE13915)9HC(9918)9 FLINF1391‘)9TAU(Q916)980LTZ(h916)9HFVJ1Q916)9SF69 CV$F6117)9CVH211719CVHF(17)9FLUXJ13915)9HFLAGC3915)9 VELAG13915)9NVAQ9NPVAR1FPRQOA9SL0p1391919TRCEP(3919) pFAL KcRyK°P9 N? PFA0119 59)F 3.21012, 10mg pEAG1391018 QEADCB910)” 3.51911 0 11 o 1: PFAD189§01CVC9CVF79CVV79CVSF69CVH29CVHF READ199201chYH9 EHPYH’, EHPYF9EHPYF2, EHPYHF _ - 9:00111290)5L09 190:: 10 FORHAT13516. 5) 20 FORMAT15F16. 5/9F1695/5F1695/2=16.59BGX) H_§QIQE§BIL-533- i”, 0. Q0 p0°MAT1F1‘1. ‘1 50 FOPHAT15F16. 51 ______ 60 FOPHAT1551§:S) _ .1 C PPPPP DAPTIAL PRFSSUPES (TORR) RF?=0. -FHPQL RH?- QHF 3&0. 9N2_ = 00_ EAR : 00 KEY: 78 __gs_z_6=~. , ‘CTR=.53h9c-7 PLUx310E'1“ 11.19. ates-6 H1=Oo DELHTN = 10E-15 PUP 9-6059131314100- pronr-nPa—I H = 0'. E‘Ip HHTN DFLVIN ”MAX 20F-7 rPS=1oFP1 HAXDFP = 8 RL:0‘ R0399; LNTH=1690 TSKIP=5.F-9 TIHFL=390FP6 TSKIpaolF-6 TSTEP 2 TSKIP _NRVAR : P0 NVAP = NPVAR NOT 8 O KKK’Q 106 Table F.8. (Continued) POWER = 0. ENGY 8 0. 9L 8 n. T 3 0. JSVAPT = 0 HF = Q FLVL‘I1 3 no 00 990 J 3 1,1; FLUXJ‘T9J1 3 rLUX M1'-l.A';(T9J1 3 0 NFLA5119J1 2 0 pLTNFCI9J1 8 09 FLINF(I9J1 8 09 990 ALDHA119J1 = 09 no 1000 I 3 191?? 1000 VH0!!!) 8 195.0 DO 1003 181116? fiCTF111 3 00 9678(1) = 0. 100: RFC!) s 0. HE 3 our * FCY? N? = PM? ‘ FCTP AP : FAD ‘ PCT? SFS 8 95:6 . FCTD THCAIN = -ALOG(°0 ’ RL)/(2. ‘ LNTi) 00 1010 121,125 1010. ”.1211 ? 0- v11,73) = 002 2 FCTR V(1,7u) = 0. 1.11.115) A: '3- 7. v11,751 = 951‘ FCTR v11,77) = 2F2 ' FCTR _7 _;!(1,70) = 300. AK? = (961.701 - 100.1/25. + 1.5 KT=IFIX(AKT) 7__ __,. JLQAEZ- -131,» KT _. 9610112.101KFQ 1022 READ€79301 KBR AIKT = (v11,7n) - 100.1125. + 1.5 11-2.1157. =21 If DU 415'! 1030 IF!KT.EO.IKTI so To 1050 KTOIF = IKT - KT __". I‘lfiIfllfigfiTsfll EQ ID 1039 90101 1033,v11,7a1 1033 roqnar1/lv TEHPFRATUQE HAS ozappeo new TEMPERATURE 1s¢,r10.5//) “___C_NIIQBQP?ITQPOPQL GO To 1050 1035 KT = 1x? 9.00 1050ugiz‘1lgtpr PEAD112,?0)KFD 1000 951017.30) «a? _toSLegLaxuz761.4100- + 1-5 IBT=IFIX(ABT) 00 1051 1:1,0 . _ 00 1051 J=191§”._M_ _~—-._ - 107 Table F.8. (Continued) 00LT7(I.J)=(12.'FLOAT(J)-1.)I01I,IBT))‘EXP1-1.h387686' 1 E(I,J)/Y(1,78)) _1051”553311,J1=v11.11 1 001121I,J1 TAUA=2. 5-0/v11,7n1i'.5 x1=1.E- -3 2920-2105: 21:12.23." 00 1051 J = 1.16 1050 71011.»: ”1111'1 FXP(1. 9387086‘XI‘18119J+1)~E1I9J111Y1197811 IF1T. F0. 0.1 so To 1119 LFLAG=0 C..........SINGLF LINE 0N P1293) 2291 911.11.12.31. =G_9_.I.NG1 1. 3 . 1251 v1 00 1100 I=193 IV = I - 1 _ _ 00_1,100-2-=-.-1.1- _ . , . . ,, _ ._ C 1Y95219J9VI1GNIAG = )J9I1AH3 C.........CHECK 90° LINES NEAR 0R ABOVE THRESHOLD 31112191111112.1125 .1 1 19011111 1050,1000221000 1060 IF1FLUXJ1I9J1 - 1. E- 5) 10709108091080 1070 ~91001I,J1 = 0 2FL91411141 = ‘LUX PLINF(I,J) = 0. GO TO 1090 _L_11.NFLAG111 1 = 1 _ - 2-. . ,. 2 .- . . .._2 1090 IF1NFLAG1I9J1 .FO. HFLAG1I9J11 50 TO 1100 LFLAG = LFLAG + 1 211109 QQEIIBU522.H. _ --. . , , -. - -_ C.........TF THE SET OF VARIABLES HAS NOT CHANGEO9PROCEED TO INTEGRATION C.ooo.oo..IF THE SFT HAS CHANGED, DEFINE NEH VARIABLES AND REINITIALIZE. IELLFLB§.;LE: 01 GQ I3-1130 JSTAQT = 0 NVAP: NOVA? 2_,9021110 1:1,3 00 1110 J=1915 IF1NVLAG1I9 J1 .LE. 01 GO TO 1110 NVAP _=_Nvg9 1 1 Y119NVAR1 = FLUXJ1I9 J1 1110 CONTINUE C Co...o.oo.INIEGPAT€ C H1130 CALL DIFSUB1NVA°9T,Y9SAVE,H,HHIN9HHAX9EPS9HF9YHAX9ERROR9KFLAG9 1 JSTAPT9NAXDEQ9PH9KEY1 NCT = NOT 9 1 _ _ GO 70 1136 _ 1130 H = HHIN HHIN = HNIN/20. 21522725". = 2‘1 _ .. 1 - . . . . .-. . ..__..- - GO_ TO 11_30 -1 __ 1136 CONTINUE IF1KFLAG. LP. 01 GO TO 1150 _DO 1&10_I:Zfl9NVAP IF1Y119I1. LT.FLUX1 Y119 I1'FLUK 1610 CONTINUE ,_>__ .-. 1H1_=_T_-1_-__T_QLQ2 CONST= ?. 8591? ‘ THGAIN POHF° =0. .EXAE2EWQBVAB-_ 108 Table F. 8. (Continued) 00 1120 1:113 00 1120_g=1,15 HFLAG(I,J) = NFLAG1I J1 I91N=LAG1I.J1 .LE. 01 60 TO 1120 w__ nvn9 = NVA9 1 1 2-2222- 1_ . 1_ FLUXJ1I,J1 = v11, NVA91 9LINF1I.J1 = ‘LUXJ1I, J1 1 uc1I,J1 1 CONST 1 5.105 90959 = 90959 .L 9LINE1I1J1 212222 -222_222_22.2 ELINE‘I’J) 3 FLINEIIQJ, O PLINEII’J—L‘. ' H1 ELVLII1 = ELVL1I1 1 ELINE1I,J1 1120 CONTINUE 2212-12---212222-22 FNGY = FNGY 1 POWER 1 91 IF1T.GI.IINEL1 GO TO 1139 I917.LE.ISIE°1 00 TO 1199 15159 = TSTEP 1 TSKI9 1139 99INI 1190.1,v11,7a1,9c1,01 1190 9099111111IHF=1uf11. 5.1 SEC,1,10x,1IENPEparu9E=1,E11.5.1 K.1 1 10x.1oIFsus CALLED1,I5.1 TIMES,‘ .10x,1STE9‘SIZE=1.EI1.5,1‘§Ec11 99INT 1510, 90999, ENGY 1510 909NAI1/1 LASING 909§9=1,E11.5,1 NAITSIcc1,10x,1 PULSE ENERGVs1, 1 511.5,1 JOULESICC') 99INT 1510 1510 FO9HAI11/1 CQNQfNIRATIONS_IN N0LE§/cc11 99INI 1191 ‘"" 1151 909NAI1/ 11x,191,15x.151,1ux,1521,12x.1921011.12x,1H21111,12x, 1 1_9212111 __ 1 PPINT 110?.Ytie7211Y1116), Y 1,771,Y11,731,Y11,7u1’?(“77’T"“"’““ 1152 90990115915. 51 222 99INI 1195 ._.2_H_2_12_ _ _ 1153 909NAI1/qu,1J1,7x,1H=1011{idi;1951111i’101:1fi57211?10x,1951311, 1 10x,1NF1511,10x,1NF1511,10x,1NF15111 PRINT 2001, (V‘lzllzlit1lL2 _- . ...-——-. --... _—..__—___.V-__. 2 V.._.._.___. _. _....._ -._-- -_.. .1- 9..__- _erOOI FORMAT15X11§1§L5)1 no 2010 K=1915 J=K"1 _ _PPINT211991J1Y11171K),Y(1_,231<1_,Y(1,391K),V(1,551K1 11k“ FORHOT1I5,NE15. 5) 2010 CONTINUE PRINT 15?0_ n” 1520 FORHOT1/l/7X9'pONER 0F LASING LINES 1H5TTS’CC"910N,‘ENERGT OF ER: 1H LINE (JOULFS/CCI'91119550IN 3F ENCH POSSIBLE IP‘NSIIION 11/0N0.1 _QEIEI_ Lilfl. WI530 FORMAT1/3X,‘J‘95X9‘H5109J"97Xp’HF11’J)'g7X,‘HF(2,J)'97X9 1 .HF109J1‘9 7X9'H5119J’5,7X9'4F129J"95X,'NLPHA(09J0‘9BX, _m_mu_i___ “m .ALPW0113J)‘,NX,‘ALPHA(Z J)'/)_ 00 1500 J31915 PRINT 15509J91pLIN6119J’9I=1’3’QIELINEII’J),I=193)’ (ALPHA(IJJ,!I=1D3, -. -——-——--—— 1560 CONTINUE 1550 FORMAT1I§g9F16.51 _ . pRINT 3000 - 3000 EORHAT1/II‘ ROTATIONAL RELAXATION TIHF13E01’1 FRINT ‘010 _3010 FORMAT(/3X,'J’,F¥,'HF(0,J)‘,7X,‘H‘11,J)‘ 7X9'NF129J)’, 'X9'HF139J1’ 1’) 00 3020 J81916 -1--2_2JJ:J:LL.2-2_ 2 22 109 Table F.8. (Continued) oaINT 3010,JJ,(TAU(I.J)1181,BD 3020 CONTINUF 39}Q,EQRH!T!I§:hFlb.F! KKK=KKK¢1 00 1’30 J=1y15 -- ______.-.-_ .HLQL __-. 1 3.91.1?! 1.‘ £21.43. _ XXX(150J) spLINF(?.JI 1230 CONTINUE up- .wlllillL. %!(12783 XXX (32! =°OHFP xxx133) = Y(1,?6) . XXXSJQLPE In” HPITE(13,E")XXX PRINT 10.xvx .IFLI-EDQQ-!_GQ *0 1130 TF(T.GT.1.5-6.AND.POHEQ.LT.5.E-3)30 To 1170 1169 JSTART a 1 .13'Q_LELI;LfisTIfi§L) 69-10 1025 mu_ .-_ 99 ID Lllfl _..- 1 -11 . , _ 1160 IFCHHIN .cr. ALIP) GO to 1130 PRINT 1160,KFLAG,T _w.1150n5935511111f5168“UNSyQCESSEgLf//151E16,51 CALL DIFFUNCT,V,DERV1Y) PQINT 1169,0E9V1Y “116515Q35511L111561619!- so TO 1100 1170 °RINT 1130 1130 ‘09"‘T(/’/1XJEQEE§8 LI”II OE INI§§RATIONHPF‘CEEQI)_._"_,-_ —_. —.-, PRINT IIRSoKKK 1135 FOQHA7(II‘ KKK='.IS) 1190 CONTINQ§_J.._ ___‘Mw_1 “p _1__-m-. IF(ITD?0P.GT.0)P°INT 1033,v11,7av FND OKjOOC) 110 Table F. 9. Subroutine GAING 1H FUNCTION 001~5(IV,J,N,Y) THIS FUNCTION RALCULATES THE GAIN OF A p-BQANCH TRANSITION ”It” LOHFO LFVEL He‘IV9J) THI§ SURROUTINF TO BE USED HIT4 9RDGRAH MODELC. aranha-fipnn DIMENSION V(R,1?S) COMMON F110,31). R(Q,18),O(Q,61).‘(SO,30),KF?(1&2),KBR(IAZ),ALPHA(3,1S), ELVL(3),CVF(17),CVF2(17),EH’YH(1?)1FH°YH2(17),EH?YF(LZQ, FH9V92117),EHPYHF(17),THGAIN,PL,FLUX,CVN2(17D,AR,HE,N2, DCTFtih2),DCT8(1h2),?F(1h2),SIGNL(Q),PLINE(3,15),HC(9,18), FLINF13,19),TAU(N,16),ROLTZ(Q,16),HVVJ(N.16),SF6, CV°C6(17),FVH2(17),CVHF(17),FLUXJ(3,15),PFLAG(3,15), NFLAC(?,1‘),NVA?,NRVQR,¢PPODA,SLOP(3,19),TRCEPC3919) DFQL NUP,NPN,N7,LONTZ Yén = V(1,78) AI? TEH/tofl. 1 .9 IT = IFIX(HIT) A! = 0.0709210730 A? = 0.0h2’879123 ng = 0,009,709??? Ah = 0.00015201h3 AS = 0.000?78567? 06 = 0.0000h30638 N95”: 20;01 0099 = 3.581133RF-7 ‘HC(IV+1,J) ‘ SOPT(TFMIHHF) HT“!=U. "7.47:0. 0n 1" T=195 -“I”1=H’“1 * Y!111) 1 1 HTWZ = HTM9 1 Y(1,I171) WT”? = HTH? 1 JP 1 N2 + A? 1 Sta th = 1.79 1 (NTN1 1 v11,711 1 .065 1 HTP? f LONT? = 02.067 1 SQGT(TEH) 1 (urn + v11,1v 1 1) 1 F(IT,J) 1 (1 ?%?.Isnpttrsno) - 1.70)) yggwshz 0.13155051 1 LONTZ / 0099 IC¢VLINF.GT.(?.u)) so 10 20 pH! = (0.06071160100091 1 Fxp1vLINE1vLINE) PHI = our 1 11. 1 A1 1 YLINE + 02 1 YLINF112 1 03 1 VLIN£113 1 an 1 YLTNF“h 1 as 1 YLINE115 1 as 1 VLINE1161111-16) ' no TO 30 20 cgNST = 2, 1 11115112 1 PH? = (1. - CONST“(-1’ O 3. ‘ COVST"(-2) - 15. ' CJNST"(-3) 0 10;. ‘ C"NST"(-h))/(3.1h1q926535 ‘ LONT?) 10 IMH=IV‘1G l 23 O J 1 INL = Iv11s 1 a 1 J cnrNc = 1.17936065-11 1 uc11v+1,J) 1 9H1 1 011v+1,J1 1 1112. 1 1100710) +1.1/12. 1 1101710) - 1.)) 1 Y(1,IHU) - v11.IHL)» _PFTUDN FNW 111 Table F. 10. Subroutine DIFFUNC: §O8ROUTINF DIFFUNtT,Y1OERV1Y) C C THY? SUBSOUTINC TO PF USE? HITH MODELS. C OIHFNSION Y(8,1?5),DERV1Y(125),VVHF(6),VTHF(6),VTHZCZ) COMMON E(10,31)1 3(9115‘10(9951)17(59230’3KFR§1%ZLJK“°(1“2’9‘L91513115Qt-_J FLVL(3),RVF(17),CVFZCI7),EHPYH(17),FVPYH2(17),EHPYF(17), FHDY‘?(17),EHPY4¢(17),THSAiflgpL,FLUX,CVN2(17),ARgHfigNZ, PCTF(1QZ),FCTB(1h?)11F(Lh2113LGNLflhlgpLINEC3112),"C(9115), ELINE(?,1S),TAU(N,16),BOLTZ(Q,16),HFVJ(Q,16),SF6, CVSFS(17),CVH2(17)1CVHF(17),FLUXJ(3,1S),HFLAG(3,15)9 “NFL95(3919’9NVA?)NRVARyFPRODA2§LQBS3219’1TPCEB£31191 PERL N1.N?.M‘,Hh,H5,N6,H7,48,H9,H101KFRyKBP,N? "‘5 1' Y(lol, 1 DD 10 I = 227 10 "5 “.r) O Y(IyI, "h Y(1,7P) .1fl1_,Y(1217) f Y(11791 t V51175) FSUM = Y(1176) 1 Y(1177) HFAOF = HF 1 a: 1 ‘SUN 1 SF6 .y; ANS 1 7.1 1 Fsun 1 HE 1 AR 1 H9 *_"§,* 515 N2 HS 1 "FAQ? 1 2.5 ’ N9 1 20. ‘ H9 1 N2 H3 H5 1 HEAQF 1 M9 1 H9 1 N2 _,§5 -X(!1751 1 ._-~_.H.. H7 Y(1,77) 1 09 1 sca H8 HF .319 équAP‘ * “a ’ "9 * “1 '.19 *1“? upra~1»r-»p I unnu "IL" 6“ C THF FOLLOWING ARE THF COLD PUHPING REACTIONS C 00 20 imé”1 A RCTF(101I) .B£I§(lfl+llw -_. _~-7—_..—._-. .-.v.- WV. .71 1..” .m- M7. 7 v 7 .fi. 1 . , . - v11.73) 1 v11.76) 1 (1&11011) X'1172) f Y(1111I) ‘ KSBI1011’ )" H "‘ km ‘ I FOLLOHING AP: THE V-T OEACTIVATION REACTIONS K700 ' 4 I m 00 11"I= 1,6 RC7F(2§1I) Efitfilzs‘ll. PCTF(331T) RCTB(331I) wBEIEIEUU, 19 .55 N5 N5 15 Y(1,I11) 1 KFR(251I) v11,I),1,K39(zs1I) Y(1,I11) 1 KFQ(331I) Y(1,!) 1 «9913311) Y(1)I*1’ ‘ KFR‘QLfIl ,1 1111 a I O‘ u "1)" u x V «‘14-. c .-18918¢§111? RCTF1N91T) oc781491!) Y(1,I) ' K8Q(h11I) Y(11I11) 1 KFR(99 11) Y(1.I) 1 K8?(991I) _RCTF59311) Y(1,!11) ' K‘R(931I) RCTEt931I) Y(11I) ‘ (81(9311) RCTF(1011I) = ‘19 1 Y(11I11) ‘ KFR11011I) 150,90T9(101*I’ = HO 1 Y(1,!) ‘ K3R<1011I) 3 3 c c THE ‘OLLOHING nor «ULTI-nNANTA v-r DEACTIVATIONS c 00 60 I = 116 Rcrrc7211) = Na 1 v11,7) 1 «1117211) 60.851§!?21Yl "9 ’1Y(117-I) ’.K8R(7?1I) 00 70 I = 1 : 907117311) nu 1 Y(1,6) 1 KFQ(781I) 70wggrgy7a11). an 1 Y(1,6-I) 1 80 9n181831r)_= nu 1 Y(1,S-I) 1 K091031I) no 90 I = 191 OCTFlH71I) = Nb ’ Y(1,“, ‘ KF?(37OI) fip_RCI3fifi7fI) C “9 ' YIIQQ'I) f KBQ(87+I) DO 100 I = 19? RCTF(901I) = “h ‘ Y(1,?) ‘ KFQ(901I) IOQ 9CTR!§§}I) = flh ’ Y(193‘I) ' K88£90l1l11 ‘9c11191) = NL i’Y11,2)1KF9(93) 9073193) = N1 1 Y(1,1)1Ka9193) c é'THE'Ebtléifuf 09E «Four v-v EXCHANGE zfincrxons c '30 110 1:1,; -_ _ ‘fi ' .1__ . _ ”' ‘"’901E71091f)‘é 111,111) 1 v11,111) 1 KFR(1001I) 110 9cra11091z) = Y(1,I) 1 Y(1,I12) 1 999110911) 99 1?0 I-?-‘1“ a- . . " 'T9crr111711) = v11,111) 1 Y(1,I12) 1 KFR(1171I) 120 9019111711) = 111,1) 1 v11,r13) 1 KEP(1171I) 1-199 4?" I=11‘ 1 9nrrx12111) = Y(1.I11) 1 v11,r13) 1 KFP(12h1I) 130 9019112111) 2 Y(1,I) 1 v11,11u) 1 K0911211I) 00-1‘511111? 1 1 1. _1 1 “"'96T111301I) = v11,t11) 1 v11.11u) 1 KFQ(1301I) 139 90111130 11) = 111.1) 1 v11,115) 1 K8R(1301I) c C T”; FOLLOWING OFF WF-H? V’V EXCHANGE REOCTIONS C 00 1“" I319“ 1 RCTF113FQI! : Yl IQO pC'Pll‘h‘I, = V‘ 00 150 1:197 "QTF!133fI’ RCTP'1381I) lgI) ' Y(1,7N) ' (FP(13h1I) 1,111) ‘ Y(1973) ' K8R(13Q1I) Y(1,I) ' Y(1,?5) ‘ KFR(1381I) Y(11I11) ‘ Y(1,?h) ' K9R(13B1I) C C THE FOLLOfiZNG ROE VfT OEACYIVATIOVS FOR H2(V) C PCTF(1NO1Il M10 ’ Y(1,?31I) ‘ KFR(1QO 11) ,1151 BCTP(JHO*IX _VIO ‘ Y(117211) 1 K3R(1ho 11) "O 160 I = ligih? 160 PFlI) = RCTF(I) - RCT3(I) C 1 1 . -1 c VTHF(I) IS THE v-r DEACTIVATION 9an OF HF room HF(I) to HF(I-1) c .90 165 11116, '*”166’VTNF(I) = 9112511) 1 9113311) 1 9111111) 1 9111911) 1 9F<931I)’1' 1 99110111) C . _. . H 1 . H. c VTH211) IS THE V-T PFACTIVATION 9115 OF H2 F9on H211) to H21I-1) c VTH7l1) =.c‘(135l 1 RFJ136) 1 RF(13Z) 1 RFI138) ! RFSlhll VT9212) = 91'1139) 1 911110) 1 911112) c .CWIHF nuNPTNQ TEPHS 9UNMED 0 GOLD = RFIII) 1 PF(12) 1 RF(13) 1 QF(1Q) 1 RF(15) 1 RF(16) 'O 113 Table F. 10. (Continued) 0 vvu911) 199 90~9 v-v EXCHANGF 19913 909 09-99. 100 10 HF(I-l) 1990 0 va911) = 991110) 1 991126) 1 991131) - 991139) - 991139) vv~9121 = 991119) 1 991126) 1 991132) - 991136) - 991110) vvn911) = -991110) 1 291120) 1 991127) 1 991133) - 991137) yy9911) ; -991119) 1 991121) - 991125) 1 991120) 1 991131)-991130) vvn916) = -991121) 1 991123) - 991127) - 991132) ' vvu919) = -991120) 1 991122) - 991126) 1 991129) - 991131) 0 c 1119 0991v011v9 09 199 00~09N190110~s 009 10 099013191 0911 1‘, 999v1v11) = 99117) 1 99165) 1 99172) 1 99170) 1 99103) 1 99107) 1 1 99190) 1 99192) 1 99193) 1 991110) 1 v19911) 1 vvu911) 099v1v12) = 99110) 1 99161) 1 99171) 1 99177) 1 99102) 1 1 99106) 1 99109) 1 99191) - 99193) - 2. 1 991110) 1 1 991111) - v10911) 1 v1H912) - va911) 1 vvw912) 099v1v13) = 99119) 1 99163) 1 99170) 1 99176) 1 99101) 1 1 99109) 1 99190) - 99191) - 99192) 1 991110) - 1 1. 1 991111) 1 991112) - VTHF(2) 1 VTHF(3) - vv0912) 1 1 vv9913) 099v1v11) = 99120) 1 99162) 1 99169) 1 99179) 1 99100) 1 1 99101) - 99100) - 99109) -99190) 1 991111) - 1 2. 1 991112) 1 991113) 1 v19911) - v10913) 1 vv0911) - 1 vv9913) 099v1v15) = 99111) 1 99121) 1 99161) 1 99160) 1 99171)_1 99179) - 1 99101) - 99109) - 99106) - 99107) 1 991112) - 1 2. 1 991111) 1 991111) 1 v19919) - VTH911) 1 vvn919) - 1 vvu911) 099v1v16) = 99119) 1 99122) 1 99160) 1 99167) 1 99173) - 99179) - 1 99100) — 99101) - 99102) - 99103) 1 991113) _ 1 - 2. 1 991111) 1 991115) 1 VTHF(6) - v1H915) 1 VVHF(6) 1 - vv9919) n99v1v17) = 99116) 1 99123) 1 99199) 1 99166) - 99173) - 99171) - 1 -~H 99179) - 99176) - 99177) - 99170) 1 991111) - 1 2. 1 991115) 1 991116) - v1H916) - VVHF16) 0 ----— 9 OISSCC'ATICN = .091596 909 99900=1.E-2 ”FPEOPA=1GBGE-‘ FPQOD=FP°OQA1FLASH(T)"2 0 ----- 9N9901 19901 = 2.13JJULES/971 00 909 90:3.7093 P‘FI-99f? no 170 1:111 00 170 J=1116 1 11.1!31‘16*49° -1 . - 170 099v11119)=999v1111)19011211.J)1119VJ11,J)-v11,19))/11011.)) 00 170 1:211 KF§Q§I K=1-1 00 17h J=1916 Tflglfifi 1 1 - q . 171 099v1v11u) = 099v1v119) 1 (SLJP(K,J)'Y11178)1T9CFP(K,J))‘RF(KF) 17H 099v1v11) = 999v1v11) 1 99199) 099v1v172) 0010 099v11171) VTH?(1) - 0010 099v11171) VTH?(?) - v19211) 993v1y175) -v1H212) HFQV1Y(76) =-OEFV1Y(72)+FPROD Table F. 10. 0030 O<3CIO £300 114 (Continued) 0E9V1Y(77) = —fi. EFCECT or =AOISTTON SIGNL(I) "N HF(V’J, 9L = U. KVAp = NOV“? 00 1‘0 1:103 c:I'3NL(.T) = n. no 190 3:1,1q IF(N‘LAG(!9J) KVQP :: IVL TFU .lF. GO TO 180 KVA? § 1 T ‘ 16 + J - 8 I ' 16 0 J + 7 ALP ALOHA(I,J) ‘ Y(1.KVA2) OL = PL * Y(1,KVAR) ' WC(I’J) DFQV1Y(KVA°) = (ALPHA(I,J) - T45AIN) QF9V1Y(I”L) = “F9V1Y(IHL) + Asp OF9V1Y(I“U) = nF°V1Y(I~‘4U) - ALP SISNL(I) + ALP 0)