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 MONTE CARLO SIMULATION OF NONLINEAR RADIATION 
 INDUCED PLASMAS 
 
 by 
 
 Benjamin Shaw-hu Wang 
 
 May, 1972 
 
 IH6: imm* <X f* e 
 
 JUL 5 1972 
 
 .IVERSmf OF ILLINOIS 
 
 UBmmGkMmmtij 
 
MONTE CARLO SIMULATION OF NONLINEAR RADIATION 
 
 INDUCED PLASMAS 
 
 BY 
 
 BENJAMIN SHAW-hu WANG 
 B.S., National Taiwan University, 1960 
 M.S., University of Missouri (Rolla), 1964 
 
 THESIS 
 
 Submitted in partial fulfillment of the requirements 
 
 for the degree of Doctor of Philosophy in Computer Science 
 
 in the Graduate College of the 
 
 University of Illinois at Urbana- Champaign, 1972 
 
 Urbana, Illinois 
 
MONTE CARLO SIMULATION OF NONLINEAR RADIATION 
 
 INDUCED PLASMAS 
 
 Benjamin Shaw-hu Wang, Ph.D. 
 Department of Computer Science 
 University of Illinois at Urbana-Champaign, 1972 
 
 A Monte Carlo simulation model for radiation induced 
 plasmas with nonlinear properties due to recombination has 
 been developed, employing a piecewise linearized predict- 
 correct iterative technique. Several important variance 
 reduction techniques have been developed and incorporated 
 into the model, including an antithetic variates technique 
 (negatively correlated variates) which has proven to be 
 most useful. This approach is especially efficient for 
 plasma systems with inhomogeneous media, multidimensions , 
 and irregular boundaries, where other analytic or numerical 
 solutions are either unrealistic or impractical. This model 
 is quite general in scope and should be applicable to a 
 variety of similar particle transport and diffusion-type 
 plasma problems . 
 
 The Monte Carlo code developed in this work has been 
 applied to the determination of the electron energy dis- 
 tribution function and related parameters for a noble gas 
 plasma created by a-particle irradiation. The radiation 
 induced plasma involved was characterized by the following 
 features : 
 
 1) a continuous internal volume-source of high energy 
 electrons 
 
2) an applied electric field (optional) 
 
 3) secondary electron production via ionization pro- 
 cesses 
 
 4 ) losses via recombination and leakage across boundaries 
 
 The calculations were specifically carried out for a helium 
 
 14 22 
 
 gas medium with an electron source rate from 10 to 10 
 
 3 
 (#/cm -sec), initial electron energies from 70 to 1000 eV , 
 
 pressures from 10 to 760 tcrrs and E/P (electric field/pres- 
 sure) ratios from 1 to 10 V/cm-torr. 
 
 For the lower source rates, it is observed that, in the 
 zero electric-field case, the low-energy portion of the 
 distribution function is quite close to a standard Maxwellian 
 distribution. However, the high-energy portion (above the 
 ionization potential) of the distributions is a rapidly 
 decaying parabolic-shaped tail which can be crudely repre- 
 sented by a 1/ (Energy) distribution. The addition of the 
 electric field causes the distribution function in the low- 
 energy region to shift towards Smit ' s (or Druyvesteyn' s ) 
 distribution while the high-energy protion essentially 
 
 retains its zero field shape. As the source rate increases 
 
 18 — 3 — 1 
 beyond certain value (roughly 10 cm -sec in the pre- 
 sent case), the low-energy region begins to depart from the 
 standard Maxwellian or Smit distributions and this depar- 
 ture can become quite large. 
 
A major emphasis in this development was placed on 
 computation efficiency. The present computation scheme 
 with its variance and accelerated convergence techniques 
 is thought to be quite efficient compared to alternate 
 approaches . 
 
Ill 
 
 ACKNOWLEDGEMENT 
 
 The author wishes to express his sincere appreciation 
 and gratitude to Professor George H. Miley for his patient 
 guidance and invaluable help throughout the course of this 
 work, without which this study would have been impossible. 
 The author wishes to also thank Mr. Ronne Lo for his most 
 valuable and helpful discussions. 
 
 The author would also like to thank Professor J. N. 
 Snyder, the head of Computer Science Department, and the 
 Nuclear Engineering Program for their financial support. 
 This work was supported in part by the Physics Branch of 
 the AEC Research Division under Grant No. AT(11-1 )-2007 
 and by the NASA-AEC Space Power Office under Grant No. 
 NGR 14-005-183. 
 
 Finally, the author wishes to thank Mrs. Sharon Taylor 
 for her excellent typing of the final thesis manuscript. 
 
IV 
 
 TABLE OF CONTENTS 
 
 CHAPTER Page 
 
 I . INTRODUCTION 1 
 
 A. Statement of the Problem and Objectives . 1 
 
 B. The Importance of Nuclear Radiation 
 Produced Plasmas and the Physical Model . 1 
 
 C. General Approaches and Methods of 
 
 Solution 5 
 
 II. THE MATHEMATICAL MODEL: NUMERICAL SCHEME 
 
 AND PHYSICAL PROBLEM 8 
 
 A. General Description of the Physics of 
 
 the Problem 8 
 
 1. Physical Processes Involved 8 
 
 2. Relation to the Tiansport Equation .. 10 
 
 B. The Stochastic Model - Numerical Scheme . 12 
 
 1. The Geometry and the Coordinate 
 
 System 13 
 
 2 . Tracing Procedure 20 
 
 a. The System Without Electri'c 
 
 Field (E =0) 21 
 
 b. With Electric Field Present 25 
 
 c. Types of Collisions and Cross 
 Sections 27 
 
 3. The Prediction and Correction of 
 Dominant System Parameters 29 
 
 III. VARIANCE REDUCTION TECHNIQUES 33 
 
 A. General Background 33 
 
 B . Importance Sampling 38 
 
 C. Russian Roulette and Splitting 42 
 
 D. Initial Guess 44 
 
 E. Antithetic Variates 46 
 
 F. Double History (Double Processing) 
 Technique 50 
 
CHAPTER Page 
 
 IV. MONTE CARLO SIMULATION CODE 52 
 
 A. Introduction 52 
 
 B. Calculational Procedure and Program 
 Description 52 
 
 V. RESULTS 63 
 
 A. Convergence and Reliability 63 
 
 B. Error Estimation and Analysis 64 
 
 1. Error Analysis 64 
 
 2. Data Filtering 69 
 
 C. Solutions of the Problem 71 
 
 1 . Preliminary Checks 71 
 
 2. The Electron Energy Distribution 
 Function without Electric Field .... 77 
 
 3. The Electron Energy Distribution 
 Function with an Applied Electric 
 
 Field 90 
 
 D. Conclusion on Monte Carlo Results 104 
 
 VI . CONCLUS IONS 108 
 
 A. Prelude ' 108 
 
 B. Summary of Present Work 109 
 
 C. Practical Limitations and the Cures .... 112 
 
 D. Future Extensions 116 
 
 E. Concluding Remarks 119 
 
 LIST OF REFERENCES 121 
 
 APPENDIX 125 
 
 VITA 126 
 
CHAPTER I 
 INTRODUCTION 
 
 A. Statement of the Problem and Objectives 
 
 Two major goals were set for this study. The first was 
 to develop an efficient Monte Carlo simulation technique for 
 solving radiation-induced plasma problems. This required 
 adapting several variance reduction techniques in order to 
 increase the Monte Carlo efficiency. The second goal was to 
 apply the model developed herein to study the characteristics 
 of a noble-gas plasma created by alpha-particle irradiation. 
 
 The major point of interest is the determination of the 
 electron energy distribution in the nuclear radiation induced 
 plasma. This distribution is of direct importance to 
 current experimental studies at the University of Illinois 
 where the possibility of directly pumping* gas lasers with 
 nuclear radiation is under study (6). Recent results from 
 these studies also show that nuclear radiation can be used 
 to enhance the output from some type of electrically pumped 
 lasers (5 ) . 
 
 B . The Impo rt ance of Nuclear Radiation Produced Plasmas 
 and th e Ph ysical Model 
 
 Nucleai radiation can be used in two ways: to directly 
 
 pump the laser, or alternately, where an electric discharge 
 
 *" Pumping" of a laser is defined as the introduction of 
 energy into the lasing medium required for population inver- 
 sion. The electrical current which creates the discharge in 
 the conventional gas laser is the pump in that case. Flash 
 lamps are used for many solid state lasers. 
 
serves as the main pump, to enhance the operation of lasers. 
 The possibility of efficient coupling of lasers and nuclear 
 reactors has generated an interest in the use of high-energy 
 ion beams to pump lasers. To avoid radiation damage problems, 
 gas lasers are generally thought of in this connection. 
 
 There are several important reasons for searching for 
 more direct methods of coupling energy from reactor sources 
 into lasers (8). One is the possibility that the resulting 
 laser may offer unique characteristics, e.g., new frequencies 
 and new methods of modulation and control. Simplified 
 coupling could also offer weight reductions and improved 
 reliability, which may be of advantage in space applications. 
 Also, it may eventually become feasible to simultaneously 
 extract several energy forms, e.g., laser and electrical 
 power from a single nuclear station. 
 
 In most gas lasers the excited states, and consequently 
 the inversion of population necessary for laser operation, 
 are formed directly or indirectly by electronic collisions 
 in a plasma generated by an electrical discharge. However, 
 an alternate way to accomplish this goal is via nuclear 
 radiation which can produce ionization and excitation through 
 both direct interactions and interactions due to secondary- 
 electron production. Given the high neutron fluxes available 
 from nuclear reactors, the particles from neutron- induced 
 nuclear reactions (e.g., a-particles from a Boron coating on 
 the wall of a laser tube) can furnish the nuclear radiation 
 
 i 
 

 source. The description of the transport of both the primary 
 radiation and the secondary electrons, from their point of 
 birth throughout their path in the gas, is a key problem in 
 the quantitative evaluation of the plasma. Therefore the 
 development of nuclear -radiation pumping requires a thorough 
 understanding of the radiation-induced plasma. 
 
 The following physical model is applied to the situation 
 of a radiation-induced plasma. Pertinent quantities of 
 interest include: electron energy distribution spectrum; 
 average electron energy; and rate of energy loss. 
 
 The heavy-charged particles (a-particles) are born in 
 coating and enter an adjacent gas. In the present calcula- 
 tions, the cylindrical geometry of the actual laser tube 
 is approximated by a slab geometry (cf. Figure 1) with two 
 parallel plane boron coatings separated by a gaseous medium. 
 The a-particle radiation continuously creates high energy 
 electrons throughout the gas volume (with energy E ) . The 
 resulting plasma is a weakly ionized, three-component plasma 
 (electrons, positive ions, and neutral gas atoms). It is 
 assumed that the alpha-particle trajectory can be treated 
 analytically using methods such as developed in (7). 
 Further, this reference shows how to obtain the primary 
 secondary-electron (6-ray) energy spectrum. The difficult 
 problem, however, is to follow the thermalization and 
 further secondary production of these primary electrons. 
 
Thus, the Monte Carlo simulation starts off by following 
 these electrons on their life time histories, i.e., by 
 simulating the course of their thermalization. Electrons 
 go through various processes of collisions and scattering, 
 resulting in a complex process of diffusion, energy degrada- 
 tion and secondary production, but finally, they recombine 
 with positive ions or escape from the system , ending their 
 histories. 
 
 With the radiation rates of interest here, the plasma 
 is only weakly ionized and electrons mostly thermalize by 
 collisions with the neutral background-gas atoms. In other 
 words, collisions with ions and other free , electrons can 
 be neglected. In this sense, the problem is simplified in 
 that it remains linear. However, some non-linearity enters 
 in the following way. The recombination process depends 
 on the ion density and spatial distribution. However, the 
 ions are originally formed by ionizations which occur as the 
 electrons slow down. 
 
 Another important characteristic is that the plasma is 
 highly non-equilibrium. The thermalized electrons and ions 
 form an approximately Haxwellian background, but superimposed 
 on this is the high energy tail due to the electrons in the 
 process of slowing. Also, the alpha particles themselves 
 represent a non-equilibrium component. 
 
 In summary, the special features of the plasma system 
 are: 
 
1) Continuous internal volume source. 
 
 2) An applied electric field may or may not be present. 
 
 3) Finite cylindrical geometry — approximated here by 
 slab geometry. 
 
 4) Electron-ion pair production and recombination 
 processes. 
 
 5) The energy range of the electrons range from 
 kilovolts to thermal (eV.). 
 
 6) The system represents a non-equilibrium plasma, 
 but in a steady-state condition (or stationary). 
 
 7) Due to relatively small densities of electrons and 
 ions compared with that of gas atoms, direct 
 interactions between the electrons themselves or 
 electron-ion collisions are neglected. 
 
 C. General Approaches and Methods of Solution 
 
 Possible approaches towaid solutions of these types of 
 problems (particle transport and diffusion) can be cate- 
 gorized as follows: 
 
 1) Analytical solution of the governing Boltzmann 
 transport equation. 
 
 2) Numerical solution of the Boltzmann transport 
 equation. 
 
 3) Direct analog simulation (Monte Carlo). 
 
 4) Mixtures of the above approaches. 
 
 The governing equation for this type of problem is the 
 Boltzmann equation, and most classical methods have involved 
 approximate analytic solutions of it. Solutions are possible 
 only after various approximations and simplifications are 
 made, and such solutions are often very restrictive and/or 
 difficult to extend to actual situations of interest. More 
 
or less the same drawbacks apply to numerical solutions. 
 However, in this case approximations are necessary or the 
 computation time becomes too long, especially as the 
 dimensionality and the complexity of the problem increases. 
 In fact, the computation time increases roughly exponentially 
 with the dimensionality of the problem. Usually the govern- 
 ing partial differential and integral equations (coupled 
 Boltzmann equations) are reduced to finite difference 
 equations, and the requirements for stability and convergence 
 severely limit the maximum step sizes of the independent 
 variables. Further difficulties with the practical applica- 
 tion of such approaches arise because it is often difficult 
 to include realistic boundary conditions. 
 
 Direct simulation by the application of Monte Carlo 
 techniques seems to be the better approach; sometimes the 
 only approach for such problems. It by-passes the direct 
 solution of the partial differential-integral equation, and 
 generally no approximations and/or simplifications are 
 necessary. Complicated boundary conditions can be handled 
 easily, and the computation time increases at the most 
 linearly with the dimensionality of the problem. However, 
 computer time still may be a problem so that judicious 
 selection of numerical schemes and variance reduction 
 techniques are required. 
 
 Many methods have been devised in the past which involve 
 a mixture of analytic, numerical, and Monte Carlo approaches. 
 
In such situations, Monte Carlo is usually applied to the 
 most difficult part of the problem. For instance, in the 
 evaluation of the non-linear, five-dimensional collision 
 integral in the Boltzmann equation, straight forward 
 numerical quadrature would require months of computation 
 time even on the present day fast computers. The Monte Carlo 
 (statistical sampling techniques which closely resemble the 
 actual collision phenomena) techniques can produce results 
 within reasonable computation time (24). The Monte Carlo 
 method may be defined as a numerical device (numerical 
 experimentation) for studying an artificial stochastic 
 model of a physical or mathematical process. This study 
 may be approached by one of two methods. The first is the 
 process of simulation (direct analog simulation) in which 
 the particles - electrons and ions in the case of plasma 
 systems, or photons in the case of radiative transfer - are 
 represented by a game of chance in the computer. In the 
 second method, which is non-simulation, the Monte Carlo 
 technique is strictly used as a numerical stochastic device 
 for the solution of a given integral or partial differential 
 equations (24,25). 
 
8 
 
 CHAPTER II 
 THE MATHEMATICAL MODEL: NUMERICAL SCHEME AND 
 PHYSICAL PROBLEM 
 
 A. General | Description of the Physics of the Problem 
 
 The plasma under study is a weakly-ionized three- 
 component plasma, and is non-equilibrium as described in 
 Chapter I. The o.-particle radiation continuously creates 
 high-energy electrons throughout the gas volume. The Monte 
 Carlo simulation is to trace these electrons' life histories 
 and their complex processes of diffusion and energy degrada- 
 tion. Since the diffusion process of the electrons through 
 the medium is affected by the density of the positive ions 
 and the other electrons present (background electrons), the 
 resulting problem is non-linear. 
 
 1. Physical Processes Involved 
 
 The mechanisms by which particles interact with the 
 atoms of the medium are elastic and inelastic collisions, 
 by which the incident particle is slowed and dissipates some 
 or all of its kinetic energy to the medium by the combined 
 action of the above mentioned processes. The particle can 
 also experience so called superelastic collision by which the 
 particle collides with the excited atoms gaining some energy. 
 Finally, the particle can collide and recombine with positive 
 ions or can leak out of the system thus ending its life 
 
history. 
 
 As described in Chapter I, the type of interaction 
 which is involved in any particular collision is governed 
 by the laws of chance. For large numbers of interactions, 
 the frequency of occurrence of any given interaction is 
 determined by the relative magnitudes of the interaction 
 cross sections. Secondary knock-on electrons due to the 
 ionization process may be capable of further interacting 
 with the atoms and producing additional secondary electrons. 
 
 As the system reaches steady-state, a balance is 
 reached such that the birth rate for new particles, i.e., 
 the rate at which primary or source electrons enter the 
 system plus the production rate of secondary electrons due 
 to ionization should approximately equal the loss rate due 
 to leakage and absorption or recombination. 
 
 Since the transport of radiation through a medium 
 affects its absorption properties, the transport of individual 
 particles is indirectly affected by the density of the 
 other particles present, and this introduces some non- 
 linearities into the system. We shall adopt a piecewise 
 predict-correct technique to approximate the nonlinear 
 system with a linear model. 
 
 We are concerned with the simulation, by random 
 sampling of the scattering of the charged particles. The 
 direct simulation of the physical scattering processes in 
 such a way that the diffusion process is imitated by letting 
 
10 
 
 the particles carry out a random walk, each step of which 
 takes into account the combined effect of many collisions. 
 The mechanisms by which electrons interact with the atoms of 
 the medium are elastic, inelastic, and superelastic colli- 
 sions as well as recombination with positive ions. The incident 
 source electron is slowed and dissipates some or all of its 
 energy to the medium by the combined action of the various 
 scattering processes, until it escapes, or recombines with 
 an ion (radiative energy losses are neglected). The 
 probability that an interaction occurs is determined by the 
 total collision cross-section. The type of interaction is 
 then determined by the relative magnitudes of the individual 
 cross- sect ions . 
 
 2. Relation to the Transport Equation 
 
 Although no direct use will be made of the governing 
 Boltzmann transport equation in the sequel, we shall write 
 it down briefly, in order to indicate the mathematical 
 problems to be solved implicitly by the Monte Carlo method. 
 It is a nonlinear integro-dif f erential equation of the form 
 
 i 
 
 ■]£- + u . 7 f - ^ (E + u x B) . Vf = (-ir) ,n • 
 
 2t v r m T v u' 3t collisions 
 
 wheie f: number density distribution function 
 
 B: magnetic field 
 
 u: velocity of the particles 
 
 E: electiic field strength 
 
 (1) 
 
 
11 
 
 t: time 
 
 r: spatial variable 
 In our situation B=0 since the magnetic field does not exist. 
 Further, if we assume an infinite medium and uniform source, 
 \? f=0, the distribution function f depends only on energy. 
 Then the above equation reduces to 
 
 3t m v u v 3t ; collisions (2) 
 
 The right hand side term is the collision term, which is the 
 most difficult part of the equation. The effects of all 
 relevant particle interactions must be included in this 
 term. For the case of electron collisions with stationary 
 gas atoms of uniform density, we can write the collision 
 term as: 
 
 ^ d N. - d N . 
 / d± \ _ / m put , , ~» 
 
 K }t' collision ~ v dV-dt ; K J 
 
 where dN . gives the number of phase space points (electrons) 
 
 that are scattered into the phase- space volume dV by 
 
 collision in time dt . Similarly, dN , represents the number 
 
 J ' out ^ 
 
 of points scattered out of dV by collisions in time dt . This 
 PJq. 2 states that the change in the distribution function f 
 are caused by collisions of various kinds and the external 
 force due to the electric field. 
 
 The assumptions that enter into the derivation of the 
 transport equation which carry over to the Monte Carlo 
 
12 
 
 simulation include: 
 
 a) the scattering centers (background atoms and ions) 
 are distributed at random, i.e., possible correlations 
 between the positions of various atoms and ions are not 
 taken into account. 
 
 b) In the absence of an electric field, the trajectory 
 of the particle is idealized as a zig-zag path, consisting 
 of flights interrupted by sudden collisions in which the 
 energy and direction of the particle changed. The quantum- 
 mechamical interference (electron diffration) resulting 
 
 from the coherent scattering by several centers are neglected, 
 and a particle, in the course of traversing the medium, 
 interacts with one scattering center at a time. 
 
 The diffusion process in terms of the transport equa- 
 tion is analogous to the use of Eulerian coordinates in 
 hydrodynamics, in that one asks about the flux at given 
 points in space. By contrast, the Monte Carlo method uses 
 Lagrangian coordinates; one attaches a label to a particular 
 bit of fluid, i.e., a diffusing particle, and follows its 
 history. Then, by sampling many histories, one is able, at 
 least in principle, to solve any diffusion problem. 
 
 B. The Stochastic Model - Numerical Scheme 
 
 The essential appioach to the nonlinear problem adopted 
 here is the use of piecewise linearized steps. In each 
 such step (an amount of time &t along the time axis), the 
 
 
13 
 
 dominate parameters, which characterize the nonlinear 
 properties , are predicted and considered to be fixed with 
 respect to the independent variables. During the time At, 
 the system can be considered as more or less linear, so the 
 conventional Monte Carlo simulation technique is used. 
 The quantities pertaining to the given system are computed 
 for each of a reasonably large number of independent 
 particle histories. The desired parameter values are 
 obtained by averaging the results over all of the histories. 
 At the end of the step, the dominate parameters are 
 corrected before beginning the next step. In each step 
 (time interval At, or cycle), instead of computing the 
 trajectory and angular deflection of an electron after each 
 interaction, we perform a random walk calculation. Since 
 each step of the random walk takes into account the combined 
 effects of many individual interactions, we form a so-called 
 condensed history by this approach. Let us describe the 
 model in some detail, step by step. 
 
 1 . The Geometry and the Coordinate System 
 
 The initial spatial geometry in this problem is a one- 
 dimensional, finite slab (from z=0 to z=D) which is sub- 
 divided into uniform intervals called zones as shown in 
 Figure 1. The energy range is from to E (=lkeV) which 
 is also subdivided into three regions: ^ 1 (thermal 
 energy region), 1 to ED2, and ED2 to E . Each of these 
 
14 
 
 / 
 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 • 
 / 
 / 
 
 / 
 / 
 / 
 
 / 
 / 
 / 
 / 
 / 
 / 
 / 
 A 
 
 2 = 
 
 / 
 
 / 
 / 
 / 
 / 
 
 / 
 / 
 / 
 / 
 
 / 
 / 
 / 
 
 '<, 
 ', 
 
 / 
 
 / 
 
 ', 
 
 / 
 / 
 
 ', 
 
 / 
 / 
 
 / 
 / 
 
 / 
 / 
 / 
 / 
 / 
 / 
 
 (' 
 
 z = d 
 
 Figure 1. Slab geometry of the medium with subdivided 
 zones. 
 
15 
 
 regions is further divided into smaller uniform or logar- 
 ithmic spacing (intervals). The particle number density 
 distribution f(E,z) is a function of energy E and spatial 
 position z and is generated such that particles with 
 energy and spatial position falling into the same intervals 
 are grouped together and normalized by the total particles 
 processed. Although the system is stationary, we deliber- 
 ately introduce the time variable for the convenience of 
 recording output information and carrying out the calcula- 
 tion steps. The time serves as a clock , or we can imagine 
 that the system begins in a transient state. Then, as time 
 goes on it will eventually reach steady state. The storage 
 of the timing information for individual particles is not 
 necessary. Instead of following a single particle from the 
 beginning to the end of its history, we follow a large 
 number of particles, say N (which represent the sample 
 particles of the whole system population) through a small 
 time interval £t (called the cycle time). Within this 
 time interval, the system is considered to be linear, and 
 the particles are processed more or less independently as 
 in a conventional linear Monte Carlo model. At the end of 
 £t, called census time, we record data information for each 
 particle (record particle state information) . During this 
 time interval (or present time cycle) some particles may 
 experience collisions of some kind, some may not, and new 
 particles (source particles, secondary particles produced 
 
16 
 
 by ionization) enter the system. Some particles may be 
 absorbed or escape from the system; their histories are 
 therefore ended. This represents a random walk calculation, 
 and each step of the random walk takes into account the 
 combined effect of many individual interactions. The 
 trajectory of a particle which is computed by this approach 
 is called a condensed history. The residue particles 
 (ionization produced secondary particles) are also set 
 into motion, and their histories are followed separately. 
 They are like all the other source particles in all respects. 
 The complete description of particle histories is approxi- 
 mated by the condensed histories, or we take sequences of 
 snapshots of all the particles in the system at various 
 times to provide a moving picture of the history that can 
 be used to estimate the quantities of interest. The path 
 lengths travelled by the particles are also used as a clock 
 to measure time, or equivalently, the variable t may be 
 
 replaced by S(t) = J v(x)dx. The trajectories of the 
 
 o 
 particles (the condensed histories) are described by the 
 
 arrays as indicated in Figured 2. The superscript denotes 
 the cycle time number; the subscript denotes particle 
 number where the index refers to the initial state of 
 the particle. Other notation includes E: the particle 
 energy , z: spatial position, and U: the direction, and 
 wt: the weight. When the particle has traveled a path- 
 length DCEN (in the same duration At), a condensed history 
 
 
17 
 
 U 
 
 wt 
 
 .0 
 1 
 
 ,0 
 
 1 
 
 
 
 1 
 
 
 
 u 
 
 wt 
 
 '1 
 
 -1 
 J l 
 
 1 
 1 
 
 1 
 
 -I 
 
 u 
 
 wt 
 
 m 
 
 "i 
 
 wt 
 
 m 
 
 E 
 
 U 
 
 wt 
 
 u 
 
 wt 
 
 '2 
 
 .1 
 '2 
 
 1 
 2 
 
 1 
 
 U; 
 
 wt; 
 
 
 "S 
 
 wt 
 
 m 
 
 N 
 
 U 
 
 wt 
 
 n 
 
 n 
 
 
 n 
 
 z 1 
 
 n 
 
 
 n 
 
 u 1 
 
 n 
 
 
 
 ^1 
 
 n 
 
 wt' 
 
 U 
 wt 
 
 n 
 
 fi 
 
 J n 
 
 J 
 
 n 
 
 J 
 n 
 
 Z 
 
 u 
 
 wt 
 
 ,m 
 'n 
 
 m 
 
 n 
 
 m 
 n 
 
 m 
 "n 
 
 Figure 2. Snapshops of the particles' condensed histories 
 
18 
 
 is sampled. Letting the particle carry out a random walk 
 in each step (from state M to state M+l) takes into account 
 the combined effect of many collisions. The sizes of the 
 steps of the random walk, i.e., the pathlength intervals 
 DCEN (equivalent to At) are chosen such that the total 
 number of steps should be kept as small as possible. Too 
 large a step size (At) may cause inaccuracies in the 
 approximation. However, the length of the Monte Carlo 
 calculation is directly proportional to the total number 
 of steps. Thus, a compromise must be reached such that we 
 choose as big a step size as possible without introducing 
 intolerable inaccuracies. A reasonable initial value is 
 chosen such that the particle with the average energy of 
 
 the time cycle travels one mean free path, i.e., At=D /V 
 
 3 * ' ' avg avg 
 
 Different time cycles have different E (V • ) and D so 
 
 J avg avg avg 
 
 At may vary, but as steady-state is approached, At will 
 
 not change much. By this technique we not only randomize 
 
 and average the different particle samples, ensemblewise, 
 
 but we also do the same (on the particle ensemble sets) 
 
 along the time axis (averaging independent but equivalent 
 
 time cycles). Trial and error results indicate that the 
 
 optimum values of At (or DCEN) lie between 0.5 (D „_/V _.._) 
 
 avg avg 
 
 2.0 (D /V ). 
 avg' avg 
 
 The maximum number of particles (N) that can be 
 processed in each cycle is limited by the amount of storage 
 available in the computer system. For the IBM 360/75 system 
 
19 
 
 at the Illinois Computer Center, the storage of the data 
 for 5,000 particles is easily accomplished. By playing 
 Russian roulette, splitting, or similar techniques, the 
 total number of artificial particles (N) can be increased 
 by an order of magnitude. The end of one time cycle t. is 
 the start of the next time cycle t. .. While keeping track 
 of the particles in a current time cycle at one time, we 
 simultaneously record the data information after each 
 particle reaches the census and extract the output para- 
 meter values after all the particles have been processed. 
 In this way the storage requirements are reduced to a minimum 
 level possible. 
 
 The limitation on N by the storage requirement may be 
 further relaxed by the following technique. Instead of 
 recording and tallying the output information and clearing 
 or reinitializing all parameter values at the end of every 
 cycle, we record the accumulated information for K contin- 
 uous, and similar but statistically independent time cycles, 
 then renormalize and reinitialize. In this way the total 
 number of particles processed is KN, and if processing of 
 a large number of particles is desired, we can use a larger 
 value of K(K>1) at the expense of more processing time. The 
 reason for this is based on the fact that the system being 
 simulated is time independent. We have introduced the 
 time variable artificially so that ergodicity holds, namely 
 ensemble averaging is the same as time averaging. 
 
20 
 
 2. Tracing Procedure 
 
 The simulation is initialized by introducing N particles 
 with energy E. [sampled from an initial guess for the 
 number density distribution vs. energy], with spatial 
 position z. (uniformally distributed throughout the spatial 
 region), with direction u. (u=cosfi; 9: angle of flight with 
 z axis) which is uniformly distributed between -1 and +1 
 and with weight wt . . Volume particles are emitted uniformly 
 throughout the medium and isotropically in direction; namely, 
 
 at t=t : 
 o 
 
 
 
 E° 
 
 1 
 
 = 
 
 E. 
 
 in 
 
 Z° 
 
 1 
 
 = 
 
 r • D 
 i 
 
 1 
 
 = 
 
 2r.-l 
 
 l 
 
 / 
 
 (4a) 
 
 (4b) 
 
 (4c) 
 
 where r.: random number, uniformly distributed between 0, 
 
 1, or x e (0, 1) 
 
 d: slab width or cylinder diameter 
 
 wt.,E. : weight and particle energy determined from 
 
 initial distribution. 
 
 Once the particle data are determined, the geometry 
 
 routine is entered. This includes the determination of the 
 
 distance to boundary d„ , the distance to collision d _ , and 
 
 J B' col' 
 
 the distance to census time d (clock distance equivalent 
 
 cen ^ 
 
 to At). These distances are defined by: 
 
 
21 
 
 d B = [(j+l)AZ - Z]/u if u > (5) 
 
 = (Z - jAZ)/ |u| if u < 
 = DLB (a large number DLB » AZ) if u^r 
 
 d CQl = A(E) |lnr| = |lnr| / J t (E) (6) 
 
 cen 
 
 D • DI = V • At* DI (7) 
 
 avg avg 
 
 where r 
 3 
 
 It 
 
 D 
 
 avg 
 
 random number uniformly distributed 
 
 zone number 
 
 spatial interval size 
 
 the total cross-section evaluated at the particles' 
 energy 
 
 the average value of D ^ of N particles of the 
 v col * 
 
 previous cycle. It is approximately one mean 
 
 free path corresponding to E^ tt/ ^: the average 
 
 energy of the cycle. 
 
 avg 
 
 DI : input parameter for controlling the cycle time 
 interval. 
 
 The next step depends on which of these distances is the 
 
 smallest. 
 
 a . The System Without Electric Field (E =0 ) 
 
 i) If the smallest distance is d : the particle 
 
 cen ^ 
 
 is advanced in position and time to census time. 
 
 z 1 = z + u.d 
 
 u' = u 
 E« = E 
 
 cen 
 
 (8a) 
 (8b) 
 (8c) 
 
 The particle data is stored, the energy is tallied, and 
 
22 
 
 the density counter is incremented. Then we go on to 
 process the next particle. 
 
 ii) If the smallest distance is d the particle is 
 advanced to the boundary such that: 
 
 z' = z + (d + 0.001«Az) . u 
 j = INT (z»/Az) + 1 
 
 d T 
 
 d' , = d , 
 
 col col 
 
 B 
 
 d' = d - d n 
 cen cen B 
 
 (9a) 
 (9b) 
 (9c) 
 
 (9d) 
 
 where j is the zone number. If j>JZ or j<l , then the 
 
 particle is considered lost or has escaped from the system, 
 
 and its history is ended. Otherwise the calculation and 
 
 selection is repeated until the particle reaches the census 
 
 time . 
 
 iii) If d t is the smallest, the particle is first 
 col 
 
 advanced to collision point and 
 
 z ' = z + u «d 
 d' 
 
 cen 
 
 col 
 = d - d 
 
 cen col 
 
 (10) 
 
 Then the type of collision is specified. For specifying 
 the collisional processes, we divide the whole energy range 
 into two regions. Region I: energy range to EDI eV. 
 Region II; energy range EDI eV to E eV. Edl and E are 
 specified by input parameters and are referred to as the 
 energy dividing line and source particle energy, respectively. 
 
. 23 
 
 If the particle's energy is in the range of Region I, 
 three kinds of collisions are possible: elastic collisions, 
 electron- ion recombination, and superelastic collisions 
 (where electrons hitting an excited atom pick-up the 
 excitation energy). If the particle's energy falls in 
 Region II, it can experience four kinds of collisions; 
 namely, elastic collisions and three types of inelastic 
 collisions — excitation to metastable level, excitation to 
 other levels, and ionization. For ionization, the secondary 
 electron so produced shares energy with the primary electron, 
 and the energy of primary electron before collision less 
 the ionization potential is shared between the two electrons. 
 The partition of the shared energy could be determined 
 according to Goodrich's experimental distribution curve (31). 
 But based upon Thomas' (1) calculation experience, the way 
 in which the energy is partitioned does not have a strong 
 influence on any of the calculated mean properties. Thus, 
 for computational simplicity, as Thomas did in all his 
 calculations, we choose the partition ratio 9:1. The 
 secondary electrons produced via ionization are considered 
 as new particles introduced into the system. Thus they are 
 followed from their point of birth just like other particles. 
 This is equivalent to the generation method for processing 
 branched trees of samples. 
 
 The type of collision is determined according to 
 Figure 3 using a random number (uniformly distributed 
 
24 
 
 Elastic 
 
 Excitation 
 to 
 
 metastable 
 level 
 
 Excitation 
 
 to 
 
 other 
 
 levels Ionization 
 
 A 
 
 B 
 
 A = P 
 
 el 
 
 B = P . + P . 
 el met 
 
 C = P 1+ P ,, + P 
 
 el met exc 
 
 P ' + P ,, + P + P. =1 
 el met exc ion 
 
 Figure 3. The probability model for determining the 
 types of collision. 
 
25 
 
 between and 1) to play the law of chances. 
 
 b. With Electric Field Present 
 
 The cooidinate axis z is defined as antipaiallel to 
 the electiic field E . In the.piesence of electiic field 
 E the state of motion of an electron is specified by its 
 speed V and the direction cosine cos A , where A is the 
 angle made by the line of flight with the z-axis at the 
 time t. If an electron is in a state . [E(V), u(cosA), z, t] , 
 it will assume (after a flight along the orbit by At) the 
 new state [E'(V'), u'(cosA'), z+Az, t+At] . These para- 
 meters are connected by the following relations: 
 
 V -sinA' = V-sinA (11) 
 
 mV'cosA' = mVcosA + eE f *At (12) 
 
 1/2 mV 2 = 1/2 mV 2 + eE f .£z § (13) 
 
 E' = E + eE «Az (14) 
 
 where m and -e are the mass and the electric charge of the 
 
 electron, respectively. Here Az is the projection of the 
 
 path length L on the z axis. It can be obtained as follows; 
 
 fAz (At • E 
 Az = V dt a (V-u + -=- At) dt 
 ^0 z Jo m 
 
 E f 2 
 = V-u-At + l/2~« At (15) 
 
 In carrying out the geometry routine, it is suitable to 
 
 use time directly as the clock. The determination of the 
 
 distance to collision d ,(DCOL) and the distance to census 
 
 col ' 
 
26 
 
 d (DCEN) is accomplished as follows 
 
 cen r 
 
 DCOL = A-|lnr| (16) 
 
 " 2/(l/ A(E) + V A(E() ) (17) 
 
 DCEN =/ V y 2 + V z 2 . At (18) 
 
 V y = V.sinB (19) 
 
 E f 
 V z = V.cosB + ^ At (20) 
 
 where E: the initial energy of the particle 
 
 E' : the energy of the particle after flight time At. 
 The next step is to determine which of the distances 
 
 is smaller. If DCEN<DCOL, then the particle is advanced 
 
 to the new state 
 
 E' = E + E f .Az 
 
 z' = z + Az 
 
 u' = aJe/E . u + eE f -At (21) 
 
 If E'<0 or z'>D r z'^0 then the particle is considered to 
 have escaped from the system. If DCEN > DCOL, the particle 
 is advanced to collision point. 
 
 At' = At - DCOL-At/DCEN 
 
 DCEN = DCEN - DCOL 
 
 Z' = z + Az-DCOL/DCEN 
 
 E« = E + Az.E -DCOL/DCEN (22) 
 
 then the specific collision type must be determined. 
 
27 
 
 c. Types of Collisions and Cross Sections 
 For the elastic collisions, the particle scatters off 
 with angle of deflection 6 and energy loss &E, the follow- 
 ing formula is used (36). 
 
 AE = 4r < E ~ E J (1-cosG ) (23) 
 
 M n s 
 
 where 9 : scattering angle (angle of deflection) 
 
 E: particle energy 
 
 E : target gas atom energy 
 
 m: electron mass 
 
 M: gas atom mass 
 
 E is selected according to a Maxwellian distribution, and 
 
 n * ' 
 
 the AE so formed may have either positive or negative 
 
 values, indicating the loss or gain of energy respectively 
 
 For calculational simplicity, instead of selecting the 
 
 deflection from the elaborate distributions, we use 7T/4 
 
 or tt/2 deflection approximations, namely 9 = X/4 or V2, 
 
 then cosB = or 0.70711. Another reasonable alternative 
 s 
 
 is to assume that the direction after collision is equally 
 probably in all directions (isotropic scattering). We 
 select this direction at random with uniform distribution, 
 and based on Eq. 23 calculate cos9 and AE, respectively, 
 
 where 
 
 cos9 = 2r.-l (24) 
 
 l 
 
 6=9 --6 . 
 s 1 
 
28 
 
 where 9 . : incident angle 
 
 1 ' 
 
 9 : outgoing angle after collision 
 
 For the three kinds of inelastic collision cross sections 
 
 we use approximate formula based on experimental data as 
 
 correlated by Itoh and Musha (2). The formulas are expressed 
 
 in the form: 
 
 B(E-E. ) 
 £( E ) = 1 j + C (25) 
 
 A+(E-E ± ) Z 
 
 where the constants A, B, C, and E. depend on the kind of 
 collision and scattering angle. For the elastic cross 
 section in helium, we use a single approximate formula for 
 whole effective energy range based on Heylen and Lewis's 
 paper (10), namely: 
 
 T el (E) = 26exp(-0.04E) (cm -1 ) , (26) 
 
 For the recombination cross section for helium (35), we 
 assume the simple form: 
 
 1rcb (e) =^<e).n+ 
 
 jol(E) = 10" 9 /V(E) (27) 
 
 where N : ion density 
 
 V: electron velocity 
 the superelastic cross section is 
 
 \ fp % 0. 109(19. 8-E) N^ (28) 
 
 Zsel = 0.25 + (E-19.8) 2 3 - D n 
 
29 
 
 where N + : positive ion density 
 
 D : gas atom density 
 
 n ' J 
 
 E: electron's energy 
 
 3. The Prediction and Correction of Dominant System 
 Parameters 
 
 There exists a single parameter, the ion density 
 
 distribution N (E,z), which completely characterizes the 
 
 nonlinearity of the system. This simplifies the problem 
 
 and suggests the following predict-and-correct scheme. 
 
 We assume that the average mean free path (or At) and the 
 
 ion density distribution N + (E,z) in the present time cycle 
 
 also applies to the next time cycle, In other words , in 
 
 each time cycle we use the predicted values of At and 
 
 N (E,z) from previous time cycle. At the end of present 
 
 time cycle, we use the up to date information to renormalize 
 
 and correct the value to be used for the next time cycle. 
 
 As steady state is reached, the predicted and corrected 
 
 values of the parameters converge. From the continuity 
 
 equation 
 
 y-J = S - L (29) 
 
 where J: the particle current flow in or out of the system 
 
 S: the source 
 
 L: the sum of losses 
 
 To start with, we assume that V*J =0, so that 
 
 S = L = a(E) N + (E,z) N"(E,z) (30) 
 
30 
 
 Here N~(E,z) is the electron number density, the key 
 quantity we are calculating. To initialize the run we have 
 to predict N + (E,z). It is reasonable to initially assume 
 this distribution is uniform in z, and a maxwellian dis- 
 tribution in E, i.e., N + (E,z) = N n M(E). We also assume 
 that N"*(E,z) is the same as N + (E,z). Then we obtain 
 
 S = L = a(E) N + (E,z) N"(E,z) 
 
 = ct(E) [N + (E,z)] 2 (31) 
 
 or S 6(E-E ) = a(E) N 2 M 2 (E) 
 
 Integration of both sides gives 
 ■E, 
 
 S = N b . 
 
 J ° 2 
 a(E) M (E) dE 
 
 
 
 y 
 
 w;° ■ 
 
 . N n = [S n / a(E) M^E) dE] 
 
 (32) 
 
 This gives a starting value for the calculations. In later 
 cycles the renormalization process involves integration of 
 Eq. 29, with V-J £ 0, i.e., 
 
 JJ dJiE.z) dzdE m IT s(E/Z)dzdE 
 
 - /J a(E) N + (E,z) N~(E,z)dzdE - J egc + J g 
 which reduces to 
 
 J(Z=d) - J(z=0) = % d - Jja(E) N + (E,z) N~(E,z)dzdE 
 
 - J + J 
 esc s 
 
31 
 
 s p d + j s - [j(»-d) -j(z = o)] -j esc 
 
 N . = . ■ '■ ' — 
 
 1 JJa(E) M(E) rT(E,z) dzdE (33) 
 
 where i refers to ith time cycle 
 
 J : current due to escaped electrons 
 
 esc c 
 
 J : current due to secondary electrons produced 
 by ionizations 
 
 M(E): Maxwellian distribution (normalized) 
 
 n~(E): electron number density distribution 
 (normalized) 
 
 S n : source rate 
 
 N.: ion density distribution (normalized) 
 The value of N. replaces N„ in equation (33) at later 
 time cycles. The value of N. (N + ) enters the formula for 
 the recombination cross section and superelastic cross 
 section, thus it directly affects the rates of recombination 
 and superelastic collision. Indirectly, it affects the 
 balance of the system, or equivalently the rate of conver- 
 gence of the Monte Carlo calculation. An adaptive correc- 
 tion procedure is used here. It is based on the value of 
 the "balance factor" BF, defined as the ratio of the 
 particles introduced into the system by source and ioniza- 
 tion processes to the particles removed from the system due 
 to recombination and leakage. We adopt a over-relaxation 
 type of correction technique, the over-correction is used 
 to speed up the over-all convergence rate. As defined 
 before, if BF > 1, the system has more particles introduced 
 
32 
 
 into than eliminated, but if BF < 1, the opposite is true. 
 The over-correction factor is proportional to (BF) t i.e., 
 to the kth power of the balance factor. The value of k. 
 initially chosen was k=2, but an optimum value can be 
 obtained by trial and error. 
 
33 
 
 CHAPTER III 
 VARIANCE REDUCTION TECHNIQUES 
 
 A. General Background 
 
 The Monte Carlo method (or method of statistical 
 trials) consists of solving various problems of computational 
 mathematics by the construction of some random process for 
 each such problem, with the parameters of the process set 
 equal to the required quantities of the problem. These 
 quantities are then approximated by observations of the 
 random process and the computation of its statistical 
 characteristics, which are approximately equal to required 
 parameters [from Schreider (15)]. Every Monte Carlo 
 computation that leads to quantitative results may be 
 regarded as estimating the value of a multiple integral. 
 Suppose we have M random numbers in the computation system. 
 The results will be a vector valued function R(^ 1 ,£?, ... £ M ) 
 involving the sequence of random numbers £.. , £~, ... c M . 
 
 This is an unbiased estimator of ... R(x, ,x~. ... x„) 
 
 Jo Jo 1 2 M 
 
 dx. ... dx . For the sake of simplicity, we take the one- 
 dimensional integral as a standard example. Let 
 
 r 
 
 Jo 
 
 f(x)dx (34) 
 
 
 
 2 f x 2 
 
 where f e L (0,1) or [f(x)] dx exists. 
 
 / 
 
 Define the relative efficiency of two Monte Carlo methods 
 as follows. Let the methods call for n, and n ? units of 
 computing time, respectively, and let the resulting 
 
34 
 
 2 2 
 
 estimates of 9 have variances aC and <T1 . Then the 
 
 efficiency of method 2 relative to method 1 is defined as 
 
 n 2 • a> 2 
 
 (35) 
 
 Equation 35 is the product of two terms, the variance ratio 
 
 n. 
 
 CT^ 2 / 6^2 and the labor ratio "l/n 2 . If % , %, 
 
 ?N 
 
 are independent random numbers with uniform distribution 
 between 0, 1 or *1 . e(0,l); then the quantities f.=f(^.) 
 are the independent random variates with expectation 8 . 
 Therefore 
 
 N 
 
 f4l f- 
 N i=l X 
 
 is an unbiased estimator of 6 , and its variance is 
 
 (36) 
 
 i r 1 (f(x)- 
 
 J 
 
 B) 2 dx= £ 
 
 (37) 
 
 Accordingly the quantity f~, which has been determined by 
 observation of the random process, is approximately equal 
 to the required quantity 8 (or in other words, it is an 
 unbiased estimator of 8 ) with .a probability which can be 
 made as close as required to unity if a large number of 
 trials is practical. We refer to the estimator 7 as the 
 crude Monte Carlo estimator of 8 (commonly referred to as 
 Monte Carlo estimation) . 
 
 Let us introduce another even less efficient method, 
 namely hit or miss Monte Carlo. Suppose that < f (x) < 1 
 
 
35 
 
 when < x < 1. We may draw the curve y = f(x) in the unit 
 square < x, y < 1, and 9 is the proportion of the area 
 of the square beneath the curve. Stating this formally, 
 we write 
 
 f(x) = g(x,y)dy (38) 
 
 Jo 
 
 where g(x,y) = if f(x) < y 
 
 = 1 if f (x) > y '■:: . 
 
 Then 
 
 fl 
 
 Jo JO 
 
 The estimator 
 n 
 
 g(x,y) dxdy (39) 
 
 5-5? w'^n, <2i ) -5 
 
 1 = 1 
 
 (40) 
 
 n* is the number of occasions where f( £ ~ . .. ) > £_. 
 
 Y 2i-l — 7 2i 
 
 In other words, we take n trials at random in the unit 
 square, and count the proportion of them which lie below 
 the curve y = f(x). 
 
 Historically, a hit or miss method was first propounded 
 in the explanation of Monte Carlo techniques. It is the 
 easiest to understand but one of the least efficient 
 techniques. The rate of convergence and the variance 
 (statistical error) is in general proportional to /JN 
 where N is the number of samples. It is obviously imprac- 
 tical to gain a significant improvement of accuracy by 
 merely increasing N. Therefore more efficient techniques 
 for reducing the variance and/or increasing the efficiency 
 
36 
 
 are very important. Monte Carlo experimentalists need 
 
 wide experience and background in both the mathematics and 
 
 the physics of the problem, and they have to exercise 
 
 considerable ingenuity in distorting and modifying problems 
 
 in the pursuit of variance reduction techniques. Statistical 
 
 and inferential procedures are also important in order to 
 
 extract the most reliable conclusions from the observational 
 
 data. 
 
 The basic requirement of the Monte Carlo calculations 
 
 is to establish how random numbers may be used to sample 
 
 a function that describes, in a probabilistic fashion, a 
 
 physical event. The procedure by which this is done 
 
 follows from the fundamental principle of Monte Carlo (16 ), 
 
 which may be stated as follows. If p(x)dx is the probability 
 
 of x lying between x and x+dx, with a < x < b 
 
 -b 
 
 p(x)dx = 1 (41) 
 
 a 
 
 Then r = P(x) = j p(y)dy (42) 
 
 Ja 
 
 determines x uniquely as a function of the random number r 
 which is uniformly distributed on range (0, 1). The quan- 
 tities p(x), p(x) are the probability density and the 
 probability distribution functions, respectively. 
 
 With the development of large computers, the use of 
 the Monte Carlo method in a wide range of problems has 
 
 
37 
 
 increased rapidly. In particular, the method is applied to 
 various physical problems such as plasmas (15), radiative 
 transfer problems (21), neutron transport and nuclear 
 reacter physics (29), etc. 
 
 The random walk type simulation can be briefly described 
 in the following manner. One first introduces the basic 
 physics of the problem into the computer in a probabilistic 
 fashion. A system of coordinates and boundaries are defined 
 and then, as a computer experiment, particles are released 
 from the source. These particles are traced as they diffuse 
 through a prescribed medium following the probabilistic 
 interaction laws. The particles are followed until they 
 escape from the medium or are absorbed or are converted to 
 the thermal field. 
 
 The parameters pertinent to the evaluation of the 
 desired quantities are recorded. One continues processing 
 additional particles until adequate statistical estimates 
 of the quantities of interest have been obtained. 
 
 When doing a Monte Carlo problem one focusses attention 
 on three main topics. They are: 
 
 1. The development of the analogy for the probability 
 processes (simulation). 
 
 2. The generation of sample values of the random variables 
 on a given computing machine. 
 
 3. The design and use of variance reduction techniques. 
 
38 
 
 The variance reduction methods are often strongly 
 dependent on the probabilistic model, thus the greatest 
 gains are often made by exploiting specific details of the 
 problem, rather than by routine application of general 
 principles. 
 
 B. Importance Sampling 
 
 The general idea of importance sampling is to draw 
 samples from a distribution other than the one suggested 
 by the problem. Then an appropriate weighting factor is 
 introduced to correct the biasing caused by changing the 
 original distribution. If done correctly the final results 
 are essentially unbiased. The object is to concentrate the 
 distribution of sample points in the parts of the region 
 that are most "important" instead of spreading them out 
 evenly. Thus, the probability of sampling from an interest- 
 ing region is increased; the probability of sampling from 
 unimportant or less interesting region is correspondingly 
 decreased. Stating this formally, we have 
 
 e = j% <x)dx = £ §£} g(x)dx = £f{!j. dG(x) (43) 
 
 for any functions g and G satisfying 
 
 r x 
 
 G(x) = g(y)dy (44) 
 
 Jo 
 
 where g must be a positive valued function such that 
 
 
 
39 
 G(l) = g(y)dy (45) 
 
 Jo 
 
 if ^ is a random number sampled from the distribution G, 
 then f(9)/ /*)) has the expectation A and variance 
 
 We notice that if g(x) = cf(x), or if g is proportional to 
 
 1 2 
 
 f and if c = r then C„ , = 0. This perfect situation does 
 6 f/g 
 
 not exist, since we do not know 9. We always obtain an 
 unbiased estimate for positive function g. Our object is 
 to select g to reduce the standard error of our estimate. 
 
 Because the estimate is the average of observed values 
 of f/g, we choose g such that f/g is as constant as possible 
 in order to achieve a small sampling variance. While we 
 intend g to mimic f, we generally restrict our choice of g 
 to functions simple enough to allow analytic integration. 
 
 For the present plasma problem, the importance sampling 
 technique is developed as follows: during the particle's 
 (electron's) random walk, it suffers a large number of 
 collisions with gas atoms, and these collisions may be 
 elastic or inelastic or of other kinds. The probabilities 
 of their occurrence depend on their cross section 2- • We 
 have 
 
 2 t - I x + 2 2 + . . . Z M (47) 
 
40 
 
 where 2 . : the cross section for ith process 
 
 2 .: total cross section or the summations of the 
 cross sections of all the possible processes 
 that the particle may experience. 
 
 Dividing both sides of equation 47 by £ we obtain: 
 
 1 = P ± + P 2 + • • • P M (48) 
 
 where P. = ^-i. y the probability of ith process occurring 
 l / £ t 
 
 as the particle experiences a collision. If we introduce 
 
 weights w. into equation 47 we obtain 
 
 2m (49) 
 
 J t -w 1 .2 1 + w 2 -2 2 + . . . w M _ M 
 
 Similarly we have 
 
 1 = p 1 + p 2 + . . . p M (50) 
 
 After the application of the weights, we put more weights 
 on the interesting or more important events. Then their 
 artificial probabilities are greater, therefore we can get 
 better estimates of the important parameters. Coincidently, 
 such a linear weighting scheme can only be applied to the 
 elastic collision process in our system. The other colli- 
 sion processes involve appreciable changes in the particle's 
 energy (catastrophic collision). Also cross sections are 
 energy dependent, these nonlinear effects prevent the 
 application of above mentioned weighting scheme to inelastic 
 collision processes. By looking at the cross section values, 
 in the low energy region, the elastic cross section is by 
 
41 
 
 far the largest by order of magnitude. Others have shown 
 that the elastic process does not strongly influence the 
 estimated parameters that we are interested in (1, 2). 
 As an electron makes an elastic collision, it loses only a 
 small amount of energy while making a small angle deflec- 
 tion. In fact, most other workers simply neglect elastic 
 loses if an electric field is present (1, 2, 10). In our 
 case, with E =0, elastic effects must be included. In the 
 low energy region (~0 eV to 20 eV), the energy of a majority 
 of the particles lies below 1 eV and over 90% collisions 
 are elastic collisions, and only a few collisions are 
 inelastic . 
 
 The collision process subroutines are a time-consuming 
 part of the simulation calculations. If we can supress 
 the elastic processes by a factor of k . the chances of 
 all other processes are increased by approximately the same 
 factor. Therefore we convert more computing time to 
 processes which contribute more strongly to the quantities 
 of interest. The following formulation provides the 
 theoretical support for this. . In lieu of equations 47-50, 
 the p. are the original probabilities (undistorted proba- 
 bilities) while the p! are the weighted probabilities. 
 Then the correction factors are introduced as follows; at 
 energy E, we have 
 
 C 1 = p l/p- , C 2 = p 2/p- . . . C M = PM/pjJj (51) 
 
42 
 
 n P- / . ?1 * It 7t 1 /c , , 
 
 or C. = *i/p! = - - = — = (51a) 
 
 *-t -^l l ^t i 
 
 1 
 
 If we let w. = ~ — w = w_ = . . . w M = 1 
 1 k _ ' 2 3 M 
 
 el 
 
 then from ( 51 ) we have 
 
 1 * t z t 1 zt 
 
 let £, = J , namely we introduce weight on elastic cross 
 
 section only. Since 2 -.(E) » Z • (E) . i > 1, V- then 
 
 el i ' ' 1 
 
 2|' E> 24l (E ) i 
 
 77^ ~T£ro "i = Kl (53) 
 
 and c « w i- l.. C i-TT- (i ^> (53a) 
 
 1 el 
 
 Equation 53 holds for low energy region, which is approxi- 
 mately independent of energy. 
 
 • 
 
 C. Russian Roulette and Splitting 
 
 In general, the sampling is done such that we examine 
 the sample and classify it as being in some sense "inter- 
 esting" or "uninteresting" . We are willing to spend more 
 than average amount of work on the interesting samples and 
 on the contrary, we want to spend less effort on the 
 uninteresting ones. This can be done by splitting the 
 interesting samples into independent branches, thus 
 resulting in more of them, and by killing off some of the 
 uninteresting ones. The first process is splitting and the 
 second Russian Roulette. 
 
43 
 
 The "killing off" is done by a supplementary game of 
 chance. If this game is lost, the sample is killed; if it 
 is won, the sample is counted with an extra weight to make 
 up for the fact that some other samples have been "killed". 
 The game has a certain similarity to the Russian game of 
 chance played with revolvers - whence the name. 
 
 For the present plasma system we divide the whole 
 energy spectrum into two regions, the low energy region I 
 (0 eV to EDI eV) and the high energy region II (EDI to E ). 
 Here E_ is energy of source particles, which is the highest 
 energy the particle can have. EDI is the energy region 
 dividing line, which can be chosen as any value. between 5 eV and 
 100 eV depending on the.initial guessed energy distribution. 
 In region I, the number density is high, and the elastic 
 process is the dominant process. As we mentioned earlier, 
 the elastic process does not contribute much, so Region I 
 is classified as uninteresting. Each particle in this 
 region (particles with energy less than EDI) is assigned a 
 weight kl(kl > 1). In other words, each particle plays 
 the role of kl particles. . 
 
 In region II, classified as interesting region, each 
 particle carries a weight w ? (w ? = /k ? , w ? < 1). Namely, 
 each particle in this region is split into k ? particles 
 with weight w~. 
 
 As particles cross over to other regions, care must 
 be taken to account for killing off and splitting effects. 
 

 44 
 
 For instance, a particle in region I may acquire enough 
 energy, either from electric field or by superelastic 
 collision with the excited atoms, to cross over to region 
 II. Then we should generate k~ particles (k~ = k. • k ? ) in 
 region II with the same energy. 
 
 Particles from the high-energy region may lose part 
 of their energy by inelastic collision such that they 
 cross over to the low-energy region. Then killing off should 
 be effective. For every k~ such particles that cross over 
 from region II to region I, only one particle survives. 
 The values of k, and k~ can be chosen according to the 
 total number of particles in the system to be processed, 
 the source rate, and related accuracy requirements. They 
 are input parameters. 
 
 D. Initial Guess 
 
 Monte Carlo methods comprise that branch of experimental 
 mathematics which is concerned with experiments on random 
 numbers. The user, like the experimental physicist needs 
 theory and knowledge to give structure and purpose to his 
 experiments, and as experimental work provides growing 
 insight into the nature of a problem and suggest appropriate 
 theory. Good Monte Carlo practice should keep this relation- 
 ship as a general maxim. 
 
 The basic procedure of the Monte Carlo method is the 
 manipulation of random numbers, but these numbers should not 
 
45 
 
 be employed prodigally. Each random number is a potential 
 source of added uncertainty in the final result. Thus it 
 will usually pay to scrutinize each part of a Monte Carlo 
 experiment to see whether that part can not be replaced 
 by an exact theoretical analysis contributing no uncertainty. 
 In other words, exact analysis should replace Monte Carlo 
 sampling wherever possible. Sometimes reliable intuition 
 would aid in increasing the efficiency of Monte Carlo 
 calculations. For the plasma system we are simulating, the 
 straight-forward approach would be to follow N source 
 particles on their life time history, until they lose energy 
 sufficient by collisions to reach a thermal equilibrium. 
 Over 50 inelastic collisions generally occur, and the 
 probability (inelastic cross sections) for inelastic 
 processes is low. Consequently, the particles travel a 
 relatively large distance which consumes a good amount of 
 time. Thus a large number of time cycles would be required 
 before the system reaches steady state. To avoid this, we 
 use our intuition, to start off with a guessed initial 
 distribution. The electron energy distribution f(E) (or 
 electron number density distribution) is a function of the 
 electron's energy E. Initial particle data is generated 
 using this distribution, and this can save considerable 
 irrelevant computations. This idea is simple, yet very 
 useful and effective. Without an initial distribution, 
 the problem would require hundreds of time cycles to 
 
46 
 
 converge. However, with a reasonable guess, good results 
 have been obtained in as little as tens of time cycles. 
 Of course, the better the initial guess, the faster the 
 convergence. 
 
 E. Antithetic Variates 
 
 When we estimate an unknown parameter 9 by means of 
 an estimator t, we may seek another estimator t 1 having 
 the same (unknown) expectation as t but a strong 
 negative correlation with t. Then l/2(t + t') will be an 
 unbiased estimator of 8 , and its variance is 
 
 Var[-|(t+t' )]= ~var(t) + jvar(t') + icov(t,t«) (54) 
 
 where cov(t,t») is negative, and 
 
 var(t) = £ [(t-u) 2 J 
 
 cov(t,f) = £ f (t-uMt'-u«)) 
 u,u' are the mean of t,t' respectively, and£(x) is the 
 expectation of the random variable x. 
 
 It is possible to make var'(tft' ) smaller than var(t) 
 by suitably selection t'. For example, 1-r is uniformly 
 distributed whenever r is, and if f(r) is the unbiased 
 estimator of 6, so is f(l-f). When f is a monotone function, 
 f(r) and f(l-r) will be negatively correlated. Thus we 
 could take 
 
 ■|(t+t« ) = \ t(r) + -| f (1-r) (55) 
 
47 
 
 as an estimator of 9 . 
 
 The name and idea of antithetic variates was first 
 introduced by Hammer sley and Morton (13, 14) during attempts 
 to improve Buffon's needle experiment for estimating the 
 value of n (3.1415926) . 
 
 The main idea of antithetic variates is based on the 
 underlining theorem [from Hammersley and Morton (12)] 
 which can be stated informally as follows: "Whenever we 
 have an estimator consisting of a sum of random variables, 
 it is possible to arrange for a strict functional dependence 
 between them, such that the estimator remains unbiased, while 
 its variance comes arbitrarily close to the smallest that 
 can be attained with these variables". As the name implies, 
 antithetic variates are the set of estimators which mutually 
 compensate each other's variations. Essentially, we 
 rearrange the random variables by permuting finite subinter- 
 vals in order to make the sum of the rearranged functions 
 as nearly constant as possible. From the practical viewpoint, 
 the mathematical conditions imposed by antithetic variates 
 to the Monte Carlo calculation are quite loose and flexible. 
 It is relatively easy to find a negatively correlated 
 unbiased estimator to produce an efficient variance reduc- 
 tion scheme. 
 
 A system of antithetic variates is obtained by simple 
 transformations based on a stratification of the interval 
 (12). The transformations are constructed in such a way 
 
48 
 
 that the efficiency of the method will increase as a higher 
 power of M (the number of antithetic variates taken). The 
 transformations considered are linear combinations of the 
 values of the function f at a number of points. Thus a 
 correlated stratified sampling technique is developed, and 
 
 its efficiency generally exceeds that of crude sampling by 
 
 3 
 a factor M . 
 
 The following useful transformations are obtained in 
 
 this manner: 
 
 t. = - T\ -faf(ar.) + (1-a) f[a+(l-a)r.]} 
 1 n *— J , <• l l * 
 
 i=l 
 
 (56) 
 
 n 
 
 n <-■ n a i 
 i=l 
 
 1 n 
 2 = n f± 1 {af(ar. ) + (1-a) f [l-Q-a)^]} (57 
 
 t„ = 
 
 n 
 
 l-l 
 
 The value of a is between and 1, the variance 
 var(t.) has a minimum at some value of a. An adequate 
 rule of thumb is to choose a by finding the root of 
 
 f(a) - (1-a) f(l) + af(0) 
 
 (58) 
 
 Another useful transformation is 
 
 ^ m 
 
 t _ = if f(£±l) = U f(r) 
 3 m 4-" ^ m m 
 
 m r=o 
 
 (59) 
 
49 
 
 In general, the variance of crude Monte Carlo is 
 
 var(t) = 0(m~ ) while the variance of t~ is var(t-) = 
 
 — 2M 
 0(m ), (M "> 1). The simplest form of antithetic 
 
 transformations have proved quite successful in neutron 
 
 transport calculations. 
 
 The general notion and a simple form of antithetic 
 variates are applied in our present calculations. High 
 efficiency gain can be obtained easily if it is used 
 sparingly and at judicious points in the computation. We 
 have used the random number r.(r. e(0,l)) most of the time 
 for playing the law of chances, and 1-r . is another random 
 number of the same kind which is the image of r.. The 
 two are used together in good many places where repeated 
 use of r. is required. This serves the purpose of two 
 antithetic variates and cuts the labor in half (same 
 random number r. being used twice). Equations 55 and 59 
 have been used in the initial data preparation program and 
 source routine where initial particles states (E . ,u . , z . , wt . ) , 
 E. are generated according to input initial distribution. 
 For instance: 
 
 u. = 2r.-l, u i+1 = 2(l-r.)-X 
 
 z i " r i d ' z i+l = <l-*i>'<» 
 
 E i " \ + r i- iE k' E i + 1 " E k + <l- r i>-A E k <60) 
 E . = E n for source particles. 
 
 Here k is an energy index, or kth energy interval. 
 
 Wherever r. appears, r. and 1-r. are used interchangeably. 
 
50 
 
 Thus we are generating source particles from diametrically 
 opposite directions, or from symmetric locations. 
 
 F. Double History (Double Processing) Technique 
 
 In the particle tracing routine, using the general 
 idea of antithetic variate technique, we have adopted a 
 "so-called" double history processing technique which is 
 described as follows; 
 
 a) In the course of tracing the particle's history, 
 
 in each time cycle, we use the transformed random number 
 
 X* I lnr to decide the distance to collision d n , and go 
 1 col 1 r 
 
 on until the particle reaches census time. However, we 
 
 process the same particle twice under the identical 
 
 initial condition except that we use >^«|ln(l-r)| as the 
 
 distance to collision d ., „. Both sets of history are 
 
 col 2 J 
 
 ta lied but only one set of particle data is 'stored. 
 This way each particle is processed twice independently 
 but correlated in such a way that they compensate each 
 other's variation (variance). 
 
 b) Another alternative for double history processing 
 is, to play a dual direction random walk instead of dual 
 collision distances. We process the same particle twice 
 
 in such a way it looks like there are two identical particles 
 emerging into the system in two diametrically opposite 
 directions. Again both histories are tallied, but only 
 the original particle's data is stored. 
 
51 
 
 The two versions of the double history processing 
 technique increase the labor (computation time) by a factor 
 of 1.2 to 1.5. The efficiency gain is difficult to specify, 
 At least it is certain that the gain is more than double 
 the total number of the particles in the system. 
 
 The purpose of dual processing is to compensate each 
 particle's variation and minimize the variance (statistical 
 errors). One set of particle data in dual processing is 
 thrown away, while the other set is stored just like normal 
 processing. Which one of the two resulting sets is saved 
 is completely random. 
 
52 
 
 CHAPTER IV 
 MONTE CARLO SIMULATION CODE 
 
 A. Introduction 
 
 This chapter piesents the complete Monte Carlo simula- 
 tion program, including an input, output, main flow chart, 
 and the background theory and techniques incorporated. 
 This program implements the condensed history approach that 
 was described in Chapter II together with various variance 
 reduction techniques described in Chapter III. 
 
 The basic Monte Carlo calculation can be described most 
 easily by reference to the simplified flow chart shown in 
 Figure 4. By means of this flow chart we shall follow the 
 particle tracing process, making relatively brief comments 
 on the various steps implied by each box, on each routine, 
 and on the main program variables. The program is written 
 in FORTRAN IV and is designed to be run on IBM 360/75 under 
 the HASP-MVT system. 
 
 B • Calculational Procedure and Program Description 
 
 To obtain an overall view of the calculations, we note 
 that there are three basic steps involved in the simulation 
 process. The first is the initialization of all parameter 
 values, i.e., prepare the particle data and predict the 
 unknown dominate parameter values. The second is the main 
 loop of particle tracing which generates census information. 
 The third uses the census information and takes tallies to 
 estimate the histograms (electron energy or number density 
 
53 
 
 Input control 
 
 
 
 ' * 
 
 
 
 
 Initialization and 
 particle data generation 
 (E, Z, U, WT) 
 
 
 
 < ' 
 
 
 
 
 Follow initial source particles 
 to census, tally particle states 
 
 
 
 
 i 
 
 
 " 
 
 
 
 
 last source 
 particle? 
 
 No 
 
 
 
 
 
 * 
 
 Yes 
 
 
 
 Follow residue particles 
 to census and tally states 
 
 
 
 i 
 
 . 
 
 
 1 
 
 f 
 
 
 
 
 last residue 
 particle? 
 
 
 
 No 
 
 
 
 • 
 
 Yes 
 
 
 From state tallies (particle histories) 
 calculate estimatd parameter values 
 
 + 
 
 Renormalize and correct the ion density, 
 reinitialize parameter values increment 
 t n to tn+1 time pycles and print the 
 output quantities 
 
 
 > 
 
 ' 
 
 
 If a prescribed number of time cycles 
 have been processed, output and stop. 
 
 
 Figure 4. Flow chart of the Monte Carlo simulation code. 
 
54 
 
 frequency distribution vs. energy), and parameter values. 
 Finally the dominate parameter va ues are corrected and the 
 program returns to the first step. 
 
 Let us follow the logic flow diagram (Figure 4) in the 
 order of the number marked on the left hand side of the boxes, 
 and describe their individual functions. 
 
 1. Input Control: The main program is usually compiled 
 and the loaded version is then stored on disk storage, 
 ready for execution. In order to offer a wide flexibility, 
 the following input parameters are used to control the 
 program runs. 
 a) NDISK : program start control. The program can 
 
 start from the beginning (cycle 0), or it can 
 start from the stopping point of the previous 
 run. This feature enables long runs to be 
 broken up into several short runs. It also 
 offers debugging flexibility and economy of 
 computer time. While a long run may run 
 up to 150 time cycles, we can always break 
 it up into multiple numbers of fives and tens 
 cycles. Thus we can adjust other parameter 
 values on a cut and try basis without wasting 
 much computing time. The input parameter 
 for this control purpose is NDISK: input 
 for storing from very beginning; input 1 for 
 starting from the latest tenth cycle of last 
 
55 
 
 run; input 2 for starting from the latest 
 fifth cycle. 
 
 b) Kl, K2, and EDI: Regional weights and dividing 
 
 energy. The Russian Roulette and splitting 
 variance reduction techniques use three 
 input parameters to control the amount of 
 killing off and/or splitting of particles. 
 This also determines where to divide energy 
 regions for the Russian Roulette and 
 splitting, assuming that the option to use 
 Roulette is selected. The input parameters 
 are Kl: the weight of particles in region I 
 (Kl '■ 1) , K2: the inverse of the weight w2 
 of particles in region II where w2 = 1/K2 
 (K 2 '• 1) , EDI: the energy dividing line, 
 namely (0 to EDI) is region I, (EDI to E ) 
 region II where the energies are in electron 
 volts. 
 
 c) NTC : The total number of time cycles to be run. 
 DTC : The input parameter to control the length 
 
 of census time At of the time cycles. In 
 terms of distance travelled by the particles, 
 At is equal to a fraction or multiple of 
 one mean free path , corresponding to 
 average energy of particles in the system. 
 
56 
 
 d) NTP: The total number of particles that actually 
 
 exist in the simulation system. 
 NTW: The artificial* total number of particles 
 
 in the system during Russian Roulette and 
 
 splitting games. 
 NS : The number of source particles injected into 
 
 the system during each time cycle. 
 
 e) KELC: The input parameter that controls the impor- 
 
 tance sampling option of the program, i.e., 
 the multiplying factor used to suppress the 
 elastic cross section (reduces the probability 
 of elastic process by a factor KELC). 
 
 f) EDP : The input value of the ratio of the electric 
 
 field strength to the pressure. 
 PDPO: The input value for the reduced pressure of 
 the system ( P n where p« = 1 torr). 
 
 g) NIP: The input parameter used to control double 
 
 processing: NIP = 2 for double processing, 
 NIP = 1 for single processing. 
 IUSN: IUSN s 1 for dual-collision type double 
 processing 
 
 IUSN = -1 for dual-direction type double 
 processing 
 
 
 *The total number modified by scaling and weighting 
 factors. 
 
57 
 
 h) D: The input value of the width of the slab 
 (diameter of the cylinder) which defines 
 the external boundaries of the medium. 
 JZ: The number of zones the slab is divided 
 into, or the number of internal zones. 
 
 i) NCUM: Inp t control to give cumulative results 
 (time average) over the time cycles or a 
 cumulative average over NCUM time cycles. 
 NAVP : Input constant to control the power of At, 
 (or &D) namely At.**NAVP is the weight of 
 the cycle to be used for cumulative results. 
 These two parameters are used to implement 
 antithetic variates techniques along the 
 time axis. 
 
 j) EO: The energy of source particles. 
 
 SR: The rate of the source particles entering 
 the system. 
 
 k) FID: An array that contains the initial guess 
 distribution (number density vs. energy). 
 
 1) NUSC: An input parameter used to control the up- 
 scattering elastic collision process; NUSC = 
 1: with upsacttering, NUSC = 0: without 
 upscattering. 
 2. Data preparation: The initialization is entered only 
 
 once during the calculation, where the time cycle is 
 
 (or at time t„). After the input phase, all the pertinent 
 
58 
 
 parameters are set to their proper permanent or temporary 
 values. Before entering the next phase to trace the 
 particle histories, several dominate parameter values 
 must be predicted. The data preparation routine is 
 called in to generate particle data (E.: energy, u..: 
 direction of travel, Z.: position, Wt . : weight) based 
 on the guess for the initial distribution (generate 
 machine particles). Then the average energy of the 
 system is calculated and is used to predict the census 
 time (At or equivalently DCEN) as well as the number of 
 ions(N + ) . 
 3. Particle tracing: With the point of origin, direction 
 of propagation, and energy (velocity) known for the 
 initial source particles, the next phase follows the 
 particles through a series of stochastic interactions 
 until the census time is reached, and the particle data 
 and parameters values are tallied and updated. During 
 the course of tracing the particle histories, all the 
 relevant events are recorded and later on can be inter- 
 rogated to provide the output quantities of interest. 
 This section of the program is the central part of the 
 main program. It contains the main loop for tracing all 
 the particles in the system in a self consistent way. 
 Most of the variance reduction techniques are incorporated 
 here, including double processing, Russian Roulette, 
 and splitting, as well as the various versions of 
 
59 
 
 importance sampling techniques. 
 
 4. Secondary particle tracing: The residue particles, i.e., 
 particles generated by ionization collision process 
 (the secondary electrons so produced) are treated in all 
 respects like the regular source particles, except that 
 their starting positions and time coordinates as well as 
 directions are those associated with the particles that 
 generated them. These residue particles may go further 
 and produce other residue particles (secondary electrons) 
 if their energy is high enough, thus forming a short 
 production chain. The probability of such chain produc- 
 tion is small however, since the average energy of the 
 secondaries is quite low. The end of the main loop 
 comes after all such residue particles are processed. 
 
 5. Data calculations: The details of the phase of the 
 computations depend upon the specific goals and charac- 
 ter of study. Our primary interest is the particle 
 energy distribution (or number density vs. energy). 
 This frequency distribution and other pertinent para- 
 meters are estimated (or calculated) using the stored 
 data from various parameter counters in the main loop 
 (tallied information) as well as tallied histories. 
 The quantities of interest include local values (for 
 the current time cycle only) and the cumulated values 
 from several time cycles. After these calculations, the 
 next phase is to prepare for the processing of new time 
 
60 
 
 cycle. 
 6. Normalization and printout: Up to this stage, all the 
 output quantities have been calculated. Next the output 
 information is printed out, and at the same time, if the 
 cycle number is the multiple of five, all the particle 
 data and related parameter values are stored on the disk. 
 Then the next run can be read off of the disk and the 
 process can proceed from where we left off the last time. 
 All the parameter values are again initialized, and 
 new source particles are generated for the next time 
 cycle. Particles that have escaped from the system or 
 have been absorbed create holes in the particle data 
 storage which are filled up by the new source particles 
 and the residue particles produced during the time cycle. 
 Before going on to the next cycle, the ion density is 
 renormalized and corrected in a semiadaptive way accord- 
 ing to a balance factor BF. The ion density distribution 
 parameter N is a dominate parameter since it affects 
 several processes such as the rate of recombination and 
 the rate of super-elastic collision (refer to Equations 
 27 and 28 ) . It is corrected in such a way as to 
 accelerate convergence, till the system reaches a bal- 
 anced situation. Then increment time from t to t . 
 The output quantities are the following: 
 a) The kind of gas in the medium, source particle 
 
 energy, pressure, slab thickness, and At the time 
 
 interval of this cycle. 
 
61 
 
 b) The number of time cycles, the time in seconds, 
 
 the number of source particles generated during the 
 cycle time, and the average energy of the particles 
 in the system. 
 
 c) The number density distribution vs. energy, the 
 single cycle distribution, the cumulated distribu- 
 tion, the number-density deviation, the average- 
 energy deviation, the average energy, and the energy 
 interval vs. energy. 
 
 d) The number density and the average energy vs. z 
 (position zone) . 
 
 e) The ion distribution (N+), the number of ionizations 
 (NIZN), the number of escaped particles (NES), the 
 number of absorbed or recombined particles (NAS), the 
 number of super-elastic collisions (NSE), the number 
 of excitations to metastable levels (NMET), and the 
 number of excitations to other levels (NEXC). 
 
 f) All the input control parameters as listed in 
 section IV-1. 
 
 g) The electric field strength (EF), the source rate, 
 the W-value, the average energy of escaped particles, 
 and the average energy of absorbed particles. 
 
 h) The number check on the density disbributions. 
 All the above outputs include both current cycle 
 values and the cumulated values over the past cycles. 
 
62 
 
 Final checks: The last stage is to check whether the 
 prescribed number of time cycles have been processed, 
 adequate statistics have been collected, and all the 
 useful data have been stored on the disk. If so, the 
 program stops. 
 Further programming details are provided in Appendix 
 
63 
 
 CHAPTER V 
 RESULTS 
 
 A. Convergence and Reliability 
 
 The present Monte Carlo Model is a predict-correct 
 iterative scheme, with semiadaptive control ability. For 
 such an iterative procedure , our main criteria for the 
 evaluation and determination of convergence are based on 
 the following inferences: 
 
 1. The balance factor (BF) as defined in Chapter II 
 (Section II-B-3) signifies whether or not the system 
 has reached steady state, i.e., converged to a final 
 solution. At steady state, the value of BF should be 
 approximately equal to unity. It may approach to unity 
 in both directions, and due to semiadaptive control 
 procedure used, it may oscillate around unity. Larger 
 amplitude oscillations are observed at first, but they 
 rapidly damp out. 
 
 2. The estimated values of the major parameters, e.g. the 
 average energy and the W-value show a similar fluctua- 
 tion, and then they converge to nearly constant values 
 as steady state is approached. 
 
 3. The validity of the convergence has been checked using 
 test problems where analytic results are available. 
 
 A good comparison has been obtained, showing that the 
 program and the model are working correctly, free of 
 obscure errors. 
 
64 
 B. Error Estimation and Analysis 
 
 1. Error Analysis 
 
 1. As mentioned in early chapters, Monte Carlo simula- 
 tion is equivalent to performing mathematical experiments, 
 the results being based on observations of such experiments. 
 From this analogy we can derive the following procedures 
 for the analysis of errors or accuracies of our calculations. 
 
 It is convenient to subdivide experimental errors into 
 three broad types, namely: random errors, systematic errors, 
 and blunders. In general, the experimental error is some 
 additive function of all three, while blunders can hopefully 
 be eliminated. We shall describe them in detail in follow- 
 ing sections. 
 
 a. Random errors - such errors are of great concern in 
 Monte Carlo type calculations. They generally involve 
 statistical fluctuations or deviations, representing 
 the difference between the singly measured value and 
 the best value of a set of measurements, i.e., the 
 difference between the arithmetic mean and the true 
 mean. Sometimes such errors are referred to as ele- 
 mentary (or inherent) errors in the measurements. The 
 commonly used measure of random errors is the variance 
 or standard deviation. 
 
 b. Systematic error - a systematic error tends to have 
 the same algebraic sign, ± . e. it' is either an additive 
 or subtractive quantity introduced in the measurement 
 
65 
 
 process. It is an unpleasant and insidious contribu- 
 tion which is not generally amenable to statistical 
 treatment. Thus such errors seriously impair the 
 reliability of the estimation. Typical examples 
 include: 
 
 i) Incorrect assumptions or approximation in the 
 representation of certain processes, 
 ii) Constructional faults and mistakes in the 
 algorithms or subroutines, 
 iii) Inadequate regard of constancy of experimental 
 conditions or inadequate sampling techniques 
 (biased) . 
 c. Blunders - these are outright mistakes which should be 
 corrected by all means. Possible examples include: 
 
 i) Incorrect logic, misunderstanding of the problems, 
 ii) Errors in transcription of data, 
 iii) Mistakes in constants used, 
 iv) Confusion of units. 
 In any type of calculation, the systematic error as well 
 as blunders should be removed. One way for detecting such 
 errors is to compare the program against some known reliable 
 solutions. Only the random errors are subject to reduction 
 by the various treatments amenable to attack by variance 
 reduction technqiues. 
 
 The precision in an estimated (mean) value is propor- 
 tional to the reciprocal of the statistical error and is 
 
66 
 
 high if the statistical error is small, i.e. the accuracy 
 is high if the net systematic error is small. Usually, 
 but not necessarily, high accuracy implies small statistical 
 errors as well. 
 
 As discussed in later sections, our model and program 
 has been checked against several test problems, and good 
 agreement has been obtained. Therefore, it can be assumed 
 that any serious systematic errors have been removed and 
 that the program is free of blunders. Thus, we shall con- 
 centrate on the analysis of random errors. 
 
 No thorough analysis of variance has been made. Due 
 to the intricate way that the histories calculated during 
 a given time cycle depend on previous time cycles, a 
 meaningful variance estimate beyond the first time cycle 
 seems to be out of the question. However as the system 
 reaches steady state, the deviations or variations (statis- 
 tical fluctuations) of the estimated parameters can be 
 estimated by means of standard deviation (or variance) 
 over subsequent time cycles. Namely, we use the following 
 formulation : 
 
 S. = 
 
 1 
 
 £. (f . . - f .) 
 
 n 
 
 V i 
 
 (61) 
 
 where s . : 
 l 
 
 f . 
 ij 
 
 f . : 
 .i 
 
 the standard deviation at the ith energy interval 
 over n consecutive time cycles. 
 
 the parameter value (density distribution func- 
 tion of ith energy interval, at jth time cycle. 
 
 the average parameter value of ith interval over 
 n time cycles. 
 
67 
 
 A heuristic approach was adopted to measure local deviations 
 or statistical fluctuations at each time cycle. This pro- 
 vides a relative magnitude of deviation or dispersion which 
 can be used to compare the various variance reduction tech- 
 niques. The approach is described as follows: 
 
 The estimated major frequency distribution (the elec- 
 tron energy density distribution function in our case) is 
 obtained in the form of histogram. The abscissa of the 
 histogram is divided into so-called class intervals (energy 
 intervals in our calculations). In each such interval an 
 
 erect rectangle (block) of heigh f. and width E. is formed. 
 
 1 1 
 
 This block-area type of distribution is the fitted fre- 
 quency distribution curve. We devise a numerical descrip- 
 tion of local deviations by defining: 
 
 i) the location index of the center of the erected 
 
 rectangles in the histogram as i and with abscissa 
 
 value E . . 
 
 l 
 
 ii) the spread or dispersion around the center. 
 The total number of particles in each class interval 
 (energy interval) is proportional to the area of the 
 rectangular flock (f. AE.) / the particle's energy in this 
 class interval falls in the range (E. - AE . ._ f E. + AE. /? ) 
 with overall average energy E~. = E.. Then the following 
 relationship holds (refer to Figure 5): 
 
68 
 
 - 
 
 k 
 
 
 
 
 
 
 
 UJ 
 
 
 AE, 
 
 
 «*- 
 
 
 
 
 
 
 i 
 
 > 
 
 AEj*| 
 
 
 o 
 
 
 
 c 
 
 Z3 
 
 , 
 
 . 
 
 
 c 
 o 
 
 
 
 »l 
 
 
 f M 
 
 
 
 3 
 
 5 
 
 
 
 
 
 < 
 
 ' 
 
 i 
 
 ' 
 
 
 ^_ 
 
 E| 
 
 E i + I E l+2 
 
 E-Energy, eV 
 
 Figure 5. Rectangular block representation of the electron 
 energy distribution histogram. 
 
69 
 
 ifi til 
 
 AE. " f . 
 
 AE 
 
 or A£ i - f i AE7 (62) 
 
 where AE~. is the deviation of the average energy FT. from E. 
 
 or AE~. = E\ - E.. Also Af. is the corresponding deviation 
 111 1 r -a 
 
 of the frequency distribution at location index i. 
 
 Equation (62) provides a relative measure of the ampli- 
 tude of the deviation of density distribution curve at energy 
 E. which can be used for comparing the efficiency of 
 different variance reduction techniques as well as provide 
 an heuristic clue to the modification of the fitted density 
 distribution curve. 
 
 2. Data Filtering 
 
 The random errors (statistical fluctuations) result 
 in bumps and ridges in the estimated frequency distribution 
 function. It may be possible to alleviate many of these 
 distortions by means of data smoothing techniques (data 
 filtering) . 
 
 Two types of data smoothing techniques commonly used 
 in the processing of time series might be used here: 
 a. Moving averages - this technique operates by replacing 
 each point of the frequency distribution function 
 (height of the histogram at E.V.) at time cycle t. 
 with an average value of several subsequent points 
 in the time cycle series. Thus if f . . is the value 
 
70 
 
 of frequency function at ith energy interval and jth 
 time cycle, then f . . is replaced by 
 
 
 J/M 
 
 where W., the weighting factor for the jth time 
 cycle, usually has the value of 1 or we can define it 
 as a function of &t . . The value of M is the number of 
 successive points of the time cycles to be included 
 in each average, 
 b. Parabolic filtering - this is similar to the moving 
 
 averages technique, but M is selected as an odd number, 
 the point to be replaced is located in the center of 
 these M points, and a parabola is fitted to these points 
 by means of least squares. Thus it is replaced by the 
 corresponding point f . on the derived parabola. The 
 process is repeated by shifting the center point to the 
 next one in the time cycle series. 
 
 The technique of moving averages (a) is employed in 
 the program, because of it's simplicity and the fact that 
 the parabolic smoothing technique is too sensitive to the 
 changes in the distribution function. Note that this 
 smoothing (filtering) technique only becomes effective as 
 the simulated system reaches steady state. 
 
71 
 
 C. Solutions of the Problem 
 
 1. Preliminary Checks ■ 
 
 In order to verify that the model is valid and correctly 
 working, and the computer program is free of bugs f blunders, 
 and systematic errors, we solved a specific problem first 
 reported by J. A. Smit (11) and later extended by Heylen and 
 Lewis (10). Their solutions for the electron energy distri- 
 bution were obtained analytically based on the Boltzmann 
 transport equation for the special case characterized as 
 follows: 
 
 i) electric field present 
 ii) no external sources 
 iii) infinite size medium of background noble gases. 
 
 The comparisons are made for helium gas at E . =4, 5, 
 
 10 V-. cm~ . (E^_ . : electric field strength to pressure ratio), 
 ' f/p 
 
 The results are shown in Figures 6, 7 and 8. Both Smit and 
 
 Heylen' s results have taken into account all the collision 
 
 processes including those due to elastic loss as well as 
 
 inelastic losses, Heylen' s results is more up-to-date, for 
 
 lowE^., values onlyE,. ', = 5 is given, which was chosen for 
 f/p J f/p 
 
 the present comparison. In so far as possible, the same 
 cross sections as Smit and Heylen and Lewis used were 
 incorporated in the present calculations. The slight 
 discrepancies apparant at the low-energy end is probably 
 due to differences in treating the scattering process, and 
 
72 
 
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75 
 
 the many approximations required in their analytic solutions. 
 Also, some of the cross sections used in this calculations 
 are experimental values, which may differ by some amount 
 from that used for analytic solutions. 
 
 For the high energy region, (near the source energy) 
 checks have been made against the analytic solutions 
 reported by Lo and Miley (32). As shown in Figure 9, the 
 general shape of the electron energy flux distributions are 
 in reasonable agreement. In this case it was not practical 
 to use the same cross sections as incorporated in the 
 analytic solutions, and this may account for some of the 
 differences observed. Also the analytic solutions are 
 obtained for an infinite medium vs. a 1 cm slab in the 
 present case. Thus the analytic solutions do not allow 
 leakage, although this becomes important in the present 
 calculation for low pressures. Lo and Miley' s results are 
 plotted in Figure 9 for the source energy E n = 1500 eV 
 and E = 500 eV, whereas the present results are for E n = 
 1000 eV. The plot shown is for the flux density (normalized) 
 instead of energy number density. They are related as 
 follows : 
 
 CyP(E) = f(E)-V(E) (63) 
 
 where ^(E) 
 f(E) 
 V(E) 
 
 electron flux density 
 electron number density 
 Electron kinetic velocity 
 
76 
 
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77 
 
 Sines we only intend to compare general shapes of distribu- 
 tion, the normalizations used for the various curve, shown 
 here are arbitrarily choosen for optimum display of informa- 
 tion. 
 
 2. The Electron Energy Distribution Function without 
 Electric Field 
 
 The calculations for electron energy distribution func- 
 tions for helium with a high energy electron source, but no 
 applied electric field (Figures 10 to 14) were carried out 
 at two different pressures and various source rates. The 
 slab thickness D is chosen to be fixed at D = 1 cm, which 
 is equal to the actual diamter of cylindrical tube used in 
 the laboratory (7). The two pressures are chosen to be 
 P = 10 torr and P = 760 torr, corresponding to a typical 
 low-pressure case and a normal atmospheric pressure. 
 
 We have chosen two different source energies (E n ), 
 one at E = 1,000 eV and the other at E = 70 eV. These 
 values roughly correspond to the highest energy and the 
 average energy in the primary electron energy spectrum for 
 MeV alpha-particle irradiation of helium as done in the 
 experimental studies conducted at the University of Illinois (6) 
 
 The range of interest for the source rates for a boron 
 
 coated tube irradiated in the Illinois TRIGA reactor lies 
 
 14 18 "} 
 between 10 to 10 (#/cm -sec). However, results shown 
 
 for source rates above about 10 are questionable since 
 
 the resulting ionization density becomes large enough that 
 
78 
 
 electron-electron collisions become important, and this 
 process was neglected here. 
 
 The major quantity of interest here is the electron 
 energy distribution function which were defined as n~(E) 
 in Chapter II. For simplicity, we represent the normalized 
 distribution corresponding to n"~(E) by f(E). This 
 normalization is defined by: 
 
 J 
 
 E 
 o 
 
 f(E) dE = 1 (64) 
 
 o 
 
 For the plasma system without electric field (E = 0), 
 
 the distribution functions with source energy E = 1000 eV 
 
 ^ J o 
 
 are shown in Figures 10 and 11 for the pressure p = 10 torr, 
 and p = 760 torr respectively. Similar distribution func- 
 tions for E = 70 eV are shown in Figures 12 and 13. 
 
 o 3 
 
 An idealized Maxwellian distribution plus 1/E tail 
 are shown in Figures 10 to 13 for reference. (The Maxwellian 
 plus 1/E distribution was used as the initial distribution 
 for these calculations. Note that since this curve is for 
 reference only, it is arbitrarily normalized to a point on 
 the Maxwellian curve and does not satisfy Equation 64.) 
 For the range of source rates mentioned, the calculated 
 energy distribution for helium (He) are reasonably Maxwellian 
 at low energy region. The differences due to variations in 
 the source rates show up at the high energy end, especially 
 near the source energy, where the magnitudes seem to depend 
 linearly on the values of source rate. This is somewhat 
 
79 
 
 deceptive, however, since the absolute magnitude at any 
 point is due to the normalization. (This point is stressed 
 again later in connection with Figure 24). 
 
 The fact that the distributions drop off rapidly for 
 energies above about 20 eV is attributed to the fact that 
 the threshold energy for inelastic processes is 20 eV, 
 above which energy the ionization and excitation processes 
 bring more electrons into the low-energy region. 
 
 In order to investigate the change in distribution 
 
 function at very high source rates, a calculation for S = 
 
 22 3 
 
 10' particles/cm -sec was carried out. As mentioned 
 
 earlier, the neglect of electron-electron collisions makes 
 the accuracy of this calculation questionable. Still the 
 trends are of interest, and it is seen from Figure 14 that 
 the distribution curve changes drastically. In fact, it is 
 no longer Maxwellian at the low-energy end. Due to the 
 high source rate, more high-energy particles and more 
 particles with energy in the range from 0.3 eV to 20 eV 
 are present in the system. 
 
 As discussed earlier, the distribution curve drops 
 sharply for energies above the excitation threshold energy 
 (at E = 20 eV) due to the dominance of ionization and excita- 
 tion collision processes. At the higher pressure, the 
 distribution curves decrease more rapidly compared to low 
 pressure, since increased scattering slows the electrons 
 down more rapidly. This effect is shown in Figures 15 and 16, 
 
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87 
 
 16 "3 
 for the same source rate S =10 (#/cm -sec) and source 
 
 o 
 
 energy, but two different pressures. The reduction in 
 magnitude of the high energy tail as the pressure increases 
 is clearly illustrated. 
 
 In these calculations, the rate of both leakage and 
 absorption (recombination) are strongly dependent on the 
 pressure. At higher pressures (e.g., p = 760 torr), very 
 few particles leak out of the system, and the loss are 
 dominated by the absorption process. For lower pressure 
 (p = 10 torr), fewer particles are absorbed, while more 
 escape or leak out of the system. Also, more particles 
 escape in the high-energy region, but the absorped particles 
 are mainly in the low-energy region. 
 
 The estimated major parameters are summarized in Table 
 l f which includes the average energy of the system (E~) , 
 the number of positive ions (N + ) and the W-values for various 
 combinations of source rate S , source energy E and pressure 
 
 P. 
 
 The W-value is defined as the amount of energy 
 
 required (input) to produce an ionization pair (positive 
 
 ion and electron), or 
 
 (N .E - Em - E m , ) 
 T , , so -Les +ao ires 
 
 W-value = (65) 
 
 z 
 
 Where N : the number of source particles introduced into 
 the system during the period At. 
 
 E m : the total energy lost with escaping particles 
 during the time At. 
 
88 
 
 Table 1. Calculated Parameter Values for the Plasma 
 Without a Electric Field 
 
 * 
 
 p 
 
 (torr) 
 
 E 
 o 
 
 (eV) 
 
 S 
 
 °3 
 
 (#/cm -sec) 
 
 E 
 
 (eV) 
 
 10 
 
 1000 
 
 10 18 
 
 0.171 
 
 10 
 
 1000 
 
 10 16 
 
 0.157 
 
 10 
 
 1000 
 
 10 14 
 
 0.111 
 
 760 
 
 1000 
 
 10 18 
 
 0.167 
 
 760 
 
 1000 
 
 10 16 
 
 0.123 
 
 760 
 
 1000 
 
 10 14 
 
 0.104 
 
 10 
 
 70 
 
 10 18 
 
 0.104 
 
 10 
 
 70 
 
 10 16 
 
 0.082 
 
 10 
 
 70 
 
 10 14 
 
 0.075 
 
 760 
 
 . 70 
 
 10 18 
 
 0.082 
 
 760 
 
 70 
 
 10 16 
 
 0.075 
 
 760 
 
 . 70 
 
 10 14 
 
 0.075 
 
 Maxwell 
 
 ian di 
 
 stribution 
 
 
 
 
 
 
 0.05 
 
 (#) W-value Range 
 
 0.125 x 10 14 41 -^- 63 
 
 0.125 x 10 13 35 — ^ 53 
 
 0.125 x 10 12 49 — — 58 
 
 0.583 x 10 15 32 —^ 38 
 
 0.446 x 10 14 40 ■ 65 
 
 0.67 x 10 13 35 —w 54 
 
 0.125 x 10 14 50 -^^ 62 
 
 0.125 x 10 13 47 —— 63 
 
 0.124 x 10 12 40 --—57 
 
 0.16 x 10 15 47 ~- 65 
 
 0.141 x 10 14 42 *-w 58 
 
 0.157 x 10 13 39 -55 
 
 *P: pressure, E : source energy, S : source rate, 
 E~: average energy of the particles, N : number of positive 
 ions. 
 
89 
 
 E T ,: the total energy lost with recombining particles 
 during the time At 
 
 N : the total number of ionizations produced during 
 the time period At. 
 
 As seen from Table 1, the average energy E~ tends to be 
 higher for higher source rates and higher source energy / 
 and it is lower for higher pressures. However, the differ- 
 ences are in general very small. (All the average energies 
 are consistently greater than that of a Maxwellian distribu- 
 tion). The values for ion density N + are roughly proportional 
 to the square root of source rate, and are higher for higher 
 pressures, virtually independent of source energy. 
 
 The W-values are quite sensitive to statistical fluctua- 
 tions, and the ranges of variation obtained over the last 10 
 time cycles in a given calculation are shown in Table 1. 
 The W-values obtained here do not show any clear dependence 
 on the system parameters, thus variation indicated may be 
 totally due to the statistical fluctuations. This is not 
 too surprising since the W-value is known to be roughly 
 independent of particle energy, etc. The W-value for 
 helium obtained experimentally by Jesse and Sadouski (34) 
 is 43. A reasonable value to choose for the W-value from 
 Table 1 would be 50, which is little bit higher than the 
 experimental result. The reason for this discrepancy is 
 not clear, but it may be due to small inaccuracies in the 
 ionization-excitation cross section employed here. The 
 W-value is, by definition, quite sensitive to the details 
 
90 
 
 of the ratio of the ionization to the excitation cross 
 section as a function of energy. 
 
 3. The Electron Energy Distribution Function with an Applied 
 Electric Field 
 
 The calculated electron energy distribution functions 
 
 in He with an applied electric field (E /p = 10) are plotted 
 
 in Figures 17 to 21 , in a manner similar to that in the 
 
 previous section. All these calculations are carried out 
 
 at the electric field to pressure ratio E /p = 10. Smit's 
 result (without source or S =0) are plotted together with 
 present calculations at given pressure and source energy 
 (p = 10, 760 and E = 1000 eV, 70 eV), with three different 
 source rates. It is observed that the distribution func- 
 tions are highly non-Maxwell ian. For the source rate in the 
 range 10 #/cm -sec to 10 #/cm -sec, the distribution 
 curves at low energy end are quite close to the Smit's 
 results (11). At the high energy end, they have a shape 
 similar to the curves of no electric fields. Again differ- 
 ences due to the_ source rate show up at high-energy region. 
 Below 10 eV, all the distribution functions fall on the 
 top of each other, and they begin to spread out as the energy 
 increases above 10 eV. For high pressures or low source 
 
 energy, more difference (between S =0 and S jk 0) show 
 
 ^ J ' o o ^ 
 
 up at high-energy end than that of low pressure and/or high 
 
 22 
 
 source energy. Again at very high source rates S =10 
 
 #/cm -sec) the distribution function changes appreicably . 
 
 
91 
 
 In Figure 21, the comparison is made with a very high source 
 
 22 3 
 rate (S =10 /cm -sec) and with one in the region 
 o 
 
 of interest (S =10 ,p = 760 torr) . For very high source 
 rate, the distribution function changes quite a lot; it 
 still preserves the general shape, but at low-energy end, 
 it is much smaller in amplitude. (However, as stressed 
 earlier, the high source rate calculation is questionable 
 due to theneglect of electron-electron interactions.) 
 
 The changes in these distribution functions due to 
 different pressures are also shown in Figures 22 and 23. 
 The observed trends are similar to those in the previous 
 section for the electric field E = case. 
 
 So far, the normalized distribution functions f(E) 
 
 have been displayed. However, to stress the fact that the 
 
 absolute magnitudes of the curves depends strongly on the 
 
 source (shapes have received our interest to this point), 
 
 Figure 24 shows the unnormalized electron energy distribution 
 
 functions for He (with electric field or E f / 0) are plotted, 
 
 with E_/p =10, p = 10 torr, E = 1000 eV, and three differ- 
 f / ^ ' ^ ' o ' 
 
 ent source rates (S = 10 14 , S = 10 16 , S = 10 18 ). The 
 
 o ' o ' o 
 
 magnitude (unnormalized) distribution is defined as 
 
 N~ (E) = N + • f(E) (66) 
 
 where N + : number of positive ions 
 f(E): normalized distribution 
 
 The values of N + are 0.24 x 10 15 , 0.24 x 10 14 , and 0.24 x 10 13 
 
 l ft 1 fi 
 
 corresponding to the three source rates S =10 , S =10 
 
92 
 
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101 
 
 14 
 and S = 10 . respectively. 
 o 
 
 In order to show the effect of recombination, a normal 
 case is compared in Figure 25 to a calculation where the 
 absorption cross section was arbitrarily increased 100 times. 
 It is observed that the major effect comes in at low-energy 
 region. More and more electrons recombine as they thermalize 
 and hence, the lower energy region of the distribution is 
 depleted. Since recombination does not occur at high 
 energies, this region is not affected. 
 
 The calculated major parameter values for the system 
 with an electric field are summarized in Table 2. As expected 
 on physical grounds the values of the average energies for 
 this case are much higher than the corresponding values with 
 no electric field (compare to Table 1). However, other 
 general trends are the same, e.g. the value of the average 
 energies are slightly higher for higher source rate and/or 
 higher source energy, but are lower for higher pressures. 
 
 For comparison the average energy (7.7 eV) found by 
 
 Smit for an electric field and E , . =10 but with no 
 
 f/P 
 
 source present, is included in Table 2. It is in general 
 higher than the present values. The reason is not entirely 
 clear. At first thought the additional high-energy source 
 electrons might be expected to raise the average energy. 
 However, as found by T. Ganley (37), the presence of high- 
 energy electrons to produce ionization allows the electrons 
 in the Maxwellian region to decrease in energy, and as a 
 result the overall average may actually decrease. 
 
102 
 
 Table 2. Calculated Parameter Values for Plasma with 
 Applied Electric Field (E . = 10) 
 
 * P o o 3 E N + Modified 
 (torr) (eV) (#/cm -sec) (eV) (#j W-value range 
 
 0.24 x 10 15 35-V-44 
 
 0.21 x 10 14 34 — 54 
 
 0.24 x 10 13 37.5-— 60 
 
 0.23 x 10 17 37-- 50 
 
 0.204 x 10 16 30-^51 
 
 0.24 x 10 15 34^-57 
 
 0.21 x 10 15 31 — 35 
 
 0.21 x 10 14 34 — 50 
 
 0.213 x 10 13 31 ~ 56 
 
 0.284 x 10 16 35 46 
 
 0.26 x 10 15 39 — 48 
 
 0.24 x 10 14 32-^-49 
 
 10 
 
 1000 
 
 io 18 
 
 8.1 
 
 10 
 
 1000 
 
 10 16 
 
 7.5 
 
 10 
 
 1000 
 
 io 14 
 
 7.3 
 
 760 
 
 1000 
 
 10 18 
 
 7.0 
 
 760 
 
 1000 
 
 io 16 
 
 6.6 
 
 760 
 
 1000 
 
 io 14 
 
 5.4 
 
 10 
 
 70 
 
 IO 18 
 
 5.86 
 
 10 
 
 70 
 
 io 16 
 
 5.18 
 
 10 
 
 70 
 
 io 14 
 
 4.46 
 
 760 
 
 . 70 
 
 io iB 
 
 5.7 
 
 760 
 
 70 
 
 io 16 
 
 5.0 
 
 760 
 
 70 
 
 io 14 
 
 4.3 
 
 Smit's distribut 
 
 ion 
 
 
 
 
 
 
 7.7 
 
 tp_: pressure; E = source energy; S : source rate; 
 E: average energy; N : number of positive ions. 
 
103 
 
 The values of N" 1 " are roughly proportional to the square 
 root of the source rate (recall Equation 33) and are inde- 
 pendent of the pressure and source energy. 
 
 Strictly speaking, the definition of the W-value for 
 the field case should include an allowance for energy 
 gained to electrons in the electric field which then 
 result in ionization. There is no way to include this 
 energy transfer in the present case, so the values listed 
 in Table 2 are based on the ion pair production divided 
 by the energy input with the high-energy source electron 
 alone (no field contribution). This is, then the same 
 formulation (Eq. 65) as that of no field case, and we 
 lable the result as the "effective" or "modified" W-value. 
 Again these values fluctuate considerably, so the ranges 
 observed in the last 10 time cycles are given f As would be 
 expected from the neglect of the field contribution, the W- 
 values in this table are in general smaller than that of 
 no field case. As in the earlier calculation, no general 
 trend can be observed, as so far as the dependence of the 
 
 parameters (E , S , p) are concerned, and this is consistent 
 
 ^ o ' o ' ^ ' 
 
 with the earlier observations and the literature which indicate 
 that the W-values are fairly insensitive to these parameters. 
 
 We have concluded so far that, the distribution func- 
 tions for non-field case (E_ = 0) are more or less assume 
 
 r 
 
 a Maxwellian distribution in the low-energy region for 
 
 18 
 source rates below 10 . However, when an electric field 
 
104 
 
 is added, the distribution shifts to that found by Smit in 
 the low-energy region. This is,. in effect, the well-known 
 Druyvesteyn distribution. It is significant that the addi- 
 tion of a source adds a tail to this distribution, but it 
 does not drastically change the low-energy region. By 
 intuition, it might be expected that Smit's distribution 
 (with electric field or E / 0) would change back to a 
 Maxwellian distribution as the electric field approaches 
 zero. In order to display the transition region, the 
 
 distribution functions for smaller electric field (E_, = 1) 
 
 f /P 
 
 are plotted in Figure 26 together with the calculated 
 
 distribution functions at E,. = 10 and E_ = 0. The high- 
 
 f/p f 
 
 energy tail of the distribution curve are not significantly 
 affected by reducing the electric field E but the distri- 
 bution function in the low energy region starts to approach 
 a Maxwellian distribution as expected. 
 
 D. Conclusion on Monte Carlo Results 
 
 Based on the Monte Carlo solutions we obtained so far, 
 
 ,we can make the following comments. 
 
 18 
 
 1. For the normal range of source rate (S < 10 ), 
 
 where the neglect of electron-electron interactions remains 
 valid, the Maxwellian plus 1/E distribution is a good initial 
 approximate distribution for the no electric field case. 
 Likewise the Smit (or Druyvesteyn) plus 1/E distribution is 
 a good approximate initial distribution to be used for the 
 system with an electric field present.' 
 
105 
 
 K£> 
 
 £ O 
 En »W 
 
 OH uoipunj uojinquis.iQ 
 
 M 
 
 •H 
 P4 
 
106 
 
 2. The dependence of the distribution functions (with 
 or without electric field) upon the pressure and source rate 
 only shows up in the high-energy region, where the distri- 
 bution functions have higher tails for higher source rate, 
 and have smaller values for higher pressures. 
 
 3. The range of variations observed in the W-value . 
 calculation reflects statistical fluctuations which limit 
 the calculational accuracy of the absolute numerical W- 
 values. Still the results demonstrate that the W-value is 
 more or less independent or insensitive to the changes of 
 pressure, source rate and source energy. 
 
 4. Due to a tight budget, the primary emphasis was 
 on increasing the computation efficiency and reducing 
 the computation time to a possible minimum. Thus most of 
 the calculations were carried out with 2000 to 5000 actual 
 particles and for any particular set of parameter values, 
 and the results were generally obtained within 10 minutes 
 of (IBM 360/75) computer time. Compared with hours of 
 computer time and 5 to 10 times more particles used by 
 others for the similar calculations (e.g., see References 
 3, 4, and 9) the gain in computation efficiency seems to be 
 significant. Of course, a complete comparison is not 
 possible since all of these calculations achieved different 
 accuracies in the final results. 
 
 5. The present results also show close agreement 
 with the analytic solutions of Lo and Miley (32,33) on the 
 
107 
 
 following points: 
 
 a. The shape of high-energy portion of the distribu- 
 tion curves are in reasonable agreement as shown 
 in Figure 9. 
 
 b. The high-energy part of the distribution function 
 
 is not changed by the presence of an electric 
 
 field for field values E_ , < 10 . 
 
 f/p - 
 
 c. With an electric field, the low-energy part of 
 the distribution function agrees with Smit's 
 result; and this part is not affected appreciably 
 
 TO T 
 
 by source rates < 10 particles/cm -sec. 
 
108 
 
 CHAPTER VI 
 CONCLUSIONS 
 
 A. Prelude 
 
 The application of Monte Carlo techniques to solve 
 particle transport problems is of fundamental interest in 
 many fields of physics and engineering; especially when 
 analytic and/or numerical solutions of the basic governing 
 equations are too complex to be practical. Yet direct 
 application of Monte Carlo techniques to simulate individual 
 interactions is sometimes prohibitively costly because of 
 the large amount of computational time required. Thus 
 techniques for improving the efficiency are required. 
 Various existing variance reduction techniques can be used, 
 but they are problem dependent. Thus techniques that are 
 efficient for one type of problem might not be as effective 
 for other types of problems. Thus such techniques must be 
 judiciously selected and modified for the particular problem 
 under consideration. One basic underlying principle which 
 applies to any Monte Carlo calculation is: Apply as much 
 known information as one can (given in analytic and/or 
 numerical form) to reduce the uncertainties of the problem. 
 Whenever possible Monte Carlo experiments should be checked 
 and replaced by exact theoretical analysis to reduce uncer- 
 tainties . 
 
 The Monte Carlo experimentalist has to exercise ingen- 
 uity in distorting and modifying problems in the pursuit 
 
109 
 
 of variance-reduction techniques. Although Monte Carlo 
 methods are general devices, still a great deal of work 
 depends upon the individual's originality to create special 
 methods or numerical schemes to suit their needs. Unlike 
 the physical experimentalist, the mathematical experimentalist 
 using Monte Carlo Methods has the advantage that his 
 experimental material consists of mathematical objects which 
 can be distorted, controlled, and modified more easily. 
 
 The greatest successes of the Monte Carlo method have 
 arisen where the basic mathematical problem itself consists 
 of the investigation of some random processes. However, 
 there are exceptions involving deterministic problem- solution 
 of boundry value problems and partial differential equations. 
 The solutions of these problems are closely connected with 
 the characteristics of certain random diffusion type 
 processes (or can be converted to such type of processes), 
 therefore these problems are reduced to the modeling of 
 such processes. Thus the model and the special techniques 
 developed in present research are quite general in scope, 
 and it should be possible to apply them to many classes of 
 Monte Carlo calculations with a certain amount of modifica- 
 tion. 
 
 B. Summary of Present Work 
 
 In this research study, we have devised a general 
 mathematical model for the particle transport or diffusion 
 
110 
 
 type plasma problems which have inherent nonlinear proper- 
 ties due to recombination. Many special variance reduction 
 techniques are incorporated into the system. The model can 
 be used for calculating both time-dependent and steady-state 
 nonlinear transport problems provided that suitable methods 
 exist to calculate, predict, and correct the important 
 parameters which dominate the nonlinearities. A wide range 
 of nonlinear problems can be formulated and handled in this 
 way. The principle idea, the piecewise linearized predict- 
 correct model, is a general, simple, and efficient approach 
 for solving nonlinear problems. In principle, most nonlinear 
 problems can be cast in terms of the basic ideas and algo- 
 rithms of present work. This represents an extension of 
 earlier works of Musha and Itoh (8), Thomas (9), and Fleck 
 Jr. (11). As for the amount of computation time and effi- 
 ciency involved, the present study has made significant 
 improvements. Some of the improvements are a natural con- 
 sequence of the large storage space available in modern 
 computer systems. Additional improvements come from a few 
 simple, but effective, special techniques incorporated into 
 the present model. As described in Chapter II, these include: 
 1) The initial distribution - straightforward particle 
 tracing procedures would require following every 
 simulated particle from its birth (from source) with 
 
 initial energy E at t = t to the end of its life time 
 3J o o 
 
 history. This type of simulation would require hundreds 
 
Ill 
 
 of cycle times to achieve a final steady state condition. 
 Instead, we start with an apprixmately steady-state 
 system by guessing an initial particle energy distribu- 
 tion. Then we simulate the system by tracing particles 
 starting with energy corresponding to this input 
 distribution. This is eguivalent to skipping over the 
 initial transient time cycles and greatly reduces the 
 computation time and increases the calculation efficiency 
 
 2) The application of the negatively-correlated variates 
 (antithetic variates) technique greatly reduces 
 statistical fluctuations with a minimum number of 
 simulating particles (machine particles) . 
 
 3) The introduction of a weighting scheme and an artificial 
 game of chance (using weighted cross sections) contri- 
 buted some reduction in calculation time and resolved 
 some practical difficulties. 
 
 4) For the system with an applied electric frield, the 
 exact formulation for a parabolic flight path was used 
 to advance the particles in larger steps than would have 
 been possible using the small-step linear approximation 
 employed by Musha and Itoh (8) and also by Thomas (9). 
 This improves the efficiency appreciably. 
 
 Throughout the entire calculation, one basic balance 
 equation (Eq. 30) played a very important role. It merely 
 states that at steady state, the source is balanced by losses 
 
112 
 
 By means of this equation, we predict the nonlinear parameter 
 values and provide checks for convergence of the solution. 
 
 The need for large amounts of storage space for Monte 
 Carlo calculations no longer creates serious problems or 
 drawbacks on modern large computer systems , such as was the 
 case as recent as seven or ten years ago. In addition, 
 techniques exist for further reducing the storage require- 
 ment. In the present study, we suggest a technique to 
 eliminate major storage requirements, and use only a small 
 amount of storage as a temporary buffer storage (described 
 later) . 
 
 In other types of numerical solutions, e.g., in finite 
 difference solutions of the diffusion equation, the step 
 size At is restricted by stability requirements. However, 
 in this present model, At is only restricted by the degree 
 and amount of nonlinearities present in the system. This 
 restriction is far less stringent. 
 
 C. Practical Limitations and the Cures 
 
 It was necessary to solve several difficulties in the 
 present simulation study in order to provide practical 
 solutions. We shall describe them item by item. 
 1) The electron energy distribution is the quantity of 
 
 major interest. The difference between the magnitudes 
 
 of its maximum and minimum non-zero values is of the 
 
 9 
 . order 10 . This is due to high density in the low-energy 
 
 region (thermal energy) and the relatively low density 
 
113 
 
 in the high- energy region (near the source energy) . 
 
 It is unrealistic to hope to produce reasonable numbers 
 
 of particles in the low-density (high-energy) region 
 
 9 
 
 and at the same time have 10 times as many in the low- 
 energy region. This problem is resolved by combining 
 the following techniques: 
 
 a) Russian Roulette and splitting. 
 
 b) Weight factors which produce fractional particles. 
 
 c) Logarithmic energy intervals. The number of particles 
 in each energy class interval AE . is directly propor- 
 tional to f(E.)*AE. (f(E.): number density at E.). 
 
 By using logarithmic energy intervals, we have small 
 intervals at low energy region and large intervals 
 at high energy so that the product f(E.)*AE. assumes 
 a practical magnitude. 
 2) The total number of simulating particles (machine 
 
 particles) N T is limited by the storage requirement and 
 the economy of the calculation (computing time is propor- 
 tional to N ) . We have to set an upper limit (Nmax) 
 on the total number of simulating particles in any time 
 cycle.. During each time cycle of duration At, N source 
 particles are introduced into the system, and N secon- 
 dary particles are produced (by ionization) . At the 
 same time N particles escape from the system, and N , 
 particles are lost due to recombination. 
 
 There are two conditions to be satisfied at all times: 
 
114 
 
 a) N + N + N <r N <r N 
 
 c z s — T— max 
 
 b) N + N ~ N + N AD 
 z s es T AB 
 
 (67a) 
 (67b) 
 
 where N : the number of census particles (background 
 particles) . 
 
 Condition a) insures that total number of particles 
 in the system is always under the limit (Nmax) . Condi- 
 tion b) indicates that a particle balance should be 
 satisfied at steady state. 
 
 In the program, there are two equivalent problems to 
 be considered. One is the balance of the machine 
 particles [corresponding to condition (a)], and the 
 other is the balance of simulated particles [weighted 
 particles, corresponding to condition (b)]. Both condi- 
 tions must be satisfied simultaneously. Condition (a) 
 allows a certain amount of leeway since the number of 
 particles that survive the integration cycle (time cycle) 
 and hence require storage will be less than N + N + N . 
 
 w «u w 
 
 Here N is chosen to be an input number, the weights of 
 
 these particles are determined by the source rate. If 
 
 condition (a) can not be met, N is taken to be 
 
 ' s 
 
 N = N -N -N -JZ-1 
 s max c z 
 
 (68) 
 
 where JZ is the number of spatial zones. It has been 
 shown (12) that Equation 68 insures the stabilization of 
 census population without overly reducing the number of 
 source particles. 
 
115 
 
 The proper selection of the energy dividing line 
 (EDI) for Russian Roulette and splitting can insure the 
 satisfaction of condition (b) . Also as mentioned in 
 earlier chapters, the balance factor (3F) is used to 
 speed up convergence and force condition (b) to be 
 satisfied sooner. 
 
 3) In any Monte Carlo calculation, finite samples are 
 used to estimate or simulate the behavior of a large 
 parent population. This is the main source of statistical 
 fluctuations (variance or uncertainty) . In the present 
 
 simulating system, less than 5000 particles were used to 
 
 14 
 simulate approximately 10 physical particles. This 
 
 creates the problem of matching the values of input and 
 estimated output quantities. Due to the huge scaling 
 factor (10 /5000), a small amount of statistical fluctua- 
 tion or estimation error in the estimated value will be 
 amplified about 10 times. To avoid this, we use the 
 estimated (or equivalent) input quantities which carry 
 the same order of magnitude of statistical fluctuation 
 and weight. 
 
 4) There are several reasons that make the logarithmic 
 energy scale the logical choice for the group intervals 
 in the energy axis of phase space. These include: 
 
 a) the wide energy span (0 to 1500 eV) 
 
 b) the relatively high number density at low energies 
 combined with a low rate and small amount of energy 
 
116 
 
 exchange due to collision processes. In the high- 
 energy region, the situation is just the opposite. 
 c) to produce reasonable number of particles in each 
 
 energy interval [for the reason mentioned in the first 
 section (II-B-1)]. The size and number of intervals 
 are mainly dictated by the range of data (energy 
 range in this case) and number of observations 
 available. Excessive fragmentation of the data 
 may produce many intervals with few occupants. On 
 the other hand, insufficient divisions may obscure 
 important dispersions. Thus an optimum number of 
 intervals will be a compromise between these extremes. 
 
 D. Future Extensions 
 
 The Monte Carlo methods are applicable to the most 
 widely diverse branches of computational mathematics includ- 
 ing particle physics (neutron transport for instance), 
 operations research problems such as the investigation of 
 servicing processes, modelling the processes of information 
 -transmission . in communication theory, the evaluation of 
 definite integrals, the solution of partial differential 
 equations (e.g. boundry-value problems); the solution 
 of systems of linear equations - inversion of matrices, and 
 many other applications . Problems handled by such methods 
 are in general of two types, the probabilistic and the 
 deterministic, according to whether or not they are directly 
 concerned with the behavior and outcome of random processes. 
 
117 
 
 Their basis lies in simulating statistical experiments by 
 means of computational techniques. Based on the underlying 
 principles described earlier / various variance reduction 
 techniques can be applied. The present work serves as an 
 example and guide for using the important new techniques 
 such as correlated sampling. 
 
 1) The code developed can readily be used to calculate 
 electron energy distribution functions for other gaseous 
 media by merely changing the cross sections input. The 
 model can also be modified to include the electron- 
 electron and electron-ion interactions although this 
 adds more nonlinearities into the system. This would 
 allow calculation for higher source rates and cases 
 where the fractional ionization is large. 
 
 2) As for the complex transport problems encountered in 
 areas such as plasmas physics and astrophysics, the 
 model and techniques can readily be extended to the 
 problems with more complex geometry and/or boundry 
 conditions, with inhomogeneous media and multi-dimensions 
 Also many kinds of nonlinear features can be studied in 
 
 a realistic fashion. 
 
 3) More elaborate error checking and correction procedure 
 (based on the known physical phenomena or theoretical 
 formulations) such as the correction matrix used by 
 Berger (9) should be developed to treat the combined 
 effect of statistical fluctuations and other types of 
 
118 
 
 
 uncertainties. This is essential for generating good 
 results for Monte Carlo calculation. 
 
 4) More elaborate algorithms or techniques for playing 
 the artificial games of chance by means of weighted 
 cross sections would be a good direction for future 
 research. This must be based on the investigator's 
 intuition and experience, but still it is a promising 
 approach for increasing the efficiency. 
 
 5) The present model and techniques can be also implemented 
 
 on smaller computer systems such as PDP-11 or other 
 
 similar systems where large amount of fast storage space 
 
 are not available. The following techniques could be 
 
 used to process the calculation without storing all the 
 
 particle data (namely E., u., z., Wt . V.). At the 
 * l ' i' i' l l 
 
 beginning of each time cycle, we generate a small number 
 (some reasonable amount, £t: 100 for instance) of particles 
 The corresponding particle data can be generated from 
 the initial input distribution or from last time cycle. 
 After these particles have been processed, only the 
 accumulated information is stored. A new group of 
 particles come into the system by generating a new set 
 of data, based on the same distributions, which are 
 stored in the same temporary storage space. The pre- 
 vious particle data is destroyed after they have been 
 processed. The same process is repeated until all the 
 necessary prespecified number of particles have been 
 
119 
 
 processed, and the new distributions are obtained. This 
 eliminates the necessity of storing all the particle 
 data at each time cycle, and this eliminates the large 
 arrays that demand the major amounts of storage space. 
 Due to the random nature of the processing algorithms, 
 and the Markovian behavior of the time cycle process 
 (these time cycle processes are a kind of Markov process), 
 this storage saving algorithm can accomplish the desired 
 results . 
 
 E . Concluding Remarks 
 
 In developing any Monte Carlo models and performing 
 calculations, as mentioned in Chapter V the systematic errors 
 and outright blunders must be detected and removed by 
 testing and comparing with known results. 
 
 The various variance reduction techniques are used to 
 increase the relative computation efficiency, reduce inher- 
 ent random errors or statistical fluctuations. For better 
 calculational results, more samples are favorable even though 
 ■not economic. In any Monte Carlo calculation there exists 
 a minimum number of samples below which the results or 
 statistics are no longer valid and dependable regardless 
 of the variance reduction schemes employed. In the other 
 words, variance reduction techniques help to reduce statistical 
 fluctuations, but they can not be used to remove all the 
 fluctuations caused by small samples. 
 
120 
 
 The minimum sample size. is, of course, problem dependent 
 There are no definite rules or formula for it, thus a cut 
 and try method is necessary until the calculated variance 
 is acceptable. 
 
 Monte Carlo calculations are perhaps best in the prelim- 
 inary stages where they help to give a general idea of the 
 situation and give hints or feelings for trends. If more 
 accurate results are necessary, Monte Carlo calculations 
 may not always be practical good method. In any case, the 
 Monte Carlo method is an important tool in the inferential 
 analysis of the mathematical-physics problems, but it 
 should be viewed as supplementing, rather than replacing, 
 analytic and other numerical methods. 
 
121 
 
 LIST OF REFERENCES 
 
 1. R. W. L. Thomas and W. R. L. Thomas. "Monte Carlo 
 Simulation of Electrical Discharges in Gases," 
 J. Phys. B (Atom. Molec. Phys .) Ser. 2, Vol. 2, 562, 
 
 (1969 
 
 E 
 
 2. Tomizo Itoh and Toshimitsu Musha. "Monte Carlo Calcula- 
 
 tions of Motion of Electrons in Helium," J. Phys. 
 Soc. Japan , Vol. 15, 9, 1675 (1960). 
 
 3. J. A. Fleck, Jr. "The Calculation of Nonlinear Radia- 
 
 tion Transport by a Monte Carlo Method," Methods in 
 Com putational Physics , Vol. 1, 43. New York: 
 Academic Press, (1963 ) . 
 
 4. J. A. Fleck, Jr. and J. D. Cummings. "An Implicit Monte 
 
 Carlo Scheme for Calculating Time and Frequency 
 Dependent Nonlinear Radiation Transport," UCRL - 
 72924 (1970). 
 
 5. G. H. Miley, J. T. Verdeyen, T. Ganley, J. Guyot and P. 
 
 Thies. "Pumping and Enhancement of Gas Lasers via 
 Ion Beams," Sy mp. on Electron, Ion & Laser Beam 
 Technology , University of Colorado, May, 1971. 
 
 6. J. C. Guyot, G. H. Miley, J. T. Verdeyen and T. Ganley. 
 
 "On Gas Laser Pumping Via Nuclear Radiations," Symp . 
 on Research on Uranium Plasmas and Their Technological 
 Applications , University of Florida, 1970. 
 
 7. J. C. Guyot, G. H. Miley and J. T. Verdeyen. "Applica- 
 
 tion of a Two-Region Heavy-Charged Particles Model 
 to Noble-gas Plasmas Induced by Nuclear Radiations," 
 Intern. Rept . Nuclear Engineering Program, University 
 of Illinois (1970) . 
 
 8. L. Herwig. "Summary of Preliminary Studies Concerning 
 
 Nuclear Pumping of Gas Laser Systems," Report C- 
 110.053-5, United Aircraft Corp., East Hartford, 
 Connecticut (1964). 
 
 9. M. J. Berger. "Monte Carlo Calculation of the Penetra- 
 
 tion and Diffusion of Fast Charged Particles," Methods 
 in Computational Physics , Vol. 1, 135-213. New York: 
 Academic Press, 1963. 
 
 10. A. E. D. Heylen and T. J. Lewis. "Electron Energy Dis- 
 tribution Functions and Transport Coefficients for 
 the Rare Gases," Proc. R. Soc. A ., Vol. 271, 531 (1963) 
 
122 
 
 11. J. A. Smit. "Berechnung der Geschwindigkeits Verteilung 
 
 der Elektronen bei Gasentladungen in Helium," 
 Physica , Vol. 3, 543 (1936). 
 
 12. J. M. Hammersley and D. C. Handscomb. Monte Carlo 
 
 Methods . New York: Wiley, 1964. 
 
 13. J. M. Hammersley and K. W. Morton. "A New Monte Carlo 
 
 Technique: Antithetic Variates," Proc. Camb. phil . 
 Soc. , Vol. 52,449 (1956). 
 
 14. J. M. Hammersley and J. G. Mauldon. "General Principles 
 
 of Antithetic Variates," Pioc. Camb. phil. Soc , 
 Vol. 52, 476 (1956) . 
 
 15. Y. A. Schreider. The Monte Carlo Method . London: 
 
 Pergamon Press, 1966. 
 
 16. E. D. Cashwell and C. J. Everett. A Practical Manual 
 
 on the Monte Carlo Method for Random Walk Problems . 
 New York: Pergamon Press, 1959. 
 
 17. M. A. Uman. Introduction to Plasma Physics . New York: 
 
 McGraw Hill, 1964. 
 
 18. L. G. Parratt. Probability and Experimental Errors in 
 
 Science . New York: Wiley, 1961. 
 
 19. T. D. Sterling and S. V. Pollack. Introduction to 
 
 Statistical Data Processing . New Jersey: Prentice- 
 Hall, 1968. 
 
 20. S. Karlin and W. J. Studden. Tchebycheff Systems : 
 
 With Applications in Analysis and Statistics . New 
 York: Wiley, 1966. 
 
 21. L. L. House and L. W. Avery. "The Monte Carlo Technique 
 
 Applied to Radiative Transfer," J. Quant. Spectrosc . 
 Radiat. Transfer , Vol. 9, 1579 (1969) . 
 
 22. H. Kahn. "Use of Different Monte Carlo Sampling Tech- 
 
 niques," Symposium on Monte Carlo Methods. New 
 York: Wiley, 1956. 
 
 23. R. J. Cicerone and S. A. Bowhill . "Monte Carlo and 
 
 Thomson-scatter plasma-line studies of Ionospheric 
 Photoelectrons, " Aeronomy Rept., No. 39, University 
 of Illinois, Urbana, Illinois (1970). 
 
123 
 
 24. A. Nordsieck and B. L. Hicks. "Monte Carlo Evaluation 
 
 of the Boltzmann Collision Integral," CSL Repoi t 
 R-307, University of Illinois (1966). 
 
 25. S. M. Yen and B. L. Hick? "On the Accuracy of Approxi- 
 
 mate Solutions of the Boltzmann Equation," CSL 
 Report R-308, University of Illinois, Urbana, 
 Illinois (1966). 
 
 26. M. Y. Youakim, C. H. Liu and K. C. Yeh. "Generation of 
 
 Random Hyper surf aces, " Intern. Rept., Ionoshere 
 Radio Laboratory, University of Illinois, Urbana, 
 Illinois (1970). 
 
 27. M. Rich and S. S. Blackman. "A Method for Eulerian 
 
 Fluid Dynamics," LAMS- 2826, UC-32, Mathematics and 
 Computers (1963). 
 
 28. W. D. Little. "Hybrid Computer Solutions of Partial 
 
 Differential Equations by Monte Carlo Methods," 
 Proc. FJCC, 181 (1966). 
 
 29. L. B. Miller and G. H. Miley. "Some Monte Carlo 
 
 Techniques for the Calculation of Doppler Coefficients," 
 Nuclear Science and Engineering , Vol. 40, 438 (1970). 
 
 30. C. K. Birdsall and D. Fuss. " Cloud- In-Cell Computer 
 
 Experiments in Two and Three Dimensions," Proc. APS 
 Topical Conference on Numerical Simulation of Plasma . 
 Los Alamos Scientific Laboratory, University of 
 California (1968). 
 
 31. M. Goodrich. "Electron Scattering in Helium," Physical 
 
 Review , Vol. 54, 259 (1937). 
 
 32. G. H. Miley and R. H. Lo . "Calculation of Electron 
 
 Energy Flux Distribution in Noble Gases," To be 
 published in Proc. 18th Annual Meeting of ANS, June, 
 1972. 
 
 33. R. H. Lo . Department of Nuclear Engineering, Univer- 
 
 sity of Illinois, Private Communication, April, 1972. 
 
 34. W. Jesse and J. Sadouski. "Ionization in Pure Gas- and 
 
 the Average Energy to Make an Ion Pair for Alpha 
 
 and Beta Particles," Phys. Rev ., Vol. 97, 1665 (1955). 
 
 35. S. C. Brown. B asic Data of Plasma Physics . Cambridge, 
 
 Massachusetts: The M.I.T. Press, 1966. 
 
124 
 
 36. T. Holstein. "Energy Distribution of Electrons in 
 
 High Frequency Gas Discharges," Phys. Rev . Vol. 70, 
 5, 367 (1946) . 
 
 37. J. T. Ganley. Effect of a Heavy Particle External 
 
 Ionization Source on CO_ Laser Discharges, Ph.D. 
 Thesis, University of Illinois, 1972. 
 
125 
 
 APPENDIX 
 
 FORTRAN IV G LEVEL T! mTTO \ DATE - fJl2* OS/31/02 
 
 C MAS T ER PR05RAH DP HON TE OHIO 5I W ULI T 10N OP HUNU W tAR AL P HA PLASMS UL P HA »» 
 
 C TICLES INOUCEO PLASM* ELECTRON ENERGY DISTRIBUTION STUDY 
 
 C DEFINITIONS AND TERMINOLOGY ~~ 
 
 C D: DI AMETER OF THE CYLINOER THE LENGTH OF THE FINITE REGION 
 
 C EO» INITIAL ENERGY OF THE SOURCE PARTICLES SOINSM SOURCE RATE 
 
 C JZtTOTAL NUMBER OF ZONES JEt TOTAL NUMBER OF ENEROV RANGES 
 
 c i*Si array for S T OrIng T hE I NO EX flf AB50R P EP P AW TI CLES HAS! T HE MO. OP T HW 
 
 C DBJ DISTANCE_TO BOUNDRY PCOL: OISTANCE TO COLLISION OCENt OISTANCE TO CENSUS 
 
 C OH ZONE INTERVAL 
 
 C ARRAYS E. Z. I) STORING E NERGY. OISTANCE. ANO DIRECTIO N OF PARTI CLES 
 
 C NTP: TOTAL~NUMBER OF PARTICLES TO BE PROCESSED EACH CYCLT^ 
 
 C NTJ TOTAL NUMBER OF T IME CYCLES , 
 
 C NSY NUMB E R OP SOURCE PAR TI CL E S PER UN IT TI HE»P I 5 T ANCE OB SOURC E RA T E 
 
 C NIZN: NUMBER OF IONIZATION PROCESS OCCURREO EACH TIME CYCLE 
 
 0001 REAL NEI55I, NEZI 55, 101 ,EZ( 111 ,0X1 31 , EE( 531 , CEZI 11 ) t EKZC55 
 1,10> ,FID<60l,NREZtlll,EZI< UI.NTPM 
 
 Or02 DIMENSION FNEI55 I , FNREZIU I, NRZ 1 1 111 ,CNE< 35 I , CNREZI 11 1 ,TNE (551 
 
 000 3 COMMON 0, E(5100),Z15100I, UI5100) , I AS< 500 I , IRRI 8501 ,NRR , IES( 600) 
 
 X,WT(5100I,SWF 
 
 0004 COMMON /AREA1 / NIZN , NOZN, NAS, 01 ZNI5 I ,N SE, NMET, NEXC, ETAS , 
 
 XWNME'T", WNEXC, WNIZN," ELS, UNAS ,WNSE 
 0C05 COMMON /AREAZ/ CI ,MUO, E I ,EI S, SF, ANEAM,PQP0,PI ,E0,NI0N, J E , JZ tNASWl , 
 
 1NSEW ,AKT, AIKT , AMI , AMC1 , EDI , PDMUC, P0?6,K1 , K2~~t K3. MN2.EDZ,ED3 
 0C06 COM MON /AREA3/ XSEL , XELC , XRCB, XMET. XEXC ,XIQN.XT0L,KELC 
 
 0007 " COMMON /AREA4/ CONST, 0EL2, ELT,IE, Oil, DIZ, 013 ,DELI 
 
 0008 COMMON /A5/ EG <_55 I , E0EI33I, EBDI55I 
 
 0009 COMMON >A6/ ESEI50I , NUP ,NDNR,NDN, WSETSOI 
 
 0010 REAL MUO. NION.ND 
 
 C STATEMENT FUNCTION FOR EVALUATING MAXWELL IAN AND RELATEO FCN? 
 
 0011 F M1(£,FNEM(-EXPI-E»AIKT I »SQRT<EI»FNEM 
 
 0012 "DATA TMS, NT, DTOT,ETOT,NIN,NI ZNT.NAST/O. 0,1, 0.0, 0.0, 0,0,0/ 
 
 0013 DATA W5VLE,CN5IZ.CN5S,CNES,CNAS,CNMET,CNEXC,C ESAVC,CE ASG/»»0 .0/ 
 
 0014 OATA CNE/ 55*6.0/, CEZ/11*0.0/, CNREZ/il*0.0/,7NT735*0. /,EKZ7J35*5~ 
 1.0/.05AVS/0.0/ __ _ 
 
 0015 DATA NEZ/ 550*0. /,NE/55*0. /.EZ/1 1*6. /.NREZ/ 11*0. / ,EE/55*0. 0/ 
 
 0016 R EAD(5,1I NT APE 
 
 0017 READ(5,151I Ki , K2 , NTC. NTP.NTW.NPM, KEL^J ,JZ, IM.NCUM, NOPN.NTVP 
 X, MC ,IUSN,_NySTi NBF _ 
 
 0018 REA0(5,i52) E01 ,ND , EO.ANSM, D, EOP, POPO, SR 
 
 0019 READ(5,153J FJJ) _ 
 
 0020 1 FORMAT ( HOI 
 
 00 21 151 FORM ATt 16151 
 
 0022 152 HORMATi BF10.4I 
 
 0023 153 FORMAT! 8 E10. 3I _ _ 
 
 0024 DELE-1.0 
 
 002 5 DELI-UO/OELE _ 
 
 0026 DEL2-2.0 
 
 002 7 AKT=0.05 
 
 0028 AIKT- l.O/AKT 
 
 0029 PI«3. 141593 _ 
 
 0030 EMASSI»l.60203E*16/9. 10B 
 
 0031 K3«K1*K1 
 
 0032 WN2»1.0/K2 
 
126 
 
 FORTRAN IV G LEVEL 18 
 
 MAIN 
 
 DATE - 72126 
 
 09/JT75F 
 
 0033 
 
 003* 
 0035 
 0036 
 0037 
 0038 
 0039 
 0040 
 0041 
 00 42 
 0043 
 0044 
 0045 
 0046 
 0147 
 0048 
 0049 
 005C 
 0051 
 00 52 
 00 53 
 0054 
 0055 
 0056 
 005 7 
 0058 
 0059 
 0060 
 0061 
 0062 
 0063 
 0064 
 0065 
 0066 
 0067 
 0068 
 0069 
 0070 
 0071 
 00 72 
 0073 
 0C74 
 0075 
 0076 
 00 77 
 0078 
 0079 
 0080 
 0081 
 0082 
 0083 
 0084 
 OC85 
 
 CJO-0.0 
 
 EF»EDP*PDP0 
 
 £IS-Ut_* 
 
 01-2.0*0.000549/4.003873 
 
 FTV S0RTI2. 0*1. 60203/9. 108 l»1.0E»08 
 
 VTE- l.O/(ETV*ETV» 
 
 CALL JMO|RH'QFF«_L 
 
 PD26-PDP0*26.0 
 
 MU0"LL._0JL-C91/JLy .__ 
 
 CMU0-1.0E-09 
 C HUPPP.CPI)Q*P0PQ 
 
 IF ( PDPO.GT. 10.01 
 
 PDMUQ-CMUPOP/ETV 
 
 ELS-l.O 
 
 NUP-Q ..._ . 
 
 NDN-0 
 
 .N DNR-Q 
 
 INO-0 
 
 NRR»Q ._ _ ._ 
 
 NAS-0 
 NESsflL. . 
 NSE-0 
 
 nq:n-o 
 
 NIZN-0 
 
 NIN-0_.-_. 
 
 LK«0 
 
 INiy-999999 
 
 NMET»0 
 
 CMUP0P-CMU0*10.0 
 
 ETAS-0.0 
 
 ET£P-0_t0 
 
 NTP5-0 
 
 N*?Q 
 
 N5TP-0 
 
 NCQL-0 
 
 HNMET«0.0 
 WNI2N-Q.0 
 WNEXC-0.0 
 WNAS?Q.Q. ._ 
 WNES-0.0 
 
 ■WHSE-Q«0 
 
 CLOSS»0. 
 
 CSF2-Oj0 
 
 CNI02«0.0 
 
 CiFl»0.0 
 
 CETAS-0.0 
 
 CETEP-Q.O 
 
 OTAVS»0.0 
 JE-48 
 IE«JE 
 
 D E N- 1m 5.4 E*l 6 * PJLP5 
 DETZ=D/JZ 
 
127 
 
 FORTRAN IV G LEVfL Tl 
 
 Tnrnr 
 
 pa t e ■ 78 12* 
 
 C5/51/OZ 
 
 0C86 
 
 00S7 
 
 0C88 
 
 0089 
 
 0090 
 
 0091 
 
 0092 
 
 0093 
 
 0094 
 
 0095 
 
 0C96 
 
 0097 
 
 0098 
 
 0099 
 
 0100 
 
 0101 
 
 0102 
 
 0103 
 
 0104 
 
 0105 
 
 0106 
 
 0107 
 
 0108 
 
 0109 
 
 0110 
 
 0111 
 
 0112 
 
 0113 
 
 Oil* 
 
 0115- 
 
 0116 
 
 0117 
 
 0118 
 
 0119 
 
 0120 
 
 0121 
 
 0122 
 
 0123 
 
 0124 
 
 0125 
 
 0126 
 
 0127 
 
 0128 
 
 0129 
 
 0130 
 
 0131 
 
 0132 
 
 0133 
 
 0134 
 
 0135 
 
 0136 
 
 0137 
 
 0138 
 
 DMV- 1.0/BETZ 
 
 CONST-1.0/ALOGI2.0) 
 
 ED2-0.44T513 
 
 ED 3_- 19.8 
 
 PH-ED2*AIKT 
 AMl-2.0*AIKT/SQRTtPI*AKT> 
 
 Tall — RNJfUZlWU&SSllI 
 
 S0»P0P0*SR*1.0E*16 
 
 SF-l.O 
 
 AITG* CMUPDP /t AKT»PI> 
 
 OE1-0.6T6 
 
 DE2-0.355 
 
 DE3-0.5 
 
 DII-1./0.016 
 
 D12-1./DE2 
 
 D13-1./0EJ ■ _ 
 
 AMC1- SQRTf2.0*2.7l83*AIKtl*E02 
 
 DO 11 1-6, 36 
 
 EBL «( I-5l*DE2 -4.0 
 
 EMV- EBL - 0.1685 
 
 EBD,( I I- 2.0**EBL 
 EGU-1]^ 2.0**EMV 
 
 11 EDEl : ll-EBDUI*I2.C**DE2 -1.01 
 
 JE1-JE+1 
 
 DO 12 1-37. JE1 
 
 EBL" 7.J>05_*Jl-36I*DE3 _ 
 
 EMV- EBL - 0.1405 
 
 EBD( 1J^ 2.0**EBL_ 
 
 EG(I-1»- 2.0**EMV 
 
 12 EOEI I>- EBD1 1 >»(2.0»»0E3 - 1.01 
 DO "10 I "It 5 
 
 EDE( I )» 0.016 _ 
 
 EG(I )» EDEl 11*1 -0.008 
 10 EBD< II-_EDE(II*( I-l> 
 EDEI36I-29.69 
 
 E GI48I- 173 7.482 _ 
 
 IF (NTAPE.NE.U GO TO 2 
 
 READ(10» EAVG. DAVG, TME,_NTW, NT, 
 
 HEADI10I Z, WT, ANI0S.SF3 
 
 NS«NSW*K2 _ 
 
 WGHT-1.0 
 
 REWIND 10 
 
 NTP, NION. DOEG. NSW. SWF, E,U 
 
 NT«NT*1 
 2 CONTINUE 
 
 IF (NTAPE.NE.2I 
 READI9 I EAVG, 
 RE AD (9 I Z, WT, 
 
 .. NS_ -HSW«K2 
 
 WGHT-1.0 
 
 REWIND. 9 
 NT-NT*1 
 5 CONIJNUE_ _ 
 
 IF (NTAPE.NE.Ol 
 
 GO TO 5 
 
 DAVG, THE, NTW, NT, NTP, 
 ANI0S.SF3 
 
 NION, DDEG, NSW, SWF, E, U 
 
 GO TO 122 
 
128 
 
 FORTRAN IV. G LEVEL 10 
 
 MAIN 
 
 DATE • 7*126 
 
 05 tiUSi 
 
 0139 
 0140 
 0141 
 0142 
 0143 
 0144 
 0145 
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 0147 
 0148 
 0149 
 0150 
 0151 
 0152 
 0153 
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 0160 
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 0165 
 
 0166 
 0167 
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 0169 
 
 0170 
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 0172 
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 0175 
 C176 
 0177 
 0178 
 0179 
 0180 
 0181 
 0182 
 0183 
 0184 
 0185 
 0186 
 
 NION- SQRTIS0*10/AITGI 
 AN£AM-N10N»0.33333/OEN 
 
 CALL S0RCE2INTP, EAVG, NTH, PM, FID» 
 A L 'A I QA( EAYG) 
 
 OAVG-AL 
 DOEG-DAVG«ND/SORT(EAVG) 
 
 SF2-1.0 
 IF (NION.NE.O) 
 
 1EHMJM/H1W. 
 
 SN-S0*AL»ND*SF2/!ETV*SQRTIEAVGI I 
 NSH- A NSW »0.5 
 
 IF (SN.GT.ANSWt 
 
 NS.NSM»K2 
 
 NSW-SN +0.5 
 
 SWF-NSW/SN 
 
 SF-SMF 
 
 SF1-SF2 
 
 SF3-SF2 
 
 CALL SOURCEINS.NTP) 
 TflE-QiO. 
 
 ANIOS-NION 
 
 HGHT-1.0 , 
 
 122 CONTINUE 
 C GENER ATE SOURCE _P AR T J CL_ES_ J 0_ INITIALIZE THE PROGRAM 
 ANEAM-NION*0. 333 33/OEN 
 WRITE16.70) 
 
 70 FORMAT!' 'IS "PROGRAM INITIALIZATION AND THE INITIAL PARAMETERS* ,//> 
 H R1TE16.71) A 1TG. NI ON, NTP . AL.6AVG 
 
 71 FORMAT*' ', 'AITG -'.E14.6, 3X,'NI0N - • , E 14. 6, 3X , *NTP -•,112,31, 
 l 'OCENlEOi -'. F14.6.3K. 'EAV G- '.F14.4.n . 
 
 72 FORMAT!' ' , 8X , • OCOL • , 1 3X , • OCEN' , 13 X , • ENERGY' , I IX , • I • , 14X , ' U • , 10X, 
 
 . „,;. X'MEIGHT'./I 
 
 75 FORMAT!* *,F14.6,3X, F14.6, 3X, F14.5.3X, F12. 4.3X, F12. 4,3X, El 3. 41 
 
 M R1TEJ6.72I . 
 
 RIP-RAN3Z<0> 
 
 C WITHIN THE OUTER I LOOP THE~ I TH PARTICLE HISTORY 
 
 r r.FOMFT Rir. rhutinf ENTERING here 
 
 7777 IF IIEiII.LE.O.OI.OR.IEin.GT.EO*1.5M GO TO 9008 
 ... TCEN-DDEG/.EJX __ 
 
 7778 IF UNIZN.GT.OI.ANO.iI.GT.NTPM TCEN- DIZN!N0ZN*1I 
 
 Ul I ) -U!II»IUSN ... ...__ ._. 
 
 LFG2-0 
 
 ETA.G1-1.Q 
 
 IF !E( D.GT.EDlt ETAGi — 1.0 
 
 SQEje _iQBJ!ElIJ> ... _ . _ 
 
 DVA-EF*TCEN*EMASSI*0. 5 
 
 VZA"Sfl£*E.TV*Ul !)_..♦ DV.A . 
 
 VYS-E! I»*ETV*! 1.0-U! I»*U! 1 1 »*ETV 
 nr.FN- S0RT!V?A»«2+VYSI»TCEN 
 
 
 
 NFLG-0 
 
 qnn TF IFdl .LF. O.OI £0 TO _ 9Q08 
 
 IF (NFLG .EQ.OI GO TO S09 
 
 DVA-EF*TCEN*EMASSI*0. 5 
 
129 
 
 FORTRAN iv G LEVEL It , MAIN DATE • 72124 OTmTO 
 
 0187 VZA-50E*ETV*Um ♦ DVA ' 
 
 0188 809 DZ- VZA»TCEN 
 
 0189 ENEW" E(II ♦ EMDZ 
 
 0190 IF ( ENSW.LE.O.OI GO TO 9008 
 
 0191 SQEP-SQRT(ENEM) 
 
 0192 UN EW» SQE«U<n/SQEP *0VA«2. / <SQEP*ETV» 
 
 0193 ALt-iLbimm ' 
 
 0194 A L2-AL0A1ENEWI ' 
 
 0195 ALI-l.O/ALi ♦ 1.0/AL2 
 
 0196 AL0» 2.00/ALI 
 
 0197 RI-RAN3ZI0I 
 
 0198 ALR-ABStALOG<Rl> > 
 
 0199 "" DCflL- ALIUALD 
 
 0200 IF (NFL G.EQ.lt GO TO 901 
 
 0201 ~ IFTmod ( I , I M I . EO. » lrf"RTTTf677TT DCOL.DCEN.EU ) ,Z( I ) ,U( ! ITVTfTI 
 
 0202 DC 0L2-ABS(AL0G(1 < -RI l)*ALD 
 
 0203 " 0T0T-0T0T ♦DC0L»0C0L2 
 
 02 04 ET0T-ET0T*E1II 
 
 0205 IF (N0PN.LT.2I 615 TT5 5oT 
 
 02 06 TCEN2»TCEN 
 
 0207 DCEN2-0CEN 
 
 0208 EIO^EIU 
 
 0209 UIO» U(I»*IUSN 
 
 0210 ZIQ-Ztll 
 
 0211 ENMO-ENEW 
 
 0212 UNW0-UNEM_ 
 
 0213 OZO-DZ " 
 
 0214 901 NFLG"! 
 
 0215 902 IF (OCEN. LT.DCOU GO TO 905 
 
 0216 TCEN-TCEN - 0COL»TC£N/0CEN 
 
 0217 DZC-DZ*DCOL/DCEN 
 
 0218 E(IL"E(_I»^EF»_DZC 
 
 0219 zm-zii > *ozc 
 
 0220 DC EN- OCEN -DCOL 
 
 0221 NC0L«NC0L*1 
 
 0222 I F (MO0INCO LiMC».EQ. 0» RIP-RAN3ZI0) 
 
 0223 IF (MOO(NCOLfMC).NE.OI RIP-l.O-RIP 
 
 0224 CALL COLPROU, TCEN, NTP, RIP, NUST, &9008, C9045I 
 
 0225 IF((E( I).LE.6.0).ANO.(LFG2.EQ.l) t GO TO 9008 
 
 0226 GO TO 900 _ 
 
 0227 905 E(I»»ENEW 
 
 0228 U<I>- UNEW _ 
 
 0229 Z<I )- Z( II ♦ DZ 
 
 0230 IF ((Z<n.LE.0.0>.OR.(Z(n.GT._0_n GO_TO 9008 
 
 0231 j-INT(Z(I)*DZIVi*l 
 
 0232 IF (J.GT.JZI J»JZ 
 
 0233 K» INOEXE(E(III 
 
 02 34 __ IF (K.GT.JEI K- JE 
 
 0235 IF (K3.E0. II GO TO 904 
 
 0236 ETAG2 -1.0 
 
 0237 IF lEIII.GT.EDir ETAG2— 1.6 
 
 0238 IF <(ETAGl*ETAG2I.GT.0.0J GO _T0 _?04 
 
 0239 IF (ETAG2.GT.0.0I GO TO 903 
 
130 
 
 FORTRAN 
 
 IV G LEVEL 18 MAIN 
 
 DATE - 7212* 
 
 05/31/62 
 
 0240 
 0241 
 
 NUP-NUP*1 
 ESEINUPI-EUI 
 
 
 
 0242 
 0243 
 
 WTU I-WTMI/K3 
 WSEINUPI-WTUI 
 
 
 
 0244 
 0245 
 
 GO TO 904 
 903 IFULFG2.EQ.il. OR. IN0PN.LT.2M 
 
 NONR-NDNR*! 
 
 
 0246 
 0247 
 
 IFKLFG2.EQ.il. OR.INDPN.LT. 211 
 NDN-NDN+1 
 
 IRR(NDNRt-I 
 
 
 0248 
 0249 
 
 IF (M0D(NDN t K3).NE.0l GO TO 
 UTIII«MTm*K3 
 
 9045 
 
 
 0250 
 0251 
 
 904 NEZtK.JI- NEZ(K.J) ♦ MTdt 
 EKZ(K,JI-EK{(KiJI*Em*WTm 
 
 
 
 0252 
 0253 
 
 9045 IFK LFG2.EQ.il. OR. (NDPN.LT.2M 
 LFG2-1 
 
 GO TO 9009 
 
 
 0254 
 0255 
 
 EK1-EI0 
 
 um«uio 
 
 
 
 0256 
 0257 
 
 ZK l-ZIO 
 TCEN-TCEN2 
 
 
 
 0258 
 02 59 
 
 DCEN-DCEN2 
 0C0L-0C0L2 
 
 
 
 0260 
 0261 
 
 ENEW-ENWO 
 UNEW*UNWO 
 
 
 
 02 62 
 0263 
 
 DZ-DZO 
 
 IF KUSN.LT.O) GO TO 900 
 
 
 
 0264 
 02 65 
 
 GO TO 902 
 9008 NES-NES^l 
 
 
 
 0266 
 02 67 
 
 IF (E( II.LT.ED1I HTK I-WTK1/K3 
 IF IWTlII.GT.il WTm-1.0 
 
 
 
 0268 
 0269 
 
 HNES>WNES ♦WT(I) 
 IESINESI-I 
 
 
 
 0270 
 0271 
 
 IF (EIII.GT.O.O) ETEP-ETEP*E(I»»MTIII 
 9009 I« 1*1 
 
 
 02 72 
 0273 
 
 LFG2-0 
 
 IF (I.LE.NTPI GO TO 7777 
 
 
 
 0274 
 0275 
 
 IF K I.GT.NTP).AND.(NIZN.GT.O) t 
 IF INOZN.LT.OI NIN-1 
 
 NOZN* NOZN-1 
 
 
 02 76 
 0277 
 02 78 
 0279 
 
 IF KNIZN.GT.Ot . AND. (NI N. EQ.O II 
 . _ JLLZW»H_NJjt_N__»Q.5 . 
 
 NWS-HNES +0.5 
 NASW-WNAS *0.5 
 
 GO TO 7778 
 
 
 02 80 
 0281 
 
 WNAS-MNAS/KELC 
 WNSE-WNSE/KELC 
 
 
 
 0282 
 0283 
 0284 
 0285 
 
 NSEH-WNSE +0.5 
 
 _QQ L9JL_ N-LrJE. - _ .._.. 
 
 00 191 N-l.JZ 
 
 EE(HI-EE<MI ♦ EKZ(M.N) 
 
 — 
 
 
 0286 
 0287 
 
 191 NE(H)= NEIMI ♦ NEZIM.N) 
 
 IF KK3. EO. ll.OR. (NUP.EQ.OM GC 
 
 TO 24 
 
 
 0288 
 0289 
 0290 
 0291 
 0292 
 
 N3«K3/JZ 
 
 IF LN3.-LT.H N3.-1 
 
 NK3-K3-1 
 
 IF 1NK3.LT.JZI IZ-NK3 
 IF (N3.GE.1) M3-N3 
 
 
 
131 
 
 FORTRAN IV G LEVEL 18 WTU DATE - 72124 ZW7JT7VT 
 
 0293 "'"'• 00 23 W-1,nUp 
 
 0294 )CINOEXE(ESE<NH _ __ , 
 
 0295 ~ EE(K»-EE(K» ♦ U3-1 )*ESE<n4«WSE(N) : 
 02 96 NE IKI-NE(K) ♦ < K3-U»WSE t N> 
 
 0297 22 IF (NK3.GE.JZt IZ-JZ 
 
 0298 DO 21 L-ltIZ 
 
 0299 NREZ(LI-NREZ«L» ♦ M3*WSE(Ni 
 
 0300 21 EZ(Lt«EZ(L> ♦ ESE t N>«N3*MSE(N» 
 
 0301 NK3-NK3 -JZ*H3 
 
 0302 IF (N K3.LT.JZ> IZ-NK3 
 
 0303 N3-1 
 
 0304 IF (NK3.GT.0) GO TO 22 
 
 0305 23 CONTINUE 
 
 0306 2* CONTINUE 
 
 0307 DO 192 M-l, JZ 
 
 0308 DO 1 915 N-l.JE 
 
 0309 NREZ(MI-NREZIM) ♦ NEZIN.MI 
 
 0310 1915 EZ(H>-EZ(HI ^EKZIN,*) 
 
 0311 192 CONTINUE 
 
 0312 NTPN-NTP*NIZN-NES- NAS 
 
 0313 NTPW»0 
 
 0314 E TOL-0. 
 
 0315 00 14 M-l.JE 
 
 0316 ETOL» ETOL »EE(H> 
 
 0317 14 NTPW-NTPW *NE(M» 
 
 0318 EWTOL-ET OL 
 
 0319 NTPT-NTPN 
 
 0320 DAVGO- OAVG»NO _ 
 
 0321 DAVS-(DAVGO*1000.»**NTVP 
 
 322 05AVS»0SAVS* DAVS 
 
 0323 DTAVS-OTAVS *OAVS 
 
 324 _ DI5G"1.0/05AVS _ 
 
 0325 ANTPO - 1.0/NTPW 
 
 0326 N5TP- N5TP ♦ NTPW »0.5 
 
 0327 ~NTP5«NTP5* NTPN*DAVS 
 
 0328 N TPO-NTP 
 
 0329 EWAVG»ENTOL*ANTPO 
 
 0330 NSO»NS 
 
 0331 NSW0>NSO*WN2 *0. 5 
 
 0332 NTP- NTP+NIZN 
 
 0333 NTPP*NTP 
 
 0334 DAVG'0T0T»0.5/NTP 
 
 0335 EAVG- ETOT/NTP 
 
 0336 DE G-DO EG/ETV _ ____ 
 
 0337 TME"TME*DEG 
 
 0338 T HS-THS*PEG 
 
 C IF SOURCE PARTICLES SOINSK NES fHEN ELIMINATE AS KANV RESIOUE PARTICLES AYTH 
 
 0339 NTPS»NTPW* NWS +NASW 
 
 0340 NTU^NTPM ♦. 5 
 
 0341 PCE N1 - AL0A1 EAVG I _ 
 
 0342 DDEG*DAVG*ND/SQRT( EAVGI 
 
 0343 BF-1 .0 _ 
 
 0344 IF HWNES+MNASI.GT.O.OI 8F« ( HN I ZN+SN I / (HNES+WNA S» 
 
132 
 
 
 FORTRAN IV G LEVEL IB »UIN OATE • 72126 05/31/02 
 
 0345 SF 2*1.0 
 
 0346 u IF (AN10S.NE.0I Sf 2-NTPS/ANIOS , 
 
 0347 ESftVG-0.0 
 
 0348 IFIHNES.NE.O.OI ESAVG- ETEP/MNES 
 
 0349 EASAVG-0.0 
 
 0350 IF IWNAS.NE.Ot EASAVG-ETAS/WNAS | 
 
 0351 SHFO-SWF 
 
 0352 CN5S-CN5S* SN»DAVS»WGHT 
 
 0353 CSF1- CSF1 ♦< ETEP*ETAS*40. *MNIZN l/TNS 
 
 03 54 SFi»CSFl«DI 5 G/tS0»E0> 
 
 0355 CSF2- CSF2 ♦ SF2*0AVS 
 
 0356 IF ((H0D(NT,5).EQ.0».AN0. (NT.NE.Ot) SF3-CSF2»Q 136 
 
 0357 WGHT-SF1/SF3 
 
 0358 SN«S0»0DEG*SF3/ETV 
 
 0359 IF (NTP.GT. 49201 ANSW-2 
 
 0360 NSWANSW »0.5 
 
 0361 IF (SN.GT.ANSMI NSM-SN +0.5 
 03 62 SWF-NSH/SN 
 
 0363 NS-NSM*K2 
 
 0364 SF-SHF 
 
 0365 NMEM>MNMET *0.5 
 
 0366 NSXW-HNEXC +0.5 
 
 0367 M5VLE-M5VLE* HVALUE*0AVS 
 
 0368 CETEP-CETEP «■ ETEP»DAVS 
 
 0369 CETAS-CETAS *ETAS*0AVS 
 
 0370 C N 5IZ-C N 5 U»WN It N»PAVS - 
 
 0371 CNES*CNES*MNES*DAVS 
 
 0372 CNAS.CNAS»WNAS»DAVS 
 
 0373 CNMET«CNNET*WNNET*OAVS 
 
 0374 CNEXC-CNEXt>MNEXC»DAVS 
 
 0375 CESAVG-CESAVG* ESAVG*0AVS 
 
 0376 _ CEASCCEASG*EASAVG»DAVS 
 
 0377 CLOSS- CL0SS*IWNIZN-WNES»*0AVS/<TMS*SF3» 
 
 0378 . H VLE1-H5VLE»0I5G 
 
 0379 NSAVG»CN5S*DI5G *0. 5 
 
 03 80 NIZAVG»CN5IZ»D1SG »0. 5 
 
 0381 ANIZVG»CN5IZ*DI5G 
 
 0383 AETEP-CETEP/DTAVS 
 
 03 84 AEIAS-CETAS/DTAVS 
 
 0385 IF (ANIZVG.NE.OI MVLE2*< EO*ANSVG-AETEP-AETASI/ANIZVG 
 
 386 ALQSS-CIQSS*PI5G 
 
 0387 ANES-CNES/DTAVS 
 
 0388 .. ANAS-CNAS/DTAVS 
 
 0389 AESAVG*CESAVG*0I5G 
 
 0390 ACAS6-CEASG»DI5G . 
 
 0391 ANMET-CNNET*DI5G 
 
 0392 ANEXOCNEXC»D15G 
 
 0393 ANS-ANES+ ANAS -ANIZVG 
 
 0394 IF IANS.LF.0.0 1 AHSj-N£H/£«F_ 
 
 0395 IF (WNIZN .NE.O.OI MVALUE- (SN*EO*WGHT-ETEP- ETASI/MNIZN 
 
 03 96 IF <AN17tffi.NF.01 HVOI ANS»EQ-AETEP-AETAS 1/ANI ZVG 
 
 0397 IF (NS.GE.lt TMS-0.0 
 
 
133 
 
 FORTRAN IV 
 
 0398 
 0399 
 
 0400 
 0401 
 04C2 
 0403 
 0404 
 
 0405 
 0406 
 0407 
 0406 
 0409 
 0410 
 0411 
 0412 
 0413 
 0414 
 0415 
 0416 
 0417 
 0418 
 0419 
 0420 
 0421 
 0422 
 0423 
 0424 
 0425 
 0426 
 0427 
 0428 
 0429 
 0430 
 0431 
 0432 
 0433 
 0434 
 0435 
 0436 
 0437 
 0438 
 0439 
 0440 
 0441 
 0442 
 0443 
 0444 
 0445 
 0446 
 0447 
 
 i G LEVEL IB MAIN BATE - 72124 
 
 05/31/05 
 
 NET-NES*NAS 
 
 CALL S0URCE<NS,NTPI 
 
 IF (NET.GT.O) CALL F ILHOL (NTP.NES, NASI 
 90 NIZNT- NIZNT ♦NIZW 
 
 IF (NUP .GT.OI CALL SEPROCINTP, K3I 
 IF (NONR.GT.OI CALL RRPROC (NTP,K3 1 
 
 NASI- NAST +NUS 
 C INTRODUCE THE SOURCE PARTICLES FOR THE NEXT CYCLE 
 
 C EVERY FIFTH CYCLE RENORMALIZE NION ANO OUTPUT QUANTITIES 
 C RENORMALIZATION PROCESS OF NION 
 
 ARE PRINTED 
 
 AN5TP- 1.0/N5TP 
 ANTP5-D5AVS/NTP5 
 
 N5TPN-N5TP 
 
 DO 16 M-l.JZ 
 
 CNREZ<M)> CNREZIM! ♦ NREZ(MI*DAVS 
 16 CEZ(M|- CEZ(M) ♦ EZ(MI*OAVS 
 
 
 00 18 M-l.JZ 
 FNREZ(M)«NREZ(M)*ANTPO 
 
 18 EZ(N)« EZ(M(/ NREZ(M) 
 SUMNE-0.0 
 
 DO 193 M-l.JE 
 SUMNE- SUMNE ♦ NE(M» 
 
 TNEIMI- TNEIMI ♦ NEIMI/EOEIMI 
 193 FNE(MI-NE«M)*ANTPO/EDE<M> 
 
 
 EAV3-0. 
 
 DO 1755 M-l, JE 
 
 EAV3- EG(MI»FNE(MI*EDEIMI *EAV3 
 1755 CNEIMI- TNE(M»«AN5TP 
 
 
 CNION-CNUPDP*2.0*AIKT/SQRT(AKT*PI» 
 ANION-0.0 
 
 
 ANI02-0.0 
 
 DO 20 H-l.JE 
 
 ANI02- ANI02 * FMHEGINI ,CNE<MII*EDE(MI 
 ANION- ANION ♦ FM1 (EG( Ml ,FNE(MI l*EDE(MI 
 
 
 20 CONTINUE 
 
 ONION" ANION*CNION 
 
 
 DNI02«ANI02*CNION 
 CJO-ALOSS 
 
 IF ( ALOSS.LT.O.OI CJO-O.O 
 
 IF <ABS<CJD».GT.110.*S0» CJ0-110.*S0 
 
 TNI0N-ABSIS0+CJD1/DNI0N 
 TNI02-ABS(SO»CJDI/DNI02 
 
 TNI02*SQRT(TNI02I 
 TNION-SQRTITNION) _ 
 CNI02-CNI02 ♦ TNIO2*0AVS 
 ANIOS-CNI02»DI5 G _ 
 WF-1.0 
 
 IF ( ANES»ANAS.NE.O. > HF«t ANIZVG*ANSVG»/( ANES+ANAS > 
 
 IF (MODCNT.NPMI.NE.OI 
 N ION-ANIOS 
 
 GO TO 1790 
 
 IF (IWF.GT. 0.0051. AND. IWF.LT. 200.11 NION-ANIOS*WF**NBF 
 
 1790 CONTINUE 
 
 DAVT«DAVG*ND 
 
13U 
 
 FORTRAN IV G LEVEL 18 
 
 MAIN 
 
 DATE - 72126 
 
 09/31/02 
 
 0448 
 0449 
 0450 
 
 0451 
 0452 
 0453 
 
 0454 
 0455 
 
 0456 
 0457 
 
 0458 
 0459 
 
 0460 
 0461 
 0462 
 0463 
 0464 
 0465 
 0466 
 0467 
 0468 
 0469 
 0470 
 0471 
 
 0472 
 
 0473 
 
 0474 
 0475 
 
 04 76 
 
 0477. 
 
 0478 
 
 0479 
 0480 
 
 0481 
 0482 
 0483 
 0484 
 
 ALPHA-UN IZN/(NTPM*DAVT I 
 ANEAM-NIQN»0.33333/DEN 
 
 1785 FORMAT CO' ,' ANION- '.E 13.4, 3X,'DNI ON-', El 3.4. 3X.-TNI0N-' ,E 13.4, 3X, 
 
 X'ALPHA-'.F12.5.4X.»LEAK-'.E13.4./I : 
 
 URITEI6. 17851 ANION. 0NI02, TNI02. ALPHA* ALOSS 
 
 WRITEI6.171I EO. PDPO. D. OETZ. PEG. NDPN 
 
 171 FORMAT!'!' ,'GASt HE •, 2X, • SOURCE ENERGYi ', F5. 0, 3X ,' PRESSURE i • ,F4. 0, 
 
 X 'T0RR'.3X.'SLAB TH1C K NESS> j .E*. 1 . ' CM',' DXt',F4.1,« CH' ,4 X,'PEL T 
 X=',E12.5,3X,'»PROC-',I2,/ I 
 WRITEI6.18 0I NT. THE. NS.NSU 
 
 183 FORMAT!' •, "NUMBER OF TIME CYCLES-' . 16, 3X , 'TIME -', E12.5,' SEC, 
 13X. 'GENERATED SOURCE PART -', 15. 3X, 'MGTED * OF SOURCE PART.-', 
 
 115 I 
 
 L8___l _B AVT __ 0CEN1.EAVG.EWAVG. EAV3 
 
 1805 FORMAT!' ','DAVG -'.F12.5.3X, 'OCEN -• ,F 12. 5, 3X, • OVERALL EAVG-'. 
 
 1F10.5.3X.' WGTEO E AVG-' .F10. 5.3X. 'EAVG C HECK-' .F10. 5 ./I 
 
 WRITE(6,181I 
 181 FORMAT!' '. IX. 'NUMBER 01 STR1 BUTION' ,2X. 'FLUX DENS 1TV .4X. ' AVERAGE 
 IE ENERGY', 3X, 'CUMULATED # D1STR' , 3X, 'ENERGY INTERVAL' ,2X, 'CUM FLUX 
 
 XDEN' ,4X.'«PEN-PEVIATI0N './I . 
 
 CESUM-0.0 
 
 DO 281 M-l. JE . 
 
 CESUM-CESUM ♦ CNEIM)*EDE!M) 
 
 IF INE1MI.GT.0I PEAY6«eG<'H-EE<H^Ng<M) 
 
 IF (NE(M).LE.O) DEAVG-O.O 
 IF INEIMI.EQ.OI _DN__G_Oj . 
 
 FNDV-CNEIMI-FNE!MI 
 
 VFE»SORT!EG!HII»ETV 
 FLX1«FNE!MI*VFE 
 FLX2-CNE(MI»YFE 
 
 IF !NE!MI.EQ.0l GO TO 281 
 
 IF1HH.EQ. 1I.AN0. I OEAVG.GT.O.O II.OR. ! IH. EQ. JE I. AND. I OEAVG.LT. 0. OH 
 
 X) 0NAVG--DEAVG*FNE(MI*0.5/E0E(Mt 
 
 IF UDEAVG.LT.O.OI.AND.ttM+ll.LE.JEII DNAVG— DEAVG*! P NEIH»1 l-FNE(H 
 xn/(EG(M*ll-EGIMM 
 IF M 0EAW6.6T.0. t . AND. 1 M. GT. 1 M DNAVG-OE AV6« ( FNEI MI-FNE < M-l >> / < EGI 
 
 w yk i _. C C I U _» 111 
 
 ? B1 URITE16.182I H.FNE I H I . FLX1 . EGI Ml .CNE! HI . EDE ! H I .FLX2 . PNAVG 
 
 182 FORMAT!' • , 13 ,E12. 5.5X.E12.4, 5X, F12.6.8X, E12.5.6X, F12.6.4X.E12 
 X.5.4X.E12.4I 
 
 DET-ODEG/ETV 
 
 MRITE16.183I NTPP. N5TPN.N TPW .PET. DAVGO. NCOL 
 
 183 FORMAT!' ',// ,3X, • TOTAL • OF PARTICLES PROCESSEO -• , I5.5X, 'CUMULAT 
 _ 1FP i OF PARTICLES -' . I10.3X . ' WGTED TOTL • OF PAR T ICLES-' .F12. 3 .//. 
 
 X' NEW DT-',E12.5,4X,'0LD DAVG-' ,F12. 5.4X, • • OF COLLISIONS-' , 15 , //I 
 
 yRlTE16.187l 
 
 187 FORMAT!' • ,1X, 'ZONE NUMBER'. 5X, 'AVERAGE ENERGY' ,5X, 'NUMBER 
 
 IF FLFCTR0NS'.3X. 'CUMULATIVE » PEN ( I I ' . 3X . 'CUMU. A VG ENERGY'. 3X. 
 
 l'ZONE » OENSITY' ,/l 
 
 .DO 282 M-l. -J-Z 
 
 ANZ-CNREZ!MI«AN5TP*PI5G 
 
 PC7-CF7IHI/CNREZ!HI 
 
 282 WRITE(6,184I M, EZIMI, NREZ(M), ANZ. PCZ,FNREZ(M» 
 
135 
 
 FORTRAN IV G LEVEL 18 
 
 mmr 
 
 DATE - 72124 
 
 05/31/0r 
 
 04t85 
 
 0486 
 0487 
 
 0488 
 0489 
 
 0490 
 0491 
 
 0492 
 
 0493 
 0494 
 
 0495 
 
 0496 
 
 04 97 
 0498 
 0499 
 
 0500 
 0501 
 0502 
 0503 
 0504 
 0505 
 05 06 
 0507 
 0508 
 
 184 FBKHrm '. **♦ I S, SX. FU.6,M,FU.t, fT, F U.8. M.FU.l. T I, — 
 1 F12.5) - 
 
 " WRITE(6,183I NION,WN!ZN,NIZN, ANIZV6,WNES. NES.ANES, MN A S.N AS, 
 XANAS t NTP, Kl t K2, HF.BF 
 
 185 FORMATC'O", 'NUMBER OF POSITIVE ION-' ,E 14.6,// , • NUMBER OF ION I Z AT 
 1I0NS 0CCURE0-',F9. 3 , 5X, 'ACTUAL !■', 15, 3X, 'CUMULATIVE AVC»» ,F10. 3, 
 
 x//,' » of ESCAPED PARTICLES-', F9. 3,8y,. ' AC T UALI- ' . I S, *X, ' CUMULA TI VE 
 
 X AVG«',F9.2,//.' # OF ABSORPEP PART1C-' ,E12. 4 , 3X, 'ACTUAL t-',I 5,3X 
 X, 'CUMULATIVE AVG- ,F9.2 ,//, • TOTAL I OF PARTICLES IN THE NEW CVC1T 
 
 XE -',15,//,' K1-',I2, 3X,'K2-',12, 
 
 4X,» BALANCE FACTOR WF-", 
 
 XF10.5,3X,'BF2«', F10.5,/» 
 
 WRITE(6,188> E01. NPM, NO, KELC 
 
 Ta e FORMATS ENERGY PlvlPlNG LINE FOR RR PLAV E - ' . F 10.4,4X, ' EV E RYMZ 
 
 X,'TH_ CYCLE CHA NGE N »' ,//,' TIME INTERVAL MU LTIPLY FACTOR-' ,F9. 4, 
 
 X3X, 'ELASTIC XSEC / BY', I 2,/ I 
 
 NOR - N DN-NDN/K3 
 
 WRITEI6.186I NSO, NSW, ANSVG,NOR,WNSE,WNMET, ANMET.WNEXC, ANEXCNUP, 
 
 XWGHT 
 
 186 FORMAT!' • • • SOURCE PARTICLES GENERATEO OLP -',110,//, 
 
 1' SOURCE PARTICLES (WEIGHTED! IN THE NEW CYCLE -',U0,3X. 
 
 'CUMULAT 
 
 XIVE AVG-',F10.3,// 
 
 1 ,' * OF PARTICLES KILLED OFF IN RR PROC-' , IIP,//, 
 
 1' « OF SUPER ELASTIC COLLI SION-' ,F10.4, // , 
 
 1 ' • OF EXCITATIONS TO META STABLE LEVEL-' 
 
 X.F10.4, 3X, 'CUMULATIVE AVG-' , F9.3, 
 
 I //,' i OF EXCITATIO NS TO OTHER LEVELS-' ,F 1Q. 4, 
 
 X4X, 'CUMULATIVE AVG-' , F9.2 ,// , ' • OF PARTICLES CROSS OVER TO Hl'SH 
 XE NERGY REGION NUP- ' , I 5.4X, ' OVERALL WT-' ,E12.4, / I 
 
 WRITE(6,195I EF, EOP, POPO ,SF1, SF2, SF3 
 195 FORMAT I' E-F1ELD-', F9.2,'V/CM', 4X, ' E/P-' , F9. 2, 4X, 'P/PO -',F7.1, 
 X3X, 'SF1-', E12.4,3X, • SF2-' , E12. 4, 3X, 'SF3-', E12.4./I 
 
 MRITE(6,170I S0,WVC,SWF0,WVALUE,WVLE1, WVLE2,ESAVG, AESAVG,EASAVG, 
 
 I-', 
 
 XAEASG 
 170 FORMAT! ' SOURCE RATE -',E11.3,« PER SEC ' ,4X, 'W-VALUEI LOCAL 
 XF10.3, 3X, 'SOURCE REP. FACT-' ,613.4,//, • W-VALUE-', 
 
 XF10.3.4X,'CUM AVG WV1-' ,F9. 3,3X, 'CUM AVG WV2-',F9.3, 
 
 1 //,' AVERAGE ENERGY OF ESCAPEO PARTICLES-' , F12. 5, 3X, 'CUMULAT 
 XIVE AVG-'.F11.4,//,' AVERAGE ENERGY OF ABSORPEP P A RT 1 CLES-' ,F12. 5, 
 X3X, 'CUMULATIVE AVG-' ,F11.4,/ I 
 CKFNE-SUMNE+ANTPO 
 
 WRITE(6,189>CKFNE, CESUM, TNION, TNI02 
 189 F ORMATC ' , ' THE SN » CHECK ■' , F12. 4, 3X, ' THE CN « CHECK-' ,F12.4,//, 
 1« CALCULATED N* USE F-' ,E14. 6,3X, 'CALC. N* USE CN -',E14.6,3X, »AR 
 
 X TIF CN+-'.E14.6,//) 
 
 IF ((M00(NT,5».NE.0».0R. ( MODI NT, 10 ». EQ.O M GO TO 3 
 
 WRITSI9) EAVG, OAVG, TME , NT W, NT, NTP, NION.OPEG, NSW, SWF, E, JJ 
 
 WRITEI9I Z, WT ,ANIOS, SF3 
 
 EWFUE 9 
 
 REWIND 9 
 
 3 CONTINUE 
 
 IF (MOO(NT,10).NE.O) GO TO 4 
 
 WRITEdOl E AVG, DAVG. TM E, NTW, NT, NTP , NION, PPEG, NSW, SWF, E, U 
 
 WRITEUOI Z, WT ,ANIOS, SF3 
 
136 
 
 FORTRAN IV G LEVEL 18 
 
 MAIN 
 
 DATE • 72126 
 
 OS/31/02 
 
 0309 
 0510 
 0511 
 0512 
 0513 
 
 051* 
 
 0515 
 
 0516 
 
 0517 
 
 0518 
 
 0519 
 
 0520. 
 
 0521 
 
 0522 
 
 0523 
 
 0524 
 
 0525 
 
 0526- ... 
 
 0527 
 
 0526 
 
 0529 
 
 0530 
 
 0531 
 
 Q5J2 — 
 
 0533 
 
 0534 
 
 0535 
 
 0536 
 
 0537 
 
 0538 
 
 0539 
 
 0540 
 
 0541 
 
 0542 
 
 0543 
 
 054it. 
 
 0545 
 
 0546 
 
 0547 
 
 0548. 
 
 0549 
 
 0.5.50 
 
 0551 
 
 0552 
 0553 
 0554 
 0555 
 0556 
 0557 
 0558 
 C559 
 0560 
 
 ENOFILE 10 
 RFMlNO 10 
 
 * CONTINUE 
 
 1F <MnDINT.Nf.UH)«NE.O» 
 
 N5TP-0 
 
 NTPS-Q 
 
 r.p TQ 6666 
 
 05AVS-0.0 
 -NAST-0 
 
 NIZNT-0 
 
 myj g-o.o 
 
 CN5IZ-0.0 
 rwiS-0.0 
 
 CNMET-0.0 
 
 CESAVG-0.0 
 fFASG-O.O 
 
 CSF2»0.0 
 nnss.Q.O 
 
 CN102-0.0 
 
 00 54 M-l.JZ 
 
 rNP F7im.o.o 
 
 54 CEZ(M»"0.0 
 
 nn 55 M«1.JE 
 
 TNE<M»"0. 
 ^ fMPlMfO.O 
 
 CNES-0.0 
 S-O.Q 
 
 OTAVS-0.0 
 rf TCP»0.0 
 
 CETAS-0.0 
 
 if tn nniNTt> n| - wg - Q> 
 
 6666 CONTINUE 
 
 NT- MT ♦ 1 
 
 ftp Tfl 6666 
 
 00 56 MM 1 500 
 nWW tMi-0.0 
 
 IES(N»"0 
 JABJJUafl 
 
 56 IASIMI-0 
 
 p p si M-1.SO 
 
 HSE(NI"0.0 
 11 F^F' MI "°- Q 
 
 I«l 
 
 norHT l Al WF ™* OUTPUT QUANTITIE S 
 DO 52 M»li JE 
 
 FFIHI»0. 
 
 52 NE(MI- 0.0 
 
 nn S3 BslaJl 
 
 NREZ(M>« 0.0 
 
 C7IMI-0.0 
 
 DO 53 N«lfJE 
 
 FKPM.MfO.O 
 
 53 NEUNtM»»0.0 
 
 
137 
 
 FORTRAN IV 6 LEVEL 18 
 
 MAIN 
 
 DATE - 72126 
 
 05/31/02 
 
 0561 NIZN-0 
 
 0562 NOZN-0 
 
 0563 NIN-0 
 0560 NES-0 
 
 0565 NSE-0 
 
 0566 NCOL-0 
 
 0567 NAS«0 
 
 0568 NASW1-0 
 
 0569 OTOT-0.0 
 
 0570 NNET-0 
 
 0571 NEXC-0 
 
 0572 WNES-0.0 
 
 0573 WNAS-0.0 
 
 0574 WNEXC-0.0 
 
 
 
 
 0575 WNIZN-0.0 
 
 0576 WNMET-0.0 
 
 
 
 
 0577 WNSE-0.0 
 
 0578 ETAS-0.0 
 
 
 
 
 0579 ETEP-6.6 
 
 0580 ETOT-0.0 
 
 
 
 
 0581 NW-0 
 
 0582 NUP-0 
 
 
 
 
 05 83 NON-0 
 05 84 NONR-0 
 
 
 
 
 0585 INIV-INIV* 1100 
 
 0586 CALL RN3INZIINIVI 
 
 
 
 0587 WRITEI6.72I 
 
 0588 IF (NT.LE.NTCI 
 
 GO TO 7777 
 
 
 0589 9999 STOP 
 
 0590 DEBUG SUBCHK 
 
 
 
 0591 END 
 
 
 
138 
 
 
 FORTRAN IV 6 CEVET Tf" 
 
 TOTETF 
 
 DAT! - TlI2i 
 
 09/31/02 
 
 0001 
 0002 
 0003 
 0004 
 0005 
 0006 
 0007 
 
 oooe 
 
 0009 
 0010 
 0011 
 
 function indexeiei 
 
 COM MON /AREA*/ CONST. 0EL2. ELT,tE, Oil. 012. 013 ,DEL1 
 
 IF (E.LT. 0*081 GO TO 
 IF (E.GT. 158. 1335 1 GO 
 
 11 
 
 TO 
 
 13 
 
 12 tNOEXE- 5*(AL0G(E»*C0NST ♦4.01*012 
 RETURN 
 
 11 INDEXE-l ♦ E*DI1 
 RETURN 
 
 13 INOEXE-36 ♦ CALOG<E»*CONST-T. 005 1*013 
 RETURN 
 
 END 
 
 
139 
 
 Fl")«T«AN IV G LEVEL 1M 
 
 SflKCE? 
 
 OATI • 7?1>6 
 
 r*>n\n 
 
 oou? 
 
 ODOS 
 
 <*«■> 
 ■>'• : 7 
 
 00 J« 
 0010 
 
 r -11 
 
 -'■12 
 
 t-'^n 
 
 CM* 
 
 0015 
 CU 
 0017 
 
 c .: 1 H 
 0019 
 0020 
 r 21 
 0022 
 0023 
 
 0025 
 CC26 
 0027 
 OC26 
 0C29 
 0030 
 Of 31 
 0032 
 0033 
 00 34 
 0035 
 0036 
 0037 
 0038 
 0039 
 0040 
 CC41 
 0042 
 0043 
 0044 
 0045 
 0046 
 0C47 
 0048 
 
 SUGKOUTlNfc Sr)PCE2(NS0t EAVG, NTWtPM, FIOI 
 
 "LAL Nl(55),Fin(50> ,WI(55» 
 
 Common I), r(5ie>l,2(5lCCI, UtSlOOl . 1 AS1 5CC » , IftUI 850 I t N»H , I* S< 6 * : I 
 X,«T( 51001 ,SWF 
 
 COMMON /AKEA2/ C1,MU0,EI,FIS,SF,ANEAM,P0I>C,PI ,frf , NION , JF , JZ ,NA*MJ , 
 INSEM ,AKT, AIKT ,AMl , AMC 1 ,E01 , PONUO, Pf)26,K1, «?. Kit N^iEDNEOl 
 
 COMMON /A5/ EG(55I, E0H55I, E80(55> 
 
 FMt(t)« tXP(-E*A!KT)*SOt-T(E»«AHl 
 
 FM2(M»AA/f 
 
 r : M3(F»-EX(M-»:*AIKn*SURT(E» »AMC1 
 KEAL MIJO. NION 
 <>04 FURMATd «,F12.6,3X, 13, It, E12.4, 2X, F 12. 6, 2X, H2. 5 , 3X, H 2 4 I 
 605 F(I«MAT(« •,• Jl-« ,I3,3X,«E01»»,F9.3 t ?X , »NSO • , 1 *, 3X ♦ «K1»» , 
 
 1M.3X, 'K2*«, I«,//,' INITIAL DI STR 1 WITHIN* ,//, • #0F PTS • , 7X, • I NOCX 
 l',5X, •« OfcNSITY* ,4X,»AVG ENERGY* ,6X , • OFL-t • ,*X, • *F IGHT* ,/ I 
 6"6 FORMAT! • •,«NSl« , ,I5,*X,«NS2" , ,I5t3X,»NSO- , .l5,3X, , NTW»« . 15,3X, 
 
 X«NTW<CALC»«».I5,//) 
 felt? FORMAT <• EAVG-', F 12. 5, 3X, • WGHTED E AVG-' ,F12. 5 ,// » 
 AA-? 0*PM»SORT(PM/Pt >*EXP(-PM| 
 Jl«lNDEXE(Eni-0.5l 
 
 J2»Jl*l 
 
 ISUM-0 
 
 HRIT5I6tf05l 
 
 00 1"? 1-l.Jl 
 
 ONI -F10II »*EOE(I )-NTw/Kl 
 
 Mil l-DNI 
 
 wl( I 1-1.0 
 
 IF IFIDm. 
 
 (ONI.GT. 
 
 (ONI, 
 
 (DM, 
 
 f UNI, 
 
 Jl, EDI, ►'SO, Kl, K2 
 
 112 
 
 IF 
 IF 
 IF 
 IF 
 
 r,T, 
 
 LT, 
 ,LT, 
 
 112 
 
 100 
 
 202 
 
 20: 
 
 LE.C.O) GC TO 
 
 220. I Nil 11*200 
 
 200.1 WI(l l-DNI/ 200. 
 
 l?l NKII-12 
 
 12) MI(II«0NI/12.0 
 NRITE(o,604) NKII, I, FIDdl.EGdl, EDEIII.Mllll 
 I SUM- I SUM* NI (I I 
 JSUM-0 
 
 DO 20^ I-J2, JF. 
 DNI «FIOm*EDE(I »*NTW*K2 
 NI (I MOM 
 Mid 1-1.0 
 
 IF (FlOd I.Lc.O.Ot GO TO 202 
 IF (ONI.GT. 203. I NKU-200 
 IF (ONI.GT. 200. I Ml ( ! l-ONI/200. 
 IF (0NI.LT.12) NIdl-12 
 IF (ONI. LT. 121 Wl (I l-ONI/12.0 
 
 WRITE(6,604I NKII, I, FlOd ),EG(II, EDFdl, Midi 
 JSUMx jsum ♦ NKII 
 NSUM«ISUM*JSUM 
 NT-2-ISUM-K1 +JSUM-WN2 *0.5 
 MRITEI6.606I ISUM, JSUM, NSdM, NTW, NTM2 
 NTW-NTW2 
 IE-i 
 EAG-0.0 
 00 300 I -I, JE 
 
lUO 
 
 
 FORTRAN IV li LEVEL Id 
 
 SDRCE2 
 
 OATF - 72126 
 
 05/31/02 
 
 00*o 
 
 
 IF (Nim.LT.ll GO TO 300 
 
 C0 5C 
 
 
 JI-NKII ♦!£ -1 
 
 0051 
 
 
 00 301 J-IE, JI, 2 
 
 0052 
 
 
 R!afUN3Z(<M 
 
 OC53 
 
 
 R2M.0-RI 
 
 005* 
 
 
 f ( J»» E*0( I 1 ♦ EDEU l*Rl 
 
 0^55 
 
 
 EU»1>* FBOU 1 ♦ FOE( I»*R2 
 
 0056 
 
 
 wT( JI'HM I 1 
 
 0057 
 
 
 -T(JM I'UII [) 
 
 0C5P 
 
 
 F.AG -EAG ♦ F(J) +EIJUI 
 
 005<J 
 
 
 IF( ( MU0( ( JI-IF l.?t .EO.OK ANO. (J.GE. JI ) I 
 
 C n br 
 
 3^1 
 
 CONTINUE 
 
 OCbl 
 
 
 IE*IE*NI< I 1 
 
 00*2 
 
 300 
 
 CONTINUE 
 
 0C6 3 
 
 
 NSO-NSUM 
 
 0064 
 
 
 EAVG«EAG/NSO 
 
 0065 
 
 
 I2-NS0 
 
 C066 
 
 
 ETW»0.0 
 
 0067 
 
 
 00 *00 IM.NSO 
 
 0068 
 
 
 IF ( E< I 1. LE.ED1 » KT(II-Kl *WTC 1 1 
 
 0069 
 
 
 IF (E< II.GT.EDU WT(I l»WN2*WT(I» 
 
 00 70 
 
 
 IF (E( n.LE.O.CI EID-EI 1-1 » 
 
 0C71 
 
 
 FTWETW ♦ Ed >*WTU) 
 
 00 72 
 
 too 
 
 CONTINUE 
 
 0073 
 
 
 EWAVG-ETW/NTW2 
 
 007* 
 
 
 HRITE(6,607» EAVG, EMAVG 
 
 0075 
 
 
 QDZ«D*0. 25 
 
 0076 
 
 
 I»l 
 
 0077 
 
 101 
 
 RZ«RAN3Z(0I 
 
 0078 
 
 
 RU-RAN3ZI0) 
 
 0079 
 
 
 zm»o*Rz 
 
 0080 
 
 
 UU)-2.0*RU -1.0 
 
 0081 
 
 
 I- I* 1 
 
 0082 
 
 102 
 
 Z(II«D»RI 
 
 0083 
 
 
 um»i.o- 2.0*RU 
 
 008* 
 
 
 l«_l..»l ... . . . 
 
 0085 
 
 103 
 
 HI »«0*(1.0-R2» 
 
 0086 
 
 
 UII)« 2.0*RZ-1." 
 
 0087 
 
 
 I- I *l 
 
 0088 
 
 10* 
 
 ZII)»D*I1.0-RZI 
 
 0089 
 
 
 um» l.-2.*RZ 
 
 Q090 
 
 
 1* It 1 ... . . 
 
 0091 
 
 
 IF(( 12 -I-3».GE.0» GO TO 101 
 
 0092 
 
 
 IF (I2.LT.I) GO TO 105 
 
 0093 
 
 
 RZ»RAN3Z(0) 
 
 0094 
 
 
 RU-RAN3ZI0) 
 
 0095 
 
 
 IF(( 12 -M.EQ.2I GO TO 102 
 
 0096 . 
 
 . . ... . 
 
 I£iJ-12_ill*ifl.li_ GO TO Lfl3.__ 
 
 0097 
 
 
 IF(< 12 -II.EQ.OI GO TO 104 
 
 0098 
 
 105 
 
 RJrTUBJ* 
 
 0099 
 
 
 END 
 
 EAG»FAG-EU»n 
 
 
1U1 
 
 FORTRAN IV G LEVEL 18 SOURCE DATE » 72126 05/31/02 
 
 0001 SUBROUTINE SOUPCE I NSt NTP I 
 
 0002 COMMON 0, E<5100). ZI51001, UI5100I . I A S( 500 > , IRR < 950 » ,NRR , I E S( 600 J 
 X,WT<51CP|,SWF 
 
 00m COMMON /AREA2/ CI . MUO , E I , E I S , SF , ANEAM.PDPO ,01 , FO , NION , JE , JZ ,NA SWl , 
 1NSEW ,AKT, AIKT .AMI ,AMC1,ED1,P0MU0, PD26.K1, K2 . K3, WN2,feD2,E03 
 
 0004 KEAL MUO, NION 
 
 C SOURCE PARTICLE PARAMETERS GENERATING PROCEDURE 
 
 C II: THE INITIAL INDEX VALUE FOR THE SOURCE PARTICLE 
 
 0005 I*NTP*1 
 
 0006 DO 100 K-li NSt 2 
 
 0007 E(II"EO 
 
 0008 RZ«RAN3Z(0I 
 
 0009 RU-RAN3ZICI 
 0C10 Z(M«D*RZ 
 
 0011 U(II«RU*RU-1.C 
 
 0012 HT( I l»WN2/SWF 
 
 0013 IF UNS-IO.LE.OI GO TO 100 
 
 0014 E(I«1)-EC 
 
 0015 Z I I ♦ 1 » » RZ»D 
 
 0016 U< 1*1 1— Ul I I 
 0C17 taTI I*ll*rtN2/SMF 
 
 0018 I»I*2 
 
 0019 100 CONTINUE 
 
 0020 NTP«NTP*NS_ 
 
 0021 105 RETURN 
 
 0022 END 
 
1U2 
 
 FORTRAN IV G LEVEL 18 
 
 MAIN 
 
 DATE • 72126 
 
 05/M/02 
 
 0001 
 0002 
 
 0003 
 0004 
 0005 
 0006 
 0007 
 0008 
 0009 
 0010 
 0011 
 0012 
 0013 
 0014 
 0015 
 0016 
 0017 
 0018 
 0019 
 0020 
 0021 
 0022 
 0023 
 00 24 
 
 C WHENEVER THERE ARE" MORE HOLES THAN" SOOTCT P ARTICLE'S, FTLL IN THE EXTRA HOLES W 
 C WITH PARTICLES AT THE END 
 
 SUBROUTINE FILHOLtNTP, NES, NASI 
 
 COMMON 0, EI5100),Z(5100I, UI5100I , I AS< 500 I , I RRI 850 I ,NRR , I ESI 600 I 
 X,wTI5l00l,SWF 
 
 100 
 
 105 
 
 300 
 
 200 
 
 K-NTP 
 
 IF (NES.Lt.OI 
 
 DO 100 1*1 • 
 
 J-IESI II 
 
 E(JI- EIKI 
 
 Z(J>* ZIK) 
 
 U( Jl« U( Kl 
 
 HTIJI'WTIKI 
 
 K*K-1 
 
 NTP-NTP-NES 
 
 IF INAS.LE.OI 
 
 K*NTP 
 
 DO 300 I»l, 
 
 J-IASI II 
 
 e< j i -e ( k i 
 
 Z( J>*Z(K> 
 
 UIJI'UIKI 
 
 WT(JI>MT(KI 
 
 K*K-l_ 
 
 NTP-NTP-NAS" 
 
 RETURN 
 
 END 
 
 GO 
 NES 
 
 tO 105 
 
 RETURN 
 
 NAS 
 
1U3 
 
 FORTRAN IV G LEVEL 18 RRPROC DATE » 72126 05/31/02 
 
 0001 SUBROUTINE RRPROC INTP.K3I '"" 
 
 0C02 COMMON D, E ( 5 100 I , I ( 5 100 ( , U(5100» , US < 500 I , IRP ( 850 > ,NRR , I ES ( 600 » 
 X,WT(5100I,SWF 
 
 0003 COMMON /A6/ ESEI50I , NUP .NDNR.NON, WSEI50I 
 
 0004 NR-NDNR 
 
 0005 K-NTP 
 
 0io 6 IF (K3.LE.ll RETURN 
 
 0007 100 CONTINUE 
 
 0008 I«IRRINR» 
 0C09 E<II«E(K> 
 
 0010 UIII'UIKl 
 
 0011 Z(I)«Z(K) 
 
 0012 WT(It>Wt(K) 
 
 0013 K«K-l 
 
 0014 200 NR-NR-1 
 
 0015 IF (NR.GT.OI GO TO 100 
 
 0016 NTP«K*l 
 
 0017 300 RETURN 
 0013 END 
 
Ikk 
 
 FORTRAN IV G LEVEL 18 
 
 ALDA 
 
 DATE - 72126 
 
 05/31/02 
 
 00C1 
 00 2 
 
 00C3 
 0004 
 0005 
 0006 
 0007 
 0008 
 0009 
 0010 
 0C11 
 0012 
 CC13 
 0014 
 0015 
 0016 
 0017 
 0018 
 0019 
 0020 
 0021 
 0022 
 0023 
 0024 
 0025 
 0026 
 0027 
 0028 
 0029 
 0030 
 0031 
 0032 
 0033 
 0034 
 0035 
 00 36 
 0037 
 
 FUNCTION ALOA(E) 
 
 COMMON /AREA2/ CI , MUO, E I , F I S , SF , ANEAM, PDPO, P I , EO , NI ON , JF , J* ,NA $W 1 , 
 1NSEW iAKT, AIKT .AMI , AMC1 , EDI , PDMUO, P026.KI, K2 , K3. WN2.ED2.ED3 
 COMMON /ARFA3/ X SEL , XELC ,XPC B, XMET , XEXC , X ION, XTOL ,KELC 
 REAL MUO, NION 
 
 WELCM.O/KELC 
 
 IF (E.LE.5.T XELC-23.6*PDP0 
 
 IF< (F.GT.5. I. AND. (E.LE.19.BI I XEL C-23. *PDPO« SQfi T ( 5 . /E I 
 
 IF (E.GT. 19.81 XELC-EXP(-0.02*E»*16.76*P0P0 
 
 GO TO 5001 
 - 19.801**2 
 
 - 1 9 ;i oT7 rei ♦ <i.03n" 
 
 XELC*XELC*WELC 
 
 IF (E.LE.E03 I 
 4001 AE1« 0.2J_*(E 
 
 XMET= 0.109*<E 
 
 XMET=XMET*PDPO 
 
 E19» E - 21.4 
 
 AE2« 1850. ♦ E19**2 
 
 XEXC- 68.80*E19*PDP0/AE2 
 
 IF (E.LT.21.4J XEXC«0.0 _ 
 
 E24« E - EI 
 
 AE3- 4900. ♦ E24**2 
 
 XION* 1~72.*E24/AE3 
 
 XION»XION*PDPO 
 
 IF (E.LT. 24.461 XION-0.0 
 
 X TOL" XMET ♦XEXC ♦ XION* X ELC 
 
 XRCB-0.0 
 
 XSEL-0.0 
 
 GO TO 6001 
 5001 BE1-0.250* (E-19. 81**2 
 
 XSEL-6. 109*119. 8 -E )/BEl 
 
 XSEL -XSEL » PDPO*A NEAM 
 
 XRCB»PDMUO*NION/SORT("E» 
 
 XION-0.0 
 
 XEXC-0.0 
 
 XME T-0. 
 
 XTOL- XSEL ♦XELC +XRCB 
 6 001 ALDA-l.O/XTOL 
 
 RETURN 
 
 END 
 
 
1U5 
 
 FORTRAN IV G LEVEL 18 
 
 COLPRO 
 
 DATE • 72126 
 
 05/31/02 
 
 0001 
 0002 
 
 0003 
 
 0004 
 
 0005 
 0006 
 0007 
 OCOB 
 0000 
 
 0010 
 0011 
 
 0012 
 0013 
 0014 
 0015 
 0016 
 0017 
 0018 
 0019 
 00 20 
 0021 
 0022 
 0023 
 0024 
 00 25 
 
 0026 
 0027 
 0028 
 0029 
 0030 
 0031 
 0032 
 
 0033 
 _0O_34._ 
 0035 
 0036^ 
 0037 
 5038 
 0039 
 
 SUBROUTINE COLPROd, DCEN, NT~P\**IP, TOST; '*"," * ) 
 
 COMMON 0, £ (5100 ) , 2(61001 , U(5100» , I AS ( 500 ( , IRR ( 850 ) ,NRR , I ES ( 60C ) 
 X,WT(5100),SWF 
 
 COMMON /AREA1/ NIZN, NOZN, NAS, 0! ZN ( 500 » ,NSE , NMET, NEXC, ETAS , 
 XMNMET, WNEXC, WNIZN, ELStMNAS tMNSE 
 
 COMMON /AREA2/ CI ,MUO ,E1 , EI S, SF. ANEAM,PD PO, Pi , EO. NION, JE, JZ .NASM1 , 
 INSEW ,AKT, AIKT ,AM1 , AMCTTEDT, PDMUO, PD26,KfT~K? t ' KIT WNUi 1152 .163 
 
 COMMON /AREA3/ XSEL , XELC ,XRCB, XMET^XEXC, X ION, XTOL ,KELC 
 
 COMMON /A6/ ESEI50) , NUP ,NONR,NDN, WSE<50> 
 
 DIMENSION PX(4t 
 
 FM3(EI-EXP(-E*AIKT»*SQRT(E)»AMC1 
 
 REAL MUOi NION 
 C DECIDE WHICH KINO OF COLLISION 
 
 XTI -ALDAIEdll 
 
 if (em.LE.eo3 » go to 5006 
 
 C ENERGY REGION 1 19.0EV TO I KEV 
 4000 PMET* XMET*XTI 
 
 PEXC* XEXC*XTI _ 
 
 PION* XION*XTI 
 PELC«XFLC*XTI 
 
 PX<1 1* 
 
 PEXC 
 
 PX<2)» 
 
 PEXC *PMET 
 
 PX(3»* 
 
 PXI2I +PION 
 
 IP»1 
 
 
 DO 45 
 
 IX«1,3 
 
 45 IF(RIP.GT.PX( IX) I 1P-IX+1 
 
 GO TO (4015.4013*4017,40111, 
 4011 Rl-RAN3Z(0t 
 
 U(I>« 2.0*R1 -1.0 
 
 RETURN _ 
 
 C INELASTIC COLLISION PROCESS EXCITE 
 C EXCITATION TO METASTABLE LEVEL 
 4013 WNMET* WNMET ♦MTU I 
 
 Ed)- Ell l - 19*80 
 
 IF (E(II.LE.O.O) RETURN1 
 NMET»NMET»1 
 
 IP 
 
 TO META STABLE STATE 
 
 RI 1> RAN3Z(0I 
 Udl - Rl l ♦K11-— - 
 
 1.0 
 
 RETURN 
 C EXCITION TO O THER LEVELS 
 4015 WNEXC -WNEXC +WT ( I I 
 
 Em» mt -2it»o 
 
 IF <€< II. LE. 0.01 
 NEX C»NEXC*1 _ 
 RU* RAN3Z(0I 
 U(II - Rll ♦*!! ■ 
 
 RETURN 
 
 IPJUlIIIOfl P«QtSSS 
 
 RETURNl 
 
 l*Q 
 
 0040 
 
 0041 
 
 0042 
 
 .M43_ 
 
 0044 
 
 4017 WNIZN- WNIZN* WTdt 
 
 _EJ_I1*_ Ed) - £J 
 
 IF (EUI.LE.O.OI RETURNl 
 N1ZN«NIZN»1 
 
 PRODUCE SEC ONORV ELECTIONS 
 
 NOZN-NOZK ♦! 
 
1U6 
 
 FORTRAN IV G LEVEL 18 
 
 COLPRO 
 
 DATE ■ 72124 
 
 05/31/02 
 
 00*5 
 00*6 
 0C*7 
 00*8 
 00*9 
 0050 
 0051 
 005? 
 0053 
 005* 
 C055 
 0056 
 0C57 
 0058 
 0059 
 0060 
 
 0061 
 0062 
 0063 
 006* 
 0065 
 0066 
 0067 
 0068 
 0069 
 
 0070 
 0071 
 00 72 
 0073 
 007* 
 0075 
 0076 
 00.77 
 0078 
 0079 
 0080 
 0081 
 0062 
 008* 
 008* 
 0085 
 0086 
 0087 
 0088 
 0089 
 0090 
 0091 
 0092 
 0093 
 
 C IONIZATION PRODUCED PART" "lCL"BS~ A"lWAVS~PuT fS't'HF ENO~ OfHlfSIDUES 
 DIZN(N01N»- OCEN . _ . 
 
 RI-RAN3ZC0I 
 
 UIII»KI*R|-l.fl ..._ . 
 
 NNTP «NTP *NIZN 
 
 E(NNTF.i- £111*0,1 
 
 Z<NNTP>»Z< I I 
 
 U(NNTP)— U(I» 
 
 E( I) ' E( I »*0.9 
 
 WT(NNTP)- MT(I) 
 
 IF ((E(NNTP) .GT.EDD.OR. ( (E( I I»1.11UE1 I.LT.F01 ) I GO TO 4011 
 
 NDN«N0N*l 
 
 NDNR"NDNP«-1 
 
 WT(NNTP>"WTU»*K3 .. 
 
 IRRINDNR)-NTP*NIZN 
 *018 CONTINUE 
 RETURN 
 C ENERGY _RE_G_JJJJL I LQl<JNE_RGY RANGE P0WN..T0 JHERHAL REGION 
 5000 PELC- XELC*XTI 
 PSEL* XSEL*XTI 
 PRCB- XRCB*XTI 
 PX» L»« PRCB 
 PX(2»« PSEL *PRCB 
 
 IP^l ..._ 
 
 DO 55 IX»1,2 
 55 IFIRIP.GT.PXI UM I P« I X + 1 
 GO TO 15013, 5015,50111, IP 
 
 C ELASTIC £DJAJ_S_LQN_PRQCJSJL_ . 
 
 5011 IF (ELS. LE. 0.01 GO TO 50125 
 
 ... IF ((NUST.EQ.0).0R.mn.CT.2.3H GO TO 50124 
 
 50115 RE-RAN3ZJ0I 
 
 E1»ED2*RE 
 
 ~>Rl«FMIEl7"~ 
 FR2«FH3(£2I 
 
 IF (RE.LE.FR2I RS-RE 
 
 . 1F__CP2.IE. FP 1 ) g feHI 
 
 IF <<RE.LE.FR2I.0R.(R2.LE.FR1U 
 
 GQ— ID 5DJLL5 
 
 5012 ENN«RS*ED2 
 5Q12* IFtlNUST.EQ.QI.QR. I£II1.GT.2.?II 
 
 GO TO 5012 
 
 ENWQtP. 
 
 UM-SORTI Cl. -Ul**2»*( l.-UI I 1**2 M 
 
 US-__Q»PJ-»rl_-3 
 
 C0SST-U1*U(I» +SIGN(UM,US> 
 DELE-Cl»(Em-ENNI»(1.0-C0SSTt*ELS»KELC 
 Ell I- EUI -DELE 
 
 um-ui 
 
 50125 IF( ELS. LE. 0.01 
 
 BJLtUJLN. 
 
 C RECOMBINATION PROCESS 
 
 U ( I I - 2.0*RAN3Z(0» -1.0 
 
1U7 
 
 FORTRAN IV 
 
 G LEVEL 
 
 IS 
 
 009* 
 
 5013 
 
 NAS-NAS+1 
 
 0095 
 
 
 IASINASI-l 
 
 0C96 
 
 
 WNAS- WNAS*WT(l) 
 
 0097 
 
 
 IF (E(I).GT.O.OI 
 
 0098 
 
 
 PETURN2 
 
 
 C SUPERELASTIC COLLISIC 
 
 0099 
 
 5015 
 
 E ( I ) » EI 1 1 ♦ E I S 
 
 0100 
 
 
 WNSE-MNSE ♦WTI II 
 
 0101 
 
 
 NSE«NSE+1 
 
 01C2 
 
 
 ftP« RAN3ZI0) 
 
 C1C3 
 
 
 um- RP *RP -1.0 
 
 0104 
 
 
 RETURN 
 
 0105 
 
 
 ENO 
 
 COLPRO DATE ■ T2126 03/31/02 
 
 ETAS«ETAS*E(II*WT(I I 
 PROCESS 
 
1U8 
 
 FOR TRAN 
 
 lv a li 
 
 :vul in 
 
 
 c 
 
 MASTE" P 
 
 
 L 
 
 TICL r S I 
 
 
 c 
 
 DEPIN1TI 
 
 
 r 
 
 0: l)IA«E 
 
 
 c. 
 
 i 1 : INI T I 
 
 
 c 
 
 JZ : Ti'Tai 
 
 
 c 
 
 IAS: AfcP 
 bn: IJ 1ST 
 
 
 c 
 
 V: JUNE 
 
 
 c 
 
 ASSAYS 6 
 
 
 r 
 
 NTR: T IT 
 
 
 c 
 
 NT; TOTA 
 
 
 c 
 
 c" 
 
 mi MIMH 
 NIZN: Nil 
 
 occi 
 
 
 kE 
 1*101 
 
 0002 
 
 
 DIME 
 
 or 3 
 
 
 COMrt 
 
 CO 04 
 
 
 
 X,wJ( 
 
 ~ comm" 
 
 XwNME. 
 
 000? 
 
 
 COMM 
 INSEw 
 
 0006 
 
 
 COMM 
 
 0CG7 
 
 
 COMM 
 
 OOOfi 
 
 
 COMM 
 
 0009 
 
 
 COMM 
 
 00 10 
 
 
 REAL 
 
 OX 11 
 
 c 
 
 STATEMEN 
 FMU 
 
 0012 
 
 
 DATA 
 
 MAIN 
 
 DATE - 72125 
 
 C1/2B/29 
 
 0013 
 .?CJ.i. 
 
 0015 
 0016 
 0017 
 
 DATA 
 
 DATA 
 
 1.0/7 
 
 OATA 
 
 KilliOAM OF MONTE CARLO SIMULATION OF NONLINEAR ALPHA «>LASM»S (ALOHA 
 
 NfMJCED PLASMA FL tCfRON" (tNTRBY D ISt«" I ftUT ION STUDY 
 
 ONS AND TFKMINOLOGY 
 
 TER OF THE CYLINOEft. THE LFNGTH OF THE FINITF. REGION 
 
 4L ENERGY OF THE SOURCE PARTICLES SC(NSM SOURCF RATE 
 
 NUMBER OF ZONES JEJ TOTAL NUMM.R OF ENERGY RANGES 
 AY FOK STORING THE INDEX OF ABSORPEO PARTICLES NAS« T HE NO. OF THFM 
 ANCF TO ROUNDPY ' OCOL~: 0"lST ANCf'TO COLLfSloKT 0"CE"N~: OtsrANCE TO C£N 
 
 INTERVAL 
 . Z. li STOPING ENERGY. DISTANCE. AND OIRECTION OF PARTICLES 
 At NUMHER OF PARTICLES TO BE PROCESSED EACH CYCLF 
 L NUMBER OF TIME CYCLES 
 
 f-K OF SOURCE PARTICLES PERJJNIT T IM?*DI STANCE OR SOURCF PATE 
 MBER OF IONIZATION PROCFlS OCCURRED tACH TIME CYCLE 
 AL NF(55(, NEZt 55, 10 » , F I i 1 1 » , DX ( 3 ) , EE< 55 I , CEZl 111 tEKZf 19 
 
 ,FlC(6d» ,NRFZ(U).EZK11»,NTPW 
 NSIPN FNf(55», FNREZ(ll), NRZ I (1 1 ) , CNE( 55 » t CNRE/ I 1 1 ) , TNF « 55 » 
 ON 0. F<510C >,Z(5100I, U(5100» , I AS ( 500 ) , IRR( «550 I .NRP , IF S< 600 » 
 51001, SWF 
 
 ON /ARFA1/ NIZN, NOZN, NAS, 01 ZN( 50C j ,NSE% NMET, NEXC, ETAS , 
 T. wNfXC, WNIZN, ELS,MNAS ,WNSE 
 ON /AREA2/ Cl,MU0,fcI,EIS,SF,AN£AM,POPC,PI,EO,NION,JE,JZ,NASKl, 
 
 ,AKT, AIKT ,AM1 .AMC1.E01, PDMUU, PD26.KI, K2 , K3, WN2.E02.ED3 
 ON /AREAV XSEL,XELC,XRCB,XMET,XEXC,XION,XTOL,KELC 
 ON /AftFA4/^ CONST, DEL2, ELT.IE, Oi l, DI2, DI3 .DELI 
 ON /A5/ EG{55»,~ EDEI55I, EB0<55» 
 ON /A6/ ESEI5CI , NUP ,NDNR,NON, WSE(50I 
 
 MUO, NION.ND 
 T FUNCTION FOR EVALU ATING MAXWELL IAN jAND RELATED FCNS 
 E,FNtM»*EXP<-b»AIKTf*SQRTfE)*FNEM 
 
 TM S, NT, DTOT, E TO T, NI N, N I ZNT,NAST/0. 0,1,0.0. 0.0 ,0,0,0/ 
 
 w5VLE,CN5IZ,CN5S,CNES,CNAS,CNMET,CNEXC,CESAVG,CEASG/9*0."*7~ 
 
 oio 
 
 SUS 
 
 CNF/ 55»0.C/, CEZ/U+0.3/, CNR E Z/l 1*0. 0/.TNE/55»0./,Elt?/5S0»Q 
 D5AVS/0.b/ 
 
 NEZ/ 550*0. /,NE/55 *0./. EZ/11»0./.NRSZ/ 11*0,/ , EE/S5*0.0/ 
 READ! 5,1 I NTAPE 
 P EA D(5,15 II Kl, K2, NTC, NTP,NTW,NPM, KELC ,JZ, 1M,NCUM, NDPN,NTMP 
 
 001<L 
 
 0019 
 
 0C20 
 
 0021 
 0022 
 
 0023 
 0024 
 
 0025 
 PQ 26 
 
 0027 
 
 Q02 a 
 
 0029 
 0030 
 
 0031 
 0032 
 
 X, MC ,IUSN, NUST, NBF 
 PEADI 5 ,152» EDI ,ND , EO, ELS, D.ANSW, 
 
 PEAD<5,153) FID 
 1 FORMA T _(_ J10J 
 
 151 FORMAT! 16151 
 
 152 FORMAT! 8F10.4) 
 
 PDPO, SR 
 
 153 FORMAT! BE10.3) 
 AKT-0.05 
 
 AIKT- 1.0/AKT 
 PI-3.1M5 93 
 
 EMASSIM.60203E + 16/9.108 
 
 K3-m»K2 
 
 WN2-1.0/K2 
 CJD-0.0 
 
 EI"24.46 
 EIS-21,4 
 
lU9 
 
 F'IRTKAN IV t; LTV eL LB 
 
 MAIN 
 
 DATE - 7212b 
 
 01/28/29 
 
 00 3 3 
 00 34 
 0035 
 o " 36 
 0037 
 3*3 8 
 0CJ9 . 
 0040 
 r -(. 't\ 
 0042 
 
 C44 
 
 U045. . 
 
 0C.46 
 
 0047 
 
 C<"48 
 
 C-4S 
 
 0050 
 
 CC5.1__ 
 
 005? 
 
 005? 
 
 or: :>4 
 
 0055 
 0056 
 QQ5 7 
 
 Cl»2.'*0. 010549/4. 003873 
 
 ETV* SORT (2* 0*1. 60203/9. 108 1 *l.0E*08 
 
 VTf« 1.0/«FTV*ETVI 
 CALL UNDERZI 'OFF* > 
 P026=PnPU*26.0 
 MU? = < 1.0F-C9I/ETV 
 r.rtlJ0 = 1. 0F-09 
 
 CMJPDP=CMU0*PDP1 
 IF ( POPr-.CT. 10. C > 
 P9MU0*C"ijPDP/ETV 
 
 CLOSS=". r 
 NUP*0 
 
 _"i!iN3j .... _ 
 
 CMUP0P«>CHIJQ*1Q,Q 
 
 0058 
 CC59 
 0060 
 OQfil 
 0062 
 
 0063 
 
 C064 
 
 -O&Ai.. 
 
 0066 
 
 -0C67 
 
 0068 
 006.9 
 
 00 70 
 _OQJJ_ 
 
 0072 
 
 0073 
 
 0074 
 OPTS 
 
 00 76 
 0077 
 
 0078 
 0079 
 
 0080 
 
 nofli 
 
 0082 
 0083 
 
 0084 
 0085 
 
 NDNP ■? 
 INO»0 
 
 NAS = 
 NES*0 
 
 NSb-J... 
 
 NG7N*0 
 
 NIN»0 
 LK = 
 
 iMv^^o^pgg 
 
 NMET=0 
 
 NEXC=0 
 ETAS = r>.C 
 ETEP-0.0 
 NTP 5-0 
 
 NW»0 
 N5TP*0 
 
 DTAVS=0." 
 _C_SJ=2i0^0 
 CNI02=0.0 
 
 CETAS=0.0 
 
 f.FTFPsP.P 
 
 NCOL-0 
 . HNME T'Q .O. 
 WNSE«0.0 
 WNEXC^O.0 
 
 WNES»0.0 
 MNAS^fKP 
 
 WNIZN'O. 
 JE«4d _. 
 
 IE*JE 
 ,DEN»3.-5AE*L6»RDPQ_ 
 
 DETZ=D/JZ 
 
 CONST=1.0/ALOG(2.0I 
 _ED2»0. 447513 . 
 
 E03«19.8 
 PM«Fn?»A!KT 
 
150 
 
 FUPTBAN IV G LEVEL lfl 
 
 MAIN 
 
 OATE - 72125 
 
 01/2H/29 
 
 C086 
 
 oca'1 
 
 00R8 
 QC89 
 0090 
 0C91 
 
 ecu 
 
 C093 
 
 0095 
 SO 96 
 Of 9 7 
 009A 
 TC99 
 0100 
 0101 
 
 cn? 
 
 0103 
 010* 
 0105 
 0106 
 C1C7 
 0108 
 01C<5 
 0110 
 0111 
 0112 
 0113 
 0114 
 0115 
 0116 
 
 r 1 1 7 
 
 0118 
 C119 
 C12T 
 0121 
 0122 
 0123 
 0124 
 0125 
 0126 
 0127 
 0128 
 
 AMl-7.r*AIKWSQRT(PI»AKT) 
 
 CALL f>M3INf(?"«»<»T632rn — 
 
 SO-POP^* SK*1.0E*16 
 
 AITG- TMUPOP /< AKT»PI I 
 
 DEl-O.Olb 
 
 Pt2« - .?55 
 
 DE3*0.3 
 
 11 
 
 0129 
 0130 
 
 0131 
 13? 
 
 0133 
 9P * 
 
 0135 
 0136 
 
 011*1. /(J. 016 
 DI2-1./OE2 
 
 cn«i./ne3 
 
 AMC1- SOkT(2.r'.*2.71fl3*AIKT>»ED2 
 
 00 11 1=6, 36 
 
 EBL = < I-i>)*D£2 -4.0 
 
 EmWTBL" - r .l685 
 
 FBO( I (- ?.0*»EBL 
 
 rG(I-l)« 2.0**EMV 
 
 FDF( 1 )rfn0( I )*(2.0**0E2 -I. 01 
 
 JE1*JE*1 
 
 00 12 1=37, JE1 _ _ 
 ""pr'l'b" 7.005 ♦(I-36I»DE3 
 
 FMV= Cf'L - C.140 5 
 
 FHDl I )= 2.0»*EBL 
 
 FT,( 1-1 >» 2.0-**E*V 
 12 EDE<I>= EBP< I »*<2.0**DE3 
 
 00 10 1=1, 5 
 
 FOE( I 1= 0.016 
 
 EG< I )= EDL( I )+I -C.008 
 10 EBD( I t= EDt( I »*( I — 1 » 
 
 E0E( 3bl=29.6V 
 
 EG(4ri»» 1737.482 
 
 _IF_ (NTAPJE.NE.il GO 
 
 PtADIlM EAVG, DAVG* 
 
 PEADI10I Z, wT 
 
 NS*NSW*K2 
 _J*EWINO 10 
 
 NT«NT*1 
 2 CONTINUE 
 
 1.01 
 
 TO 
 
 TME, NTW, NT, NTP, NION, DDEG, NSW, SWF, E,U 
 ANI0S.SF3 
 
 IF (NTAPE.NE.2l GO TO 5 
 
 REA019 ) EAVG, DAVG, TME, NTW, NT, NTP, NION, DOEG, NSW, SWF, E, U 
 
 i)EAD(9 | Z, WT, ANI0StSF3 
 
 NS=NSW*K2 _ _ 
 
 REWIND 9 
 
 NT*NT*1 
 
 CONTINUE 
 
 IF _iNTA_PE.NE.O) GO _T0__ 122 
 
 NION- SQBT<S0*10/AITG» 
 
 ANEAM«NIUN»0.33333/0EN 
 CALL S0RCE2INTP, EAVG, NTW, PM, FIO» 
 AL=ALOA«EAVG> 
 
 OAVG=AL 
 
 OOEG = DAVG»ND/SOP.T I EAVG) 
 
 0137 
 0138 
 
 SF2=l.O 
 IF (NION.NE.Ot 
 
 SF2-NTW/ N10N 
 
151 
 
 FORTRAN IV G LEVEL 18 .. MAIN DATE "72125 01/28/29 
 
 0139 SN»S0»AL*ND»SF2/(ETV«SQRTt£AVGI ) 
 
 0140 NSW«ANSW*0.5 
 
 0141 .... IF ISN.GT.ANSw.J . N5W«SN_.*0j5_ 
 
 0142 NS»NSW*K2 
 
 0143 SKF?.NS-W/SN 
 
 0144 SF-SWF 
 0145 ANIQS'NKN 
 
 0146 CALL SOURCE(NS,NTP> 
 
 0147 SF1-SF2 
 
 014H SF3-SF2 
 
 0149 TME=O.C 
 
 0150 122 CONTINUE 
 _D1A1 SF=S*F 
 
 GENERATE SOUkCE PARTICLES TO INITIALIZE THE PROGRAM 
 
 0152 ... . ANFAHx,mion*0.333?3/DE l N .,_._■_ 
 
 0153 WRITE(6,70) 
 
 0154 70 FORMATCl't'PROGRAM INI T 1 AL 12 AT IO N AND THE INJTIAJ. PARAMETERS',//) 
 015^ WMITE<6,71I AITG, NtON, NTP, ALiEAVG 
 
 ill 56 71 FORMAT!' '. 'AITG = '.E14.6, 3X.'NIQN - ' , E 1*.6, 3X, 'NTP -MlZ .n, 
 
 l'DCbN(EO) =', F14.6.3X, 'EAVG» ',F14.4,/> 
 0157 72 FQKMATC • , 8X , 'OCOL' , 13*, •D£EJ4A»ilX J .' ENERGY* , 11X, • I • , 14X t «U' , 10X, 
 
 X • WEIGHT', / I 
 C15H 75 FORMATC »,F14,6,3X, F14.6, Mi F14.5,3_X, F12. 4 ,3X ,Fl-2,4, 3X , Fl 3. 4) 
 0159 WRITE(6,72I 
 .0160. K1P = »AN3Z(0J 
 
 0161 1=1 
 
 I WITHIN THE OUTER 1 LOOP THE I. T_H. PART ICLf HISTORY 
 
 C GEOMETRIC ROUTINE ENTERING HERE 
 
 0162 _ .....77J.7...1F.i.LLlll.LE,0.0UQA*iE(IlAiiIxlfl.U GQ LO 9008 
 
 0163 DCEN*OOEG*SORT(E(I I) 
 
 ri64 7776 IF MNI/N.GT.OI.AND.U.GT.NTPH DCEN» PI ZN< NOZNM I 
 
 0165 NFLG=0 
 
 0166 0JJJ=U11J*1USN _.. . 
 
 0167 LFG2-0 
 
 ETAG1-UQ_ . .. 
 
 0169 
 ni 70 
 
 moo 
 
 IF !E< II.GT.ED1I ETAGl—l.O 
 
 IF 1(711 l.l F.O.OI.nR.(ZtII.GT.DI.QR.(E(I I.LE.O.OI 1 GO TO 9008 
 
 0171 
 01 7? 
 
 
 J»INT(Z(I)*DZIV» *1 
 RI.RAN17C0) 
 
 0173 
 r»t 74 
 
 
 ALD«ALDA(E( It I 
 0COL*ALD*ABS(ALQGIRI 1 1 
 
 0175 
 0176 
 
 
 IF (NFLG.EO.l) GO TO 809 / 
 
 IFIMnnil.lNI.EQ.OI WRITEI6.75I t>COL. DCEN.EI I 1 .Zf I 1 .Uf I 1 .HTI I 1 
 
 0177 DC0L2*ABS( ALOGI 1. -Rll l«>AL0 
 
 01 7ft DTOT-DTOT t0C0L»DCQL2 
 
 0179 ET0T-ET0T*E(I I 
 
 ni«n IF 1N0P N .LT.2I GO. 10 609 
 
 0181 DCEN2-DCEN 
 nift? Fin-El II 
 
 0183 UI0« UIII*IUSN 
 
 0184 ZIDx/lll 
 
 0185 809 NFLG-i 
 
 0186 900 IF IDCF N .LT^DCQL) GO 10 902 
 
152 
 
 FORTRAN IV G LEVEL 18 MAIN DATE ■ 72125 01/2H/24 
 
 _01B7 901 OCFN'OCFN- DCOL 
 
 0188 ZIII-ZUI ♦ OCOL*UUI ' '. 
 
 0189 NC0L«NC0L»1 
 
 0190 IF <«.OD(NCOLiMC).EO.0l RIP"RAN3Z(0I 
 
 0191 IF (MOD(NCOLtMCI.NE.OI R IP* 1 ■ O-MP 
 
 0192 CALL COLPROUt DCENt NTP, R'T>t NUST, C9608, H9C09I 
 
 1 9? Gn TO ?oe o , 
 
 0194 902 2(11" Z < I » ♦OCFN*inTT 
 
 0195 IF <</< I I.LE.f.Ol.OR.mil.GT.DI I GO TO 9008 
 
 0196 J«INT(Z( II*DZIVI*1 
 0H7 IF (J.GT.JZI J-J7 
 Cl-iH K> INOfcXfc(E< I II 
 
 0199 IF (K.GT.JE) K-JE 
 
 0200 "' IF"(K3.E0.n GO "TO "SO* 
 
 02"! ETAG2-1.0 _ 
 
 020? IF (E( I I.GT.E01I ETAG2--1.0 
 
 0203 , IF < (ETAG1*ETAG2).GT.0.C| GO TO 9C4 
 
 C2G4 IF (ETAG2.GT.0.0) GO TO 903 
 
 0205 NUP»NUP*1_ 
 
 0206 " estiN"n>i«E(n " ' •--•—-- 
 
 0207 WT(I l*WTU I/K3 
 020A HSF(NUP)«wT( I I 
 C2C9 GC TO 90* 
 
 0210 903 IF< < LFG2.EQ.il. OR. JNDPN.LT. 211 NDNR-NONR^l 
 
 C 2 1 1 _J FKLF G? . b . 1 I . R .(NDPN .LT.2II IRR(NONRI«I 
 
 0212 "" NDN«N0N*1 
 
 0213 IF ( MnD(NON,K3).NE.OI GO TO 9045 
 
 0214 WY(I l=WT(I l»K3 
 
 0215 904 NEZ(K,JI» NEZIK.JI ♦ WT(M 
 
 0216 Ek7<K,JI»6KZ(K,JI*E(i l*Wf III ""' 
 
 2 17_ 9C45 IF( I L F G2 . E Q. Jl .OP . (N0PN.LT.2II GO TO 9 009 
 
 0218 "" LFG2=1 
 
 0219 E(I)=EIO 
 
 0220 uiii-uio 
 
 02 21 Z(l)«ZlO 
 
 2 22 0CEN»0CFN2 
 
 0223 DCUL-DC0L2 
 
 224" GO TD 900 
 
 0225 9QQ3 NES»NES*1 
 
 0226 IF" (6UY.LT.E0il WT(I l«WT(il/K3 
 
 0227 _ _W N E S_? * NF. S ♦ W T ( 1 1 
 
 0228 " IES(NES)*I 
 
 0229 IF (Edl.GT.O.OI ETEP-ETEP»E ( I l*WT ( I I 
 
 023C 
 0231 
 
 9009 
 
 I- 1+1 
 
 LFG2»0 
 
 
 0232 
 0233 
 
 
 IF ( I.LE.NTPI GO TO 7777 
 
 IF (( I.GT.NTP).AND.(NI2N.GT.0H 
 
 NOZN- NOZN-1 
 
 0234 
 0235 
 
 
 IF (NOZN.LT.OI NIN-l 
 
 IF (<NIZN.GT.0l .AND.(NIN.Efl.O) 1 
 
 GO TO 7778 
 
 0236 NIZW-WNIZN +0.5 
 
 _0237_ NWS'WNES *0.5 
 
 0238 NASW-WNAS +0.5 
 
 0239 WNAS-NNAS/KELC 
 
153 
 
 FORTRAN TV f.JJVEjL _.18_ MAIN D ATE » T212S 01/28/29 
 
 02<»0 HNSE"WNSE/KELC ' ; 
 
 02*1 NSEW»WNSE ♦O.S 
 
 024? P.O. 191 M«liJE 
 
 C?43 00 191 N«1,JZ 
 
 024* E6.<MJ«EE(.NL ♦ EKZjMiNl 
 
 0245 191 NE(M)s NEIMI ♦ NEZ(M,NI 
 
 JlZ±b. IP ((K3.EQ.H.0a.(NUP.EQ.0I) CO TO 24 
 
 0247 N3-K3/JZ 
 
 C?48 ...... . IF (N3.LT.1I H3-1 _ 
 
 0249 NK3»K3-1 
 
 0250 IF (NK3.LT. JZI IZ-NK3 
 C251 IF (N3.GE.1) M3-N3 
 
 _Q2 52. nu 23 K»1,NUP 
 
 0253"" K«INO*=XE(fcSE(N»» 
 
 0234 EE(KI-fE(K) ♦ <K3-1)*ESF(N»«WSE(N) 
 
 0255 N£(K»»NE(K| ♦IK3-l»*KSE«Nt 
 
 02!>6 11 IF (NK3.GS.JZI IZ-JZ 
 
 0257 00 21 L-l.IZ 
 
 023% NRFZ(L1«NREZ(LI ♦ M3»wSF(N> _ 
 
 r2"j9 ?1 ^Z ( L » -E7 I L » ♦ FSE(N»*«*3*WSE(NI 
 
 0260 NK3-NK* -JZ*M3 
 
 0261 IF (NK3.LT.JZl IZ-NK3 
 
 0262 M3«l 
 
 0263 IF (NK3.GT.0) GO TO 22 
 
 C26f. ,?3 CONT INUE 
 
 0265 24 CONTINUE 
 
 f'66 DO V->2 M»i, JT 
 
 0267 DU 1915 N»1,JE 
 
 0?6d N»F.Zm»NHFZ(MI ♦ NEZ(N.M) 
 
 0269 1915 EZ <M > =CZ ( M I ♦FK/JN.MI 
 
 D2.7.Q LK CU-NTI^UX . ._ 
 
 C271 NTPN*\<Tf>*NIZN-NFS-NAS 
 
 0272 NTP.J-0.0 
 
 027* FTUL-0.0 
 
 0274 L)Q 14 V=1,JE 
 
 0275 ETUL» FTCL ♦C-FJMI 
 
 . 02_7a. 14 NTPd.NTPW »NF(H1 , 
 
 0277 EWT(JL*ETCL 
 
 0278 ..N.TPT»NTPN 
 
 0279 UAVr,n= 04V0»N0 
 
 0280 DAVS'IOAVGOMOOO. )**NTMP 
 
 0281 D5AVS=JSAVS* OAVS 
 
 A2.a2 D_L5_G = U1ZQ1AY5 
 
 0283 ANTPO = 1.0/NTPW 
 
 0234 N5TP- N5TP ♦ NTPW *C.5 
 
 0285 DTAVS-DTAVS tOAVS 
 
 0286 NTP5*NTP5» NTPN*OAVS 
 
 0287 NTPO=NTP 
 
 Q2B8 cwAvr.«F^ Tni*ANTPn . 
 
 0289 NSO-NS 
 
 C29C NSWQ*NS0*wN2 *0.5 
 
 0291 NTP«NTP*NIZN 
 
 G292 NTPP*NTP 
 
15U 
 
 
 FORTRAN IV G LEVEL 18 
 
 HMN 
 
 DATE • 72125 
 
 01/28/29 
 
 0293 
 "0294" 
 0295 
 0296 
 0297 
 
 0298 
 02 99 
 C3 0C 
 03->l 
 302 
 C3C3 
 0304 
 "0 3 05 
 C3 06 
 0307 
 306 
 *■> "3 
 0310 
 0311 
 0312 
 0313 
 0314 
 0M5 
 '■316 
 
 -"317 
 02 18 
 
 on 9 
 
 T320 
 0321 
 0??? 
 0323 
 0324 
 ^3?5 
 0326 
 0327 
 3 2 8 
 0329 
 0330 
 0331 
 0332 
 33J 
 0334 
 0335 
 0336 
 0337 
 0338 
 -0J3JL 
 0340 
 0341 
 342 
 0343 
 
 DAVG"OTOT*0.5/NTP 
 
 EAvfc- eToT>nT^ 
 
 DEG-ODFG/ETV 
 
 TMF"TME*DEG 
 
 TMS«TMS*OEG 
 C IF SOURCE PARTICLES SOINSX NES THEN ELIMINAtC AS MANY RESIDUE PARTICLFS ' 
 NTPS«NTP* ♦ NWS »NA5W 
 
 DCFN1- ATOM' EAVGI 
 
 NTW-NTP* *,5 
 
 OOEG«DAVG*ND/SQRT< EAVGI . 
 
 SF2-1.0 
 
 IF (ANIOS.NE.O) SF2-NTPS/ANI0S 
 ESAVG-0.0 
 
 IF(WNES.KE.O.O) 
 
 EfAvTT-~l"«P/WNP5 
 FASAVG-ETAS/WNAS 
 
 BF»(WNIZN*SNI/IWNES*WNASI 
 
 EASAVG-O.C 
 
 IF ( WNAS.ImE.O) 
 
 irtFD»SwF 
 
 PF«1.0 
 
 IF ( (wNtS + WNASKGT.OjiOl 
 
 CN5S*CN5S+ SN*DAVS 
 
 CSF2- CSF2 +(ETEP*ETAS*40.*WNIZN»/TMS 
 
 IF(IM00(NT,5t.E0.C). ANO.INT.NE.CN SFI-CSF2 /(SO*EC*S.I 
 
 CSF1- CSf-1 ♦ SF2*DAVS 
 IF I <M0CMNT,5).E0.0I.AND.(NT.NE.C M ' SF?«CS Fl*0 I5G 
 IF ( r*CD(_NT,5».FQ.0I.AND. <NJ.NF.0) JL AN0.JS_F3.LT.SF1«0.:\1) » SF?« 
 XS0RT(SFl"*SF3» 
 
 •.GHTxSFl/SF* 
 
 SN«SJ*00£f.*SF3*W(iHT/FTV 
 
 NSw»ANS**0.5 
 
 IF ( SN.GT.ANSrtl NSW-SN *0.5 
 _ SwF-NSw/SN _ 
 
 NS»NSw*K» 
 
 SF«SwF 
 
 NME»i*WNMtT *0.5 
 
 NFXw«WNEXC +0.5 
 
 W5VLE*w5VLF* WVALUE*OAVS 
 
 CETEP«CETEP ♦ FTEP'OAVS 
 
 CETAS«CETAS *ETAS*DAVS 
 CN5IZ = CNMZ*WNIZN*PAVS 
 CNES=CNPS + kJNES*DAVS 
 Ci\JAS*CNAS*wNAS*DAVS 
 CNMET=CNMfcT*WNMET*OAVS 
 
 _CNFXC = CNEXC*WNEXC*OAVS 
 
 CESAVG=CESAVG* ESAVG*OAVS 
 
 CtASG=CEASG*EASAVG*DAVS 
 
 CLOSS* CL0SS*(WNIZN-WNESI*0AVS/(TMS*SF3*WGHT» 
 
 wVLFl**5VLE*r)I5G 
 
 ALl)SS*CL0SS*0I5G 
 
 ANSVG=CN5S*0I5G 
 
 
 ANIZVG»CN5 IZ*DI5G 
 AETEP=CETEP/DTAVS 
 AETAS*CETAS/DTAVS 
 IF ( ANIZVG.NE.O) WVLE2»(E0*ANSVG-AETEP-AETASJ/ANIZVG 
 
155 
 
 FORTRAN -IV G LEVEL 18 . _ MIR. BiU.2 JtU! -.__ Ol/2|/2« 
 
 Q3»» ANES-CNFS/OTAVS ■ 
 
 03*5 ANAS-CNAS/DTAVS 
 
 0346 A£SAVr,-CESAV5*0IVG .... .. ... ^ 
 
 0*47 AEASG"CEASG«DI5G 
 
 034b .. ANJ4£T«CNMtI»DI5G__ _ . _.. J _.. 
 
 03*9 ANEXC»CNfXC*0I5G 
 
 ____ ANS-ANES+ ANAS -ANIZVG 
 
 0351 IF (4NS.Lt. 0.0 I AN$-NSH/S*F 
 
 C'52 _ _IF_ (ANj7VG.NEt0t.Ql -VALUE- ISN »E0- AETEP-AETAS) / ANIZV6 
 
 0353 IF (WNI7N.NE.0.) WVC»IANS»EO -ETEP-ETAS »/WNI ZN 
 
 0354 IF (NS.GE.ll TMS-0.0 . .... _ 
 
 0355 NET=NES+NAS 
 42.5.6. CALL SOURC E < NS t NTP) 
 
 0357 IF (NET.GT.O) CALL F ILHOL < NTP. NES, NASI 
 
 035e 90 NIZNT- NIZNT »NIZW 
 
 0"»59 IF (NUP .GT.OI CALL SEPR0C(NTP, K3» 
 
 036C '.. If- (NONB.GT.OJ CALL RRPROC INTJ>jK? > 
 
 0361 NAST« NAST ♦NWS 
 
 C INTRODUCE THE SOURCE PARTI CLES FOR THE NEXT CYCLE 
 
 C EVFBY FIFTH CYCLE RENORMALIZE NION ANO OUTPUT QUANTITIES ARE PRINTED 
 
 C RfcNUkMALI/ATICN PROCESS OF NION 
 
 i?6? AN5TP* 1.0/N5TP 
 
 0363 ANTP5-05AVS/NTP5 
 
 ji&A N5TPN*N5TP 
 
 r>3oi... J2a._i6-._M-l i JZ 
 
 0166 CNHEZ(M)* CN«EZ(*> ♦ NREZ(M»*DAVS 
 
 0367 16 C E _ ( ^ I = 0C7(M) ♦ EZ(MI*OAVS 
 
 0J6J 00 l« M-l.JZ 
 
 0?69 FNREZ<MI»N*EZ<M)»ANTPO 
 
 C37C 18 6Z(i*»« F71 Ml/ NREZIMI 
 
 0371 . __i__i_«_.Q ... _ ... ....... 
 
 •■•372 00 19 i M«1,JE 
 
 0373 SUMNfc* SUMNE ♦ N?(M» 
 
 0374 TNEMI- TNF(M» ♦ NE(MI/5DE(M» 
 
 0375 193 F_E.IMI*Nfc(M)*ANTPO/EOE(M| . . 
 
 0376 RAV3*0. 
 
 C377. _______ 5__M_lt _£_ 
 
 0378 t£AV3* =G(M)*FNE(MI*EOE(M) »EAV3 
 
 0379 1755 CN_("I= TNE«M»«AN5TP 
 
 0?6 n C^I0J*CMUPnP*2.0*AIKT/S0RT«AKT*PI) 
 
 0381 ANION=0.0 
 
 0382 ANIO?*:.r 
 
 Q_._3 CU _____."it_£ ... 
 
 0384 ANI02* ANI02 ♦ F Ml ( EG <M I .CNE ( M |»*EOE (M ) 
 
 0385 .. A.NIQN- ANION ♦ F Ml ( EG( M > , FNEl M) >*EDE (M » 
 
 0386 20 CONTINUE 
 
 0387 . DNION*ANICN*CNIJN 
 
 0388 DNI02=ANin2*CNION 
 
 _3fl9 ___Q_AL__S . ._.._. 
 
 0390 IF ( ALCS-.LT.C.OI CJD-C.0 
 
 0391 _■_ IF 1 A_S(C_DI.GT.11Q.*SQ) CJD-ii__^SQ 
 03<»2 TNI0N*ARS(S0*CJ0)/DNI0N 
 
 0393 ... _ _INID2-ABS(S0*CJDI/DNIC2 
 
156 
 
 
 FORTRAN IV G LEVEL IB MAIN OATI ■ 71125 Cl/21/2* 
 
 03 94 TNI02-SQRT(TNIQ2I . 
 
 ~0~393 '' TNION-SORTJTNlONI ' '. 
 
 C396 CNI02-CN102 ♦ TNI02*OAVS 
 
 0397 ANr0S»CNI02*0I3G 
 
 0^98 wF-1.0 
 
 0399 IF (ANES4ANAS.NE.0.0I WF-I AN I Z VG*ANSV£ 1 1 \ ANIS*AN AS I 
 
 04 00 IF (MOO!NT .NPM}.NE.O) G O TO 1790 
 
 C4C1 NION-ANIOS 
 
 040? IF ! (wF.GT. 0.005 LAND. (WF.LT. 200.1) N10N-ANI0S»WF**N*F 
 
 0*03 1790 CONTINUE 
 
 0404 OAVT«DAVG»ND 
 
 0405 ALPHA«wNlZN/lNTPW*DAVT> 
 
 04C6 ANEA M »NICN»Q. 33 333 /PEN ^_^ 
 
 0407 1795 FORMAT! '0* , 'ANION-' ,TT3.4,3X, 'ONION- * ,1 IT. 4 t TxT 7 t"NI0N« , ",ETi^*", i*t 
 
 X'ALPHA-' , F 12. 5, 4X,' LEAK-', El 3. 4,/ I 
 lv)8 ' hP lTfc(6,1785 ) ANIONi ON 1 02, TNI02, ALPHA, ALOSS 
 
 0409 wklTt (6,171 > EO, POPO, 0, DETZ, OEG, NOPN 
 
 0410 171 FORMATCl' ,'GAS: ME ', ?X ,' SOURCE ENERGY J ', F5.0, 3X ,' PRESSURE! ', F4.*, 
 
 X_ •jn°« , t3X 1 _« SLAB THICKNESS: '.F^Jj,' CM',' 0XI',F4.1 L » CW ,4X, 'OfcLT 
 
 "~ X-"' ,E12.5,3X,'»PROC«' , 12,/ I 
 
 C411 WRITE!6,18GI NT, TME, NS.NSW 
 
 0412 140 FORMAT!' ', 'NUMBER OF TIME CYCLES- ', t*>, 3X, • TIME •', *12.5,' SEC, 
 
 1»X, 'GENERATED SOURCE PART -', 15, 3X, 'HGTtO « OF SOURCE PART.-', 
 115 I 
 
 0413 wRITE<6tl905l OAVT, DCEN1 , EAV G, Ew AVG, EAV3 
 
 -41^ 1H:5 FORMATC ','OAVG - • , F12.5, 3X , •DC'EN - • ,~F"1T. 5", ?X, • OVERALL PAVG- • , 
 
 1F10.5.3X,' WGTEO E AVG-' ,F10. 5,3X, 'EAVG CMFCK-' ,F 10. 5 ,/ I 
 
 r'415 WRITE!*, ISl) 
 
 0416 141 FORMAT!' ', IX, 'NUMBER D I STR IBUT ION • ,2X . 'FLUX DENS ITV • ,4X, • AVFRAGF 
 It ENERGY' ,?X, 'CUMULATED « DISTrt ', 3X, • ENERGY INTER VAL ' ,?X, 'CUM FLUX 
 XDEN' ,4X, '«DEN-DEVIATION%/l ' 
 
 04 17 - - - - (• ESUMs0#0 
 
 OMk 00 281 M-l, JE 
 
 n^i<; CESUM-CGSUM ♦ CNE ! Ml *EOE (Ml 
 
 0420 TF (Nc(M).GT.OI OEAVG-EGI MI-EE( Ml /NE<M I 
 
 C421 IF (NE(M).LE.OI DEAVG-0.0 
 
 0-,22 IF !NEi_MI.EO.OI DNAVG-Oj 
 
 0423 FNDV«CNE!M|-F~NE(M) 
 
 0424 VFE=SQPT(EG(M» |*ETV 
 
 0425 FLX1=FNE(MI-VFE 
 
 0426 _ FLX2*CNF<M)»VFE 
 
 C427 IF (NE(M).EQ.ftl GO TO 281 
 
 04 28 IF( ! (M.EO. 1 > . ANO. ! QE AVG.GT .Q. 0) I .OR. ( (M. EO. JE I . ANO. ( OEAVG.LT.0.0 I I 
 
 X) DNAVG--0EAVG*FNE(M|*0.5/EDE(M| 
 0429 IF ( <DfcAVG.LT.0.0LAN0.!(M*l I.LE.JEII ON AVG— DE4VG*( FNF I M*l J-FNE (M 
 
 X) l/(EG(H«l )-EG«M) ) 
 043C IF 1 (DEAVG.GT.C. ) . ANO. ( M.GT. 1 ) ) ONAVG-DE AVG* « FNE! M)-FNE <M-l II / ( EG( 
 
 XM|-EG!H-il I 
 
 0431_ 281 WMTS!6 .1 B?I P .FNE ( M) ,FL X1 , EG( M) , CNE !M ) , EPE ( M) ,FLX2 , DN AVG 
 
 "0432 182 FORMAT!' ' , I 3 ,E1 2. 5, 5X,F 12. 4, 5X, F12.6.8X, E12.5.6X, F12.6.4X.E12 
 
 X .5,4X,E12.4) _ 
 
 0433 DET-DDEG/ETV 
 
 0434 WRIT_E!6,183I NTPP, N5TPN ,NTPW_, PET , PAVGOj_NCC)L 
 
157 
 
 FORTRAN IV G LEVEL Id MA1X BATE. ■ 11111. 01/28/29 
 
 0435 1B3 FORMAT!' ',//, 3X .' TOTAL I QF PARTICLES PROCESSED »« , 15, SX, 'CUMULAT 
 
 1E0 » OF PARTICLES ■• , 1 10, 3X, 'WGTEO- TOTL • OF PART ICLES-' ,F12. 3,//, ~ 
 ...._ X« .N£ y_ QT-' ,E12* 5 .AX^'ALP DAVE" * if 12iSi»x , • « 0F._ CfllUiiflM'* j 19 , // 1 
 
 0436 WRITEI6.187I 
 
 C437 187 _F_URMAJC • .IX. 'ZO NE NUMBER' . 5X. 'AVERAGE ENERGY ' f SX f 'NUMBER rv 
 
 IF ELECTRONS', 3X, 'CUMULATIVE • DEN( Z » • ,3X, 'CUMU. AVG ENERGY', 3X, 
 
 I'/DNF « UbNSITY' ./) ; 
 
 G43R OD 282 M«l, JZ 
 
 043Q . _ _JWZ_»Ci*R.EZlM)*AN5TP*QJ5G . _ _.._ 
 
 G44C PCZ«CEZ(MI/CNREZ<MJ 
 
 0441 282 WRITCJ6,1R4| M, EZ <M » , NREilMJ i ANI, P«, FNR.EZ (M J _.. 
 
 C4? 1*4 FORMAT!' ', 5X, 15, 5X, F 14.6 , 8X, F14. 6 , 5X, F14.5, 7X,F14.6,7X, 
 
 1F17.5I . . 
 
 0<-<.3 WP ITt(6,l95» NION.WNIZN.NIZN, ANIZVG.WNES, NES.ANES. WNAS.NAS, 
 
 .... _ JSANAS.-NTP, Kl, K2. WF.BF -._. 
 
 0*44 185 FOPMATCO' , 'NUMrtER OF POSITIVE ION-' ,E 14.6 ,//, • NUMBER OF IONIZAT 
 
 1I0NS OCCURED=' iF9. 3, 5X,'ACTUAL_f»'jl5,3X, 'CUMULATIVE AVG»' ,F10.3, 
 
 X//,' « OF ESCAPED PARTICLES"', F9. 3, 5X,'ACTUAL#»' ,15, 3X, 'CUMULATIVE 
 X AVG»'.Fq.2. //.» » UP A ftSQRPEP PARTIC«'. E12.4.3X. 'A CTUAL «"'.I5.3X . 
 
 X, 'CUMULATIVE AVG"' ,F9.2,//, • TOTAL * OF PARTICLES IN THF NEW CYCL 
 
 XC =',15,//,' Ki*',Ii, 3Xj'X2»'*Ii« 4X,' BALANCE FACTOR WF-', 
 
 XFlu.5,3X,'BF2«', FlC.5,/1 
 C445 KPITE(h,l88) EDI, NPM, NO, KELC 
 
 044* 13" roRMATC ENERGY DIVI'JING LINE FOR RR PLAY E" « ,F1 .0. 4,4X , • EVER Y» , 1 2 
 . K..' T H,-CJtXLE-ChAflGE N*'.//.' TI ME INTERVAL. MULT I PI. Y. PAC.IQRf.Li.FQ. 4 , 
 
 X^X, 'ELASTIC XSEC / BY', 12, /I 
 044? N0R=NDN-NDN/K3 
 
 C<-P *'MTL : <6,lfc6) NSO, NSW, AN SVG ,NOR ,WNSE ,WNMET , ANMET , WNEXC , ANEXC ,NUP, 
 
 XwGriT 
 0*49 1S6 FORMATC «, • SOURCE PARTICLES GENEPATEO OLD -',110,//, 
 
 -.Li-SauiCE.-] P.Aiit.TI.CLES.J.w,EJGHTEPl IN THE_JjE,w_ CYCLE .■UlUi3XL.'_CUMULAT 
 
 XI vt AvG«',F10,3,// 
 
 1,' « t)F PARTICLES KILLED OFF LN RR PRUC«' ,1 10,// , 
 
 1« * OF SURER ELASTIC COLL IS ION« • ,F 10.4, // , 
 
 1 • # OF EXCITATIONS TD MFTA STABL* LEVEL*' 
 
 X.FIO.h, 3X, 'CUMULATIVE AVG"', F9.3, 
 ! //. ' i OF EXCITATIONS T O OtHFR LE V FLS-' . f-\ 0.4 . 
 
 X4X, 'CUMULATIVE AVG"' ,F9.2 , // , • # OF PARTICLES CROSS OVER TO HIGH 
 
 XfcNERGY RElilUN NUP*' , 15, 4X,.' OVERALL WT>! .E12.4,/ | 
 0t5C WRITEJ6,195) EF, EDP, PDPO ,SF1, SF2, SF3 
 
 0451 195 FORMAT C E-FIELD"', F9.2,'V/CM'» 4X, • t/P«' , F9. 2, 4X, • P/PO .',F7,1, 
 
 X?X, »SF1 = ', !!12. 4, -*X, 'SF2«',F12.4,3X, 'SF3-', El?. 4, /I 
 .C<l52 WF IT.-tt^UQJ so. wvC.SHFfj. lvalue. nvLFl. WVLiitEiA^G x .AEiAY/i J _E.AS_AVG, 
 
 XAf ASG 
 C453 17C FUR.MATC SOURCE RATE -SE11.3,' PER SEC ' ,4X, • W- VALUE < LOCAL »■•» 
 
 XF10.3, 3X, 'SOURCE RED. FACT" • , F 1 3. 4 ,// , • W-VALUE"', 
 
 XF10,3,4X.'CUM AVG WV1»',F9.3.3X,'CUM AVG WV2»',F9.3, 
 
 I //,' AVERAGE ENERGY OF ESCAPED PARTICLES- • ,F12. 5, 3X, • CUMULAT 
 XI Vf AVG=Ufl L+Jt* LL, ' AVERAC.F ENERGY OF ABSnaPEfl P AR T I CLES" ' . F 1 2. 5 , 
 
 X3X, 'CUMULATIVE AVG"' ,F11.4,/ I 
 .0454 CKFNE"SUMNE*ANTPO 
 
 0455 *RITE<6,le<5)CKFNE, CESUM, TNION, TNI02 
 
 0456--- -189 FORMAT! • • , • THE SN # CHECK »• ,£12.4 , 3X, • THE CN » CHECK?' ,F12.4, // , 
 
158 
 
 FORTRAN IV G LEVEL II 
 
 MAIN 
 
 DAT! - 7H1J 
 
 01/11/2* 
 
 1' CALCULATED N» USE »■« «tU«*»»*t »CAtC» N» Utl CN ■',>!»,>,», »AR 
 XTIF CN*-',E14.6,//I •' 
 
 IF nHOD<NT,5t.NE.Ot.O».<WOO <NT.l O>.lQ.OH 60 TO 1 
 WRITE!*! EAVGi BAVOt Tll» NTM, NT, NTfi "NTONifiOIC, NSW.fwf, f, u 
 
 0*57 
 0*5 8 
 
 0*59 
 
 0460 
 
 0461 
 
 0462 
 
 0463 
 
 0464 
 
 0466 
 
 0466 
 
 0467 
 
 0468 
 
 0469 
 
 C470 
 
 0471 
 
 0472 
 
 047? 
 
 04 74 
 
 0475 
 
 047t 
 
 0477 
 
 04 78 
 
 0479 
 
 C4 8C 
 
 0481 
 
 04H2 
 
 0-4 83 
 
 0484 
 
 0485 
 
 04 86 
 
 0487 
 
 04«8 
 
 0489 
 
 049C 
 
 04Q1 
 
 04 92 
 
 0493 
 
 0494 
 
 0495 
 
 04 96 
 
 0497 
 
 0499" 
 
 0499 
 
 0500 
 
 0501 
 
 0502 
 
 05P3 
 
 WRITE<9> I, MT ,AN10S, 
 ENOFILE 9 
 
 REWIND 9 
 
 3 CONTINUE 
 
 SF3 
 
 IF (M00_(NT,10I.NE.0I GO TO _*_ 
 
 MITeTfCI EAVG, DAVG, TM£' ( NTw, NT, NTR, NION, OOEG, NtW.SNF, E, U 
 
 WRITE(IO) I, WT .ANIOS, $FS_ 
 
 ENOFILE" 10 
 
 PEWlNO IC 
 
 0504 
 0505 
 0506 
 0507 
 
 4 CONTINUE 
 
 IF (MOOtNT,NCUP).NE.0> 
 
 N5TP-0 
 
 NTP5-0 
 
 D5AVS«C.C 
 NAST-0 
 
 NIZNTaC 
 
 K5VLE*C.r 
 
 CLOSS«0.0 
 
 CSFl-i)." 
 
 CNI02«0.0 
 __CN5IZ-0.0 
 
 CN5SiC*. > ' 
 
 CNM?T»0.0 
 
 CNfcxc*c .c 
 
 CESAVG«0,0 
 
 CEASG-0.0 
 
 00 54 M«1,JZ 
 
 CNREZ(MI«0.0 
 
 54 CEZ(M)-C.7 
 
 OU 55 M«1,JE 
 TNE(MJ=0. 
 
 55 CNFMI*r." 
 
 IF MUD(NT i 10).NE._0> 
 '"'UfAVS«0."0' 
 CfcTEP*C.O 
 rETAS-0.0 
 CNtrSO.n 
 CNAS-0.0 
 6666 _CONTJ NUE 
 
 NT« NT" Vl 
 00 56 **1,500 
 OIZNJMJxC.r 
 JRRIM)*? 
 IES(M|«0 
 
 56 I A S I ^ > = " _ 
 N«l,50 
 
 GO TO 6666 
 
 GO TO 6666 
 
 00 51 
 __WSf <NI= r .? 
 5i ESEINI»r.O 
 
 I«l 
 
159 
 
 FORTRAN |V G LEVEL 16 j_A_J_ PftTj ■ TMli 01/21/2* 
 
 C REINITIALH. THE OUTW QUANT 1M It ' 
 
 0508 00 52 M-l, J| 
 
 0509 _ _ £E(H 1«Q. ■ _ 
 
 0510 52 NE(N>" 0.0 
 
 C511 --00.. _51...M-lt.il 
 
 0512 NRE2(M|« 0.0 
 
 csn __iH)-0_C- ___ 
 
 0514 00 53 N-l.JE 
 
 0515 n_iii_.«j»_j_a 
 
 051b 53 NE2<N. "1-0.0 
 
 0517 ...... NI2N.-0 . _..._ 
 
 0518 NOZN»P 
 
 sax? NiJi-Li : 
 
 0*»_0 NES-0 
 
 0521 . NSf-^ 
 
 0522 NCl)L«0 
 
 0523 NAS"0 .... . . ._ 
 
 r52«» • OTOT-0.0 
 
 Oi25 UhLL^i _. 
 
 052e NEXC*'' 
 
 C.27 .._... .. *N.A.$"' , _1 2 
 
 052* HNSP«0.0 
 
 052"* wNtS»0.* .... 
 
 0530 WNI2N«U.O 
 
 0511. i_N^ET-0.0 
 
 C5'2 WNtXO^.O 
 
 053? fcTAS«0.0 
 
 053h FTEP-C." 
 
 . 0535 t-TUT"",C 
 
 5 36 NUP-0 
 
 ^537 ____*»__.. ..._ 
 
 ^533 NONH«0 
 
 053<5 INtV-INIV* 1100 
 
 i)5V CALL «N->IN_(INIV) 
 
 Ob** IF (NT.Lh.NTO GO TC 7777 
 
 C543 9°99 STflP 
 
 0544 DEBUG SliUCHK 
 
 C545 END 
 
i6o 
 
 VITA 
 
 Benjamin Shaw-hu Wang was born in Inner Mongolia, 
 China, on March 25, 1937. He is the son of Mi. C. H. Wang, 
 the former Chinese Army General now the Congressman of 
 the Republic of China. He received the B.S. degree in 
 Electrical Engineering from the National Taiwan University 
 of Taipei, Taiwan, China in 1960, he then served as an 
 ensign in Chinese Naval Academy for one and a half years. 
 He came to the United States in September, 1962 and enrolled 
 in the University of Missouri at Rolla, Missouri. After 
 he received the M.S. degree in Electrical Engineering in 
 1964, be joined the Westi nghouse Research Laboratories and 
 served as a research engineer for three and a half years. 
 
 In September 1967, he came to the University of Illinois 
 at Urbana-Champaign to undertake studies for a Ph.D. in 
 Computer Science Department. He has been a research assis- 
 tant and teaching assistant in the Department of Computer 
 Science and a research assistant in the Nuclear Engineering 
 program. He is the member of Kappa Mu Epsilon and Pi Mu 
 Epsilon the two, national honor societies of mathematics. 
 
 
LIOGRAPHIC DATA 
 ET 
 
 1. Report No. 
 
 UIUCDCS-R-72-519 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 May, 1972 
 
 Itle and Subtitle 
 
 Monte Carlo Simulation of Nonlinear Radiation 
 Induced Plasmas 
 
 uthor(s) 
 
 Benjamin Shaw-hu Wang 
 
 8. Performing Organization Rept. 
 
 fllUCDCS-R-72-519 
 
 irforming Organization Name and Address 
 
 Department of Computer Science 
 
 University of Illinois at Urb ana-Champaign 
 
 Urbana, Illinois 6l801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 Sponsoring Organization Name and Address 
 
 Department of Computer Science 
 
 University of Illinois at Urb ana-Champaign 
 
 Urb ana, Illinois 618OI 
 
 13. Type of Report & Period 
 Covered 
 
 Ph.D. Thesis 
 
 14. 
 
 iupplementary Notes 
 
 ibstracts 
 
 A Monte Carlo simulation model for radiation induced plasmas with nonlinear 
 properties has been developed, which employs a piecewise linearized predict-correct 
 iterative scheme to resolve nonlinear characteristics. Several variance reduction 
 techniques have been developed and incorporated into the model which include the 
 antithetic variates technique. The model has been applied to the determination of 
 electron energy distribution functions in a noble gas plasma created by a-particle 
 irradiation. The model can be applied to other types of complex plasma problems 
 with nonlinear properties. 
 
 ey Words and Document Analysis. 17a. Descriptors 
 
 /•lonte Carlo Simulations, Monte Carlo Calculation of electron-energy distribution 
 
 functions, Monte Carlo method for radiation induced plasmas, electron energy 
 
 distribution functions in noble gas plasmas, 
 
 toite Carlo Techniques for plasma problems, 
 
 •lonte Carlo Simulation model for nonlinear plasmas. 
 
 identifiers /Open-Ended Terms 
 
 =• OSATI Field/Group 
 
 /ailability Statement 
 
 "elease unlimited 
 
 "►■ms-35 ( 10-70) 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED. 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No, of Pages 
 
 22. Price 
 
 USCOMM-DC 40329-P7 1 
 
JUL 2 6 WW 
 


 
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