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Spatial and energetic aspects of electron energy deposition

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Spatial and energetic aspects of electron energy deposition
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Jackman, Charles Herbert, 1950-
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Approximation ( jstor )
Economic inelasticity ( jstor )
Elastic scattering ( jstor )
Electrical phases ( jstor )
Electron energy ( jstor )
Electrons ( jstor )
Energy ( jstor )
Energy value ( jstor )
Ionization ( jstor )
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Collisions (Physics) ( lcsh )
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Thesis--University of Florida.
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Includes bibliographical references (leaves 220-226).
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Typescript.
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Vita.
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by Charles Herbert Jackman.

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SPATIAL AND ENERGETIC ASPECTS OF ELECTRON ENERGY DEPOSITION


By

CHARLES HERBERT JACKMAN














A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY












UNIVERSITY OF FLORIDA

1978




SPATIAL AND ENERGETIC ASPECTS OF ELECTRON ENERGY DEPOSITION
By
CHARLES HERBERT JACKMAN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF
IN PARTIAL FULFILLMENT OF THE
DEGREE OF DOCTOR OF
FLORIDA
REQUIREMENTS FOR THE
PHILOSOPHY
UNIVERSITY OF FLORIDA


ACKNOWLEDGEMENTS
Dr. A.E.S. Green has helped the author a great deal in his efforts
to complete this work. The author sincerely appreciates this guidance.
He also wishes to thank Dr. R.H. Garvey and Dr. R.A. Hedinger for their
helpful discussion about the dissertation. David Doda, David Killian,
E. Whit Ludington, George Sherouse, and Ken Cross were instrumental in
providing assistance with computer problems and other dissertation-related
work.
Woody Richardson, Marjorie Niblack, and Wesley Bolch were extremely
helpful in drafting the figures. The final manuscript was then typed and
refined by Adele Koehler. The author is grateful to Adele for her prompt
and professional assistance.
The author wishes to thank Joseph Pollack for aiding in editorial
matters concerning the dissertation. A thorough reading and criticism
of the dissertation by the author's committee (including Dr. A.E.S. Green,
Dr. L.R. Peterson, Dr. T.L. Bailey, Dr. S.T. Gottesman, and Dr. G.R.
Lebo), Dr. II.L. Chameides, and Dr. A.G. Smith was extremely helpful.
The author is especially grateful to his parents, Rev. and Mrs. H.W.
Jackman, and to his sister, Kathi Jouvenat, for their encouragement and
support throughout graduate school.
The author gratefully acknowledges financial support from the De
partment of Physics and Astronomy and the Graduate School of the Univer
sity of Florida and from NASA grant number NGL-10-005-008.


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS
ABSTRACT vi
CHAPTER
I INTRODUCTION 1
II A SHORT REVIEW OF ENERGY DEPOSITION TECHNIQUES 5
A. Energy Deposition Techniques 5
B. Monte Carlo Energy Deposition Techniques 14
IIIELASTIC AND INELASTIC DIFFERENTIAL AND TOTAL CROSS
SECTIONS FOR N2 18
A. Elastic Differential and Total Cross Sections for N2 18
B. Inelastic Differential and Total Cross Sections
for N2 88
C. Total Cross Section (Elastic Plus Inelastic) 45
IVTHE MONTE CARLO METHOD OF ENERGY DEPOSITION BY ELECTRONS
IN MOLECULAR NITROGEN 47
A. Brief Discussion of the Monte Carlo Calculation. ... 48
B. Computer Programs and Machinery Used in the Monte
Carlo Calculation 51
C. Detailed Discussion of the Monte Carlo Electron
Energy Degradation Technique 52
1. First Random Number, R.j 53
2. Second and Third Random Numbers, R^ and . 57
3. Fourth Random Number, R^ 61
4. Fifth Random Number, R^ 61
iii


Page
5. Sixth Random Number, Rg 67
6. Multiple Elastic Scattering Distribution Used
Below 30 eV 67
7. Value of the Cutoff Energy, 2 eV 74
D. Statistical Error in the Monte Carlo Calculation ... 75
VMONTE CARLO INTENSITY PLOTS AND COMPARISON WITH EXPERIMENT 77
A. Excitation of the N^ State 77
B. Range of Electrons 80
C. Previous Experimental and Theoretical Work on the
3914 A Emission of N^ 81
D. Range Results and Longitudinal Intensity Plots from
the Monte Carlo Calculation 84
E. Intensity Plots in the Radial Direction 87
VI SENSITIVITY STUDY OF THE ELECTRON ENERGY DEGRADATION ... 98
A. Effects of Ionization Differential Cross Section on
the Intensity Distributions 99
B. Influence of Inelastic Differential Cross Sections
on the Intensity Distributions 104
C. Comparison of Different Elastic Phase Functions on
the Electron^ Collision Profile 104
D. Influence of Different Elastic Phase Functions on
the Intensity Profiles 112
E. Effects of the Total Elastic Cross Section on the
Electron Energy Degradation 121
VII MONTE CARLO ENERGY LOSS PLOTS AND YIELD SPECTRA 125
A. Energy Loss of Electrons in N2 125
B. Spatial Yield Spectra for Electrons Impinging on N2> 130
1. Three Variable Spatial Yield Spectra 132
2. Four Variable Spatial Yield Spectra 143
VIII CONCLUSIONS 152
iv


Page
APPENDIX
A MONTE CARLO PROGRAM 155
B GETDAT PROGRAM 200
REFERENCES 220
BIOGRAPHICAL SKETCH 227
v


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
SPATIAL AND ENERGETIC ASPECTS OF ELECTRON ENERGY DEPOSITION
By
Charles Herbert Jackman
August, 1978
Chairman: A.E.S. Green
Major Department: Physics and Astronomy
The spatial and energetic aspects of the electron energy degradation
into molecular nitrogen gas have been studied by a Monte Carlo method.
Perpendicularly monoenergetic incident electrons with energies from 0.1
through 5.0 KeV were injected into the gas. This Monte Carlo de
gradation scheme employed previously developed N^ cross sections with new
phenomenological differential elastic and doubly differential ionization
cross sections. All these agree quite well with experimental work and
are consistent with the higher energy theoretical total cross section
fall-off with energy.
Information has been generated concerning the following topics:
o
1) range values and 3914 A intensity profiles for the longitudinal and
radial directions which can be easily compared with experimental work;
2) a sensitivity study characterizing the influence of the input cross
sections on the spatial energy deposition of the electrons; 3) the rate
of energy loss by the electrons as they interact with the N2 gas; and
4) spatial yield spectra for incident electron energies in the range
from 0.1 to 5.0 KeV (evaluated between 2 eV and the incident energy)
which are analytically characterized for future work on atmospheric prob
lems dealing with incident energetic electrons.
vi


CHAPTER I
INTRODUCTION
Calculating the spatial and energetic aspects of the energy deposi
tion of intermediate energy electrons (with incident energies from 100
to 5000 eV) in the earth's atmosphere is a difficult, yet intriguing,
problem. These intermediate energy electrons (hereafter called IEEs)
include the highest energy photoelectrons, a large bulk of the auroral
electrons, and many secondary electrons produced by solar protons and
cosmic rays.
These electrons lose most of their energy through ionization events,
electronic excitations, vibrational excitations, and rotational excita
tions. Elastic collisions reduce the electron energy slightly, but
mainly these interactions influence the direction of motion of the
electron.
The atmosphere is dominated by the presence of molecular nitrogen
up to a height of about 150 kilometers. Even above this altitude (at
least up to 300 km), continues to play a substantial role in the
atmospheric processes. For this reason the study of the influence of
impinging electrons on molecular nitrogen is the major thrust of this
paper.
One aspect of this study is the formulation of a complete cross
section (differential and total) set for IEEs impacting on N£. The very
difficult problem of modeling the interactions of the impinging IEEs in
-1-


-2-
the upper atmosphere is then reduced in complexity. Since N2 interacts
with electrons similar to the way that other atmospheric gases interact
with electrons, it follows that differential and total cross section sets
for these gases could be assembled in a like manner.
Another aspect of this work is a sensitivity comparison among several
of the influences on the electron energy deposition. The spatial energy
degradation is vitally linked to the elastic phase function used. Since
there are data available on the elastic differential cross sections of
N2 as well as the energy degradation resulting from electron impact on
N2 a comparison illustrating the effects of a variation of the elastic
phase function is quite useful. Other influences on the spatial energy
deposition, including ionization and excitation differential cross
sections and the total elastic cross sections, are also considered in
this work.
In order to deal with these spatial and energetic aspects of elec
tron energy degradation, a Monte Carlo (which may be abbreviated MC)
calculation is used. The MC technique, depending on how it is used, can
be the most accurate energy deposition approach. Use of this MC method
at various incident energies helps in the assemblage of the best cross
section set for N2 and provides the easiest way of comparing some of
the influences on the spatial energy deposition.
The details of this undertaking are discussed in Chapters II through
VIII. A paragraph summary of each chapter is given below.
The second chapter presents a brief review of some of the standard
electron energy deposition methods. The continuous slowing down approxi
mation, discrete energy bin, Fokker-Planck equation, two-stream equation
of transfer, and the multi-stream equation of transfer are all included


-3-
in section II.A. The MC method which was used in this study along with
three other MC approaches are briefly described in section II.B.
This MC approach requires knowledge of differential and total cross
sections. The third chapter discusses the cross sections that were used
for Ng* Section III.A includes the elastic differential and total cross
sections. The inelastic differential and total cross sections are next
discussed in section III.B. Section III.C, then, considers the total
(inelastic plus elastic) cross section of N2-
In Chapter IV, the MC calculational procedure is considered. A
brief discussion of the approach is given in section IV.A. The computer
programs and machinery used in this work are discussed in section IV.B
with the programs listed in appendices A and B. A detailed discussion
of the MC electron energy degradation technique is presented in section
IV.C. Finally, the statistical error resulting from the Monte Carlo
calculation is given in section IV.D.
The MC three-dimensional intensity plots with comparison to experi-
+ 2 +
ment are given in Chapter V. The excitation of the N2 B i: state is
discussed in section V.A with the concept of range being defined in
O
section V.B. Previous experimental and theoretical work on the 3914 A
emission from N2 is considered in section V.C and section V.D presents
some range results and intensity plots in the longitudinal direction
from this MC calculation. Section V.E, then, concludes the chapter with
a comparison between the MC intensity plots in the radial direction and
the experimental data.
The main concern of Chapter VI is a sensitivity study. The effects
of the ionization differential cross section on the intensity distribu
tions are considered in section VI.A. Section VI.B, then, discusses the


-4-
influence of inelastic differential cross sections on the intensity
distribution. A comparison of different elastic phase functions on the
electron-N2 collision profile (no energy loss) is given in section VI.C.
Next, the influence of different elastic phase functions on the electron
energy deposition is presented in section VI.D. Finally, section VI.E
considers the influence of the total elastic cross section on the
electron energy deposition.
The energy loss plots and yield spectra from the MC calculations
are given in Chapter VII. Section VII.A presents the energy loss plots
and section VII.B includes a discussion of the yield spectra.
Chapter VIII concludes this paper with a summary of this work and
its impact on the spatial and energetic aspects of the electron energy
deposition problem.


CHAPTER II
A SHORT REVIEW OF ENERGY DEPOSITION TECHNIQUES
Several standard energy deposition techniques will be discussed in
this chapter. In the first section, II.A, several general ways for
treating the degradation of the energy of charged particles will be re
viewed briefly. The second section, II.B, includes a brief sketch of
four Monte Carlo energy deposition schemes: The MC approach applied in
this work and three other MC techniques.
A. Energy Deposition Techniques
Since the turn of the century, researchers have been studying the
energy degradation of rapidly moving particles in a medium. Initial work
in this area was carried out by Roentgen, Becquerel, Thompson, Bragg,
Rutherford, Bohr, and other founders of modern physics.
These pioneers in the energy degradation process were mainly con
cerned with the medium affecting the particle. In order to solve this
complex energy degradation problem, most of the early workers used a
local energy deposition approximation. Even today many degradation
problems can be solved fairly accurately with this local approximation.
One of the earliest local energy deposition methods is that of the
continuous slowing down approximation (hereafter called CSDA) first
used by Niels Bohr (1913, 1915). Bethe (1930) expanded on this work and
used an approximate theoretical treatment (valid at high energies) to
describe the slowing down of particles in a medium.
-5-


-6-
This work of Bethe (1930) required knowledge of the stopping power,
- ^ (the rate at which energy E is lost from the impinging particles
incident along the x axis). This stopping power is given by
-£ -n | **t 1 <2->>
-y
(see Dalgarno, 1962, p. 624) where the summation S includes integration
i
over the continuum (thus allowing for energy loss through ionization),
W.j is the energy loss for the ith state, and o^E) is the cross section
for excitation to the ith state at energy E. Knowledge of the stopping
power then leads to information about the mean distance traveled by the
particle (referred to as the range). This range R(E) of a particle of
energy E is simply given by
R(E)' l rw (2-2)
u dx
Atmospheric researchers are more interested in the effects that the
particles have on the medium rather than the medium affecting the par
ticles. These effects could include both spectral emissions by the con
stituents and heating of the atmosphere.
Green and Barth (1965) were the first workers to adapt a variation
of the CSDA to the problems in aeronomy. In this approach the total
energy loss function L(E) = -(1) ^ is determined by
(E-IO/2 doT.(E,T)
L(E) = l W.a. (E) + l I.oy (E) + l f T J dT (2.3)
k K K j 3 j 0 dT
where is the threshold for excitation to the kth state, crk(E) is the
cross section for excitation to the kth state at energy E, I. is the
J daj (E,T)
threshold for ionization and excitation to the jth state, and 4=


-7-
is the secondary differential ionization cross section for the creation
of a secondary electron of energy, T, from a primary electron of energy
E. The loss function with detailed atomic cross sections (hereafter
called DACSs) was used to make reasonable estimates of the ultraviolet
emissions resulting from an aurora event. In this approach, the excita
tions J|^(E) of the kth state resulting from an electron of energy E were
simply represented as
E o.(E')
VE> I -TTFT dE' <2-4
k
Green and Dutta (1967) built on this work and used the Born-Bethe
approximations, the Massey-Mohr-Bethe surface, the Bethe sum rule, and
a "distorted" oscillator strength to lay the groundwork for extension
of the DACS approach to other gases. Jusick, Watson, Peterson, and
Green (1967), Stolarski, Dulock, Watson, and Green (1967), and Watson,
Dulock, Stolarski, and Green (1967) applied this approach to helium,
molecular nitrogen, and molecular oxygen, respectively.
Stolarski and Green (1967) used this CSDA to calculate auroral
intensities with these DACSs and Green and Barth (1967) applied this
method to the problem of photoelectrons exciting the dayglow. Other
atmospheric physicists (namely, Kamiyami, 1967; and Rees, Stewart, and
Walker, 1969) started around this same time and also employed a CSDA type
approach to that problem of energetic electrons depositing their energy
in the atmosphere.
The oldest discrete energy apportionment method is that of Fowler
(1922-23) which is directly related to the Spencer and Fano (1954)
approach (see Inokuti, Douthat, and Rau, 1975). The Fowler equation is


-8-
solved by building on the lower-energy solutions to obtain the higher
energy solutions. The Spencer-Fano method introduces the electron at
the highest energy and solves the equation at successively lower ener
gies. An approach similar to the Spencer-Fano method was developed by
Peterson (1969) and is called the discrete energy bin (hereafter called
the DEB) method.
Peterson (1969) pointed out certain differences between the CSDA and
the DEB results. In particular, he noted that the DEB method tends to
predict higher populations of some excited states than does the CSDA.
In the modification of the DEB method by Jura (1971), Dalgarno and
Lejeune (1971), and Cravens, Victor, and Dalgarno (1975), the equilibrium
flux or degradation spectrum f(E,EQ) (for incident energy Eq and electron
-2 -1 -1
energy E and in units of # cm sec eV ) of Spencer and Fano (1954)
is obtained directly. Douthat (1975), in an effort to make the degrada
tion spectra suitable for applications, proposed an approximate method
of "scaling." Unfortunately, this method is quite cumbersome and not
very accurate. This impelled Garvey, Porter, and Green (1977) to seek
an analytic representation of the degradation spectra and, despite its
complex nature, they found an analytic expression to represent this
spectra for W^.
The concept of the "yield spectra" U(E,Eq) was first initiated
through a modified DEB approach given in Green, Jackman, and Garvey
(1977) in an effort to find a quantity with simpler properties than the
degradation spectra. By utilizing the product
U(E,Eq) = 0T(E) f(E,Eo)
where Oy(E) is the total inelastic cross section for an electron of
energy E, one defines a quantity U(E,EQ) which not only has a simpler


-9-
shape than f(E,EQ) but also has approximately the same magnitude for
all substances. This yield spectrum can also be represented analytically.
It effectively embodies the non-spatial information of the degradation
process.
Jackman, Garvey, and Green (1977a), using this modified DEB,
elaborated on the differences between the DEB method and the CSDA. The
more accurate modified DEB method was found to produce consistently more
ions per energy loss while at the same time producing less excitations
of some of the low lying states when compared with the CSDA. The CSDA,
although inexpensive to use, appears to be ill-suited for calculations
requiring an absolute accuracy better than about 20%. Since auroral and
dayglow intensities are rarely measured to better than this accuracy, the
CSDA has been adequate for most purposes of concern in aeronomy. How
ever, with future improved measurements it should be purposeful to
utilize more accurate deposition techniques.
Several recent spatial electron energy deposition studies have been
concerned with the spatial aspects of auroral electron energy deposition.
Walt, MacDonald, and Francis (1967) employed the Fokker-Planck diffusion
equation to give a detailed description of kilovolt auroral electrons.
The Fokker-Planck equation, as given in the Strickland, Book, Coffey,
and Fedder (1976) paper, is written
+ ^ET^ [L(EMz,E,w)] (2.5)
(2.5)
-2 -1 1 -1
where <¡>(z,E,v) is the flux (in units of cm sec eV sr ), z is the
distance into the medium along the z axis, E is the electron energy,


-10-
and y is the cosine of the pitch angle associated with the direction of
motion of the electron. The symbols n, a, Q, and L are the number den
sity of the scatterers, the total cross section (both elastic and in
elastic), the momentum transfer cross section, and the loss function,
respectively.
The momentum transfer cross section, given in terms of the differ
ential elastic cross section, is written as
(2.6)
0 0
This Fokker-Planck equation may be thought of as a CSDA approach to
the spatial energy degradation problem. Its solution, therefore, is only
accurate in the higher energy regime.
Banks, Chappell, and Nagy (1974) were able to calculate the emission
as a function of altitude for a model aurora using the Fokker-Planck
equation for electrons with energy E > 500 eV along with a two-stream
equation of transfer for electrons with energy E < 500 eV. The two-
stream equation of transfer solves the transport of electrons in terms
of the hemispherical fluxes of two electron streams +(E,z), the electron
flux upward along z, and The steady state continuity equations for <|>+ and ~ can then be written
as
+

2
(2.7)
and


-11-
1
r l nk[o
k kn
a + pee] *
1 r k k + q
£ nk^eae^ 2

where
q+(E,z)
+ [1
= Ink(z) J (P^(E,)cakj(E'-)*-(E,z)
E>E
- P^E'Jla^E'-^J^E'.z)}
aj dj
q"(E,z)
+ [1
= 1 n (z) l {p^ (E* )a^ .(E'^E)d, (E' ,z)
k k j aj aj
E>E
paj(E,)]aaj(E'"E)<¡,"(E,z)}
(2.8)
(2.9)
(2.10)
(2.11)
and z is the distance along a magnetic field line (positive outward);
n^z) is the kth neutral species number density; Pe(E) is the electron
backscatter probability for elastic collisions with the kth neutral
species; a (E) is the electron total scattering cross section for the
kth neutral species; q(E,z) is the electron production rate in the
range E to E+dE due to ionization processes (both electron ionization
+
and photoionization); q" is the electron production in the range E to
E+dE due to cascading from higher-energy electrons undergoing inelastic
i/
collisions; p is the electron backscatter probability for collisions
J
with the kth neutral species resulting in the jth inelastic process; and
a is the inelastic cross section for the jth excitation of the kth neutral
species.


-12-
This approach combined these two methods of electron energy deposi
tion in order to find a reasonable solution to the very difficult auroral
energy deposition problem. The Fokker-Planck method is accurate only
at large incident energies; therefore, it should only be used at ener
gies above 500 eV. The two-stream equation of transfer approach, on the
other hand, is more accurate at energies below 500 eV. This combination
then provided a very reasonable solution to the auroral electron spatial
deposition problem for a reasonable amount of calculation.
The Fokker-Planck equation and the two-stream equation of transfer
may both be derived from the Boltzmann equation or the general equation
of transfer. This general equation of transfer, in one of its simpler
forms, is written as (from Strickland et al., 1976)
u -nUMEMz.E.v)
+ ntz)o(E) / Rfu'.u.E'.Eltfz.E'.p'ldE'dp' (2.12)
(assuming a steady state condition and no external fields) where
I o^y'.y.E'.E)
Rtw'.y.E'.E) = j^ (2.13)
with the sum over all processes. The symbols p and p1 are the cosines
of the pitch angles e and e' which are associated with the directions n
and n' given in Figure 2.1.
The first term on the right hand side of Eq. (2.12) represents the
scattering out of p. The R(p',p,E',E) in the second term is the proba
bility (eV \2wsr) ^) that a collision of an electron of energy E' and
direction p1 with some particle will result in an electron of energy E


-13-
Z
Figure 2.1 The directions denoted by n' and n are the incident and
scattered directions of the electron, respectively.


-14-
and direction y. The integral in Eq. (2.12) is over all possible ener
gies E' and directions of motion y'.
Strickland et al. (1976) studied the auroral electron scattering
and energy loss with a multiangle equation of transfer. Their approach
is one of the most accurate yet applied to auroral electrons. This multi-
angle method of solution is more realistic than the two-stream approach
and it is computationally more difficult as well.
The methods discussed above are the "state of the art" approaches
(excluding the Monte Carlo methods which are discussed in section II.B)
to the I EEs degrading in the atmosphere. Other approaches used by
Jasperse (1976, 1977) and Mantas (1975) are mainly concerned with photo
electrons and will not be discussed here.
The Monte Carlo approach can rival any of these electron energy
deposition methods in accuracy when used in the proper manner. This
stochastic technique for solving the deposition problem will be con
sidered next in section II.B.
B. Monte Carlo Energy Deposition Techniques
Another method of solving the spatial energy deposition problem is
the use of the Monte Carlo approach. The MC technique, which is used
in this paper, is a stochastic method of degrading an energetic electron.
The approach can be one of the most exact methods of electron energy
deposition. Briefly, one electron is taken at a time and allowed to
degrade in energy collision by collision. This deposition attempts to
mimic the randomness of the actual degradation process that occurs in
nature.


-15-
Many MC schemes have been applied in all areas of physics. Some
are more exact and more detailed than others. Since virtually all the
MC methods are run on the computer, the most exact approaches cost the
most computer time and money. The precision of the technique must be
balanced against a finite computer budget.
Three approaches using the MC deposition scheme, that have been
applied to electrons impinging on the atmosphere, are discussed below.
Brinkmann and Trajmar (1970) applied experimental differential electron
impact energy loss data in a MC computation for electrons of 100 eV
energy. Because of the large amount of input cross sections in numerical
form, only electrons of 100 eV incident energy were degraded with this
method.
In the lower electron energy regime (below 25 eV), Cicerone and
Bowhill (1970, 1971) used a MC technique to simulate photoelectron dif
fusion through the atmosphere. This method, which included both elastic
and inelastic processes, predicted escape fluxes from the atmosphere.
Berger, Seltzer, and Maeda (1970, 1974) (hereafter called BSM)
worked with high energy electrons (with energies from 2 KeV to 2 MeV).
They employed a MC approach that has two variations which are pointed out
below. They treat inelastic collisions in a continuous slowing down
manner. The energy deposited by the electrons along their path is
assumed to be equal to the mean loss given by the loss function, L(E),
from Rohrlich and Carlson (1954).
The angular deflection resulting from elastic collisions has been
evaluated by two separate methods in BSM. One approach employed the
multiple scattering distribution of Goudsmit and Saunderson (1940)
applied to the screened Rutherford cross section given in BSM. The


-16-
other approach involved a sampling of each elastic collision. Appli
cation of the BSM technique to a constant density medium and no
magnetic field gave good agreement with laboratory experiments (Griin,
1957; and Cohn and Caledonia, 1970).
In this study, a MC method was needed that could be applied to IEEs.
The basic equation of transfer is solved with the use of the MC approach.
This equation can be rewritten as
dU(y,z,E,E )
11 oTtE) dz n(z)U(u,z,E,E0)
E+4EElas 1
* nU) / / p iu'.u.E'.EMu'.z.E'.EJdp'dE'
E -1 e 0
Eo +1
+ n(z)I / / pTn,..(^i,,y,El,E)U(^l,z,El,En)d^ildE,
i 2E+Ii -1 lum 0
Eq
+ n(z) l I p -(u1,y,E',E)U(u',z,E,E )du'dE' (2.14)
o j
No external fields are included here and a steady state is assumed. The
U(y,z,E,E ) is the "yield spectra" (in eV_1 sec-1 sr-1) and it is assumed
that there is only one neutral scattering species. In this equation
Oy(E) is the total cross section (elastic + inelastic)for the species,
AEE, = 2E(1 cose) /-1-ect-ro-ri (2.15)
tlas ^neutral
species
is the energy loss during an elastic collision, p (y',y, E',E) is the
probability during an elastic collision with a neutral specie that an
electron with energy E' and direction y' will result in an electron of


-17-
energy E and direction y, P¡qn(y*y E',E) 1S the probability during an
ionization collision with a neutral species that an incident electron
with energy E1 and direction y1 will result in a secondary electron of
energy E and direction y, and p .(y',y,E',E) is the probability during
an inelastic collision (excitation or ionization) with a neutral specie
that an incident electron with energy E1 and direction y1 will result in
the incident electron being scattered into direction y with energy E.
Some techniques from each of the other three MC methods were in
cluded in this work. Some new approximations and assumptions were made,
however, to enhance the accuracy of the computations as well as simplify
some of the calculations. These assumptions are discussed in detail in
Chapter IV.
In this MC work the information is stored in a collision by collision
manner on a magnetic tape. Once all the case histories are generated,
then, the tape is scanned and any correlations of interest may be deter
mined. The choice of altitude and energy intervals is specified only
during the scanning of the tape. This method allows for greater flexi
bility in minimizing the statistical uncertainties of the results,
while, at the same time retaining good spatial or energy resolution
(Porter and Green, 1975).
All the degradation methods mentioned in this chapter require cross
sections as input. The cross sections used in this MC work are, there
fore, discussed in the next chapter.


CHAPTER III
ELASTIC AND INELASTIC DIFFERENTIAL AND TOTAL CROSS SECTIONS FOR N2
In this chapter differential and total cross sections for electron
impact on N^ will be discussed. Section III.A reviews the elastic cross
sections of N2 and discusses three models for representation of these
properties. In section III.B the inelastic cross sections of N2 are
presented with their relationship to theory and experiment. Section
III.C, then, concludes this chapter with a discussion of the total
(elastic plus inelastic) cross section for N2> Any energy degradation
technique requires knowledge of these cross sections for a complete
evaluation of the energy loss by electrons in a gas.
A. Elastic Differential and Total Cross Sections for N2
One of the most common differential elastic cross section forms is
the screened Rutherford cross section which can be expressed in the form
da r
dn .2. 2
Z2e4
p v (1 cose + 2n)
2] Krei
(3.1)
where K^ie) is the spin-relativistic correction factor.
The familiar Rutherford cross section (unscreened case) which can be
derived from classical scattering theory is given by
do
dp
Z2e4
? 0 p
p V (1 cose;
(3.2)
-18-


-19-
where
cose
Here, an electron is elastically scattered by a nucleus of charge Z
using the Coulomb potential
2
(3.3)
with r being the distance between the two particles.
Treating scattering in a quantum mechanical approach with the use
of the Born approximation and a potential of the form
(3.4)
where y is a positive but small quantity approaching 0, Eq. (3.2) can
again be derived. The Born approximation, using the potential in Eq.
(3.4), is only valid in certain angle and energy regimes (Mott and
Massey, 1965, pp. 24 and 466). At sufficiently high angles and low
energies, the Born approximation fails. The range of validity varies
for different substances and for the Born approximation is probably
not accurate at all angles for energies less than 100 eV and at large
angles for energies less than 500 eV.
Equation (3.2) does, however, go to infinity when 9 approaches 0.
This differential cross section also leads to an infinite value in the
total elastic cross section. Both of these results are unreasonable for
elastic scattering of electrons by atoms and molecules. The most popu
lar way of correcting this unreal behavior is to add a screening param
eter n. Equation (3.1) portrays this resulting form.


-20-
Equation (3.1) has a maximum at 0 = 0 and a minimum at e = 180.
At energies below 200 eV, experimental results indicate a minimum in the
elastic differential cross sections at about 90 with a strong forward
scattering peak at e = 0 and a secondary backward scattering peak at
9 = 180.
In Figure 3.1 experimental data for energies at E = 30 and 70 eV are
presented. These data are taken from Shyn, Stolarski, and Carignan (1972)
with the small circles denoting 30 eV points and the crosses denoting
the 70 eV data.
Later on in this section the screened Rutherford cross section and
another analytic model of the differential elastic cross section are
compared with experimental data. Before discussing the differential
cross section in more detail, first, consider the total elastic cross
section.
Several experiments have been performed deriving the total elastic
cross sections for N^. There have also been several theoretical studies
on the N2 elastic total cross sections. Two recent reviews of the data
available on this subject are Kieffer (1971) and Wedde (1976).
A plot of all this data would obscure the analytic total cross
sections specifically considered in this work. Consequently, only data
from Sawada, Ganas, and Green (1974) (theoretical), Shyn, Stolarski, and
Carignan (1972) (experimental), and Herrmann, Jost, and Kessler (1976)
(experimental) are plotted in Figure 3.2. The sets of data overlap to
a degree such that the disagreement in absolute magnitude of the total
cross sections is readily apparent.
In view of this disagreement, no experimental or theoretical data
are assumed to be absolutely correct and some average of this data is


-21-
6 (Degrees)
Figure 3.1 N2 experimental electron impact elastic cross section data
from Shyn, Stolarski, and Carignan (1972). o's denote data
from E = 30 eV and the x's denote data from E = 70 eV.


Figure 3.2 N2 electron impact total elastic cross section data from Sawada,
Ganas, and Green (1974), x; Shyn, Stolarski, and Carignan (1972),
0; Herrmann, Jost, and Kessler (1976), and Banks, Chappell, and
Nagy (1974), v. Equation (3.6) is represented by the dash-dot
line, Eq. (3.8) by the dashed line, and Eq. (3.9) by the solid
line.




-24-
desirable. An analytic function representing the total elastic cross
section is most easily used in a computer program. Consider now an
analytic form derived from the differential screened Rutherford cross
section.
Knowledge of the differential cross section implies knowledge of the
total elastic cross section as they are simply related by
a(E) = / / 7T7 sineded (3.5)
0 0 as
where is the azimuthal angle. Substituting Eq. (3.1) into Eq. (3.5),
the total elastic cross section, o^(E), resulting from the screened
Rutherford cross section is very simply given as
Z2 51.8
/ r\ L- JI.O II
aR{E) n'(T~n)
-16 2
If E is given in eV then a^(E) is in units of 10 cm .
parameter
(3.6)
The screening
n
1.70 x 10~5 Z2/3
nc t(t + 2)
(3.7)
according to the Moliere (1947, 1948) theory. Berger, Seltzer, and
Maeda (1970) assumed that nc was a constant value and decided on nc = 1
as its best value. The t in Eq. (3.7) is the electron energy in units
2
of the electron rest energy so that t = E/mc In the energy regime of
interest (E s 5 KeV), t 2, and 1=1. Noting these observations,
1 6
Eq. (3.7) can be rewritten as n % 4-,
The total cross section from Eq. (3.6) is plotted in Figure 3.2
as the dash-dot line. A noticeable difference is evident between this
model and the experimental values at practically all energies.


-25-
Using the form
q F[1 -
o(E) 1
EC W
(3.8)
on 9
implemented first by Green and Barth (1965), where qQ = 651.3 A£ eV ,
the total elastic cross section for Ng was characterized fairly well in
the range from 30 to 1000 eV using the parameters a = 1, 3 = 0.6,
c = 0.64, F = 0.43, and W = 2.66. The e"0'64 dependence of Eq. (3.8)
at the larger energies is similar to that seen by Wedde and Strand (1974)
for N^. This form does not represent the data as well below 30 eV and,
in fact, is not defined below an energy of 2.66 eV.
Porter and Jump (1978) have used a more complex total elastic cross
section form which is written as
r-X
a(E)
n(n + 1)[V2+X + E2+X]
figi
f/2
(E-E,)2-^ (E-E2)+G2
(3.9)
Here, n = j
T =
2.5 x 10"6 cm2
Fi
= 7.33
U =
1.95 x 10 3 eV
Ei
= 2.47
eV
V =
150 eV
G2
= 24.3
eV
X =
-0.770
F2
= 2.71
G1 :
= 0.544 eV
E2
= 15.5
eV
In the large energy limit, the total cross section falls off as 1/E,
similar to the screened Rutherford cross section. This form does con
tain two other terms (the second and third terms) which were introduced


-26-
phenomenalogically to describe the low energy shape and Feshbach reso
nances.
If either Eq. (3.8) or (3.9) is used as the total elastic cross
section, the differential elastic cross section must be normalized such
that:
2ir tt
/ / P(e,E) sineded 0 0
where P(e,E) is the phase function and the differential cross section can
be written as
o(E) P(e,E) (3.11)
With this in mind consider now three separate phase function forms.
The first phase function form is very similar to the screened Rutherford
cross section and it is written here as
PM1(e,E) = IT-^ (3.12)
m 2ir[(2 + a(E)) 1 a(E) 1 ][1 cose + a(E)]
This is known as model 1. The parameter "a" acts in a similar manner to
the "2n" term in the denominator of the screened Rutherford cross sec
tion form and is written
a(E) = a1
_E_\a2
1 eVj
The second phase function form (model 2) includes the forward
scattering term of Eq. (3.12) along with a backscattering term and is
given as


-27-
PM2(e>E)
=id
2tt[(2 + a(E))-1 a(E)_1][l cose + a(E)]2
where
(1 f(E))
2tt[(2 + c(E))'1 c(E)_1][l + cose + c(E)]2
f(E) =
f2
(E/f-j)
f2
(E/f-j) + f3
(3.13)
a(E) al ClEeV^
and
c{E) = c,[l 3]
Irvine (1965) was one of the first researchers in scattering prob
lems to use a phase function containing forward and backward scattering
terms. He applied a sum of two Henyey-Greenstein functions to the prob
lem of photon scattering by large particles. Porter and Jump (1978)
also have used a sum of two terms (one for forward scatter and one for
backward scatter). They fitted experimental data at several separate
energies with their form. Use of their differential cross section form
in a deposition calculation probably would require the use of spline
functions or other interpolative techniques.
The third phase function (model 3) is now considered. At small
angles the differential cross section shows a near exponential-like fall
off. This behavior has been pointed out by several experimenters (see,
for example, Shyn, Carignan, and Stolarski, 1972; and Herrmann, Jost,
and Kessler, 1976). It was this experimental observation that led to


-28-
model 3 which is written as
f-j(E)[l b2(E)]e"9/b(E)
Ph3<9E) = 2, b2(E)[l + e-*'"'1']
f2(E)
2tt[(2 + a)"1 a_1][l cose + a]2
[1 f-j (E) f2(E)]
2tt[(2 + c(E))"1 c(E)_1 ][1 + cose + c(E)]2
(E/fn)f]^
C(E/fll)fl2 + f13]
(E/f2i )f?2
[(E/f2i J^2 + f23]
where
f-, (E) =
f(E) =
f2(E) = 1 f-j (E)
for E > 200 eV
f2(E) = f(E)[l f-j (E)]
for E < 200 eV
b(E) = b1(^if)l>2
c c3
c(E) = Cl[l (f) ]
The parameters used for the rest of this study in Eq. (3.14) are
a = 0.11
fn = 100 eV
f12 = 0.84
f 13 = 1-92
b] = 0.43
(3.14)
b2 = -0.29


-29-
f21 = 10 eV
f 22 = 0,51
f23 = 0.87
c2 = 12 eV
c3 = 0.27
c, = 1.27
This form is more complex than the other phase function models but
it does describe the experimental differential cross section data the
most realistically. It includes an exponential term for the near ex
ponential-like forward scattering as well as a backward scattering term
for electron energies below 200 eV.
Comparisons of the screened Rutherford and model 3 cross sections
are given in Figures 3.3 and 3.4 at the two energies of 30 eV and 1000 eV.
Both forms are normalized to the total elastic cross section form of
Eq. (3.9). This modified screened Rutherford cross section vastly under
estimates the forward scattering from e = 0 to 30, overestimates the
scattering in the range from 0 = 30 to 120, and underestimates the
scattering at the larger angles with 0 = 120 to 180. Model 3 does a
fairly reasonable job of representing the differential cross section data
at both of these representative energies and the other energies as well.
Although there is not a large amount of energy loss during an
elastic collision, there is some. Using classical considerations (see
Green and Wyatt, 1965), the energy loss is approximately given by Eq.
(2.15). For molecular nitrogen and 0 = 90, the energy loss is about
8 x 10"5 E.
The MC approach, being a stochastic process, uses the concept of
probability for scattering within a given angle interval. In order to
compare phase functions, the probability for backscatter may be compared.


Figure 3.3 N? electron impact elastic differential cross sections,
(a and b) The screened Rutherford (dashed line) and the model 3
(solid line) are compared with the experimental data of
Shyn et al. (1972), x, and Herrman et al. (1976), o, at
the energies of 30 eV (Figure 3.3a) and 1000 eV (Figure
3.3b).


(A /Sr)
Q (Degrees)
Figure 3.3a


-32-
Figure 3.3b


Figure 3.4 N2 electron impact elastic differential cross sections
(a and b) between 0 and 20. The screened Rutherford (dashed
line) and the model 3 (solid line) are compared with the
experimental data of Shyn et al. (1972), x (the ¡a's
denote extrapolated points), and Herrmann et al. (1976),
0, at the energies of 30 eV (Figure 3.3a) and 1000 eV
(Figure 3.3b).


-34-
9 (Degrees)
Figure 3.4a


-35-
9 (Degrees)
i
Figure 3.4b


-36-
This probability, Pg(E), is simply calculated with
l / ^sineded*
pb
/ / sin0ded 0 0
In Figure 3.5, Pg(E) from the screened Rutherford and model 3 are
compared with other theoretical (Wedde and Strand, 1974) and experi
mental (Shyn et al., 1972) values. Model 3 does have a tendency to
estimate less backscatter than the screened Rutherford at the larger
energies. (The Pg(E) curves for model 3 and the screened Rutherford do
tend to converge at 5 KeV however.) The dominant exponent!'a 1-1 ike for
ward scattering is the reason behind this behavior. The discontinuity
observed at 200 eV in model 3 values results from the lack of the back
scatter characteristic above this energy.
The elastic scattering collisions influence mostly the direction of
travel of the electrons. There is some energy loss during an elastic
collision (as pointed out above), but this loss is not important for
electrons with energies above 2 eV colliding only with N particles.
Inelastic collisions, on the other hand, result in a fairly sub
stantial energy loss with some scattering. Consider now the differential
and total cross sections for these inelastic events.
B. Inelastic Differential and Total Cross Sections for N2
Inelastic collisions are divided into two types: 1) electron ex
citation and 2) electron ionization. In the excitation process the
electron is excited to a higher state which may either be an optically


Backscatter Probability
-37-
10
E (eV)
Figure 3.5 Backscatter probabilities for the screened Rutherford
(dashed line) and the model 3 (solid line) are compared
with Wedde and Strand (1974), x; and Shyn et al. (1972), o.


-38-
all owed or an optically forbidden transition. This transition for many
molecules leads to a repulsive state which can dissociate the molecule.
In N^, dissociation of the molecule in this manner is virtually non-
existant because is a very stable homonuclear molecule in which both
the singlet and triplet states are found to be strongly bound. As a
consequence of this, the main process for dissociation is through pre
dissociation of stable electronic terms by repulsive states that are
themselves strongly optically forbidden in direct excitation.
Porter, Jackman, and Green (1976) (hereafter called PJG) compiled
branching ratios for dissociation of using several experimental and
theoretical papers (see, for example, Winters, 1966; Rapp, Englander-
Golden, and Briglia, 1965; Polak, Slovetskii, and Sokolov, 1972; and
Mumma and Zipf, 1971). In PJG the efficiencies for the production of
atomic nitrogen from proton impact were determined.
This study does not include a calculation of the atomic nitrogen
production; however, the PJG branching ratios with the yield spectra,
discussed in section VII.B, can be applied to calculate this atomic
yield. The excitation events, not resulting in the dissociation of the
Ng molecule, are either electronic or vibrational transitions. Cross
sections for these transitions are taken from both PJG and Jackman,
Garvey, and Green (hereafter called JGG) (1977b).
In the ionization event an electron is stripped from the molecule
and given some kinetic energy. The ionization cros^s section is a sub
stantial portion of the total inelastic cross section above 50 eV
(compare Figures 3.6 and 5.1) and the total amount of energy loss is
always > the lowest ionization threshold (which is 15.58 eV for N^).
Subsequently, most of the energy loss of the electrons (for energies


-39-
Figure 3.6 N2 electron impact cross sections. The total inelastic,
Eq. (3.16) (solid line), total elastic, Eq. (3.9) (dashed
line), total inelastic plus elastic, Eq. (3.16) plus
Eq. (3.9) (dash-dot line), and the experimental inelastic
plus elastic values (Blaauw et al., 1977), x, are pre
sented here.


-40-
> 50 eV) is from the ionization collisions. These ionization cross
sections were also taken from PJG and JGG. The total inelastic cross
section found by summing these inelastic cross sections was fit with
the function
an(E) -
q0F[l (fn
Wxa.S ,4EC
WE
(3.16)
This form has the characteristic Born-Bethe In E/E fall off behavior at
the large energies. The parameters a = 1, 6 = 4.81, C = 0.36, F = 4.52,
and W = 11 were found with the use of a nonlinear least square fitting
program which fit Eq. (3.16) to the sum of all the inelastic cross
sections. From 30 eV up to 5 KeV this form was used for the total
inelastic cross section.
Below 30 eV much structure in the total inelastic cross section is
evident. At these low energies, the total inelastic cross section can
be read numerically into the MC program. This total inelastic cross
section is illustrated by the solid line in Figure 3.6.
Consider now the scattering of the two electrons involved in an
electron impact ionization collision. In reality, only the incident
electron is scattered. The other electron is simply stripped from the
molecule and given kinetic energy in a certain direction of travel.
Experiments are unable to distinguish between the incident electron and
the electron stripped from the molecule. In this paper, the ionization
event is assumed to cause scattering of both electrons. The scattering
angle of either is then measured with respect to the incident electron's
path.
After the collision event the electron with the higher energy is
designated the primary electron and the electron with the lower energy


-41-
is called the secondary electron. There should be a correlation between
the primary and secondary scattering, but this mutual influence is dif
ficult to quantify. The impinging electron ionizes a many body par
ticle, a molecule of nitrogen, thus momentum and energy can be conserved
without all the momentum and energy shared by the two resulting elec
trons. Only recently has work been done on triply differential cross
sections for and this interaction was measured only at one energy
E = 100 eV (see Jung, Schubert, Paul, and Ehrhardt, 1975). More ex
perimental and theoretical work needs to be done in this area before
any generalization can be made concerning the correlation between the
primary and secondary scattering.
The primary and secondary scattering will thus be treated sepa
rately in this work. In dealing with the scattering of the primary
electron after an ionization collision, a differential ionization cross
section form based on the Massey-Mohr-Bethe surface of hydrogen, is
used. The form (with aQ, the Bohr radius, and Rg, the Rydberg energy) is
2
= 9_i o JiM/2 p(x\ (3 i7)
dTdfi Wx U E' ^l/'
2
where x = (KaQ) is the momentum transfer, W is the energy loss in the
collision process which is equal to the ionization potential, I, plus
the kinetic energy of the secondary, T, and F(x) is a complex function
given in PJG.
Equation (3.17) is very highly forward peaked for small energy
losses but becomes less forward peaked as the energy loss becomes sig
nificant when compared with the incident energy, E.
The secondary electron is also scattered in the ionization event.
Probably the most comprehensive work that exists on secondary doubly


-42-
differential cross sections is that of Opal, Beaty, and Peterson (1972).
(More recent data by DuBois and Rudd (1978) agrees with their work.)
This data indicates a preferred angle range in the scattering process
(usually between 45 and 90) at all primary and secondary energies.
A Breit-Wigner form has been chosen to represent the data. Here,
, ICE,_T)C2 (3.18)
dTdfi [C2 + B(cose coseo)2]Nf
where
B(E) = 0.0448 + (
72900 eV
0.91
)
r/T\ 36.6 gV \
(T + 183 eV '
, ve)
0q E = *873 + (T + 0b(E))
0a(E) =20 eV + 0.042 E
and
0b(E) =28 eV + 0.066 E
o r r (1 + cose ) E (1 cose ) ^
f --f {fn-D c *4 tan- [ T *-]}
S(E,T) = = A(E)r2(E)/[(T Tq(E))2 + r2(E)]
(3.19)
is from Green and Sawada (1972) with
a(e) = a0 V2 '"[dh?]
T (E) 5 4 71 eV .^000 (e^)^
o^t; 4-/l ev (E + 31.2 eV)


-43-
r(E) = 13.8 eV E/(E + 15.6 eV)
oQ = 1 x 10 "16 cm2
Equation (3.18) may seem highly complicated; however, integration
over the solid angle is given very simply as Eq. (3.19) which is the
singly differential ionization cross section. The total ionization cross
section is then
TH
aI0N^ dT dT (3.20)
with
TM = \ (E ^
so that
aI0N(E) = Ar{tan_1[(TM TQ)/r] + tan_1(TQ/r)} (3.21)
The fit to Opal, Beaty, and Peterson's (1972) data is given in Figure 3.7
at several primary and secondary energies. The x's represent the ex
perimental data and the solid line represents the analytic expression,
Eq. (3.18).
Other inelastic processes include the simple excitation events.
The scattering of the incident electron due to the excitation of a par
ticular state has been studied by Silverman and Lassettre (1965), and
more recently by Cartwright, Chutjian, Trajmar, and Williams (1977) and
Chutjian, Cartwright, and Trajmar (1977).
In order to account for this scattering, the Silverman and Lassettre
(1965) generalized oscillator strength data for the 12.85 eV peak (cor
responding to the optically allowed b ^ir state) were fit with the use of
a phase function similar to model 1. The very sharply forward scattering


-44-
6 (Degrees)
Figure 3.7 N2 experimental ionization doubly differential cross sections
from Opal, Beaty, and Peterson (1972) are represented by x's.
The solid line ( ) denotes the cross section resulting from
the use of Eq. (3.18). The incident electron energy is de
noted by E (eV) and the secondary electron energy is denoted
by T (eV).


-45-
peak indicated in these data was used in a MC calculation. The electron
scattering which results using this inelastic scattering approximation
in a computation was so small as to be virtually undetectable.
Cartwright et al. (1977) and Chutjian et al. (1977) have studied a
more comprehensive list of states and have observed significant electron
scattering (especially due to the optically forbidden states) in the
range from 10 eV to 60 eV. Characterizing this data in some way appears
to be a rather endless task.
Dealing with this type of inelastic scattering is thus still a
problem. Above 100 eV the optically allowed excitations are the most
important; thus it is safe to conclude that the inelastic scattering
events will not affect the energy degradation process. Below 100 eV,
as a first approximation, it is assumed in this work that the inelastic
excitation events scatter as much as the elastic events. This is
probably a reasonable approximation to the very complex inelastic ex
citation scattering. In section VI.B the effects of this assumption
are discussed.
C. Total Cross Section (Elastic Plus Inelastic)
Elastic and inelastic processes have been considered in sections
III.A and III.B. Another aspect of the cross sections is the total
(elastic plus inelastic) cross section.
Blaauw, de Heer, Wagenaar, and Barends (1977) have recently pub
lished experimental data on the total cross section values of Ng. These
experimental values are compared with the cross section values from this
work in Figure 3.6.


-46-
Throughout the energy range the cross sections used in this study
compare favorably with those of Blaauw et al. (1977). For an easy
reference, the total inelastic and total elastic cross sections are
also given in Figure 3.6 as separate curves.
All the major influences on the IEE energy loss and scattering have
been accounted for in this chapter. The next chapter presents the MC
energy deposition scheme which employs these cross sections.


CHAPTER IV
THE MONTE CARLO METHOD OF ENERGY DEPOSITION BY ELECTRONS
IN MOLECULAR NITROGEN
A Monte Carlo calculation is used here for energy degradation by
energetic electrons in Ng. This stochastic process is probably the most
accurate method for obtaining the energy loss of particles in a medium.
The generalizations about electron impact on N£ that are made through the
use of this technique can be applied in other energy deposition approaches
to electrons impinging on the atmosphere. (This is true even though the
magnetic field is neglected in these MC calculations. The three dimen
sional yield spectrum U(E,z,EQ) [see Chapter VII] is most useful for
applications to the atmosphere and changes in the magnetic field will
not affect the yield spectra greatly below altitudes of about 300 km
where the gas density is fairly high [see Berger, Seltzer, and Maeda,
1970 and 1974].)
Building on the MC work done in this paper, more exact models of
auroral electrons and photoelectrons can be established. In the first
section, IV.A, a brief discussion of the MC calculation outlines the
general procedure involved in the degradation process. The computer
program and machinery used are briefly described in section IV.B. Sec
tion IV.C relates in detail the various aspects of the calculation.
Finally, section IV.D discusses the statistical error that arises from
the use of the MC calculation.
-47-


-48-
A. Brief Discussion of the Monte Carlo Calculation
In Figure 4.1 a short version of the MC calculation is presented.
Briefly, each electron is degraded in a collision by collision manner
down to 30 eV. Below 30 eV the electrons are degraded with the use of
a multiple scattering distribution. This multiple scattering approach
characterizes the resultant coordinates of the electron which goes
through several elastic collisions between each inelastic collision.
The incident electron has an energy E To begin with, the running
total of the electron energy, E, is set equal to Eq. At the position
START, this energy E is checked against a cutoff energy, Ec. If the E
is more than Ec, then, first the distance traveled by the electron to
the collision is calculated.
Second, the type of collision which occurs is determined. If a
collision is elastic then the electron is scattered with the use of a
phase function, the appropriate energy aE,-,p is lost, and the electron
goes back up to the START of the degradation process. Whether a coll i si on is
inelastic it is determined if the collision is an ionization event or an
excitation event. In the excitation process, scattering occurs if the
energy E is less than 100 eV, E is reduced by the threshold, W, for
excitation of this state, and the electron goes back up to the START of
the degradation process.
Ionization collisions are the most complex occurrences to compute.
The energy loss, W, by the incident electron is equal to the kinetic
energy, T, of the secondary electron produced plus the ionization thresh
old, I. The primary electron is then scattered and reduced in energy by
W. If the secondary electron has a kinetic energy greater than Ec, then,


Figure 4.1 Flowchart of the Monte Carlo degradation of an incident
electron of energy E0.


-50-
MONTE CARLO CALCULATION


-51-
it is scattered and sent back to the START to be degraded further. In
the meantime, the primary electron's properties are stored.
If a secondary produces a tertiary electron with a kinetic energy
greater than Ec, then that tertiary is completely degraded before any
further degradation of the secondary is considered. Like the primary,
the secondary electron's properties are stored in the meantime. No other
generations were included in this study as their contribution would be,
at most, a couple of tenths of a percent of the incident electron's
energy.
After the tertiary is entirely degraded below Ec> then the secondary
is again sent back to the START to be degraded further. The secondary
is next entirely degraded below Ec> and, finally, the primary is again
sent back to the START to be further degraded. This whole process may
then again repeat itself.
B. Computer Programs and Machinery Used in the
Monte Carlo Calculation
In the previous section a brief discussion was given of the electron
energy degradation process. A brief discussion will be given below about
the MC computer codes and the computing machinery used. The MC computer
program employed in this work evolved from an original MC code written
by R.T. Brinkmann (see applications in Brinkmann and Trajmar, 1970). This
program was revised for use in Heaps and Green (1974), Kutcher and Green
(1976), and Riewe and Green (1978). The author has further modified this
MC technique for energetic electron impact into N£ to be used in the
energy range from 2 eV to 5 KeV.


-52-
This MC technique was applied to several incident electron energies.
The vast majority of the MC program runs used the Amdahl 470/175 computer
at the Northeast Regional Data Center at the University of Florida.
There were, however, several MC runs using the PDP 11/34 of the Aeronomy
group of the University of Florida.
It should be noted here that running the same program on both
machines at the same energy, EQ =1 KeV, showed a factor of 240 dif
ference in the execution time. Thus a program that takes four hours on
the PDP 11/34 will take one minute on the Amdahl 470/175. This time
advantage plus the ability to store each collision of the electrons on
magnetic tape does make the Amdahl 470/175 a more desirable "number
crunching" machine. The PDP 11/34 is only able to produce intensity plots
in the longitudinal direction. This mini-computer is thus mainly useful
in deriving a range (to be described in the next chapter).
Two programs were used in deriving the MC results. The first pro
gram (listed in Appendix A), the modified version of Brinkmann's code,
degraded the electrons in energy from their initial EQ down to the Ec
and recorded each collision and its properties on the tape. The second
program (listed in Appendix B) coalesces the data from the tape into an
array of ordered output. This output contains information for three
dimensional intensity plots, energy loss plots, and yield spectra.
C. Detailed Discussion of the Monte Carlo Electron
Energy Degradation Technique
Now, a more detailed discussion is given for the MC method of
degrading an electron's energy. An electron will start off with an
energy of Eq and coordinates xQ, yQ, zq, eQ, and Q. The symbols x, y,


-53-
and z are the Cartesian coordinates of the electron. The polar angle 0
is measured with respect to the z-axis and the

angle measured with respect to the x-axis (see Figure 4.2). In this
approach, the initial coordinates xQ, yQ, zq, eQ, and o were all set
equal to zero. The coordinates xb, yb, zb, eb> and 4>b of the electron
before starting on its journey to a collision are, therefore, initially
established as xb = xQ, yb = yQ, zb = zQ, eb = 0Q, and b = The MC approach relies on the random number, R, between 0.0 and 1.0
to aid in the deposition calculation. For each collision several R's
are needed and for each R a new property of the collision is gained. In
order to explain this MC approach, an accounting of the random numbers
and their subsequent usefulness is now made. The multiple elastic scat
tering distribution used below 30 eV and the lowest energy cutoff 2 eV
are also described.
1. First Random Number, R]
The first random number, R^, is used to find the path, Py, traveled
by the electron before it collides with a molecule of N^. Calculation
of Py proceeds in the following manner. The mean free path, A, is
defined as
(4J)
3
where n is the density of N2 molecules in #/cm and ay(E) is the total
2
(inelastic plus elastic) cross section of N2 in units of cm at an energy
E. The densities used at the various initial input energies are given
in Table 4.1.


-54-
Figure 4.2 Schematic representation of the coordinates and directions
of motion of the electron in its travel between collisions
with the Ng molecules.


-55-
Table 4.1 The energy E is presented in the first column with the number
density n, used in the MC calculation, being given in the
second column. (8.0 E+ 14 means 8.0 x 10^)
E (KeV)
n (#/cm^)
0.1
8.0 E+14
0.3
2.0 E+15
1.0
8.2 E+15
2.0
2.8 E+l6
5.0
1.2 E+l 7


-56-
All electrons are forced to be degraded in a 30 cm long cylinder;
thus an increase in the density is required for an increase in the energy.
There are 10 cm allowed in the negative z direction and 20 cm allowed in
the positive z direction. The x and y axes extend to infinity. Some
electrons actually escape from the cylinder, but the energy lost due to
these electrons is only a few tenths of a per cent of the incident elec
tron energy. The path length is then given as
PT = -xln(R1) (4.2)
using the relation that
(4.3)
Figure 4.2 represents a schematic of the electron traveling and
colliding with three Ng molecules. The P^, P.^, and Py3 are the path
lengths traveled by the electron between the initial coordinates and
the first collision, the first and second collisions, and the second and
third collisions, respectively.
The x,, y and z, coordinates at this collision can now be found
a a a
from PT, xb, yb, zb> eb> and <¡>b using
x
a
y
a
z
a
2b
+ P-j. sineb cosb
(4.4)
+ Pj sineb sinij>b
(4.5)
+ PT coseb
(4.6)
In Figure 4.2 the coordinates of the first and second collisions are
represented to illustrate how the electron's direction of motion might
change during its collisions with Ng. So far emphasis has been only on


-57-
the Cartesian coordinates. Now, calculate the azimuthal angle and the
a
polar angle of the electron after a collision.
2. Second and Third Random Numbers, R2 and R3
In actuality the type of collision must be specified before the
scattering can be calculated. It is assumed, however, that the type of
collision is already known (see subsection IV.C.4). The second, R2, and
third, R^, random numbers are not chosen if the collision is an excita
tion event and E is greater than 100 eV. They are chosen for all other
collisions.
The R2 is used to calculate the azimuthal scattering angle, 41, of
the electron from its direction of motion. The premise is that the
azimuthal scattering is isotropic; therefore,
4>' = R2 2h (4.7)
(Note that the ' angle is the only angle not represented in Figure 4.2.
Inclusion of 4' adds too much complication to an already cluttered
figure.)
The third random number, R^, is employed to calculate the polar
scattering angle e' of the electron from its direction of motion. (The
angle e1 is represented twice in Figure 4.2: Once as the scattering due
to the first collision and once as the scattering due to the second
collision.)
For elastic collisions, Eq. (3.1), (3.12), (3.13), or (3.14) are
used in determining 01. In all but one of these phase functions, an
analytic expression can be used to determine 0' from the random number,
R3. These analytic expressions are given below.


-58-
Using the screened Rutherford differential cross section form (see
Eq. (3.1)), it follows that
e1 = cos"1 [1 + 2n inlV-n'R ] (4.8)
3
For model 1 (see Eq. (3.12))
0'
cos
-1
[ ^T -p + 1 + a]
R3C(2 + a) 1 a '] + a 1
and for model 2 (see Eq. (3.13))
0'
cos
-1
\/b2 4AC n
2A J
(4.9)
(4.10)
with
A =
R3 +
a[(2 + a)'
- a"1]
0 f)
(2 + c)[(2 + c)_1 c'1]
B = -A(a c) +
[(2 + a)"1 a"'] [(2 + c)"' c]
-1
(1 f)
-1
1
and
C = -A(l +a)(l +c) + HIjLc]
[(2 + a)'1 a'1]
(1 f)(1 + a)
[(2 + c)"1 c"1]
Model 3 (Eq. (3.14)) is not so easy to write in such a convenient
form. The equation for primary scattering after an ionization event
(Eq. (3.17)) is, also, not easily inverted.
For these two differential cross sections, the following approach
is taken. The angular range from 0 to 180 is divided up into angular
intervals. A certain probability for scattering at angles less than the
end of each angle interval is calculated from the differential cross


-59-
section form. The angle e' is then found through the correct placement
of R3 into an angular segment whose beginning and ending point scattering
probabilities bracket Ry
For this work twenty-four angular segments were chosen. Their end
points are given in Table 4.2. With twenty-four angular intervals, the
results from the Monte Carlo calculation came out to be the same as with
the use of forty angular intervals. If sixteen or even twenty segments
were used, the MC computation gave results that were 5% to 10% different.
The coordinate system, but represent the azimuthal and polar scattering of
the scattered electron from the direction of travel of the incident elec
tron. In order to calculate

d a
representing the motion of the electron after the collision, some spheri
cal trigonometry must be used. The following relations hold in this
transposition:
cosa = [coseb cosb sine' cos'
- sinb sine' sin' + sineb cosb cose']/sinea (4.11)
sina = [coseb sin4>b sine* cos
+ cosb sine1 sin<¡>' + sineb sinb cose']/sinea (4.12)
(4.13)
cosea = coseb cose' sineb sine* cos^'
(4.14)
(4.15)


-60-
Table 4.2 Twenty-four angle intervals are given here that were used in
the Monte Carlo calculation. First column lists the index of
the segment and the second and third columns give the begin
ning and end points for each segment with units of radians
(degrees).
Index
Beginning
End
1
0.00
(0.00)
0.01
(0.57)
2
0.01
(0.57)
0.05
(2.87)
3
0.05
(2.87)
0.11
(6.30)
4
0.11
(6.30)
0.20
(11.46)
5
0.20
(11.46)
0.40
(22.92)
6
0.40
(22.92)
0.60
(34.38)
7
0.60
(34.38)
0.80
(45.84)
8
0.80
(45.84)
0.90
(51.57)
9
0.90
(51.57)
1.00
(57.30)
10
1.00
(57.30)
1.10
(63.03)
11
1.10
(63.03)
1.20
(68.75)
12
1.20
(68.75)
1.30
(74.48)
13
1.30
(74.48)
1.40
(80.21)
14
1.40
(80.21)
1.50
(85.94)
15
1.50
(85.94)
1.60
(91.67)
16
1.60
(91.67)
1.80 (103.13)
17
1.80 (103.13)
2.00
(114.59)
18
2.00 (114.59)
2.20
(126.05)
19
2.20 (126.05)
2.40 (137.51 )
20
2.40 (137.51 )
2.60 (148.97)
21
2.60 (148.97)
2.80 (160.43)
22
2.80 (160.43)
3.00
(171.89)
23
3.00
(171.89)
3.07 (175.90)
24
3.07 (175.90)
3.14 (180.00)


-61-
and
0 = cos1(cose )
a a 1
(4.16)
Now the azimuthal angle and the polar angle 9^ have been established
a a
for the collision with respect to the fixed coordinate system. These
angles are also represented in Figure 4.2. The two angular coordinates
b and eb of the electron before traveling to the next cllision are then
set as d>. = and e, = e .
b a b a
3. Fourth Random Number, R4
A fourth random number, R^, is required if a secondary is produced
and if that secondary has an energy above the cutoff energy, Ec- This
R^ is chosen to determine the polar angle, 0', of scattering of the
secondary. Again, an analytic formula can be employed to define 0'.
This equation was derived from Eq. (3.18) and is written as
e'
r i + cose )
cos" [tan [R.{tan" ( ~ -)
VT 4 0
- tan
+ tan
i/B(l coseQ)
C
/B(l coseQ)
C
)}
-)] + coseo]
(4.17)
The a
and 0 result from the use of Eqs. (4.11) through (4.16).
a
4. Fifth Random Number, Rg
The fifth random number, Rg, determines the type of collision that
occurs. Here, the type may be either elastic or inelastic. If the


-62-
type is inelastic then the individual excitation or ionization event is
found as wel1.
There are cross sections for thirty-four states of N2 employing the
papers of Jackman, Garvey, and Green (1977) and Porter, Jackman, and
Green (1976). Using all these states in the MC calculation would greatly
increase the cost. It was therefore decided to reduce these thirty-four
states to nine states. Two allowed states, the b \ and the b'
and the six ionization states were kept the same as given in the papers.
For the ninth state, all the Rydberg and forbidden states were combined.
Above 200 eV, the forbidden states are contributing only a minuscule
amount to the total cross section. Since the other states have roughly
the same In E/E fall-off at high energies, it is assumed that the pro
babilities for excitation to any of these states will be constant. These
probabilities were simply found from the ratio of the cross section of
the state in question to the total inelastic cross section at the elec
tron energy of 5 KeV.
In Table 4.3 these states, their probabilities, and thresholds are
presented. The probability, pc> of the composite state is simply
m
pc = l pi
c i=l 1
(4.18)
where m = the total number of Rydberg and forbidden states and p^ is the
probability for excitation of the ith Rydberg or forbidden state. The
average threshold, Wc> for exciting the composite state is found easily
with the following equation
m
l P- W.
_ i=l 1 1
'c m
l Pi
i=l 1
W
(4.19)


-63-
Table 4.3 N2 inelastic states,
W, taken for electron
below.
their probabilities
energies above 200
, p, and thresholds,
eV are presented
State
P
W (eV)
It, bu
0.092
12.80
N2 b''<
0.042
14.00
Composite
0.233
15.40
N2
0.289
15.58
N2 a2u
0.127
16.73
N2
0.066
18.75
N2 D%
0.044
22.00
N2
0.044
23.60
N* 40 eV
0.063
40.00


-64-
with being the threshold of the Rydberg or forbidden
state.
Below 200 eV, the probabilities for excitation to the various
inelastic states are changing quite rapidly. The parameters for the
eight individual states are taken from Jackman et al. (1977b) and
Porter et al. (1976). The composite state's properties are found in
the same manner that they were above. In these lower energy regimes
the probability and energy loss are changing fairly rapidly, thus Table
4.4 illustrates these probabilities and threshold values at several
energies.
With the background on the inelastic cross sections and their
subsequent probabilities, consider now the collision type. The
random number determines the type of collision that occurs in the
following manner: If
aTE(E)
R5 cT (E) for electron energies (4.20)
where jE(E) is the total elastic cross section, then the collision is
elastic. If
TE(E) PiTI^E^ + TE^
" r5 ^¡TET
^[ET
and E > 200 eV
(4.21)
where 0jj(E) is the total inelastic cross section and p^ is the proba
bility for exciting the first inelastic state (in Table 4.2 the first
state is the b^ thus p-j = 0.092), then the inelastic collision results
in the excitation of the first state.
A relation follows from Eq. (4.21) that holds true for j = 2 to 9
such that: If


-65-
Table 4.4 N2 inelastic composite state with its characteristic proba
bility, p, and average energy loss, W, given for several
energies between 2 and 200 eV.
E (eV)
P
W (eV)
2
1.000
0.57
3
1.000
1.03
4
1.000
0.922
5
1.000
0.835
6
1.000
0.772
7
1.000
0.728
8
1.000
0.696
9
1.000
7.00
10
1.000
7.21
12
1.000
8.25
14
1.000
8.91
16
0.971
9.12
18
0.866
9.34
20
0.745
9.68
30
0.426
11.70
40
0.344
12.80
50
0.296
13.30
60
0.271
13.70
70
0.255
13.90
100
0.229
14.30
150
0.214
14.60
200
0.234
14.80


-66-
^ PiCTTI(E) + CTTE(E)
5^1)
rc
and E > 200 eV
(4.22)
then the inelastic collision results in the excitation of the jth state.
Thus the Rg random number for an electron of energy E > 200 eV will
determine which type of collision occurred when satisfying Eq. (4.20),
(4.21), or (4.22).
For energies below 200 eV, the following relations must be con
sidered: If
aTE(E) -j(E) + aTE(E)
^lT K5 ^T)
and E < 200 eV
(4.23)
where ^(E) is the cross section for exciting the first inelastic state,
then the inelastic collision results in the excitation of this state.
A relation similar to Eq. (4.22) can now be established for j = 2
to 8 such that: If
I a-j(E)+0te(E) J ai(E)+aTE
aT(E) < R5 oT(E)
(E)
and E < 200 eV
(4.24)
then the jth inelastic state is excited. If
l ^(E) + aTE(E)
Rr > nn and E < 200 eV
5 ot(E)
(4.25)
then the excitation of the composite state is assumed and the energy loss,
Wc, in this case is found through a linear interpolation with the use of
the values given in Table 4.3.


-67-
5. Sixth Random Number, Rg
The sixth random number, Rg, is computed only if the collision type
is an ionization event. This Rg determines the energy lost by the
primary in creating a secondary of energy, T Using the S(E,T) from
Eq. (3.19) the following relationship is established:
Ts
/ S(E,T) dT
= 0
I0N(E^
(4.26)
Integrating the numerator in Eq. (4.26) and using Eq. (3.21) to solve
for T Eq. (4.27) is derived.
Ts = r(E)[tan{Rgtan"1[(TM TQ(E))/r(E)]
+ (Rg l)tan_1[To(E)/r(E)]}] + TQ(E) (4.27)
The energy loss, W, is then found by the relation:
W = Ik + T$ (4.28)
where 1^ is the ionization threshold for the kth ionization state.
6. Multiple Elastic Scattering Distribution Used Below 30 eV
The MC calculation can be used to degrade an electron down to
practically any energy. Even below the lowest threshold for excitation
to any vibrational level, the electron will still lose energy via elastic
collisions with molecules of nitrogen as well as other electrons. This
energy loss to other electrons is fairly low unless a substantial frac
tion of the gas has been ionized (see Cravens, Victor, and Dalgarno,


-68-
1975). In this study the fraction of ionization is assumed to be
negligible; therefore, this loss is ignored.
Unless there is a very large amount of money available for computer
time, an electron can not be followed to its thermal energy with any
practicality. This implies that a multiple elastic scattering distri
bution (hereafter referred to as MESD) must be used below some given
energy. In this work the MESD will be used below 30 eV.
Bethe, Rose, and Smith (1938) used the Fokker-Planck differential
equation, neglecting energy loss, to consider the penetration of elec
trons through thick plates. This, however, leads to a Gaussian solution
in the small-angle approximation so that the tail of the angular dis
tribution was omitted. The large-angle multiple scattering has been
studied by Goudsmit and Saunderson (1940) [hereafter referred to as GS]
who used a series of Legendre polynomials to determine the resultant
angle of scattering.
Lewis (1950) studied the integro-differential diffusion equation of
the multiple scattering problem in an infinite, homogeneous medium,
without the usual small-angle approximation. He obtained the GS solu
tion for the scattering angle and also derived certain moments for the
longitudinal and transverse distributions.
Berger (1963) applied a MESD for condensed case history MC cal
culations. His application of the MESD is at the energies above 200 eV
and probably is not accurate for electrons with energies less than about
100 eV. Furthermore, Berger's (1963) work contains approximations that
are only good for the sharply forward peaked cross sections of higher
energy electrons.


-69-
In this work a different problem exists. The MC calculation is
used to degrade electrons in a collision by collision manner all the way
down to 30 eV. At this energy, the elastic collisions are occurring
with twice the frequency of the inelastic events, and at energies below
30 eV the number of elastic collisions between inelastic events may be
up to several hundred or thousand. Keeping track of all these elastic
collisions would be very costly.
Kutcher and Green (1976) [hereafter referred to as KG] studied the
radial, longitudinal, and polar angle distributions for elastic scatter
ing by in the energy range from 2 to 50 eV. An approach similar to
KG's could be applied to Ng. Since such a project would require a
substantial amount of time and computer money, the possibility of adapt
ing the KG results was first considered.
With this in mind, consider the differences between and Hg.
First of all, there are some dissimilarities between the differential
cross sections. There is more backscatter observed experimentally in
N2 at all energies. Secondly, the total inelastic and elastic cross
sections are different. The second difference is no real problem because
the MESDs are given in terms of the mean free path lengths (hereafter
referred to as MFPs). The first dissimilarity does pose a minor problem
which is solved in a simplistic way below.
Above 5 or 6 MFPs the polar angle is approximately random. At most
energies below 30 eV, the number of MFPs between inelastic collisions
is above 5 or 6. Since the distribution found in KG is not easily in
verted, a reasonable assumption is that the polar angle is oriented
randomly.


-70-
Knowledge of the radial distribution is not crucial for our pur
poses. The most interesting radial distribution output from this MC
o
calculation is that of the 3914 A emission. Electrons below 30 eV make
little contribution to this profile because the cross section for ex-
citation to this B e^ state is fairly low (see Figure 5.1). Thus
knowledge of the radial distribution of these electrons multiply scat
tered is not extremely important.
An approximation, however, is employed in most MC computations to
calculate a fairly reasonable radial distance. The average radial
distance, as observed from the calculations in KG, for most energies and
at the longer path lengths is approximately one-sixth of the total path
lengths, thus
Pave = s/6 (4.29)
The most important spatial displacement is the longitudinal dis
tance z. In order to calculate z, the total path length s must be known.
This length s is calculated from the random number, the total elastic
cross section, o^E), and the total inelastic cross section, aTI(E),
by using
aip(E)
s'-^ETln(Rl> <4-30>
The ratio a^^Ej/a^jiE) is simply a fairly accurate approximation
of the number of elastic collisions occurring per inelastic collision.
The value -ln(R-j) [see Eq. (4.2)] is the path length (in units of MFPs)
traveled by the electron between collisions. Thus knowing the number
of elastic collisions occurring and the path length traveled between
collisions allows one to write Eq. (4.30) as the expression for the


-71-
total path length s (in units of MFPs) traveled between inelastic
collisions.
In KG an equation which can be easily inverted to calculate the
z distance (in units of MFPs) from some random number, R2 and path
length s, is written
f^-1] 1
-m Wr1/V.-J]; (4.31)
-u
where
v(s) = 1 exp[-(s/sv)D]
F(0) = K{1 exp[-(s/sf)0,75]}
and
u(s) = (H + sI)/sJ
where K = 0.425.
Since there is more backscatter during N2 elastic collisions
(because of its differential backscatter contribution), it seems reason
able that the parameters for Eq. (4.31), which are useful for N2, are
different than those derived in KG. One approach to this dilemma might
be to correlate the elastic differential cross section (hereafter called
EDCS) from at some energy E1 with the EDCS from H2 at some energy E.
This would work if the H2 EDCSs showed more backscatter than the
EDCSs; however, the opposite is observed experimentally. Thus the
EDCSs from some E1 (around 6-7 eV) values correlate with the H2 EDCSs
at E values less than 2 eV (where the Kutcher and Green, 1976, MESD is
not defined).


-72-
Another straightforward and simplistic approach is to do the
following. Calculate the approximate backscatter at three energies,
the two endpoints and the middle (2 eV, 15 eV, and 30 eV), from the KG
H2 EDCS form and the experimental data on N2 EDCSs (given in Sawada,
Ganas, and Green, 1974). At these energies the backscatter with the KG
H2 EDCS form is less than that of the N2 EDCS by the following values:
2 eV ^ 5%, 15 eV ^ 10%, and 30 eV ^ 10%. An average of these three
values is about 8%. Since the major influences of the backscatter in
Eq. (4.31) is the value of K, this parameter is the only one that is
changed from the KG formulation. It is, therefore, increased by ^8% so
that in these MC calculations K = 0.46. The other parameters in Eq.
(4.31) are listed in Table 4.5.
Actually it appears that the value of K makes little difference in
the MC computational results. Two MC calculations at an incident elec
tron energy of 100 eV with K = 0.46 and with K = 0.425 were undertaken
(all other parameters and inputs were the same). The yield spectra
(described in Chapters II and VII) changes substantially only at fairly
large longitudinal distances (where the distances are about 1.5 times
the range). At these large distances there are relatively few electrons
anyway, thus there is little effect on the major aspects of the spatial
electron energy deposition process.
The Cartesian coordinates x y and z, are found from the coor-
dinates x^, y^, and in the following manner. After z is calculated
in units of MFPs with the use of Eq. (4.31), it can then be written in
units of cm or km by multiplying by the MFP, X (calculated from Eq.
(4.1)), thus zg = zb + zx.


-73-
Table 4.5 Parameters from Kutcher and Green (1976) for several energy
intervals used in Eq. (4.28).
Energy
Interval
(eV)
H
I
J
D
s
V
SF
2-5
12.
1.37
1.71
1.75
5.05
8.5
5-10
9.6
1.32
1.67
2.50
4.25
8.5
10-20
15.5
1.28
1.67
2.31
6.29
10.3
20-30
23.5
1.24
1.69
1.98
9.65
13.6


-74-
As established earlier, the polar angle 0, and azimuthal angle ,
a a
representing the motion of the electron after the collision, can be
chosen in a random way from the two random numbers, and R^, using
e =
a
$
a
21*4
(4.32)
A reasonable approximation of x, and y
a S
Eqs. (4.29) and (4.32) such that
*a xb + pave x cos+
and
ya yb + pave x sin*
can then be made usi
a
a
ng
In the MESD the fifth random number, Rg, is used to determine the
inelastic collision type. A method similar to that illustrated in sub
section IV.C.4 is employed, the only difference is the fact that the
collision is only inelastic.
7. Value of the Cutoff Energy, 2 eV
The E used in this work has been set at 2 eV because the lowest
c
threshold for excitation to an inelastic state is 1.85 eV. With this
cutoff energy the yield spectra can be defined down to 2 eV at all
longitudinal distances. Subsequently, a reasonable calculation of the
excitation to any Ng state may be made.


-75-
D. Statistical Error in the Monte Carlo Calculation
The statistical error inherent in the MC computation can be derived
by considering the following. Since the MC calculation is a probabilis
tic method of degrading an electron in energy, the multinomial distribu
tion can be used to find the statistical standard deviation for each bin
considered. This discussion of the statistical error employed the work
of Eadie, Dryard, James, Roos, and Sadoulet (1971).
The probability of getting an excitation of a certain state j in bin
k is Pj^. The p^ is normalized such that
m n
l l Pik = 1
k=l j=l JK
(4.33)
In this MC study the multinomial distribution is an array of histograms
containing N events distributed in n states and m bins with r.^ events
in state j and bin k. The r.^ values are normalized such that
m n
l l rjk
k=l j=l JK
(4.34)
Thus, the r^ observations can be considered somewhat conditional
on the fixed observational value of N. The variance of the calculation
is represented as
v(rjk> = N pjk < Pjk>
(4.35)
In this work the m x n variables r^ can all be correlated. For the
specific example of electron deposition represented in Figure 5.2,
jk
1. This is true because there are total almost 5 x 10 col
lisions (i.e., N = 5 x 10) to consider in this degradation scheme and


-76-
at maximum r^
approximated by
% 4000.
Using this information, Eq. (4.35) can then be
v N pjk rjk <4-36>
+ 2 +
and the statistical standard deviation of the number of B zu events
in a bin becomes
jk '^jk ^4-37^
Equation (4.37) holds true for the specific example represented in
Figure 5.2 and it also holds true for all the intensity plots, energy
loss plots, and yield spectra that were studied in this work. Thus,
in order to obtain the approximate standard deviation for any MC generated
number, the square root of this value is its standard deviation. The
error bars found in the rest of this paper are determined in this
manner.
Now that the MC calculational technique has been outlined, this
method will be used in the next three chapters to deal with the spatial
and energetic aspects of electron energy degradation.


CHAPTER V
MONTE CARLO INTENSITY PLOTS AND COMPARISON WITH EXPERIMENT
Incident electrons with energies between 0.1 and 5.0 KeV are de
graded in Ng using the MC method described in Chapter IV with the cross
o
sections given in Chapter III. The intensity plots of the 3914 A
emission are described in this chapter.
+ 2 +
Emission intensity plots of the 3914 A radiation from the N2 B zu
state are used in describing the range (found by extrapolating the linear
portion of the longitudinal 3914 A intensity plot to the abscissa) for
incident electrons. Section V.A describes the excitation of the
+ 2 +
N2 B state. In section V.B the range of the electrons is defined
more completely. Previous experimental and theoretical work on the
3914 emission of N^ is given in section V.C. The range results from
the MC calculation are then discussed in section V.D. Finally, section
V.E describes the intensity plots resulting when plotted as functions of
the radial direction.
A. Excitation of the NB £u State
The main concern of this chapter will be the intensity plots showing
the emission of the 0-0 first negative band (B state) of N^ at 3914 A.
Experimentally (see Rapp and Englander-Golden, 1965; McConkey, Woolsey,
and Burns, 1967; and Borst and Zipf, 1969), it has been shown that the
number of photons at 3914 produced for each ionization of N2 is
-77-


-78-
independent of the energy of the exciting electron for energies from
30 eV at least up to 3 KeV.
In Figure 5.1 the N2 total ionization cross section and cross sec-
2 + +
tion for ionization and excitation to the B Zu state of are presented.
The curves are approximately parallel thus even if the absolute values
for the two cross sections are slightly in error, the shapes of the in
tensity plots that result from this MC calculation should be fairly
accurate.
The total ionization curve lies nicely in the middle of an array of
experiments (namely, Opal, Beaty, and Peterson, 1972; Tate and Smith,
1932; Rapp and Englander-Golden, 1965; and Schram, de Heer, Wiel, and
Kistenaker, 1965) but the B cross section values may be high when
compared to experiments (see Holland, 1967; and McConkey, Woolsey, and
Burns, 1967).
2 +
The threshold for excitation to this B Zu state is 18.75 eV, thus
any electron above that energy can excite and ionize a ground state
molecule up to this level. The cross section for excitation and ioniza-
2 +
tion to the B £u state is not large when compared with the total in
elastic cross section. In fact, the probability for exciting this state
is only 0.066 for electron energies above 200 eV. The accuracy of the
MC calculation is dependent on the number of excitations in each bin
(see section IV.D). In order to enhance the precision of the MC results,
o + o 2 |
excitations of the X £ and A n states of N0 are added to the B e
g u 2 u
excitations. The ionization cross sections for these two states are
2 +
found to be proportional to the B state for electron energies above
30 eV.


-79-
E (eV)
Figure 5.1 Total loss function L(E) from N2, denoted by the solid line;
total ionization cross sections for N^, denoted by the dash-
dot line; and the Nj B cross section, denoted by the
dashed line, are given as functions of energy, E.
t-


-80-
Previous workers (Barrett and Hays, 1976; Cohn and Caledonia, 1970;
O
and Gran, 1957) have used the 3914 A emission as a measure of the energy
deposited. In these works it is assumed that since the 3914 radiation
is proportional to the number of ionizations in a given volume and if
the number of ionizations is proportional to the energy deposited in
that volume, then the 3914 A intensity is proportional to the energy
deposited in that volume. These experimenters, therefore, measured the
3914 A radiation at several energies, extrapolated their intensity plots
to find a range (to be described in section V.B), and derived an empiri
cal expression for the range that could be used to find the energy loss
function.
This idea of using the 3914 A emission to derive the energy loss
scheme is useful for energies above 2 KeV. In Figure 5.1, compare the
loss function, L(E), used in this work and the N2 B £u state cross
section.
The two curves are not parallel below 2 KeV. This implies that the
energy loss function can not be derived directly from the range results
at incident energies below 2 KeV. The energy loss plots from this MC
study are given in section VII.A and more will be discussed in that
section about them.
B. Range of Electrons
The concept of the mean range must be defined next. For each
monoenergetic primary electron impinging into a gas, a range can be
calculated. In general (at least above 100 eV), the higher the electron
energy the further the electron will penetrate into the medium. If an
+ 2 +
electron is incident along the z-axis, the excitations of the N2 B E


-81-
state can be graphed in an intensity plot with the z-axis as the
abscissa.
In Figure 5.2, the intensity plot from 5000 incident 1 KeV electrons
is graphed (the model used in this MC calculation should only be taken
as an illustrative example) in histogram form. Bins along the z-axis
are taken to be 0.5 cm in width for these incident electrons. The linear
portion of the curve may be extrapolated, as illustrated by the dashed
line, to define a mean range of the beam.
All the intensity plots are normalized in this paper so that the
beam starts out at z = 0 cm along the z-axis. The intensity in Figure
5.2 seen at negative values of z is brought about by backscattered
electrons. The error bars given near the peak of the histogram are
found simply from a method described in section IV.D.
From Figure 5.2, the range is seen to be 16 cm for these 1 KeV
2
electrons. Range values, R^, in units of gm/cm are written
Rg = Rce (5.1)
3
where R is the range in cm, p = n MN (in gm/cm ), n is the number
O
density of molecules (in #/cm ), and is the weight (in gms) of an
15^ 3
molecule. In this case, n = 8.2 x 10 molecules of ^/cm =
_ p p ^ e
4.651 x 10 gm/^ molecule, and Rc = 16 cm; therefore, Rg = 6.06 x 10
gm/cm^.
C. Previous Experimental and Theoretical Work on the
3914 A Emission of n£
Grn (1957) measured for air the total luminosity of the 3914
radiation in planes perpendicular to the axis of the electron beam with


o
Figure 5.2 The intensity of 3914 A radiation is given as a function of z
for 5000 electrons with incident energies of 1 KeV. The range,
found by extrapolating the linear portion of the histogram
(illustrated by the dashed line) to the abscissa, is 16 cm. The
I indicates the standard error involved in the MC calculation.


Z (cm)
3914 A Intensity (I03exc.)
i O ro oj ^ ui
-£8-


-84-
an initial energy of 5 to 54 KeV. Cohn and Caledonia (1970) measured
intensity profiles of electron beams with incident energies from 2 to
5 KeV impacting into Ng. Barrett and Hays (1976) then extended the
incident electron range down to 300 eV by measuring the radiation pro-
O
files of 3914 A resulting from electron beams with energies from 0.3 to
5.0 KeV impinging on Ng.
Spencer (1959) used the Spencer and Fano (1954) method of spatial
energy deposition and found good agreement between his energy loss plots
and the 3914 ft intensity plots of Griin (1957). The Berger, Seltzer, and
Maeda (1974) [BSM] MC calculation provided energy loss plots down to
2 KeV. These plots are also in fairly good agreement with the experi
ments mentioned above.
Comparisons will be made in this paper between the available experi
mental electron energy loss data and the MC calculations done here.
Since this MC calculation follows the incident electrons, as well as its
secondaries and tertiaries down to 2 eV, this MC computation is one of
the most detailed ever employed for electron impact energy degradation.
It is, therefore, of interest to compare the results from this study
with experimental results for incident electrons with energies from
300 eV up to 5 KeV.
D. Range Results and Longitudinal Intensity Plots
from the Monte Carlo Calculation
Range data at several incident electron energies are calculated
with the use of the screened Rutherford and the model 3 differential
elastic cross sections. The screened Rutherford model is used because
it is the most widely used form for elastic scattering in theoretical


-85-
studies and, also, because BSM were quite successful in using this form
down to incident energies of 2 KeV. Model 3 was used because of its
very close agreement with experimental differential cross section data
in the range from 30 eV up to 1 KeV.
Table 5.1 presents the range data (for perpendicularly incident
electrons) from three different experiments, the theoretical calculation
by BSM, and two sets of theoretical computations from this study. The
values in parentheses from BH (Barrett and Hays, 1976), CC (Cohn and
Caledonia, 1970), and G (Griin, 1957) are simply calculated from the
empirical formulae given in these works.
For the rest of this chapter, the results of this work will be com
pared with those of BH. This is the most recent experimental study and
is probably the most reliable experimental work. They also use the same
incident electron energy regime as that employed in this work. In
Table 5.1 it is apparent that the BH values have the largest ranges of
the experimental studies.
The two separate MC calculations in this study seem to bracket the
BH results at all energies. The model 3 ranges are consistently larger
than those of BH. These results are 10% higher at 5 KeV and about 19%
higher at 0.3 KeV. The screened Rutherford ranges, on the other hand,
are about 9% lower at 5 KeV and about 10% lower at 0.3 KeV.
If it can be assumed that the BH results are indeed the most re
liable data, then the following conclusion can be made: The screened
Rutherford phase function scatters the electron too much while the
model 3 phase function provides too little scattering. This conclusion
is made assuming that the total cross sections described in Chapter III
are fairly accurate.


-86-
-6 p
Table 5.1 Range data (in 10 gm/cm ) at several energies, E (in KeV),
are given below. The second column M3 (model 3), third
column SR (screened Rutherford), fourth column BH (Barrett
and Hays, 1976), fifth column CC (Cohn and Caledonia, 1970),
sixth column G (Griin, 1957), and seventh column BSM (Berger,
Seltzer, and Maeda, 1974) range values are presented.
: (KeV)
M3
SR
BH
CC
G
0.1
0.37
0.34
(0.53)
(0.07)
(0.08)
0.3
1.25
0.95
1.06
(0.51)
(0.56)
1.0
6.45
5.57
5.72
(4.17)
(4.57)
2.0
18.6
16.8
17.7
14.0
(15.4)
5.0
91.5
75.9
83.0
69.7
76.4
71.9


-87-
In this work model 3 is the result of a careful investigation of the
detailed molecular nitrogen cross sections. Therefore no attempt will
be made here to change the cross sections compiled in Chapter III. Model
3 will be used in most of the MC calculations in the rest of this
chapter and also in Chapter VII (BSM have, however, chosen nc used in
the screened Rutherford cross section, to be a constant value whose value
was selected so as to obtain the best agreement between their MC calcula
tion and the experimental results of G and CC).
In Table 5.1 the importance of the elastic phase functions is clearly
illustrated. Up to a 25% change in the range is observed when com
paring the screened Rutherford with the model 3 phase functions. More
elaboration on the effects of various phase functions on the energy
deposition process will be given in Chapter VI.
Figures 5.3 and 5.4 give intensity plots for the 3914 A radiation
resulting from 2 KeV and 0.3 KeV incident electrons, respectively. The
experimental work of BH and the calculations using model 3 and the
screened Rutherford are presented in these figures. The shapes appear
to be somewhat similar; however, the BH results at both energies pre
dict a range that is between the two theoretical calculations.
E. Intensity Plots in the Radial Direction
Most attention, so far in this study, has been concentrated on the
intensity plots in the longitudinal direction. There is experimental
data available on the intensity of the 3914 & radiation as a function
of p (the axis perpendicular to z). Experimentally, G, CC, and BH all
present data of this type.


Figure 5.3 An intensity plot for electrons of energy 2 KeV is presented as a
function of the longitudinal direction. The x's represent relative
experimental values from Barrett and Hays (1976) and the histograms
present the data from model 3 (heavy line) and the screened
Rutherford (light line). The straight solid and the straight
dashed lines represent extrapolations to find the range for the
model 3 and the screened Rutherford elastic differential cross
sections, respectively.


(ujo)z
3914 A Intensity (I03 exc)
p o o p p
, b ho b oo b ho
-68-


Figure 5.4 An intensity plot for electrons of energy 0.3 KeV is presented as a
function of the longitudinal direction. The x's represent relative
experimental values from Barrett and Hays (1976) and the histogram
presents the data from model 3 (heavy line) and the screened
Rutherford (light line). The straight lines extrapolated to the
z-axis are all measures of the range. The solid line indicates
the model 3 range; the dashed line indicates the screened Rutherford
range; and the dash-dot line indicates the Barrett and Hays (1976) range.


3914 A Intensity (I02 exc)
, o ro oj £> en en ->icd


-92-
This study uses the experimental data of BH as a comparison with
the results of this study. The next three graphs, Figures 5.5, 5.6,
and 5.7,portray sample results for incident electrons with energies
5.0, 1.0, and 0.3 KeV, respectively. The z and p values given in these
three figures are in units of fractions of the total range.
Fairly good agreement between the MC calculation (using model 3
cross sections) and the experimental work of BH and Barrett (1975) is
observed at all three incident energies. The largest differences be
tween the two sets of data are noted at 0.3 and 1.0 KeV.
For the 1.0 KeV case, the MC calculation tends to predict more in
tensity at the lower values of p for z values of 0.3 and 0.4. A similar
result is apparent for the z values of 0.36, 0.48, and 0.60 for an
energy of 0.3 KeV. At a z value of 0.12, however, the experimental data
tend to predict more intensity at all values of p.
Two conclusions can be drawn from these comparisons, if it is
assumed that the experimental data of BH and Barrett (1975) are correct.
+ 2 +
First, the cross section for excitation to the B Eu may be under
estimated in the energy regime between 0.3 and 1.0 KeV. Raising this
cross section in this energy regime could bring about an increase in the
intensity observed early in the electron's degradation process with a
subsequent decrease in intensity later in the electron's degradation
process. Second, more scattering from the elastic collisions would
help to reduce the total intensity at low p values and raise it at the
higher p values. The screened Rutherford differential cross section has
more scattering than model 3. Use of this set of cross sections in the
MC calculation did result in a little better agreement at 1.0 KeV, but
only about the same type of agreement at 0.3 KeV.


Figure 5.5 Intensity plots for electrons with incident energy 5.0 KeV
are presented at two z values as functions of p. The solid
line histogram indicates the results using model 3. The
x's denote the experimental data of BH and the o's denote
the theoretical work of BSM.


Full Text


-127-
In Figure 7.1 an energy remaining plot is given for three separate
calculations with 1 KeV incident electrons. At z = 0.0, which is the
point of incidence for the 1 KeV electrons, all calculations assume that
no energy has been lost. Thus the amount of "energy left" is simply
1 KeV.
Stolarski (1968) integrated the universal energy loss curve derived
from Grn's (1957) data to obtain the mean energy. Barrett and Hays
(1976) used their empirical range formula to calculate the mean range
and, subsequently, the energy remaining in the incident electron at
various distances into the medium. In the MC calculation of this study
only the energy lost for positive z values was employed to find the
energy remaining.
The Stolarski (1968) values are closest to the MC calculation values.
The major differences between Stolarski's results and this MC computa
tion are due to three factors: First, some energy is lost by backscatter
electrons to negative z values; second, some energy is lost by electrons
which penetrate to z values greater than 1.0 (straggling electrons); and
third, the universal energy loss curve may not be as accurate as a MC
computation.
Graphing the energy loss data in Figure 7.1 is really not very
informative. Figure 7.2 illustrates a more lucid way of representing
the energy loss data. In Figure 7.2, the fraction of the primary energy
lost is plotted as a function of z for four energies (0.1, 0.3, 1.0, and
5.0 KeV). As the incident electron's energy is reduced, the relative
backscatter is increased. The most backscatter (21%) and the most energy
lost in straggling (6%) is observed for the 0.1 KeV electrons.


Figure 3.4 N2 electron impact elastic differential cross sections
(a and b) between 0 and 20. The screened Rutherford (dashed
line) and the model 3 (solid line) are compared with the
experimental data of Shyn et al. (1972), x (the ¡a's
denote extrapolated points), and Herrmann et al. (1976),
0, at the energies of 30 eV (Figure 3.3a) and 1000 eV
(Figure 3.3b).


-34-
9 (Degrees)
Figure 3.4a


SPATIAL AND ENERGETIC ASPECTS OF ELECTRON ENERGY DEPOSITION
By
CHARLES HERBERT JACKMAN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF
IN PARTIAL FULFILLMENT OF THE
DEGREE OF DOCTOR OF
FLORIDA
REQUIREMENTS FOR THE
PHILOSOPHY
UNIVERSITY OF FLORIDA


o
Figure 5.2 The intensity of 3914 A radiation is given as a function of z
for 5000 electrons with incident energies of 1 KeV. The range,
found by extrapolating the linear portion of the histogram
(illustrated by the dashed line) to the abscissa, is 16 cm. The
I indicates the standard error involved in the MC calculation.


-19-
where
cose
Here, an electron is elastically scattered by a nucleus of charge Z
using the Coulomb potential
2
(3.3)
with r being the distance between the two particles.
Treating scattering in a quantum mechanical approach with the use
of the Born approximation and a potential of the form
(3.4)
where y is a positive but small quantity approaching 0, Eq. (3.2) can
again be derived. The Born approximation, using the potential in Eq.
(3.4), is only valid in certain angle and energy regimes (Mott and
Massey, 1965, pp. 24 and 466). At sufficiently high angles and low
energies, the Born approximation fails. The range of validity varies
for different substances and for the Born approximation is probably
not accurate at all angles for energies less than 100 eV and at large
angles for energies less than 500 eV.
Equation (3.2) does, however, go to infinity when 9 approaches 0.
This differential cross section also leads to an infinite value in the
total elastic cross section. Both of these results are unreasonable for
elastic scattering of electrons by atoms and molecules. The most popu
lar way of correcting this unreal behavior is to add a screening param
eter n. Equation (3.1) portrays this resulting form.


Fraction of Primary Energy Lost
Figure 7.2 Energy loss plot for four incident electron energies: 100 eV, solid line;
300 eV, dashed line; 1000 eV, dash-dot line; and 5000 eV, dash-dot-dot line.
-129-


-28-
model 3 which is written as
f-j(E)[l b2(E)]e"9/b(E)
Ph3<9E) = 2, b2(E)[l + e-*'"'1']
f2(E)
2tt[(2 + a)"1 a_1][l cose + a]2
[1 f-j (E) f2(E)]
2tt[(2 + c(E))"1 c(E)_1 ][1 + cose + c(E)]2
(E/fn)f]^
C(E/fll)fl2 + f13]
(E/f2i )f?2
[(E/f2i J^2 + f23]
where
f-, (E) =
f(E) =
f2(E) = 1 f-j (E)
for E > 200 eV
f2(E) = f(E)[l f-j (E)]
for E < 200 eV
b(E) = b1(^if)l>2
c c3
c(E) = Cl[l (f) ]
The parameters used for the rest of this study in Eq. (3.14) are
a = 0.11
fn = 100 eV
f12 = 0.84
f 13 = 1-92
b] = 0.43
(3.14)
b2 = -0.29


14 JULY 1978
NERDC
CARD LIST UTIL ITY
SUBROUTINE WRTDAT
C SUBROUTINE WRTDAT ***
C THIS SUBROUTINE WRITES OUT THE MATRICES THAT HAVE BEEN FILLED
C WITH INFORMATION ABOUT THE DEGRADATION PROCESS
C THIS SUBROUTINE IS ACCESSED BY THE MAIN PROGRAM
COMMON ALAB< 201.ANEI{ 5.2 0. 100) ,ALT(20> .EAVE<3) ,EIP(100) .
1 EXC550).JES(SO ).J IS(50 ) .JZE(3,50,80).JSEC< 50.80) ,KREX( 10.40.40)
2 NIE(40).NZX(20) NQFE(3,80),NCRO(10),RHO(40),SA(3,80).SIG(100).
3 TOTE(3.80), TN 1 < 3,30).TN5( 3,80) ,TN10< 3,80) ,WIE(50).WIS(50),
4 YSPEC!100).2NUM(10).ZALT(80),NUMGAS.NAR1,NVR,NJMST,NUMG.NRM,
5 NSEC,NSPZ.NEIP.NE XC NTOP,NAR,NSPEC,NRHO,NAVEE,NPRIM.EIN,
6 ZVAL.EMIN,DENGS,M,NSPUN, NSPR, ANER( 1, 10,20.100),RYS{ 10) .NSPRO
C
C
C WRITE OUT ALL THE STATISTICS YOU NEED HERE
DO 1 K1 ,3
DO 1 1= 1 NAR 1
DO 1 J=1 NU MG
C WE HAVE SUPPLEMENTED THE 3914A EMISSION WITH 3RD AND 4 T H STATES
IF ( J EQo 3 ) GO TO 6655
IF ( J EQo 4 ) GO TO 6655
IF(J oEQo 5)GO TO 6655
GO TO 767
6655 TN 5(K,I )=TN5(K.1 )+JZE < K,J I)
757 IF (J NEa 10 )G0 TO 1
TNI 0 ( K I )=TN10(KI )+J ZE(K,J,I)
1 CONTINUE
DO 4 K=2.3
DO 4 1 = 1,NA R1
00 4 J= 1,NTOP
TN5(K,I )=TN5(K,I )+ JSEC(J. I )*EXC5(J )
4 CONTINUE
DO 46 K=1,3
IF(K EQo 1 )WRIT E(6 ,780)
IF ( K EQo 2)WRITE<6,781)
IF{K oEGo 3 )WRITE(6,782)
780 FORMAT{1PRI MARY COLLISIONS*,/)
781 FORMAT C 1SECONDARY COLLISIONS',/)
782 FORMATC1PRIMARY PLUS SECONDARY COLLISIONS*,/)
WR ITE(6,755)
755 FORMAT( ALT INTERVAL ,QX, ALT VALUE*,9X,3914A EXC*.8X,
* 'ELASTIC EXC ,/)
DO 46 1=2.NAR
I M1 = I 1
ZM V=(ZALT( I)+ZAL T( I Ml ) )/2o
WRTDAT 01
WRTDAT02
WRTDAT03
WRTDAT04
WRTDAT05
WRTDAT 06
,WRTDAT 07
WRTDAT08
WRTDAT09
WRTDAT10
WRTDAT11
WRTDAT12
WRTDAT13
WRTDAT14
WRTDAT15
WRTDAT16
WRTDAT17
WRTDAT18
WRTDAT 19
WRTDAT20
WRTDAT21
WRTDAT22
WRTDAT23
WRTDAT 24
WRTDAT25
WRTDAT26
WRTDAT27
WRT0AT28
WRTDAT29
WRTDAT 30
WR TDAT31
WRTDAT 32
WRTDAT 33
WRTDA T34
WRTDAT35
WRTDAT36
WRTDAT37
WRTDAT38
WRTDAT39
WRT DAT 40
WRTDAT 41
WRTDAT42
WRT DAT 43
WRTDAT44
WRTDAT 45
-214-


-144-
The U(E,p,z,EQ) was computed in a manner similar to the way that
U(E,z,E0) was computed. In subsection VII.B.1 a rectangle AEby AZjnt
was taken as the area of interest. Here, a volume AZjnt by AEjnt by AP^rea
2 2
is used. The APApea is in units of area [(gm/cm ) ] and is defined as
Area
tt[(p +
Ap
Int\2
) (p -
ApIntx2
n
where p is the mid-point of the area of interest and Apjnt is the radial
interval of interest.
If the spatial yield spectrum U(EE, p£, ze,Eq) at a certain energy
value Ee, longitudinal distance z^, and radial distance p£ for an elec
tron of incident energy Eq is desired, then the volume with energy width
endpoint coordinates E^ (AEInt/2) and E^ + (^ longitudinal
endpoint coordinates z^ (AZjnt/2) and z£ + (AZjnt/2), and radial end
point coordinates p^ (Ap^/2) and p£ + (APjnt/2) is established. If
the longitudinal distance, z, is between z£ (AzInt/2) and z£ + (AZjnt/2)
the radial distance, pc, is between pe (ApInt/2) and p£ + (AP¡nt/2),
and the energy before the collision, E^ is between E£ (aEj^/2) and
EE + ^Int^ for an ''ne^as'tlc collision; then the number of electrons
in that volume, N(E^,p^,z^) is incremented by one.
2 3
The spatial yield spectrum [in #/eV/(gm/cm ) ] is then written as
u(EE,pEzEE0) ~ aE
N(E^,p^,z^)
Int APArea AzInt
(7.11)
This process then continues for each small volume across the entire volume
of interest. Again, it should be recognized that this yield spectrum is
normalized to one electron.


-68-
1975). In this study the fraction of ionization is assumed to be
negligible; therefore, this loss is ignored.
Unless there is a very large amount of money available for computer
time, an electron can not be followed to its thermal energy with any
practicality. This implies that a multiple elastic scattering distri
bution (hereafter referred to as MESD) must be used below some given
energy. In this work the MESD will be used below 30 eV.
Bethe, Rose, and Smith (1938) used the Fokker-Planck differential
equation, neglecting energy loss, to consider the penetration of elec
trons through thick plates. This, however, leads to a Gaussian solution
in the small-angle approximation so that the tail of the angular dis
tribution was omitted. The large-angle multiple scattering has been
studied by Goudsmit and Saunderson (1940) [hereafter referred to as GS]
who used a series of Legendre polynomials to determine the resultant
angle of scattering.
Lewis (1950) studied the integro-differential diffusion equation of
the multiple scattering problem in an infinite, homogeneous medium,
without the usual small-angle approximation. He obtained the GS solu
tion for the scattering angle and also derived certain moments for the
longitudinal and transverse distributions.
Berger (1963) applied a MESD for condensed case history MC cal
culations. His application of the MESD is at the energies above 200 eV
and probably is not accurate for electrons with energies less than about
100 eV. Furthermore, Berger's (1963) work contains approximations that
are only good for the sharply forward peaked cross sections of higher
energy electrons.


14 JULY 1978
NERDC
CARD L1ST UTILITY
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
S3
SUBROUTINE RDTAPE
SUBROUTINE RDTAPE ***
THIS SUBROUTINE IS ACCESSED BY THE MAIN PROGRAMo
THIS SUBROUTINE READS OFF OF THE TAPE AND PLACES THE EVENTS INTO
THEIR PROPER INTERVALS AND BOXES*
COMMON ALABI 20) ANEK 5*2 0, 100) ,ALT(20) EAVE(3> ,EIP( 100) .
1 EXC5(50)JES(50 )J IS(50 ),JZE(35080),JSEC(5080)KREX(1040.40)
2 NIE(40).NZX20) ,NOF =(3.80).NCRO(10)RHO(40).SA(3,80).S1G(100),
3 TOTE(3,80 ) .TNI( 3.80).TN5( 3,80) TN10(3.80) .WIE(50) .WI S(50).
4 YSPEC(100),ZNUM(IO).ZALT(80).NUMGAS.NAR1,NVR,NUMST.NUMG.NRM,
5 NSEC.NSPZ.NEIP.NE XC,NTOP.NAR,NSPEC,NRHO,NAVEE.NPRIM.ElN,
6 ZVAL.EMIN,DENGS.M,NSPUN,NSPR, ANER(1.10.20,100),RYS(10).NSPR0
CONST=DENGS*DENGS*2l 6 3E-3 5
CONST IS USED IN FINDING THE RADIAL YIELD SPECTRA WHICH MUST
BE DIVIDED BY AN AREA.
(4*651E23 GM/MOLECULE)**2 (l*E+5 CM/KM)**2 = 2el63E-35
N1 Z 0
DO 30 1=1 NEXC
ISA V=I
ELOSS=0o OE 0
ELOSS IS THE ENERGY LOSS BY EACH ELECTRON
RE AD(M.END = 3 33)NCHE
IF (N CHE EO 0) GO TO 1131
THIS IF STATEMENT DETERMINES IF THE ELECTRON HAS DROPPED BELOW THE
ENERGY CUTOFF*
IF NCHE=1. THEN WE KNOW THERE IS AN ELECTRON TO BE PUT INTO ITS
PROPER BIN BELOW THE ENERGY CUTOFF*
READ(M ) WL S,Z
EL OSS =ELOS S +WLS
WL S=ENE RGY OF THE ELECTRON
Z= ALTITUDE IN KM*
DO 83 II=1.NTOP
I1P1=I1+1
IFWLS *GT. WIS(Il) AND* WLS *L T WIS( I IPl ) )I 1 = 1 1
CONTINUE
IF(Z *GTt ZALT( 1))JSEC(I I 1)=JSEC{ I I .1 ) + l
IF ( Z *LT* ZALT(NAR) )JSEC( I I.NARl )=JSEC( II.NARl )+1
DO 84 12=2.NAR
I2M1= 12-1
IF(Z *GT* Z ALT(12) AND* Z *LT ZALT( I 2M1 ) )JSEC( I I. 12 )=
* JSEC I I I 2)+l
CONT INUE
JI S( I I>=JIS( I I) + 1
RDTAPE01
RDTAPE 02
RDTAPE03
RDTAPE04
RDTAPE05
RDTAPE06
,RDTAPE 07
RDT APEOS
RDTAPE09
RDTAPE10
RDT APE 1 1
RDTAPE12
RDTAPE13
RDTAPE14
RDT APE 15
RDTAPE16
RDTAPE17
RDTAPE18
RDTAPE19
RDTAPE20
R DT APE 21
RDTAPE22
RDTAPE23
RDT APE 24
RDTAPE25
RD TAPE 26
RDTAPE27
RDTAPE28
RDTAPE29
RDTAPE30
RDTAPE31
RDT APE 32
RDTAPE33
RDTAPE 34
RDTAPE35
RDTAPE36
RDTAPE37
RDTAPE38
RD TAPE 39
RDTAPE40
RDTAPE41
RDTAPE42
RDTAPE43
RDTAPE44
RDTAPE45
84
-208-


Page
APPENDIX
A MONTE CARLO PROGRAM 155
B GETDAT PROGRAM 200
REFERENCES 220
BIOGRAPHICAL SKETCH 227
v


-11-
1
r l nk[o
k kn
a + pee] *
1 r k k + q
£ nk^eae^ 2

where
q+(E,z)
+ [1
= Ink(z) J (P^(E,)cakj(E'-)*-(E,z)
E>E
- P^E'Jla^E'-^J^E'.z)}
aj dj
q"(E,z)
+ [1
= 1 n (z) l {p^ (E* )a^ .(E'^E)d, (E' ,z)
k k j aj aj
E>E
paj(E,)]aaj(E'"E)<¡,"(E,z)}
(2.8)
(2.9)
(2.10)
(2.11)
and z is the distance along a magnetic field line (positive outward);
n^z) is the kth neutral species number density; Pe(E) is the electron
backscatter probability for elastic collisions with the kth neutral
species; a (E) is the electron total scattering cross section for the
kth neutral species; q(E,z) is the electron production rate in the
range E to E+dE due to ionization processes (both electron ionization
+
and photoionization); q" is the electron production in the range E to
E+dE due to cascading from higher-energy electrons undergoing inelastic
i/
collisions; p is the electron backscatter probability for collisions
J
with the kth neutral species resulting in the jth inelastic process; and
a is the inelastic cross section for the jth excitation of the kth neutral
species.


END
SDIFM046
-195-


CHAPTER VI
SENSITIVITY STUDY OF THE SPATIAL ELECTRON ENERGY DEGRADATION
In section V.D the ranges from two separate models of the elastic
differential cross section have been compared. A sensitivity study of
the influence of other differential cross sections on the electron energy
deposition is the subject of this chapter. The effects of the ionization
differential cross sections on the intensity distribution are considered
in section VI.A. Section VI.B then discusses the influence of the in
elastic differential cross sections on the intensity distribution.
In sections VI.C and VI.D, several different elastic phase func
tions are compared. (The elastic collisions cause more scattering than
the inelastic collisions at any electron energy.) Section VI.C includes
a calculation with no energy loss, while section VI.D discusses the
influence of several variations of the model 1 phase function on the
electron energy deposition.
As illustrated in sections VI.A through VI.D, the scattering phase
functions are quite important in determining the electron energy
deposition intensity or collision profiles. The total elastic cross
sections are also of some significance in determining the intensity
profiles and will be discussed in section VI.E.
-98-


-76-
at maximum r^
approximated by
% 4000.
Using this information, Eq. (4.35) can then be
v N pjk rjk <4-36>
+ 2 +
and the statistical standard deviation of the number of B zu events
in a bin becomes
jk '^jk ^4-37^
Equation (4.37) holds true for the specific example represented in
Figure 5.2 and it also holds true for all the intensity plots, energy
loss plots, and yield spectra that were studied in this work. Thus,
in order to obtain the approximate standard deviation for any MC generated
number, the square root of this value is its standard deviation. The
error bars found in the rest of this paper are determined in this
manner.
Now that the MC calculational technique has been outlined, this
method will be used in the next three chapters to deal with the spatial
and energetic aspects of electron energy degradation.


-106-
0 30 60 90 120 150 180
9 (Degrees)
Figure 6.2 Differential cross section graph for model 1 trials
A1, A2, A3, A4, and A5.


-128-
Figure 7.1 Energy remaining plot for electrons with energies incident
at 1 KeV. The A's give the calculation of Barrett and Hays
(1976), the o's give the values from Stolarski (1968), and
the solid line gives the average energy left at various z
values from this MC work.


-44-
6 (Degrees)
Figure 3.7 N2 experimental ionization doubly differential cross sections
from Opal, Beaty, and Peterson (1972) are represented by x's.
The solid line ( ) denotes the cross section resulting from
the use of Eq. (3.18). The incident electron energy is de
noted by E (eV) and the secondary electron energy is denoted
by T (eV).


0.2 0.4 0.6 0.0 0.2 0.4 0.6
3914 A Intensity (|03 Exc.)
-176"


-108-
Figure 6.3 Collision plots for MC trials A1, A3, and A5. The histograms
represent the MC data while the os represent the fit using
Eq. (6.2).


-63-
Table 4.3 N2 inelastic states,
W, taken for electron
below.
their probabilities
energies above 200
, p, and thresholds,
eV are presented
State
P
W (eV)
It, bu
0.092
12.80
N2 b''<
0.042
14.00
Composite
0.233
15.40
N2
0.289
15.58
N2 a2u
0.127
16.73
N2
0.066
18.75
N2 D%
0.044
22.00
N2
0.044
23.60
N* 40 eV
0.063
40.00


-51-
it is scattered and sent back to the START to be degraded further. In
the meantime, the primary electron's properties are stored.
If a secondary produces a tertiary electron with a kinetic energy
greater than Ec, then that tertiary is completely degraded before any
further degradation of the secondary is considered. Like the primary,
the secondary electron's properties are stored in the meantime. No other
generations were included in this study as their contribution would be,
at most, a couple of tenths of a percent of the incident electron's
energy.
After the tertiary is entirely degraded below Ec> then the secondary
is again sent back to the START to be degraded further. The secondary
is next entirely degraded below Ec> and, finally, the primary is again
sent back to the START to be further degraded. This whole process may
then again repeat itself.
B. Computer Programs and Machinery Used in the
Monte Carlo Calculation
In the previous section a brief discussion was given of the electron
energy degradation process. A brief discussion will be given below about
the MC computer codes and the computing machinery used. The MC computer
program employed in this work evolved from an original MC code written
by R.T. Brinkmann (see applications in Brinkmann and Trajmar, 1970). This
program was revised for use in Heaps and Green (1974), Kutcher and Green
(1976), and Riewe and Green (1978). The author has further modified this
MC technique for energetic electron impact into N£ to be used in the
energy range from 2 eV to 5 KeV.


nnnnn**
14 JULY 1978
NERDC
CARD LIST UTILITY
C
c
c
c
c
c
793
850
79 5
80 0
C
C
C
803
790
1 44
C
EDEL=EIP(J)-EIP(JM1 >
THIS IS FOR THE LONGITUDINAL YIELD SPECTRA
YSPEC(JM1 )=ANEI< 1. I.JM1 )/EDEL/NPRIM/ZDEL
NORMALIZE THE YIELD SPECTRA TO ONE ELECTRON
YSPEC( JM 1 ) IS IN UNITS OF #/EV/(GM/CM**2)o
ANEI (1 I, JM1 ) HAS THE EXTRA DIMENSION OF THE MATRIX IN CASE A
DEGRADATION SPECTRA OR SOME OTHER FLUX IS DESIRED
IF{J EO N EIP ) YSP EC(JM1)=Y SPEC(JMI) *EDEL
THIS EXPRESSION TAKES CARE OF THOSE ELECTRONS IN THE SOURCE TERM
WR I TEC 6,793)EMID .YSPEC(JMI)
FORMAT(1 PE113*5X. 1PE11.3)
IF ( YSPEC (JMI ) .L To 1 o E3 0 ) GO TO 795
IF(NSPUN o EQ 1 )WRITE( 7. 85 0)E MID.ZRNG.EIN.YSPECCJMl )
FORMAT(2X.4EI25)
CONTI NUE
IF (NSPRO oNE. 1 )GO TO 790
DO 803 IR=2NSPR
IR 1=IR- 1
RM ID= ( RYS ( IRI + RYSC IP1) )/2,*lE+5
WRITE(6.80 0 ) ZMV.RMID
FORMAT( 1AT A Z VALUE = 1 PE 1 1 3 CM ,
*2X,*AND AT A RHO V ALU E= 1 PE 1 1 3. CM WE HAVE THE FOLLOWING *.
* / ELECTRON SPECTRA, YIELD SPECTRA = UCE) IN 0/EV/(GM/CM**2)**3
*.//, MID-ENERGY ,8X, U( E ) .//)
DO 803 JR=2.NEIP
JRM1=JR-I
WRTDAT73
WRTDAT74
WRTDAT75
WRTDAT76
W RT DAT 77
WRTDAT78
WRTDAT79
WRTDAT80
WRTDAT81
WRTDAT82
WRTDAT 83
WRTDAT84
WRTDAT85
WRTDAT86
WRTDAT 87
WRTDAT 88
WRTDAT 89
W RT DAT 90
WRTDAT91
WRTDAT92
WRTDAT93
WRTDA T 94
WRTDAT95
WRTDAT9
WRTDAT97
WRTDAT98
EMID=(EIPCJRMl)+EIPCJR))/2
EDEL=E IP(JR) E IP ( JR Ml )
THIS IS FOR THE RADIAL YIELD SPECTRAo
YSRHO=ANER(1 *IR1 .I.JRMl)/EDEL/NPRIM/ZDEL
YSRHO HAS THE DIMENSIONS OF H/E V/ ( GM/CM* *2 ) ** 3
IF (J R .ECU NEIP)YSRHO = YSP.HO*EDEL
THIS EXPRESSION TAKES CARE OF THE SOURCE TERM
WRITE C 6,793)EM ID .YSRHO
CONT I NUE
CONTINUE
CONTINUE
WRITE OUT A QUICK ANO DIRTY ENERGY CONSERVATION COMPARISON
THIS IS NOT ACCURATE FOR REASONS GIVEN BELOW
ESI=NPRIM*EIN
ESI IS THE INCIDENT ENERGY
ESF=O.OEO
WRTDAT99
WRTDA100
WRTDA101
WRTDA102
WRTDA103
WRTDA104
WRTDA1 05
WRTDA106
WRTDA107
WR TDA108
WRTDA109
WRTDA 110
WRTDA111
WRTDA112
WRTDA 113
WRTDA114
WRTDA115
WRTDA 1 16
WRTDA117
-216-


>0 n
605
1
609
C
c
50 4
1 1
61 3
C
C
WRITE(6.605)G1.G2.G3.G4,G5,ALPE(I) ,BETE(I),CE( I),FE ( I ).WE(I)
FORMAT(IX.5A4,9F8 3)
CONT INUE
WRITE(6,609)
FORMAT!////, THE P(ET) PARAMETERS FOLLOW*//6X,'GAS',14X,
7X,J 1
TA* 7Xi
6X,*JB*,6X,*JC',7X,'GAMMAS
TB* ,/)
).
THE SECONDARY
1 'THRESH*,6X,*K,5X,KB*
1 2X,'GAMMAB*,6X,TS*,SX,
DO 3 1=1 NUMGAS
RE AD( 5,502) G1 .G2*G3,G4,G5,THRESH( I) ,(AK(K,I),K=1,2
1 (AJ(J, I ),J = 1,3 )
RE AD(5.504) (GAMA(K.I) ,K = 1 ,2),
READ IN THE P(E,T) PARAMETERS,, THESE ARE USED TO FIND
ELECTRON ENERGY AFTER AN IONIZATION EVENT,
FORM AT(20X,5E8o0)
WRITE(6.611 ) G1.G2.G3, G4.G5.THRE SH( I ) , (AJ(J,I) *J=1 ,3 ),(GAM A(K I),K=l,2),(TO(K,I),K = l,3)
FORMA T( IX. 5A4,F8o 3, 1 X. 5F 3, 3.2X.2F8, 3 ,F1 Oo 3 ,F8o 1 ,F8,3)
CONT INUE
WRITE (6.613)
FORMAT(////, THE DOUBLY DIFFERENTIAL IONIZATION CROSS SECTION
AMETERS ARE READ IN NOW *.//,6X. GAS,14X, I/B1 *.7X, AT/B2 *,
l 7X, *A 1/B3* ,5X A2/B4 .6X A3/B5 6X A4/G1 6X A5/G2 7X .
1
1 *A6/G3*.7X,*/G4*,8X, */G5,//)
DO 4 1=1.NUMGAS
RE AD (5. 56 2) Gl,G2,G3.G4,G5.PION(I) ,AT(I) ( A(K I ) .
READ IN THE DOUBLY DIFFERENTIAL IONIZATION CROSS SE
FOR THE PRIMARY SCATTERING,
K=1,4)
CTION PARAMETERS
DAT A0046
OATA0047
DATA0048
DATA0049
DAT A0050
DATA0051
DATA0052
DATA0053
DA T A 00 54
DAT AOO 55
DATA0056
DATAO057
DA TA 0058
DAT A 0 059
DAT A0060
DA TA 0061
DAT A0062
0ATA0063
DA T A 0 064
PAR DAT AO065
DATA0066
DAT AO 067
OAT A0068
DATA0069
DAT A0070
DA TA 0071
DATA0072
-69L-


-130-
The fraction of energy backscattered is also of interest. The
backscattering of electrons from the ionosphere has been observed in
rocket experiments by McDiarmid, Rose, and Budzinski (1961) and in the
Injun III satellite experiments of O'Brien (1964). Berger, Seltzer, and
Maeda (1974) [BSM] have calculated backscattering coefficients for mono-
energetic electrons incident on a semi-infinite air medium at energies
from 104 KeV down to 2 KeV.
One quantity calculated by BSM is R^, the energy albedo (computed by
summing the energy backscattered). Since the incident energy range used
in this work overlaps the incident energy range used by BSM from 2 to 5
KeV, a comparison of the R^'s from both calculations is of interest.
Table 7.1 presents the results from the model 3 (hereafter called
M3) and the screened Rutherford (hereafter called SR) phase functions
and the work of BSM. The M3 energy albedos are lower than those energy
albedos resulting from the SR and the BSM calculations (at least where
there are values available) down to the energy of 0.1 KeV. At this
energy the M3 phase function reveals a fairly substantial backscatter
with approximately one-fifth of the incident energy lost in backscatter.
Although little consideration is given to the backscattered elec
trons in this study, there is much information that can be derived from
studying these backscattered particles in detail. This detailed spatial
MC technique would be an appropriate method of studying these back-
scattered particles.
B. Spatial Yield Spectra for Electrons Impinging on N2
The yield spectrum for an electron energy degradation process con
tains all the information necessary for computing excitations from that


-60-
Table 4.2 Twenty-four angle intervals are given here that were used in
the Monte Carlo calculation. First column lists the index of
the segment and the second and third columns give the begin
ning and end points for each segment with units of radians
(degrees).
Index
Beginning
End
1
0.00
(0.00)
0.01
(0.57)
2
0.01
(0.57)
0.05
(2.87)
3
0.05
(2.87)
0.11
(6.30)
4
0.11
(6.30)
0.20
(11.46)
5
0.20
(11.46)
0.40
(22.92)
6
0.40
(22.92)
0.60
(34.38)
7
0.60
(34.38)
0.80
(45.84)
8
0.80
(45.84)
0.90
(51.57)
9
0.90
(51.57)
1.00
(57.30)
10
1.00
(57.30)
1.10
(63.03)
11
1.10
(63.03)
1.20
(68.75)
12
1.20
(68.75)
1.30
(74.48)
13
1.30
(74.48)
1.40
(80.21)
14
1.40
(80.21)
1.50
(85.94)
15
1.50
(85.94)
1.60
(91.67)
16
1.60
(91.67)
1.80 (103.13)
17
1.80 (103.13)
2.00
(114.59)
18
2.00 (114.59)
2.20
(126.05)
19
2.20 (126.05)
2.40 (137.51 )
20
2.40 (137.51 )
2.60 (148.97)
21
2.60 (148.97)
2.80 (160.43)
22
2.80 (160.43)
3.00
(171.89)
23
3.00
(171.89)
3.07 (175.90)
24
3.07 (175.90)
3.14 (180.00)


-26-
phenomenalogically to describe the low energy shape and Feshbach reso
nances.
If either Eq. (3.8) or (3.9) is used as the total elastic cross
section, the differential elastic cross section must be normalized such
that:
2ir tt
/ / P(e,E) sineded 0 0
where P(e,E) is the phase function and the differential cross section can
be written as
o(E) P(e,E) (3.11)
With this in mind consider now three separate phase function forms.
The first phase function form is very similar to the screened Rutherford
cross section and it is written here as
PM1(e,E) = IT-^ (3.12)
m 2ir[(2 + a(E)) 1 a(E) 1 ][1 cose + a(E)]
This is known as model 1. The parameter "a" acts in a similar manner to
the "2n" term in the denominator of the screened Rutherford cross sec
tion form and is written
a(E) = a1
_E_\a2
1 eVj
The second phase function form (model 2) includes the forward
scattering term of Eq. (3.12) along with a backscattering term and is
given as


-114-
The screened Rutherford and model 3 phase functions have somewhat
different forms. Thus it is difficult to compare them in ways other
than the way they were compared in section V.D.
A more convenient phase function form to use for comparison is that
of model 1. Model 1 depends on the one screening parameter, a, which
a2
can be written as a function of the energy such that a(E) = a^(E/l eV) .
When a.j = 32 and a2 = -1, this form simulates the screened Rutherford
phase function and a(E) in this case is represented in Figure 6.6 by the
sol id 1ine (trial Cl).
Four other representations of the parameter a(E) are given in
Figure 6.6. These five trials represent attempts to characterize the
influence of this screening parameter on the energy deposition process.
All five trials (Cl, C2, C3, C4, and C5) used 1000 perpendicularly in
cident electrons with energies of 1 KeV and the trials and their para
meters are given in Table. 6.3.
Trials C2 and C3 were attempts to detect the influence of the
starting screening parameter a(1000 eV) on the energy deposition. For
trial C2 a(1000 eV) [from Table 6.3] is a factor of ten lower than
a(1000 eV) for trial Cl. For trial C3 a(1000 eV) [from Table 6.3] is a
factor of ten higher than the a(1000 eV) for trial Cl. However, for
trials Cl, C2, and C3 the a(30 eV) values are the same.
The range and fraction of incident energy backscattered obtained
from trials Cl, C2, and C3 (and also trials C4 and C5) are given in
Table 6.4. The range from trial Cl is 38% lower than the range from
trial C2 and 38% higher than the range from trial C3. The symmetry of
these results is remarkable and probably fortuitous.


-142-
This yield of 0 atoms was then used to predict the variation of polar
mesospheric oxygen and ozone during auroral events.
Shemansky, Donahue, and Zipf (1972), however, showed that Maeda's
spectra are deficient in low-energy degraded primary electrons. This
conclusion was also supported by BSM.
The spatial yield spectrum calculated with the use of this MC approach
should be quite accurate statistically from 2 eV up to the incident
energy E There may be some errors inherent in the assumptions and
approximations used in these MC calculations however. The analytic
spatial yield spectrum given by Eq. (7.6) does represent fairly well the
actual spatial yield spectrum. Thus the analytic spatial yield spectrum
can be applied to some of the problems in aeronomy involving impinging
electrons into the atmosphere.
Consider now the use of Eq. (7.6) with an incident electron energy
p
flux of <¡>(Eq) [in units of #/cm /sec/eV]. A yield Jj[z,(Eo)] in units
3
of #/cm /sec can be calculated using
Eui
Uz.tE )] = / / 4>(E )U(E,z,E )p(z)dEdE (7.9)
J 0 W. W, 0 0 0
J U
3
where p(z) is the density (in gm/cm ) of the air at altitude z. Equation
(7.9) is not applied by the author to any given flux 4>(E ) in this work.
Future studies can make use of Eq. (7.9) in applications to aurorae and
their effects on the atmosphere.
The spatial yield spectrum for ^ can be used fairly accurately for
problems dealing with the atmosphere in spite of the fact that the
atmosphere is not entirely molecular nitrogen. In Green, Jackman, and
Garvey (1977) the two variable yield spectrum U(E,Eq) was observed to be


(A /Sr)
Q (Degrees)
Figure 3.3a


-38-
all owed or an optically forbidden transition. This transition for many
molecules leads to a repulsive state which can dissociate the molecule.
In N^, dissociation of the molecule in this manner is virtually non-
existant because is a very stable homonuclear molecule in which both
the singlet and triplet states are found to be strongly bound. As a
consequence of this, the main process for dissociation is through pre
dissociation of stable electronic terms by repulsive states that are
themselves strongly optically forbidden in direct excitation.
Porter, Jackman, and Green (1976) (hereafter called PJG) compiled
branching ratios for dissociation of using several experimental and
theoretical papers (see, for example, Winters, 1966; Rapp, Englander-
Golden, and Briglia, 1965; Polak, Slovetskii, and Sokolov, 1972; and
Mumma and Zipf, 1971). In PJG the efficiencies for the production of
atomic nitrogen from proton impact were determined.
This study does not include a calculation of the atomic nitrogen
production; however, the PJG branching ratios with the yield spectra,
discussed in section VII.B, can be applied to calculate this atomic
yield. The excitation events, not resulting in the dissociation of the
Ng molecule, are either electronic or vibrational transitions. Cross
sections for these transitions are taken from both PJG and Jackman,
Garvey, and Green (hereafter called JGG) (1977b).
In the ionization event an electron is stripped from the molecule
and given some kinetic energy. The ionization cros^s section is a sub
stantial portion of the total inelastic cross section above 50 eV
(compare Figures 3.6 and 5.1) and the total amount of energy loss is
always > the lowest ionization threshold (which is 15.58 eV for N^).
Subsequently, most of the energy loss of the electrons (for energies


-122-
P
Figure 6.8 At the cut z = 0.3, the 3914 A intensity distribution is
given as a function of p for the three trials Cl, x; C2, A;
and C2, 0.


-131-
Table 7.1 Energy albedos presented at five energies E0 with the use of
the model 3 (column labeled M3) and the screened Rutherford
(column labeled SR) phase functions and the work of BSM.
Eq (KeV)
M3
SR
BSM
0.1
0.210
0.188
--
0.3
0.072
0.105
--
1.0
0.051
0.078
--
2.0
0.041
0.068
0.062
5.0
0.039
0.045
0.052


-78-
independent of the energy of the exciting electron for energies from
30 eV at least up to 3 KeV.
In Figure 5.1 the N2 total ionization cross section and cross sec-
2 + +
tion for ionization and excitation to the B Zu state of are presented.
The curves are approximately parallel thus even if the absolute values
for the two cross sections are slightly in error, the shapes of the in
tensity plots that result from this MC calculation should be fairly
accurate.
The total ionization curve lies nicely in the middle of an array of
experiments (namely, Opal, Beaty, and Peterson, 1972; Tate and Smith,
1932; Rapp and Englander-Golden, 1965; and Schram, de Heer, Wiel, and
Kistenaker, 1965) but the B cross section values may be high when
compared to experiments (see Holland, 1967; and McConkey, Woolsey, and
Burns, 1967).
2 +
The threshold for excitation to this B Zu state is 18.75 eV, thus
any electron above that energy can excite and ionize a ground state
molecule up to this level. The cross section for excitation and ioniza-
2 +
tion to the B £u state is not large when compared with the total in
elastic cross section. In fact, the probability for exciting this state
is only 0.066 for electron energies above 200 eV. The accuracy of the
MC calculation is dependent on the number of excitations in each bin
(see section IV.D). In order to enhance the precision of the MC results,
o + o 2 |
excitations of the X £ and A n states of N0 are added to the B e
g u 2 u
excitations. The ionization cross sections for these two states are
2 +
found to be proportional to the B state for electron energies above
30 eV.


CHAPTER IV
THE MONTE CARLO METHOD OF ENERGY DEPOSITION BY ELECTRONS
IN MOLECULAR NITROGEN
A Monte Carlo calculation is used here for energy degradation by
energetic electrons in Ng. This stochastic process is probably the most
accurate method for obtaining the energy loss of particles in a medium.
The generalizations about electron impact on N£ that are made through the
use of this technique can be applied in other energy deposition approaches
to electrons impinging on the atmosphere. (This is true even though the
magnetic field is neglected in these MC calculations. The three dimen
sional yield spectrum U(E,z,EQ) [see Chapter VII] is most useful for
applications to the atmosphere and changes in the magnetic field will
not affect the yield spectra greatly below altitudes of about 300 km
where the gas density is fairly high [see Berger, Seltzer, and Maeda,
1970 and 1974].)
Building on the MC work done in this paper, more exact models of
auroral electrons and photoelectrons can be established. In the first
section, IV.A, a brief discussion of the MC calculation outlines the
general procedure involved in the degradation process. The computer
program and machinery used are briefly described in section IV.B. Sec
tion IV.C relates in detail the various aspects of the calculation.
Finally, section IV.D discusses the statistical error that arises from
the use of the MC calculation.
-47-


-25-
Using the form
q F[1 -
o(E) 1
EC W
(3.8)
on 9
implemented first by Green and Barth (1965), where qQ = 651.3 A£ eV ,
the total elastic cross section for Ng was characterized fairly well in
the range from 30 to 1000 eV using the parameters a = 1, 3 = 0.6,
c = 0.64, F = 0.43, and W = 2.66. The e"0'64 dependence of Eq. (3.8)
at the larger energies is similar to that seen by Wedde and Strand (1974)
for N^. This form does not represent the data as well below 30 eV and,
in fact, is not defined below an energy of 2.66 eV.
Porter and Jump (1978) have used a more complex total elastic cross
section form which is written as
r-X
a(E)
n(n + 1)[V2+X + E2+X]
figi
f/2
(E-E,)2-^ (E-E2)+G2
(3.9)
Here, n = j
T =
2.5 x 10"6 cm2
Fi
= 7.33
U =
1.95 x 10 3 eV
Ei
= 2.47
eV
V =
150 eV
G2
= 24.3
eV
X =
-0.770
F2
= 2.71
G1 :
= 0.544 eV
E2
= 15.5
eV
In the large energy limit, the total cross section falls off as 1/E,
similar to the screened Rutherford cross section. This form does con
tain two other terms (the second and third terms) which were introduced


-20-
Equation (3.1) has a maximum at 0 = 0 and a minimum at e = 180.
At energies below 200 eV, experimental results indicate a minimum in the
elastic differential cross sections at about 90 with a strong forward
scattering peak at e = 0 and a secondary backward scattering peak at
9 = 180.
In Figure 3.1 experimental data for energies at E = 30 and 70 eV are
presented. These data are taken from Shyn, Stolarski, and Carignan (1972)
with the small circles denoting 30 eV points and the crosses denoting
the 70 eV data.
Later on in this section the screened Rutherford cross section and
another analytic model of the differential elastic cross section are
compared with experimental data. Before discussing the differential
cross section in more detail, first, consider the total elastic cross
section.
Several experiments have been performed deriving the total elastic
cross sections for N^. There have also been several theoretical studies
on the N2 elastic total cross sections. Two recent reviews of the data
available on this subject are Kieffer (1971) and Wedde (1976).
A plot of all this data would obscure the analytic total cross
sections specifically considered in this work. Consequently, only data
from Sawada, Ganas, and Green (1974) (theoretical), Shyn, Stolarski, and
Carignan (1972) (experimental), and Herrmann, Jost, and Kessler (1976)
(experimental) are plotted in Figure 3.2. The sets of data overlap to
a degree such that the disagreement in absolute magnitude of the total
cross sections is readily apparent.
In view of this disagreement, no experimental or theoretical data
are assumed to be absolutely correct and some average of this data is


Figure 7.4 The three variable spatial yield spectrum U(E,z,E0) is
plotted as function of Er. The MC calculations are repre
sented by symbols for each z (in fractions of the range)
and E0 (in KeV): o, z = 0.126, E0 = 0.1; A, z = 0.316,
E0 = 0.3; x, z = 0.606, E0 = 1.0; v, z = 0.928, E0 = 2.0;
and z = 1.052, E0 = 5.0. The analytic fit using Eq. (7.6)
is represented by the solid line with the source term con
tribution represented by a.


14 JULY 1978
NEROC
CARD LIST UTILITY
SUBROUTINE DFDW C SUBROUTINE DFDW ***
C THIS SUBROUTINE CAN BE USED TO HELP CALCULATE THE POLAR SCATTERING
C ANGLE OF THE PRIMARY ELECTRON IN AN IONIZATION EVENT*
C THIS SUBROUTINE IS NOT ACCESSED BY MC AND IS INCLUDED ONLY
C BECAUSE IT CAN BE EASILY ADAPTED TO BE ACCESSED BY MC*
C THIS SUBROUTINE CALLS THE FOLLOWING FUNCTION:
C l ) DC S
NG=NSCS*2 1
SUM2=0.0
NA=NX-1
IF(N X1)44 33, 11
t 1 DO 22 1=2,NX
SUM1 = SUM2
XL=T HET C I 1 )
XU=THET(I)
AA = *5*( XU+XL )
B8=XU-XL
CC=* 4305682 *BB
Z= AA+CC
G1=DCS(E,W,NG,Z)
Z=AACC
G2=DCS(E,W,NG,Z)
Y=.1739274 *(G1+G2)
CC=o1699905*BB
Z= AA + CC
G1=DCS(E,W,NG,Z)
Z-AA-CC
G2 =DCS (E,WNG, Z)
Y=BB* SUM2=SUM2+Y
22 F(1-1)=SUM1
33 F(NX)=SUM2
44 CONTINUE
DO 4 J=1 .NX
F( J) = F( J)/F(NX)
4 CONTINUE
F< 1) = 00
F(NX)=1*0
C THE ARRAY F(I> OF SCATTERING PROBABILITIES WITHIN CERTAIN
C ANGULAR BINS IS ESTABLISHED*
RETURN
END
DFDW0001
DFDW0002
DF D WOO 03
DFDW0004
DFDW0005
DFDW00G6
DFDW0007
DFDW0008
DFDW0009
DFDW0010
DFDW0011
DFDWOO12
DFDW0013
DFDWOO14
DFDWOO15
DFDWOO16
DF DW 0 017
DFDW0018
DFDW0019
DFDW0020
DFDW0021
DFDWOO 22
DFDWOO 23
DFDW0024
DFDWOO 25
DFDW0026
DFDW0027
DFDWOO 28
DFDW0029
DFDWOO 30
DFDWOO 31
DFDW0032
DFDW0033
DFDW0034
DFDW0035
DFDW0036
DFDWOO 37
DFDWOO 38
DFDW0039
DFDW0040
DFDW0041
DFDW0042
DFDW0043
-86 L-


-118-
Using only these three trials it is found that essentially
VFc^V^wg1 (6-4)
where R (F ) is the range of the electrons using the screening parameter
9 c
a(1000 eV) = Fc aQ(1000 eV). The Fc is some factor (in the case of trial
C2, Fc = 0.1) and RgQ is the range of the electron using the screening
parameter aQ(1000 eV). Again (see Eqs. (6.2) and (6.4)) there appears
to be a logarithmic type dependence on the screening parameter. In sec
tion VII.C shapes of the collision profiles were found to be proportional
to ln(a). Here, the range appears to be approximately proportional to
ln(a) for the three cases studied.
Next, use two more new trials, C4 and C5. The energy dependence
of "a" in Cl, C4, and C5 is the same and is illustrated in Figure 6.6.
As seen from Table 6.4, the screening parameter for the incident energy
has the most influence on the spatial energy deposition. The energy
dependence of "a" influences the energy deposition such that if the
parameter, a, is lower throughout the entire energy regime for a given
trial than the "a" used in another trial (for example, C4 compared with
C2), then the electrons will penetrate further during the course of that
trial (the C4 range is greater than the C2 range).
The radial distribution of the 3914 intensity profile is another
quantity of interest. Figure 6.7 illustrates this radial distribution
for trials Cl, C2, and C3 at three set distances into the medium. The
sharpest forward peaked phase function (hereafter the last four words can
be called FPPF) trial, C2, has an intensity distribution clustered close
to the z-axis throughout the x regime. The least FPPF trial, C3, has


-72-
Another straightforward and simplistic approach is to do the
following. Calculate the approximate backscatter at three energies,
the two endpoints and the middle (2 eV, 15 eV, and 30 eV), from the KG
H2 EDCS form and the experimental data on N2 EDCSs (given in Sawada,
Ganas, and Green, 1974). At these energies the backscatter with the KG
H2 EDCS form is less than that of the N2 EDCS by the following values:
2 eV ^ 5%, 15 eV ^ 10%, and 30 eV ^ 10%. An average of these three
values is about 8%. Since the major influences of the backscatter in
Eq. (4.31) is the value of K, this parameter is the only one that is
changed from the KG formulation. It is, therefore, increased by ^8% so
that in these MC calculations K = 0.46. The other parameters in Eq.
(4.31) are listed in Table 4.5.
Actually it appears that the value of K makes little difference in
the MC computational results. Two MC calculations at an incident elec
tron energy of 100 eV with K = 0.46 and with K = 0.425 were undertaken
(all other parameters and inputs were the same). The yield spectra
(described in Chapters II and VII) changes substantially only at fairly
large longitudinal distances (where the distances are about 1.5 times
the range). At these large distances there are relatively few electrons
anyway, thus there is little effect on the major aspects of the spatial
electron energy deposition process.
The Cartesian coordinates x y and z, are found from the coor-
dinates x^, y^, and in the following manner. After z is calculated
in units of MFPs with the use of Eq. (4.31), it can then be written in
units of cm or km by multiplying by the MFP, X (calculated from Eq.
(4.1)), thus zg = zb + zx.


Z (cm)
3914 A Intensity (I03exc.)
i O ro oj ^ ui
-£8-


-141
Table 7.3 Comparison between the yield of the 3914 A emission [#/(.5 cm)]
from the MC calculation [column labeled MC] and with the use
of Eq. (7.6) in Eq. (7.8) [column labeled AF] for several in
cident energies [column labeled E0 (in KeV)] and longitudinal
distances [column labeled z (in fractions of the range)].
E
z
MC
AF
0
0.1
0.01
280
287
0.1
0.2
240
272
0.1
0.5
155
168
0.1
0.8
70.0
82.3
0.1
1.0
35.2
49.7
0.3
0.01
452
406
0.3
0.2
626
523
0.3
0.5
490
429
0.3
0.8
174
177
0.3
1.0
74.2
73.2
1.0
0.01
550
556
1.0
0.2
740
797
1.0
0.4
860
908
1.0
0.7
500
456
1.0
1.0
100
91.9
2.0
0.01
600
578
2.0
0.2
780
841
2.0
0.4
1050
995
2.0
0.7
600
517
2.0
1.0
130
85.5
5.0
0.01
1300
1323
5.0
0.2
1760
1927
5.0
0.4
2100
2337
5.0
0.7
1380
1214
5.0
1.0
200
160


p
Figure 5.7 Intensity plots for electrons with incident energy 0.3 KeV are presented at four z values
as functions of p. The solid line histogram indicates the results using model 3. The
x's denote the experimental data of BH.


-100-
most deflections of the primary electron are at small angles while most
deflections of the secondary electrons are clustered near 90.
Strickland, Book, Coffey, and Fedder (1976) used a variety of dif
ferent secondary doubly differential ionization cross sections. Their
solutions to the equation of transfer show little dependence on the
functional form being used. Thus it is valid to ask if different primary
or secondary ionization differential cross sections will have any in
fluence on the intensity plots resulting from a MC calculation.
The primary electron is scattered the least. Therefore, for com
parison, it is assumed that no scattering of the primary electron was
incurred during the ionization event. The results of this comparison,
o
using the 3914 A intensity plots for incident electron energies of 2000
and 300 eV, were not too surprising: There was virtually no observed
difference in the two intensity plots. This simply means that the
scattering of the primary during an ionization event is minuscule com
pared to the much larger scatterings inherent in the elastic collisions.
For most of the calculations following this comparison, it was assumed
that no scattering of the primary electron occurred in an ionization
event. This resulted in a factor of eight savings in the computer time
and cost.
In Figure 6.1, the effect of the primaries and secondaries on the
total 3914 radiation is clearly seen at the incident energy of 1 KeV.
The major contribution of the secondaries is early in the history of the
incident electron, when it has sufficient energy to create high energy
secondaries capable of producing the 3914 emission. Also, the con
tribution by the primaries is a more sharply peaked curve than that of
the secondaries. The backscatter contribution from the secondaries is


APPENDIX B
GETDAT PROGRAM
The Getdat program is listed in this appendix. This program was
written entirely by the author. This program (written in Fortran IV)
collects the collision data from the magnetic tape and coalesces and
systematizes it.
-200-


-6-
This work of Bethe (1930) required knowledge of the stopping power,
- ^ (the rate at which energy E is lost from the impinging particles
incident along the x axis). This stopping power is given by
-£ -n | **t 1 <2->>
-y
(see Dalgarno, 1962, p. 624) where the summation S includes integration
i
over the continuum (thus allowing for energy loss through ionization),
W.j is the energy loss for the ith state, and o^E) is the cross section
for excitation to the ith state at energy E. Knowledge of the stopping
power then leads to information about the mean distance traveled by the
particle (referred to as the range). This range R(E) of a particle of
energy E is simply given by
R(E)' l rw (2-2)
u dx
Atmospheric researchers are more interested in the effects that the
particles have on the medium rather than the medium affecting the par
ticles. These effects could include both spectral emissions by the con
stituents and heating of the atmosphere.
Green and Barth (1965) were the first workers to adapt a variation
of the CSDA to the problems in aeronomy. In this approach the total
energy loss function L(E) = -(1) ^ is determined by
(E-IO/2 doT.(E,T)
L(E) = l W.a. (E) + l I.oy (E) + l f T J dT (2.3)
k K K j 3 j 0 dT
where is the threshold for excitation to the kth state, crk(E) is the
cross section for excitation to the kth state at energy E, I. is the
J daj (E,T)
threshold for ionization and excitation to the jth state, and 4=


14 JULY 1978
NERDC
CARD LIST UTILI TY
DO 6 I =1 NU MGAS
RE AD(5.522)SA( 1,1)
C READ IN THE MODEL ATMOSPHERE.
C RIGHT NOW THIS IS SET UP FOR ONLY ONE DENSITY AND ONLY ONE GAS BUT
C IT CAN BE EASILY MODIFIED.
6 CONTINUE
WRITE6.637)
637 FORMAT!////, THE STANDARD ATMOSPHERE IS NOW PRINTED*,
1 WITH DENSITIES IN 1/CM>**3',//,3X,
1 ALT I TUDE 1QX ,*GAS 1' ,10X,'GAS 2* ,1 OX, GAS 3*,/)
WR ITEl 6.639)ZALT( 1 >,SA<1, 1)
639 F0RMAT(4G15.7)
RETURN
END
DA TA 0217
DAT AO 218
DATA0219
DAT A 0 220
DAT A0221
DATA0222
DATA0223
DAT A0224
DATA0225
DAT AO 226
DAT A02 27
DA TA 0228
DAT AO 2 29
DATA0230


( UJ0/UJ6)/A8/# U! (3z -147-
Figure 7.5a


o n
14 JULY 1978
NERDC
CARD LIST UTILITY
ENM£ = ENR< J)-E
ENM£N=ENR(J)ENR(JM1 )
WLOSS = (WCFA(ItJ)*EMEN+ WCFA(I,JM1)*ENME)ENMEN
ENERGY LOSS FOR THE COMBINED STATE IS CALCULATED
CONTINUE
GO TO 15
12 WLQSS=WFA(K)
NSTA T = K
GO TO 15
13 WLOSS=THRI(KIO)
NSTAT=K+KIO
15 NSCS=N
RETURN
END
CTB20073
CTB20074
CTB20075
CTB20076
C TB 20077
CTB20078
CTB20079
CTB20080
CTB20081
CTB20082
CTB20083
CTB20084
CTB20085
CTB20086
-167-


-92-
This study uses the experimental data of BH as a comparison with
the results of this study. The next three graphs, Figures 5.5, 5.6,
and 5.7,portray sample results for incident electrons with energies
5.0, 1.0, and 0.3 KeV, respectively. The z and p values given in these
three figures are in units of fractions of the total range.
Fairly good agreement between the MC calculation (using model 3
cross sections) and the experimental work of BH and Barrett (1975) is
observed at all three incident energies. The largest differences be
tween the two sets of data are noted at 0.3 and 1.0 KeV.
For the 1.0 KeV case, the MC calculation tends to predict more in
tensity at the lower values of p for z values of 0.3 and 0.4. A similar
result is apparent for the z values of 0.36, 0.48, and 0.60 for an
energy of 0.3 KeV. At a z value of 0.12, however, the experimental data
tend to predict more intensity at all values of p.
Two conclusions can be drawn from these comparisons, if it is
assumed that the experimental data of BH and Barrett (1975) are correct.
+ 2 +
First, the cross section for excitation to the B Eu may be under
estimated in the energy regime between 0.3 and 1.0 KeV. Raising this
cross section in this energy regime could bring about an increase in the
intensity observed early in the electron's degradation process with a
subsequent decrease in intensity later in the electron's degradation
process. Second, more scattering from the elastic collisions would
help to reduce the total intensity at low p values and raise it at the
higher p values. The screened Rutherford differential cross section has
more scattering than model 3. Use of this set of cross sections in the
MC calculation did result in a little better agreement at 1.0 KeV, but
only about the same type of agreement at 0.3 KeV.


non no n non non
S 5 + pj
XVN=RT*SINPA*COSPHl +XV
YVN= RT*SINP A *SINPH I+ YV
CALCULATE THE COLLISION TYPE
CALL RANDU
IR 1= I R 2
CALL COLTYP
NCHE= (NSCS/2),it2
CALCULATE THE AZIMUTH ANGLE HERE
CALL RANDU
IR1=1R2
AZAV=6283185*R
AZAV IS THE AZIMUTHAL SCATTERING ANGLE
SI NAZ A = SI N ( A ZA V)
COSAZA= COS( AZAV )
IF (NIEtNST AT ) .NEo 2)
GD TO 70 0
CALCULATE THE SECONDARY ELECTRON ENERGY T
CALL RANDU
IR1=IR2
MC 000118
MC 00 0119
MC000120
MC 000121
MC000122
MC000123
MC000124
MC000125
MC 000126
MCOOO127
MC000128
MC 000129
MCOOO130
MC000131
MCOOO132
MC000133
MCOOO 134
MCOOO135
MC000136
MC 000137
MC000138
MC000139
MCOOO140
MCOOO141
MCOOO142
MCOOO143
MC000144


o n
14 JULY 1978
NERDC
CARD LIST UTIL ITY
C
C
C
c
c
c
45
c
SUBROUTINE ZVAL
SUBROUTINE ZVAL ***
THIS SUBROUTINE IS ACCESSED THROUGH MC*
THIS subroutine calculates the path length to the collision and
THE LONGITUDINAL DISTANCE TO THE COLLISION
THE FOLLOWING SUBROUTINE IS CALLED HERE:
1) CRSEC
COMMON ALPE (6) ,BETE (6 ) ,CE(6).FE(6>,W£(6), ALFA ( 15) BEFAt 15 ).
1 CFA( 15), FFA(15) ,WFA( 15) .WF(15) FACI(15) NFA .NAR.SA3 ,80)
2 ZALT(80),ENR(30),PCFA(3,30),WCFA(3,30),NENR,FDG(3),PSE(3),
3 PI ON (3) AT (3) A (6,3 ) ,B(5,3),G(5,3),UH(10) ,UI(10),UJ(10),
4 U D( 10),USN U( 10) ,USF( 10) ,UEIN( 11) ,IUTP.IUTM1 ,DG{3) ,NSG(3) ,NIG(3) ,
5 NUMGAS,NUMSTWIS(5 0) ,NS EC.MUNIT,I SEED,I STOP,ILAST,EIN,NPRIM,C0SI
6 H,EMIN,ZSTART,TMIN.THRESH(3) AK ( 2 ,3 ) A J ( 3,3 ) GAMA (2, 3),
7 T0(3,3),FAC(6),A0E(6),A1E(6) B OE( 6) ,B1E( 6),C0E(5) ,D0E(6) .
8 D1E(6) ,PR0B(40) ,W(40) ,NIE(40),A21 (3),A22(3) ,A31(3),A32(3),
9 A 33( 3),B11( 3) ,B 12 ( 3) ,B13( 3) ,C1(3) ,C21(3).C22(3),C31(3),
A C32(3),D1 (3)D2(3)F1 (3 ),F2(3)THR I ( 16).AKI(2,16),AJI(3,16),
B GAM AI (2,16) ,TOI (3,16) ,SIGT(6) THET (40 ) NX CS I E ( 2 0 ) E IT ( 20 ) ,
C IEILS. IEILMl,P,R,ZDIS,NPIN,NOP,CO SPA,RT F(4Q) ,R1 ,R2,R3,R4,
D R5.XVYV,ZV,XVN ,Y V N ZVN P A PH I EV WL OSS N ST AT NS CS NPHF ,
E IR1,IR2.NG,EVPRI,RAN,FMSDEINEL,EXC5(50),COSPAN,T,FOVAL
R= l.-R
AC=ABS(COSPA )
STUN=1E-16
FR AC=0.0
ZCHE = ZV
ZA DD=ZCHE*1 E-5
ZCHE=ZCHEZADD
J=1
IF ( PA oGTo 1 570 8) J=NAR
ZDIS = 0.0
DG(1)=SA(1.1)
CALL CRSEC
DO 45 IDS=1 NUMGAS
JU=2*IDS
JL = JU 1
SUM=0o 0
DO 45 JDS=JL,JU
SUM=SUM + DG(IDS )*SIGT(JDS )*STUN
CONTINUE
RM FP= l ./SUM
RMFP IS THE MEAN FREE PATH OF THE ELECTRON IN THE MODEL ATMOSPHERE
ZV ALO0 01
ZVALO 002
ZVAL0003
ZVAL0004
ZVAL0005
ZVAL0006
ZVAL0007
Z VAL 0008
ZVAL0009
ZVAL0010
ZVALOO11
ZVALOO12
,ZVALOO13
ZVALOO14
ZVALOO15
ZVALOO 16
ZVAL0017
ZVAL0018
ZVALOO19
ZVALOO 20
ZVAL 0021
ZV AL0022
ZVAL0023
ZVAL0024
ZV ALO 0 25
ZVAL0026
ZVAL0027
ZVAL0028
ZVAL 0 0 29
ZV AL00 30
ZVAL0031
Z VAL 00 32
ZV AL00 33
ZVAL0034
ZVAL 00 35
ZVAL0036
ZVAL0037
ZV AL 00 38
ZVAL0039
ZVAL 0040
ZVAL0041
ZVAL0042
ZVAL 0043
ZVAL0044
ZVAL0045
-96L-


-80-
Previous workers (Barrett and Hays, 1976; Cohn and Caledonia, 1970;
O
and Gran, 1957) have used the 3914 A emission as a measure of the energy
deposited. In these works it is assumed that since the 3914 radiation
is proportional to the number of ionizations in a given volume and if
the number of ionizations is proportional to the energy deposited in
that volume, then the 3914 A intensity is proportional to the energy
deposited in that volume. These experimenters, therefore, measured the
3914 A radiation at several energies, extrapolated their intensity plots
to find a range (to be described in section V.B), and derived an empiri
cal expression for the range that could be used to find the energy loss
function.
This idea of using the 3914 A emission to derive the energy loss
scheme is useful for energies above 2 KeV. In Figure 5.1, compare the
loss function, L(E), used in this work and the N2 B £u state cross
section.
The two curves are not parallel below 2 KeV. This implies that the
energy loss function can not be derived directly from the range results
at incident energies below 2 KeV. The energy loss plots from this MC
study are given in section VII.A and more will be discussed in that
section about them.
B. Range of Electrons
The concept of the mean range must be defined next. For each
monoenergetic primary electron impinging into a gas, a range can be
calculated. In general (at least above 100 eV), the higher the electron
energy the further the electron will penetrate into the medium. If an
+ 2 +
electron is incident along the z-axis, the excitations of the N2 B E


COSPLF=COSPA*COSPAN-SINPA*SINPAN*COSA ZA
MC 0002 62
IFCCOSPLF o G To 0o999) C0SPLF=0o999
MC 000 263
IFCCOSPLF ,LTo -Oo 999) COSPLF=-0.999
MC000264
ARGC=1. COS PLF* C OS PLF
MC 000265
IF(AR GC oLTo OoOEOJGO TO 6979
MC000266
SINPLF=S QRT(ARGC )
MC 0002 67
GO TO 6970
MC 000268
6979
SI NPLF =laOE-6
MC000269
6970
CONT INUE
MC000270
PLF=ARCOS( COSPLF )
MC 000 271
IF (COSPLF LEo 0,99) GO TO 60
MC000272
ALF=0.0
MC 000273
GO TO 61
MC000274
60
COSALF=(COSPA*COSPHl*SINPAN*COSAZA-SINPHI *SINPAN*SINAZA +
MC 0002 75
t SINPA*COSPHI*COSPAN)/SINPLF
MC000276
SI NAL F=(CO SP A*SINP HI*SINPAN *COSA ZA + COSPHI*SINP AN* SINAZA
MC000277
1 +SINPA + SINPHI* COS PAN)/SINPLF
MC 000278
IF (COSALF oLTo Oo 999) C0SALF=-0o 999
MC000279
IF{COSAL F GT 0 .999)COSALF = Oa999
MC0002 80
ALF=ARC0S(COSALF)
MC 000281
IF ( S I NALF oLTo 0.0) ALF= 6o 2831 85-A LF
MC000282
61
CONT I NUE
MC000283
COSPA=COSPLF
MC000284
PH 1= ALF
MC000285
PA=PLF
MC000286
GO TO 27
MC000287
C
MC000288
-182-


-43-
r(E) = 13.8 eV E/(E + 15.6 eV)
oQ = 1 x 10 "16 cm2
Equation (3.18) may seem highly complicated; however, integration
over the solid angle is given very simply as Eq. (3.19) which is the
singly differential ionization cross section. The total ionization cross
section is then
TH
aI0N^ dT dT (3.20)
with
TM = \ (E ^
so that
aI0N(E) = Ar{tan_1[(TM TQ)/r] + tan_1(TQ/r)} (3.21)
The fit to Opal, Beaty, and Peterson's (1972) data is given in Figure 3.7
at several primary and secondary energies. The x's represent the ex
perimental data and the solid line represents the analytic expression,
Eq. (3.18).
Other inelastic processes include the simple excitation events.
The scattering of the incident electron due to the excitation of a par
ticular state has been studied by Silverman and Lassettre (1965), and
more recently by Cartwright, Chutjian, Trajmar, and Williams (1977) and
Chutjian, Cartwright, and Trajmar (1977).
In order to account for this scattering, the Silverman and Lassettre
(1965) generalized oscillator strength data for the 12.85 eV peak (cor
responding to the optically allowed b ^ir state) were fit with the use of
a phase function similar to model 1. The very sharply forward scattering


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Alex E.S. Green, Chairman
Graduate Research Professor of Physics
and Astronomy
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Thomas L. Bailey f
Professor of Physics ajnd Electrical
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Qjcpkii*. | xli' U-h-Mr-
Stephen T. Gottesman
Associate Professor of Astronomy


-48-
A. Brief Discussion of the Monte Carlo Calculation
In Figure 4.1 a short version of the MC calculation is presented.
Briefly, each electron is degraded in a collision by collision manner
down to 30 eV. Below 30 eV the electrons are degraded with the use of
a multiple scattering distribution. This multiple scattering approach
characterizes the resultant coordinates of the electron which goes
through several elastic collisions between each inelastic collision.
The incident electron has an energy E To begin with, the running
total of the electron energy, E, is set equal to Eq. At the position
START, this energy E is checked against a cutoff energy, Ec. If the E
is more than Ec, then, first the distance traveled by the electron to
the collision is calculated.
Second, the type of collision which occurs is determined. If a
collision is elastic then the electron is scattered with the use of a
phase function, the appropriate energy aE,-,p is lost, and the electron
goes back up to the START of the degradation process. Whether a coll i si on is
inelastic it is determined if the collision is an ionization event or an
excitation event. In the excitation process, scattering occurs if the
energy E is less than 100 eV, E is reduced by the threshold, W, for
excitation of this state, and the electron goes back up to the START of
the degradation process.
Ionization collisions are the most complex occurrences to compute.
The energy loss, W, by the incident electron is equal to the kinetic
energy, T, of the secondary electron produced plus the ionization thresh
old, I. The primary electron is then scattered and reduced in energy by
W. If the secondary electron has a kinetic energy greater than Ec, then,


ANUM=-lo
ADEN=-1o
IF (CHI o GT 55o 0 ) A NUM= R2 (JNU I N-1 o
IF ( CH2 ,GT-550 )A D EN= FO **UNU I N-l.
Z=-ALOG(ANUM/ADENJ/UMU
C Z IS IN MEAN FREE PATHS
S I GEE=S I GT ( 1 )+SIGT(2)
FMFP=1o/SIGEE/SA(1.1>/l.E-16
C FMFP IS IN CMg
ZADD=Z*FMFP*1.E-5
C ZADD IS IN KM.
ZVN= ZV+ZADD
C CALCULATE THE Z COORDINATE
PA=3*141592*R4
PHI=6,283185*R5
GA M=2
IF (PL oLTo 400 )GAM = 2o*( l-EXP(-PL/4o ) )
DEL=(22.+SQRT(PL/03>)/(PL+0.3)*415
RHO= 0.0
IF ( GAM o GT o 0.2 )RHO = (-ALOG( lo-R4)/DEL) *( lo/GAM)
XADD = C0S(PHI )* RHO*F MFP + 1 oE-5
YADD = SIN(PHI )*RH0*FMFP*laE-5
XVN=XV+X ADD
YVN=YV+YADD
C CALCULATE THE X AND Y COORDINATESo
P=0.0
PSE( 1> = 0o 0
MESD0046
MESD 0 047
MESD0048
MESD0049
ME SD 00 50
MESDOO 51
MESD0052
MESD0053
MESD0054
ME SD 0055
MESDOO56
MESD0057
MESD0058
MESD0059
MESD0060
MESD0061
MESD0062
MESD0063
MESD0064
ME SD0065
MESD0066
MESD0067
ME SD0068
MESD0069
MESD0070
MESD0071
MESD0072
-187-


-143-
quite similar in all the gases considered. Thus it is expected that
the spatial yield spectrum U(E,z,EQ) will also be similar for electron
energy degradation into the other atmospheric gases.
2. Four Variable Spatial Yield Spectra
The last subsection (VII.B.l) was only concerned with the spatial
yield spectra in the longitudinal direction. This subsection deals with
the four variable spatial yield spectra U(E,p,z,Eq) which is also a
function of the radial direction p.
The MC calculation appropriately accounts for the coordinate p down
to the energy of 30 eV, below which the multiple elastic scattering dis
tribution is used. In subsection IV.C.6 an approximation was made which
assumed that p was about one-sixth of the total path length. A better
approximation would be to simply assume that the p distribution for Ng
is similar to that of Hp.
With this assumption and inverting Eq. (8) from Kutcher and Green
(1976), the expression for p is
where
p = [-In (1 R)/6]1/y
. <22 + (0^3>VZ)
(s+0.3)'-5
(7.10)
Y = 2[1 exp(- J)]
and R = a random number. The parameters are found by averaging those
parameters in Table I of Kutcher and Green (1976).
Use of Eq. (7.10) in the MC computations resulted in the spatial yield
spectra U(E,p,z,Eo) which is fairly accurate down to 2 eV.


-121-
its intensity distribution spread out the most from the z-axis throughout
the z regime.
In Figure 6.8, a cut is taken through each intensity profile at the
distance z = 0.3 (units are fraction of range). This type of distribu
tion continues for all the longitudinal distances throughout the range.
The 3914 intensity profile of the sharpest FPPF, C2, again hugs the
z-axis, whereas the profile of the least FPPF, C3, again shows the
greatest spread from the z-axis.
E. Effects of the Total Elastic Cross Section on the Electron
Energy Degradation
As illustrated in the previous sections of this chapter, the spatial
electron energy degradation is governed mainly by the elastic differen
tial cross section. Consider the effects of the total elastic cross
section on the electron energy degradation.
The total elastic cross section given by Eq. (3.9) is used in
practically all the MC calculations of this study. This analytic form
agrees quite well with experiment.
Berger, Seltzer, and Maeda (1970, 1974) used the integrated Ruther
ford cross section for their total elastic cross section. This cross
section is somewhat different from the experimental data and is plotted
in Figure 3.2.
If a MC calculation is made with this lower elastic cross section,
it is expected that the 1 KeV electrons will penetrate further into the
medium. Figure 6.9 illustrates the results of this calculation where
the screened Rutherford phase function was used.


-42-
differential cross sections is that of Opal, Beaty, and Peterson (1972).
(More recent data by DuBois and Rudd (1978) agrees with their work.)
This data indicates a preferred angle range in the scattering process
(usually between 45 and 90) at all primary and secondary energies.
A Breit-Wigner form has been chosen to represent the data. Here,
, ICE,_T)C2 (3.18)
dTdfi [C2 + B(cose coseo)2]Nf
where
B(E) = 0.0448 + (
72900 eV
0.91
)
r/T\ 36.6 gV \
(T + 183 eV '
, ve)
0q E = *873 + (T + 0b(E))
0a(E) =20 eV + 0.042 E
and
0b(E) =28 eV + 0.066 E
o r r (1 + cose ) E (1 cose ) ^
f --f {fn-D c *4 tan- [ T *-]}
S(E,T) = = A(E)r2(E)/[(T Tq(E))2 + r2(E)]
(3.19)
is from Green and Sawada (1972) with
a(e) = a0 V2 '"[dh?]
T (E) 5 4 71 eV .^000 (e^)^
o^t; 4-/l ev (E + 31.2 eV)


NERDC CARD LIST UTILITY
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-75-
D. Statistical Error in the Monte Carlo Calculation
The statistical error inherent in the MC computation can be derived
by considering the following. Since the MC calculation is a probabilis
tic method of degrading an electron in energy, the multinomial distribu
tion can be used to find the statistical standard deviation for each bin
considered. This discussion of the statistical error employed the work
of Eadie, Dryard, James, Roos, and Sadoulet (1971).
The probability of getting an excitation of a certain state j in bin
k is Pj^. The p^ is normalized such that
m n
l l Pik = 1
k=l j=l JK
(4.33)
In this MC study the multinomial distribution is an array of histograms
containing N events distributed in n states and m bins with r.^ events
in state j and bin k. The r.^ values are normalized such that
m n
l l rjk
k=l j=l JK
(4.34)
Thus, the r^ observations can be considered somewhat conditional
on the fixed observational value of N. The variance of the calculation
is represented as
v(rjk> = N pjk < Pjk>
(4.35)
In this work the m x n variables r^ can all be correlated. For the
specific example of electron deposition represented in Figure 5.2,
jk
1. This is true because there are total almost 5 x 10 col
lisions (i.e., N = 5 x 10) to consider in this degradation scheme and


"6 2
Figure 6.7 At three longitudinal distances (given in 10 gm/cm ), the
intensity distribution is given as a function of p for the
three trials Cl, x; C2, A; and C3, o.


UNIVERSITY OF FLORIDA
3 1
262 08667 026 1


-221-
Cartwright, D.C., A. Chutjian, S. Trajmar, and W. Williams, Electron
impact excitation of the electronic states of N2. I. Differential
cross sections at incident energies from 10 to 50 eV, Phys. Rev. A,
16, 1013, 1977.
Chutjian, A., D.C. Cartwright, and S. Trajmar, Electron impact excitation
of the electronic states of N2. III. Transitions in the 12.5-14.2 eV
energy-loss region at incident energies of 40 and 60 eV, Phys. Rev. A,
16, 1052, 1977.
Cicerone, R.J. and S.A. Bowhill, Photoelectron escape fluxes obtained by
a Monte Carlo technique, Radio Sci., J5, 49, 1970.
Cicerone, R.J. and S.A. Bowhill, Photoelectron fluxes in the ionosphere
computed by a Monte Carlo method, J. Geophys. Res., 76, 8299, 1971.
Cohn, A. and G. Caledonia, Spatial distribution of the fluorescent radiation
emission caused by an electron beam, J. Appl. Phys., 41, 3767, 1970.
Cravens, T.E., G.A. Victor, and A. Dalgarno, The absorption of energetic
electrons by molecular hydrogen gas, Planet. Space Sci., 23, 1059, 1975.
Dalgarno, A., 15. Range and energy loss, Atomic and Molecular Processes: 13,
pp. 622-662, edited by D.R. Bates, Academic Press, New York, 1962.
Dalgarno, A. and G. Lejeune, The absorption of electrons in atomic oxygen,
Planet. Space Sci., 19, 1653, 1971.
Douthat, D.A., Energy deposition by electrons and degradation spectra,
Radi at. Res., 64, 141 1975.
DuBois, R.D. and M.E. Rudd, Absolute doubly differential cross sections
for ejection of secondary electrons from gases by electron impact.
II. 100-500-eV electrons on neon, argon, molecular hydrogen, and
molecular nitrogen, Phys. Rev. A, 17, 843, 1978.
Eadie, W.T., D. Dryard, F.E. James, M. Roos, and B. Sadoulet, Statistical
Methods in Experimental Physics, pp. 44-47, North-Holland Publishing
Company, Amsterdam, 1971.
Ehrhardt, H., M. Schulz, T. Tekaat, and K. Willmann, Ionization of helium:
Angular correlation of the scattered and ejected electrons, Phys.
Rev. Lett., 22, 89, 1969.
Fowler, R.H., Theory of motion of a-particles through matter, Proc. Camb.
Philos. Soc., 121, 521, 1922-23.
Garvey, R.H., H.S. Porter, and A.E.S. Green, An analytic degradation
spectrum for J. Appl, Phys., 48, 190, 1977.
Goudsmit, S. and J.L. Saunderson, Multiple scattering o.f electrons,
Phys. Rev., 57, 24, 1940.
Green, A.E.S. and C.A. Barth, Calculations of ultraviolet molecular nitro
gen emissions from the aurora, J. Geophys. Res., 70, 1083, 1965.


-134-
E (eV)
Figure 7.3 Three variable spatial yield spectrum for an incident energy
of 1 KeV given at three longitudinal distances (in fractions
of the range): z = 0.0739 represented by x, z = 0.429
represented by o, and z = 0.961 represented by A.


CHAPTER I
INTRODUCTION
Calculating the spatial and energetic aspects of the energy deposi
tion of intermediate energy electrons (with incident energies from 100
to 5000 eV) in the earth's atmosphere is a difficult, yet intriguing,
problem. These intermediate energy electrons (hereafter called IEEs)
include the highest energy photoelectrons, a large bulk of the auroral
electrons, and many secondary electrons produced by solar protons and
cosmic rays.
These electrons lose most of their energy through ionization events,
electronic excitations, vibrational excitations, and rotational excita
tions. Elastic collisions reduce the electron energy slightly, but
mainly these interactions influence the direction of motion of the
electron.
The atmosphere is dominated by the presence of molecular nitrogen
up to a height of about 150 kilometers. Even above this altitude (at
least up to 300 km), continues to play a substantial role in the
atmospheric processes. For this reason the study of the influence of
impinging electrons on molecular nitrogen is the major thrust of this
paper.
One aspect of this study is the formulation of a complete cross
section (differential and total) set for IEEs impacting on N£. The very
difficult problem of modeling the interactions of the impinging IEEs in
-1-


14 JULY 1978
NERDC
CARD LIST UTILI TY
C
C
C
C
c
562
61 5
506
61 7
4
643
C
C
564
64 5
647
15
619
C
540
621
5
THIS SCATTERING IS NO LONGER INCLUDED IN THE MONTE CARLO
CALC ULATI ONa
THE SUBROUTINE DFDW AND THE FUNCTION DCS ARE INCLUDED AT THE
END OF THIS LISTING AND CAN BE MODIFIED A LITTLE TO BE PUT INTO
THE MONTE CARLO COMPUTATION
FO R M A T ( 5 A45 E1 0 0.E10.2)
RE AD(5522) (ACK ,1) ,K=5,6)
WRITEI6.615) Gl. G2, G3. G4.G5,PIONtI ),AT(I ),(A(K,I> K=1 ,6)
FORMAT(1X,5A4,2X F6j23X 1 PEI03*2XF8o23XF7.33X* 1PE13.3 IX
l 1PE 10. 3. 3X.F 8 2, 1 X.F 8. 2)
READ(5.506) (B.K=1,5 ) (G FORMAT( 1 0E8. 0)
WRITE (6,617 ) ( B( K, I ) K= 1 5)(G(K,I ),K= 1, 5)
FORMAT(21X.F8o2.3X.Fo3.3X.F8.3.2X,1 PE 10.3.2X,F8a 3. 3X.1PE 10*3
1 2X ,F8a 3, 3X.F 8 3. 2X. 1 PE 1 Oo 3.2X ,F8. 3 .//)
CONT I NUE
WRITE (6.643)
FORMAT(////. THE DOUBLY DIFFERENTIAL SECONDARY IONIZATION
* CROSS SECTIONS ARE READ IN,//,6X,GAS,14X,FI/C1,6X,F2/C21
* 4X A 2/C 22 3X A22/C31 ,3X.' A31 /C32 3X A3 2/D1 4 X, A33/D2 .
* 4X. B11/.8X, B 1 2 / *.8X.B 13/,//)
DO 15 1=1,NUMGAS
RE AD ( 5*562)G 1.G2.G3.G4 .G5.F1 ( I ) .F2(I) .Bll(I) ,B12(I ) ,B13(I )
RE AD (5.564 ) Cl (I) C2 1 ( I ) C22( I ) C31 ( I ), C32( I),D1(I),D2(I)
READ IN THE DOUBLY DIFFERENTIAL SECONDARY SCATTERING PARAMETERS.
READ IN THE STATES AND THEIR THRESHOLDS.
FORMAT(8E10.0)
READC 5,564)A 21( I ),A 22( I) ,A 31(I ) ,A3 2( I) ,A33 WRITE (6.645 )G1 ,G2. G3. G4. GS.F1(I),F2( I),A21(I),A22(I),
* A31(I) ,A32(I ) ,A33(I ) .Bll ( I ), B12( I),013(1 )
FORMAT ( 1 X.5A4,7( F80 3. 2X ) ,F 8. 3,2 X .F 1 0. 0 ,2X,F8o 3)
WRITE(6,647)C1 ( I ).C21 ( I ),C22( I),C3 1 ( I ),C32( I ) ,D 1 ( I ),D 2( I )
FORMAT(21 X,7( F8a 3.2X) )
CONT INUE
WRITE6.619)
FORMA T(/ 1 THE STATES WITH THEIR PROBABILITIES AND THRESHOLDS ARE
1 NOW READ IN,//,5X.'STATE .16X. 'PROBABILITY* ,5X.THRESHOLD ,3X,
1 IONEXC NO.,3X INDEX NUMBER./)
DO 5 1=1, NUMST
READ(5,540) G1,G2,G3,G4,G5,PROB(I),W( I).NIE(I)
READ IN THE PROBABILITIES AND THRESHOLDS FOR THE STATES ABOVE 200
FO RM AT(5A4,2E8.0, 15)
WRITE(6.621) G1.G2,G3.G4,G5.PROS(I).W(I),NIE(I),I
FORMA T( IX, SA4.5X ,F 1 Oa 5.6X.F 8. 3.4X, 15,9X, 15 )
CONTINUE
DAT A0073
DAT A0074
DATA 0075
DAT A 0 0 76
DAT AO 077
DATA0078
DATA0079
DAT A0080
DATA0081
DAT A0082
DA TA 0083
DATA0084
DAT A0085
DA TA 0 0 86
DATA0087
DATA0088
DATA0089
DATA0090
, DA TA 0091
DAT A0092
DAT A0093
DATA0094
DAT A0095
DATA0096
DA TA 00 97
DAT A0098
DATA0099
DATAOIOO
DATA01 01
DATAO102
DAT AO 103
DATA01 04
DA TA 0105
DATAO106
DA TAO107
DATAO108
DATA0109
DATAO 1 10
DAT AO 111
DATAOl 12
EVDATA01 13
DATA0114
DATA01 15
DATA0116
DATA0117
-170-


c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
r
C
9002
C
C
c
14 JULY 1978
NERDC CARD LIST UTILITY
SUBROUTINE MC
SUBROUTINE MC ***
THIS IS THE MAJOR SUBROUTINE
THIS SUBROUTINE IS ACCESSED THROUGH THE MA
IN THIS SUBROUTINE THE ELECTRON IS DEGRADE
BY COLLISION MANNER DOWN TO THE CUTOFF ENE
THE FOLLOWING SUBROUTINES ARE CALLED FROM
1)RANDU
2)ZVAL
3 ) PHF
4)PHFEL
5) PE TI
6)SD IFM
7)CR SEC
8)COL TYP
9)MESD
COMMON ALPE (6) BET E (6 ) CE ( 6 ) FE ( 6 > WE ( 6 )
1 C FA ( 15).FFA(15) WF At 15) ,WF( 15) .FACK15)
2 ZALT (80 ) > ENR(30 ), PCFA (3,30 ) WCFAt 3,30 ) ,
3 PION3) iAT(3) i A (6 <3 ) B(53) G ( 5 3 ) UH ( 1
4 UD( 10 ) USNU( 1 0 ) ,USF{ lO).UEIN(ll), IUTP ,1
5 NUMG AS NUMSTiWIS(50) .NSEC.MUNIT, I SEED. I
6 H.EMIN.ZSTART,TMIN.THRESH(3),AK(2,3),AJ
7 TO(3,3),FAC(6),AOE(6),A1E(6),BOE(6),BIE
8 DIE(6) iPROB(40) ,W(40 > ,N IE(40),A21 (3),A2
9 A 33( 3),B11(3).B12( 3>,B13(3) C 1 ( 3) ,C21 (3
A C32(3).D1(3),D2(3).F1(3),F2(3 ) THR I ( 16)
B GAMA I(2.16) .TOI (3.16 ) ,SIGT(6) THET(40),
C IEILS.IEILMl.P.R, ZD I S NPI N NOP CO SP A R T
D R5.XV.YV.ZV.XVN ,Y V N Z VN P A PH I EV.WLSS
E IR1. IP. 2 NG EVPR I ,R AN EM SD E I NEL ,E XC5(50
IN PROGRAMo
D IN ENERGY IN A COLLISION
RGY EMINo
THIS subroutine:
,ALFA( 15 ) i B EF A ( 15).
iNFA iNAR*S A(3 ,30) ,
NENR.FDGt 3) P SE ( 3 ) .
0),UI ( 10 ),UJ( 10 ) .
UTMl .DG(3) ,NSG(3) ,NIG(3) ,
STOP,ILAST.EIN.NPRIM.COSI
(3,3).GAMA(2,3 ) ,
(6),C OE(6) ,DOE(6) ,
2(3) A 3 1 ( 3 ) A 32( 3),
) ,C22(3) ,C31(3) ,
.AKI ( 2.16) ,AJI( 3.16) ,
NX ,CSIE(20 ),E IT(20).
F(4 0) ,R1 .R2.R3.R4,
,NSTAT,NSCS.NPHF,
),COSPAN,T,F0VAL
WR IT E(6,9002 )
FORMAT( 1 )
HEADING FOR ELECTRON
COUNT CAN BE PUT HERE
INITIALIZE SOME INTEGERS
NA=NX- 1
IR1 = I SEED
NI 2=0
NSPA= 0
NG200=0
IEL=0
M=9
MC 000001
MC000002
MC 000003
MC000004
MC000005
MC 000006
MC 0000 07
MC000008
MC 000 0 09
MC000010
MC 000011
MCOOOO 12
MCOOOO13
MC 0 00014
MCOOOO15
MC 0 00016
MCOOOO17
MCOOOO18
MC 000019
MCOOOO 20
MCOOOO 21
MC000022
MC000023
MC000024
MC 000025
MCOOOO 26
MC 000027
MC000028
MCOO 0029
MC000030
MC000031
MC 000032
MCOOOO 33
MC000034
MC000035
MC000036
MC000037
MC000038
MCOOOO39
MC 000040
MC000041
MC000042
MC000043
MC000044
MC 000045
-175-


14 JULY
1978
NERDC
CARD LI ST UTILI TY
C
C
RAN=R3
CALL CTB 200
WLGSS = WLOSS + RT*EV*7 .776E-5
CALCULATE THE ENERGY LOSS DUE
AS THE ENERGY LOSS DUE TO THE
RE TURN
END
TO THE INELASTIC COLLISION AS WELL
ELASTIC COLLISIONS*
MES00073
MESD0074
MESD0075
MESDOO 76
MESD0077
ME SD 0078
MESD0079
-188-


-Til
lable 6.2 Model 2 parameter values (columns "a," "c," and "f") and
phase function properties for various trials. The PFFO
and AAS columns
. are
described in
Table 6.1.
Trial
a
c
f
PFFO
AAS
B1
0.1
0.3
0.8
1.0
52.3
B2
0.033
0.38
0.92
2.0
29.0
B3
0.012
0.46
0.97
3.0
16.3


-69-
In this work a different problem exists. The MC calculation is
used to degrade electrons in a collision by collision manner all the way
down to 30 eV. At this energy, the elastic collisions are occurring
with twice the frequency of the inelastic events, and at energies below
30 eV the number of elastic collisions between inelastic events may be
up to several hundred or thousand. Keeping track of all these elastic
collisions would be very costly.
Kutcher and Green (1976) [hereafter referred to as KG] studied the
radial, longitudinal, and polar angle distributions for elastic scatter
ing by in the energy range from 2 to 50 eV. An approach similar to
KG's could be applied to Ng. Since such a project would require a
substantial amount of time and computer money, the possibility of adapt
ing the KG results was first considered.
With this in mind, consider the differences between and Hg.
First of all, there are some dissimilarities between the differential
cross sections. There is more backscatter observed experimentally in
N2 at all energies. Secondly, the total inelastic and elastic cross
sections are different. The second difference is no real problem because
the MESDs are given in terms of the mean free path lengths (hereafter
referred to as MFPs). The first dissimilarity does pose a minor problem
which is solved in a simplistic way below.
Above 5 or 6 MFPs the polar angle is approximately random. At most
energies below 30 eV, the number of MFPs between inelastic collisions
is above 5 or 6. Since the distribution found in KG is not easily in
verted, a reasonable assumption is that the polar angle is oriented
randomly.


-112-
The results of these MC calculations are illustrated in Figure 6.5.
There is no sharp discontinuity at the origin. The peak of the dis
tribution moves along the z-axis as the forward scattering increases.
The reason for this continuity in the collision distribution arises from
the backscatter peak.
The average scattering angle is given by
2lT TT
0 = / / 6 P(0) sined0d4> (6.3)
ave 0 0
where P(@) is the phase function. The average scattering angle is nearly
the same in trials A2 and B1; A3 and B2; and A4 and B3 (compare Tables
6.1 and 6.2). This means that the shape of the backward scattering part
of the phase function is also very important in determining the spatial
energy deposition of an electron.
D. Influence of Different Elastic Phase Functions on
the Intensity Profiles
A sensitivity study involving several different, but constant,
phase functions was the subject of section VI.C. The elastic scattering
phase function of electrons changes with energy. Generalizing the influ
ence of the energy dependent phase function is the subject of this
section.
Section V.D includes a comparison of the screened Rutherford and
the model 3 phase functions and their influence on the range values.
It was learned that in the electron energy regime of interest, 2 eV to
5000 eV, the screened Rutherford causes more scattering than the model 3
phase function. The screened Rutherford range values, therefore, tend
to be lower than the model 3 range values (see Table 5.1).


-55-
Table 4.1 The energy E is presented in the first column with the number
density n, used in the MC calculation, being given in the
second column. (8.0 E+ 14 means 8.0 x 10^)
E (KeV)
n (#/cm^)
0.1
8.0 E+14
0.3
2.0 E+15
1.0
8.2 E+15
2.0
2.8 E+l6
5.0
1.2 E+l 7


on
14 JULY 1978
NERDC
CARD L1ST UTIL I TY
C
C
C
C
C
THE SECONDARY ELECTRON
SUBROUTINE PET I
SUBROUTINE PETI ***
THIS SUBROUTINE IS ACCESSED THROUGH MCo
THIS SUBROUTINE CALCULATES THE ENERGY OF
AFTER AN IONIZATION EVENT
COMMON ALP E 16) B ETE ( 6 ) ,CE(6),FE<6) ,WE(6) ,ALFA(15) BEF A( 1 5 )
1 CFA{15)FFA(15)WFA(15),WF(15)FACI( 15).NFA.NAR, SA(3.80) t
2 ZALT< 80) .ENR30 ) .PCFA (3 .30) WCFA ( 3 .30 ) NENR. FDG ( 3 ) PSE ( 3 ),
3 PI0N(3),AT(3),A(63),8( 53).G(5.3). UH ( 10) ,UI ( 10) .UJ( 10) .
4 UD ( I O) .USNU10 ) .USFll 0) .UEIN(ll). IUTP. IUTM1 DG( 3 ) NSG( 3) ,N IG( 3)
5 NUMGAS.NUMST.WIS(5 0) .NSEC.MUNIT.I SEED.1 STOP.1 LAST.EIN.NPRIM.COS
6 H.EMI N, ZST ART T M IN T HRES H ( 3 ) AK ( 2.3 ) A J ( 3. 3 ) GAM A { 2. 3) .
7 TO(3.3) .FAC(6) ,AOE(6) .A1E{6),BOE(6).B1E(6).COE(6).DOE(6),
8 D1E(6),PR0B(40) ,W( 40),NIE(40),A 21(3) .A22(3) ,A 31(3) .A323),
9 A33(3),B11 (3 > ,B12(3 >,B13(3),C1(3).C21(3 >.C22(3).C31(3).
A C 32 ( 3) ,D1( 3),D2(3) c 1 l 31 F2(3) THR I (16) .AKH2.1S) ,AJI( 3,16 ) ,
B GAMA I (2.1 6 ) ,T O I (3, 16),SIGT(6),THET(40).NX,CSIE(20> EIT(20) .
C IEILS.IEILMl.P.R.ZOIS.NPIN,NOP,COSPA.RT,F(40).R1.R2,R3,R4,
D R 5.XV,YV,Z V.XVN,YVN,Z VN,PA,PH I ,EV,WLOSS,NSTAT,NSCS NPHF,
E I R1 IR2 ,NG, EVPR I, RAN, EMSD, EINEL .EXC5( 50 ) .CD SPAN, T F OVAL
E=EV
RN=R
N=NG
TSE T = TO(1 ,N)-TO(2,N)/(E*TO<3,N) )
GS ET = GAM A ( l ,N ) *E/( E+GAMA < 2,N ) )
TM={ETHRESH(N))/2o
TN 1P = A TAN2 ( { TM-TSE T) G SE T )
TN2P= ATAN2(TSET GSET)
TN1 = TAN( RN*TN1 P+ (RN-1 o )*TN2P)
T=GSET*TN1 + TSET
IS THE KINETIC ENERGY OF THE SECONDARY
RE TURN
END
ELECTRON,
PET10001
PETI 0002
PET 10003
PETI 0004
PET 10005
PET 10006
PE TI0007
PET 10008
PET 10 009
. PETI00 10
I PET 100 11
PETI 0012
PET 100 13
PET 10014
PE T I 0 0 15
PET 10016
PETI 0017
PETI0018
PET 100 19
PE TI0020
PET 10021
PETI0022
PETI 0023
PET 10024
PETI 0025
PE TI 00 26
PET 10027
PE TI 00 28
PET 10029
PET 100 30
PETI 0031
PET 100 32
PETI 0033
PET 10034
PET 10035
-189-


nno
14 JULY 1978
NERDC
CARD LIST UTILITY
SUBROUT INE
C SUBROUTI
C THIS SUBROUTI
C THIS PROGRAM
C THE NECESSARY
COMMON ALAB
1 EXC5(50)J
NIE < 40) ,NZ
TOTEO,80 )
YSPEC100 )
NSEC ,NSPZ ,
ZVAL.EMIN,
RDCARD
NE RDCARD ***
NE IS ACCESSED BY THE MAIN PROGRAM*
READS IN THE DATA FROM THE CARDS AND SETS UP ALL
2
3
4
5
6
40)
THE NECESSARY DATA CARDS
C
7
C
2248
2249
C
c
c
c
c
c
c
c
c
c
c
c
71 0
2246
c
c
c
c
THE TAPE.
THE AVERAGE ENERGY OF THE
THE
Z VALUES*
READ IN
M=9
M IS THE LOGICAL UNIT NUMBER FOR
RE AD(5,2 24 8) (ALA B( I ). 1=1,20 )
FIRST CARD IS THE TITLE CARD*
FORMAT(20A4)
WRITE(6.2249)(ALABl),1=1,20)
FORMAT(1,20A4,///)
READ(5.2244)NAVEE,NRHO,NV R,ND M,NSP EC,NSPUN,N SPR3
1= NAVEE=1 THEN WE WANT TO CALCULATE
PRIMARY AT A GIVEN DISTANCE
IF NRHO=l THEN WE WANT TO CALCULATE THE DISTRIBUTION OF
EXCITATIONS AS A -UNCTION OF RHO FOR A GIVEN NVR NO* OF
NV R IS THE NO. OF Z VALUES AT WHICH YOU WANT THE DISTRIBUTION
NRM IS THE NUMBER OF RHO VALUES IN THE MATRIX IN WHICH THE
EXCITATIONS WILL BE INPUT
IF NSPE C = 1 THEN WE WANT TO GET OUT THE SPECTRUM OF ELECTRONS
IF NSPUN = 1 THEN WE WANT TO PUNCH OUT THE YIELD SPECTRUM OF THE
EL ECTRONS
1= NSPRO = 1 THEN WE WANT THE YIELD SPECTRUM AS A FJNCT ION OF RHO
IF(NVR oEQo 0)NVR=1
NV R=1 FOR DEFAULT
WRITE(6,710) NAVEE.NRH O,NVR,NRM.NS PEC,NS PUN.NS PRO
FORMAT! NAVEE= 15,/, NRHO= ',15,/,' NVR=,I5./, NRM=,I5./,
* NSPEC=* 15 ,/, NSPU N= 15,/ NSPR0=, 15,///)
READ ( 5.224 6) NE XC ,NPRI M ,NAR NSEC NU MST .NUMGAS, NEIP, NSPZ, NS PR
FORMAT(110,1415)
NE XC IS THE TOTAL NUMBER OF COLLISIONS
NPRIM IS THE NUMBER OF PRIMARY ELECTRONS
NAR IS THE NUMBER OF ALTITUDE INTERVALS
NSEC IS THE NUMBER O" LOWER ELECTRON ENERGY BIN ENDS*
RDCARD01
RDCARD 02
RDCARD03
RDCARD04
RDCARD05
RDCARD06
.RDCARD07
RDCARD08
RDCARD09
RDCARD10
RDCARD11
RDCARD12
RDCARD13
RDCARD14
RDCARD 15
RDCARD16
RDCARD17
RDCARD 18
RDCARD19
RDCARD20
RDCARD 21
RDCARD22
RDCARD 23
RDCARD 24
RDCARD25
RDCARD 26
RDCARD27
RDCARD28
RDCARD29
RDCARD30
RDCARD31
RDCARD32
RDCARD33
RDCARD34
RDCARD35
RDCARD36
RDCARD37
R DCARD38
RDCARD 39
RDCARD40
RDCARD41
RDCARD42
RDCARD43
RDCARD44
RDCARD45
00
o
ro
i


-104-
increases the MC computation by 2-3%. Therefore it was left in all the
calculations.
B. Influence of Inelastic Differential Cross Sections
on the Intensity Distributions
Model 3 includes scattering from inelastic excitation collisions.
Because of the very highly forward peaked nature of most optically
allowed excitations, inelastic excitation scattering is only used below
100 eV and then only in an ad hoc manner. Below 100 eV, scattering due
to inelastic excitation collisions is assumed to be the same as that due
to elastic collisions (see section III.B).
The main purpose of this section is to determine whether this ad hoc
excitation collision scattering makes a significant difference in the
spatial energy deposition. When a MC calculation is run assuming
no excitation scattering at any energy, no difference is detectable in
the 3914 A intensity plots at energies above 300 eV.
For electrons of energies 300 and 100 eV, a difference is detected.
-6 2
The range (in units of 10 gm/cm ) changes from 1.25 to 1.34 for an
electron energy of 300 eV and from 0.365 to 0.391 for an electron energy
of 100 eV. This means that the extra scattering due to the inelastic
excitation events reduces the range by about 7% at these two energies.
C. Comparison of Different Elastic Phase Functions on the
Electron-N2 Collision Profile
The large influence of the phase functions on the spatial energy
deposition has been pointed out in section V.D and will be further dis
cussed in section VI.D. These phase functions all have some type of


AU=1/(1,+AA-TU)
CU=1./(lo+C+TU)
RL U=1EXP(THET( I )/Q 2 ) { S IN ( THE T( I) )/Q2+COS(THET( I )} )
F3=Q1/(1. + EXP(-3*1 416/02) )
F3RF 3*RLU
Ft I 1 = FF1 *( AU-AL )-FF2*( CU-CL )+F3R
1 CONTINUE
F( 1)=00
F(NX ) = 1 .0
C THE ARRAY F(I) OF PROBABILITIES FOR SCATTERING IN CERTAIN ANGULAR
C REGIMES HAS BEEN SET*
RETURN
END
PHF00046
PHF00047
PHF00043
PHF00049
PHF00050
PHF00051
PHF000S2
PHF00053
PHF00054
PHF00055
PHF00056
PHF00057
PHF00058
-191-


NI Z=0
RDTAP190
13
CO NT I NUE
RD TAPI 91
6769
CONTI NUE
RDTAP192
IF (NCHE EQ a 1 )GO TO 30
RDT API 93
GO TO 6622
RD TAPI 94
30
CONTINUE
RDTAPI 95
333
CONTINUE
RD T API 96
WRITE(6.334)ISAV
RDTAP197
334
FORM AT( ISA V=, 19 )
RDTAP198
RETURN
RDTAP199
END
RDTAP200


CHAPTER VIII
CONCLUSIONS
There are several different theoretical approaches now being em
ployed to study the auroral electron energy deposition problem. Re
searchers using these methods have concentrated for the most part on the
details of the computation and on the input atmospheric parameters.
One of the concerns of this work was the cross sections, both dif
ferential and total, and their impact on the spatial and energetic
aspects of the electron energy deposition. This research has shown that
the input cross sections have a very large influence on the resulting
electron energy deposition.
Perpendicularly incident electrons with energies from 0.1 through
5.0 KeV were degraded in molecular nitrogen using a Monte Carlo spatial
energy deposition technique. This degradation method followed each elec
tron, its secondaries, and its tertiaries in a collision by collision
manner down to 30 eV. Below 30 eV, a multiple elastic scattering dis
tribution was used to describe the energy deposition process down to the
cutoff energy of 2 eV.
This Monte Carlo calculation employed new phenomenological differen
tial elastic and doubly differential ionization cross sections which agree
quite well with experimental data. Other cross sections previously
developed for N^ were also used in these computations.
-152-


xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EPN31E71Q_ZYZQ08 INGEST_TIME 2017-07-13T22:10:20Z PACKAGE AA00003911_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES


14 JULY
1978
NERDC
CARD LIST UTILITY
2 CONTINUE
STOP
END
MAINO146
MAINOl47
MAINOl48
-09 L-


-50-
MONTE CARLO CALCULATION


(ujo)z
3914 A Intensity (I03 exc)
p o o p p
, b ho b oo b ho
-68-


14 JULY 1978
NERDC
CARD _1ST UTIL ITY
C £XC5(I) ARE THE NUMBER OF EXCITATIONS THAT CAUSE THE 3914 A
C EMISSION THAT ARE EXCITED BY EACH ELECTRON IN THESE LOWER ENERGY
C BINS.
2255 FORMAT ( 8E 10.0)
WRITE(6.3509)
3509 FORMATl1EXC. OF 3914A BY SECONDARIES.5X.ENERGY BIN ENDS*./)
DO 4509 1=1. NTOP
IP1=I+1
45 09 WR ITE ( 6. 351 0)EXC5( I).WIS(I).WISUPl)
3510 FORM AT(G15716XG15.75XG15.7)
READ(5,2 2 66) (NIE(I ) .1=1. NUMG)
C N IE( I ) INDICES INDICATE WHICH STATES ARE IONIZATION AND WHICH
C ARE ELASTIC.
2266 FORMAT! 161 5)
WRIT E(6.351 1 )
3511 FORMAT!///, STATE N0.*,5X.*NIE VALUE*,/)
DO 4511 1=1,NUMG
*511 WRITE6.3512)1,N IE( I)
3512 FORMAT(17,112)
RETURN
END
RDCAR145
RDCAR146
RDCAR147
RDCAR148
RDCAR149
RDCAR150
RDCAR151
RDCAR152
RDCAR153
RDCAR154
RDCAR155
RDCAR156
ROCAR157
RDCAR158
RDCAR159
RDCAR160
RDCAR161
RDCAR162
RDCAR163
RDCAR164
RDCAR165
i
ro
o
cn


-150-
E (eV)
Figure 7.5d


-62-
type is inelastic then the individual excitation or ionization event is
found as wel1.
There are cross sections for thirty-four states of N2 employing the
papers of Jackman, Garvey, and Green (1977) and Porter, Jackman, and
Green (1976). Using all these states in the MC calculation would greatly
increase the cost. It was therefore decided to reduce these thirty-four
states to nine states. Two allowed states, the b \ and the b'
and the six ionization states were kept the same as given in the papers.
For the ninth state, all the Rydberg and forbidden states were combined.
Above 200 eV, the forbidden states are contributing only a minuscule
amount to the total cross section. Since the other states have roughly
the same In E/E fall-off at high energies, it is assumed that the pro
babilities for excitation to any of these states will be constant. These
probabilities were simply found from the ratio of the cross section of
the state in question to the total inelastic cross section at the elec
tron energy of 5 KeV.
In Table 4.3 these states, their probabilities, and thresholds are
presented. The probability, pc> of the composite state is simply
m
pc = l pi
c i=l 1
(4.18)
where m = the total number of Rydberg and forbidden states and p^ is the
probability for excitation of the ith Rydberg or forbidden state. The
average threshold, Wc> for exciting the composite state is found easily
with the following equation
m
l P- W.
_ i=l 1 1
'c m
l Pi
i=l 1
W
(4.19)


-39-
Figure 3.6 N2 electron impact cross sections. The total inelastic,
Eq. (3.16) (solid line), total elastic, Eq. (3.9) (dashed
line), total inelastic plus elastic, Eq. (3.16) plus
Eq. (3.9) (dash-dot line), and the experimental inelastic
plus elastic values (Blaauw et al., 1977), x, are pre
sented here.


ACKNOWLEDGEMENTS
Dr. A.E.S. Green has helped the author a great deal in his efforts
to complete this work. The author sincerely appreciates this guidance.
He also wishes to thank Dr. R.H. Garvey and Dr. R.A. Hedinger for their
helpful discussion about the dissertation. David Doda, David Killian,
E. Whit Ludington, George Sherouse, and Ken Cross were instrumental in
providing assistance with computer problems and other dissertation-related
work.
Woody Richardson, Marjorie Niblack, and Wesley Bolch were extremely
helpful in drafting the figures. The final manuscript was then typed and
refined by Adele Koehler. The author is grateful to Adele for her prompt
and professional assistance.
The author wishes to thank Joseph Pollack for aiding in editorial
matters concerning the dissertation. A thorough reading and criticism
of the dissertation by the author's committee (including Dr. A.E.S. Green,
Dr. L.R. Peterson, Dr. T.L. Bailey, Dr. S.T. Gottesman, and Dr. G.R.
Lebo), Dr. II.L. Chameides, and Dr. A.G. Smith was extremely helpful.
The author is especially grateful to his parents, Rev. and Mrs. H.W.
Jackman, and to his sister, Kathi Jouvenat, for their encouragement and
support throughout graduate school.
The author gratefully acknowledges financial support from the De
partment of Physics and Astronomy and the Graduate School of the Univer
sity of Florida and from NASA grant number NGL-10-005-008.


-81-
state can be graphed in an intensity plot with the z-axis as the
abscissa.
In Figure 5.2, the intensity plot from 5000 incident 1 KeV electrons
is graphed (the model used in this MC calculation should only be taken
as an illustrative example) in histogram form. Bins along the z-axis
are taken to be 0.5 cm in width for these incident electrons. The linear
portion of the curve may be extrapolated, as illustrated by the dashed
line, to define a mean range of the beam.
All the intensity plots are normalized in this paper so that the
beam starts out at z = 0 cm along the z-axis. The intensity in Figure
5.2 seen at negative values of z is brought about by backscattered
electrons. The error bars given near the peak of the histogram are
found simply from a method described in section IV.D.
From Figure 5.2, the range is seen to be 16 cm for these 1 KeV
2
electrons. Range values, R^, in units of gm/cm are written
Rg = Rce (5.1)
3
where R is the range in cm, p = n MN (in gm/cm ), n is the number
O
density of molecules (in #/cm ), and is the weight (in gms) of an
15^ 3
molecule. In this case, n = 8.2 x 10 molecules of ^/cm =
_ p p ^ e
4.651 x 10 gm/^ molecule, and Rc = 16 cm; therefore, Rg = 6.06 x 10
gm/cm^.
C. Previous Experimental and Theoretical Work on the
3914 A Emission of n£
Grn (1957) measured for air the total luminosity of the 3914
radiation in planes perpendicular to the axis of the electron beam with


4
o
X
cu
>v
co
c
c
o<
CT)
ro
1 1 1 1
A Z =0.667
1 1 1 1
Z = 1.81
1 1 1
Z = 2.96
0
A
*
*
"O
y
A
o
X
o
o
O
A
A
X
X o
o
X o
o
A XXyXXy O00
AA x xxxx
Aa x x i
XX
A
i .A 1 r^xn,
.** X**
00 oaA^xxxxv
,6^aAa)a>>
0.0 2.0 4.0 6.0 8.0 0.0 2.0 4.0 6.0 8.0
p (cm)
0.0 2.0 4.0 6.0 8.0
-120-


Figure 3.2 N2 electron impact total elastic cross section data from Sawada,
Ganas, and Green (1974), x; Shyn, Stolarski, and Carignan (1972),
0; Herrmann, Jost, and Kessler (1976), and Banks, Chappell, and
Nagy (1974), v. Equation (3.6) is represented by the dash-dot
line, Eq. (3.8) by the dashed line, and Eq. (3.9) by the solid
line.


14 JULY 1978
NERDC
CARD LIST UTILITY
NG=(NSCS+1)/2
CALL PET I
EVPR1=E V
WL =T+WLOSS
KED = '(IL/EV
IF ( E V LEo WL) GO TO 2213
700 CONTINUE
IF (NCHE oEQo NSCS1GO TO 751
C IF NCHE = NSCS THEN THE COLLISION WAS ELASTIC
IF ( N IE ( NST AT ) EOo 2 > GO TO 2213
IF { E V GT a EINEUGO TO 2213
C IF THE ENERGY EV IS LESS THAN E INEL AND THE COLLISION IS INELASTIC
C AND ONLY AN EXCITATION THEN ALLOW IT TO SCATTER BY THE ELASTIC
C PHASE FUNCTION,,
C THE PRIMARY SCATTERING ANGLE IS NOT COMPUTED AND THE SUBROUTINE
C THAT IS USEFUL FOR THIS IS NOT INCLUDED IN THIS PROGRAM. THE
C SCATTERING FROM THE IONIZATION EVENTS IS MINISCULE FOR THE
C PRIMARY ELECTRONS
C
c
C CALCULATE THE SCATTERING ANGLE AFTER AN ELASTIC COLLISION
75 1 CONTINUE
CALL RANDU
IR 1=IR2
C
C CALCULATE THE SCATTERING ANGLE
IF(NP HF EOa 0)GO TO 771
CALL PHF
DO
DO 5 J=1 NA
J2 = J4-1
IF(RR F(J2 ) >4,6,5
5 CONTINUE
WRITE{ 6.50 0 )
500 FORM AT(1,******-COS INE CELL NOT FOUND-ERROR******** )
6 COSPAN=COS ( THET ( J2 ) )
GO TO 761
4 J1=J2-1
FJ2=F(J2)
FJ 1=F( J 1 )
CJ2=THET(J2)
CJ 1= THE T{ J 1 )
PAN=(CJ1*(FJ 2-RR J+CJ2*(RR-FJ1))/(FJ2-F J1)
C PAN IS THE POLAR SCATTERING ANGLE
COSPAN=COS(PAN)
MCOOO145
MC000146
MCOOO147
MC 000148
MC 000149
MCOOO150
MC 0001 51
MCOOO152
MCOOO153
MC000154
MCOOO155
MC000156
MC 000157
MCOOO158
MC000159
MCOOO160
MC 000161
MC000162
MCOOO163
MC000164
MCOOO165
MC000166
MCOOO167
MCOOO168
MC 000169
MCOOO170
MCOOO171
MCOOO172
MCOOO173
MC000174
MC 000175
MCOOO176
MCOOO177
MCOOO178
MCOOO179
MC 000180
MCOOO181
MC 000182
MCOOO183
MCOOO184
MCOOO 185
MCOOO186
MC000187
MC 000188
MCOOO189
-179-


-53-
and z are the Cartesian coordinates of the electron. The polar angle 0
is measured with respect to the z-axis and the

angle measured with respect to the x-axis (see Figure 4.2). In this
approach, the initial coordinates xQ, yQ, zq, eQ, and o were all set
equal to zero. The coordinates xb, yb, zb, eb> and 4>b of the electron
before starting on its journey to a collision are, therefore, initially
established as xb = xQ, yb = yQ, zb = zQ, eb = 0Q, and b = The MC approach relies on the random number, R, between 0.0 and 1.0
to aid in the deposition calculation. For each collision several R's
are needed and for each R a new property of the collision is gained. In
order to explain this MC approach, an accounting of the random numbers
and their subsequent usefulness is now made. The multiple elastic scat
tering distribution used below 30 eV and the lowest energy cutoff 2 eV
are also described.
1. First Random Number, R]
The first random number, R^, is used to find the path, Py, traveled
by the electron before it collides with a molecule of N^. Calculation
of Py proceeds in the following manner. The mean free path, A, is
defined as
(4J)
3
where n is the density of N2 molecules in #/cm and ay(E) is the total
2
(inelastic plus elastic) cross section of N2 in units of cm at an energy
E. The densities used at the various initial input energies are given
in Table 4.1.


-67-
5. Sixth Random Number, Rg
The sixth random number, Rg, is computed only if the collision type
is an ionization event. This Rg determines the energy lost by the
primary in creating a secondary of energy, T Using the S(E,T) from
Eq. (3.19) the following relationship is established:
Ts
/ S(E,T) dT
= 0
I0N(E^
(4.26)
Integrating the numerator in Eq. (4.26) and using Eq. (3.21) to solve
for T Eq. (4.27) is derived.
Ts = r(E)[tan{Rgtan"1[(TM TQ(E))/r(E)]
+ (Rg l)tan_1[To(E)/r(E)]}] + TQ(E) (4.27)
The energy loss, W, is then found by the relation:
W = Ik + T$ (4.28)
where 1^ is the ionization threshold for the kth ionization state.
6. Multiple Elastic Scattering Distribution Used Below 30 eV
The MC calculation can be used to degrade an electron down to
practically any energy. Even below the lowest threshold for excitation
to any vibrational level, the electron will still lose energy via elastic
collisions with molecules of nitrogen as well as other electrons. This
energy loss to other electrons is fairly low unless a substantial frac
tion of the gas has been ionized (see Cravens, Victor, and Dalgarno,


14 JULY 1978
NERD
CARD LIST UTILITY
READ(5,2255 ) (ALT(I ). 1=1.NSPZ)
C ALT(I) ARE THE Z VALUES AT WHICH THE YIELD SPECTRUM IS CALCULATED
RE AD (5* 2255 XRYS I >,I=1.NSPR)
C RYS(I) ARE THE RHO VALUES AT WHICH THE YIELD SPECTRUM IS CALCULATED
READ ( 5, 2255) (E IP ( I ) .I = 1.NEIP)
C EIP(I) ARE THE ENERGY VALUES AT WHICH THE YIELD SPECTRUM IS CALCU-
C LATED
NE IP 1 = NE I P 1
WRITE(6,3499)
3499 FORMAT( MALTITUDES WHERE SPECTRUM WILL BE CALCULATED*,//.
* INDEX*,5X.ALT IN KM.*,/)
DO 3497 1=1,NSPZ
WR ITE(63504)NZX(I ) ,ALT( I )
3497 CONTINUE
WRITE(6.3599)
3599 FORM AT(1R HO VALUES WHERE SPECTRUM WILL BE CALCULATED*.//,
* RHO ( I N KM, ) ./ )
DO 3597 1=1,NSPR
WR ITE(6,3604 )RYS( I )
3604 FORMAT!1 PEI 1.3)
3597 CONTINUE
WRITE(6,3515)
3515 FORMAT!///, SPECTRA 1NDEX*.8X,ENERGY INTERVAL ENDS'.///)
DO 4515 1= 2 NE IP
IM1=I-1
WRITE(6.3517)1 Ml,EIP(I Ml) ,EIP(I)
3517 FORMAT(17,12X,1PE11.3,3X,IPE113)
4515 CONTINUE
1142 CONTINUE
NAR1 =NAR + 1
NUMG=NUMST + NUMGAS
NT0P=NSEC-1
IF (NRHO oNEo 1) GO TO 1141
READ(5,2244)(NCR0(J).J=1,NVR)
C NCRO(J) ARE THE INDEX VALUES OF THE Z DISTANCES AT WHICH A RHO
C INTENSITY PLOT IS DESIRED.
READ(5,2255)(ZNUM( J).J = 1,NVR)
C ZNUM(J) ARE THE Z VALUES AT WHICH A RHO INTENSITY =>_0T IS DESIRED
WRITE(63503)
3503 FORMAT (///,* 1ALT ITUDES WHERE RHO EXC WILL 3E CALCULATED',//,
* I NDEX ,5X,* ALT IN KM',/)
DO 4503 J=1,NVR
4 5 03 WRITE(6,3504 )NCRO(J). ZNUM(J)
3504 FORMAT!I5.5X.1PE113)
READ(5,2255)(RHO(I).1=1,NRM)
RDCARD 73
RDCARD74
RDCARD75
RDCARD 76
RDCARD77
RDCARD78
RDCARD79
RDCARD80
RDCARD81
RDCARD82
RDCARD83
RDCARD84
RDCARD85
RDCARD86
RDCARD 87
RDCARD88
RDCARD89
RDC ARD90
RDCARD91
RDCARD92
RDCARD93
RDCARD94
RDCARD 95
RDCARD96
RDCARD97
RDCARD98
RDCARD99
RDCAR100
RDCAR101
RDCAR102
RDCAR103
RDCAR104
RDCAR105
RDCAR106
RDCAR107
RDCAR108
RDCAR109
RDCAR110
RDCAR111
RDCAR112
RDCAR113
RDCAR114
RDCAR115
RDCAR1 16
RDCAR117
-204-


-145-
The four variable spatial yield spectrum is presented in Figure 7.5
(a, b, c, and d) for an incident electron energy of 1 KeV. It is given
at four radial distances at each longitudinal cut (all in units of
fractions of the range). The U(E,p,z,Eq) from other incident electron
energies are not presented here but show a similar type of behavior.
The shape of U(E,p,z,EQ) is observed to be quite similar to U(E,z,Eq)
[see Figure 7.3] and, indeed even to U(E,Eo) [see Green, Jackman, and
-1 52
Garvey, 1977, Figure le]. The lower energy power fall-off is<*ER *
in all three yield spectra. All three spectra also exhibit a constant
component in the middle energies with the source term feature at the
incident energy (ER = 1.0).
The four variable and three variable spatial yield spectra illustrate
an increasing tendency at higher values of energy (ER % 0.9 -* 1.0) and
at the lower values of z and p. This feature is not as prominent in the
non-spatial yield spectrum U(E,E ), which is calculated by integrating
over the spatial component of the spatial yield spectra. In the inte
gration process the higher energy spectra increase is averaged out by
the equally important higher energy spectra decrease exhibited at the
higher values of z and p.
Knowledge of U(E,p,z,EQ) implies more detailed information about
the entire spatial degradation process. Once the U(E,p,z,Eq) is known
then the number of excitations J^Cp.^2) f the jth state can be found.
This Jj(p.|^2z) is a result of an incident electron flux 4>(EQ) and is the
number of excitations at altitude z in the ring between p^ and p?. Thus
00
Ji(p-u9>z) = ir(p?-P?) I I (E )U(E,p,z,E )P(z)dEdE
J 1 W. W. 0 0 0
(7.12)


Figure 6.9 An intensity plot for electrons of energy 1 KeV is presented as a function of z. The
differential and total screened Rutherford elastic cross section are used in this
calculation. The range, found from the dashed line is 18.3 cm or 6.98 x 10-b gm/cnr.
-123-


14 JULY 1978
NERDC
CARD LIST UTILITY
C
c
c
c
c
c
c
30
c
20
C
C
1
c
SUBROUTINE COLTYP
SUBROUTINE COLTYP ****
THIS SUBROUTINE IS ACCESSED THROUGH MC AND MESDo
THIS SUBROUTINE COMPUTES THE TYPE OF COLLISION THAT OCCURS.
SOME OF THIS SUBROUTINE IS SET UP TO TAKE MORE THAN ONE GAS
COMMON ALPE(6).BETE(6),CE( 6).FE(6) WE ( 6) ALF A ( 1 5) ,BEFA(15).
1 CFA(15),FFA(1S) ,WFA<15),WF(15),FAC I( 15).NFA,NAR,SA(3.80> .
2 ZALT(80).ENR(30> .PCFA(3 .30> .WCFA(3 ,30),NENR,FDG(3 >,PSE(3 >,
3 PION(3),AT(3),A(6 3)B( 5 3)G(S3)UH( 10) *UI( 10) U J ( 10)
4 UD(10).USNU(10) aUSF(lO) UEIN( 1 1 ) IUTP. IUT Ml ,DG(3).NSG(3),NIG(3),
5 NUMGAS.NUMST,WIS( 5 0) .NSEC.MUNIT.ISEED.1 STOP.I LAST.EIN.NPRIM,COS I
6 H.EM IN.ZST ART, T MIN .THRESH! 3 ).AK( 2.3). AJ ( 3. 3 ) GAM A ( 2 3) ,
7 TO(3 3) FAC(6) ,A0E<6> .A1E(6),BOE(6),B1E(6),COE(5),DOE(6),
8 D1E6 ),PR0B(40 >,W(40),NIE(40),A21(3) A22( 3) .A 31 ( 3) ,A32( 3 ) .
9 A 33 ( 3 ) B1 1 (3) B1 2 ( 3 ) .61 3(3),Cl (3) C2 1(3), C22C 3), C3 1( 3) ,
A C32( 3 ) D 1 ( 3 ) D 2 ( 3) ,F 1(3) ,F2(3) THR 1(16) ,AKI(2,15) ,AJI(3,16) ,
B GAMA I (2,16 ) .TO I (3, 16 ) S I GT ( 6 ) THE T ( 40 ) NX C SI E( 20) ,E IT( 20) ,
C IEILS.IEILM1.P.R.ZDIS.NPIN,NOP,COSPA,RT,F(40),R1.R2,R3,R4,
D R 5.XV,YV,ZV.XVN,YVN,Z VN,PA,PH I ,EV,WLOSS.NSTAT,NSC S NPHF,
E I R1 I P.2 NG, EVPR I .RAN EMSD EI NEL E XC 5 ( 50 ) CO SPAN, T, FOVAL
E=E V
EV IS THE ENERGY FROM THE MAIN PROGRAM
DGT=0.0
DO 30 1=1,NUMGAS
DG T=DGT+DG( I)
CALCULATE DGT, THE TOTAL DENSITY OF THE GASES
DO 20 1=1,NUMGAS
FDG( I )=DG( I ) /DGT
CALCULATE FDG(I). THE FRACTION DF THE TOTAL MODEL ATMOSPHERE THAT
IS THE ITH SPECIE.
SI GE E = 0 o 0
K=0
DO 1 1=1.NUMGAS
DO 1 J= 1,2
K=K+1
SI GEE = S I GE E+SIGTK) *=DG( I )
CALCULATE SIGEE, THE TOTAL CROSS SECTION.
PSE(1)=SIGT(2)/SIGEE
P= PS E( 1 )
IF (R LT P) GO TO 40 3
GO TO 404
NSTAT=NUMST+1
GO TO 4
COLTYPOl
COLTYP02
COLTYP03
C0LTYP04
COLTYP05
C0LTYP06
COLTYP07
C0LTYP08
C0LTYP09
COLTYP10
,COLTYPI1
COLTYP12
COLTYP 13
C0LTYP14
COLTYP15
COLTYP 16
C0LTYP17
COLTYP 18
C0LTYP19
C OLTYP 20
C0LTYP21
CCLTYP22
COLTYP 23
COLTYP24
C0LTYP25
C0LTYP26
COLTYP 27
COLTYP28
COLTYP29
C0LTYP30
C0LTYP31
COL TYP 32
C0LTYP33
COLTYP34
COLTYP35
C0LTYP36
COLTYP37
COLTYP38
C0LTYP39
COLTYP 40
C0LTYP41
COLTYP42
COLTYP 4 3
COLTYP44
C0LTYP45
403
-L9L-


-17-
energy E and direction y, P¡qn(y*y E',E) 1S the probability during an
ionization collision with a neutral species that an incident electron
with energy E1 and direction y1 will result in a secondary electron of
energy E and direction y, and p .(y',y,E',E) is the probability during
an inelastic collision (excitation or ionization) with a neutral specie
that an incident electron with energy E1 and direction y1 will result in
the incident electron being scattered into direction y with energy E.
Some techniques from each of the other three MC methods were in
cluded in this work. Some new approximations and assumptions were made,
however, to enhance the accuracy of the computations as well as simplify
some of the calculations. These assumptions are discussed in detail in
Chapter IV.
In this MC work the information is stored in a collision by collision
manner on a magnetic tape. Once all the case histories are generated,
then, the tape is scanned and any correlations of interest may be deter
mined. The choice of altitude and energy intervals is specified only
during the scanning of the tape. This method allows for greater flexi
bility in minimizing the statistical uncertainties of the results,
while, at the same time retaining good spatial or energy resolution
(Porter and Green, 1975).
All the degradation methods mentioned in this chapter require cross
sections as input. The cross sections used in this MC work are, there
fore, discussed in the next chapter.


-66-
^ PiCTTI(E) + CTTE(E)
5^1)
rc
and E > 200 eV
(4.22)
then the inelastic collision results in the excitation of the jth state.
Thus the Rg random number for an electron of energy E > 200 eV will
determine which type of collision occurred when satisfying Eq. (4.20),
(4.21), or (4.22).
For energies below 200 eV, the following relations must be con
sidered: If
aTE(E) -j(E) + aTE(E)
^lT K5 ^T)
and E < 200 eV
(4.23)
where ^(E) is the cross section for exciting the first inelastic state,
then the inelastic collision results in the excitation of this state.
A relation similar to Eq. (4.22) can now be established for j = 2
to 8 such that: If
I a-j(E)+0te(E) J ai(E)+aTE
aT(E) < R5 oT(E)
(E)
and E < 200 eV
(4.24)
then the jth inelastic state is excited. If
l ^(E) + aTE(E)
Rr > nn and E < 200 eV
5 ot(E)
(4.25)
then the excitation of the composite state is assumed and the energy loss,
Wc, in this case is found through a linear interpolation with the use of
the values given in Table 4.3.


-58-
Using the screened Rutherford differential cross section form (see
Eq. (3.1)), it follows that
e1 = cos"1 [1 + 2n inlV-n'R ] (4.8)
3
For model 1 (see Eq. (3.12))
0'
cos
-1
[ ^T -p + 1 + a]
R3C(2 + a) 1 a '] + a 1
and for model 2 (see Eq. (3.13))
0'
cos
-1
\/b2 4AC n
2A J
(4.9)
(4.10)
with
A =
R3 +
a[(2 + a)'
- a"1]
0 f)
(2 + c)[(2 + c)_1 c'1]
B = -A(a c) +
[(2 + a)"1 a"'] [(2 + c)"' c]
-1
(1 f)
-1
1
and
C = -A(l +a)(l +c) + HIjLc]
[(2 + a)'1 a'1]
(1 f)(1 + a)
[(2 + c)"1 c"1]
Model 3 (Eq. (3.14)) is not so easy to write in such a convenient
form. The equation for primary scattering after an ionization event
(Eq. (3.17)) is, also, not easily inverted.
For these two differential cross sections, the following approach
is taken. The angular range from 0 to 180 is divided up into angular
intervals. A certain probability for scattering at angles less than the
end of each angle interval is calculated from the differential cross


14 JULY 1978
NERDC
CAPD LIST UTILITY
C
C
c
c
c
c
c
c
c
c
SUBROUTINE PHF
SUBROUTINE PHF
THIS SUBROUTINE IS ACCESSED THROUGH MC
THIS SUBROUTINE HELPS CALCULATE THE POLAR ANGLE OF SCATTERING
AS A RESULT OF AN ELASTIC COLLISION
ANY PHASE FUNCTION FORM MAY BE PLACED FEREo
RIGHT NOW MODEL 3 IS EXPRESSED HERE
this subroutine is mainly for phase functions which are not
EASILY INTEGRATED AND SOLVED FOR THE POLAR ANGLE OF SCATTERING
COMMON ALPE (6) B ETE ( 6 ) ,CE(6),FE(6) WE(6) .ALFA ( 15) ,BEFA{15) .
1 CFA(15),FFA(15).WFA(15),WF(15).FACI( 15),NFA.NAR SA(3.80) .
2 ZALT 80) .ENR(30 ) PCFA (3 ,30) WCFA(3 30 ). NENR, FDG(3 ) PSE( 3 ),
3 PION (3 ) AT ( 3) ,A (6.3) ,B( 5,3).G(5.3).UH(10) *UI( 10) .UJ( 10) ,
4 UDilO).USNU(lO) ,USF(10) .UEIN(ll), IUTP. IUTM1 DG( 3) NSG( 3) ,N IG( 3) ,
5 NUMGASNUMST.Wl S(50) ,NSEC.MUNI T,I SEED.I STOP.I LAST.ElN,NPRIM.COSI
6 H.EMIN.ZSTART,TMIN,THRESH!3>,AK{2,3),AJ(3,3),GAMA(2.3),
7 TO(3.3) ,FAC(6) ,AO 5 (6) ,AlE(6) ,BO E( 6 ) ,B1E(6 ),CO E(6)DO E(6)
8 DIE!6),PRO B(40),W( 40) ,NIE(40),A 21(3) ,A22< 3) ,A 31(3) ,A 32(3),
9 A33(3) ,B11 (3),B12(3).B13(3),C1(3) C21 (3),C22(3),C31C 3),
A C 32 ( 3 ) D 1 ( 3) D 2 ( 3) ,F 1(3) ,F2{3) THR I(16),AKI(2,16) ,AJ 1(3,16 ) ,
B GAMA 1(2,16). TO I (3, 16),SI GT (6 ) THE T ( 4 0 ) NX C S I E ( 20) ,E I T( 20) ,
C IEILS.IEILM1 ,P, R ZDIS.NPIN, NOP.COSPA.PT F ( 40 ) R 1 R 2. R3 R 4,
D R 5.XV,YV,ZV.XVN,YVN.Z VN.PA,PHI ,EV,WLOSS.NSTAT,NSCS ,NPHF,
E I R1 IR2 ,NG, EVPR I, RAN, EMSD, E INEL EXC5I 50 ), CO SPAN, T,F OVAL
E=EV
FT={ E /1 0 )**(.5)
FM=FT / ( FT +0.87)
C= l,2 7*(l,-(12o/E)**0o 27 )
02=0. 43*E**(-0. 29)
QT=(E/100o)**(.84)
01=0 T/(QT+1.92)
AA = 0.1 1
FC = 1 -Ql
FCP= 0 OE 0
IF ( E .LT. 200 )FC=FC*FM
IF ( E LT 20 0. ) F CP= 1 Q1-F C
FFI=FC/( l/( 2.+AA)- lo/AA )
FF ZFCP/ (l./(2. + C)-l./C)
AL=1a/AA
CL = l./(C+2. )
NX Ml = NX 1
DO 1 1=2,NXM 1
TU=COS(T HET( I ) )
PHF 00001
PHF00002
PHF00003
P HF00004
PHF00005
PHF00006
PHF00007
PHF 00008
PHF00009
PHFOOO10
PHF 00011
PHFOOO12
PHFOOO13
PHF 000 14
,PHFOOO15
PHF 00016
PHFOOO 17
PHFOOO18
PHF 00019
PHF00020
PHF 00021
PHF 00022
PHFOOO 23
PHF 00024
PHFOOO 25
PHF00026
PHF 00027
PHFOOO 28
PHF00029
PHF 00030
PHF00031
PHF00032
PHF00033
PHF00034
PHF 00035
PHF00036
PHF00037
PHF00038
PHF00039
PHF 00040
PHF00041
PHF00042
PHF 000 43
PHF00044
PHF 00045
-190-


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
SPATIAL AND ENERGETIC ASPECTS OF ELECTRON ENERGY DEPOSITION
By
Charles Herbert Jackman
August, 1978
Chairman: A.E.S. Green
Major Department: Physics and Astronomy
The spatial and energetic aspects of the electron energy degradation
into molecular nitrogen gas have been studied by a Monte Carlo method.
Perpendicularly monoenergetic incident electrons with energies from 0.1
through 5.0 KeV were injected into the gas. This Monte Carlo de
gradation scheme employed previously developed N^ cross sections with new
phenomenological differential elastic and doubly differential ionization
cross sections. All these agree quite well with experimental work and
are consistent with the higher energy theoretical total cross section
fall-off with energy.
Information has been generated concerning the following topics:
o
1) range values and 3914 A intensity profiles for the longitudinal and
radial directions which can be easily compared with experimental work;
2) a sensitivity study characterizing the influence of the input cross
sections on the spatial energy deposition of the electrons; 3) the rate
of energy loss by the electrons as they interact with the N2 gas; and
4) spatial yield spectra for incident electron energies in the range
from 0.1 to 5.0 KeV (evaluated between 2 eV and the incident energy)
which are analytically characterized for future work on atmospheric prob
lems dealing with incident energetic electrons.
vi


Figure 5.3 An intensity plot for electrons of energy 2 KeV is presented as a
function of the longitudinal direction. The x's represent relative
experimental values from Barrett and Hays (1976) and the histograms
present the data from model 3 (heavy line) and the screened
Rutherford (light line). The straight solid and the straight
dashed lines represent extrapolations to find the range for the
model 3 and the screened Rutherford elastic differential cross
sections, respectively.


-40-
> 50 eV) is from the ionization collisions. These ionization cross
sections were also taken from PJG and JGG. The total inelastic cross
section found by summing these inelastic cross sections was fit with
the function
an(E) -
q0F[l (fn
Wxa.S ,4EC
WE
(3.16)
This form has the characteristic Born-Bethe In E/E fall off behavior at
the large energies. The parameters a = 1, 6 = 4.81, C = 0.36, F = 4.52,
and W = 11 were found with the use of a nonlinear least square fitting
program which fit Eq. (3.16) to the sum of all the inelastic cross
sections. From 30 eV up to 5 KeV this form was used for the total
inelastic cross section.
Below 30 eV much structure in the total inelastic cross section is
evident. At these low energies, the total inelastic cross section can
be read numerically into the MC program. This total inelastic cross
section is illustrated by the solid line in Figure 3.6.
Consider now the scattering of the two electrons involved in an
electron impact ionization collision. In reality, only the incident
electron is scattered. The other electron is simply stripped from the
molecule and given kinetic energy in a certain direction of travel.
Experiments are unable to distinguish between the incident electron and
the electron stripped from the molecule. In this paper, the ionization
event is assumed to cause scattering of both electrons. The scattering
angle of either is then measured with respect to the incident electron's
path.
After the collision event the electron with the higher energy is
designated the primary electron and the electron with the lower energy


c
c
c
c
c
c
c
c
c
c
c
c
16
c
M IS THE UNIT NUMBER FOR THE TAPE
DO 8 M1=1,NPIN
NPIN = I IN OUR CASE IT CAN BE VARIED HOWEVER To
ENT GROUPS OF ELECTRONS TO BE DEGRADED
N= 0
N KEEPS TRACK CF THE NUMBER OF COLLISIONS
DO 18 IPRI= 1 NOP
NOP= THE NUMBER OF PRIMARIES
IEL=IEL+1
IEL KEEPS TRACK OF THE NUMBER OF PRIMARY ELECTRONS
ALREADY BEEN DEGRADED
WRITE OUT IEL IF DESIRED
STARTING PARAMETERS
ZV=ZST ART
XV =Oo 0
YV=00
COSPA=COSI
PH 1=0o 0
PA-= ARCOS (COSI )
S=0.
EV=EIN
IF(EV oLE EMIN)GO TO 27
EMIN IS THE CUTOFF ENERGY
IESCP=0
MC000046
MC000047
MC000048
ALLOW FOR DIFFER-MC 00 0049
MC000050
MC000051
MC 0000 52
MC000053
MC000054
MC 0000 55
THAT HAVE MC000056
MC 000057
MC 000 058
MC000059
MC000060
MC000061
MC000062
MC0D0063
MC000064
MC 0000 65
MC0000 66
MC000067
MC000068
MC000069
MC000070
MC000071
MC000072
-176-


203-
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<^0'0'0\0'0'0'0'0'O'0NNN
QQQQQQQQQOQQQOQQQQQQQOQOOQQ
aacraaaaaaacraaaaaQiaixaaaaaccaa
UUUUUUUUUUUUUUUUOUUUUUUUUUU
QQQQQQQQQQOQOQOQOQQOQQOOQQO
crcnrcrcxcxcrcrttcrtrcxcrQcQccrircrcrcracircxaacra:
o
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iDSUJCDUja UJ ¡1.ILM N SDcascnui zj o u. >o:huzm z -* o
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ujz zjin-io uj a u)Z_i nwiz^iu
jX uj uj o i-a o ir uj cr id >- ujov w u^-
II-UJIUJ>-i'£>N UJ S Ul UJ UJ CU UJ Ul I N X CM 01
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011- I- --ZZZDS *1-11- M3-SSla
ui*-< uj uji-'-i-o DzDin t- i/)w(- *(jQ.in< s
uicruix (/) 1-t < I-Z H E O I-(/>-' hj zq-q:
h-< awtr -v < *-* uioaMcri-oiro x uj_j zcoqio u. ui u
£zaazusii o. < z ~ z 3 u. *->axuj
DDUJ0)01<
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ZZZZZ*
N IU UJ 0
Z 01
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m
cu
0
0
uuuuuuw
N
uuou
N
0 u


-52-
This MC technique was applied to several incident electron energies.
The vast majority of the MC program runs used the Amdahl 470/175 computer
at the Northeast Regional Data Center at the University of Florida.
There were, however, several MC runs using the PDP 11/34 of the Aeronomy
group of the University of Florida.
It should be noted here that running the same program on both
machines at the same energy, EQ =1 KeV, showed a factor of 240 dif
ference in the execution time. Thus a program that takes four hours on
the PDP 11/34 will take one minute on the Amdahl 470/175. This time
advantage plus the ability to store each collision of the electrons on
magnetic tape does make the Amdahl 470/175 a more desirable "number
crunching" machine. The PDP 11/34 is only able to produce intensity plots
in the longitudinal direction. This mini-computer is thus mainly useful
in deriving a range (to be described in the next chapter).
Two programs were used in deriving the MC results. The first pro
gram (listed in Appendix A), the modified version of Brinkmann's code,
degraded the electrons in energy from their initial EQ down to the Ec
and recorded each collision and its properties on the tape. The second
program (listed in Appendix B) coalesces the data from the tape into an
array of ordered output. This output contains information for three
dimensional intensity plots, energy loss plots, and yield spectra.
C. Detailed Discussion of the Monte Carlo Electron
Energy Degradation Technique
Now, a more detailed discussion is given for the MC method of
degrading an electron's energy. An electron will start off with an
energy of Eq and coordinates xQ, yQ, zq, eQ, and Q. The symbols x, y,


C RV
c zv
7
RMFP=RMFP*1E5
RV=-ALOG(R)*RMFP
RT = RV
AND RT ARE THE PATH LENGTHS TO THE COLLISION*
RVCOS=RV*AC
IF (PA o GT a 1.570 0) GO TO 7
ZVN=ZVRVCQ S
N IS THE NEW Z COORDINATE OF THE ELECTRON.
RE TURN
ZVN=ZV+R VCOS
RETURN
END
ZVAL0046
Z VAL 0047
ZVAL0048
ZVAL0049
ZV AL 00 50
ZVAL0051
Z VAL 00 52
ZVAL0053
ZVAL0054
ZVAL0055
ZVAL0056
Z V AL 0 0 57
-197-


771
76 1
8979
8970
20
21
GO TO 761
CALL PHFEL
SI NPAN=SQRT( lo -COSPAN* COSPAN)
COSPL F=COSPA *COSPANSINPA *SINPAN*COSAZA
IF( CO SPLF a GTa 0a999) COSPLF=Oa999
IF { CO SPLF oLTo -0o 999) C OSPLF =-0.> 9 99
AR GC=1o COS PLF#COSPLF
IFARGC oLT, 0 o 0 EO)GO TO 8979
SINPLF=SQRT(ARGC)
GO TO 8970
SINPLF=laOE-6
CONTINUE
pl f= arcosicosplf:
IF (CO SPLF oLEo Oa 99 ) GO TO 20
ALF=0.0
GO TO 21
COS AL F= ( C0SPA*C0 SP H I *S INPAN*CO SAZA -SI NPHI SI NP AN* SI NAZA +
1 SINPA*COSPHI*COSPAN)/SINPLF
SINALF=(CO SPA* SI NPHI*SINPAN*COSA2A+COSPH I*SINPAN*SINAZA
1 +SINPA*S I NPHI*CO SPA N)/SINPLF
IF(COSALF oLTo -Oo999)COSALF=-0999
IF ( CO SALF .GTa 0 99 9) C O SALF = Oo 999
ALF= A RCOS(COSAL F)
IF ( SI NALF oLT. OoO) ALF=6 a 2831 85-A LF
CONTINUE
COSPA=COSPLF
PH I=ALF
MC000190
MC 0 0 0191
MC000192
MC000193
MC 0 00194
MC000195
MC 000196
MC 000197
MC000198
MC 000199
MC000200
MC000201
MC 000202
MC000203
MC 0002 04
MC 000205
MC000206
MC000207
MC 000208
MC000209
MC 000 210
MC000211
MC 000212
MC0J0213
MC000214
MC 000215
MC000216
-180-


14 JULY 1978 NERDC CARD LIST UTILITY
JU=NSG COLTYP73
DO 10 J=JL,JU
COLTYP74
P=PROB( J )*-DG( I ) *< 1 PSE ( I ) ) +P
C0LTYP75
IF (P G To R) GO TO 13
C0LTYP76
NSTAT = NSTAT + 1
C0LTYP77
c
NSTAT IS THE INDEX OF THE STATE.
C0LTYP78
10
CONTINUE
C0LTYP79
JL=JU+1
CCLTYP80
11
CONTINUE
COLTYP 81
13
WLOSS=W(NSTAT)
C0LTYP82
C
XL OSS IS THE ENERGY LOSS OF THE STATE*
C0LTYP83
12
RE TURN
CCLTYP84
END
C0LTYP85
-163-


-99-
A. Effects of Ionization Differential Cross Sections on
the Intensity Distributions
The primary and secondary differential ionization forms represented
in Eqs. (3.18) and (3.19) are convenient for calculating the scatter of
the electrons during an ionization event. Here, the influence of these
forms on the intensity plots will be considered.
Other MC calculations have computed the scattering of the electrons
during an ionization event. Brinkmann and Trajmar (1970) calculated the
primary scattering angle from experimental energy loss differential
cross section data. They then employed an empirical simplification of
the coincidence data obtained by Ehrhardt, Schulz, Tekaat, and Willmann
(1969), in which half of the secondary electrons were presumed to
scatter at four times the primary scattering angle and the other half at
tt radians plus four times the primary scattering angle.
In another MC approach, Berger, Seltzer, and Maeda (1970, 1974)
used the Moller cross section for the scattering of secondary electrons
as a result of an ionization collision. The angular deflection e is
given such that
sin2e = T(i 2e) + t + 4
where e is the energy transfer in units of E, and t is the kinetic energy
in units of the rest mass.
At the maximum incident energy of 5000 eV used in this work,
t % 0.01. Using this value of x for primary scattering, in which e < 0.5,
all scatterings are between 0 and 45. The secondary scattering turns
out to be between 45 and 90, since 0.5 < e < 1.0. This means that


-10-
and y is the cosine of the pitch angle associated with the direction of
motion of the electron. The symbols n, a, Q, and L are the number den
sity of the scatterers, the total cross section (both elastic and in
elastic), the momentum transfer cross section, and the loss function,
respectively.
The momentum transfer cross section, given in terms of the differ
ential elastic cross section, is written as
(2.6)
0 0
This Fokker-Planck equation may be thought of as a CSDA approach to
the spatial energy degradation problem. Its solution, therefore, is only
accurate in the higher energy regime.
Banks, Chappell, and Nagy (1974) were able to calculate the emission
as a function of altitude for a model aurora using the Fokker-Planck
equation for electrons with energy E > 500 eV along with a two-stream
equation of transfer for electrons with energy E < 500 eV. The two-
stream equation of transfer solves the transport of electrons in terms
of the hemispherical fluxes of two electron streams +(E,z), the electron
flux upward along z, and The steady state continuity equations for <|>+ and ~ can then be written
as
+

2
(2.7)
and


-117-
-6 2
Table 6.4 Range, Rg (in 10 gm/cm ), and fraction of energy back-
scattered, Fg, are given for trials Cl, C2, C3, C4, and C5.
Trial
R
9
fb
Cl
5.57
0.078
C2
7.70
0.012
C3
3.43
0.236
C4
8.16
0.009
C5
3.32
0.211


-164-
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-226-
Wedde, T. and T.G. Strand, Scattering cross sections for 40 eV to 1 KeV
electrons colliding elastically with nitrogen and oxygen, J. Phys.
B: Atom. Molec. Phys., 1091 1974.
Winters, H.F., Ionic adsorption and dissociation cross section for
nitrogen, J. Chem. Phys., 44, 1472, 1966.


-46-
Throughout the energy range the cross sections used in this study
compare favorably with those of Blaauw et al. (1977). For an easy
reference, the total inelastic and total elastic cross sections are
also given in Figure 3.6 as separate curves.
All the major influences on the IEE energy loss and scattering have
been accounted for in this chapter. The next chapter presents the MC
energy deposition scheme which employs these cross sections.


END
WRTDAl <30
-219-


-21-
6 (Degrees)
Figure 3.1 N2 experimental electron impact elastic cross section data
from Shyn, Stolarski, and Carignan (1972). o's denote data
from E = 30 eV and the x's denote data from E = 70 eV.


1978 NERDC CARD LIST UTILITY
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-225-
Shemansky, D.E., T.M. Donahue, and E.C. Zipf, Jr., N2 positive and No
band systems and the energy spectra of auroral electrons, Planet.
Space Sci., 20, 905, 1972.
Shyn, T.W., R.S. Stolarski, and 6.R. Carignan, Angular distribution of
electrons elastically scattered from No, Phys. Rev. A, 6, 1002,
1972.
Silverman, S.M. and E.N. Lassettre, Generalized oscillator strengths and
electronic collision cross sections for nitrogen at excitation
energies above 10 eV, J. Chem. Phys., 42, 3420, 1965.
Spencer, L.V., Energy dissipation by fast electrons, National Bureau of
Stand. Monoqr., 1_, 1959.
Spencer, L.V. and U. Fano, Energy spectrum resulting from electron
slowing down, Phys. Rev., 93, 1172, 1954.
Stolarski, R.S., Calculation of auroral emission rates and heating
effects, Planet. Space Sci., 16, 1265, 1968.
Stolarski, R.S., V.A. Dulock, Jr., C.E. Watson, and A.E.S. Green, Elec
tron impact cross sections for atmospheric species. 2. Molecular
nitrogen, J. Geophys. Res., 72, 3953, 1967.
Stolarski, R.S. and A.E.S. Green, Calculations of auroral intensities
from electron impact, J. Geophys. Res,, 72, 3067, 1967.
Strickland, D.J., D.L. Book, T.P. Coffey, and J.A. Fedder, Transport
equation techniques for the deposition of auroral electrons, J.
Geophys. Res., 81, 2755, 1976.
Tate, J.T. and P.T. Smith, The efficiencies of ionization and ionization
potentials of various qases under electron impact, Phys. Rev., 39,
270, 1932.
Walt, M., W.M. MacDonald, and W.E. Francis, Penetration of auroral
electrons into the atmosphere, Physics of the Magnetosphere, p. 534,
edited by R.L. Carovillano, J.F. McClays, and H.R. Radoski,
D. Reidel, Dordrecht, Netherlands, 1967.
Watson, C.E., V.A. Dulock, Jr., R.S. Stolarski, and A.E.S. Green, Elec
tron impact cross sections for atmospheric species. 3. Molecular
oxygen, J. Geophys. Res., 72, 3961, 1967.
Wedde, T., Scattering cross sections. A compilation of 0, 02, and N2
data between 1 and 1000 eV. I. Elastic scattering, Center for
Research in Aeronomy, Tech. Rep. CRA 76-1, Utah State University,
Logan, Utah, 1976.


-32-
Figure 3.3b


-70-
Knowledge of the radial distribution is not crucial for our pur
poses. The most interesting radial distribution output from this MC
o
calculation is that of the 3914 A emission. Electrons below 30 eV make
little contribution to this profile because the cross section for ex-
citation to this B e^ state is fairly low (see Figure 5.1). Thus
knowledge of the radial distribution of these electrons multiply scat
tered is not extremely important.
An approximation, however, is employed in most MC computations to
calculate a fairly reasonable radial distance. The average radial
distance, as observed from the calculations in KG, for most energies and
at the longer path lengths is approximately one-sixth of the total path
lengths, thus
Pave = s/6 (4.29)
The most important spatial displacement is the longitudinal dis
tance z. In order to calculate z, the total path length s must be known.
This length s is calculated from the random number, the total elastic
cross section, o^E), and the total inelastic cross section, aTI(E),
by using
aip(E)
s'-^ETln(Rl> <4-30>
The ratio a^^Ej/a^jiE) is simply a fairly accurate approximation
of the number of elastic collisions occurring per inelastic collision.
The value -ln(R-j) [see Eq. (4.2)] is the path length (in units of MFPs)
traveled by the electron between collisions. Thus knowing the number
of elastic collisions occurring and the path length traveled between
collisions allows one to write Eq. (4.30) as the expression for the


noon
ZMV=Z VAL ZM V
WR ITE(6. 32) I.ZMV,TN5(K I ) ,TN10(K.I)
32 FaRMAT(I10.11X,lPE12o4,6X,G15o7,4X,G15o7)
46 CONTINUE
WRITE OUT THE YIELD SPECTRA INFORMATION
IF ( NSPEC o NEa 1)GO TO 1144
ZDEL= ( ZALT ( 1 )-ZALT ( 2) ) *1 E+05*DENG S*4 65 IE-2 3
DO 790 I=1 NSPZ
N 1 =NZX( I )
N1 Ml = N1-1
ZMV = ( ZALT ( N1 ) +ZALT INI Ml ) )/2
ZMV= < ZVAL-ZMV) lo E + 05
ZRNG=ZMV*DENGS*4o651E-23
C ZRNG IS THE PATH LENGTH IN GM/CM**2
C THERE ARE 4o651E-23 GM/MOLECULE
WRITE(6.791}ZMV.ZRNG
791 FORMA T( 1AT AN ALTITUDE= 1PE11o3. CMo WITH A VALUE OF*.
1 1 PEI 1 o3 GM/CM**2 *./. WE HAVE THE FOLLOWING ELECTRON SPECTRA,
2 4 X, YI ELD SPECTRA = U(E) IN #/EV/ (GM/CM**2) //,
3 MID-ENERGY*.8X.*U(E)//)
NEIP1 =NEI P 1
DO 795 J=2,NEIP
JM1=J-1
EMID=(EIP(JM1)+EIP(Jl)/2,
WR TDAT 46
WRT DAT 47
WR TOAT48
WRTDAT49
WRTDAT50
WRTDAT51
WRTDAT52
WRTDAT53
WRTDAT54
WRTDAT55
WRTDAT56
WRT DAT 57
WR TDAT58
WRTDAT59
WRTDAT 60
WRTDAT61
WRTDAT 62
WRTDAT 63
WRTDAT 64
WRTDAT 65
,WRTDAT 66
WRTDAT67
WRT DAT68
WRTDAT69
WRTDAT70
WRTDAT 71
WRTDAT72
-215-


-36-
This probability, Pg(E), is simply calculated with
l / ^sineded*
pb
/ / sin0ded 0 0
In Figure 3.5, Pg(E) from the screened Rutherford and model 3 are
compared with other theoretical (Wedde and Strand, 1974) and experi
mental (Shyn et al., 1972) values. Model 3 does have a tendency to
estimate less backscatter than the screened Rutherford at the larger
energies. (The Pg(E) curves for model 3 and the screened Rutherford do
tend to converge at 5 KeV however.) The dominant exponent!'a 1-1 ike for
ward scattering is the reason behind this behavior. The discontinuity
observed at 200 eV in model 3 values results from the lack of the back
scatter characteristic above this energy.
The elastic scattering collisions influence mostly the direction of
travel of the electrons. There is some energy loss during an elastic
collision (as pointed out above), but this loss is not important for
electrons with energies above 2 eV colliding only with N particles.
Inelastic collisions, on the other hand, result in a fairly sub
stantial energy loss with some scattering. Consider now the differential
and total cross sections for these inelastic events.
B. Inelastic Differential and Total Cross Sections for N2
Inelastic collisions are divided into two types: 1) electron ex
citation and 2) electron ionization. In the excitation process the
electron is excited to a higher state which may either be an optically


-126-
This is one important reason why the range values of the electrons
are under investigation. Assuming a continuous slowing down of the
electron, the range R(EQ) for an electron of incident energy EQ may be
defined as
(employing Eq. (2.2)).
R(U I
dE
dE/dx
(7.1)
Since nL(E) = 4^ (as noted in Chapter II), then
dx
R U m
dE
(7.2)
In the Born-Bethe approximation L(E) is proportional to In E/E.
- 75
At the higher energies, In E/E can approximately be written as E *
(see Green and Peterson, 1968). Using this approximation in Eq. (7.2),
the range is
R(Eq) = C EoK75 (7.3)
where C is a constant.
Griin (1957) and Cohn and Caledonia (1970) have shown that such an
expression is correct for electron energies from 2 to 54 KeV. Barrett
and Hays (1976), on the other hand, extended this energy range down to
0.3 KeV and derived a slightly more complicated empirical formula for
the electron energies from 0.3 to 5 KeV.
Much of the significance of the range is built on the idea that all
the energy of the electrons is lost between z = 0.0 and z = 1.0 where
z = the fraction of the range traveled. Such an approximation is quite
good above 2 KeV; however, at energies below 1 KeV this is not such a
good approximation.


Figure 7.5b


APPENDIX A
MONTE CARLO PROGRAM
The Monte Carlo program which is a modified version of a program
used in Brinkmann and Trajmar (1970) is listed in this appendix. This
program (written in Fortran IV) degrades electrons in the range from 2
to 5000 eV in a spatial manner. Each collision with its characteristics
are placed on a magnetic tape.
-155-


Figure 6.1 An intensity plot for electrons of energy 1 KeV is presented
as a function of z. The solid line represents the primary or
incident electron's contribution to the 3914 A emission while
the dashed line illustrates the contribution from the secondary
electrons.


14 JULY 1973
NERDC
CARD LIST UTILITY
c
MC000289
c
MC 000290
31
IF(T LTo TMINIGO TO 27
MC000291
c
TM IN IS CUTOFF OF THE TERTIARIES
MC000292
NG 20 0=2
MC 000293
c
SAVE THE SECONDARY PARAMETERS
MC000294
EVSAVS=EV
MC000295
EV=T
MC 000296
CPSAVS=COSPA
MC000297
PASAVS=PA
MC000298
PHISVS=PHI
MC 000 299
XVSVS=XV
MC000300
YVSVS=YV
MC 000301
ZVSVS=ZV
MC000302
c
CALCULATE THE TERTIARY ELECTRON SCATTERING
MC 0003 03
CALL RANDU
MC000304
IR 1 = 1 R2
MC 000305
CALL SDIFM
MC000306
CO SPLF=CQS PA*COS PAN-SINPA*SINPAN*COSAZA
MC 000307
IF(COSPLF G T C o 999) COSPLF=Q>999
MC000308
IF{COSPL F .LT. 0 o 9 99) COSPLF=-0o999
MC 000 309
ARGC = 1 oCOSPLF COSPLF
MC 0003 1 0
IFCARGC LT 0o0E0)G0 TO 5979
MC000311
SINPLF=S QRT (ARGC)
MC 0003 1 2
GO TO 5970
MC 000 313
5979
S INPLF= 1 .0E-E
MC000314
5970
CONTINUE
MC 0003 15
PLF= A RCOS( COSPLF )
MC0003 16
IF (COSPLF oLEo 0*99) GO TO 70
MC000317
ALF=0,0
MC0003 18
GO TO 71
MC000319
70
COSALF=(COSPA*COSPHI*S IN PAN *C0SAZA S INPHI *SINPAN*SI NAZA +
MC 000320
1 SINPA*COSPHI*COSPAN)/SINPLF
MC000321
SI NAL F=(COSPA*SINPHI4SINPAN*COSAZA+COSPHI SINPAN*SINAZA
MC000322
1 +SINPA*SINPHI COS PA N)/SINPLF
MC 000323
IF ( C 0 SALF oLTo Oo 999) CO SALF=0 99 9
MC000324
IF (COS ALF ,GT. 0 <.999 ) COS AL F= 0 999
MC 000325
ALF = ARCOS(CO SALF)
MC 000326
IF (S INALF o LTo 0.0) AL F= 6 2831 85-A LF
MC000327
7
1 CONTINUE
MC 000328
CO SPA = COSPLF
MC000329
PH 1= ALF
MC000330
PA=PLF
MC 000331
C
MC000332
c
MC 0003 33
-183-


-65-
Table 4.4 N2 inelastic composite state with its characteristic proba
bility, p, and average energy loss, W, given for several
energies between 2 and 200 eV.
E (eV)
P
W (eV)
2
1.000
0.57
3
1.000
1.03
4
1.000
0.922
5
1.000
0.835
6
1.000
0.772
7
1.000
0.728
8
1.000
0.696
9
1.000
7.00
10
1.000
7.21
12
1.000
8.25
14
1.000
8.91
16
0.971
9.12
18
0.866
9.34
20
0.745
9.68
30
0.426
11.70
40
0.344
12.80
50
0.296
13.30
60
0.271
13.70
70
0.255
13.90
100
0.229
14.30
150
0.214
14.60
200
0.234
14.80


-105-
energy dependence. Another way to approach a sensitivity study of the
elastic phase functions is the following: 1) Fix the number of collisions
allowed in the MC calculations at some set number, say 25000; 2) allow
only elastic collisions; and 3) assume that there can be no energy loss
during a collision (the electron energy remains fixed at 300 eV).
Employing all the above assumptions, the scattering problem is very
similar to the photon scattering process. The aspects of this section
may, therefore, be of interest both to researchers in photon scattering
as well as electron scattering.
One of the simplest ways to represent elastic scattering phase func-
_2
tions is with model 1 [Eq. (3.12): P^gie.E) {l cose + a(E)} ]. This
scattering form, as noted earlier, is very similar to the screened
Rutherford cross section. Figure 6.2 illustrates five trial phase func
tions, designated as A1 through A5, whose properties are indicated in
Table 6.1.
These five phase function trials were then run in a MC computation
using the three restrictions given above. The collision plots for three
of these trials are given in Figure 6.3. The input number density in all
15 3
cases was 4 x 10 molecules/cm and electrons entered the Ng gas until
the total number of collisions was 25000.
From Figure 6.3 several observations can be made. Generally, the
number of backscatter collisions decreased as the forward peaking of the
differential cross section increased. The number of collisions occurring
close to the origin of the perpendicularly incident electrons also
decreased, while the number of collisions at distances forward from
the origin increased.


CHAPTER III
ELASTIC AND INELASTIC DIFFERENTIAL AND TOTAL CROSS SECTIONS FOR N2
In this chapter differential and total cross sections for electron
impact on N^ will be discussed. Section III.A reviews the elastic cross
sections of N2 and discusses three models for representation of these
properties. In section III.B the inelastic cross sections of N2 are
presented with their relationship to theory and experiment. Section
III.C, then, concludes this chapter with a discussion of the total
(elastic plus inelastic) cross section for N2> Any energy degradation
technique requires knowledge of these cross sections for a complete
evaluation of the energy loss by electrons in a gas.
A. Elastic Differential and Total Cross Sections for N2
One of the most common differential elastic cross section forms is
the screened Rutherford cross section which can be expressed in the form
da r
dn .2. 2
Z2e4
p v (1 cose + 2n)
2] Krei
(3.1)
where K^ie) is the spin-relativistic correction factor.
The familiar Rutherford cross section (unscreened case) which can be
derived from classical scattering theory is given by
do
dp
Z2e4
? 0 p
p V (1 cose;
(3.2)
-18-


-116-
Table 6.3 Parameters a-|, a?, a( 1000 eV), and a(30 eV) for trials Cl,
C2, C3, C4, and C5.
Trial
al
a2
a(1000 eV)
a(30 eV)
Cl
32
-1.0
0.032
1.07
C2
303
-1.66
0.0032
1.07
C3
3.44
-0.344
0.32
1.07
C4
3.2
-1.0
0.0032
0.107
C5
320
-1.0
0.32
10.7


-132-
calculation. Green, Jackman, and Garvey (1977) [hereafter called GJG]
have discussed the use of the yield spectrum U(E,EQ), which was de
scribed in Chapter II of this work. This yield spectrum can be used
to calculate the yield of any state by means of the equation
Eo
0,(E J = / U(E,E )p-(E)dE (7.4)
JO w u J
J
where p.(E) = a.(E)/oTT (E) is the probability for excitation of the
J J I *
jth state with excitation energy W..
In these MC calculations information about the yield spectrum can
be attained at any longitudinal and radial distance. The three variable
yield spectrum U(E,z,Eq), which is a function of the longitudinal dis
tance z, as well as the incident electron energy Eq and the energy E, is
considered in subsection VII.B.l. The four variable yield spectrum
U(E,p,z,E ), which is also a function of the radial distance p, is then
considered in subsection VII.B.2.
1. Three Variable Spatial Yield Spectra
The three variable spatial yield spectrum U(E,z,EQ) is found in the
following manner. A MC calculation takes place for a certain incident
energy Eq which places all the collisions with their characteristics
on a magnetic tape. The longitudinal- or z-axis is divided up into
2
several equal intervals of AZjnt gm/cm each and the energy regime from
2 eV up to the incident energy Eq is divided up into several intervals
of AEjnt eV each (not all intervals being equivalent in energy).
If the spatial yield spectrum U(Ee,ze,Eq) for a certain energy
value Ee and longitudinal distance z£ is desired, then the two-dimensional


CHAPTER V
MONTE CARLO INTENSITY PLOTS AND COMPARISON WITH EXPERIMENT
Incident electrons with energies between 0.1 and 5.0 KeV are de
graded in Ng using the MC method described in Chapter IV with the cross
o
sections given in Chapter III. The intensity plots of the 3914 A
emission are described in this chapter.
+ 2 +
Emission intensity plots of the 3914 A radiation from the N2 B zu
state are used in describing the range (found by extrapolating the linear
portion of the longitudinal 3914 A intensity plot to the abscissa) for
incident electrons. Section V.A describes the excitation of the
+ 2 +
N2 B state. In section V.B the range of the electrons is defined
more completely. Previous experimental and theoretical work on the
3914 emission of N^ is given in section V.C. The range results from
the MC calculation are then discussed in section V.D. Finally, section
V.E describes the intensity plots resulting when plotted as functions of
the radial direction.
A. Excitation of the NB £u State
The main concern of this chapter will be the intensity plots showing
the emission of the 0-0 first negative band (B state) of N^ at 3914 A.
Experimentally (see Rapp and Englander-Golden, 1965; McConkey, Woolsey,
and Burns, 1967; and Borst and Zipf, 1969), it has been shown that the
number of photons at 3914 produced for each ionization of N2 is
-77-


-3-
in section II.A. The MC method which was used in this study along with
three other MC approaches are briefly described in section II.B.
This MC approach requires knowledge of differential and total cross
sections. The third chapter discusses the cross sections that were used
for Ng* Section III.A includes the elastic differential and total cross
sections. The inelastic differential and total cross sections are next
discussed in section III.B. Section III.C, then, considers the total
(inelastic plus elastic) cross section of N2-
In Chapter IV, the MC calculational procedure is considered. A
brief discussion of the approach is given in section IV.A. The computer
programs and machinery used in this work are discussed in section IV.B
with the programs listed in appendices A and B. A detailed discussion
of the MC electron energy degradation technique is presented in section
IV.C. Finally, the statistical error resulting from the Monte Carlo
calculation is given in section IV.D.
The MC three-dimensional intensity plots with comparison to experi-
+ 2 +
ment are given in Chapter V. The excitation of the N2 B i: state is
discussed in section V.A with the concept of range being defined in
O
section V.B. Previous experimental and theoretical work on the 3914 A
emission from N2 is considered in section V.C and section V.D presents
some range results and intensity plots in the longitudinal direction
from this MC calculation. Section V.E, then, concludes the chapter with
a comparison between the MC intensity plots in the radial direction and
the experimental data.
The main concern of Chapter VI is a sensitivity study. The effects
of the ionization differential cross section on the intensity distribu
tions are considered in section VI.A. Section VI.B, then, discusses the


UNIVERSITY OF FLORIDA
3 1
262 08667 026 1


-84-
an initial energy of 5 to 54 KeV. Cohn and Caledonia (1970) measured
intensity profiles of electron beams with incident energies from 2 to
5 KeV impacting into Ng. Barrett and Hays (1976) then extended the
incident electron range down to 300 eV by measuring the radiation pro-
O
files of 3914 A resulting from electron beams with energies from 0.3 to
5.0 KeV impinging on Ng.
Spencer (1959) used the Spencer and Fano (1954) method of spatial
energy deposition and found good agreement between his energy loss plots
and the 3914 ft intensity plots of Griin (1957). The Berger, Seltzer, and
Maeda (1974) [BSM] MC calculation provided energy loss plots down to
2 KeV. These plots are also in fairly good agreement with the experi
ments mentioned above.
Comparisons will be made in this paper between the available experi
mental electron energy loss data and the MC calculations done here.
Since this MC calculation follows the incident electrons, as well as its
secondaries and tertiaries down to 2 eV, this MC computation is one of
the most detailed ever employed for electron impact energy degradation.
It is, therefore, of interest to compare the results from this study
with experimental results for incident electrons with energies from
300 eV up to 5 KeV.
D. Range Results and Longitudinal Intensity Plots
from the Monte Carlo Calculation
Range data at several incident electron energies are calculated
with the use of the screened Rutherford and the model 3 differential
elastic cross sections. The screened Rutherford model is used because
it is the most widely used form for elastic scattering in theoretical


673
26
572
C
C
651
65 3
30
DO 26 J= 1 I E IL S
WR ITE(6,673 )EIT( J )CSIE( J )
FORMAT(2G15o7)
CONTINUE
READ! 5.572) IUTP, FOVAL
FORMA T(I 5 4E100)
IUTM 1= IUTP- l
READ(5,522)(UEIN(I).1=1,IUTP)
READ ( 5.522 ) ( UH ( I ) I =1 IUTM1 )
RE AD (5.522 ) (UI( I ). 1=1. IUTM1)
READ ( 5,522 ) { U J ( I ) I =1 IU T M1 )
READ(5,522)(UD(I).1=1,IUTM1)
READ(5,522 ) (USNU (I ) 1=1. IUTM1 )
RE AD ( 5.522 ) ( USF ( I ) 1=1 ,1 UTM1 )
READ IN THE MULTIPLE ELASTIC SCATTERING PARAMETERS FOR THE LONGITU
DINAL DISTRIBUTION
WRITE(6,651 )
FORMAT(1PARAMETEPS FOR THE LONGITUDINAL OR Z CHARACTERIZATION OF
* THE ELECTRONS WITH ENERGIES BELOW 30 EV ARE NOW PRESENTED*,//.
*4X,*E M IN 4X E MAX', 6X ,*H* ,8X' I 8X J ,8X,*D* 5X *S SUB NU .
*2X,*S SUB F ,3X, 'FOVAL *,/)
DO 30 1=1.1UTM1
IP 1= 1+1
WR ITE (6.653 )UEIN( I ) ,UE IN ( I PI ) UH( I ) ,U I ( I ) U J ( I ), UD( I ) ,USNU( I ) .
*USF( I ) .FOVAL
FORMAT (9 F9.2)
CONTINUE
DAT AO 190
DAT AO 191
DATA0192
DATAOl93
DATA0194
OATAOl95
DAT AOl96
DATA 01 97
DATAO198
DAT AO 199
DA TA 0 200
DAT AO 201
DATA02 02
DA TAO 203
DAT AO 204
DA TA 0205
DAT AO 206
DATA02 07
DATA02 08
0ATA0209
DATA0210
DAT AO 211
DAT A0212
DATA0213
DAT AO 214
DAT A0215
DATA0216
-173-


uuuu
14 JULY 1978
NERDC
CARD LIST UTILITY
C AT A GIVEN ALTITUDE AS A FUNCTION OF RHO.
151 DO 6149 IZ V = 1NVR
IC=IZV
NC=NCRO(IZV)
NCRM1=NCRO( IZV)- 1
I F ( Z *GTo ZALT{NC > AN Da Z ,LTa ZALT(NCRM 1 >>GO TO 155
6149 CONTINUE
GO TO 1161
155 RHOV=ABS(X)
DO 159 I 3=2NRM
I3M1= 13-1
IF(RHOV oLTo RHO(13) o AND RHOV .GT RHOII3M1))
* KREXt IC N I 3) =KRE X( I C, N, I 3) + 1
159 CONTINUE
GO TO 1161
THIS PART OF THE PROGRAM CALCULATES THE AVERAGE ENERGY AT A GIVEN
ALTITUDE,
6611 CONTINUE
IF(Z aLTo ZALT(1))GO TO 6767
NOFE(1.1 ) = NOFE(1 ,1 ) *1
TOTEt 1 .1 )=TOTE( 1 ,1 J+ELOSS
IF(P A ,LE. 157)NOFE(2,1)=NOFE(2,1)+1
IF(PA ,LE. 1 ,57)TOTEt2*1 >=TOTE(2,1 l+ELOSS
IF(P A GT o 1 o 57)NOFE( 31>= NOFE(3,1 )+1
IF(PA ,GT o 157)TOTE(3,1)=TOTE(3,1>+ELOSS
GO TO 6769
6767 IF{Z GT, ZALT{NAR))GO TO 6768
NOFE(1 NAR1 )=NFE(1 N AR1 ) + l
TOTEt 1 .NAR1 )=TOTE(1 .NAR1 )+E LOSS
IF (P A .LEo 1 a57)N0FE(2,NARl)=N0FE( 2.NARD + 1
IF < PA ,LE, 1,57) TOTE ( 2, NAR1 )=TOTE (2. N AR 1 ) +ELOSS
IF (PA .GT. 1,57) NOFEt 3.1 )=NOFEI 3.1 ) + l
IF(PA a GT a 1o57)T0TE(3,1 ) = TOTE(3.1 >+ELOSS
GO TO 6769
6768 DO 13 14=2,NAR
14 Ml = 14 1
IF ( Z .GT# ZALTt 14) .AND Z LTo ZA LT ( 14 Ml ) ) N1 Z = 1
IFtNIZ EOo l)NOFE( 1I 4) = NOFEt1,I 4)+1
IFtNIZ oEQc 1)TQTE(1 .14)=T OT E(1,14 ) +ELOSS
IFtNIZ oEQ o 1 oANDo =>A oLE, 157)NOFE(2.I 4)=NOFE(2,I 4) +1
IFtNIZ a E Q, 1 .AND) PA .LEo 1 a 57 ) TOTE ( 2, I 4 > =TOTE ( 2 I 4 ) + EL OSS
IFtNIZ oEQo 1 AND, PA a GT o 1,57) NOFE(3.I 4) = NOFE(3.I 4) + 1
IFtNIZ EQ a 1 oANDo PA GT a 1,57) TOTE ( 3,1 4)-=T0TE( 3.1 4) +E LOSS
RDTAP145
RDTAP146
RDTAP147
RD TAPI 48
RDTAP149
RDTAP150
RDTAP151
RDTAP152
RDTAPI 53
RDTAP154
RDT API 55
RDTAP156
RDTAPI 57
RDTAP158
RDTAP159
RDT API 60
RD TAP 161
RDTAP162
RDT API 63
RDTAP164
RDT API 65
RD TAPI 66
RDTAP167
RDT API 68
RDTAPI 69
RDTAPI 70
RDT API 71
RDTAP172
RDTAP173
RD TAPI 74
RDTAP175
RDT API 76
RDTAP177
RDTAPI78
RDTAPI 79
RDTAP 1 80
RDTAPI 81
RDTAP 1 82
RDTAP183
RDT API 84
RDTAP185
RDT AP186
RD TAPI 87
RDTAP188
RDT API 89
-212-


-135-
It should be noted that at the small longitudinal distances a
fairly large "source" term persists at energies E = EQ. In the inter
val from about 4 eV to about 10 eV there is a noticeable dip in the yield
spectra. In this range (see Figure 3.6), the total inelastic cross
sections show a very large dip, thus many of the electrons in this range
do not interact inelastically with the Ng gas in the region of interest.
For the purposes of many applications it is useful to represent the
yield spectra by
(7.6)
(following the notation of GJG), where e is the Heaviside function with
EQ, the minimum threshold of the states considered, and 6(E E) is the
Dirac delta function which allows for the contribution of the source
itself. The U (E,z,E ) is represented approximately by
a 0
and
^-ZR a2^Eo^ + a3^Eo^
B,U.E0) -
[zr b12(E0)]2 + b]3
e
+ 1


no
14 JULY 1978
NERDC
CARO LIST UTILITY
C
C
C
C
C
C
C
C
C
c
c
c
c
***GETDAT PROGRAM***
THIS PROGRAM TAKES THE COLLISION DATA FROM THE MAGNETIC TAPE
AND COALESCES AND SYSTEMATIZES ITo
THIS PROGRAM IS RUN IN CLASS = L
THE FOLLOWING PROGRAMS ARE CALLED IN THIS MAIN PROGRAM:
11RDCARD
2) MATIN
3) RDTAPE
4)WRT DAT
THIS COMMON BLOCK CONTAINS ALL THE VARIABLES OF
SUBROUTINES. IT IS SO LONG TO PERMIT THE OPTIMI
H COMPILER TO BE USED FOR MOST EFFICIENCY.
COMMON ALAB(20)iANEI( 5.2 0. 100),ALT(20) .EAVE C 3)
1 EXC5(50),JES(50 ) JIS(50) .JZE(3.50.80),JSEC(50
2 NIE(40).NZX(20) .NOFE3.80) .NCRO(IO) ,RHO(4 0),S
3 TOTE(3.60 ) ,TN1( 3.30).TN5( 3.80) ,TN10( 3.80) ,WIE
4 YSPEC(IOO) ZNUM (10). Z AL T ( 80 ) NUMG AS N AR1 N V R ,
5 NSEC.NSPZ.NEIP.NEXC.NTOP,NAR.NSPEC,NRHO.NAVEE
5 ZVAL.EM IN,DENGS,M,NSPUN,NSPR.ANER( 1, 10.20,100
CALL RDCARD
CALL MATIN
CALL RDTAPE
CALL WRTDAT
STOP
END
USE BY THE
ZE = 2 IN THE
,EIP( 100) ,
,80).KR EX( 10.40.40)
A ( 3,8 0 ) .SIG(IOO) ,
(50) WI S( 5 0 ) ,
NJMST.NUMG.NRM,
NPRIM.ElN,
) ,R YS( 1 0) NSPRO
MAI NO 0 01
MAIN0003
MAIN0004
MAIN0005
MAIN0006
MAI NO 0 07
MAIN0008
MAINO O 09
MAIN0010
MAIN0011
MAI NO 012
MAIN0013
MA INOO 14
MAIN0015
,MAINOO16
MAINOO 17
MAINOO18
MAINOO 19
MAINOO 20
MAIN0021
MAINOO 22
MA I NO0 23
MAI NOO24
MAINOO 25
MAIN0026
MA IN 0 0 27
MAIN0028
MAI NO029
-201-


-64-
with being the threshold of the Rydberg or forbidden
state.
Below 200 eV, the probabilities for excitation to the various
inelastic states are changing quite rapidly. The parameters for the
eight individual states are taken from Jackman et al. (1977b) and
Porter et al. (1976). The composite state's properties are found in
the same manner that they were above. In these lower energy regimes
the probability and energy loss are changing fairly rapidly, thus Table
4.4 illustrates these probabilities and threshold values at several
energies.
With the background on the inelastic cross sections and their
subsequent probabilities, consider now the collision type. The
random number determines the type of collision that occurs in the
following manner: If
aTE(E)
R5 cT (E) for electron energies (4.20)
where jE(E) is the total elastic cross section, then the collision is
elastic. If
TE(E) PiTI^E^ + TE^
" r5 ^¡TET
^[ET
and E > 200 eV
(4.21)
where 0jj(E) is the total inelastic cross section and p^ is the proba
bility for exciting the first inelastic state (in Table 4.2 the first
state is the b^ thus p-j = 0.092), then the inelastic collision results
in the excitation of the first state.
A relation follows from Eq. (4.21) that holds true for j = 2 to 9
such that: If


-136-
al(Eo> all Co
12
a2(E0) = a2i o % )
a3(Eo)=a31 +lf
. ,b112n (1 ^o^-i
blll 5o [1+ ]
b-n(z>U
113
lr o; [expL(zR f-j (Eq) >/f2(Eq) J + 1]
b12^Eo) = bl21 (1 ^22)
*o
c112
VzEo> clll 0 / VEo))/f2 d12
dl Vo
fl fll(1 +7i)
0
f2 f21 &
Rg(Eq) = ri + r2 V = Range of an electron of
where the parameters and their values are all given
0 = EO/,00- ER E/E0- and ZR Z/Rg(E0>-
primary energy EQ
in Table 7.2. Also,


-153-
To the author's knowledge, this was the first theoretical calculation
resulting in three dimensional intensity profiles for incident electron
energies below 2 KeV which could easily be compared to experimental work.
+ 2 +
The B intensity profiles and range values for incident energies
from 0.3 to 5.0 KeV showed reasonable agreement with experimental electron
energy degradation work in both the longitudinal and the radial direction.
A sensitivity study was included in this work which characterized
the influence of 1) the differential ionization cross sections, 2) the
differential inelastic cross sections, 3) the different shaped elastic
phase functions, and 4) the total elastic cross sections on the energy
deposition process. In particular, it was shown that: 1) Differential
ionization cross sections have very little influence on the degradation
process; 2) inelastic scattering appeared to be somewhat important for
incident electrons with energies below 0.3 KeV; 3) the shape of the
o
electron collision profiles and the range from the 3914 A intensity
profiles were functions of the screening parameter of the model 1 elastic
scattering phase function; and 4) the total elastic cross section had a
significant influence on the electron's spatial degradation process.
The resultant energy loss plots are used to help determine the
energy albedo of the incident electrons and also the rate at which energy
is lost in the medium. The spatial yield spectrum is easily employed to
find the excitation profiles for any state at any position in the
medium. For this reason the three variable spatial yield spectrum
U(E,z,Eq) is analytically characterized. The four variable spatial
yield spectrum U(E,p,z,Eq) is even more complex than U(E,z,EQ); never
theless the systematics of this quantity are described qualitatively.


Figure 5.5 Intensity plots for electrons with incident energy 5.0 KeV
are presented at two z values as functions of p. The solid
line histogram indicates the results using model 3. The
x's denote the experimental data of BH and the o's denote
the theoretical work of BSM.


-74-
As established earlier, the polar angle 0, and azimuthal angle ,
a a
representing the motion of the electron after the collision, can be
chosen in a random way from the two random numbers, and R^, using
e =
a
$
a
21*4
(4.32)
A reasonable approximation of x, and y
a S
Eqs. (4.29) and (4.32) such that
*a xb + pave x cos+
and
ya yb + pave x sin*
can then be made usi
a
a
ng
In the MESD the fifth random number, Rg, is used to determine the
inelastic collision type. A method similar to that illustrated in sub
section IV.C.4 is employed, the only difference is the fact that the
collision is only inelastic.
7. Value of the Cutoff Energy, 2 eV
The E used in this work has been set at 2 eV because the lowest
c
threshold for excitation to an inelastic state is 1.85 eV. With this
cutoff energy the yield spectra can be defined down to 2 eV at all
longitudinal distances. Subsequently, a reasonable calculation of the
excitation to any Ng state may be made.


-56-
All electrons are forced to be degraded in a 30 cm long cylinder;
thus an increase in the density is required for an increase in the energy.
There are 10 cm allowed in the negative z direction and 20 cm allowed in
the positive z direction. The x and y axes extend to infinity. Some
electrons actually escape from the cylinder, but the energy lost due to
these electrons is only a few tenths of a per cent of the incident elec
tron energy. The path length is then given as
PT = -xln(R1) (4.2)
using the relation that
(4.3)
Figure 4.2 represents a schematic of the electron traveling and
colliding with three Ng molecules. The P^, P.^, and Py3 are the path
lengths traveled by the electron between the initial coordinates and
the first collision, the first and second collisions, and the second and
third collisions, respectively.
The x,, y and z, coordinates at this collision can now be found
a a a
from PT, xb, yb, zb> eb> and <¡>b using
x
a
y
a
z
a
2b
+ P-j. sineb cosb
(4.4)
+ Pj sineb sinij>b
(4.5)
+ PT coseb
(4.6)
In Figure 4.2 the coordinates of the first and second collisions are
represented to illustrate how the electron's direction of motion might
change during its collisions with Ng. So far emphasis has been only on


-138-
The yield of any state is then found from
(7.8)
where
E
o
[e{ZR gl(Eo)}/92 +
and
The upper limit of integration in Eq. (7.8) is not Eq but is Eul-.
As the electrons penetrate further and further into the medium, they lose
more and more of the high energy particles. The energy E^ is thus a
cutoff energy which must be invoked.
Equation (7.6) represents the yield spectra data fairly well in
this regime of incident electron energies. The fit can be seen in
Figure 7.4 for five incident energies at five longitudinal values.
A comparison is given in Table 7.3 between the yield using Eq. (7.8)
and the yield using the MC calculation for several incident energies and
longitudinal values for the yield of the 3914 emission. The two cal
culations are in fair agreement throughout the entire range considered.
It should be noted, however, that Eq. (7.8) is not accurate at longi
tudinal values in the backscatter direction.
Maeda and Ailcin (1968) attempted to apply an analytic degradation
spectrum to problems of the atmosphere. They calculated the number of
oxygen atoms resulting from the dissociation of 02 from auroral events.


non
14 JULY 1978
NERDC
CARO LIST UTILITY
C
C
C
26
139
140
142
1 43
144
145
SUBROUTINE MATIN
SUBROUTINE MATIN ***
THIS SUBROUTINE IS ACCESSED BY THE MAIN PROGRAMo
THIS SUBROUTINE INITIALIZES ALL THE MATRICES
COMMON ALABt 20) ,ANE1(5 .20,100) .ALT(20) ,EAVE(3),EIP{100),
1 EXC5(50),JES(50 ).J IS(50 ).JZE(3.50.80)* J SEC(50,80) ,KREX( 10,40,40)
2 NIE(40) ,NZX(2 0) .MOFE (3,80),NCRO(10)RHO(40),SA(3.80).SIG(100).
3 TOTE (3,80) ,TNI( 3,8 0) TN5( 3,80) .TNI 0(3 .80) ,WIE(50) ,WIS(50),
4 Y SPEC(100),ZNUM(10 ),ZAL T{80).NUMGAS,NARl.NVR,NUMST.N UMG,NRM,
5 N SEC NSPZ,NEIP NEXC NTOP,NA R.NSPEC.NRHO.NAVEE,NPRIM,EIN ,
5 ZVAL.EM IN.DENGS.M,NSPUN.NSPR.ANER( 1 1 0. 20,1 00 ) R YS ( 1 0) .NSPRO
INITIALIZE THE MATRICES
DO 26 K=1,3
DO 26 I=1,NAR1
T OTE(K,I) = 00E0
NOFE(K I >=0
TN10(K I ) = 0 OEO
TN5 ( K I)=0,OEO
DO 139 1 = 1 NVR
DO 139 J=1,NUMG
DO 139 K=1,NRM
KR EX ( I, J ,K )= 0
DO 140 K=1 .3
DO 140 1=1.NUMG
DO 140 J=1,NAR1
JZE ( K I J) =0
DO 142 1=1,NSEC
DO 142 J=1,NAR1
JSEC(I,J)=0
DO 143 J= 1 NSEC
J I S( J ) = 0
DO 144 J=1,NUMG
WIE(J)=0 0E0
JES( J ) =0
DO 145 K= 1 1
DO 145 1=1,NSPZ
DO 145 J=1,NEIP
AN EI ( K I, J ) = 00E0
DO 145 L=1,NSPR
ANER (K.L.I J ) = 0o OEO
RETURN
END
MATI N001
MATIN002
MATIN003
MATIN004
MATIN005
.MA TINO 06
MAT IN 007
MATIN008
MA TI NO09
MAT INO 10
MAT I NO 11
MATINO 12
MATINO 13
MATIN014
MATIN015
MAT INO16
MATIN017
MATINO18
MAT I NO19
MATIN020
MATINO 21
MATIN022
MATIN023
MATIN024
MATIN025
MATINO 26
MATIN027
MATIN 0 28
MATIN029
MATIN030
MATINO 31
MAT I NO 32
MATIN033
MATIN034
MATIN035
MATIN036
MATIN037
MA TI NO 38
MATIN039
MATIN040
MATINO 41
MATIN042
MATIN043
MATINO 44
-207-


Figure 3.3 N? electron impact elastic differential cross sections,
(a and b) The screened Rutherford (dashed line) and the model 3
(solid line) are compared with the experimental data of
Shyn et al. (1972), x, and Herrman et al. (1976), o, at
the energies of 30 eV (Figure 3.3a) and 1000 eV (Figure
3.3b).


14 JULY 1978
NERDC
CARD LI ST UTILI TY
B E MSD = ,1 PEI Oo4i 1 EV EINEL=',1 PE10o4, EV',//>
NA=NX-1
WR ITE(6,250 )NX
250 FORMAT!/////.3X,THE DISTRIBUTION FUNCTION IS EVALUATED AT *,I5,
A* ANGLES BETWEEN ZERO AND PI'.////)
WRITE(6,622)
622 FORMAT!' INDEX OF ANGLE* ,5XANGLE VALUE IN RADIANS',//)
READ(5,131 ) (T HE T( I ) I = 1 ,NX)
C THET(I) ARE THE ANGULAR ElN ENDS USEFUL IN CALCULATING THE ANGLE
C OF SCATTER,
DO 3 1=1,NX
WR I TE ( 6,624 ) I.THET(I)
624 FORMAT!18,14X,1PE10,4)
3 CONTINUE
READ!5.105) NAR
C NAR IS THE NUMBER OF ALTITUDE INTERVALS
RE AD(5,131 ) (ZALT( I ) ,1=1 ,NAR)
C ZALT(I) ARE THE ARRAY OF Z VALUES,
C THIS ARRAY IS MAINLY USEFUL IF PRELIMINARY OUTPUT IS DESIRED.
C ZALT(l) AND ZALT(NAR) ARE THE LIMITS ON THE Z DIMENSION
C OF THE BOX FOR DEGRADATION,
C IF AN ELECTRON REACHES OUTSIDE THE DIMENSIONS (IN Z) OF THE BOX,
C THEN THE ELECTRON HAS ESCAPED.
131 FO RM AT(8E10.0)
WRITE(6 ,632 )
632 FORMAT!///, 1INDEX OF ALTITUDE RANGE ,5X,'ALTITUDE IN KILOMETERS*
* /)
DO 10 J=1,NAR
WRITE(6.634) J,ZALT(J)
634 FORMA T(I8,21X.lPE12o4)
10 CONTINUE
RE AD (5,131 ) (W IS! I ), 1=1. NSEC)
C WIS(I) ARE THE LOWER ELECTRON ENERGY BIN ENDS.
NT OP= NSEC 1
READ(5,131 ) (EXC5(I),1=1.NT OP)
C EXC5(I) ARE THE NUMBER OF EXCITATIONS OF THE 3914A EMISSION STATE
C RESULTING FROM THE LOCAL DEGRADATION OF THE SECONDARIES IN THAT BIN,
C THIS ARRAY IS USEFUL ONLY IF PRELIMINARY OUTPUT IS DESIRED,
WRITE(6,640)
640 FORMAT(//,' BOUNDARIES OF BINS FOR SECONDARIES IN EV ,
* 5X, NO. OF 391 4A EMISSIONS EXCITED*,/)
DO 20 1=1,N TOP
IP1=I+1
WRITE(6.645)WIS! I) .WIS(IPI ) ,EXC5{I )
FORMAT!IPEl 2.3,5X,1PE 12,3,14X, 1PE12 3)
MAI NO0 74
MAIN0075
MAIN0076
MAIN0077
MAIN0078
MAIN0079
MAIN0080
MAIN0081
MAIN0082
MAIN0083
MAIN0084
MAIN0085
MAIN0086
MAIN0087
MAIN0088
MAIN0089
MAIN0090
MA IN0091
MAIN0092
MA IN0093
MAIN0094
MAIN00 95
MAIN0096
MAIN0097
MAIN0098
,MAIN0099
MAI NO 100
MAINO101
MAI NO 102
MAI NO 103
MAINOl04
MAIN0105
MAINOl06
MAI NO 107
MAI NO 108
MA INO109
MAI NO 1 10
MAINOl11
MAINOl 12
MAIN01 13
MAINOl14
MAIN0115
MAIN0116
MAINO1 17
MAINOl18
645
-158-


14 JULY 1978
NERDC
CARD LIST UTILITY
C
c
c
c
CROSS SECTI
C
C
C
C
c
c
500
69 1
695
693
22
601
r
502
SUBROUTINE DATA
SUBROUTINE DATA ***
THIS SUBROUTINE IS ACCESSED THROUGH THE MAIN PROGRAM,
ALL THE REST OF THE DATA IS READ IN HERE INCLUDING THE
PARAMETERS AND ALSO ALL THE OTHER PARAMETERS NEEDED,
COMMON ALPE(6) .BETE6) ,CE(6).FE(6> .WE(6) ,ALFA<15) ,REFA(15 ).
I CFA(15)FFA(15)*WFA(15)WF( 15) FACI( 15).NFANARSA(380> .
ZALT 80) ENR( 30 > PCFA (3 ,3 0 ) ,WCFA(3,30 ) NENR FDG( 3 ) ,PSE( 3 ),
PION(3)AT(3J .A(6.3) ,B< 5, 3> ,G(5.3) ,UH(10) ,UI(10) ,UJ10) ,
UD(IO), US NU(10 ) U S F( 1 0) U E IN ( 1 1 ). IUTP IUTM1 DG( 3) NSG( 3> ,N IG( 3)
NUMGAS.NUMSTWl S(50) .NSEC.MUNIT.I SEED.I STOP.I LAST,EIN.NPRIM.COS
H, EM IN.ZSTART.TMIN,THRESH!3)AK(2.3).AJ(3,3)* GAMA(2,3) ,
TO (3 ,3) .FAC (6) AOE (6 ) ,Al E (6 ) B0E(6 ) B1E(6 ) CO E( 6 ) D0E( 6 > ,
DIE! 6),PR OB(40) ,W(40),NIE(40),A21(3),A22(3) ,A31 (3) ,A32(3 ) ,
A 33 (3) Bl 1 ( 3 ) B 1 2 ( 3), B13(3),C1 (3) ,C21 (3)C22( 3) C 3 1 ( 3) ,
C 32( 3) D1 ( 3 ) D2 ( 3 ) ,F1 (3>,F2<3) ,THRI(16),AKI(2.16).AJI(3, 16).
GAMA I(2.16).TO 1(3.16),SIGT(6) .THE T(40),NX.CSIE(20).EIT(20).
IEILS,IEILM1,P.R.ZDIS,NPIN,NOP,C0SPA,RT.F(40).R1.R2,R3.R4.
R 5.XV.YV,Z V.XVN,Y VN. Z VN .PA .PHI ,EV.WLOSS,NSTAT,NSCS .NPHF,
IR1 IR2 .NG.EVPR I, RAN, EMSD.E INEL ,EXC5( 50) CO SPAN T ,F OVAL
NUMGAS),(NI3(I>,1=1.NUMGAS)
2
3
4
5
6
7
3
9
A
B
C
n
E
ITH GAS,
NUMBER*,/)
READ(5.500) MJMG AS,NUMST, (NSG( I ),1 = 1
NUMGAS IS THE NUMBER OF GASES,
NUMST IS THE TOTAL NUMBER OF STATES,
NSG(I) ARE THE NUMBER OF STATES IN THE ITH GAS,
NIG(I) ARE THE NUMBER OF IONIZATION STATES IN THE
FORMAT(815)
WR I T£ (6,691) NUMGAS,NUMST
FORMAT(/* THE NUMBER OF GASES= *,I5.//,
1 THE TOTAL NUMBER OF ST AT £S= *,I5.//>
WRITE (6.695)
FORMAT!* NO, STATES *. 3X. 'NO, ION STATES* 3X. GAS
DO 22 1=1,NUMGAS
WRITE(6 69 3 ) NSG(I) ,NIG( I ) I
FORMAT(I6.7X,16,10X, 16 )
CONTI NUE
WRIT E(6,60 l )
FORMAT!///,' TOTAL INELASTIC CROSS SECTION PARAMETERS ARE GIVEN
1 HERE ,//,6X, *GA S 1 5X, 'ALPHA* ,3X,* BETA ,5X ,* C* ,7X F* ,7X *W ,/ )
NU P=2 NU MG AS
DO 1 1=1,NUMGAS
READ(5.502) G1,G2.G3,G4.GS,ALPE(I) BE TE(I) CE(I) ,FE ( I ) ,WE(I )
READ IN THE TOTAL INELASTIC PARAMETERS.
FORMA T(5A4,7E8,0)
DAT AOO 01
DA TA0002
DATA0003
ONDAT AO 0 04
DATA0005
DAT A0006
DAT A0007
DATA0008
DATA0009
, DA TA 0010
I, DAT AOO 11
DAT AOO12
DATAOO 13
DATAOO14
DATA0015
DATAOO 16
DAT AOO17
DATA0018
DATAOO 19
DAT AO 020
DATA0021
DAT AOO 22
DA TA0023
DATA0024
DAT A0025
DA TA 0 026
DAT AOO 27
DA TA 0028
DA TA 0 029
DAT AOO 30
DATA 0031
DAT AOO 32
DAT A0033
DATA0034
DAT AOO 35
DA TA 0 036
DATAOO 37
DAT A0038
DATA0039
DAT A0040
DAT A0041
DATA0042
DAT AOO 43
DA TA 00 44
DATA0045
-89 L-


1 1 31
C
C
3
4466
C
c
20
C
c
c
c
c
c
c
c
c
c
c
IFiNAVEE oEQ. 1 ) GO TO 66 11
GO TO 30
CONTINUE
RE AD ( M> N,WL
ELOSS= EL OSS + WL
N= NUMBER OF THE STATE
WL=ENER GY LOSS OF THE
W IE ( N )=WL
JE S( N) =JES( N) + l
IF (NIE(N ) o EQ0 2 ) GO
GO TO 20
READ < M) WL
11=N S EC
PUT THE SECONDARY AND
PRIMARY ELECTRON IN EXCITING STATE N
TO 3
WE PUT THE SECONDARY AND TERTIARY HERE INITIALLY IF THE ;
ENERGY IS ABOVE THE TMI N THEN THE ELECTRON STAYS HERE
READ(M)X.Y.Z.E.PA,PHI> NG200 EBEF
X= THE X COORDINATE
Y= THE Y COORDINATE
Z= THE Z COORDINATE
E= THE ENERGY OF THE PRIMARY OR SECONDARY AT THIS POINT IN
DEGRADATION
PA = THE POLAR ANGLE OF THE ELECTRON
PHI = THE AZIMUTHAL ANGLE
PRIMARY HERE
SECONDARY HERE
TERTIARY HERE
ELECTRON BEFDRE THE COLLISION
IF NG20 0=0 THEN WE HAVE A
IF NG2 00=1 THEN WE HAVE A
IF NG200=2 THEN WE HAVE A
EBEF IS THE ENERGY OF THE
RDTAPE46
ROT APE47
RDTAPE48
RDTAPE49
RDTAPE50
RDTAPE 51
ROT APE52
RDTAPE 53
RDTAPE54
RDTAPE55
RDTAPE56
RDTAPE57
RDTAPE58
ELECTRON *SRDTAPE59
RDTAPE60
RDTAPE61
RDTAPE62
RDTAPE63
RDTAPE64
ITS RDTAPE 65
RDTAPE66
RDTAPE 67
RDTAPE68
RDTAPE69
RDTAPE70
RDTAPE71
RDTAPE 72
-209-


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS
ABSTRACT vi
CHAPTER
I INTRODUCTION 1
II A SHORT REVIEW OF ENERGY DEPOSITION TECHNIQUES 5
A. Energy Deposition Techniques 5
B. Monte Carlo Energy Deposition Techniques 14
IIIELASTIC AND INELASTIC DIFFERENTIAL AND TOTAL CROSS
SECTIONS FOR N2 18
A. Elastic Differential and Total Cross Sections for N2 18
B. Inelastic Differential and Total Cross Sections
for N2 88
C. Total Cross Section (Elastic Plus Inelastic) 45
IVTHE MONTE CARLO METHOD OF ENERGY DEPOSITION BY ELECTRONS
IN MOLECULAR NITROGEN 47
A. Brief Discussion of the Monte Carlo Calculation. ... 48
B. Computer Programs and Machinery Used in the Monte
Carlo Calculation 51
C. Detailed Discussion of the Monte Carlo Electron
Energy Degradation Technique 52
1. First Random Number, R.j 53
2. Second and Third Random Numbers, R^ and . 57
3. Fourth Random Number, R^ 61
4. Fifth Random Number, R^ 61
iii


non non
20
C
c
c
c
c
120
21 0
CONTINUE
RE AD(5.120) NPRIM.COSI .EIN.H.ZSTART
NPRIM IS THE NUMBER OF PRIMARIES
COSI IS THE COSINE OF THE INITIAL ELECTRON
EIN IS THE ENERGY OF THE PRIMARY ELECTRON
H IS THE MAGNETIC FIELD
START IS THE ALTITUDE AT WHICH THE PRIMARY ELECTRON IMPINGES
FORMAT ( I 10.F10o5.3E10oO)
WRITE(6.210 ) NPR IM.COSI.EIN,H,ZSTART
FORMAT(///.ITHE BATCH OF PARTICLES HAS THE FOLLOWING ATTRIBUTES:
A / 9X THE NUMBER OF PART ICLES = I 5,/
B9X,COSINE OF THE INITIAL POLAR AN GL E= F 6.3 ,
C /,9X,INCIDENT ENERGY= .F12o3.2X EV.
D /,9X,MAGNETIC FI ELD = ,F 1 2o3,2X. GAUSS .
E /,9X, INI TIAL ALTITUDE = .1PE123.2X. KILOMETERS* )
CALL DATA SUBROUTINE TO READ IN THE REST OF THE DATA
CALL DATA
NAR1=NAR+1
NUMG=NUMST + NUMGA S
CALL SUBROUTINE MC WHICH DEGRADES ALL THE PRIMARIES
CALL MC
IF(I STOPo E Q OIGO TO 2
GO TO 1
MA
I NO 1
19
MA
I NO 1
20
MA
I NO 1
21
MA
I NO 1
22
MA
I NO 1
23
MA
I NO 1
24
MA
I N01
25
MA
INO 1
26
MA
I NO 1
27
MA
I NO 1
28
MA
I NO 1
29
MAIN01
30
MA
INO 1
31
MA
I NO 1
32
MA
I NO 1
33
MA
INO 1
34
MA
I NO 1
35
MA
I NO 1
36
MA
I NO 1
37
MA
I N01
38
MA
INO 1
39
MA
I NO 1
40
MA
I NO 1
41
MA
INO I
42
MA
I NO 1
43
MA
INO 1
44
MA
I NO 1
45
-159-


-103-
seen to be fairly high. Secondaries are at lower energies; therefore,
more of them are backscattered.
In considering the secondaries, it is of interest to discover any
difference in the intensity plots that may be due to the use of a dif
ferent secondary scattering distribution. Consequently, a comparison
was made between Eq. (3.18) and an isotropic secondary scattering func
tion. The difference in the two resulting intensity plots was so small
that they were the same within their standard deviation error bars.
This result, although surprising at first glance, did not seem as sur
prising under careful inspection.
Consideration of Table 5.1 gives the answer. A 5 KeV electron has
9
a range of 91.5 x 10 gm/cnr while a 0.3 KeV electron has a range of
-6 2 9
1.25 x 10 gm/cm Most of the secondaries contributing to the 3914 A
emission that are produced by an incident electron of energy 5 KeV have
energies of only a few hundred eV or less. These electrons do not travel
far, relative to the total range of the incident particle. Therefore
o
their characteristic 3914 A intensity profiles do not alter the total
3914 $ profile noticeably.
15
A 0.3 KeV electron traveling in N at a density of 2 x 10 mole-
3
cules/cm (which corresponds to a height in the atmosphere of roughly
70 km) has range of about 12 cm. At 150 km (where the density is about
5 x 10 molecules/cm ) this same electron will have a range of about
5 km. The secondary doubly differential ionization cross sections may
thus have an influence on the energy deposition in applications to the
upper atmosphere. As mentioned earlier, however, Strickland et al.
(1976) do not observe such an effect. Inclusion of Eq. (3.19) only


-24-
desirable. An analytic function representing the total elastic cross
section is most easily used in a computer program. Consider now an
analytic form derived from the differential screened Rutherford cross
section.
Knowledge of the differential cross section implies knowledge of the
total elastic cross section as they are simply related by
a(E) = / / 7T7 sineded (3.5)
0 0 as
where is the azimuthal angle. Substituting Eq. (3.1) into Eq. (3.5),
the total elastic cross section, o^(E), resulting from the screened
Rutherford cross section is very simply given as
Z2 51.8
/ r\ L- JI.O II
aR{E) n'(T~n)
-16 2
If E is given in eV then a^(E) is in units of 10 cm .
parameter
(3.6)
The screening
n
1.70 x 10~5 Z2/3
nc t(t + 2)
(3.7)
according to the Moliere (1947, 1948) theory. Berger, Seltzer, and
Maeda (1970) assumed that nc was a constant value and decided on nc = 1
as its best value. The t in Eq. (3.7) is the electron energy in units
2
of the electron rest energy so that t = E/mc In the energy regime of
interest (E s 5 KeV), t 2, and 1=1. Noting these observations,
1 6
Eq. (3.7) can be rewritten as n % 4-,
The total cross section from Eq. (3.6) is plotted in Figure 3.2
as the dash-dot line. A noticeable difference is evident between this
model and the experimental values at practically all energies.


-16-
other approach involved a sampling of each elastic collision. Appli
cation of the BSM technique to a constant density medium and no
magnetic field gave good agreement with laboratory experiments (Griin,
1957; and Cohn and Caledonia, 1970).
In this study, a MC method was needed that could be applied to IEEs.
The basic equation of transfer is solved with the use of the MC approach.
This equation can be rewritten as
dU(y,z,E,E )
11 oTtE) dz n(z)U(u,z,E,E0)
E+4EElas 1
* nU) / / p iu'.u.E'.EMu'.z.E'.EJdp'dE'
E -1 e 0
Eo +1
+ n(z)I / / pTn,..(^i,,y,El,E)U(^l,z,El,En)d^ildE,
i 2E+Ii -1 lum 0
Eq
+ n(z) l I p -(u1,y,E',E)U(u',z,E,E )du'dE' (2.14)
o j
No external fields are included here and a steady state is assumed. The
U(y,z,E,E ) is the "yield spectra" (in eV_1 sec-1 sr-1) and it is assumed
that there is only one neutral scattering species. In this equation
Oy(E) is the total cross section (elastic + inelastic)for the species,
AEE, = 2E(1 cose) /-1-ect-ro-ri (2.15)
tlas ^neutral
species
is the energy loss during an elastic collision, p (y',y, E',E) is the
probability during an elastic collision with a neutral specie that an
electron with energy E' and direction y' will result in an electron of


14 JULY 1978
NERDC
CARD LIST UTILITY
C
c
c
c
c
c
c
c
c
c
c
UF SCATTERING OF THE
SUBROUTINE SDIFM
SUBROUTINE SDIFM ***
THIS SUBROUTINE IS ACCESSED THROUGH MC.
THIS SUBROUTINE CALCULATES THE POLAR ANGLE
SECONDARY ELECTRON,
COMMON ALPE( 6) ,BETE (6),CE(6)*FE(6) ,WE(6) .ALFA{15) ,BEFA(15),
1 CFA<15),FFA(15) ,WFA(1S) ,WF( 15),FAC I(15),NFA,NAR,SA(3,80) .
2 ZALTt 80) E NR(30)PCFA(3 30) WCFA(3,30).NENR.FDG(3),PSE(3 ) ,
3 PI ON(3),AT(3),A(6.3).B< 5, 3).G(5.3),UH( 10).UI(10).UJ(10),
4 UD ( 1 0) US NU { 1 0) USF( 1 0) ,UEIN( 1 1 ) IUTP, IUTM1 DG 3) NS GC 3 ) N I G( 3 ) ,
5 NUM GASNUMSTWIS(50),NSEC.MUNIT.I SEED.1 STOPI LAST.EIN.NPRIM* COSI
6 H.EMIN, ZST ART T M IN .THRESH! 3 ) AK( 2,3 ) AJ ( 3.3) GAMA ( 2, 3) .
7 TO(3.3).FAC(6) .AOE(6) ,A1E (6) ,BOEC 6),B1E(6).COE(5),DOE(6),
8 D 1 E ( 6 ) PR OB (40 ) W ( 40).NIE< 40) .A 21< 3) A22( 3) A 31 ( 3 ) A 32 ( 3 ) .
9 A33 ( 3) ,B1 1 (3) .B12 (3 ) B1 3 (3 ) Cl (3 ) C21 (3 ), C22( 3) C3K3),
A C32(3),D1( 3 ) D 2 ( 3 ) F 1 ( 3 ) F 2 ( 3 ) THR I ( 16),AKI(2,16),AJI(3,16).
B GAMA I(2.16).TO I (3, 16).SIGT(6 ) ,THET(40),NX,CSIE(20) ,EIT(20) .
C IEILS.IElLMl.P.R.ZDIS.NPIN.NOP.COSPA RT.F(40),R1.R2.R3.R4,
D R5.XV.YV.ZV.XVN.YVN.ZVN ,PA.PHIEV.WLOSS,NSTAT.NSCS.NPHF.
E IR1 IR2.NG.EVPRI.RAN,EMSD.EINEL.EXC5(50).COSPAN,T,FOVAL
E=EVPRI
D=D1(NG)/(T + D2(NG) )
D IS THE CCT ) IN TFE SEC, DIF. FORM,
BB=B11(NG)+(E/Bl2(NG))**B13(NG)
B IS THE B(E) IN THE SEC, DIF, FORM,
C2 =C21(NG)+C22(NG)*E
C2 IS THE THETA SUBSCRIPT A (E)
C3=C31(NG)+C32NG)*E
C 3 IS THE THETA SUBSCRIPT B (E)
SQB=SQRT(BB)
THO=C1(NG)+C2/(T+C3)
CST=COS(THO)
FTN=SQB*(-1,-CST)
STN=SQB*(1o-CST)
TMNM = ATAN2( FTN ,D )ATAN2< STN.D)
SL1 = AT AN2(STN D)
RA TAN=R* TMNMFSLI
TM 1= D*T AN ( R ATAN ) /SQB
CO SP AN-=TMl +CST
PAN IS THE POLAR SCATTERING ANGLE
PAN=ARCOS(COSPAN )
IF(PAN o GT, 31416)PAN=PAN3,14 16
RE TURN
SDIFM001
SDIFM002
SDIFM003
SDIFM004
SD IFM005
SDIF MO 06
SDIFM007
SDIFM008
SDIFM009
SDIFM010
,SDIFMO11
SDIFM012
SDIFM013
SDIFMO 14
SDIFM015
SDIFM016
SDIFM017
SD IFMO 18
SDIFM019
SD IFMO20
SD IFMO 21
SDIFM022
SD IFMO23
SDIFM024
SD IFM025
SDIFM026
SDIFMO27
SDIFMO28
SDIFM029
SDIFM030
SD IFM031
SDIFM032
SDIFM033
SD IFMO 34
SDIFMO35
SDIF MO 36
SDIFM037
SDIFM038
SDIFM039
SDIF MO40
SDIFM041
SDIFM042
SDIF MO 4 3
SDIFM 044
SDIFM045
-194-


Figure 7.5 Four variable spatial yield spectra for an incident
(a, b, c, and d) electron energy of 1 KeV given at four longitudinal
distances: z = 0.061 (Figure 7.5a), z = 0.305
(Figure 7.5b), z = 0.549 (Figure 7.5c), and
z = 0.793 (Figure 7.5d). At each longitudinal cut
the yield spectrum is given at four radial dis
tances: p = 0.061, o; p = 0.305, ; p = 0.549, ;
and p = 0.793, A.


-45-
peak indicated in these data was used in a MC calculation. The electron
scattering which results using this inelastic scattering approximation
in a computation was so small as to be virtually undetectable.
Cartwright et al. (1977) and Chutjian et al. (1977) have studied a
more comprehensive list of states and have observed significant electron
scattering (especially due to the optically forbidden states) in the
range from 10 eV to 60 eV. Characterizing this data in some way appears
to be a rather endless task.
Dealing with this type of inelastic scattering is thus still a
problem. Above 100 eV the optically allowed excitations are the most
important; thus it is safe to conclude that the inelastic scattering
events will not affect the energy degradation process. Below 100 eV,
as a first approximation, it is assumed in this work that the inelastic
excitation events scatter as much as the elastic events. This is
probably a reasonable approximation to the very complex inelastic ex
citation scattering. In section VI.B the effects of this assumption
are discussed.
C. Total Cross Section (Elastic Plus Inelastic)
Elastic and inelastic processes have been considered in sections
III.A and III.B. Another aspect of the cross sections is the total
(elastic plus inelastic) cross section.
Blaauw, de Heer, Wagenaar, and Barends (1977) have recently pub
lished experimental data on the total cross section values of Ng. These
experimental values are compared with the cross section values from this
work in Figure 3.6.


-137-
Table 7.2 Parameters and their values are given below which are to be
used in Eq. (7.7) for the molecular nitrogen spatial yield
spectrum.
Parameter
Value
Parameter
Value
all
587
dn
0.6 x
10s
a12
-1.63
d12
-1.68
a21
0.4
d2
0.2
a22
0.075
0.9
a31
0.1
f12
0.044
a32
0.019
f21
0.104
blll
81
f22
-0.39
b112
-1.8
911
0.85
bl 13
8.0
g12
0.07
bl 21
0.4
g2
0.2
bl 22
0.05
rl
2.27 x
10 ^ gm/cm^
*13
0.2
r2
6.22 x
10-b gm/cm^
B2
cin
-1.52
1.30 x 104
r3
1.67
C112
-1.5
C12
0.15
C2
10


-12-
This approach combined these two methods of electron energy deposi
tion in order to find a reasonable solution to the very difficult auroral
energy deposition problem. The Fokker-Planck method is accurate only
at large incident energies; therefore, it should only be used at ener
gies above 500 eV. The two-stream equation of transfer approach, on the
other hand, is more accurate at energies below 500 eV. This combination
then provided a very reasonable solution to the auroral electron spatial
deposition problem for a reasonable amount of calculation.
The Fokker-Planck equation and the two-stream equation of transfer
may both be derived from the Boltzmann equation or the general equation
of transfer. This general equation of transfer, in one of its simpler
forms, is written as (from Strickland et al., 1976)
u -nUMEMz.E.v)
+ ntz)o(E) / Rfu'.u.E'.Eltfz.E'.p'ldE'dp' (2.12)
(assuming a steady state condition and no external fields) where
I o^y'.y.E'.E)
Rtw'.y.E'.E) = j^ (2.13)
with the sum over all processes. The symbols p and p1 are the cosines
of the pitch angles e and e' which are associated with the directions n
and n' given in Figure 2.1.
The first term on the right hand side of Eq. (2.12) represents the
scattering out of p. The R(p',p,E',E) in the second term is the proba
bility (eV \2wsr) ^) that a collision of an electron of energy E' and
direction p1 with some particle will result in an electron of energy E


CHAPTER II
A SHORT REVIEW OF ENERGY DEPOSITION TECHNIQUES
Several standard energy deposition techniques will be discussed in
this chapter. In the first section, II.A, several general ways for
treating the degradation of the energy of charged particles will be re
viewed briefly. The second section, II.B, includes a brief sketch of
four Monte Carlo energy deposition schemes: The MC approach applied in
this work and three other MC techniques.
A. Energy Deposition Techniques
Since the turn of the century, researchers have been studying the
energy degradation of rapidly moving particles in a medium. Initial work
in this area was carried out by Roentgen, Becquerel, Thompson, Bragg,
Rutherford, Bohr, and other founders of modern physics.
These pioneers in the energy degradation process were mainly con
cerned with the medium affecting the particle. In order to solve this
complex energy degradation problem, most of the early workers used a
local energy deposition approximation. Even today many degradation
problems can be solved fairly accurately with this local approximation.
One of the earliest local energy deposition methods is that of the
continuous slowing down approximation (hereafter called CSDA) first
used by Niels Bohr (1913, 1915). Bethe (1930) expanded on this work and
used an approximate theoretical treatment (valid at high energies) to
describe the slowing down of particles in a medium.
-5-


14 JULY 1978
NERDC
CARD L 1ST UTIL ITY
C
C
C
C
C
C
C
c
c
SUBROUTINE PHFEL
SUBROUTINE PHFEL ***
THIS SUBROUTINE IS ACCESSED THROUGH MC
THIS SUBROUTINE CALCULATES THE SCATTERING POLAR ANGLE FROM AN
ELASTIC COLLISION WHICH IS EXPRESSED ANALYTICALLY,
RIGHT NOW THE SCREENED RUTHERFORD DIFFERENTIAL CROSS SECTION
IS EXPRESSED ANALYTICALLY
COMMON ALPE (6) B ETE (6 ) ,CE(6),FE(6) ,WE<6 ) ALF A { 15).BEFA(15).
1 CFA(15),FFA(15),WFA(15).WF(15).FACI( 15) NFA .NAR,SA(3 .80) ,
2 2 ALT (80) ENR30 ),PCFA (3 ,30 > WCFA( 3, 30 ) NENR F DG( 3) ,PSE( 3),
3 PI ON(3) AT(3) A ( 6 *3 ) ,B(5,3).G(5,3),UH(10),UI(10),UJ(10),
4 UD( 10 ),USNU( 10 ) ,USF( 1 0) ,UEIN( 1 1) I UTP ,1 UTM1 DG< 3) NSG<3> ,NIG(3 ) ,
5 N UMGA S,NUMST,WIS(50) .NSEC.MUNIT,I SEED,I STOP,ILAST,EIN,NPRIM,COSI
6 HEMINZSTART TMIN,THRE SH(3) ,AK{2 *3) ,AJ(3,3) GAMA(2,3) ,
7 TO( 3,3) .FAC (6 ) AOE (6 ) A1E( 6),B0E( 6),B1E(6),C0E(6),D0E(6) .
8 D 1E(6) ,PROS(40) ,W(40),NIE(40).A21(3),A2 2(3) ,A31(3).A323 ),
9 A33(3)B1 1 ( 3 ) B 1 2( 3).B13(3),C1(3),C21(3),C22(3),C31(3),
A C 32(3) ,D1 (3 ) ,D2(3) ,F1 (3),F2(3).THRI(16),AKI(2,16),AJI(3,16),
B GAMA 1(2,16),T0I (3, 16) ,SIGT(6).THE T(40) ,NX ,CSIE(20) ,E IT(20) ,
C IEILS,IEILMl.P.R,ZDIS.NPIN,NOP,C3SPA,RT,F(40),R1,R2,R3,R4,
D R5.XV,YV,Z V.XVN,Y VN,ZVN,PA,PHI ,EV WLQSS NST AT,NSCS,NPHF,
E IR1.IR2.NG.EVPRI,RAN,EMSD,EINEL,EXC5(50),COSPAN,T,F0VAL
E= EV
ETA=16./E
CO SPA N= 1 o+2 o *E TA2 a *E T A* ( lo + ETA)/{ lo+ETA-R)
PAN=ARCOS( COSPAN )
PAN IS THE POLAR SCATTERING ANGLEo
RETURN
END
PHFEL001
PHFEL 0 02
PHFEL 003
PHFELO 04
PHFEL 0 05
PHFELO 06
PHFEL007
PHFEL 008
PHFEL009
PHFEL010
PHFELO 11
PHFELO12
.PHFEL 013
P H F EL 014
PHFEL 015
PHFELO 16
PHFELO17
PHFEL 018
PHFELO19
PHFEL020
PHFEL021
PHFEL022
PHFEL023
PHFEL 0 24
PHFEL025
PHFEL026
PHFEL027
PHFEL028
PHFEL 029
PHFELO 30
PHFEL 031
-Z6L-


c
27 I F ( E V a GT o EMI N) GO TO 16
IF (NG 200 oEQo 0)GO TO 29
EVD=EV
ZVD=ZV
C
C
C
39 IF(NG200.EQ.1)GO TO 33
C NOW DEGRADE THE SECONDARY
EV =E V SA VS
COSPA=CPSAVS
PA=PA SA V S
PH I=P H ISVS
XV=XVSVS
YV=YVSVS
ZV=ZVSVS
NG200=1
IFCIESCP a EQ 1)GO TO 16
NI 2=1
WR ITE ( M )NI 2
WRIT E(M ) EV D ZVD
N=N+1
NI 2=0
GO TO 16
C
C
MC 000334
MC000335
MC000336
MC 000337
MC000338
MC000339
MC 000340
MC000341
MC 000342
MC030343
MC000344
MC 000345
MC000346
MC000347
MC 000 348
MC000349
MC 000350
MC000351
MC000352
MC 000353
MC000354
MC000355
MC 000356
MC000357
MC 0003 58
MC 000 359
MC000360
-184-


no no (Ti
100
11 1
51
200
201
52
C
C
c
c
11
WR1TE(6.100)
FORMAT! 1 INDEX OF STATE',6X, NO OF
* 'ENERGY LOSS',//)
DO 51 IC=1NUMG
WRITE (6,111) IC, JES(IC), WIE( IC)
FORMAT(7X,15, 10X,I 7.12X ,F12o 3)
ESF=ESF+JES{IC)*WIE(IC)
WRITE(6,200)
FORMAT!///. LOWER EN INTERVAL,7X,MID
* 'NOo OF EXCITATIONS',//)
DO 52 ID=l,NTOP
ID PI = ID+1
WMI D = (WIS(ID)+WI S( I DP1 ))/2o
WR ITE ( 6, 201 ) ID.WMID ,JI S( ID)
FORMAT!I10,11X,F123,5X.110)
WRTDA118
EXCITATIONS',6X, WRTDA119
WRTOA120
WRTOA121
WRTDA122
WRTOA123
WRTDA1 24
WRTDA125
ENERGY',?X, WRTDA126
WRTDA127
WRTDA128
WRTDA129
WRTDA 1 30
WRTDAl31
WRTDA 132
ESF=ESF+JI S( ID)* WMID
ESF IS THE ENERGY DEPOSITED
The ENERGIES ESI AND ESF WILL NOT BE EQUAL BECAUSE OF THE COMBINED
STATE THE THRESHOLD OF THE COMBINED STATE IS CHANGING BELOW
200 EV THUS YOU WOULD NOT EXPECT THIS MODE OF COMPARISON TO WORK
WRITE(6,511)ESI.ESF
FORMAT (////, INITIAL ENERGY= G 1 5.7 / ENERGY D EP OS I TE D = G1 5 7 )
O
WRTDA133
WRTDAl34
WRTDA135
WRTDAl36
WRTDAl37
WRTDA138
WRTDAl39
WRTDA140
WRTDA141
WRTDAl42
WRTDA143
WRTDA144
WRITE OUT THE INTENSITY INFORMATION AS A FUNCTION OF RHO
IF ( NR HO oNE 1 ) G O TO 3311
-217-


-140-


-109-
The shape of the collision profile at positive longitudinal dis
tances appears to have a distribution which is dependent on the "a"
parameter. This functional form for the distribution of collisions D(z)
can be represented simply as
where
and
D(z)
1
a(z ZQ)2 + 3
a = -0.148 1 n(y|g-)
3 = -0.09 ln( r)
3.2 x 10
z = 0.056/a
o
(6.2)
The o's in Figure 6.3 represent a visual fit to these data. Equation
(6.2) does a reasonable job of indicating the gross features of the
collision profile. The features illustrated with this form are the
height of the peak, the width of the distribution, and the location of
the peak.
The location of the peak is observed to be inversely proportional
to the value of the screening parameter a. The values of a and g are
directly proportional to the natural logarithm of the screening para
meter a, thereby causing the peak of the collision profile to decrease
in value as "a" decreases.
Model 2 [Eq. (3.13): PM2(e,E) (-f/{l cose + a}2) -
p
((1 f)/{l + cose + c} )] can also be used in a MC computation of a
similar nature. Figure 6.4 indicates the differential cross sections
(trials B1, B2, and B3) used in this comparison and Table 6.2 lists the
properties of these trials.


-2-
the upper atmosphere is then reduced in complexity. Since N2 interacts
with electrons similar to the way that other atmospheric gases interact
with electrons, it follows that differential and total cross section sets
for these gases could be assembled in a like manner.
Another aspect of this work is a sensitivity comparison among several
of the influences on the electron energy deposition. The spatial energy
degradation is vitally linked to the elastic phase function used. Since
there are data available on the elastic differential cross sections of
N2 as well as the energy degradation resulting from electron impact on
N2 a comparison illustrating the effects of a variation of the elastic
phase function is quite useful. Other influences on the spatial energy
deposition, including ionization and excitation differential cross
sections and the total elastic cross sections, are also considered in
this work.
In order to deal with these spatial and energetic aspects of elec
tron energy degradation, a Monte Carlo (which may be abbreviated MC)
calculation is used. The MC technique, depending on how it is used, can
be the most accurate energy deposition approach. Use of this MC method
at various incident energies helps in the assemblage of the best cross
section set for N2 and provides the easiest way of comparing some of
the influences on the spatial energy deposition.
The details of this undertaking are discussed in Chapters II through
VIII. A paragraph summary of each chapter is given below.
The second chapter presents a brief review of some of the standard
electron energy deposition methods. The continuous slowing down approxi
mation, discrete energy bin, Fokker-Planck equation, two-stream equation
of transfer, and the multi-stream equation of transfer are all included


-13-
Z
Figure 2.1 The directions denoted by n' and n are the incident and
scattered directions of the electron, respectively.


-8-
solved by building on the lower-energy solutions to obtain the higher
energy solutions. The Spencer-Fano method introduces the electron at
the highest energy and solves the equation at successively lower ener
gies. An approach similar to the Spencer-Fano method was developed by
Peterson (1969) and is called the discrete energy bin (hereafter called
the DEB) method.
Peterson (1969) pointed out certain differences between the CSDA and
the DEB results. In particular, he noted that the DEB method tends to
predict higher populations of some excited states than does the CSDA.
In the modification of the DEB method by Jura (1971), Dalgarno and
Lejeune (1971), and Cravens, Victor, and Dalgarno (1975), the equilibrium
flux or degradation spectrum f(E,EQ) (for incident energy Eq and electron
-2 -1 -1
energy E and in units of # cm sec eV ) of Spencer and Fano (1954)
is obtained directly. Douthat (1975), in an effort to make the degrada
tion spectra suitable for applications, proposed an approximate method
of "scaling." Unfortunately, this method is quite cumbersome and not
very accurate. This impelled Garvey, Porter, and Green (1977) to seek
an analytic representation of the degradation spectra and, despite its
complex nature, they found an analytic expression to represent this
spectra for W^.
The concept of the "yield spectra" U(E,Eq) was first initiated
through a modified DEB approach given in Green, Jackman, and Garvey
(1977) in an effort to find a quantity with simpler properties than the
degradation spectra. By utilizing the product
U(E,Eq) = 0T(E) f(E,Eo)
where Oy(E) is the total inelastic cross section for an electron of
energy E, one defines a quantity U(E,EQ) which not only has a simpler


-57-
the Cartesian coordinates. Now, calculate the azimuthal angle and the
a
polar angle of the electron after a collision.
2. Second and Third Random Numbers, R2 and R3
In actuality the type of collision must be specified before the
scattering can be calculated. It is assumed, however, that the type of
collision is already known (see subsection IV.C.4). The second, R2, and
third, R^, random numbers are not chosen if the collision is an excita
tion event and E is greater than 100 eV. They are chosen for all other
collisions.
The R2 is used to calculate the azimuthal scattering angle, 41, of
the electron from its direction of motion. The premise is that the
azimuthal scattering is isotropic; therefore,
4>' = R2 2h (4.7)
(Note that the ' angle is the only angle not represented in Figure 4.2.
Inclusion of 4' adds too much complication to an already cluttered
figure.)
The third random number, R^, is employed to calculate the polar
scattering angle e' of the electron from its direction of motion. (The
angle e1 is represented twice in Figure 4.2: Once as the scattering due
to the first collision and once as the scattering due to the second
collision.)
For elastic collisions, Eq. (3.1), (3.12), (3.13), or (3.14) are
used in determining 01. In all but one of these phase functions, an
analytic expression can be used to determine 0' from the random number,
R3. These analytic expressions are given below.


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-224-
Mumma, M.J. and E.C. Zipf, Dissociative excitation of vacuum-ultraviolet
emission features by electron impact on molecular gases. II. No,
J. Chem. Phys., 55, 5582, 1971.
O'Brien, D.J., High-latitude geophysical studies with satellite Injun 3.
3. Precipitation of electrons into the atmosphere, J. Geophys. Res.,
69, 13, 1964.
Opal, C.B., E.C. Beaty, and W.K. Peterson, Tables of secondary-electron-
production cross sections, Atomic Data, 4, 209, 1972.
Peterson, L.R., Discrete deposition of energy by electrons in gases,
Phys. Rev., 187, 105, 1969.
Polak, L.S., D.I. Slovetskii, and A.S. Sokolov, Dissociation of nitrogen
molecules from excited electronic states. I. Predissociation
efficiency, High Energy Chemistry, 6_, 350, 1972.
Porter, H.S. and A.E.S. Green, Comparison of Monte Carlo and continuous
slowing-down approximation treatments of 1-KeV proton energy deposi
tion in N2, J. Appl. Phys., 46, 5030, 1975.
Porter, H.S., C.H. Jackman, and A.E.S. Green, Efficiencies for production
of atomic nitrogen and oxygen by relativistic proton impact in air,
J. Chem. Phys., 65, 154, 1976.
Porter, H.S. and F.W. Jump, Jr., Analytic total and angular elastic
electron impact cross sections for planetary atmospheres, Tech. Rep,
CSC/TM-78/6017, prepared for Goddard Space Flight Center by Computer
Sciences Corporation, Greenbelt, Maryland, 1978.
Rapp, D. and P. Englander-Golden, Total cross sections for ionization
and attachment in gases by electron impact. I. Positive ionization,
J. Chem. Phys., 43, 1464, 1965.
Rapp, D., P. Englander-Golden, and D. Briglia, Cross sections for dis
sociative ionization of molecules by electron impact, J. Chem.
Phys., 42, 4081, 1965.
Rees, M.H., A.I. Stewart, and J.C.G. Walker, Secondary electrons in
aurora, Planet. Space Sci., 1_7, 1997, 1969.
Riewe, F. and A.E.S. Green, Ultraviolet aureole around a source at a
finite distance, to be published in Appl. Optics, 1978.
Rohrlich, F. and B.C. Carlson, Positron-electron differences in energy
loss and multiple scattering, Phys. Rev., 93, 38, 1954.
Sawada, T., P.S. Ganas, and A.E.S. Green, Elastic scattering of electrons
from N2> Phys. Rev. A, 9_, 1130, 1974.
Schram, B.L., F.J. de Heer, M.J. Van der Wiel, and J. Kistemaker, Ioniza
tion cross sections for electrons (0.6-20 KeV) in noble and diatomic
gases, Physica, 3T_, 94, 1965.


non
63
64
702
31
555
560
CONTINUE
IF (2 .GTo ZALT( 1 ) ) JSEC( I I, 1 ) = JSEC( I I. 1 >1
IF{Z LT o ZALTCNAR) )JSEC DO 64 I 2=2, NAP.
12 Ml=I 2 1
IF ( Z o GTo ZALTII2) -.AND Z LT, ZALT ( 12Mi > ) JS EC { I I 12 )=
* JSECIII, 12 )+l
CONTINUE
J IS< II )=JIS( II )+ 1
CO NTINUE
IFIEBEF oLTo WIS(NSEC))GO TO 30
IF { Z GTc Z ALT ( 1 > ) J ZE ( 3N,1)=JZE(3N,1 ) 4-1
IF { Z .LTo ZALT(NAR)) J ZE ( 3 N N AR )= JZE ( 3, N N AR ) + 1
DO 31 IA=2NAR
IAM 1= IA1
I A SE T = I A
IF ( Z .GTo ZALT( I A) 3 AND
CONTINUE
GO TO 30
DO 560 KA= 1,3
JZE( KA,N, I ASET 1= JZE(KA,N, I ASET ) +1
IF(NG 200 oEQ. 1)JZE(1*N#IASET)JZE(1 N I AS 5T)-1
IF(NG200 oEQ. 0)JZE(2,N,IA SET)=JZE(2,N.I ASET)-1
GO TO 30
THIS PART OF THE PROGRAM CALCULATES THE DISTRIBUTION OF EXCITATIONS
1 LT a ZALTII AMI ))GO TO 555
ROT AP1 18
RDTAP119
RDTAPI 20
RDT API 21
RDTAPI 22
RDTAP123
RDTAPI 24
RDTAP125
RDTAPI 26
RDTAPI 27
RDTAP128
RDTAPI 29
RD TAPI 30
RDTAP131
RDT API 32
RDTAP 133
RDT AP134
RDT API 35
RDTAP136
RDTAPI 37
RD TAPI 38
RDTAP139
RDT API 40
RDTAP141
RDT AP142
RDT API 43
RDTAP144
-211-


-110-
Figure 6.4 Differential cross section graph for model 2 (which contains
a forward and a backward scattering contribution) trials:
B1, B2, and B3.


-9-
shape than f(E,EQ) but also has approximately the same magnitude for
all substances. This yield spectrum can also be represented analytically.
It effectively embodies the non-spatial information of the degradation
process.
Jackman, Garvey, and Green (1977a), using this modified DEB,
elaborated on the differences between the DEB method and the CSDA. The
more accurate modified DEB method was found to produce consistently more
ions per energy loss while at the same time producing less excitations
of some of the low lying states when compared with the CSDA. The CSDA,
although inexpensive to use, appears to be ill-suited for calculations
requiring an absolute accuracy better than about 20%. Since auroral and
dayglow intensities are rarely measured to better than this accuracy, the
CSDA has been adequate for most purposes of concern in aeronomy. How
ever, with future improved measurements it should be purposeful to
utilize more accurate deposition techniques.
Several recent spatial electron energy deposition studies have been
concerned with the spatial aspects of auroral electron energy deposition.
Walt, MacDonald, and Francis (1967) employed the Fokker-Planck diffusion
equation to give a detailed description of kilovolt auroral electrons.
The Fokker-Planck equation, as given in the Strickland, Book, Coffey,
and Fedder (1976) paper, is written
+ ^ET^ [L(EMz,E,w)] (2.5)
(2.5)
-2 -1 1 -1
where <¡>(z,E,v) is the flux (in units of cm sec eV sr ), z is the
distance into the medium along the z axis, E is the electron energy,



PAGE 1

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PAGE 236

81,9(56,7< 2) )/25,'$


-124-
6 2
The range is observed to be 6.98 x 10 gm/cm as compared with
-fi O
5.57 x 10 gm/cm using the cross sections given by Eq. (3.9) (here
after called MSR). The shape of the intensity profile of the 3914 A
emission is also different. The maximum of the profile is less peaked
than that given by MSR.
Banks, Chappell, and Nagy (1974) used cross section values which are
shown in Figure 3.2. These cross section points are somewhat different
than the cross sections from MSR in the range from 100 to 500 eV. This
indicates that electrons degraded with these cross sections have a range
less than that of MSR.
This section completes the sensitivity study. The next chapter
discusses the MC calculation, the important yield spectra, and the general
energy loss scheme.


-59-
section form. The angle e' is then found through the correct placement
of R3 into an angular segment whose beginning and ending point scattering
probabilities bracket Ry
For this work twenty-four angular segments were chosen. Their end
points are given in Table 4.2. With twenty-four angular intervals, the
results from the Monte Carlo calculation came out to be the same as with
the use of forty angular intervals. If sixteen or even twenty segments
were used, the MC computation gave results that were 5% to 10% different.
The coordinate system, but represent the azimuthal and polar scattering of
the scattered electron from the direction of travel of the incident elec
tron. In order to calculate

d a
representing the motion of the electron after the collision, some spheri
cal trigonometry must be used. The following relations hold in this
transposition:
cosa = [coseb cosb sine' cos'
- sinb sine' sin' + sineb cosb cose']/sinea (4.11)
sina = [coseb sin4>b sine* cos
+ cosb sine1 sin<¡>' + sineb sinb cose']/sinea (4.12)
(4.13)
cosea = coseb cose' sineb sine* cos^'
(4.14)
(4.15)


Backscatter Probability
-37-
10
E (eV)
Figure 3.5 Backscatter probabilities for the screened Rutherford
(dashed line) and the model 3 (solid line) are compared
with Wedde and Strand (1974), x; and Shyn et al. (1972), o.


-35-
9 (Degrees)
i
Figure 3.4b


DO 4 1 = 1 NUMGAS
N= 2* I -1
JTOP=N1GCI)
DO 4 J=1 JTOP
KI0=K10+1
IF (E oLT o THRIIKIQDGO TO 4
TSET = TOI(1 iKIO)-TOI (2,KIO)/(E+TO I (3.K 10) )
GSET = GAMAI( 1,KIO)*E/ TM=(E-THRI(KIO)1/2j
ASET=(AKI(1 ,KIQ)/(E + AKI( 2.KIO) ) )*ALOG(E/AJI(1,KI0)+AJ I(2.KI0) +
1 AJ I(3,K10 )/E )
SIGI=ASET*GSET*(AT AN((TM-TSET J/GSET)+ATAN(TSET/GSET ) )
P=SIGI/SIGT(N)*FDG( I)*(lo-PSE(I ) > + P
IF (P GE. RAN) GO TO 13
C INDIVIDUAL IONIZATION STATE PROBABILITIES ARE CALCULATED,.
4 CONTINUE
KADD=K +K10
30 J= 0
DO 6 1=1 iNENR
J= J+l
IF (E L To ENR(I)) GO TO 20
6 CONTINUE
20 JM1 = J1
DO 5 1=1, NUMGAS
N=2*I -1
NSTAT=KADD+I
EM EN=E-ENR(JM1 )
CTB20046
CTB20047
CTB20048
CTB20049
CT 820050
CTB20051
CTB200S2
CTB20053
CTB20054
CTB20055
CTB200 56
CTB20057
CTB20058
CTB20059
CTB20060
CTB20061
CTB20062
CTB20063
CTB20064
C TB 200 65
CTB20066
CTB20067
CTB20068
CTB20069
CTB20070
CTB20071
CTB20072
-99 L-


14 JULY 1978
NEROC
CARD LIST UTILITY
C
C
9
10
C
C
8
633
635
1 1
522
C
24
671
1 THR I ,6X. KI ,6X, KBI 5X JI 5X, JB I ,5X,'JCI ,6X GAMS I* .
1 GAME) I 7X T S I 4X TAI.5X,TBI,/)
DO 10 1=1* NUMGAS
JU=NIG(I)
DO 9 J=JL,JU
READ ( 5,502) G1 *G2 G3 G4 G5 THR I ( J) ( AKI ( K, J) ,K = 1,2 ) ( AJ I ( K J ) .
1 K=1*3)
READ(5,504)(GAMAI{KJ).K=1,2),(T0T(KJ),K=1,3)
READ IN THE S(E*T> PARAMETERS FOR THE INDIVIDUAL IONIZATION CROSS
SECT IONS.
WRI TE (6.61 1 > G1 G2 G3 G4 G5 THR I (J ) ( AKI ( K J ) K=1,2 ) {A J I (K J ) ,
1 K=1 3)*(GAMAI(K.J) *K = 1*2) .(TOI(K.J) ,K=1 ,3)
CONTINUE
JL=JU+1
3X DAT AO145
DATAOl46
DATA 0147
DATAOl48
DATAOl 49
DAT AO 150
DAT AO 151
DA TA 0152
DAT AO 153
DATAOl54
DATAOl55
DATAOl56
DA TA 0 1 57
D AT AO 1 58
CONT INUE
RE AD ( 5 *500 ) NE NR
READ ( 5. 522 ) (ENRII ) 1=1 NENR )
DO 8 1 = 1 NU MGAS
RE AD (5,522 ) ( PCF A ( I J ) J=1 NENR )
READ(5,522) {WCFAt I .J) .J = 1 .NENR)
READ IN THE COMPOSITE STATE PARAMETERS OR ITS PROBABILITY AND
THRESHOLD VALUES FOR ENERGIES LESS THAN 200 EV a
CONTINUE
WRITE{6 ,633 )
FORMAT!////, THE COMPOSITE LEFT-OVER STATES ARE NOW READ IN,//,
1 4X,ENERGY,6X.PROBABILITY,5X.AVE,ENERGY LOSS*.3X,
l 'GAS INDEX./)
DO 11 1=1,NUMGAS
DO 11 J=1,NENR
WRITE(6,635) ENR(J) .PCFA CI ,J),WCFA(I ,J) ,I
FORM AT(3G15.7.3X,17)
CONTINUE
FORMAT(8E10a 0)
PE AD(5,500)IEILS
IE ILM1=IEILS-1
DO 24 1=1, NUMGAS
READ(5 ,522) READ{5,522)(CSIE(J),J=1,IEILS)
READ IN THE TOTAL INELASTIC CROSS SECTION AT ENERGIES BELOW 30 EV.
CONTINUE
WRITE(6,671)
FORMAT!/////,1 THE TOTAL INELASTIC CROSS SECTION IS READ IN FOR
* ENERGIES BELOW 30 EV. THE UNITS ARE
* 4X,'ENERGY*,6X CROSS SECTION,/)
DO 26 1=1,NUMGAS
10**(-16) CM**2.//,
DATAOl59
DATAOl60
DAT AO 161
DATAOl62
DATAOl63
DATAO164
DATAOl 65
DA T A O 166
DAT AO 167
DATAOl68
DAT AO 169
DATAOl70
DAT A 0171
DAT AO 172
DATAOl73
DATAOl74
DATAOl75
DATA 01 76
DAT AO 177
DATAOl78
DATA 0179
DAT AOi80
DATAOl 81
DATAO182
DATAOl83
DA TAO184
OAT AO 185
DATAOl86
DATAOl87
DA TAO188
DAT AO 189
-172-


-223-
Jasperse, J.R., Electron distribution function and ion concentrations in
the earth's lower ionosphere from Boltzmann-Fokker-Planck theory,
Planet. Space Sci., 25, 743, 1977.
Jung, K., E. Schubert, D.A.L. Paul, and H. Ehrhardt, Angular correlation
of outgoing electrons following ionization of H2 and N2 by electron
impact, J. Phys. B: Atom. Molec. Phys., 8, 1330, 1975.
Jura, M., Models of the interstellar gas, Ph.D. thesis, Harvard Univ.,
Cambridge, Mass., 1971.
Jusick, A.T., C.E. Watson, L.R. Peterson, and A.E.S. Green, Electron
impact cross sections for atmospheric species. 1. Helium, J.
Geophys. Res., 72, 3943, 1967.
Kamiyami, H., The electron density distribution in the lower ionosphere
produced through impact ionization by precipitating electrons and
through photoionization by the associated bremsstrahlung x-rays,
J. Geomag, and Geoelec., 19, 27, 1967.
Kieffer, L.J., Low-energy electron-collision cross-section data, part III:
Total scattering; differential elastic scattering, Atomic Data, 2,
293, 1971.
Kutcher, G.J. and A.E.S. Green, Multiple elastic scattering of slow
electrons: Parametric study for H2, J. Appl. Phys., 47, 2175, 1976.
Lewis, H.W., Multiple scattering in an infinite medium, Phys. Rev., 78,
526, 1950.
Maeda, K. and A.C. Aikin, Variations of polar mesospheric oxygen and ozone
during auroral events, Planet. Space Sci., 16, 371, 1968.
Mantas, G.P., Theory of photoelectron thermalization and transport in
the ionosphere, Planet. Space Sci., 23, 337, 1975.
McConkey, J.W., J.M. Woolsey, and D.J. Burns, Absolute cross section for
electron impact excitation of 3914 A N$, Planet. Space Sci., 15,
1332, 1967.
McDiarmid, J.B., D.C. Rose, and E. Budzinski, Direct measurement of
charged particles associated with auroral zone radio absorption,
Can. J. Phys., 39, 1888, 1961.
Moliere, G., Theory of scattering of fast charged particles. I. Single
scattering in a screened Coulomb field, Z. Naturf., 2a, 133, 1947.
Moliere, G., Theory of scattering of fast charged particles. II. Plural
and multiple scattering, Z. Naturf., 3a, 78, 1948.
Mott, N.F. and H.S.W. Massey, The Theory of Atomic Collisions, Clarendon
Press, Third Edition, Oxford, England, 1965.


-14-
and direction y. The integral in Eq. (2.12) is over all possible ener
gies E' and directions of motion y'.
Strickland et al. (1976) studied the auroral electron scattering
and energy loss with a multiangle equation of transfer. Their approach
is one of the most accurate yet applied to auroral electrons. This multi-
angle method of solution is more realistic than the two-stream approach
and it is computationally more difficult as well.
The methods discussed above are the "state of the art" approaches
(excluding the Monte Carlo methods which are discussed in section II.B)
to the I EEs degrading in the atmosphere. Other approaches used by
Jasperse (1976, 1977) and Mantas (1975) are mainly concerned with photo
electrons and will not be discussed here.
The Monte Carlo approach can rival any of these electron energy
deposition methods in accuracy when used in the proper manner. This
stochastic technique for solving the deposition problem will be con
sidered next in section II.B.
B. Monte Carlo Energy Deposition Techniques
Another method of solving the spatial energy deposition problem is
the use of the Monte Carlo approach. The MC technique, which is used
in this paper, is a stochastic method of degrading an energetic electron.
The approach can be one of the most exact methods of electron energy
deposition. Briefly, one electron is taken at a time and allowed to
degrade in energy collision by collision. This deposition attempts to
mimic the randomness of the actual degradation process that occurs in
nature.


-222-
Green, A.E.S. and C.A. Barth, Calculations of the photoelectron excitation
of the dayglow, J. Geophys. Res., 72, 3975, 1967.
Green, A.E.S. and S.K. Dutta, Semi-empirical cross sections for electron
impacts, J. Geophys. Res., 72, 3933, 1967.
Green, A.E.S., C.H. Jackman, and R.H. Garvey, Electron impact on atmospher
ic gases. 2. Yield spectra, J. Geophys. Res., 82, 5104, 1977.
Green, A.E.S. and L.R. Peterson, Energy loss functions for electrons and
protons in planetary gases, J. Geophys. Res., 73, 233, 1968.
Green, A.E.S. and T. Sawada, Ionization cross sections and secondary
electron distributions, J. Atmos. Terr. Phys., 34, 1719, 1972.
Green, A.E.S. and P.J. Wyatt, Atomic and Space Physics, p. 170, Addison-
Wesley Publishing Company, Inc., Reading, Mass., 1965.
Grn, A.E., Lumineszenz-photometrische messungen der energieabsorption in
strahlungsfeld von elektronenquellen eindimensionalor fall in luft,
Z. Naturforschg., 12a, 89, 1957.
Heaps, M.G. and A.E.S. Green, Monte Carlo approach to the spatial deposi
tion of energy by electrons in molecular hydrogen, J. Appl. Phys.,
45, 3183, 1974.
Herrmann, D., K. Jost, and J. Kessler, Differential cross sections for
elastic electron scattering. II. Charge cloud polarization in N2
J. Chem. Phys., 64, 1, 1976.
Holland, R.F., Cross sections for electron excitation of the 3914-A (0,0)
band of the N$ first negative system. Tech. Rep. LA-3783, Los Alamos
Scientific Laboratory of the University of California, Los Alamos,
New Mexico, 1967.
Inokuti, M., D.A. Douthat, and A.R.P. Rau, Degradation spectra and
ionization yields of electrons in gases, Proceedings of the Fifth
Symposium on Microdosimetry, pp. 977-1006, Varbania-Pallanza, Italy,
Sept. 22-26, 1975.
Irvine, W.M., Multiple scattering by large particles, Astrophys. J., 142,
1563, 1965. ~
Jackman, C.H., R.H. Garvey, and A.E.S. Green, Yield spectra and the
continuous-slowing-down approximation, J. Phys. B: Atom Molec. Phys.,
10, 2873, 1977a.
Jackman, C.H., R.H. Garvey, and A.E.S. Green, Electron impact on atmo
spheric gases. 1. Updated cross sections, J. Geophys. Res., 82,
5081, 1977b.
Jasperse, J.R., Boltzmann-Fokker-Planck model for the electron distribu
tion function in the earth's ionosphere, Planet. Space Sci., 24,
33, 1976. ~


-71-
total path length s (in units of MFPs) traveled between inelastic
collisions.
In KG an equation which can be easily inverted to calculate the
z distance (in units of MFPs) from some random number, R2 and path
length s, is written
f^-1] 1
-m Wr1/V.-J]; (4.31)
-u
where
v(s) = 1 exp[-(s/sv)D]
F(0) = K{1 exp[-(s/sf)0,75]}
and
u(s) = (H + sI)/sJ
where K = 0.425.
Since there is more backscatter during N2 elastic collisions
(because of its differential backscatter contribution), it seems reason
able that the parameters for Eq. (4.31), which are useful for N2, are
different than those derived in KG. One approach to this dilemma might
be to correlate the elastic differential cross section (hereafter called
EDCS) from at some energy E1 with the EDCS from H2 at some energy E.
This would work if the H2 EDCSs showed more backscatter than the
EDCSs; however, the opposite is observed experimentally. Thus the
EDCSs from some E1 (around 6-7 eV) values correlate with the H2 EDCSs
at E values less than 2 eV (where the Kutcher and Green, 1976, MESD is
not defined).


-154-
This work compiled a reasonably comprehensive and realistic cross
section set for in the energy range from 2 to 5000 eV. The influence
of various differential and total cross sections on the spatial and
energetic aspects of the electron energy deposition problem was also
characterized. Finally, this study presented a spatial yield spectrum
along with an analysis and analytic fit of some of its most important
properties.


404 IF {NUMGAS tEQ. 1 ) GO T 0 6
J= 2
C COME TO THIS SECTION WHEN THERE ARE MOPE THAN ONE GASES*
IT0P=2* NUMGAS
DO 2 1 = 4ITOP, 2
PSE(J ) = SIGT( I)/S IGEE
P=P+PSE C J)
IF (R jLT o P ) GO TO 5
J=J+1
2 CONTINUE
GO TO 6
4 NS CS= 2
C ELASTIC COLLISIONS ARE SENT TO STATEMENT 4*
GO TO 7
5 NSCS=J
NSTAT=NUMST+J
7 RETURN
6 NS CS = 1
RA N= R
C RAN AND R ARE THE RANDOM NUMBERS*
IF(E *LT* 200*) CALL CTB200
C GO TO SUBROUTINE C TB 200 FOR ENERGIES LESS THAN 200 EV
IF (E *LT* 200*) GO TO 12
NSTAT=1
JL = 1
DO 11 1=1,NUMGAS
NSC S=NSC S+2
C0LTYP46
C0LTYP47
C0LTYP48
COL TYP 49
C0LTYP50
COLTYP51
COL TYP 52
COLTYP53
C0LTYP54
C0LTYPS5
COL TYP 56
C0LTYP57
C0LTYP58
C0LTYP59
COLTYP60
COL TYP 61
C0LTYP2
C0LTYP63
C0LTYP64
COLTYP65
C0LTYP66
C0LTYP67
COLTYP68
C0LTYP69
C0LTYP70
C0LTYP71
C0LTYP72
-162-


non ono
14 JULY 1978
NERDC
CARD LI ST UTILI TY
PA=PLF
2213 CONTINUE
IF(NCHE dEO> NSCS)WL0SS=EV*(lo-C0SPA)T7o776E-5
C COMPUTE ENERGY LOSS DURING AN ELASTIC COLLISION^
WRIT E(M)NST AT WL OSS
C WRI TE(M)NSTATiWL OSS
IF (N IE (NSTAT ) o E Qo 2)WRITE(M)T
C WRITE(M)T
EVN=EV-WLOSS
IF (NIEINSTAT ) EOo 2) EVN=EVN-T
ZV=ZVN
XV=XVN
YV=Y VN
WRITE (M)XV ,YV.ZV .EVN.PA, PHI,NG200, EV
C WRITE(M)XV YVZViEVN,PA,PHI ,NG2 00,EV
N= N+l
C
C
C
c
c
EV=EVN
IF { ZV oGT ZAL T( 1 ) ) GO TO 801
IF ( Z V ,LT. ZALT { NA R ) ) GO TO 801
IF(NG200oEQo 2)GO TO 27
IF(NG200,EQo1)GO TO 31
IF NG2 0 0=0 THEN THERE IS A
IF N G20 0=1 THEN THERE IS A
IF NG200=2 THEN THERE IS A
IF(T LTo TMIN) GO TO 27
TM IN IS THE CUTOFF FOR THE SECONDARIES
IF T IS GREATER THAN TMIN THEN WE DEGRADE
NG 200=1
PRIMARY COLLISION
SECONDARY COLLISION
TERTIARY COLLISION
IT OTHERWISE WE DO NOT,
SAVE THE PRIMARY PARAMETERS
EVSAV=EV
EV=T
CPSAV = COS PA
PA SA V = PA
PHISV=PHI
XVSV=XV
YVSV=Y V
ZVS V=ZV
CALCULATE THE SECONDARY ELECTRON SCATTERING
CALL RANDU
IR 1=1 R 2
CALL SDIFM
MC 000 2 17
MC000218
MC 000219
MC000220
MC000221
MC 000222
MC000223
MC 000224
MC000225
MC000226
MC000227
MC000228
MC000229
MC 0 00230
MC000231
MC 000232
MC000233
MC000234
MC 000235
MC000236
MC 0002 37
MC000238
MC000239
MC 000240
MC 000241
MC000242
MC 000243
MC000244
MC 000245
MC 000 246
MC000247
MC 000248
MC000249
MC 0002 50
MC 000 251
MC000252
MC 0002 53
MC000254
MC000255
MC 000 256
MC000257
MC000258
MC 000259
MC000260
MC 000261
-181-


-97-
At 5.0 KeV a comparison is made between the theoretical calculations
from this work and those of BSM. The results from this work appear to
agree much better with the BH data than does the BSM work. In BSM,
they follow only the primary and secondaries down to 200 eV. Since
this work follows the primary, secondaries, and tertiaries down to 2 eV,
it seems straightforward that the agreement should be better in this
work.


623
62 5
C
c
52 4
627
7
NUMG=NUMST+1 DATA0118
NI E (NUMG)=4 DA TA0 1 19
WRITE(6 623) DATA0120
FORMAT (//// THE STATES WITH THEIR CROSS SECTION! PARAMETERS ARE DATA0I21
1 RE AD IN',//) DA TA 0122
WR ITE(6 625 ) DAT AO 123
FORMAT(6X,STATE ,15X. 'W ,7X,ALPHA',4X, BETA* 6X, DATA0124
1 *WBAR',7X,'C OR O .1 OX.F .9X,FAC I ./) DATA0125
K=0 DAT A0126
DO 7 1=1 NUMGAS DATA0127
JTOP=NSG( I )-1-N I G( I ) DAT AO 1 28
DO 7 J=1JTQP DATA0129
K=K+1 DATA0130
READ ( 5. 524 ) G1,G2,G3,G4,G5,WF(K),WFA(K>. FACI (K).ALFA(K), DAT AO 131
1 BEF A READ IN THE PARAMETERS USED FOR THE INDIVIDUAL CROSS SECTIONS BELOW DATA0133
200 EV. DATA0134
FORMAT!5A4,6E6o0 .E3#0) DATA0135
WF(K>=ABS(WF(K)> DATA0136
WRITE (6,627 ) G1 G2 G3 G4G5,WF(K), ALFA (K. ) QEFA ( K ) WFA(K) DATA 01 37
1 CFA(K),FFA(K),FACI(K) DATA0138
FO RM AT (IX, 5A4,3F93F10o 3,F1 1 3 F 1 5 6, 2F 10 3) DATA0139
CONTINUE DATA0140
NFA=K DAT AO 141
JL-=1 DATAOl 42
WR ITE(6,629) DATA0143
FORMAT!////' THE S(E,T) PARAMETERS FOLLOW' ,//,6X,' ION. STATES' ,7X,DATAO 144
629


3914 A Intensity (I02 exc)
, o ro oj £> en en ->icd


nnnnui
C RHO(I) ARE THE R HD VALUES AT WHICH THE PHO INTENSITY PLOT IS
C CALCULATED
WR ITE6.3505)
3505 FORMAT(////. RHO INTERVAL 10X, HO BIN ENDS',/)
DO 4 5 05 1=2 NRM
IM 1= I 1
4 5 05 WRITE(6,350 6 >IM1 ,RHO{I Ml ),RHO< I )
35 06 FORMA T < I5.llX.lPEll,3,5X.lPEllo3>
1141 CONTINUE
RE AD(5,2255) (ZALT( I ), 1=1 ,NAR)
C ZALT(I) ARE THE Z INTERVALS INTO WHICH THE LONGITUDINAL REGIME
C IS DIVIDED
C ZALTC1) AND ZALTINAR) MUST CORRESPOND TO THE Z BOJNDS SET IN THE
C MONTE CARLO PROGRAM,
C THE OTHER VALUES IN BETWEEN ARE VARIABLE,
WR ITE(6,3507)
3507 FORMAT! 1ALT INDEX',QX, 'ALT, ENDS',/)
DO 45 07 1=1 NAR
4507 WRITE(6,3508)1 ZAL T(I )
508 FORMAT(I89X,1PE11 ,3)
WIS(l) AND EXC5 ( I ) ARE MAINLY USEFUL WHEN LOCAL ENERGY DEGRADA
TION IS USED BELOW EMIN,
READ! 5,2 255 MW IS( I ), 1= 1, NSEC )
C WISt I) ARE THE ENERGY BIN ENDS FOR THE LOW ENERGY ELECTRONS,
RE AD( 5, 2255 ) { EXC 5{ I ), 1 = 1 ,NTOP)
RDCAR118
RDCAR1 19
RDCAR120
RDCAR121
RDC AR122
RDCAR123
RDCAR124
RDCAR125
RDCAR126
RDCAR127
RDCAR128
RDCAR129
RDCAR130
RDCAR131
RDCAR132
RDCAR1 33
RDCAR134
RDCAR135
RDCAR136
RDCAR137
RDCAR138
RDCAR139
RDCAR140
RDCAR141
RDCAR142
RDCAR143
RDCAR144
-205-


-54-
Figure 4.2 Schematic representation of the coordinates and directions
of motion of the electron in its travel between collisions
with the Ng molecules.


-133-
rectangle with longitudinal endpoint coordinates azjnt/2 and
ZE + AZInt^ anc* ener9y width endpoint coordinates E^. aEjnt/2 and
Ee + AEjnt/2 is established. If the longitudinal distance to any in
elastic collision (elastic collisions are excluded because of the real
lack of interest in their spatial properties and also because they are
not well defined at electron energies below 30 eV), zc, is between
ZE AzInt/2 and ZE + AzInt^2 and the ener9y of the electron before the
collision, Ebc, is between E^ AEjnt/2 and E£ + aEInt/2 then the number
of electrons in that rectangle, N(EE,zE), is incremented by one. This
process continues until all the collisions are accounted for.
2
The spatial yield spectrum [in #/eV/(gm/cm )] is then written as
U(Ee.Ze.E0) -
aEInt 4zInt
(7.5)
(This spatial yield spectrum is also normalized to one electron.) This
process then continues for each two-dimensional rectangle across the entire
plane of interest. As an example, the U(E,z,Eq) for three longitudinal
distances is given in Figure 7.3 for the incident energy of 1 KeV.
This U(E,z,Eq) [as observed in Figure 7.3], although more complex
than the U(E,EQ) of GJG, has some nice general characteristics that con
tinue throughout the entire incident energy range (from 0.1 KeV up to
5 KeV). It is, therefore, reasonable to continue the philosophy of
analytic representation (see Green and Barth, 1965; Green and Dutta,
1967; Stnlarski and Green, 1967; and GJG). The analytic properties of
U(E,z,Eq) will permit researchers to infer important spatially derived
properties of with a degree of accuracy which should suffice for many
atmospheric applications.


-87-
In this work model 3 is the result of a careful investigation of the
detailed molecular nitrogen cross sections. Therefore no attempt will
be made here to change the cross sections compiled in Chapter III. Model
3 will be used in most of the MC calculations in the rest of this
chapter and also in Chapter VII (BSM have, however, chosen nc used in
the screened Rutherford cross section, to be a constant value whose value
was selected so as to obtain the best agreement between their MC calcula
tion and the experimental results of G and CC).
In Table 5.1 the importance of the elastic phase functions is clearly
illustrated. Up to a 25% change in the range is observed when com
paring the screened Rutherford with the model 3 phase functions. More
elaboration on the effects of various phase functions on the energy
deposition process will be given in Chapter VI.
Figures 5.3 and 5.4 give intensity plots for the 3914 A radiation
resulting from 2 KeV and 0.3 KeV incident electrons, respectively. The
experimental work of BH and the calculations using model 3 and the
screened Rutherford are presented in these figures. The shapes appear
to be somewhat similar; however, the BH results at both energies pre
dict a range that is between the two theoretical calculations.
E. Intensity Plots in the Radial Direction
Most attention, so far in this study, has been concentrated on the
intensity plots in the longitudinal direction. There is experimental
data available on the intensity of the 3914 & radiation as a function
of p (the axis perpendicular to z). Experimentally, G, CC, and BH all
present data of this type.


Figure 5.6 Intensity plots for electrons with incident energy 1.0 KeV are presented at four z
values as functions of p. The solid line histogram indicates the results using
model 3. The x's denote the experimental data of Barrett (1975).


-4-
influence of inelastic differential cross sections on the intensity
distribution. A comparison of different elastic phase functions on the
electron-N2 collision profile (no energy loss) is given in section VI.C.
Next, the influence of different elastic phase functions on the electron
energy deposition is presented in section VI.D. Finally, section VI.E
considers the influence of the total elastic cross section on the
electron energy deposition.
The energy loss plots and yield spectra from the MC calculations
are given in Chapter VII. Section VII.A presents the energy loss plots
and section VII.B includes a discussion of the yield spectra.
Chapter VIII concludes this paper with a summary of this work and
its impact on the spatial and energetic aspects of the electron energy
deposition problem.


149
Figure 7.5c


BIOGRAPHICAL SKETCH
Charles Herbert Jackman was born on September 9, 1950, at David
City, Nebraska. He graduated as salutatorian of his class from O'Neill
Public High School of O'Neill, Nebraska, in May, 1968. In May, 1972, he
received the degree of Bachelor of Science with highest distinction in
physics and mathematics from Nebraska Wesleyan University in Lincoln,
Nebraska. From September, 1972, until the present time he has pursued
his work toward the degree of Doctor of Philosophy in the Department of
Physics and Astronomy at the University of Florida. During this time he
has held a graduate teaching and research assistantship.
His publications to date include:
Green, A.E.S., R.H. Garvey, and C.H. Jackman, A Thomas-Fermi-1ike
analytic independent particle model for atoms and ions, Int. J.
Quantum Chem. Symp., 9^, 43-50, 1975.
Garvey, R.H., C.H. Jackman, and A.E.S. Green, Independent-particle-model
potentials for atoms and ions with 36 < Z < 54 and a modified
Thomas-Fermi atomic energy formula, Phys. Rev. A, 12, 1144-52, 1975.
Porter, H.S., C.H. Jackman, and A.E.S. Green, Efficiencies for produc
tion of atomic nitrogen and oxygen by relativistic proton impact
in air, J. Chem. Phys., 65, 154-67, 1976.
Green, A.E.S., R.H. Garvey, and C.H. Jackman, Analytic yield spectra for
electrons on H2, Int. J. Quantum Chem. Symp., 11, 97-103, 1977.
Jackman, C.H., R.H. Garvey, and A.E.S. Green, Yield spectra and the
continuous-slowing-down approximation, J. Phys. B: Atom. Molec.
Phys., 10, 2873-82, 1977.
Jackman, C.H., R.H. Garvey, and A.E.S. Green, Electron impact on atmo
spheric gases, 1, Updated cross sections, J. Geophys. Res., 82,
5081-90, 1977. ~
Green, A.E.S., C.H. Jackman, and R.H. Garvey, Electron impact on atmo
spheric qases, 2, Yield spectra, J. Geophys. Res., 82, 5104-11,
1977.
-227-


-41-
is called the secondary electron. There should be a correlation between
the primary and secondary scattering, but this mutual influence is dif
ficult to quantify. The impinging electron ionizes a many body par
ticle, a molecule of nitrogen, thus momentum and energy can be conserved
without all the momentum and energy shared by the two resulting elec
trons. Only recently has work been done on triply differential cross
sections for and this interaction was measured only at one energy
E = 100 eV (see Jung, Schubert, Paul, and Ehrhardt, 1975). More ex
perimental and theoretical work needs to be done in this area before
any generalization can be made concerning the correlation between the
primary and secondary scattering.
The primary and secondary scattering will thus be treated sepa
rately in this work. In dealing with the scattering of the primary
electron after an ionization collision, a differential ionization cross
section form based on the Massey-Mohr-Bethe surface of hydrogen, is
used. The form (with aQ, the Bohr radius, and Rg, the Rydberg energy) is
2
= 9_i o JiM/2 p(x\ (3 i7)
dTdfi Wx U E' ^l/'
2
where x = (KaQ) is the momentum transfer, W is the energy loss in the
collision process which is equal to the ionization potential, I, plus
the kinetic energy of the secondary, T, and F(x) is a complex function
given in PJG.
Equation (3.17) is very highly forward peaked for small energy
losses but becomes less forward peaked as the energy loss becomes sig
nificant when compared with the incident energy, E.
The secondary electron is also scattered in the ionization event.
Probably the most comprehensive work that exists on secondary doubly


Z (cm)
3914 A Intensity (I03exc.)
p p ro pj
ib bo ro
-20 L-
4.0


Figure 4.1 Flowchart of the Monte Carlo degradation of an incident
electron of energy E0.


14 JULY 1978
NERDC
CARD LIST UTILITY
C
33
C
C
c
29
C
C
C
C
C
37
C
C
C
00 1
65 0
C
18
655
8
48 3
iEOo 1 )GO TQ 37
IF (NG200 EQ 0)GO TO 29
NOW DEGRADE THE PRIMARY
EV=EVSA V
COSPA=CPSAV
PA=PASAV
PH I=P HI S V
XV-XVSV
YV=Y VSV
ZV = ZVSV
IF(IE SCP
NI 2=1
WR IT E ( M )N 12
WRITE(M)NI2
IF NI2=1 THEN THE PRIMARY OR SECONDARY HAS SLIPPED BELOW EMIN AND
NEEDS TO BE ACCOUNTED FOR
IF N12= 0 THEN THE PRIMARY OR SECONDARY BEING DEGRADED IS ABOVE EMIN
WR IT E CMJEVD.ZVD
WRITE(M)EVD.ZVD
N=N+ 1
N I 2=0
IF(NG200 o E Q. 0)GO TO 18
NG200=0
GO TO 16
THE PRIMARY OR SCONDARY HAS ESCAPED IF WE REACH HERE
WRITEC6.50) ZV.EV
FORMATC THE ELECTRON HAS ESCAPED AT A LT I TUDE= 1 PE 1 2 3
1 4X 'WITH AN ENERGY=* ,1 PEI 4.6)
IESCP= 1
IESCP = 1 MEANS THE ELECTRON HAS ESCAPED
IF(NG200 s NE 0)GO TO 39
CONT INUE
WRITE6.655 ) N
FORMAT( 1 THE TOTAL NUMBER OF COLLISIONS FOR ALL ELECTRONS = .112)
CONT INUE
WRITE(6,483)IR1,IR2
FORMATC* IR1=*,I15,* IR2='.I15)
RETURN
END
MC000361
MC000362
MC 000 363
MC000364
MC 000365
MC000366
MC000367
MC000368
MC000369
MC000370
MC000371
MC000372
MC 000 373
MC000374
MC000375
MC 000376
MC000377
MC000378
MC000379
MC000380
MC000381
MC000382
MC 0003 83
MC000384
MC000385
MC 000386
MC 000387
MC000388
MC 000389
MC000390
MC000391
MC 000 392
MC000393
MC 000394
MC000395
MC000396
MC 000 3 97
MC0003 98
MC 0003 99
MC 000400
MC000401
MC 000402
MC 000403
-185-


-7-
is the secondary differential ionization cross section for the creation
of a secondary electron of energy, T, from a primary electron of energy
E. The loss function with detailed atomic cross sections (hereafter
called DACSs) was used to make reasonable estimates of the ultraviolet
emissions resulting from an aurora event. In this approach, the excita
tions J|^(E) of the kth state resulting from an electron of energy E were
simply represented as
E o.(E')
VE> I -TTFT dE' <2-4
k
Green and Dutta (1967) built on this work and used the Born-Bethe
approximations, the Massey-Mohr-Bethe surface, the Bethe sum rule, and
a "distorted" oscillator strength to lay the groundwork for extension
of the DACS approach to other gases. Jusick, Watson, Peterson, and
Green (1967), Stolarski, Dulock, Watson, and Green (1967), and Watson,
Dulock, Stolarski, and Green (1967) applied this approach to helium,
molecular nitrogen, and molecular oxygen, respectively.
Stolarski and Green (1967) used this CSDA to calculate auroral
intensities with these DACSs and Green and Barth (1967) applied this
method to the problem of photoelectrons exciting the dayglow. Other
atmospheric physicists (namely, Kamiyami, 1967; and Rees, Stewart, and
Walker, 1969) started around this same time and also employed a CSDA type
approach to that problem of energetic electrons depositing their energy
in the atmosphere.
The oldest discrete energy apportionment method is that of Fowler
(1922-23) which is directly related to the Spencer and Fano (1954)
approach (see Inokuti, Douthat, and Rau, 1975). The Fowler equation is


14 JULY 1978
NERDC
CARO LIST UTILITY
5588
5577
C
C
c
IF(NIE(N) NE 2)60 TO 5577
IF (NG 200 EQ. 2)60 TO 5588
IF(WL GTa W IS ( N SEC))GO TO 5577
EL OSS=ELOSS + WL
IF ( Z GTs ZALT(l) a ORo Z o LT
ZALT (NAP))EL0SS= EL OSS +E
C
c
c
c
22 03
2204
69
IF(EL OSS .LT. 0 4)GO TO 1143
00 NOT COUNT ELASTIC COLLISIONS IN YIELD SPECTRA
SOMETIMES ELOSS MAY BE GREATER THAN 0.4 AND STILL 3E AN ELASTIC
THE ELOSS IS GREATER ONLY IN THE CASE WHERE THE ELECTRON HAS AN
ELASTIC COLLISION OUTSIDE THE BOX AND ESCAPES
IF (NSPEC oNE. 1 ) GO TO 1143
DO 2203 IL=1 NSPZ
IN IE= IL
N1=NZX( IL)
NIM1=N11
IF ( Z GT Z ALT ( N 1 > .AND. Z oLT. ZA L T ( N 1M 1 ) ) GO TO 2204
CONTINUE
GO TO 1143
DO 69 IM=2,NEIP
I M Ml = I M 1
IF(EBEF GT, EIP(IMMl) AND EBEF .LT. El P { I M) ) JNI E = I MM1
CONTINUE
ANEI(1.INIE.JNIE)=ANEI(l.INIE.JNIE)+loO
IF (NSPRO oNE 1 ) GO TO 1143
RHOV=SQRT(X*X+Y*Y)
DO 2205 IN=2.NSPR
IN1=IN-1
IF(RHQV o GT a RYS(INl) .AND. RHOV oLTo RYS(IN)) GO TO 2207
CONTINUE
GO TO 1143
ARE A = 3o 14159*(RYS( IN)* RY S ( I N) -RYS { INI ) *RYS (INI ))* CONST
AN ER( 1 INI. INIE,JNIE)=ANER( 1,INI,INIE.JNIE)+l 0/AREA
CONTINUE
2205
2207
11 43
C
C
C
IF ( NR HO oEQ a 1 ) GO TO 151
1161 IF(NAV EE E Q< 1 )GO TO 66 11
6622 IF (NIE(N) ,NE. 2)GO TO 702
DO 63 11=1.NTOP
II P1 = I1+1
IF(WL GTo WIS(II) .AND, WL o LT WIS( I 1 PI ) ) I I = 11
RDTAPE73
RDTAPE74
RDTAPE75
RDTAPE76
RDTAPE77
RDTAPE78
RDTAPE79
RDTAPE80
RDTAPE81
RDT APE 82
COL.RDTAPE83
RDTAPE 84
RDTAPE 85
RDTAPE86
RDTAPE 87
RDTAPE88
RDTAPE89
RDTAPE90
RDTAPE91
RDTAPE 92
RDTAPE93
RDTAPE94
RDTAPE 95
ROTAPE96
RDTAPE97
R DTAPE98
RDTAPE99
RD TAP 100
RDTAPI 01
RDT API 02
RDTAP103
RDTAPI 04
RD TAPI 05
RDTAP106
RDT API 07
RDTAP108
RDTAPI 09
RDT API 10
RDTAP 1 1 1
RDTAPI 12
RDTAP1 13
RDTAP114
RDTAPI 15
RD TAP 1 16
RDTAPI 17
-210-


-15-
Many MC schemes have been applied in all areas of physics. Some
are more exact and more detailed than others. Since virtually all the
MC methods are run on the computer, the most exact approaches cost the
most computer time and money. The precision of the technique must be
balanced against a finite computer budget.
Three approaches using the MC deposition scheme, that have been
applied to electrons impinging on the atmosphere, are discussed below.
Brinkmann and Trajmar (1970) applied experimental differential electron
impact energy loss data in a MC computation for electrons of 100 eV
energy. Because of the large amount of input cross sections in numerical
form, only electrons of 100 eV incident energy were degraded with this
method.
In the lower electron energy regime (below 25 eV), Cicerone and
Bowhill (1970, 1971) used a MC technique to simulate photoelectron dif
fusion through the atmosphere. This method, which included both elastic
and inelastic processes, predicted escape fluxes from the atmosphere.
Berger, Seltzer, and Maeda (1970, 1974) (hereafter called BSM)
worked with high energy electrons (with energies from 2 KeV to 2 MeV).
They employed a MC approach that has two variations which are pointed out
below. They treat inelastic collisions in a continuous slowing down
manner. The energy deposited by the electrons along their path is
assumed to be equal to the mean loss given by the loss function, L(E),
from Rohrlich and Carlson (1954).
The angular deflection resulting from elastic collisions has been
evaluated by two separate methods in BSM. One approach employed the
multiple scattering distribution of Goudsmit and Saunderson (1940)
applied to the screened Rutherford cross section given in BSM. The


REFERENCES
Banks, P.M., C.R. Chappell, and A.F. Nagy, A new model for the interaction
of auroral electrons with the atmospheres: Spectral degradation,
backscatter, optical emission, and ionization, J. Geophys. Res., 79,
1459, 1974.
Barrett, J.L., Energy loss of electrons in nitrogen, Ph.D. thesis, Uni
versity of Michigan, Ann Arbor, Michigan, 1975.
Barrett, J.L., and P.B. Hays, Spatial distribution of energy deposited
in nitrogen by electrons, J. Chem. Phys., 64, 743, 1976.
Berger, M.J., Monte Carlo calculation of the penetration and diffusion of
fast charqed particles, Methods in Computational Physics, 1, 135,
1963.
Berger, M.J., S.M. Seltzer, and K. Maeda, Energy deposition by auroral
electrons in the atmosphere, J. Atmos. Terr. Phys., 32, 1015, 1970.
Berger, M.J., S.M. Seltzer, and K. Maeda, Some new results on electron
transport in the atmosphere, J. Atmos. Terr. Phys., 36, 591, 1974.
Bethe, H., Zur theorie des durchgangs schneller korpuskularstrahler durch
materie, Ann. Phys., 5^ 325, 1930.
Bethe, H.A., M.E. Rose, and L.P. Smith, The multiple scattering of elec
trons, Proc. Amer. Phil. Soc., 78, 573, 1938.
Blaauw, H.J., F.J. de Heer, R.W. Wagenaar, and D.H. Barends, Total cross
sections for electron scattering from N? and He, J. Phys. B: Atom.
Molec. Phys., 10, L299, 1977.
Bohr, N., On the theory of the decrease of velocity of moving electrified
particles on passing through matter, Phil. Mag., 25, 10, 1913.
Bohr, N., On the decrease of velocity of swiftly moving electrified
particles in passing through matter, Phil. Mag., 30, 581, 1915.
Borst, W.L. and E.C. Zipf, Cross section for electron impact excitation
of the (0,0) first negative band of Nj from threshold to 3 KeV,
Phys. Rev. A, 1, 834, 1969.
Brinkmann, R.T. and S. Trajmar, Electron impact excitation of No, Ann.
Geophys., 26, 201, 1970.
-220-


14 JULY 1978
NERDC
CARD LIST UTILITY
DO 160 1=1,NVR
ZM ZV= ( ZVAL-ZNUM I ) ) *1 oE+05
WRITE(6,73 0 )ZMZV ,NCfiO( I)
730 FORMA T( 1A T A Z VALUE OF 1 PE 1 1 o 3 CM0 AND NCP0=*,I5,/
* WE HAVE THE FOLLOWING EXCITATION DISTRIBUTION*.///,
4 RHO VALUE* ,6X,391 4A EXCo* 8X* ELAST IC EXC*,/)
DO 160 J=2,NRM
JM1 = J 1
RMV=(RHO(J)+ RHOJM1)>/2o
K345=KR EX ( I.3.J)+KREX( I.4,J) + KREX( 1,5. J)
WRITE(6.705)RMV.K345.KREX(I.10,J)
70 5 FORMA T( 2X,1PE104,2X.I9,9X.I10)
150 CONTINUE
C
C
C
C WRITE OUT THE ENERGY LOST INF ORMAT IO No
3311 IF(NAV EE ,NE. 1) GO TO 6633
00 4477 1=1.3
4477 EAVE( I) = OoOD0
WRITE(6,712)
712 FORMAT( 1 THE AVEo SNEPGY LOST IN EACH Z VALUE IS PRINTED OUT NOW*
* //. ALT, INTERVAL* 8X ALT. VAL UE *. 4X ,* MEAN ENERGY (ALL ) ,
4 5X,* MEAN ENERGY(PAPI/2)'./)
DO 15 1=1,NAR1
IF(I ,EQo 1)GO TO 2965
IF ( I EQo NAR1 ) GO TO 2 966
IM 1= 1-1
ZMV=(ZALT I)+ZALT( IM1 ) )/2,
ZMV=ZVAL-ZMV
GO TO 2967
2965 ZMV=ZVALZALT(1)
GO TO 2967
2966 ZMV=ZV ALZALT(NAR)
2967 DO 2961 J=1,3
EAVE(J)= EAVE(J)+TO TE(J.I )
2961 CONTINUE
EV1 = E A VE( 1 )/NPRI M
EV 2= E AV E ( 2 )/NPR I M
EV3=EAVE(3 J/NPRIM
WRITE(6,714)I.ZMV.EVl,EV2,EV3
714 FORMAT(110,1 OX,1 PE12,4,7X,G15 7,6X G15,7,10X.G15,7)
15 CONTINUE
6633 CONTINUE
RETURN
WRTDA145
WRTDA146
WRTIDA 1 47
WRTDA148
WRTDA149
WRTDA150
WRTDA151
WRTDA1 52
WRTDA153
WRTDA154
WRTDA155
WRTDA156
WRTDA1 57
WRTDA158
WRTDA159
WR TOA 1 60
WRTDA161
WRTDA162
WRTDA163
WRTDA164
WRTDA1 65
, WR TDA 166
WRT0A167
WR TOA 1 68
WRTDA169
WRTDA170
WRTDA171
WRTDA17 2
WRTDA 173
WRTDA174
WRTDA175
WRTDA176
WRT0A177
WRTOAl78
WRTDA179
WRTOAl80
WRTDA1 81
WRTDA182
WRTDA183
WRTDA184
WRTDAl85
WRTDA186
WRTDA187
WRTDAl88
WRTDAl 89
-8LZ-


nnn>i n norm
c
c
c
c
8989
8980
39
14 JULY 1978
NEPDC
CARD LIST UTIL ITY
T=0.0
MC000073
THIS T=OoO IS INITIALIZED HERE SO
THAT THE
STATEMENT
MC 000074
IF (T L To TMIN)GO TO 27 CAN WORK,
OTHERW ISE
THE T WOULD BE
MC000075
GREATER THAN TM IN FOR MORE CASES
THAN DESIRED
MC000076
WRITE(M)NI2
MC 000077
WR I TE { M ) NI 2
MC000078
AR GC=1 -COSPA *CO SPA
MC000079
IFIARGC oLTo 0 o 0 EO)GO TO 8989
MCOOOO80
SINPA=SQRT(ARGC)
MC000081
GO TO 8980
MC 000082
SINPA = 0o 0E0
MC 000063
CONTINUE
MC000084
SINPHI-=SIN(PHI )
MC 000085
CO SPHI=CQS(PHI>
MC000086
MC000087
MC 0000 88
MC000089
MC000090
IF (E V oGEo E MSD ) GO TO 7139
MC000091
WE GO HERE WHEN THE ELECTRON ENERGY IS LESS
THAN EMSD
MC000092
CALL RANDU
MC000093
IR 1=1 R2
MC000094
R1=R
MC000095
CALL RANDU
MC000096
IR 1=1 R2
MC000097
R2=R
MC 000098
CALL RANDU
MC 000099
IR 1=IR2
MC000100
R3=R
MC000101
CALL RANDU
MC000102
IR 1= IR2
MC000103
R4 =R
MC 0 00104
CALL RANDU
MC000105
IR1= I R2
MC 0001 06
R5=R
MC000107
CALL MESD
MC000108
GO TO 2213
MC 0001 09
CONTINUE
MC 0001 10
MC0001 11
MC0001 12
CALCULATE PATHLENGTH RT HERE
MC0001 13
CALL RANDU
MC 000114
IRl=Ifi2
MC 0001 15
CALL ZVAL
MC000116
ZVN IS CALCULATED IN ZVALo
MC0001 17
C
-ILL-


-79-
E (eV)
Figure 5.1 Total loss function L(E) from N2, denoted by the solid line;
total ionization cross sections for N^, denoted by the dash-
dot line; and the Nj B cross section, denoted by the
dashed line, are given as functions of energy, E.
t-


106
C
c
c
c
c
c
c
c
c
105
C
c
c
c
c
c
c
55 0
62 0
WRITE(6106) NP IN NOP NPHF
FORM AT {.//%* THE NUMBER OF PRIMARY SEGMENTS=, I 5.
* /, THE NUMBER OF PRIMARIES IN EACH SEGMENT=*. I 5,/,
* INDEX FOR PHASE FUNCTION= ,I 5//)
RE AD ( 5 1 0 5 ) ISEED, I LAST* ISTOP.NX.NSEC
I SEED IS THE INITIAL RANDOM NUMBER STARTER
ILAST = 0 IN ALL CASES
IF I ST QP= 1 THEN WE HAVE ANOTHER RUN TO MAKE
IF I STOP= 0 THEN THIS IS THE ONLY RUN
NX IS THE NUMBER OF ANGLES USED IN FINDING THE SCATTERING
ANGLE IF NO ANALYTIC EXPRESSION CAN BE USED
NSEC IS THE NUMBER OF LOWER ENERGY BIN ENDS
NSEC IS USEFUL ONLY IF A LOCAL ENERGY DEGRADATION SCHEME IS
EMPLOYED BELOW SOME ENERGY AND IF PRELIMINARY OUTPJT IS DESIRED
FO RMAT(ailO)
RE AD(5,550)EMIN.TMIN,EMSDEINEL
THE PRIMARY ELECTRON IS DEGRADED BEFORE
THE SECONDARY AND TERTIARY ELECTRONS
EMIN IS THE ENERGY TO WHICH
LOCAL DEGRADATION IS USED
TM IN IS THE ENERGY TO WHICH
DEGRADED
BELOW EMSD USE THE MULTIPLE
EINEL IS THE HIGHEST ENERGY
SCATTER INEL ASTICALLY ,
FORMAT(8E103)
WR ITE ( 6,62 0) ISEED I LAST, I STOP .EMIN,T MIN, EMSD, E INEL
FORMAT!* IS EED= 110,/,* ILAST= I 10,/, ISTOP = .I 5 /
A EMIN-* 1 PEI G4 EV /',* T M I N = 1 PE 1 0 4 EV*,/
SCATTERING DISTRIBUTION
AT WHICH ELECTRONS ARE ALLOWED
TO
MA
MA
MA
MA
MA
MA
MA
MA
MA
MA
MA
MA
MA
MA
MA
MA
MA
MA
AREMA
MA
MA
MA
MA
MA
MA
MA
MA
IN0047
I NO0 48
IN 0 0 49
IN0050
IN0051
IN0052
IN0053
I NO054
IN0055
IN0056
IN0057
IN0059
IN 0 059
IN0060
IN0061
IN0062
IN0063
IN0064
IN0065
IN0066
INO 067
IN0068
IN0069
IN0070
IN0071
IN0072
IN0073
-si-


-61-
and
0 = cos1(cose )
a a 1
(4.16)
Now the azimuthal angle and the polar angle 9^ have been established
a a
for the collision with respect to the fixed coordinate system. These
angles are also represented in Figure 4.2. The two angular coordinates
b and eb of the electron before traveling to the next cllision are then
set as d>. = and e, = e .
b a b a
3. Fourth Random Number, R4
A fourth random number, R^, is required if a secondary is produced
and if that secondary has an energy above the cutoff energy, Ec- This
R^ is chosen to determine the polar angle, 0', of scattering of the
secondary. Again, an analytic formula can be employed to define 0'.
This equation was derived from Eq. (3.18) and is written as
e'
r i + cose )
cos" [tan [R.{tan" ( ~ -)
VT 4 0
- tan
+ tan
i/B(l coseQ)
C
/B(l coseQ)
C
)}
-)] + coseo]
(4.17)
The a
and 0 result from the use of Eqs. (4.11) through (4.16).
a
4. Fifth Random Number, Rg
The fifth random number, Rg, determines the type of collision that
occurs. Here, the type may be either elastic or inelastic. If the


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Lennart R. Peterson
Professor of Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
George R. L&bo
Assistant Professor of Physics and
Astronomy
This dissertation was submitted to the Graduate Faculty of the Department
of Physics and Astronomy in the College of Arts and Sciences and to the
Graduate Council, and was accepted as partial fulfillment of the require
ments for the degree of Doctor of Philosophy.
August 1978
Dean, Graduate School


-85-
studies and, also, because BSM were quite successful in using this form
down to incident energies of 2 KeV. Model 3 was used because of its
very close agreement with experimental differential cross section data
in the range from 30 eV up to 1 KeV.
Table 5.1 presents the range data (for perpendicularly incident
electrons) from three different experiments, the theoretical calculation
by BSM, and two sets of theoretical computations from this study. The
values in parentheses from BH (Barrett and Hays, 1976), CC (Cohn and
Caledonia, 1970), and G (Griin, 1957) are simply calculated from the
empirical formulae given in these works.
For the rest of this chapter, the results of this work will be com
pared with those of BH. This is the most recent experimental study and
is probably the most reliable experimental work. They also use the same
incident electron energy regime as that employed in this work. In
Table 5.1 it is apparent that the BH values have the largest ranges of
the experimental studies.
The two separate MC calculations in this study seem to bracket the
BH results at all energies. The model 3 ranges are consistently larger
than those of BH. These results are 10% higher at 5 KeV and about 19%
higher at 0.3 KeV. The screened Rutherford ranges, on the other hand,
are about 9% lower at 5 KeV and about 10% lower at 0.3 KeV.
If it can be assumed that the BH results are indeed the most re
liable data, then the following conclusion can be made: The screened
Rutherford phase function scatters the electron too much while the
model 3 phase function provides too little scattering. This conclusion
is made assuming that the total cross sections described in Chapter III
are fairly accurate.


Figure 5.4 An intensity plot for electrons of energy 0.3 KeV is presented as a
function of the longitudinal direction. The x's represent relative
experimental values from Barrett and Hays (1976) and the histogram
presents the data from model 3 (heavy line) and the screened
Rutherford (light line). The straight lines extrapolated to the
z-axis are all measures of the range. The solid line indicates
the model 3 range; the dashed line indicates the screened Rutherford
range; and the dash-dot line indicates the Barrett and Hays (1976) range.


CHAPTER VII
MONTE CARLO ENERGY LOSS PLOTS AND YIELD SPECTRA
In this chapter two other very important energy degradation outputs
are discussed. The rate of energy loss as the electron impinges into
the medium is a very important quantity. This concept, the fraction of
energy backscattered, and the correlation between the range and the loss
function are discussed in section VII.A.
The most important output from the MC calculation is the spatial
yield spectrum. In section VII.B, this yield spectrum is calculated at
several energies and at several positions in the gas. Both the three
variable spatial yield spectrum U(E,z,Eo) and the four variable spatial
yield spectrum U(E,p,z,Eo) are considered in this section. Because the
yield or number of excitations to any Ng state is calculated quite easily
from the spatial yield spectrum, an attempt is made to represent it
analytically.
A. Energy Loss of Electrons in N2
The rate of energy loss by electrons in a medium is a useful quan
tity. For higher energy electrons (above 2 KeV), the degradation of
these electrons could be accomplished with the use of the loss function,
L(E), and the continuous slowing down approximation (discussed in
Chapter II).
-125-


-86-
-6 p
Table 5.1 Range data (in 10 gm/cm ) at several energies, E (in KeV),
are given below. The second column M3 (model 3), third
column SR (screened Rutherford), fourth column BH (Barrett
and Hays, 1976), fifth column CC (Cohn and Caledonia, 1970),
sixth column G (Griin, 1957), and seventh column BSM (Berger,
Seltzer, and Maeda, 1974) range values are presented.
: (KeV)
M3
SR
BH
CC
G
0.1
0.37
0.34
(0.53)
(0.07)
(0.08)
0.3
1.25
0.95
1.06
(0.51)
(0.56)
1.0
6.45
5.57
5.72
(4.17)
(4.57)
2.0
18.6
16.8
17.7
14.0
(15.4)
5.0
91.5
75.9
83.0
69.7
76.4
71.9


non
14 JULY 1978
NERDC
CARD L1ST UTIL!TY
C
C
C
C
C
WHOSE VALUES ARE
SUBROUTINE RANDU
SUBROUTINE RANDU ***
THIS SUBROUTINE IS ACCESSED THROUGH MCo
THIS SUBROUTINE CALCULATES THE RANDOM NUMBERS
BETWEEN Oo 0 AND 1 o Oo
RANDOM NUMBER GENERATOR (SEE IBM 360 SSP DESCRIPTION)
COMMON ALPE(6).BETS(6).CE< 6).FE{6),WE(6).ALFA 15) ,BEFA( 15).
1 CFAC 15) ,FFA(15> .WFAC15) ,WF(15> ,FACI( 15).NFA,NAR,SA(3.80).
ZALT(60).ENR(30),PCFA(3.30).WCFA(3,30).NENR,FDG(3),PSE(3),
PI0N(3) .AT(3)A(6 *3) B(5.3) G(5 3 )UH(10)*UI( 101.UJ 10),
UD( 1 0) ,USNU(1 0) ,USP( 10) ,UEI N( 1 1 ) IUTP, IUTM1 DG(3 ) NS G( 3 ) ,N I G( 3 ) ,
NUMGAS.NUMST, WI S( 5 0) NSEC .MUNIT.I SEED I STOP. I LA ST E I N NPRI M ,C OS I
H ,EMI N.ZSTART.T MIN .THRESH (3 ) AK (2.3 ) AJ ( 3.3 ) GAMA ( 2. 3) .
TO(33)FAC(6)A OE(6) A IE(6) ,B OE(6) B1E(6) ,C0E(6) ,D0E(6) .
D1E6).PROB(40) .W(40) .N IE(40),A21 (3),A22 3) ,A31(3),A32(3).
A 33( 3 ) ,B11 (3) *B12(3) .B13( 3) ,C1(3) ,C21 (3),C22(3),C31(3),
C32{ 3), D1 (3),D2(3).F1(3),F2(3),THRI(16).AKI(2,16),AJI(3,16),
GAMA 1(2 .16) .TOI (3.16),SIGT(6).THET(40),NX,CSIE(20),E IT(20) .
IEILSIEILM1PRZDISNPIN,NOP,CO SPA RT,F(40) ,R1 .R2.R3.R4,
R5.XV.YV, ZV.XVN.YVN,ZVN,PA.PHI,EV,WLOSS,NSTAT,NSC S,NPHF,
I R1 I R2 NG.EVPRI,RAN,EMSD EINEL.EXC5 (50 ) COSPAN. T,FOVAL
ARE THE RANDOM NUMBERS;
2
3
4
5
6
7
8
9
A
B
C
D
p
YFL AND R
IX= IR 1
IY = IX*6553 9
IF(IY) 5,6,6
5 IY = IY + 214748 36 47 + 1
6 YFL=IY
YFL=Y FL *4656613E-9
I R2=IY
R= YFL
RETURN
END
RANDU001
RANDU002
RANDU003
RANDU004
RANDUO 05
RANDU006
RANDU007
RANDUO 08
RANDUO 09
RANDU010
RANDUO11
,RANDUO12
RANDUO13
RANDUO14
RANDUO15
RANDUO16
RANDUO17
RANDUO18
RANDUO19
RANDU020
RANDUO 21
RANDUO 22
RANDU023
RANDUO 24
RANDUO 25
RANDU026
RANDU027
RANDU028
RANDUO 29
RANDUO 30
RANDU031
RANDUO 32
RANDU033
RANDU034
-193-


-27-
PM2(e>E)
=id
2tt[(2 + a(E))-1 a(E)_1][l cose + a(E)]2
where
(1 f(E))
2tt[(2 + c(E))'1 c(E)_1][l + cose + c(E)]2
f(E) =
f2
(E/f-j)
f2
(E/f-j) + f3
(3.13)
a(E) al ClEeV^
and
c{E) = c,[l 3]
Irvine (1965) was one of the first researchers in scattering prob
lems to use a phase function containing forward and backward scattering
terms. He applied a sum of two Henyey-Greenstein functions to the prob
lem of photon scattering by large particles. Porter and Jump (1978)
also have used a sum of two terms (one for forward scatter and one for
backward scatter). They fitted experimental data at several separate
energies with their form. Use of their differential cross section form
in a deposition calculation probably would require the use of spline
functions or other interpolative techniques.
The third phase function (model 3) is now considered. At small
angles the differential cross section shows a near exponential-like fall
off. This behavior has been pointed out by several experimenters (see,
for example, Shyn, Carignan, and Stolarski, 1972; and Herrmann, Jost,
and Kessler, 1976). It was this experimental observation that led to


-151-
No analytic expression has been derived for U(E,p,z,EQ); however, it
does have systematics that tend to point toward some type of representa
tion which would be useful for atmospheric scientists.
This concludes the discussion about the results from the MC
calculation. The most important output from the MC computations is the
spatial yield spectra because many of the other results given in
Chapters V, VI, and VII can be derived easily from this spatial quantity.


-73-
Table 4.5 Parameters from Kutcher and Green (1976) for several energy
intervals used in Eq. (4.28).
Energy
Interval
(eV)
H
I
J
D
s
V
SF
2-5
12.
1.37
1.71
1.75
5.05
8.5
5-10
9.6
1.32
1.67
2.50
4.25
8.5
10-20
15.5
1.28
1.67
2.31
6.29
10.3
20-30
23.5
1.24
1.69
1.98
9.65
13.6


-115-
E (eV)
Figure 6.6 Five trials (Cl, C2, C3, C4, and C5) of a(E) for use in
model 1.


-29-
f21 = 10 eV
f 22 = 0,51
f23 = 0.87
c2 = 12 eV
c3 = 0.27
c, = 1.27
This form is more complex than the other phase function models but
it does describe the experimental differential cross section data the
most realistically. It includes an exponential term for the near ex
ponential-like forward scattering as well as a backward scattering term
for electron energies below 200 eV.
Comparisons of the screened Rutherford and model 3 cross sections
are given in Figures 3.3 and 3.4 at the two energies of 30 eV and 1000 eV.
Both forms are normalized to the total elastic cross section form of
Eq. (3.9). This modified screened Rutherford cross section vastly under
estimates the forward scattering from e = 0 to 30, overestimates the
scattering in the range from 0 = 30 to 120, and underestimates the
scattering at the larger angles with 0 = 120 to 180. Model 3 does a
fairly reasonable job of representing the differential cross section data
at both of these representative energies and the other energies as well.
Although there is not a large amount of energy loss during an
elastic collision, there is some. Using classical considerations (see
Green and Wyatt, 1965), the energy loss is approximately given by Eq.
(2.15). For molecular nitrogen and 0 = 90, the energy loss is about
8 x 10"5 E.
The MC approach, being a stochastic process, uses the concept of
probability for scattering within a given angle interval. In order to
compare phase functions, the probability for backscatter may be compared.


14 JULY 1978
NERDC
CARD LIST UTILITY
***MONTE CARLO PROGRAM***
THIS PROGRAM IS TO BE USED IN CONJUNCTION WITH THE PROGRAM GETDATo
THIS PROGRAM PUTS THE DATA ON A MAGNETIC TAPE AND GETDAT COALESCES
THE INFORMATION ON THE TAPE
THIS PARTICULAR PROGRAM IS USEFUL FOR DEGRADING ELECTRONS IN THE
RANGE FROM 2 TO 5000 EV
RIGHT NOW THIS PROGRAM IS SET UP FOR MOLECULAR NITROGEN
THIS PROGRAM IS SET UP ONLY TO GIVE THE ELECTRONS ESCAPING AND
THE TOTAL NUMBER OF COLLISIONS THAT OCCUR NO OTHER PRELIMINARY
OUTPUT IS OBTAINED OF THE DEGRADATION SCHEME
THERE ARE PLACES IN THE PROGRAM WHERE IT IS SET UP FOR MORE THAN
ONE GAS, SOME OF THESE PLACES WILL BE USEFUL IF MORE THAN ONE
GAS IS INCLUDED IN THE MODEL ATMOSPHERE
THE MAIN PROGRAM IS LISTED BELOW THIS PROGRAM ONLY CALLS TWO
SUBRCUT INES!
1 ) DATA
2 ) MC
ALLOW FOR THE 0PTIMIZE=2
THE COMMON BLOCK FOLLOWS
ALL VARIABLES ARE PLACED IN THIS BLOCK TO
UNDER THE H COMPILER TO EE MOST EFFICIENT
COMMON ALPE (6) .BETE( 6 ),CE(6),FE(6) .WE (6) ,ALFA(15) ,BEFA(15).
1 CFA{ 15 ) .FFA(15 ) .WFA(1 5) .WF( 15). FACI ( 15) NFA N AR S A ( 3.80 ) .
2 ZALT< 80) .ENRI30),PCFA <3,30 > ,WCFA(3,30).NENR,FDG(3) ,PSE(3 ) ,
3 P ION (3),AT (3),A(6< 3),3(5, 3).G(53) ,UH( 10I.UK 10) ,UJ(10) .
4 UD (10), US N U ( 1 O ) US F ( 1 0) ,UEIN(11 ). IUTP.IUTM1. DG( 3 ) NS G< 3 ) N I G< 3)
5 NUMGAS.NUMST, WI S( 50) .NSEC MUNI T, I SEED I STOP, I LAST E I N, NPRI M C OS
6 H.EMIN.ZSTART T M IN T HRES H (3),AK(2,3),AJ(3,3) GAMA ( 2, 3) ,
7 T0(3,3).FAC(6).AOE(5),A1E(6),BCE(6),B1E(6),COE(6).DOE(6).
8 D1E(6),PR0B(40) W< 40 ) ,N IE ( 40 ) .A 21 ( 3) ,A22( 3) ,A31 ( 3) A 32 (3 ) ,
9 A33(3),B11(3).812(3).B13(3).C1(3),C21(3)C22(3),C31(3),
A C 32 ( 3 ) ,D 1 ( 3 ) D 2 ( 3 ) ,F1(3) ,F2(3) ,THFI(16),AKI(2,15) ,AJI(3,16),
B GAMAK2.16 ).TOI (3.16 ) S I GT ( 6 ) THE T ( 40 ) NX C SI E( 2 0 ) ,EIT( 20) ,
C IEILS.IEILM1,P,R,ZDIS,NPIN,NOP,COSPA,RT,F(40),R1,R2,R3,R4,
D R 5.XV,YV,ZV.XVN,YVN,ZVN,PA,PHI .EV.WLOSS.NSTAT.NSCS ,NPHF,
E IR1,IR2.NG.EVPRI,RAN,EMSD,EINEL,EXC5(50),COSPAN,T,FOVAL
READ AND WRITE THE INITIAL DATA
READ(5,105)NPIN,NOP.NPHF
NPIN = 1 IN ALL CASES
NOP IS THE NUMBER OF PRIMARIES
NPHF IS THE INDEX FOR THE ELASTIC PHASE
IF NPHF = 0 THEN USE SUBROUTINE PHFEL
IF NPHF = 1 THEN USE SUBROUTINE PHF0
FUNCTION BEING USED
MAIN0001
MAIN0003
MA IN0004
MAIN0005
MA IN0006
MAIN0007
MAIN0008
MA IN0009
MAINOO 10
MAI NO0 11
MA INOO 12
MAI NO 013
MAIN 0 0 14
MAINOO15
MAIN0016
MA INOO 17
MAI NO 018
MAIN0019
MAIN0020
MAIN0021
M A IN 00 22
MAIN0023
MAIN0024
MAIN0025
MAINOO26
MAIN0027
MAINO 0 28
MAIN0029
MA IN0030
MAIN0031
MAIN0032
MAINOO 33
MAIN0034
MAIN0035
MAINOO 36
MAIN0037
MAINOO 38
MAIN0039
MAI NO 0 40
MAIN0041
MAIN0042
MAINO 043
MAI NO 0 44
MAI NO045
MAIN0046


Page
5. Sixth Random Number, Rg 67
6. Multiple Elastic Scattering Distribution Used
Below 30 eV 67
7. Value of the Cutoff Energy, 2 eV 74
D. Statistical Error in the Monte Carlo Calculation ... 75
VMONTE CARLO INTENSITY PLOTS AND COMPARISON WITH EXPERIMENT 77
A. Excitation of the N^ State 77
B. Range of Electrons 80
C. Previous Experimental and Theoretical Work on the
3914 A Emission of N^ 81
D. Range Results and Longitudinal Intensity Plots from
the Monte Carlo Calculation 84
E. Intensity Plots in the Radial Direction 87
VI SENSITIVITY STUDY OF THE ELECTRON ENERGY DEGRADATION ... 98
A. Effects of Ionization Differential Cross Section on
the Intensity Distributions 99
B. Influence of Inelastic Differential Cross Sections
on the Intensity Distributions 104
C. Comparison of Different Elastic Phase Functions on
the Electron^ Collision Profile 104
D. Influence of Different Elastic Phase Functions on
the Intensity Profiles 112
E. Effects of the Total Elastic Cross Section on the
Electron Energy Degradation 121
VII MONTE CARLO ENERGY LOSS PLOTS AND YIELD SPECTRA 125
A. Energy Loss of Electrons in N2 125
B. Spatial Yield Spectra for Electrons Impinging on N2> 130
1. Three Variable Spatial Yield Spectra 132
2. Four Variable Spatial Yield Spectra 143
VIII CONCLUSIONS 152
iv


-113-
Figure 6.5 Collision plots for MC trials B1, B2, and B3. The histograms
represent the MC data.


-107-
Table 6.1 Model 1 parameter values (column labeled "a") and phase func
tion properties for various trials. The phase function fall-
off (column labeled PFFO) is indicated in the number of
orders of magnitude difference between the differential
cross section at 0 and its value at 180. The average
angle of scattering (column labeled AAS) is found using
Eq. (6.3).
Trial
a
PFFO
AAS
A1
0.9
1.0
64.5
A2
0.4
1.5
52.2
A3
0.095
2.5
31.6
A4
0.02
4.0
16.3
A5
0.0065
5.0
9.7