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SPATIAL AND ENERGETIC ASPECTS OF ELECTRON ENERGY DEPOSITION By CHARLES HERBERT JACKMAN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1978 ACKNOWLEDGEMENTS Dr. A.E.S. Green has helped the author a great deal in his efforts to complete this work. The author sincerely appreciates this guidance. He also wishes to thank Dr. R.H. Garvey and Dr. R.A. Hedinger for their helpful discussion about the dissertation. David Doda, David Killian, E. Whit Ludington, George Sherouse, and Ken Cross were instrumental in providing assistance with computer problems and other dissertationrelated work. Woody Richardson, Marjorie Niblack, and Wesley Bolch were extremely helpful in drafting the figures. The final manuscript was then typed and refined by Adele Koehler. The author is grateful to Adele for her prompt and professional assistance. The author wishes to thank Joseph Pollack for aiding in editorial matters concerning the dissertation. A thorough reading and criticism of the dissertation by the author's committee (including Dr. A.E.S. Green, Dr. L.R. Peterson, Dr. T.L. Bailey, Dr. S.T. Gottesman, and Dr. G.R. Lebo), Dr. UJ.L. Chameides, and Dr. A.G. Smith was extremely helpful. The author is especially grateful to his parents, Rev. and Mrs. H.W. Jackman, and to his sister, Kathi Jouvenat, for their encouragement and support throughout graduate school. The author gratefully acknowledges financial support from the De partment of Physics and Astronomy and the Graduate School of the Univer sity of Florida and from NASA grant number NGL10005008. TABLE OF CONTENTS ACKNOWLEDGEMENTS. . . ... .. ABSTRACT. . . ... .. CHAPTER I INTRODUCTION . .. . .. II A SHORT REVIEW OF ENERGY DEPOSITION TECHNIQUES . A. Energy Deposition Techniques . . B. Monte Carlo Energy Deposition Techniques . III ELASTIC AND INELASTIC DIFFERENTIAL AND TOTAL CROSS SECTIONS FOR N2. .................. .. A. Elastic Differential and Total Cross Sections for N2 . B. Inelastic Differential and Total Cross Sections for N2 '................... ..... C. Total Cross Section (Elastic Plus Inelastic) . IV THE MONTE CARLO METHOD OF ENERGY DEPOSITION BY ELECTRONS IN MOLECULAR NITROGEN. . . . A. Brief Discussion of the Monte Carlo Calculation. . B. Computer Programs and Machinery Used in the Monte Carlo Calculation. . . . C. Detailed Discussion of the Monte Carlo Electron Energy Degradation Technique . . 1. First Random Number, R. ............. 2. Second and Third Random Numbers, R2 and R3 ... 3. Fourth Random Number, R .. ..... 4. Fifth Random Number, R5. .............. Page ii vi 1 5 5 14 18 18 36 45 47 48 51 52 53 57 61 61 Page 5. Sixth Random Number, Rg. ............. 67 6. Multiple Elastic Scattering Distribution Used Below 30 eV. . .. ..... .. 67 7. Value of the Cutoff Energy, 2 eV ... 74 D. Statistical Error in the Monte Carlo Calculation 75 V MONTE CARLO INTENSITY PLOTS AND COMPARISON WITH EXPERIMENT 77 A. Excitation of the N B2 + State. .... .. 77 2 u B. Range of Electrons . .... .. 80 C. Previous Experimental and Theoretical Work on the 3914 A Emission of N ........ 81 D. Range Results and Longitudinal Intensity Plots from the Monte Carlo Calculation. . ... 84 E. Intensity Plots in the Radial Direction. .. 87 VI SENSITIVITY STUDY OF THE ELECTRON ENERGY DEGRADATION 98 A. Effects of Ionization Differential Cross Section on the Intensity Distributions. . .... 99 B. Influence of Inelastic Differential Cross Sections on the Intensity Distributions . .. 104 C. Comparison of Different Elastic Phase Functions on the ElectronN2 Collision Profile. . .. 104 D. Influence of Different Elastic Phase Functions on the Intensity Profiles . .. 112 E. Effects of the Total Elastic Cross Section on the Electron Energy Degradation. . ... 121 VII MONTE CARLO ENERGY LOSS PLOTS AND YIELD SPECTRA. .... .125 A. Energy Loss of Electrons in N2 ........... 125 B. Spatial Yield Spectra for Electrons Impinging on N2. 130 1. Three Variable Spatial Yield Spectra ...... 132 2. Four Variable Spatial Yield Spectra. .. 143 VIII CONCLUSIONS. . ... ..... 152 Page APPENDIX A MONTE CARLO PROGRAM ................... 155 B GETDAT PROGRAM .............. ....... .... 200 REFERENCES . . . .. 220 BIOGRAPHICAL SKETCH ....................... 227 v Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SPATIAL AND ENERGETIC ASPECTS OF ELECTRON ENERGY DEPOSITION By Charles Herbert Jackman August, 1978 Chairman: A.E.S. Green Major Department: Physics and Astronomy The spatial and energetic aspects of the electron energy degradation into molecular nitrogen gas have been studied by a Monte Carlo method. Perpendicularly monoenergetic incident electrons with energies from 0.1 through 5.0 KeV were injected into the N2 gas. This Monte Carlo de gradation scheme employed previously developed N2 cross sections with new phenomenological differential elastic and doubly differential ionization cross sections. All these agree quite well with experimental work and are consistent with the higher energy theoretical total cross section falloff with energy. Information has been generated concerning the following topics: 1) range values and 3914 A intensity profiles for the longitudinal and radial directions which can be easily compared with experimental work; 2) a sensitivity study characterizing the influence of the input cross sections on the spatial energy deposition of the electrons; 3) the rate of energy loss by the electrons as they interact with the N2 gas; and 4) spatial yield spectra for incident electron energies in the range from 0.1 to 5.0 KeV (evaluated between 2 eV and the incident energy) which are analytically characterized for future work on atmospheric prob lems dealing with incident energetic electrons. CHAPTER I INTRODUCTION Calculating the spatial and energetic aspects of the energy deposi tion of intermediate energy electrons (with incident energies from 100 to 5000 eV) in the earth's atmosphere is a difficult, yet intriguing, problem. These intermediate energy electrons (hereafter called IEEs) include the highest energy photoelectrons, a large bulk of the auroral electrons, and many secondary electrons produced by solar protons and cosmic rays. These electrons lose most of their energy through ionization events, electronic excitations, vibrational excitations, and rotational excita tions. Elastic collisions reduce the electron energy slightly, but mainly these interactions influence the direction of motion of the electron. The atmosphere is dominated by the presence of molecular nitrogen up to a height of about 150 kilometers. Even above this altitude (at least up to 300 km), M2 continues to play a substantial role in the atmospheric processes. For this reason the study of the influence of impinging electrons on molecular nitrogen is the major thrust of this paper. One aspect of this study is the formulation of a complete cross section (differential and total) set for IEEs impacting on N2. The very difficult problem of modeling the interactions of the impinging IEEs in the upper atmosphere is then reduced in complexity. Since N2 interacts with electrons similar to the way that other atmospheric gases interact with electrons, it follows that differential and total cross section sets for these gases could be assembled in a like manner. Another aspect of this work is a sensitivity comparison among several of the influences on the electron energy deposition. The spatial energy degradation is vitally linked to the elastic phase function used. Since there are data available on the elastic differential cross sections of N2 as well as the energy degradation resulting from electron impact on N2, a comparison illustrating the effects of a variation of the elastic phase function is quite useful. Other influences on the spatial energy deposition, including ionization and excitation differential cross sections and the total elastic cross sections, are also considered in this work. In order to deal with these spatial and energetic aspects of elec tron energy degradation, a Monte Carlo (which may be abbreviated MC) calculation is used. The MC technique, depending on how it is used, can be the most accurate energy deposition approach. Use of this MC method at various incident energies helps in the assemblage of the best cross section set for N2 and provides the easiest way of comparing some of the influences on the spatial energy deposition. The details of this undertaking are discussed in Chapters II through VIII. A paragraph summary of each chapter is given below. The second chapter presents a brief review of some of the standard electron energy deposition methods. The continuous slowing down approxi mation, discrete energy bin, FokkerPlanck equation, twostream equation of transfer, and the multistream equation of transfer are all included in section II.A. The MC method which was used in this study along with three other MC approaches are briefly described in section II.B. This MC approach requires knowledge of differential and total cross sections. The third chapter discusses the cross sections that were used for N2. Section III.A includes the elastic differential and total cross sections. The inelastic differential and total cross sections are next discussed in section III.B. Section III.C, then, considers the total (inelastic plus elastic) cross section of N2. In Chapter IV, the MC calculational procedure is considered. A brief discussion of the approach is given in section IV.A. The computer programs and machinery used in this work are discussed in section IV.B with the programs listed in appendices A and B. A detailed discussion of the MC electron energy degradation technique is presented in section IV.C. Finally, the statistical error resulting from the Monte Carlo calculation is given in section IV.D. The MC threedimensional intensity plots with comparison to experi ment are given in Chapter V. The excitation of the N2 BE state is discussed in section V.A with the concept of range being defined in 0 section V.B. Previous experimental and theoretical work on the 3914 A emission from N2 is considered in section V.C and section V.D presents some range results and intensity plots in the longitudinal direction from this MC calculation. Section V.E, then, concludes the chapter with a comparison between the MC intensity plots in the radial direction and the experimental data. The main concern of Chapter VI is a sensitivity study. The effects of the ionization differential cross section on the intensity distribu tions reconsidered in section VI.A. Section VI.B, then, discusses the influence of inelastic differential cross sections on the intensity distribution. A comparison of different elastic phase functions on the electronN2 collision profile (no energy loss) is given in section VI.C. Next, the influence of different elastic phase functions on the electron energy deposition is presented in section VI.D. Finally, section VI.E considers the influence of the total elastic cross section on the electron energy deposition. The energy loss plots and yield spectra from the MC calculations are given in Chapter VII. Section VII.A presents the energy loss plots and section VII.B includes a discussion of the yield spectra. Chapter VIII concludes this paper with a summary of this work and its impact on the spatial and energetic aspects of the electron energy deposition problem. CHAPTER II A SHORT REVIEW OF ENERGY DEPOSITION TECHNIQUES Several standard energy deposition techniques will be discussed in this chapter. In the first section, II.A, several general ways for treating the degradation of the energy of charged particles will be re viewed briefly. The second section, II.B, includes a brief sketch of four Monte Carlo energy deposition schemes: The MC approach applied in this work and three other MC techniques. A. Energy Deposition Techniques Since the turn of the century, researchers have been studying the energy degradation of rapidly moving particles in a medium. Initial work in this area was carried out by Roentgen, Becquerel, Thompson, Bragg, Rutherford, Bohr, and other founders of modern physics. These pioneers in the energy degradation process were mainly con cerned with the medium affecting the particle. In order to solve this complex energy degradation problem, most of the early workers used a local energy deposition approximation. Even today many degradation problems can be solved fairly accurately with this local approximation. One of the earliest local energy deposition methods is that of the continuous slowing down approximation (hereafter called CSDA) first used by Niels Bohr (1913, 1915). Bethe (1930) expanded on this work and used an approximate theoretical treatment (valid at high energies) to describe the slowing down of particles in a medium. This work of Bethe (1930) required knowledge of the stopping power, dE (the rate at which energy E is lost from the impinging particles incident along the x axis). This stopping power is given by dE _+ d = n S Wi. (E) (2.1) dx i 1 i (see Dalgarno, 1962, p. 624) where the summation S includes integration i over the continuum (thus allowing for energy loss through ionization), Wi is the energy loss for the ith state, and ai(E) is the cross section for excitation to the ith state at energy E. Knowledge of the stopping power then leads to information about the mean distance traveled by the particle (referred to as the range). This range R(E) of a particle of energy E is simply given by E dE R(E) = (2.2) dx Atmospheric researchers are more interested in the effects that the particles have on the medium rather than the medium affecting the par ticles. These effects could include both spectral emissions by the con stituents and heating of the atmosphere. Green and Barth (1965) were the first workers to adapt a variation of the CSDA to the problems in aeronomy. In this approach the total dE energy loss function L(E) = (!) ) is determined by (E I)/2 daI,(ET) L(E) = I Wk k(E) + + Ijal (E) + I ) T da(ET) dT (2.3) k J j j 0 dT where Wk is the threshold for excitation to the kth state, ak(E) is the cross section for excitation to the kth state at energy E, I. is the J daj.(E,T) threshold for ionization and excitation to the jth state, and dT is the secondary differential ionization cross section for the creation of a secondary electron of energy, T, from a primary electron of energy E. The loss function with detailed atomic cross sections (hereafter called DACSs) was used to make reasonable estimates of the ultraviolet emissions resulting from an aurora event. In this approach, the excita tions Jk(E) of the kth state resulting from an electron of energy E were simply represented as E ak(E') Jk(E) = I _ dE' (2.4) Wk Green and Dutta (1967) built on this work and used the BornBethe approximations, the MasseyMohrBethe surface, the Bethe sum rule, and a "distorted" oscillator strength to lay the groundwork for extension of the DACS approach to other gases. Jusick, Watson, Peterson, and Green (1967), Stolarski, Dulock, Watson, and Green (1967), and Watson, Dulock, Stolarski, and Green (1967) applied this approach to helium, molecular nitrogen, and molecular oxygen, respectively. Stolarski and Green (1967) used this CSDA to calculate auroral intensities with these DACSs and Green and Barth (1967) applied this method to the problem of photoelectrons exciting the dayglow. Other atmospheric physicists (namely, Kamiyami, 1967; and Rees, Stewart, and Walker, 1969) started around this same time and also employed a CSDA type approach to that problem of energetic electrons depositing their energy in the atmosphere. The oldest discrete energy apportionment method is that of Fowler (192223) which is directly related to the Spencer and Fano (1954) approach (see Inokuti, Douthat, and Rau, 1975). The Fowler equation is solved by building on the lowerenergy solutions to obtain the higher energy solutions. The SpencerFano method introduces the electron at the highest energy and solves the equation at successively lower ener gies. An approach similar to the SpencerFano method was developed by Peterson (1969) and is called the discrete energy bin (hereafter called the DEB) method. Peterson (1969) pointed out certain differences between the CSDA and the DEB results. In particular, he noted that the DEB method tends to predict higher populations of some excited states than does the CSDA. In the modification of the DEB method by Jura (1971), Dalgarno and Lejeune (1971), and Cravens, Victor, and Dalgarno (1975), the equilibrium flux or degradation spectrum f(E,E ) (for incident energy Eo and electron energy E and in units of # cm2 sec1 eV1) of Spencer and Fano (1954) is obtained directly. Douthat (1975), in an effort to make the degrada tion spectra suitable for applications, proposed an approximate method of "scaling." Unfortunately, this method is quite cumbersome and not very accurate. This impelled Garvey, Porter, and Green (1977) to seek an analytic representation of the degradation spectra and, despite its complex nature, they found an analytic expression to represent this spectra for H2. The concept of the "yield spectra" U(E,E ) was first initiated through a modified DEB approach given in Green, Jackman, and Garvey (1977) in an effort to find a quantity with simpler properties than the degradation spectra. By utilizing the product U(E,Eo) = aT(E) f(E,Eo) where OT(E) is the total inelastic cross section for an electron of energy E, one defines a quantity U(E,E ) which not only has a simpler shape than f(E,Eo) but also has approximately the same magnitude for all substances. This yield spectrumcan also be represented analytically. It effectively embodies the nonspatial information of the degradation process. Jackman, Garvey, and Green (1977a), using this modified DEB, elaborated on the differences between the DEB method and the CSDA. The more accurate modified DEB method was found to produce consistently more ions per energy loss while at the same time producing less excitations of some of the low lying states when compared with the CSDA. The CSDA, although inexpensive to use, appears to be illsuited for calculations requiring an absolute accuracy better than about 20%. Since auroral and dayglow intensities are rarely measured to better than this accuracy, the CSDA has been adequate for most purposes of concern in aeronomy. How ever, with future improved measurements it should be purposeful to utilize more accurate deposition techniques. Several recent spatial electron energy deposition studies have been concerned with the spatial aspects of auroral electron energy deposition. Walt, MacDonald, and Francis (1967) employed the FokkerPlanck diffusion equation to give a detailed description of kilovolt auroral electrons. The FokkerPlanck equation, as given in the Strickland, Book, Coffey, and Fedder (1976) paper, is written Snz)a(Eaz E [ 2) a.(zE,u)] "n(z)a(E)az 2a{ET aF aE + aE [L(E)4(z,E,E)] (2.5) where 4(z,E,p) is the flux (in units of cm2 sec1 eV1 sr1), z is the distance into the medium along the z axis, E is the electron energy, 10 and p is the cosine of the pitch angle associated with the direction of motion of the electron. The symbols n, a, Q, and L are the number den sity of the scatterers, the total cross section (both elastic and in elastic), the momentum transfer cross section, and the loss function, respectively. The momentum transfer cross section, given in terms of the differ ential elastic cross section, d(E)is written as 2n 7r Q(E) = i dE (1 cose)sineded4 (2.6) 0 0d This FokkerPlanck equation may be thought of as a CSDA approach to the spatial energy degradation problem. Its solution, therefore, is only accurate in the higher energy regime. Banks, Chappell, and Nagy (1974) were able to calculate the emission as a function of altitude for a model aurora using the FokkerPlanck equation for electrons with energy E > 500 eV along with a twostream equation of transfer for electrons with energy E 5 500 eV. The two stream equation of transfer solves the transport of electrons in terms of the hemispherical fluxes of two electron streams 0 (E,z), the electron flux upward along z, and 4(E,z), the electron flux downward along z. The steady state continuity equations for 0 and 0 can then be written as +L 1 n k k k + dz + + 1 kk + q +q (2.7) and 11 o = 1 Yn k k k  dz 1 kk+ + 1 n kk + + q (2.8) where ak k (2.9) a ai q (E,z) = ~ nk(z) {P (E')a (E'E)(E,z) E'>E + [1 pj(E')]c a(E'E)+(E ,z)} (2.10) k k (E,_ E)+(Ez) q(E,z) = C nk(z) I {pk E')aj( E)+(E'z) k E >E + [1 Pa(E')]a (E'E)a(E',z)} (2.11) ai a3 and z is the distance along a magnetic field line (positive outward); nk(z) is the kth neutral species number density; p (E) is the electron backscatter probability for elastic collisions with the kth neutral species; ak(E) is the electron total scattering cross section for the kth neutral species; q(E,z) is the electron production rate in the range E to E+dE due to ionization processes (both electron ionization and photoionization); q~ is the electron production in the range E to E+dE due to cascading from higherenergy electrons undergoing inelastic collisions; p k is the electron backscatter probability for collisions aj with the kth neutral species resulting in the jth inelastic process; and k. is the inelastic cross section for the jth excitation of the kth neutral species. species. 12 This approach combined these two methods of electron energy deposi tion in order to find a reasonable solution to the very difficult auroral energy deposition problem. The FokkerPlanck method is accurate only at large incident energies; therefore, it should only be used at ener gies above 500 eV. The twostream equation of transfer approach, on the other hand, is more accurate at energies below 500 eV. This combination then provided a very reasonable solution to the auroral electron spatial deposition problem for a reasonable amount of calculation. The FokkerPlanck equation and the twostream equation of transfer may both be derived from the Boltzmann equation or the general equation of transfer. This general equation of transfer, in one of its simpler forms, is written as (from Strickland et al., 1976) Sd(z E,)= n(z)o(E)O(z,E,p) + n(z)o(E) f R(v',',v,E'E)(z,E',p')dE'dP' (2.12) (assuming a steady state condition and no external fields) where Xoj(u',u,E',E) R(p',p,E',E) a G(E) (2.13) with the sum over all processes. The symbols p and y' are the cosines of the pitch angles e and e' which are associated with the directions n and i' given in Figure 2.1. The first term on the right hand side of Eq. (2.12) represents the scattering out of p. The R(p',p,E',E) in the second term is the proba bility (eV (2wsr) ) that a collision of an electron of energy E' and direction p' with some particle will result in an electron of energy E 13 0 8' / \ / W A / f  ~1 Figure 2.1 The directions denoted by n' and n are the incident and scattered directions of the electron, respectively. 0 14 and direction P. The integral in Eq. (2.12) is over all possible ener gies E' and directions of motion p'. Strickland et al. (1976) studied the auroral electron scattering and energy loss with a multiangle equation of transfer. Their approach is one of the most accurate yet applied to auroral electrons. This multi angle method of solution is more realistic than the twostream approach and it is computationally more difficult as well. The methods discussed above are the "state of the art" approaches (excluding the Monte Carlo methods which are discussed in section II.B) to the IEEs degrading in the atmosphere. Other approaches used by Jasperse (1976, 1977) and Mantas (1975) are mainly concerned with photo electrons and will not be discussed here. The Monte Carlo approach can rival any of these electron energy deposition methods in accuracy when used in the proper manner. This stochastic technique for solving the deposition problem will be con sidered next in section II.B. B. Monte Carlo Energy Deposition Techniques Another method of solving the spatial energy deposition problem is the use of the Monte Carlo approach. The MC technique, which is used in this paper, is a stochastic method of degrading an energetic electron. The approach can be one of the most exact methods of electron energy deposition. Briefly, one electron is taken at a time and allowed to degrade in energy collision by collision. This deposition attempts to mimic the randomness of the actual degradation process that occurs in nature. 15 Many MC schemes have been applied in all areas of physics. Some are more exact and more detailed than others. Since virtually all the MC methods are run on the computer, the most exact approaches cost the most computer time and money. The precision of the technique must be balanced against a finite computer budget. Three approaches using the MC deposition scheme, that have been applied to electrons impinging on the atmosphere, are discussed below. Brinkmann and Trajmar (1970) applied experimental differential electron impact energy loss data in a MC computation for electrons of 100 eV energy. Because of the large amount of input cross sections in numerical form, only electrons of 100 eV incident energy were degraded with this method. In the lower electron energy regime (below 25 eV), Cicerone and Bowhill (1970, 1971) used a MC technique to simulate photoelectron dif fusion through the atmosphere. This method, which included both elastic and inelastic processes, predicted escape fluxes from the atmosphere. Berger, Seltzer, and Maeda (1970, 1974) (hereafter called BSM) worked with high energy electrons (with energies from 2 KeV to 2 MeV). They employed a MC approach that has two variations which are pointed out below. They treat inelastic collisions in a continuous slowing down manner. The energy deposited by the electrons along their path is assumed to be equal to the mean loss given by the loss function, L(E), from Rohrlich and Carlson (1954). The angular deflection resulting from elastic collisions has been evaluated by two separate methods in BSM. One approach employed the multiple scattering distribution of Goudsmit and Saunderson (1940) applied to the screened Rutherford cross section given in BSM. The 16 other approach involved a sampling of each elastic collision. Appli cation of the BSM technique to a constant density medium and no magnetic field gave good agreement with laboratory experiments (Grun, 1957; and Cohn and Caledonia, 1970). In this study, a MC method was needed that could be applied to IEEs. The basic equation of transfer is solved with the use of the MC approach. This equation can be rewritten as dU(P,z,E,E d cYTTE dz = n(z)U(u,z,E,Eo) E+AEEl as Elas + n(z) E 1 f pe(u',p,E',E)U(u',z,E',E )du'dE'  E o +1 + n(z) J f PION1i(u',vE',E)U(i',zE',E )dp'dE' i 2E+Ii 1 + n(z) j E 0 paj (p',v,E',E)U(v',z,E',E )dp'dE' aj No external fields U(v,z,E,E ) is the that there is only aT(E) is the total are included here and a steady state is assumed. The "yield spectra" (in eVl sec1 sr1) and it is assumed one neutral scattering species. In this equation cross section (elastic + inelastic)for the species, AEElas = 2E(1 cose) electron "neutral species (2.15) is the energy loss during an elastic collision, p (v',p, E',E) is the probability during an elastic collision with a neutral specie that an electron with energy E' and direction p' will result in an electron of (2.14) 17 energy E and direction i, PIONi(p',,, E',E) is the probability during an ionization collision with a neutral species that an incident electron with energy E' and direction p' will result in a secondary electron of energy E and direction p, and p .a(',p,E',E) is the probability during an inelastic collision (excitation or ionization) with a neutral specie that an incident electron with energy E' and direction p' will result in the incident electron being scattered into direction u with energy E. Some techniques from each of the other three MC methods were in cluded in this work. Some new approximations and assumptions were made, however, to enhance the accuracy of the computations as well as simplify some of the calculations. These assumptions are discussed in detail in Chapter IV. In this MC work the information is stored in a collision by collision manner on a magnetic tape. Once all the case histories are generated, then, the tape is scanned and any correlations of interest may be deter mined. The choice of altitude and energy intervals is specified only during the scanning of the tape. This method allows for greater flexi bility in minimizing the statistical uncertainties of the results, while, at the same time retaining good spatial or energy resolution (Porter and Green, 1975). All the degradation methods mentioned in this chapter require cross sections as input. The cross sections used in this MC work are, there fore, discussed in the next chapter. CHAPTER III ELASTIC AND INELASTIC DIFFERENTIAL AND TOTAL CROSS SECTIONS FOR N2 In this chapter differential and total cross sections for electron impact on N2 will be discussed. Section III.A reviews the elastic cross sections of N2 and discusses three models for representation of these properties. In section III.B the inelastic cross sections of N2 are presented with their relationship to theory and experiment. Section III.C, then, concludes this chapter with a discussion of the total (elastic plus inelastic) cross section for N2. Any energy degradation technique requires knowledge of these cross sections for a complete evaluation of the energy loss by electrons in a gas. A. Elastic Differential and Total Cross Sections for N2 One of the most common differential elastic cross section forms is the screened Rutherford cross section which can be expressed in the form do 224 do 22 Z2 e 2] Krel(e) (3.1) p v (1 cose + 2n) where Krel(e) is the spinrelativistic correction factor. The familiar Rutherford cross section unscreenedd case) which can be derived from classical scattering theory is given by 2 4 do Z e4 ce2 (3.2) d 2 v2(1 cose)2 18 19 where sin2 1 cose Here, an electron is elastically scattered by a nucleus of charge Z using the Coulomb potential V(r) = Ze2 (3.3) r with r being the distance between the two particles. Treating scattering in a quantum mechanical approach with the use of the Born approximation and a potential of the form V(r) Ze r (3.4) V(r) r where y is a positive but small quantity approaching 0, Eq. (3.2) can again be derived. The Born approximation, using the potential in Eq. (3.4), is only valid in certain angle and energy regimes (Mott and Massey, 1965, pp. 24 and 466). At sufficiently high angles and low energies, the Born approximation fails. The range of validity varies for different substances and for N2 the Born approximation is probably not accurate at all angles for energies less than 100 eV and at large angles for energies less than 500 eV. Equation (3.2) does, however, go to infinity when e approaches 0. This differential cross section also leads to an infinite value in the total elastic cross section. Both of these results are unreasonable for elastic scattering of electrons by atoms and molecules. The most popu lar way of correcting this unreal behavior is to add a screening param eter n. Equation (3.1) portrays this resulting form. 20 Equation (3.1) has a maximum at e = 0 and a minimum at a = 1800. At energies below 200 eV, experimental results indicate a minimum in the elastic differential cross sections at about 900 with a strong forward scattering peak at e = 0 and a secondary backward scattering peak at e = 1800. In Figure 3.1 experimental data for energies at E = 30 and 70 eV are presented. These data are taken from Shyn, Stolarski, and Carignan (1972) with the small circles denoting 30 eV points and the crosses denoting the 70 eV data. Later on in this section the screened Rutherford cross section and another analytic model of the differential elastic cross section are compared with experimental data. Before discussing the differential cross section in more detail, first, consider the total elastic cross section. Several experiments have been performed deriving the total elastic cross sections for N2. There have also been several theoretical studies on the N2 elastic total cross sections. Two recent reviews of the data available on this subject are Kieffer (1971) and Wedde (1976). A plot of all this data would obscure the analytic total cross sections specifically considered in this work. Consequently, only data from Sawada, Ganas, and Green (1974) (theoretical), Shyn, Stolarski, and Carignan (1972) (experimental), and Herrmann,Jost, and Kessler (1976) (experimental) are plotted in Figure 3.2. The sets of data overlap to a degree such that the disagreement in absolute magnitude of the total cross sections is readily apparent. In view of this disagreement, no experimental or theoretical data areassumed to be absolutely correct and some average of this data is 21 I I I I I X 0 6 0 o 0 X a I 0 0 0 xo O o X X X 0 0 o X oxx  X 0 ox Ox OX X" I I I 0 30 60 90 120 150 8 (Degrees) 180 Figure 3.1 N2 experimental electron impact elastic cross section data from Shyn, Stolarski, and Carignan (1972). o's denote data from E = 30 eV and the x's denote data from E = 70 eV. 10 vl OI o< 0  10 bIC3 10   I I I "I 10 r r 0 *. d Q EC0 .oU SCn.C TO .C r *r r 4 1 0 Ln (") C 4 S. A 4t) Mc C0 CD ), U O In (I1 Ur  U IJ 1 03 cU 4.o n .) V) U) LL to 0 Q. C *r (fl (n 0r r0 'a  *' O u 0 ** *cI 4 ON ca 4. r 0o U 4) .0 u0 *c? C.D (0 C r *r . * ro V C c 5 C *0 U r nW a, ss .  M < au C CD CC CM 10 ** 10 U* *r Z CD O Z i CM a) 3r 01 iZ n (r 23 / / / / 0a O O i0 (V !) (3) 0 '0 0 0 24 desirable. An analytic function representing the total elastic cross section is most easily used in a computer program. Consider now an analytic form derived from the differential screened Rutherford cross section. Knowledge of the differential cross section implies knowledge of the total elastic cross section as they are simply related by 2w r o(E) = f sineded (3.5) 0 0 where 0 is the azimuthal angle. Substituting Eq. (3.1) into Eq. (3.5), the total elastic cross section, OR(E), resulting from the screened Rutherford cross section is very simply given as Z2 51.8 R(E) E2 n(1 + n) (3.6) 16 2 If E is given in eV then OR(E) is in units of 101 cm The screening parameter S 1.70 x 105 Z2/3 (37) n c T(r + 2) according to the Moliere (1947, 1948) theory. Berger, Seltzer, and Maeda (1970) assumed that n was a constant value and decided on nc = 1 as its best value. The T in Eq. (3.7) is the electron energy in units of the electron rest energy so that T = E/mc2. In the energy regime of interest (E s 5 KeV), T << 2, and Z = 7. Noting these observations, Eq. (3.7) can be rewritten as n % 1 The total cross section from Eq. (3.6) is plotted in Figure 3.2 as the dashdot line. A noticeable difference is evident between this model and the experimental values at practically all energies. 25 Using the form q F[1 (W)O8 o(E) = F[ ( C (3.8) Ec W implemented first by Green and Barth (1965), where q = 651.3 A2 eV2, the total elastic cross section for N2 was characterized fairly well in the range from 30 to 1000 eV using the parameters a = 1, 8 = 0.6, c = 0.64, F = 0.43, and W = 2.66. The E"064 dependence of Eq. (3.8) at the larger energies is similar to that seen by Wedde and Strand (1974) for N2. This form does not represent the data as well below 30 eV and, in fact, is not defined below an energy of 2.66 eV. Porter and Jump (1978) have used a more complex total elastic cross section form which is written as { ^E c(E) = T n(n + 1)[V+ + E2+ ] F2G2 F G2 + 1G12 2+ 22 (3.9) (E E1) + G (E E + G U Here, n = U and for N2: T = 2.5 x 106 cm2 F1 = 7.33 U = 1.95 x 103 eV El = 2.47 eV V = 150 eV G2 = 24.3 eV X = 0.770 F2 = 2.71 G1 = 0.544 eV E2 = 15.5 eV In the large energy limit, the total cross section falls off as 1/E, similar to the screened Rutherford cross section. This form does con tain two other terms (the second and third terms) which were introduced 26 phenomenalogically to describe the low energy shape and Feshbach reso nances. If either Eq. (3.8) or (3.9) is used as the total elastic cross section, the differential elastic cross section must be normalized such that: 2ir i f f P(e,E) sinededo = 1 (3.10) 0 0 where P(e,E) is the phase function and the differential cross section can be written as do = o(E) P(e,E) (3.11) With this in mind consider now three separate phase function forms. The first phase function form is very similar to the screened Rutherford cross section and it is written here as PMl(e,E) = 1 1 2(3.12) 2n[(2 + a(E)) a(E) l][ cose + a(E)] This is known as model 1. The parameter "a" acts in a similar manner to the "2n" term in the denominator of the screened Rutherford cross sec tion form and is written E a2 a(E) = al 1 eV The second phase function form (model 2) includes the forward scattering term of Eq. (3.12) along with a backscattering term and is given as 27 PM2(,E) = f(E) S2r[(2 + a(E))1 a(E)l][ chose + a(E)]2 (1 f(E)) (3.13) 27[(2 + c(E))1 c(E)1][1 + cose + c(E)]2 where (E/fl 2 f(E) = (E/fl) (E/f) 2+ f3 a2 a(E) = a l(iv) and SC3 c(E) = cl[1 ) ] Irvine (1965) was one of the first researchers in scattering prob lems to use a phase function containing forward and backward scattering terms. He applied a sum of two HenyeyGreenstein functions to the prob lem of photon scattering by large particles. Porter and Jump (1978) also have used a sum of two terms (one for forward scatter and one for backward scatter). They fitted experimental data at several separate energies with their form. Use of their differential cross section form in a deposition calculation probably would require the use of spline functions or other interpolative techniques. The third phase function (model 3) is now considered. At small angles the differential cross section shows a near exponentiallike fall off. This behavior has been pointed out by several experimenters (see, for example, Shyn, Carignan, and Stolarski, 1972; and Herrmann, Jost, and Kessler, 1976). It was this experimental observation that led to 28 model 3 which is written as fl(E)[l b2(E)]eB/b(E) M3(,E) = 2w b2(E)[ + e"/b(E) f2(E) 2w[(2 + a) a1][1 cose + a]2 [1 fl(E)  1 S2[(2 + c(E))1 c(E) fl2 (E/f l)12 fl(E) =f 21p [(E/fl1) + f13] (E/f21) 22 f2(E) = f2 [(E/f21) 22+ f23] f2(E) = 1 fl(E) f(E) = f2(E)[1 fl(E)] f2(E)] 1 ][l + cose + c(E)]2 for E > 200 eV for E < 200 eV b(E) = b (E )b2 Cc(E) = ( c(E) = c [1 (C2 The parameters used for the rest of this study in Eq. (3.14) are fl = 100 eV fl2 = 0.84 fl3 = 1.92 a = 0.11 bI = 0.43 b2 = 0.29 where (3.14) 29 f21 = 10 eV cI = 1.27 f22 = 0.51 c2 = 12 eV f23 = 0.87 c3 = 0.27 This form is more complex than the other phase function models but it does describe the experimental differential cross section data the most realistically. It includes an exponential term for the near ex ponentiallike forward scattering as well as a backward scattering term for electron energies below 200 eV. Comparisons of the screened Rutherford and model 3 cross sections are given in Figures 3.3 and 3.4 at the two energies of 30 eV and 1000 eV. Both forms are normalized to the total elastic cross section form of Eq. (3.9). This modified screened Rutherford cross section vastly under estimates the forward scattering from e = 00 to 30, overestimates the scattering in the range from 6 = 300 to 120, and underestimates the scattering at the larger angles with e = 1200 to 1800. Model 3 does a fairly reasonable job of representing the differential cross section data at both of these representative energies and the other energies as well. Although there is not a large amount of energy loss during an elastic collision, there is some. Using classical considerations (see Green and Wyatt, 1965), the energy loss is approximately given by Eq. (2.15). For molecular nitrogen and 6 = 900, the energy loss is about 8 x 105 E. The MC approach, being a stochastic process, uses the concept of probability for scattering within a given angle interval. In order to compare phase functions, the probability for backscatter may be compared. Figure 3.3 (a and b) N2 electron impact elastic differential cross sections. The screened Rutherford (dashed line) and the model 3 (solid line) are compared with the experimental data of Shyn et al. (1972), x, and Herrman et al. (1976), o, at the energies of 30 eV (Figure 3.3a) and 1000 eV (Figure 3.3b). 101 U) o< %. I i1 1 1 I I I I 1 I a I I I 0 30 60 90 120 150 180 e (Degrees) Figure 3.3a 32 102 0 bC \\ lol L. 2 4 10 4 I I I I I I I 1 1 1 1 1 I 0 30 60 90 120 150 180 9 (Degrees) Figure 3.3b Figure 3.4 (a and b) N2 electron impact elastic differential cross sections between 0 and 200. The screened Rutherford (dashed line) and the model 3 (solid line) are compared with the experimental data of Shyn et al. (1972), x (the a's denote extrapolated points), and Herrmann et al. (1976), o, at the energies of 30 eV (Figure 3.3a) and 1000 eV (Figure 3.3b). 34 (r) o< bIC! .0"a 0 4 8 12 16 20 8 (Degrees) Figure 3.4a 35 101 V) bCs *o100 I10 I0o 8 12 8 (Degrees) I Figure 3.4b 36 This probability, PB(E), is simply calculated with PB(E) = 2n ~ (3.15) f d sinededo 00 In Figure 3.5, PB(E) from the screened Rutherford and model 3 are compared with other theoretical (Wedde and Strand, 1974) and experi mental (Shyn et al., 1972) values. Model 3 does have a tendency to estimate less backscatter than the screened Rutherford at the larger energies. (The PB(E) curves for model 3 and the screened Rutherford do tend to converge at 5 KeV however.) The dominant exponentiallike for ward scattering is the reason behind this behavior. The discontinuity observed at 200 eV in model 3 values results from the lack of the back scatter characteristic above this energy. The elastic scattering collisions influence mostly the direction of travel of the electrons. There is some energy loss during an elastic collision (as pointed out above), but this loss is not important for electrons with energies above 2 eV colliding only with N2 particles. Inelastic collisions, on the other hand, result in a fairly sub stantial energy loss with some scattering. Consider now the differential and total cross sections for these inelastic events. B. Inelastic Differential and Total Cross Sections for N2 Inelastic collisions are divided into two types: 1) electron ex citation and 2) electron ionization. In the excitation process the electron is excited to a higher state which may either be an optically 37 10 o x 0 \ 2 3 g10 100 m  102 103 E (eV) Figure 3.5 Backscatter probabilities for the screened Rutherford (dashed line) and the model 3 (solid line) are compared with Wedde andStrand (1974), x; and Shyn et al. (1972), o. 38 allowed or an optically forbidden transition. This transition for many molecules leads to a repulsive state which can dissociate the molecule. In N2, dissociation of the molecule in this manner is virtually non existant because N2 is a very stable homonuclear molecule in which both the singlet and triplet states are found to be strongly bound. As a consequence of this, the main process for dissociation is through pre dissociation of stable electronic terms by repulsive states that are themselves strongly optically forbidden in direct excitation. Porter, Jackman, and Green (1976) (hereafter called PJG) compiled branching ratios for dissociation of N2 using several experimental and theoretical papers (see, for example, Winters, 1966; Rapp, Englander Golden, and Briglia, 1965; Polak, Slovetskii, and Sokolov, 1972; and Mumma and Zipf, 1971). In PJG the efficiencies for the production of atomic nitrogen from proton impact were determined. This study does not include a calculation of the atomic nitrogen production; however, the PJG branching ratios with the yield spectra, discussed in section VII.B, can be applied to calculate this atomic yield. The excitation events, not resulting in the dissociation of the N2 molecule, are either electronic or vibrational transitions. Cross sections for these transitions are taken from both PJG and Jackman, Garvey, and Green (hereafter called JGG) (1977b). In the ionization event an electron is stripped from the molecule and given some kinetic energy. The ionization cross section is a sub stantial portion of the total inelastic cross section above 50 eV (compare Figures 3.6 and 5.1) and the total amount of energy loss is always > the lowest ionization threshold (which is 15.58 eV for N2). Subsequently, most of the energy loss of the electrons (for energies 39 CM 0< 101 b d2 I(1 Figure 3.6 16C Figure 3.6 E(eV) N2 electron impact cross sections. The total inelastic, Eq. (3.16) (solid line), total elastic, Eq. (3.9) (dashed line), total inelastic plus elastic, Eq. (3.16) plus Eq. (3.9) (dashdot line), and the experimental inelastic plus elastic values (Blaauw et al., 1977), x, are pre sented here. 40 > 50 eV) is from the ionization collisions. These ionization cross sections were also taken from PJG and JGG. The total inelastic cross section found by summing these inelastic cross sections was fit with the function TI) qF[1 W (aB a ,4EC q0F[l (T) in(t + e) TI(E) = WEw (3.16) This form has the characteristic BornBethe In E/E fall off behavior at the large energies. The parameters a = 1, 8 = 4.81, C = 0.36, F = 4.52, and W = 11 were found with the use of a nonlinear least square fitting program which fit Eq. (3.16) to the sum of all the inelastic cross sections. From 30 eV up to 5 KeV this form was used for the total inelastic cross section. Below 30 eV much structure in the total inelastic cross section is evident. At these low energies, the total inelastic cross section can be read numerically into the MC program. This total inelastic cross section is illustrated by the solid line in Figure 3.6. Consider now the scattering of the two electrons involved in an electron impact ionization collision. In reality, only the incident electron is scattered. The other electron is simply stripped from the molecule and given kinetic energy in a certain direction of travel. Experiments are unable to distinguish between the incident electron and the electron stripped from the molecule. In this paper, the ionization event is assumed to cause scattering of both electrons. The scattering angle of either is then measured with respect to the incident electron's path. After the collision event the electron with the higher energy is designated the primary electron and the electron with the lower energy 41 is called the secondary electron. There should be a correlation between the primary and secondary scattering, but this mutual influence is dif ficult to quantify. The impinging electron ionizes a many body par ticle, a molecule of nitrogen, thus momentum and energy can be conserved without all the momentum and energy shared by the two resulting elec trons. Only recently has work been done on triply differential cross sections for N2 and this interaction was measured only at one energy E = 100 eV (see Jung, Schubert, Paul, and Ehrhardt, 1975). More ex perimental and theoretical work needs to be done in this area before any generalization can be made concerning the correlation between the primary and secondary scattering. The primary and secondary scattering will thus be treated sepa rately in this work. In dealing with the scattering of the primary electron after an ionization collision, a differential ionization cross section form based on the MasseyMohrBethe surface of hydrogen, is used. The form (with a the Bohr radius, and Re, the Rydberg energy) is 4a2R do oe )/2 F(x) (3.17) dTdn Wx E ( where x = (Ka )2 is the momentum transfer, W is the energy loss in the collision process which is equal to the ionization potential, I, plus the kinetic energy of the secondary, T, and F(x) is a complex function given in PJG. Equation (3.17) is very highly forward peaked for small energy losses but becomes less forward peaked as the energy loss becomes sig nificant when compared with the incident energy, E. The secondary electron is also scattered in the ionization event. Probably the most comprehensive work that exists on secondary doubly 42 differential cross sections is that of Opal, Beaty, and Peterson (1972). (More recent data by DuBois and Rudd (1978) agrees with their work.) This data indicates a preferred angle range in the scattering process (usually between 450 and 900) at all primary and secondary energies. A BreitWigner form has been chosen to represent the data. Here, do S(E,T)C2 (3.18) dTdn [C2 + B(cose cose )2]Nf where 0.91 B(E) = 0.0448 + (72o e) C() = 36.6 eV C(T) = (T + 183 eV ) 6A(E) o6(E) = 0.873 + ,(E) 0 (T + eB(E)) OA(E) = 20 eV + 0.042 E OB(E) = 28 eV + 0.066 E 2C 1 (1 + cose ) 1 (1 cose ) Nf = _ _tan  ] tanl [ cC 0 and S(E,T) =d = A(E)r2(E)/[(T T(E))2 r2(E)] (3.19) is from Green and Sawada (1972) with A(E) 5.30 ln SE n1.74 eV To(E) = 4.71 eV 1000 eV)2 o (E + 31.2 eV) 43 r(E) = 13.8 eV E/(E + 15.6 eV) a = 1 x 10 16 cm2 Equation (3.18) may seem highly complicated; however, integration over the solid angle is given very simply as Eq. (3.19) which is the singly differential ionization cross section. The total ionization cross section is then N(E) dT (3.20) 0 with T = (E I) M 2 so that OION(E) = A{tan"[(TM To)/r] + tan (Tor)} (3.21) The fit to Opal, Beaty, and Peterson's (1972) data is given in Figure 3.7 at several primary and secondary energies. The x's represent the ex perimental data and the solid line represents the analytic expression, Eq. (3.18). Other inelastic processes include the simple excitation events. The scattering of the incident electron due to the excitation of a par ticular state has been studied by Silverman and Lassettre (1965), and more recently by Cartwright, Chutjian, Trajmar, and Williams (1977) and Chutjian, Cartwright, and Trajmar (1977). In order to account for this scattering, the Silverman and Lassettre (1965) generalized oscillator strength data for the 12.85 eV peak (cor responding to the optically allowed b 1,u state)were fit with the use of a phase function similar to model 1. The very sharply forward scattering 44 *0 10  oI IlO0 10 10 0 90 180 oI 10 *00 0 90 18 0 (Degrees) 0 90 180 Figure 3.7 N2 experimental ionization doubly differential cross sections from Opal, Beaty, and Peterson (1972) are represented by x's. The solid line () denotes the cross section resulting from the use of Eq. (3.18). The incident electron energy is de noted by E (eV) and the secondary electron energy is denoted by T (eV). 45 peak indicated in these data was used in a MC calculation. The electron scattering which results using this inelastic scattering approximation in a computation was so small as to be virtually undetectable. Cartwright et al. (1977) and Chutjian et al. (1977) have studied a more comprehensive list of states and have observed significant electron scattering (especially due to the optically forbidden states) in the range from 10 eV to 60 eV. Characterizing this data in some way appears to be a rather endless task. Dealing with this type of inelastic scattering is thus still a problem. Above 100 eV the optically allowed excitations are the most important; thus it is safe to conclude that the inelastic scattering events will not affect the energy degradation process. Below 100 eV, as a first approximation, it is assumed in this work that the inelastic excitation events scatter as much as the elastic events. This is probably a reasonable approximation to the very complex inelastic ex citation scattering. In section VI.B the effects of this assumption are discussed. C. Total Cross Section (Elastic Plus Inelastic) Elastic and inelastic processes have been considered in sections III.A and III.B. Another aspect of the cross sections is the total (elastic plus inelastic) cross section. Blaauw, de Heer, Wagenaar, and Barends (1977) have recently pub lished experimental data on the total cross section values of N2. These experimental values are compared with the cross section values from this work in Figure 3.6. 46 Throughout the energy range the cross sections used in this study compare favorably with those of Blaauw et al. (1977). For an easy reference, the total inelastic and total elastic cross sections are also given in Figure 3.6 as separate curves. All the major influences on the IEE energy loss and scattering have been accounted for in this chapter. The next chapter presents the MC energy deposition scheme which employs these cross sections. CHAPTER IV THE MONTE CARLO METHOD OF ENERGY DEPOSITION BY ELECTRONS IN MOLECULAR NITROGEN A Monte Carlo calculation is used here for energy degradation by energetic electrons in N2. This stochastic process is probably the most accurate method for obtaining the energy loss of particles in a medium. The generalizations about electron impact on N2 that are made through the use of this technique can be applied in other energy deposition approaches to electrons impinging on the atmosphere. (This is true even though the magnetic field is neglected in these MC calculations. The three dimen sional yield spectrum U(E,z,E ) [see Chapter VII] is most useful for applications to the atmosphere and changes in the magnetic field will not affect the yield spectra greatly below altitudes of about 300 km where the gas density is fairly high [see Berger, Seltzer, and Maeda, 1970 and 1974].) Building on the MC work done in this paper, more exact models of auroral electrons and photoelectrons can be established. In the first section, IV.A, a brief discussion of the MC calculation outlines the general procedure involved in the degradation process. The computer program and machinery used are briefly described in section IV.B. Sec tion IV.C relates in detail the various aspects of the calculation. Finally, section IV.D discusses the statistical error that arises from the use of the MC calculation. 47 48 A. Brief Discussion of the Monte Carlo Calculation In Figure 4.1 a short version of the MC calculation is presented. Briefly, each electron is degraded in a collision by collision manner down to 30 eV. Below 30 eV the electrons are degraded with the use of a multiple scattering distribution. This multiple scattering approach characterizes the resultant coordinates of the electron which goes through several elastic collisions between each inelastic collision. The incident electron has an energy E To begin with, the running total of the electron energy, E, is set equal to E At the position START, this energy E is checked against a cutoff energy, E If the E is more than Ec then, first the distance traveled by the electron to the collision is calculated. Second, the type of collision which occurs is determined. If a collision is elastic then the electron is scattered with the use of a phase function, the appropriate energy AEElas is lost, and the electron goes back up to the START of the degradation process. Whether collision is inelastic it is determined if the collision is an ioniz tion event or an excitation event. In the excitation process, scattering occurs if the energy E is less than 100 eV, E is reduced by the threshold, W, for excitation of this state, and the electron goes back up to the START of the degradation process. Ionization collisions are the most complex occurrences to compute. The energy loss, W, by the incident electron is equal to the kinetic energy, T, of the secondary electron produced plus the ionization thresh old, I. The primary electron is then scattered and reduced in energy by W. If the secondary electron has a kinetic energy greater than E then, Figure 4.1 Flowchart of the Monte Carlo degradation of an incident electron of energy Eo. 50 MONTE CARLO CALCULATION Calculation Complete Electrons Thermalize Calculate Path Length and Coordinates of Next Inelastic Collision with the MESD 51 it is scattered and sent back to the START to be degraded further. In the meantime, the primary electron's properties are stored. If a secondary produces a tertiary electron with a kinetic energy greater than E then that tertiary is completely degraded before any further degradation of the secondary is considered. Like the primary, the secondary electron's properties are stored in the meantime. No other generations were included in this study as their contribution would be, at most, a couple of tenths of a percent of the incident electron's energy. After the tertiary is entirely degraded below Ec then the secondary is again sent back to the START to be degraded further. The secondary is next entirely degraded below E and, finally, the primary is again sent back to the START to be further degraded. This whole process may then again repeat itself. B. Computer Programs and Machinery Used in the Monte Carlo Calculation In the previous section a brief discussion was given of the electron energy degradation process. A brief discussion will be given below about the MC computer codes and the computing machinery used. The MC computer program employed in this work evolved from an original MC code written by R.T. Brinkmann (see applications in Brinkmann and Trajmar, 1970). This program was revised for use in Heaps and Green (1974), Kutcher and Green (1976), and Riewe and Green (1978). The author has further modified this MC technique for energetic electron impact into N2 to be used in the energy range from 2 eV to 5 KeV. 52 This MC technique was applied to several incident electron energies. The vast majority of the MC program runs used the Amdahl 470/175 computer at the Northeast Regional Data Center at the University of Florida. There were, however, several MC runs using the PDP 11/34 of the Aeronomy group of the University of Florida. It should be noted here that running the same program on both machines at the same energy, Eo = 1 KeV, showed a factor of 240 dif ference in the execution time. Thus a program that takes four hours on the PDP 11/34 will take one minute on the Amdahl 470/175. This time advantage plus the ability to store each collision of the electrons on magnetic tape does make the Amdahl 470/175 a more desirable "number crunching" machine. The PDP 11/34 is only able to produce intensity plots in the longitudinal direction. This minicomputer is thus mainly useful in deriving a range (to be described in the next chapter). Two programs were used in deriving the MC results. The first pro gram (listed in Appendix A), the modified version of Brinkmann's code, degraded the electrons in energy from their initial E down to the Ec and recorded each collision and its properties on the tape. The second program (listed in Appendix B) coalesces the data from the tape into an array of ordered output. This output contains information for three dimensional intensity plots, energy loss plots, and yield spectra. C. Detailed Discussion of the Monte Carlo Electron Energy Degradation Technique Now, a more detailed discussion is given for the MC method of degrading an electron's energy. An electron will start off with an energy of Eo and coordinates x y z e and o The symbols x, y, 53 and z are the Cartesian coordinates of the electron. The polar angle e is measured with respect to the zaxis and the 0 denotes the azimuthal angle measured with respect to the xaxis (see Figure 4.2). In this approach, the initial coordinates xo, y, z e and 0o were all set equal to zero. The coordinates xb, yb' zb' b', and Ob of the electron before starting on its journey to a collision are, therefore, initially established as xb = x yb = Yo, zb = zo' eb = eo, and Ob = o' The MC approach relies on the random number, R, between 0.0 and 1.0 to aid in the deposition calculation. For each collision several R's are needed and for each R a new property of the collision is gained. In order to explain this MC approach, an accounting of the random numbers and their subsequent usefulness is now made. The multiple elastic scat tering distribution used below 30 eV and the lowest energy cutoff 2 eV are also described. 1. First Random Number, R1 The first random number, R1, is used to find the path, PT, traveled by the electron before it collides with a molecule of N2. Calculation of PT proceeds in the following manner. The mean free path, A, is defined as = (4.1) naT(E) where n is the density of N2 molecules in #1cm3 and aT(E) is the total (inelastic plus elastic) cross section of N2 in units of cm2 at an energy E. The densities used at the various initial input energies are given in Table 4.1. First Collision (Xb, Yb, Zb) 7\, Second Collision (Xo, Y, Z I I %I N1 Figure 4.2 Schematic representation of the coordinates and directions of motion of the electron in its travel between collisions with the N2 molecules. 54 55 The energy E is density n, used second column. E (KeV) presented in the first column with the number in the MC calculation, being given in the (8.0 E+ 14 means 8.0 x 1014) n (#/cm3) 0.1 8.0 E+14 0.3 2.0 E+15 1.0 8.2 E+15 2.0 2.8 E+16 5.0 1.2 E+17 Table 4.1 56 All electrons are forced to be degraded in a 30 cm long cylinder; thus an increase in the density is required for an increase in the energy. There are 10 cm allowed in the negative z direction and 20 cm allowed in the positive z direction. The x and y axes extend to infinity. Some electrons actually escape from the cylinder, but the energy lost due to these electrons is only a few tenths of a per cent of the incident elec tron energy. The path length PT is then given as PT = x In(Rl) (4.2) using the relation that R1 = e T (4.3) Figure 4.2 represents a schematic of the electron traveling and colliding with three N2 molecules. The PTl' PT2' and PT3 are the path lengths traveled by the electron between the initial coordinates and the first collision, the first and second collisions, and the second and third collisions, respectively. The xa, y,, and za coordinates at this collision can now be found from PT, xb, Yb' Zb, eb' and %b using Xa = Xb + PT sineb cosCb (4.4) ya = Yb + PT sineb sintb (4.5) Za = Zb + PT coseb (4.6) In Figure 4.2 the coordinates of the first and second collisions are represented to illustrate how the electron's direction of motion might change during its collisions with N2. So far emphasis has been only on 57 the Cartesian coordinates. Now, calculate the azimuthal angle *a and the polar angle ea of the electron after a collision. 2. Second and Third Random Numbers, R2 and R3 In actuality the type of collision must be specified before the scattering can be calculated. It is assumed, however, that the type of collision is already known (see subsection IV.C.4). The second, R2, and third, R3, random numbers are not chosen if the collision is an excita tion event and E is greater than 100 eV. They are chosen for all other collisions. The R2 is used to calculate the azimuthal scattering angle, 0', of the electron from its direction of motion. The premise is that the azimuthal scattering is isotropic; therefore, = R2 2i (4.7) (Note that the 0' angle is the only angle not represented in Figure 4.2. Inclusion of 0' adds too much complication to an already cluttered figure.) The third random number, R3, is employed to calculate the polar scattering angle e' of the electron from its direction of motion. (The angle e' is represented twice in Figure 4.2: Once as the scattering due to the first collision and once as the scattering due to the second collision.) For elastic collisions, Eq. (3.1), (3.12), (3.13), or (3.14) are used in determining o'. In all but one of these phase functions, an analytic expression can be used to determine e' from the random number, R3. These analytic expressions are given below. 58 Using the screened Rutherford differential cross section form (see Eq. (3.1)), it follows that e' = cos1 [1 + 2n 2n(+n) ] (4.8) For model 1 (see Eq. (3.12)) = cos1 [ 1 + 1 + a] (4.9) R3[(2 + a) a ] + a and for model 2 (see Eq. (3.13)) S= cos1 [B B2 4AC (4.10) 2A with A=R3 + f (1 f) 3 a[(2 + a)l a"1] (2 + c)[(2 + c) c f (1 f) B =A(a c) + + (2+c f) [(2 + a)1 a1] [(2 + c)1  and C =A(l +a)(l +c) + f(l +c) (l 1 + [(2 + a)" a ] [(2 + c) c ] Model 3 (Eq. (3.14)) is not so easy to write in such a convenient form. The equation for primary scattering after an ionization event (Eq. (3.17.)) is, also, not easily inverted. For these two differential cross sections, the following approach is taken. The angular range from 00 to 1800 is divided up into angular intervals. A certain probability for scattering at angles less than the end of each angle interval is calculated from the differential cross 59 section form. The angle e' is then found through the correct placement of R3 into an angular segment whose beginning and ending point scattering probabilities bracket R3. For this work twentyfour angular segments were chosen. Their end points are given in Table 4.2. With twentyfour angular intervals, the results from the Monte Carlo calculation came out to be the same as with the use of forty angular intervals. If sixteen or even twenty segments were used, the MC computation gave results that were 5% to 10% different. The 0' and e' are not the scattering angles from the original coordinate system, but represent the azimuthal and polar scattering of the scattered electron from the direction of travel of the incident elec tron. In order to calculate 0a and ea, the azimuthal and polar angles representing the motion of the electron after the collision, some spheri cal trigonometry must be used. The following relations hold in this transposition: cOSAa = [coseb cosb sine' cos' sin b sine' sine' + sineb cos^b cose']/sinea (4.11) sin + cosb sine' sin(' + sineb sineb cose']/sinea (4.12) ;' = cos (cosia) (4.13) cosea = coseb cose' sineb sine' coso' (4.14) sinea = l cos ea (4.15) a a 60 Twentyfour angle intervals are given here that were used in the Monte Carlo calculation. First column lists the index of the segment and the second and third columns give the begin ning and end points for each segment with units of radians (degrees). Index Beginning End 0.00 0.01 0.05 0.11 0.20 0.40 0.60 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.07 (0.00) (0.57) (2.87) (6.30) (11.46) (22.92) (34.38) (45.84) (51.57) (57.30) (63.03) (68.75) (74.48) (80.21) (85.94) (91.67) (103.13) (114.59) (126.05) (137.51) (148.97) (160.43) (171.89) (175.90) 0.01 0.05 0.11 0.20 0.40 0.60 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.07 3.14 (0.57) (2.87) (6.30) (11.46) (22.92) (34.38) (45.84) (51.57) (57.30) (63.03) (68.75) (74.48) (80.21) (85.94) (91.67) (103.13) (114.59) (126.05) (137.51) (148.97) (160.43) (171.89) (175.90) (180.00) Table 4.2 61 and ea = cos1(cose ) (4.16) Now the azimuthal angle *a and the polar angle ea have been established for the collision with respect to the fixed coordinate system. These angles are also represented in Figure 4.2. The two angular coordinates Ob and eb of the electron before traveling to the next c61ision are then set as Ob = Oa and eb = ea. 3. Fourth Random Number, R4 A fourth random number, R4, is required if a secondary is produced and if that secondary has an energy above the cutoff energy, E This R4 is chosen to determine the polar angle, e', of scattering of the secondary. Again, an analytic formula can be employed to define e'. This equation was derived from Eq. (3.18) and is written as 1 C 1 v(1 +cose ) e' = cos [ tan [R4{tan ( C 1 ((1 cose ) tan C} 1 (l coseo) + tan ( C )] + chose ] (4.17) The 0' for the secondary is found with the use of Eq. (4.7) and ea and 0a result from the use of Eqs. (4.11) through (4.16). 4. Fifth Random Number, R5 The fifth random number, R5, determines the type of collision that occurs. Here, the type may be either elastic or inelastic. If the 62 type is inelastic then the individual excitation or ionization event is found as well. There are cross sections for thirtyfour states of N2 employing the papers of Jackman, Garvey, and Green (1977) and Porter, Jackman, and Green (1976). Using all these states in the MC calculation would greatly increase the cost. It was therefore decided to reduce these thirtyfour 1 1 + states to nine states. Two allowed states, the b 1 and the b' 1 , and the six ionization states were kept the same as given in the papers. For the ninth state, all the Rydberg and forbidden states were combined. Above 200 eV, the forbidden states are contributing only a minuscule amount to the total cross section. Since the other states have roughly the same In E/E falloff at high energies, it is assumed that the pro babilities for excitation to any of these states will be constant. These probabilities were simply found from the ratio of the cross section of the state in question to the total inelastic cross section at the elec tron energy of 5 KeV. In Table 4.3 these states, their probabilities, and thresholds are presented. The probability, pc, of the composite state is simply m P,= Pi (4.18) i=l where m = the total number of Rydberg and forbidden states and pi is the probability for excitation of the ith Rydberg or forbidden state. The average threshold, W for exciting the composite state is found easily with the following equation m Wc = Pi (4.19) C i=l 63 Table 4.3 N2 inelastic states, their probabilities, p, and thresholds, W, taken for electron energies above 200 eV are presented below. State p W (eV) N2 bl u 0.092 12.80 N b'I 0.042 14.00 N2 Composite 0.233 15.40 N2 X2 0.289 15.58 2 g N2 A2ru 0.127 16.73 N+ B2 + 0.066 18.75 2 u N2 D2w 0.044 22.00 N+ C2 + 0.044 23.60 N2 40 eV 0.063 40.00 N 40 eV 0.063 40.00 64 with Wi being the threshold of the Rydberg or forbidden state. Below 200 eV, the probabilities for excitation to the various inelastic states are changing quite rapidly. The parameters for the eight individual states are taken from Jackman et al. (1977b) and Porter et al. (1976). The composite state's properties are found in the same manner that they were above. In these lower energy regimes the probability and energy loss are changing fairly rapidly, thus Table 4.4 illustrates these probabilities and threshold values at several energies. With the background on the inelastic cross sections and their subsequent probabilities, consider now the collision type. The R5 random number determines the type of collision that occurs in the following manner: If aTE(E) Rg aT(E) for all electron energies (4.20) where cTE(E) is the total elastic cross section, then the collision is elastic. If OTE(E) plaTI(E) + OTE(E) aT(E) Rs a (E) T and E > 200 eV (4.21) aT(E) where aTI(E) is the total inelastic cross section and p, is the proba bility for exciting the first inelastic state (in Table 4.2 the first state is the b lu thus p, = 0.092), then the inelastic collision results in the excitation of the first state. A relation follows from Eq. (4.21) that holds true for j = 2 to 9 such that: If 65 Table 4.4 N2 inelastic composite ability, p, and average energies between 2 and state with its characteristic proba energy loss, W, given for several 200 eV. E (eV) p W (eV) 2 3 4 5 6 7 8 9 10 12 14 16 18 20 30 40 50 60 70 100 150 200 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.971 0.866 0.745 0.426 0.344 0.296 0.271 0.255 0.229 0.214 0.234 0.57 1.03 0.922 0.835 0.772 0.728 0.696 7.00 7.21 8.25 8.91 9.12 9.34 9.68 11.70 12.80 13.30 13.70 13.90 14.30 14.60 14.80 66 SpiTi(E) + TE(E) J PiaTI(E) +TE(E) iR < R =1 and E > 200 eV (4.22) oT(E) 5 aT(E) then the inelastic collision results in the excitation of the jth state. Thus the R5 random number for an electron of energy E > 200 eV will determine which type of collision occurred when satisfying Eq. (4.20), (4.21), or (4.22). For energies below 200 eV, the following relations must be con sidered: If aTE(E) ol(E) + aTE(E) T < R (5 aT(E) and E f 200 eV (4.23) where ol(E) is the cross section for exciting the first inelastic state, then the inelastic collision results in the excitation of this state. A relation similar to Eq. (4.22) can now be established for j = 2 to 8 such that: If j1 J 0i(E)+oTE(E) aii(E)+o(TEE) i=1 < R5 < i=E) and E < 200 eV (4.24) aT(E) 5 aT(E) then the jth inelastic state is excited. If 8 I oi(E) + aTE(E) Rg > T(E) and E < 200 eV (4.25) then the excitation of the composite state is assumed and the energy loss, Wc, in this case is found through a linear interpolation with the use of the values given in Table 4.3. 67 5. Sixth Random Number, R6 The sixth random number, R6, is computed only if the collision type is an ionization event. This R6 determines the energy lost by the primary in creating a secondary of energy, Ts. Using the S(E,T) from Eq. (3.19) the following relationship is established: T S I S(E,T) dT R6 0 (4.26) 6 oIoN(E) Integrating the numerator in Eq. (4.26) and using Eq. (3.21) to solve for Ts, Eq. (4.27) is derived. Ts = r(E)[tan{R6tan[(TM To(E))/r(E)] + (R6 1)tan1[To(E)/r(E)]}] + To(E) (4.27) The energy loss, W, is then found by the relation: W = Ik + Ts (4.28) where Ik is the ionization threshold for the kth ionization state. 6. Multiple Elastic Scattering Distribution Used Below 30 eV The MC calculation can be used to degrade an electron down to practically any energy. Even below the lowest threshold for excitation to any vibrational level, the electron will still lose energy via elastic collisions with molecules of nitrogen as well as other electrons. This energy loss to other electrons is fairly low unless a substantial frac tion of the gas has been ionized (see Cravens, Victor, and Dalgarno, 68 1975). In this study the fraction of ionization is assumed to be negligible; therefore, this loss is ignored. Unless there is a very large amount of money available for computer time, an electron can not be followed to its thermal energy with any practicality. This implies that a multiple elastic scattering distri bution (hereafter referred to as MESD) must be used below some given energy. In this work the MESD will be used below 30 eV. Bethe, Rose, and Smith (1938) used the FokkerPlanck differential equation, neglecting energy loss, to consider the penetration of elec trons through thick plates. This, however, leads to a Gaussian solution in the smallangle approximation so that the tail of the angular dis tribution was omitted. The largeangle multiple scattering has been studied by Goudsmit and Saunderson (1940) [hereafter referred to as GS] who used a series of Legendre polynomials to determine the resultant angle of scattering. Lewis (1950) studied the integrodifferential diffusion equation of the multiple scattering problem in an infinite, homogeneous medium, without the usual smallangle approximation. He obtained the GS solu tion for the scattering angle and also derived certain moments for the longitudinal and transverse distributions. Berger (1963) applied a MESD for condensed case history MC cal culations. His application of the MESD is at the energies above 200 eV and probably is not accurate for electrons with energies less than about 100 eV. Furthermore, Berger's (1963) work contains approximations that are only good for the sharply forward peaked cross sections of higher energy electrons. 69 In this work a different problem exists. The MC calculation is used to degrade electrons in a collision by collision manner all the way down to 30 eV. At this energy, the elastic collisions are occurring with twice the frequency of the inelastic events, and at energies below 30 eV the number of elastic collisions between inelastic events may be up to several hundred or thousand. Keeping track of all these elastic collisions would be very costly. Kutcher and Green (1976) [hereafter referred to as KG] studied the radial, longitudinal, and polar angle distributions for elastic scatter ing by H2 in the energy range from 2 to 50 eV. An approach similar to KG's could be applied to N2. Since such a project would require a substantial amount of time and computer money, the possibility of adapt ing the KG results was first considered. With this in mind, consider the differences between N2 and H2. First of all, there are some dissimilarities between the differential cross sections. There is more backscatter observed experimentally in N2 at all energies. Secondly, the total inelastic and elastic cross sections are different. The second difference is no real problem because the MESDs are given in terms of the mean free path lengths (hereafter referred to as MFPs). The first dissimilarity does pose a minor problem which is solved in a simplistic way below. Above 5 or 6 MFPs the polar angle is approximately random. At most energies below 30 eV, the number of MFPs between inelastic collisions is above 5 or 6. Since the distribution found in KG is not easily in verted, a reasonable assumption is that the polar angle is oriented randomly. 70 Knowledge of the radial distribution is not crucial for our pur poses. The most interesting radial distribution output from this MC calculation is that of the 3914 A emission. Electrons below 30 eV make little contribution to this profile because the cross section for ex citation to this N2 B2 state is fairly low (see Figure 5.1). Thus knowledge of the radial distribution of these electrons multiply scat tered is not extremely important. An approximation, however, is employed in most MC computations to calculate a fairly reasonable radial distance. The average radial distance, as observed from the calculations in KG, for most energies and at the longer path lengths is approximately onesixth of the total path lengths, thus Pave = s/6 (4.29) The most important spatial displacement is the longitudinal dis tance z. In order to calculate z, the total path length s must be known. This length s is calculated from the random number, R1, the total elastic cross section, oTE(E), and the total inelastic cross section, oTI(E), by using oTE(E) OTI) The ratio oTE(E)/oTI(E) is simply a fairly accurate approximation of the number of elastic collisions occurring per inelastic collision. The value In(R1) [see Eq. (4.2)] is the path length (in units of MFPs) traveled by the electron between collisions. Thus knowing the number of elastic collisions occurring and the path length traveled between collisions allows one to write Eq. (4.30) as the expression for the 71 total path length s (in units of MFPs) traveled between inelastic collisions. rn KG an equation which can be easily inverted to calculate the z distance (in units of MFPs) from some random number, R2, and path length s, is written [R2/ 1] z = ln [F(O)/v 1 (4.31) u where v(s) = 1 exp[(s/sV)D] F(0) = K{1 exp[(s/s f)075]} and u(s) = (H + s )/s where K = 0.425. Since there is more backscatter during N2 elastic collisions (because of its differential backscatter contribution), it seems reason able that the parameters for Eq. (4.31), which are useful for N2, are different than those derived in KG. One approach to this dilemma might be to correlate the elastic differential cross section (hereafter called EDCS) from N2 at some energy E' with the EDCS from H2 at some energy E. This would work if the H2 EDCSs showed more backscatter than the N2 EDCSs; however, the opposite is observed experimentally. Thus the N2 EDCSs from some E' (around 67 eV) values correlate with the H2 EDCSs at E values less than 2 eV (where the Kutcher and Green, 1976, MESD is not defined). 72 Another straightforward and simplistic approach is to do the following. Calculate the approximate backscatter at three energies, the two endpoints and the middle (2 eV, 15 eV, and 30 eV), from the KG H2 EDCS form and the experimental data on N2 EDCSs (given in Sawada, Ganas, and Green, 1974). At these energies the backscatter with the KG H2 EDCS form is less than that of the N2 EDCS by the following values: 2 eV t 5%, 15 eV 10%, and 30 eV % 10%. An average of these three values is about 8%. Since the major influences of the backscatter in Eq. (4.31) is the value of K, this parameter is the only one that is changed from the KG formulation. It is, therefore, increased by "8% so that in these MC calculations K = 0.46. The other parameters in Eq. (4.31) are listed in Table 4.5. Actually it appears that the value of K makes little difference in the MC computational results. Two MC calculations at an incident elec tron energy of 100 eV with K = 0.46 and with K = 0.425 were undertaken (all other parameters and inputs were the same). The yield spectra (described in Chapters II and VII) changes substantially only at fairly large longitudinal distances (where the distances are about 1.5 times the range). At these large distances there are relatively few electrons anyway, thus there is little effect on the major aspects of the spatial electron energy deposition process. The Cartesian coordinates xa, Ya, and za are found from the coor dinates xb, Yb, and zb in the following manner. After z is calculated in units of MFPs with the use of Eq. (4.31), it can then be written in units of cm or km by multiplying by the MFP, x (calculated from Eq. (4.1)), thus za = zb + zX. 73 Table 4.5 Parameters from Kutcher and Green (1976) for several energy intervals used in Eq. (4.28). Energy Interval (eV) H I J D s5 SF 25 12. 1.37 1.71 1.75 5.05 8.5 510 9.6 1.32 1.67 2.50 4.25 8.5 1020 15.5 1.28 1.67 2.31 6.29 10.3 2030 23.5 1.24 1.69 1.98 9.65 13.6 74 As established earlier, the polar angle ea and azimuthal angle a' representing the motion of the electron after the collision, can be chosen in a random way from the two random numbers, R3 and R4, using a = rR3 Ca = 2rR4 (4.32) A reasonable approximation of xa and ya can then be made using Eqs. (4.29) and (4.32) such that Xa = Xb + ave A cos a and a = Yb + Pave x sin a In the MESD the fifth random number, R5, is used to determine the inelastic collision type. A method similar to that illustrated in sub section IV.C.4 is employed, the only difference is the fact that the collision is only inelastic. 7. Value of the Cutoff Energy, 2 eV The Ec used in this work has been set at 2 eV because the lowest threshold for excitation to an inelastic state is 1.85 eV. With this cutoff energy the yield spectra can be defined down to 2 eV at all longitudinal distances. Subsequently, a reasonable calculation of the excitation to any N2 state may be made. 75 D. Statistical Error in the Monte Carlo Calculation The statistical error inherent in the MC computation can be derived by considering the following. Since the MC calculation is a probabilis tic method of degrading an electron in energy, the multinomial distribu tion can be used to find the statistical standard deviation for each bin considered. This discussion of the statistical error employed the work of Eadie, Dryard, James, Roos, and Sadoulet (1971). The probability of getting an excitation of a certain state j in bin k is Pjk. The pjk is normalized such that m n I Pjk = 1 (4.33) k=l j=l In this MC study the multinomial distribution is an array of histograms containing N events distributed in n states and m bins with rjk events in state j and bin k. The rjk values are normalized such that m n I I rjk = N (4.34) k=l j=l Thus, the rjk observations can be considered somewhat conditional on the fixed observational value of N. The variance of the calculation is represented as V(rjk) = N Pjk (1 Pjk) (4.35) In this work the m x n variables rjk can all be correlated. For the specific example of electron deposition represented in Figure 5.2, Pjk << 1. This is true because there are total almost 5 x 105 col lisions (i.e., N = 5 x 105) to consider in this degradation scheme and 76 at maximum rjk t 4000. Using this information, Eq. (4.35) can then be approximated by V(rjk) n N Pjk rjk (4.36) and the statistical standard deviation of the number of N B2 E events 2 u in a bin becomes a O WF (4.37) jk jk Equation (4.37) holds true for the specific example represented in Figure 5.2 and it also holds true for all the intensity plots, energy loss plots, and yield spectra that were studied in this work. Thus, in order to obtain the approximate standard deviation for any MC generated number, the square root of this value is its standard deviation. The error bars found in the rest of this paper are determined in this manner. Now that the MC calculational technique has been outlined, this method will be used in the next three chapters to deal with the spatial and energetic aspects of electron energy degradation. CHAPTER V MONTE CARLO INTENSITY PLOTS AND COMPARISON WITH EXPERIMENT Incident electrons with energies between 0.1 and 5.0 KeV are de graded in N2 using the MC method described in Chapter IV with the cross 0 sections given in Chapter III. The intensity plots of the 3914 A emission are described in this chapter. Emission intensity plots of the 3914 A radiation from the N2 B Z state are used in describing the range (found by extrapolating the linear portion of the longitudinal 3914 A intensity plot to the abscissa) for incident electrons. Section V.A describes the excitation of the N2 B2 E state. In section V.B the range of the electrons is defined more completely. Previous experimental and theoretical work on the 3914 A emission of N2 is given in section V.C. The range results from the MC calculation are then discussed in section V.D. Finally, section V.E describes the intensity plots resulting when plotted as functions of the radial direction. A. Excitation of the N2B 2E State The main concern of this chapter will be the intensity plots showing the emission of the 00 first negative band (B 2 state) of N2 at 3914 A. Experimentally (see Rapp and EnglanderGolden, 1965; McConkey, Woolsey, and Burns, 1967; and Borst and Zipf, 1969), it has been shown that the number of photons at 3914 A produced for each ionization of N2 is 77 78 independent of the energy of the exciting electron for energies from 30 eV at least up to 3 KeV. In Figure 5.1 the N2 total ionization cross section and cross sec tion for ionization and excitation to the B 2 state of N are presented. The curves are approximately parallel thus even if the absolute values for the two cross sections are slightly in error, the shapes of the in tensity plots that result from this MC calculation should be fairly accurate. The total ionization curve lies nicely in the middle of an array of experiments (namely, Opal, Beaty, and Peterson, 1972; Tate and Smith, 1932; Rapp and EnglanderGolden, 1965; and Schram, de Heer, Wiel, and Kistenaker, 1965) but the B 2 cross section values may be high when u compared to experiments (see Holland, 1967; and McConkey, Woolsey, and Burns, 1967). The threshold for excitation to this B 2 + state is 18.75 eV, thus u any electron above that energy can excite and ionize a ground state N2 molecule up to this level. The cross section for excitation and ioniza tion to the B 2+ state is not large when compared with the total in u elastic cross section. In fact, the probability for exciting this state is only 0.066 for electron energies above 200 eV. The accuracy of the MC calculation is dependent on the number of excitations in each bin (see section IV.D). In order to enhance the precision of the MC results, excitations of the X and A states of N are added to the B g u 2 u excitations. The ionization cross sections for these two states are found to be proportional to the B 2 E state for electron energies above 30 eV. 30 eV. 79 wb b ci' IO . 101 Figure 5.1 101 E(eV) Total loss function L(E) from N2, denoted by the solid line; total ionization cross sections for NZ, denoted by the dash dot line; and the N B zy cross section, denoted by the dashed line, are given as functions of energy, E. I103 2 10 > 0 ) CM L 0 i0 104 N 0< 102 80 Previous workers (Barrett and Hays, 1976; Cohn and Caledonia, 1970; 0 and Grun, 1957) have used the 3914 A emission as a measure of the energy deposited. In these works it is assumed that since the 3914 A radiation is proportional to the number of ionizations in a given volume and if the number of ionizations is proportional to the energy deposited in that volume, then the 3914 A intensity is proportional to the energy deposited in that volume. These experimenters, therefore, measured the 3914 A radiation at several energies, extrapolated their intensity plots to find a range (to be described in section V.B), and derived an empiri cal expression for the range that could be used to find the energy loss function. This idea of using the 3914 A emission to derive the energy loss scheme is useful for energies above 2 KeV. In Figure 5.1, compare the loss function, L(E), used in this work and the N2 B 2u state cross section. The two curves are not parallel below 2 KeV. This implies that the energy loss function can not be derived directly from the range results at incident energies below 2 KeV. The energy loss plots from this MC study are given in section VII.A and more will be discussed in that section about them. B. Range of Electrons The concept of the mean range must be defined next. For each monoenergetic primary electron impinging into a gas, a range can be calculated. In general (at least above 100 eV), the higher the electron energy the further the electron will penetrate into the medium. If an electron is incident along the zaxis, the excitations of the N2 B 2E 81 state can be graphed in an intensity plot with the zaxis as the abscissa. In Figure 5.2, the intensity plot from 5000 incident 1 KeV electrons is graphed (the model used in this MC calculation should only be taken as an illustrative example) in histogram form. Bins along the zaxis are taken to be 0.5 cm in width for these incident electrons. The linear portion of the curve may be extrapolated, as illustrated by the dashed line, to define a mean range of the beam. All the intensity plots are normalized in this paper so that the beam starts out at z = 0 cm along the zaxis. The intensity in Figure 5.2 seen at negative values of z is brought about by backscattered electrons. The error bars given near the peak of the histogram are found simply from a method described in section IV.D. From Figure 5.2, the range is seen to be 16 cm for these 1 KeV electrons. Range values, R in units of gm/cm2 are written Rg = Rc (5.1) where R is the range in cm, p = n MN2 (in gm/cm 3), n is the number density of N2 molecules (in #/cm ), and MN is the weight (in gms) of an N2 molecule. In this case, n = 8.2 x 101 molecules of N2/cm MN2 = 4.651 x 10"23 gm/N2 molecule, and R = 16 cm; therefore, Rg = 6.06 x 106 gm/cm2 C. Previous Experimental and Theoretical Work on the 3914 A Emission of N GrUn (1957) measured for air the total luminosity of the 3914 A radiation in planes perpendicular to the axis of the electron beam with N C C 0  * o 5 E E 0 1MID U3 C 0) S  U U)VIcn 4a) (n 0) *r 0 r U , *r *r a >3 > 0C C .C OC r in r 0 0 VI to *r W a C Q C 0r *r *r 0 " 0cn L0 a) , u S oC C) 4) M10 ***r *r3* C Cr 4) (n 4( 0) C4)C  a) 0 *r 0 CCO 4; *Dr S a c C) C S0 0> C r j= ( ) 0 t 0 Cl r C LL. V i Z CQ 83 0 (D co N 0 OD I CN IC)x Sol) Nuuau 0'I 84 an initial energy of 5 to 54 KeV. Cohn and Caledonia (1970) measured intensity profiles of electron beams with incident energies from 2 to 5 KeV impacting into N2. Barrett and Hays (1976) then extended the incident electron range down to 300 eV by measuring the radiation pro 0 files of 3914 A resulting from electron beams with energies from 0.3 to 5.0 KeV impinging on N2. Spencer (1959) used the Spencer and Fano (1954) method of spatial energy deposition and found good agreement between his energy loss plots and the 3914 A intensity plots of Grin (1957). The Berger, Seltzer, and Maeda (1974) [BSM] MC calculation provided energy loss plots down to 2 KeV. These plots are also in fairly good agreement with the experi ments mentioned above. Comparisons will be made in this paper between the available experi mental electron energy loss data and the MC calculations done here. Since this MC calculation follows the incident electrons, as well as its secondaries and tertiaries down to 2 eV, this MC computation is one of the most detailed ever employed for electron impact energy degradation. It is, therefore, of interest to compare the results from this study with experimental results for incident electrons with energies from 300 eV up to 5 KeV. D. Range Results and Longitudinal Intensity Plots from the Monte Carlo Calculation Range data at several incident electron energies are calculated with the use of the screened Rutherford and the model 3 differential elastic cross sections. The screened Rutherford model is used because it is the most widely used form for elastic scattering in theoretical 85 studies and, also, because BSM were quite successful in using this form down to incident energies of 2 KeV. Model 3 was used because of its very close agreement with experimental differential cross section data in the range from 30 eV up to 1 KeV. Table 5.1 presents the range data (for perpendicularly incident electrons) from three different experiments, the theoretical calculation by BSM, and two sets of theoretical computations from this study. The values in parentheses from BH (Barrett and Hays, 1976), CC (Cohn and Caledonia, 1970), and G (GrUn, 1957) are simply calculated from the empirical formulae given in these works. For the rest of this chapter, the results of this work will be com pared with those of BH. This is the most recent experimental study and is probably the most reliable experimental work. Theyalso use the same incident electron energy regime as that employed in this work. In Table 5.1 it is apparent that the BH values have the largest ranges of the experimental studies. The two separate MC calculations in this study seem to bracket the BH results at all energies. The model 3 ranges are consistently larger than those of BH. These results are 10% higher at 5 KeV and about 19% higher at 0.3 KeV. The screened Rutherford ranges, on the other hand, are about 9% lower at 5 KeV and about 10% lower at 0.3 KeV. If it can be assumed that the BH results are indeed the most re liable data, then the following conclusion can be made: The screened Rutherford phase function scatters the electron too much while the model 3 phase function provides too little scattering. This conclusion is made assuming that the total cross sections described in Chapter III are fairly accurate. 86 Table 5.1 Range data (in 106 gm/cm2) at several energies, E (in KeV), are given below. The second column M3 (model 3), third column SR (screened Rutherford), fourth column BH (Barrett and Hays, 1976), fifth column CC (Cohn and Caledonia, 1970), sixth column G (GrUn, 1957), and seventh column BSM (Berger, Seltzer, and Maeda, 1974) range values are presented. E (KeV) M3 SR BH CC G BSM 0.1 0.37 0.34 (0.53) (0.07) (0.08)  0.3 1.25 0.95 1.06 (0.51) (0.56)  1.0 6.45 5.57 5.72 (4.17) (4.57)  2.0 18.6 16.8 17.7 14.0 (15.4) 15.2 5.0 91.5 75.9 83.0 69.7 76.4 71.9 87 In this work model 3 is the result of a careful investigation of the detailed molecular nitrogen cross sections. Therefore no attempt will be made here to change the cross sections compiled in Chapter III. Model 3 will be used in most of the MC calculations in the rest of this chapter and also in Chapter VII (BSM have, however, chosen n used in the screened Rutherford cross section, to be a constant value whose value was selected so as to obtain the best agreement between their MC calcula tion and the experimental results of G and CC). In Table 5.1 the importance of the elastic phase functions is clearly illustrated. Up to a 25% change in the range is observed when com paring the screened Rutherford with the model 3 phase functions. More elaboration on the effects of various phase functions on the energy deposition process will be given in Chapter VI. Figures 5.3 and 5.4 give intensity plots for the 3914 A radiation resulting from 2 KeV and 0.3 KeV incident electrons, respectively. The experimental work of BH and the calculations using model 3 and the screened Rutherford are presented in these figures. The shapes appear to be somewhat similar; however, the BH results at both energies pre dict a range that is between the two theoretical calculations. E. Intensity Plots in the Radial Direction Most attention, so far in this study, has been concentrated on the intensity plots in the longitudinal direction. There is experimental data available on the intensity of the 3914 A radiation as a function of p (the axis perpendicular to z). Experimentally, G, CC, and BH all present data of this type. >E 4) 0) () r *O M Enu0 rO 3I MB *r 0 L ) 4) +J +4) S0 S (A 03 .C C 0 c4 ( 0 4 0 Q)  0) 0() s n or )C .C S SL.S u0 O 4. S. *r .c 0410 m X c(d0 * (J r0 C ) C SC > r *r U SI U r *r S4  S3Cr r, CO Un 0 o al 0) S 4J 0 s 41. 4.c: + 4 ) *i + .I. 4. 'S C 0 a 0 C) 0 5 C ) CL.C 4 U tO t *.r L L. r S 3 4S JC CCO e 0' U .'r i .4 "0 E o W "V 3 0 EO 0 4> S 0) 41 c >1 L r 4 E C C r 0 0) o 0 0) 4 C n LS. wn S. > o 04 0) U0 *r 49r 3 % U) +3 0 r(EC U r a3 u 4C 0) (4 Q..C= > r * L.S .C Q. S4 M 4 04Jl  r I0 > r IA (v t # 44 to ) a) 1 S n E C4 I *r 0 CC 04 O r 1 0 0 E4 4 ) C 4 *r *r C S '0 0 C4) S. 0) 0) r r * 00U in ..C .C = Q34 C 0. 0D 4) (n 0 U CX$.=io00wI <* 4 O0 0.C E in 0) 03 *r u. LL ECLr~ 89 OW 00 0 c0 N~ 0 cc N, 0t c ' b u 0 0 (3:01O) I4!su;94ul V V16 2 0 10 >E O E 4 S 0) n S. r 4J S A 0 U U  *r 4.O 4 *r C . C n S 4r r_ 01.C o >i .a u(fC G "OnCOt C. QW 4J U L W 0 ' S (Au C c l ".'O 0 r ~5. (a * C 0 Q.r U S. to LC ( 4J W Ut +. *r 0) S o0 X r.C S. X r C a) 0 4 sL. 0 o. u( m) < U c *r Hc t 0 *r r U tJ S. flo r t CO 4.COC) c1 4C 30 Av a) *r "0 > o) o)r 4 4 C 0 Ir C L0 L 4~ 4) *0 4 U 0 4) ( o a) u 0 L S C *r S. 4J 4 O " 1Q *r u 4 r r C W c *4 O "L #3 '' 4 (0o a'4 0 C x C 'u rcno "a w 'C O O+ C y." 0I  *r 0 L e r Sr E o *0 0 a 0)  4E =. 4 0 ) 0t 40 t .. Ul 0 I* r C fO 4 r r0 XC o o 01. * >r >"a o oE a) r ).C 43 r 0 4. 40 )V C (A r S to Sr 0 4 CO S O r C S N 4 C. 0) 0 E 4' 1 4 O 4J *r *fr c S. o * *r U 4U) X= X 0 c C. (U +1 to aj c C 3 X I I X= (a .I LL. 91 0 00 ct t cli ro cQ I I I Afls u V V1610 ~ 0 92 This study uses the experimental data of BH as a comparison with the results of this study. The next three graphs, Figures 5.5, 5.6, and 5.7,portray sample results for incident electrons with energies 5.0, 1.0, and 0.3 KeV, respectively. The z and p values given in these three figures are in units of fractions of the total range. Fairly good agreement between the MC calculation (using model 3 cross sections) and the experimental work of BH and Barrett (1975) is observed at all three incident energies. The largest differences be tween the two sets of data are noted at 0.3 and 1.0 KeV. For the 1.0 KeV case, the MC calculation tends to predict more in tensity at the lower values of p for z values of 0.3 and 0.4. A similar result is apparent for the z values of 0.36, 0.48, and 0.60 for an energy of 0.3 KeV. At a z value of 0.12, however, the experimental data tend to predict more intensity at all values of p. Two conclusions can be drawn from these comparisons, if it is assumed that the experimental data of BH and Barrett (1975) are correct. First, the cross section for excitation to the N2 B may be under estimated in the energy regime between 0.3 and 1.0 KeV. Raising this cross section in this energy regime could bring about an increase in the intensity observed early in the electron's degradation process with a subsequent decrease in intensity later in the electron's degradation process. Second, more scattering from the elastic collisions would help to reduce the total intensity at low p values and raise it at the higher p values. The screened Rutherford differential cross section has more scattering than model 3. Use of this set of cross sections in the MC calculation did result in a little better agreement at 1.0 KeV, but only about the same type of agreement at 0.3 KeV. 0"O a)*a O r O s.. .5 ) 0 0 * So MC) 01 4 a aQ *O C o c 4nJ 4 " 4 r 0 C N EO 5. to V) go 4 l O ) r N 0 S 0) . Osl oC E 4a x 4O 44) 0 UU 5. 0. 3* s5 + o + o0 4 > *r C U ) .S 4) 4. () ) 4 4) r d) C 4 O 0) CU) S. r .C ' r X S4) U, in ). *r 94  6o ('X3 01) Aiusualul (D 0 0 o (M d O o d " .c *. V t16 0 