A STUDY OF THE EFFECTS OF DIFFUSION AND KINETICS ON THE
SPATIAL DISTRIBUTION OF PRODUCTS CREATED BY ELECTRON
DEPOSITION
BY
DANIEL EDWARD RIO
A DISSIf-TATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PA'.TIAL FULFILLMENT OF THE REQUIREMENTS
FO-. THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1983
To my loving parents and to Chelsea for her first Christmas
ACKNOWLEDGEMENTS
I would like to extend my gratitude to Dr. Genevieve
Rosseler for her introduction to the program of biomedical
enqineerinq and her continual amenities over the years. Dr.
A. E. S. Green receives special thanks for his constant
vigilance and knowledgeable guidance. I also wish to thank
the rest of my committee members, Drs. Philip Achey,
Lawrence Fitzqerald and James Keeslinq for the time they
invested in reading this manuscript. They provided many
helpful suqqestions to improve it. Special mention and
credit is due Dr. Robert Coldwell for his endless computer
assistance. Paul Schippnick was very helpful in producing
the contour graphs for the yields and calculating the
diffusion coefficients, and I appreciate the many
enliqhteninq discussions I have had with him. I am also
grateful to Dr. R. Sinqhal for his introduction to the
computer methods associated with the Monte Carlo transport
code.
My parents Evelio and Jasmine Rio deserve exceptional
recognition for joining me on the treadmill of work
associated with this dissertation. Their devotion and love
was of the upmost importance to me. Without their faith and
indefatigable support this qoal would not have been reached.
iii
I am very thankful to my brother, Michael, for his
assistance to me when I really needed it. I am especially
grateful to my wife, Patti, for her fortitude and strength
and for her ability to support our home during the many days
and nights it took to complete this dissertation.
This work was supported by the Office of Health and
Environmental Research of the U. S. Department of Energy
(Contract DE-AS-5-76 No. V03798) grant on charge particle
deposition studies, biophysical studies related to energy
generation, awarded to A. E. S. Green. Extensive use was
made of the computing facilities of the Northeast Regional
Data Center (NERDC) of the State University System of
Florida. This manuscript was prepared at the NERDC using
the UFTHESIS program.
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS . . *
LIST OF TABLES . . . ..
ABSTRACT . . . . . .
CHAPTER
I. INTRODUCTION . . ...
Overview . . . .
History and Motivation .
Description . . .
* . . * . iii
vii
. . . . . . vii
. . . . . viii
II. THE PRODUCTION OF INITIAL YIELDS
. a . . . . 12
The Monte Carlo Method .
Basic Molecular Cross Sections
Elastic Cross Sections
Inelastic Cross Sections
Multiple Elastic Scatterinq
The Yield Spectrum and Yields
III. DIFFUSION AND KINETICS . . .
General .. ......
Kinetics . . . . . *
Development of the Diffusion Kinetic System
Computer Adaptation . . .
IV. RESULTS . . . . . ...........
Discussion . . . . *
Computational Simulation . . . .
Monte Carlo and Yields ...
Diffusion Kinetics . . .
Limitations . . . .. . . . -
. 63
. 93
93
93
96
115
122
. * A & * .
* A
V. CONCLUSIONS . . . . . . . . . .
124
APPENDIX
A. MONTE CARLO PROGRAM .. . . . . . . 127
B. DIFFUSION KINETIC PROGRAM . 133
BIBLIOGRAPHY . . . . . . . . . . 176
BIOGRAPHICAL SKETCH . . . . . . 184
LIST OF TABLES
TABLE PAGE
2.1. Differential and total elastic scattering
parameters. . ,. . .a . . . . 15
2.2. Differential elastic scattering functions. . 17
2.3. Ionization scattering algorithms . . . . 21
2.4. The total and secondary electron cross section. 23
2.5. Parameters for the excited states of H20. .. .. 27
2.6. Cumulative distribution functions for elastic
scattering. . . . . . . .. . .. 31
2.7. Parameter values for radial distribution. . 32
2.3. Parameter values for longitudinal distribution, 33
2.9. Parameter values for Polar Angles distribution. 34
2.10. Parameters for two dimensional yield spectrum. 47
2.11. Parameters for four dimensional yield spectrum. 48
2.12. Parameters for electron deposition spectrum. . 53
2.13. Normalization constants, A. .. . . . . 54
2.14. States and associated products. . . . ... 58
3.1. Chemical species included in the diffusion
kinetic system. . ... . . . . . . 68
3.2. Reaction rates at temperature equal to 400 K. .. 69
3.3. Reaction rates at temperature equal to 400 K. 70
vii
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A STUDY ON THE EFFECTS OF DIFFUSION AND KINETICS ON THE
SPATIAL DISTRIBUTION OF PRODUCTS PRODUCED BY ELECTRON
DEPOSITION
By
Daniel Edward Rio
December 1983
Chairman: Alex E. S. Green
Major Department: Nuclear Engineerinq Sciences
We have updated our analytic representations of
fundamental molecular cross sections for I20 vapor, drawing
extensively on the recent work of the Applied Physic Group
at Los Alamos. These cross sections, with slight
modifications, have been used in a Monte Carlo computer code
to build up a statistical profile of the collisional events
incurred by an electron as it is degraded in H20. The
results are described by a continuous function called the
spatial yield spectrum. This four dimensional function can
then be used to calculate the spatial distribution of the
excited and ionized states created by the electrons which in
viii
turn are used to describe the steady state input into a
Diffusion-Kinetic computer code. In addition, reaction
rates and diffusion coefficients which are not
experimentally known have been estimated usinq current
models. It is then possible to calculate the spatial
distribution of the final products for a steady state beam
of electrons. Additional results related to the yield
spectrum such as eV/ion pair and deposition contours are
also presented. The results are intended to facilitate an
understanding of the effects of radiation in biological
systems.
CHAPTER I
INTRODUCTION
Overview
Radiation physics, radiation chemistry and radiation
biology form a triad of scientific fields which address the
effects of radiation in terms of three basic and
historically different points of view. The study of
radiation effects on biological systems has in the past
looked toward chemistry to supply the fundamental concepts
necessary to understand this field. Radiation chemistry has
been able to explain many of the observed effects of
radiation with the free radical theory and the discovery of
the aqueous electron. At the same time the field of
radiation physics has evolved froa the study of macroscopic
concepts such as exposure, dose and LET (linear energy
transfer) to the study of individual atomic and molecular
processes which require the formalism of quantum mechanics.
During the past few years the progress in these fields has
increased to the point where it is possible to beqin to
und.jrst.iLd the biological effects of radiation in terms of
individual atomic events and their aqqreqate behavior! This
is due in large part to the development of hiqh speed
con puters, experimental studies of basic atomic and
2
molecular cross sections and picosecond studies of chemical
kinetics.
While the incident radiation may be gamma rays, protons
or charged particles, ultimately it is electrons which
dissipate much of the incident energy. Radiation deposition
is split into two broad categories: (1) direct effects-- in
which the incident radiation interacts directly with the
biologically sensitive sites and (2) indirect effects-- in
which the incident radiation interacts with the major
component of the system, for example H 0 which in turn forms
free radicals which interact with the important biochemical
(see Fig. 1.1).
Refering to this figure let us describe the knowledge
necessary to understand the fundamental physics and
chemistry which leads to biological damage due to incident
electrons. First, detailed kncwledqe of the medium into
which the electrons will be fired is required. This would
imply knowledge of the various molecules which populate the
medium, their spatial relationship and individual atomic and
molecular energy levels. Next, collisional cross sections
for excitation, ionization and dissociation for individual
molecules are needed. These include both total and
scattering cross sections. At this time we have spatial
distributions of "initial yields" of, for example H 0+, OH+,
H+, OH, H and H O* (excited states). Part of the radiate
energy has of course been absorbed directly by the critical
energy has of course been absorbed directly by the critical
DIRECT EFFECTS INDIRECT EFFECTS
Energy absorption
by biomolecules
Biochemicals
Energy absorption
by H20
Excited
Ionized
H20
Dissociated
Kinetics
Ionic
Primary Lesion
Reactions
Bioradicals
Reactions
with
Biomolecules
Diffusion
radical
production
Molecular
alterations
u
D Production
SH2,02,H202
Biological Damage
Figucie 1.1 Physical chemical stages in deposition
mechanism.
H20
e
H 0*
2
DIRECT EFFECTS
INDIRECT EFFECTS
biochemicals. We move now from the physical stage to the
chemical physical stage. Furthermore, since the bulk of the
damage is caused by indirect effects additional chemical and
physical properties are required. A large percentage of the
information needed in this study is drawn from literature in
the combustion field (Kee et al., 1983: Westbrook, 1981)
since hydrogen-oxygen flames form a major area of study in
this field and many of the chemical species present in this
type of flame are also present in the irradiated H20 system.
The ions and excited states quickly react or dissociate and
leave us with a distribution of neutral products. Therefore
while it is necessary to know the possible reactions and
associated reactions rates for the ion and neutral products,
knowledge of the diffusion coefficients for the ions is not
as critical as that for the neutrals. Lastly, these
molecules are free to move (within bounds) and therefore it
is necessary to know transport properties for them.
Ultimately it is necessary to know the types of chemical
reactions which result from the interaction of these
products and the nearby biomolecules. This information is
found in the radiation chemistry literature (Scholes, et
al., 1969; Myers, 1974) but often does not take into account
biological structure and function which will of course lead
to modifications of the immediate chemical effect. For
example, it is well known that DNA will repair itself if
damaged. Furthermore the ability to repair itself is also a
function of the amount and or type of damage sustained by
the molecule.
While it is not possible at this time to study a
realistic biological system in this manner, several studies
usinq this method have been applied in varyinq degrees to
study the radiolysis of H20 in both its liquid and gaseous
state. The major programs are located at Oak Ridqe National
Labs (Hamm et al., 1978; Sitchie et al., 1978), Los Alamos
National Labs (Zaider, Brenner and Wilson, 1982 (ZBW);
Zaider and Brenner, 1983; Brenner and Zaider, 1983),
Lawrence Berkeley Laboratory (Maqee and Chatteriee, 1980;
Chatterjee and Maqee, 1980) and Battelle Northwest
Laboratories (Wilson et al., 1978). Much of the work by
these groups depends heavily on the pioneering work done on
fundamental collisional cross sections of electrons with H20
at the University of Florida (UF) (Green et al., 1971;
Olivero et al,, 1972; Kutcher and Green, 1976).
Subsequently the thrust of the research at UF has involved
usinq similar cross section work to study the collisional
history of electrons in various qases and representing this
information in analytic form (Green, Garvey and Jackman,
1977; Jackman, 1978; Green and Sinqhal, 1979 (GS); Sinqhal
anr Gr-te n, 1981; Siniihal et al., 1983). Mary of the
tech-i.iques required for t1he.e calculations originated ir tne
ficld of atm~ipheric pi'rsics. With this thesis wv brini
toqcther the work oi cl li .ioinal cross sections and
6
transport calculations at UF to imply the initial yields of
ionized, excited and dissociated H20 in the gaseous state.
These are then used as input, along with chemical kinetic
rate constants and diffusion coefficients to calculate the
spatial distribution of chemical species created by a
monoenerqetic beam of electrons.
Histo _and Motivation
The field of radiation chemistry originated in the 18th
century with the study of chemical changes produced by
electrical discharges in gases. Many of the reaction
mechanisms to explain the experimental results studied in
the intervening years still remain opaque (Anderson, 1968).
The establishment of radiation chemistry as an exact science
is basically due to the work of Lind (1921, 1938) in the
U.S.A. and Mund (1935) in Belgium. Much oi this research
used alpha particles as a radiation source and many of the
quantitative measurements are still good today. For
example, it was observed that the radiolysis of pure water
vapor produced almost no significant decomposition unless
chemically reactive scavengers were introduced into the
system. The initial interpretation of the observed
chemistry was centered on the assumption that the
radiologically produced ions were surrounded by "clusters"
of neutral molecules which would ultimately recombine either
with themselves, if they were of opposite charge, or with
7
free electrons. Upon doinq so the e-necqy released would be
shared by the adjacent neutral molecules which would
instiqate further reactions.
As early as 1909 Debierne (1909) proposed that free
radicals were responsible for the observed chemistry.
Durinq the 1930's two papers by Eyrinq, Hirshfelder and
Taylor (1936a, 1936b) indicated that since the "W value"
averagee e-nerqy to form an ion pair) is much greater than
the lowest ionization energy of a molecule that the excess
energy must be channeled directly into excitation. In
addition they proposed that charge neutralization reactions
would also lead to excited states. The excited states would
then dissociate forming free radicals from which the
remainder of the chemistry would follow.
Currently it is realized that both ions and excited
molecules produced directly by the incident radiation are
important. Recent research by Anderson, Kniqht and Winter
(1964,1966a, 1966b) has clarified the neutralization process
in H 0. These reactions produce free radicals which
continue. to build up until th cir concentrations are
suff iciently hiqh that reverse cheiiicil reactions compete
with radical production aid small steady state
conrceLtra tio;s of H O and H 0 arc observ-i.
2 2 2 2
Th.- c'oicz of using F El 0 vapor instead of liquid as the
first test ca.se for tudy requires a coiftient. While th"
r.diolysis of Jl O liquid is obviously of more biological
2-
significance than H20 vapor, physical processes in the
liquid are not well known. For example, the question of
defining ionization in condensed matter (Douthat 1983) does
not need to be addressed. In comparison, the collisional
cross sections for the gas phase are known to a much greater
degree of accuracy. Often many of the liquid cross sections
are extrapolated from gaseous models (Kutcher and Green,
1976; Hamm et al., 1978, 1983) and questions of their
validity have been raised by Inokuti (1983). Furthermore as
expected the general features of both vapor and liquid water
are very similar (Thomas, 1969). The initial yields of
ionized, excited and dissociated H20 are comparable (Thomas,
1969; Kutcher and Green, 1976). However the reaction
mechanism for the liquid is more complex due to the presence
of the hydrated electron, theoretically predicted by
Platzman (1955) and experimentally observed by Boaq and Hart
(1963) and the more ambiguous role of excited molecules
(Thomas, 1969). Finally H20 vapor represents a system in
which the sensitivity to cross section inputs is most easily
studied, comparisons with experimental data are inherently
simpler and the calculations accomplished for the vapor
phase can be modified for the liquid phase orce the
appropriate input data are obtained.
Descri tion
The purposes of this dissertation are to
1. Describe multiple elastic scatterinq of low energy
electrons in H20 vapor and characterize the spatial
distribution of the resultant distribution of
electrons,
2. Extend the work on initial yields due to incident
electrons in H20 vapor as reflected by the concept of
a yield spectrum (see chapter II)
3. Use the above information to study the subsequent
chemical reactions which occur during the diffusion
o. these chemical species in H20 vapor.
This will be develope.i in three major segments. In chapter
II, usiiL- the earlier work by Kutcher and Green (1976)
multiple elastic scattering distributions for electrons
unrder.oinq only elastic collisions in H20 vapor are
calculated. These results alonq with a compilation of cross
sections basedi oin empirical modeis deve;lope] by Green et il.
(1971), Olivero et al. (1972) ada Zai- r et al. (1982) ar
used to calculate spatial yield spectra (GreOhri ai Si nqial,
1979). TIlese dce t hrn used tu calculate spditial fields for
all the important "initial products" anri a spatial
distribution. ror suhexcit atior electrons. Analytic
uepresentations of all these Juantities are ivenr. In
chicpteL III tl.e above calculated quantities a f colilowed as
they ]ifuse and react. Thic is accomplished by solvinq a
10
system of nonlinear partial differential equations. This
approach is based on the "classical" diffusion model as
developed for the radiolysis of H20 which originated with
the radical theory of Weiss (1944) and expanding spur
kinetics as hypothesized by Lea (1946, 1947) and was
ultimately formulated by Samuel and Maqee (1953). The
computations were developed by Ganquly and Maqee (1956),
Flanders and Fricke (1958), Dyne and Kennedy (1958) and
Kuppermann (1961 ,1974). Recently these calculations were
revived by Maqee and Chatterjee (1980) and Chatterjee and
Maqee (1980). Essentially, it consists of representing the
"initial" (in this instance those chemical species which are
seen approximately 10(-13) seconds after the radiation
impact) yields of radicals by a few estimated parameters and
following their evolution in time, assuming spatial
variation in only one dimension, radial in either spherical
or cylindrical coordinates, depending on the LET of the
radiation. The calculations are then compared against
experimental G-values(# produced/100 eV) of, for example,
H2, 02 and H2 02
In this study a major step is taken to elucidate the
mechanisms involved in this transition region between the
physical and early chemical physical states. The problem
will be solved in cylindrical coordinates, assuming
cylindrical symmetry and in the steady state limit. Solvinq
the temporal problem usinq this technique would have been
excessively expensive. The ma jor emphasis of this
computation is to establish initial spatial yields for the
chemical species produced by the incident radiation and to
tie the physics and chemistry together by taking into
account the ion chemistry as well as radical recombination.
The calculations are performed usinq the IMSL finite element
partial differential equation solver TWODEPEP (1981). The
system of equations is solved in the steady state case and
spatial concentrations of all species involved are
calculated. Finally, in chapter IV the quantities
calculation are appraised and applications and extensions of
the computations are discussed.
CHAPTER II
THE PRODUCTION OF INITIAL YIELDS
The Monte Carlo Method
The Monte Carlo approach to solving the transport
equation is a stochastic method of imitating the actual path
of an electron as it is degraded in energy by its
interactions with the medium. During the simulation the
movement of the electron is governed by its cross sections
which describe elastic or inelastic collisions and the
generation of a few random numbers. These numbers take into
account the statistical naturc- of th dJeqradation process.
Depending o01 the iicidut eneLy of tne electron the number
of incident elections allowed to deqrade is varii- in order
to build up t statistical profile of the actual solution of
tni trirspoct equation. Th- statistical ercror L-sultinq from
t.e- Ilonte Carlo calculation is discus.rd in the thesis by
JackiLanl (1973) where a detailed description of an earlier
v,-rsion of the proqramL use] in this calculation may7 be
fourWd. A compact overview of this particulaLr tudly mday be
foui r in Jackiaiii an,] Green (1379).
In bLief, tLie irlfor;Ja tioiL necf-sssar y to carry out thia
calculation- i3 a detailed collection of atooic or molecular r
croJs sections, the attributes of the incident electrons, in
13
particular their energy and direction and the density of the
medium which is arbitrary since the collision pattern is
simply scaled spatially by this quantity. Thus the ranqe of
the electrons which depends on the density is chosen. The
information generated at each collision for each electron is
stored on magnetic tape allowing data analysis to be more
flexible, since new items of interest may be pursued at a
later time without rerunning the program. The information
stored on the tape, at each collisional point, is the type
of collision and associated energy loss, the absolute
position of the electron collision and the incident and
final energy of the electron, its absolute direction and a
taq delineating whether it is a primary or later generation
electron. Extensive modifications of the Monte Carlo
program have been made while applying it to H20. These are
documented in appendix A.
Basic Molecular Cross Sections
Elastic Cross Sections
The production of a spatial yield spectrum requires
detailed comprehennsive data on the elastic scattering of
electrons Lfro H O. Unfortunately, such data are sparse or
2
nonexistent. To overcome this problem we rely on the
ex tens-ive work done by Porter and Jurnp (1976), 'wich was
re.ce:tly updated at Los Alamos National Labs (ZB5.') (1982)
for l 0, to characterized elastic cross sections with
convenient analytic functions.
For electron scattering in w20 we use the standard
Rutherford representation for differential elastic cross
sections modified by a screening parameter for electron
energies above 200 eV (Jackman, 1978). The functional form
is
do(eE) = Z 2e 4 2.1
d m2 V (l+2n-cos ) 2
where n (E,Z),the screening parameter, is a modified
formulation proposed by Grosswendt and Waibel (1978) and
used by Zaider et al. (1982) for H20o It is given ny
n = n k/[E(E/mc + 2)]
nc = a + P1nE 2.2
k = klmZ2/3
where Z is the atomic number of the medium and e,p,v and
theta are the charge, momentum, velocity and angular
deflection of the electron. The parameters are listed in
Taole 2.1,
Below 200 eV the empirical function proposed 'by Porter
and Jump (1978)
do(O,E)c 1 + O(E)_
dQ 1+2X(E) cos 0 [1+26(E) + cos ]2 2.3
TABLE 2.1
Differential and total elastic scattering parameters.
Elastic Scatterinq ( < 200 eV )
7.51525 2.9612 -1.7013
-0.419122 -0.26376 -1.48284
0.72017 (-2) 0.4307 (-2) 0.6331
-0.4646 (-4) -0.26895 (-4) -0.10911
-0. 10289 (-6) 0.583505(-7) 0.8358 (-2)
-0.2388 (-3)
-3.32517
0.10996
-0.45255 (-2)
0.58372 (-4)
0.58372 (-4)
-0.24659 (-6)
0.24775 (-1)
-0.296264(-4)
-0. 120655(-6)
Rutherford Scatterinq ( > 200 eV )
3 = .0825 K1
= .000017
Total Inelastic Cross Sec7tion
0 ta I~nelastic Cross Section
E0 = 1.798
X = -.77
U = .00195
V = 150.
F1 = 72.53
GI = .4734
E1 = .172
F2 = 11.093
G2 = 19.9223
E2 = 13.93
a = 1,64
16
is used where p, & and X are parameters fitted by Brenner
(1982) to experimental data. These functional forms are
qiven in Table 2.2.
The experimental data used in constructing this model are
those by Senq (1975) (0.35-10 eV, 20-108 degrees), Traimar
et al. (1973) (15-53 eV, 10-90 degrees), Nishimura (1979)
(30-200 eV, 10-140 degrees), Hilqaer et al. (1969) (60-300
eV, 35-150 degrees), and Bromberq (1975) (300-500 eV, 2-160
degrees). Examples of the functions alonq with experimental
data are presented in Fiqure 2.1.
For the total elastic cross section we aqain use a
function suggested by Porter and Jump (1978) and fitted to
experimental data by ZBW (1982)
x 2 F G2
(E)=EI 2+x 2+X + n 2 21 2.4
n(n+l)[ E n=l (E-E ) + G
where n=tU/ and the parameters are listed in Table 2.1.
Data used in fitting this function come from Senq (1975),
Nishimura (1979), Bromberq (1975) and Bruche (1929)
(4-36eV). A plot of the total elastic cross section is
presented in Fiqure 2.2.
TABLE 2.2
Differential elastic scattering functions.
6 1 1
(E) =e-l
5i-1
6(E)=e
iE 1 i 1E
X(E)=e .35eV
10 i-6
i6 XiE
=e 10eV
13 i-iL
=e 100eV
60.00 90.03
T7HETA
Figure 2.1 Differential elastic scattering cross section
function compared against experimental data
at .6eV Seng (1975), and 20 and 53 eV Trajmar
et al. (1973).
-7-
CD
o o
'- O-
O 0
C)
0 r-
-,4 m
u cnoa
o o
O-r (
CO
o O
*H -
0 -
o rN
r..
a)l (0
ti5
C 0 C
0 C o
Inelastic Cross Sections
We begin this section with a description of the
differential scattering cross sections for ionization
collisions. The theoretical work on the ionization
continuum by Massey and Mohr (1933) and Mott and Massey
(1965) provide a basic framework for the calculation of
simple atomic and molecular cross sections for ionization.
Green and Dutta (1967) have developed a practical method of
determining the doubly differential cross sections da/dQdT
(T is the energy of the secondary electron) for complex
molecules, such as H O, from sparse experimental data usinq
2
the formalism of generalized oscillator strengths (GOS) as
described by Mott and Massey (1965). A specific analytic
representation of the GOS for H O is qiven by Green et al.
2
(1971) and Olivero et al. (1972) with a correction to the
cptical oscillator strerqth part of the GOS bin:q made bv
-'utcher aLid Gr (197) This representatioL of t:he
aM:.1ul.-r depen-er. icJrc fo ioniza tio, scattering is perhapss the
most co:,.plete vailibl at ipr sent; however Lecause oif its
cocjl.-.xity it was felt that a less ambitious approach for
t:is inc-lstic scattering type should be made. Foilowigq the
work of 7,B:; (1932) we will qen-eruate the scattorinq anqles
according to the prescription oft lrossfn.Jt and ;iaiyel
(1978) th functions are qiven in Table 2.3.
TABLE 2.3
Ionization scatterinq alqorithms.
Inelastic Scattering ( > 100 eV )
cos 01 = /1-T/E(1-E(1-T/E))/C
cos 02 = V1-(1-T/E)/(1-T/C) T>200eV
cos 02 = .707R T<200eV, cos82 = -1+2R T<50eV
---_----- ---- _--------^^-^^ -^y
Inelastic Scatterinq ( < 100 eV
cos 1 = .707 + .292R
= .707R T>50eV
= -1+2R T<50eV
At low energies (<100 eV) these functions are rather
arbitrary; fortunately at these energies the elastic cross
section is beginning to dominate and we can attribute most
of the spatial characteristics of the collisional
distribution to elastic scattering.
While it is possible to derive the differential cross
section S(E,T) from the GCS a simple invertible function has
been developed by Green and Sawada (1972). It is
S(E,T) = A(E)F2(E)/([T-T (E)]2 + 2(E)}
2.5
KA E
A(E) = K E In( + J
B A
where E and T are the incident and secondary electron
energy. The adjustable parameters r and To have been taken
from Olivero et al. (1972). The function was then integrated
over T to give the total cross section and the parameters
KA, KB' JA' and JB were adjusted to reproduce experimental
data. Four ionization cross sections were considered. The
data for the 1B1+3A1 cross sections were taken from Mark and
Eqqer (1976) and for the 1B2+2A1 cross sections from
Schutten et al. (1966). The ratios of the states were
implied from the work of Paretzke and Berger (1978). The
states, their energy, and associated analytic
representations are given in Table 2.4. The functions are
plotted in Fig. 2.3 and 2.4.
TABLE 2.4
The total and secondary electron cross section.
Ion. States Threshold K K J
A B A
1B1 H20+ 12.620 2.590 173.1 1.0
3A1 H20+ 14.750 2,115 176.4 1.0
1B2 OH+,H+,0+ 18.510 2.069 116.2 1.0
2A1 H+,0+ 32.400 0.884 241.0 1.0
JB r To
1B1 H20+ -5.580 11.40 -2.37
3A1 H20+ -10.39 11.40 -2.37
1B2 OH+,H+,0+ -7.885 11.40 -2,37
2A1 H+,0+ -30,00 11.40 -2.37
Next we consider the excitation cross sections.
Inelastic scattering cross sections for excitation
collisions are assumed to be of secondary importance to the
elastic and ionization scattering distributions since at
hiqh energy (above 100 eV) ionization is the dominant
collisional process and below this energy elastic collisions
dominate (see Fiq. 2.2). The excitation cross sections are
repreAsented &'y tLe analytic function
r-q
(1l
N
e +
C tn +
V CW f
0)
0 rq 4-
*r41 (
U 131 fa-
OrW
UC
M 4- 0
.0
00) U
0 0-
0 0
E-c u
OOi
(0/ ) (a)ZUOT
(aTnoaToui/ zo) (a) *O
N
-------------
e +
een
-'
en 0 0
.,-4 ^
'O U
SW' m
'--4
*f-4C
C)
o o
H
-4 -) '
o v
d E-U C
uo 1n cL
\\P* V
^N" =! *-1
^^Y 0 'O *-
-^a .r-i o
^^s*^^- ur +
'***3~-.^ ~1- (0 0
"B -- (0 di C~
q F. w." j w.
o(E)= I [1-( (l ) + 2.6
3W E E 2 v
S(Vj3-E) + Vj2
where the resonance term is used only for the vibrational
states. In all, fifteen individual states were considered.
These are presented in Table 2.5. The functional form is
taken from Green and Stolarski (1972) and the modification
for the vibrational states is due to ZBW (1982). The
paraneterization for all excitation states is described by
ZBW (1982) which draws on work by Olivero et al. (1972),
Kutcher and Green (1976), and data by Beenakker et al.
(1974), Mohlmann and de Heer (1979), and Fujita et al.
(1977) on dissociative excitation. The parameters used in
Eq. 2.6 are presented in Table 2.5 and the functions are
plotted in Fiq. 2.5. Special note should be made of the
Rydberq states where contributions from all possible states
with n=3 or above are included according to the prescription
of Green and Stoiarski (1972) and Jackman et al. (1977).
Finally we calculate the total inelastic cross section by
summinq the above functions and above 20 eV we fit this
cross section with the function
qF E
V E
Stot(E) = In [a(E-1) +1] 2.7
where F=2.9625, W=5.5343, "=.02239, and 0=1.7737. Below 20
eV we enter the total cross section discretely. The fit
above 20 eV is virtually indistinguishable from the
TABLE 2.5
Parameters for the excited states of H20.
Exc. States Threshold a
(A1)V2 0.1980 1.0000 5.1930
(A1)V1(B1)V3 0.4600 1.0000 6,0900
(B1)V1(B1) V3 0.4600 1.0000 6.0900
TRIPLET 4.5000 1.0000 3.0000
TRIPLET 12.5000 1.0000 1.0000
DIS.CONT 7.4000 1.0000 3.0000
DIS.CONT 9.6700 1.0000 3.0000
DIFFUSE BANDS 13.3200 1.0000 3.0000
H* LYMAN ALPHA 17.0000 0.6000 6.0000
H* DALMER ALPHA 19.0000 0.7500 3.0000
OH* 3064A 10.0000 0.4500 0.6500
SERIES A 9.9998 1.0000 3.0000
SERIES B 9.9998 1.0000 3.0000
SERIES C 11.0600 1.0000 3.0000
SERIES D 11.0600 1.0000 3.0000
W Q F
(A1)V2 0.1980 0.2390 0.254 (-4)
(A1)V1(B1)V3 0.4600 0.5000 0.850 (-3)
(B1)V1(B1)V3 0.8990 0.5000 0.850 (-4)
TRIPLET 4.5000 3.0000 0.700 (-2)
TRIPLET 9.8100 3.0000 0.180 (-1)
DIS.CONT 7.4000 0.7500 0.630 (-1)
DIS.CGNT 9.6700 0.8000 0.138
DIFFUSE BANDS 14.7300 0.7500 0.626
H* LYMAN ALPHA 17.0000 1.3000 0.994 (-1)
H* BALMER ALPHA 18.5000 1.0000 0.266
OH* 3064A 9.5000 0.7250 0.538 (-1)
SERIES A 9.9998 0.7250 0.340 (-1)
SERIES B 9.9980 0.7500 0.338 (-1)
SEFIES C 11.0600 0.7500 0.771 (-1)
3EF.I 3 D 11.0600 0.7500 0.771 (-1)
----------- ^-- -Y2 Y- -,-
V1 V2 V3
(A1)V2 0.135 0.677 (-11) 0.275
(. 1)V1(BI1) V 0.495 (-1) 0.130 (-1) 0.587
(B1)V 1(E1) V3 0. 95 (-2) 0. 130 (-1) 0.587
10-16
db
DC 9.
0-17
DC 7.4
Vib .899
10-18 r riplet 9.
10-19
Triplet
10-20
10-1 100 101 102
E (eV)
Figure 2.5 Total excitation cross sections versus
incident energy.
calculated value. The total elastic and inelastic cross
sections are graphed in Fiq. 2.2.
Multiple Elastic Scattering
Below 50 eV the elastic cross section is at least twice
as large as the inelastic cross section. This ratio
continues to increase as the electron energy decreases
approaching the threshold for the lowest excitational state
of H20. The electrons will then slowly approach thermal
equilibrium with the H20 molecules. It would be uneconomical
to follow these electrons collision by collision during this
staqe of their degradation; therefore a multiple elastic
scattering distribution (MESD) is used below 20 eV.
MESDs have been used for condensed histories
calculations, for example, by Berger (1963), Jackman (1978)
and Sinqhal and Green (1981). The latter two studies used
the work by Kutcher and Green (1976), who carried out a
Monte Carlo calculation in order to follow the position of
electrons elastically scattered after s mean free
pathlenqths and characterized the results using cumulative
functions for the radial, longitudinal and polar angle
distributions.
Following the work of Kutcher and Green (1976) (KG), we
have made a detailed Monte Carlo study of the position and
direction of the electron distribution after it has traveled
1 to 500 mean free pathlengths. Using the computer code
developed by Kutcher we have essentially solved the one
velocity Boltzmann equation
as(R.z,u.4,s) + Q(4.e)*Vf(R.z.u.4,, s)
2.8
= [f(R.z.u,A.,s) f(R.z.u',A)'.,s)] 1 (JI '-nl)dQ'
where R, Z, p=coso, A = 1- and s are defined in Fiq. A.2
and omeqa is a unit vector alonq the direction of the
velocity. In particular we have calculated the distribution
functions for R(s), Z(s), and X(s)=(1- (s))/2 and
represented them with analytic functions based on those by
KG, with minor modifications, at nine energies from .1 eV to
50 eV. The functional forms used i:n fitting these
distributions are presented in Table 2.6. Representative
qraphs or the distributions are shown in Fiqs. 2.6-2.14.
The solutions generated by the Monte Carlo simulation are
represented as discrete points. We see that the analytic
functions represent these solutions quite accurately,
although for fewer than four mean free pathlenqths, the fits
systebid tical ly ui'nd Eshcot or overshoot the qentrated
solutions. This is ot miiioor concern, since these results
arc usually ppli:d only to those cases iu wnic!: the iLiumiber
of elastic collisiois bctweeC each ilelastic collision is
much rcater than tour.
TABLE 2.6
Cumulative distribution functions for elastic scattering.
F(R.S) =
X(s) = 2[1-Ae (s/s)B
sa
6(s) = (--)
S
S, ,
+ ( )
S
F(z.s) = {1-
v(s) = 1-A e
F(O) = FF[1
-uZ -1/u -uI
e [F(O)-1])
D
-(s/s ) V
- e
B
-(s/sF)
u(s) = (H + s )/s
F(X.s) = (- X) X
vr+X
-s/s
I(s) = Le
-s/s
c(s) = Me
-(s/st)N
t(s) = 1-e
x = (1-cos 0)/2
l-e-R
TABLE 2.7
Parameter values for radial distribution.
E(eV)
.1
.2
.5
S
5.24
5.37
4.74
B
Y
1.03
1.00
1.10
A
.740
,728
.739
S6
2.79
2.87
2.73
2.00
1.95
2.06
Si
.100
.100
.100
.734
.755
.755
1.0 5.94 1.00 .641 2.94 2.21 .100 .790
2.0 7.00 1.00 .596 3.09 2.28 .100 .786
5.0 6.37 1.02 .664 2.96 2.07 .100 .760
10.0
20.0
50.0
7.09
6.88
9.00
1.09
1. 11
1.07
.623
.645
.563
3.05
3.09
3.73
2.07
2.00
2.32
.100
.100
.100
.753
.731
.799
ABLE 2.8
Parameter values for longitudinal distribution.
E(eV) S D A I J H FF S B
.1 10.5 1.34 -.600 .560 .957 1.50 .435 9.22 .600
.2 9.90 1.50 -.252 .600 1,00 1.80 .435 11.5 .640
.5 9.20 1.40 .0085 .650 1.05 2.30 .470 13.0 .720
1.0 9.15 1.52 .170 .680 1.12 2.40 .447 16.5 .719
2.0 8.40 1.71 .350 ,700 1.17 2.60 .435 16.1 .725
5.0 8.00 1.87 .400 .740 1.17 2.60 .440 12.3 .704
10.0
20.0
50. 0
7.50
8.00
9.00
1.94 .451 .800 1.22 2.50 .428 10.2 .700
1.75 .500 .900 1.29 2.60 ,435 9.10 .750
1.72 .475 .940 1.39 2.60 .425 14.4 .750
TABLE 2.9
Parameter values for Polar Anqles distribution.
E(eV) S L S M St N
.1 .800 .650 .450 2.00 1.06 .650
.2 .700 .800 .600 1.60 1.46 1.07
.5 .580 1.10 .840 1.40 1.76 1.25
1.0 .528 1.28 1.13 1.13 2.58 1.27
2.0 .459 1.26 1.50 1.10 2.84 1.08
5.0 .500 1.25 1.30 1.30 1.98 1.23
10.0 .540 1.20 1.10 1.40 1.85 1.20
20.0 .770 .973 .815 1.68 1.60 1.20
50.0 1.40 .650 .550 2.00 3.75 1.19
W3 >,
W -d
._____---------------------------4-)- C: 41 4-1
'" "- ri 4P n
,0 rd
OH
CO4 C4
LW- I-i rd
II .-I 0 Q 0
4-1 0 -04
cq o
,4
-4 40 4a)
P -rl
3 s -, T -.
r 3 >C
C 0- 4 4
\ \ t>i
\r 4) 0 C)(
C) U
0 M,
0 Ct
-44 lC
(N 0
0 (C U
4-)
^ *H (C
0C)
-. CO 0
cj
N n
ro t >
m ()
C
Sq0 'J 04
H *CG
(U 3 -1) (
> [l kj
(0 C 0
'U C)E-2
N
N
C)
(U) J
C) .Q
) -
) 0
S4- 4J 4 4J
o o
i 0")
I Q1) C)
C)4
S4-0) 4-
\ ITi r--
0 0 rd
44 > o -r
4-)
aEm
\ 0 4 4
S0 4J
U) >.
r-l 04-1
S* -1) 0
*H 4-
O -
\c)Q)
-H ) *
\> e 41
. .- D o a
\4- 0 0-
S -C\ C) W aC
0 CC)
o a N
*H
Q)
*P
4-t)
U)
>-i )
E c:
4-4
S0
U)
O -Hi
-re P
4J
4*
-4 o
U3
U3
Sa0
0 4-
a)
>
N
a)
c4i
::se
FX4H
(Z) a
39
a)
4-)
4-)
0
H
r4
4I-)
Ul
-H
r-
k
C4
4-O
U
-r
C) 0
4-
IC
4-1
--4
a3
0
4- 4
-- 1
GJ^
fi ^
0 4-1
=1U*
0 ^
E QJ
a -
0 V:
40
oo
Q1)
a4
-P
H *
u M
-H Z
I zp
0,
-,H
-P ro
0 >
-rd
4-1 0
GN 4O
cl
3->
ra ro
'-4
1 *l 0 4
0 o
r-C
nc
4-0
(U
!-!
^i C
*H
(z)
41
1.0
.8
*
*0
.6
E = .2eV
SS> 4
.2
0 .2 .4 .6 .8 1.0
X = (1-COS )/2
Figure 2.12 Cumulative polar angle distributions for
multiply scattered electrons with energy
.2eV for various pathlengths, S.
1.0
.8
.6
4 E = 2eV
S.4
S>4
.2
I -
0 .2 .4 .6 .8 1.0
X = (.1-COSe)/2
Figure 2.13 Cumulative polar angle distributions for
multiply scattered electrons with energy
2.0eV for various pathlengths, S.
43
1.0
.8
.6
.4
E = 20eV
.2 S>4
0 .2 .4 .6 .8 1.0
X = (1-COS6)/2
Figure 2.14 Cumulative polar angle distributions for
multiply scattered electrons with energy
20eV for various pathlengths, S.
The Yield Spectrum and Yields
The concept of a yield spectrum was introduced by Green,
Garvey and Jackman (1977) (GGJ). This concept traces its
evolution from the work on the equilibrium flux of Spencer
and Fano (1954), Peterson (1969) and Jura (1971). The yield
spectrum has properties which are much simpler than the
equilibrium electron distribution. Its shape can be
described by a continuously varying function of incident and
spectral electron energy and as demonstrated by GGJ (1977)
this functional dependence is similar for many atoms and
molecules. As introduced by GGJ (1977) the yield spectrum
is defined by
U(E.E ) = N(E,E )/AE = Ua(E.E )H(Eo-E-E )+&(Eo-E) 2.9
where N is the total number of inelastic collisions which
occur in the energy interval AE centered at E and E is the
ener of the incident electron. II is the Heaviside
function witr. E set at the lowe( t threshold value or all
the iielastic interactions coiisidered and 6 is the Dirac
delta fur.ction waich allows thri second term to represernt the
so.irce coILtrib utiorl to the yield spectrum.
rhis det-inition has also been qentealiz _. to fcouu
di mensLions (.S) Tht'- Luictional forn is
U(E,p.X.E ) = N(E.p.z)/AEAshz
= U (E.p.z.E )H(E -E-E )+D(p.Z.E )6(E -E)
a 0 0 m 0 0
where
As = ][(p+ ) (p )2
2 2
and N(E, p,z) is the total number
which occur in the volume element
the energy interval AE centered at
for U and D are
a
3
U a(E.p.Z.E ) = 1 A (E )G (p.Z.
D(p.ZE i=1
D(p.z.Eo) = A4(Eo)G4(P.z.Eo)Do)
of inelastic collisions
As, z centered at p,z and
E. Analytic expressions
3
E x PE,E D 0
2.11
where
A.(E ) = (D J I G. (p.Z.E )pdpdz -1
1 0 -O -01 0
-Eo/e i
o) = ke
1iE, 0 1 + 1l
and D is the density of the medium with units of rams per
0
centimeter cubed. Note, by definiiiq the normalization
constant A with D3 explicitly displayed, we have
0
essentially taken all density dependence out of it since the
integrated quantity is proportional to the inverse density
cubed. The parameterization for H20 is qiven in Tables 2.10
2.10
and 2.11. Representative qraphs of the two and four
dimensional yield spectra are qiven in Fiqs. 2.15-2.17.
It is also necessary to consider the ultimate fate of the
electrons both the incident and those produced in ionization
collisions in order to carry out the next part of the
calculation. The MC code was modified to follow these
electrons below the lowest excitation threshold energy.
These electrons undergo elastic collisions, but it is a well
known result from classical mechanics that they can exchange
kinetic energy with the H20 molecule, losing energy on the
average, until they reach thermal equilibrium. Because the
average energy lost to the H 2 molecule by the electrons
approaches zero as the electron approaches equilibrium, it
is not practical to follow the electron to equilibrium in
the MC simulation. Therefore a cutoff of E = 125 eV was
chosen; that is, once the electrons energy fell below this
energy its position was recorded and it was no longer
followed. A new quantity is defined, called the deposition
spectrum which is defined as the number of electrons with
energy less than .125 which are in the volume element As, z
centered at p,z. This deposition spectrum is unlike the
yield spectrum which represents the number of collisions or
the deposition spectrum as defined by Sinqhal et al. (1983)
which represents the number of electrons with energy E after
their last inelastic collision. The deposition spectrum can
be analogously represented by a function similar to that
ABLE 2.10
Parameters for two dimensional yield spectrum.
3
U(E.Eo) = I x(E.Eo)
i=1
-E /0i
xi(EEo) C i+ e
Ek = E /1000
kY 0
.0339
0.0
3 0.0
. 8036
99.85
830.6
40,3
4.61
.074
----- -~- ~------- -c-L-
48
TABLE 2.11
Parameters for four dimensional yield spectrum.
2 2
-z6 .(z-z ) 2
Gi(p.z.Eo) = e
Pi(E ) = Ai/(E +Bi)ln(E +Ci)
-1
z o(E) = (E +Bi)/Aln- (E +C.)
A B C
1 21.43(7) -76.90 -96.91
2 20.82(7) -25.80 -92.83
3 18.30(7) -6.224 -90.97
4 91.92(7) 87.39 6.005
_---- ---_-- -----_-^.- ^ ^^^_ ^ ^ -
1 115.3(7) 104.5 105.4
2 619.6(7) 465.1 833.1
3 34.07(7) -52.95 8.82
4 88.70(7) 355.6 -66.04
-------- ------------ ~71~ ~ -"
1 .7091
2 .5902
3 .2835
.700
101
10
10-1
10-2 10-1 100 101 102
E (eV)
Figure 2.15
Two dimensional yield spectrum and electron
deposition spectrum as function of spectral
energy E at various incident energies, E .
The points represent the distribution values
obtained directly from the Monte Carlo
Transport code.
(1.25, 1.25) (cm)
E = 500eV
E(eV)
Figure 2.16
Four dimensional yield spectrum for
electron with incident energy 500eV
versus spectral energy E at the spatial
points (p, Z).
1019
o 10
4-
-4
c 10
Q)
U
1
c0
" 1016
'0
wI
1015
I '
19
(3.75, 1.25) E = 200eV
19* \
18
1018
u> 017 *
4-J
ru
-'-4
m 16
Qi)
*, i15
10 -
S10
1014
10-1 100 101 102 103
E (eV)
Figure 2.17 Four dimensional yield spectrum for electron
with incident energy 200eV versus spectral
energy E at the spatial points (p, Z).
52
used for the four dimensional yield spectrum except that it
requires only one term instead of three and has no spectral
energy dependence. The function is
DS(p.z.Eo) = N(Eo)A(E )G(p.Z,E ) 2.12
where the incident energy dependence is given by
2 2
-pD Vp +6(z-z )
o 0
G(p,z,Eo) = e 2.13
and the parameters are given in Table 2.12. Normalization
factors A for yield and deposition spectra are given in
Table 2. 13. Tae function is plotted for various
isoelectronic densities in Fiq. 2.18 and Fig. 2,19.
Having obtained the yield spectrum we can then calculate
the specific state yields (#mol. of H O in a particular
2
state per cubic centimeter due to all collisions for one
electron) as a function of position by integrating the yield
spectrum multiplied by the probability of a collision
producing a molecule in the ith state over E from the
threshold of that particular state to the incident electron
energy. Analytically, the yields are
E
J (p,z,Eo) W. Pi(E)U(E,p,z,E )dE 2.14
1
TABLE 2. 12
Parameters for electron deposition spectrum.
2 2
-PDo p +6(z-z )
G(p.z,E ) = e O
P(Eo) = A/(Eo+B)ln(E +C)
-(E /E )n
(E) = l-Ae c
o(Eo) = (EoB)/A ln- (Eo+C)
0 0 0
A=19.30 (7)
B=-22.77
C=-91. 70
A=102,8 (7)
6 A=.3809 (7)
B=213.7
E=511.9
C=-90.53
N=8.98
Energy Number of thermal electrons
(eV) produced per incident electron
100 3.81
200 7.00
500 16.62
32. 87
1000
TABLE 2.13
Normalization constants, A.
Incident Yield Spectrum Electron
Energy (eV)
1 2 3 4 Spectrum
100 3.25 (19) .399 (19) .100 (19) 3.35 (20) .3774(19)
200 1.49 (19) ,412 (19) .110 (19) 1.39 (20) .3400(19)
500 .0796(19) .0434(19) .0140(19) .2590(20) .4927(18)
1000 .112 (18) .729 (17) .250 (17) .5587(19) .9711(17)
2000 .169 (17) .120 (17) .422 (16) .1051(19) .1597(17)
5000 .144 (16) .106 (16) .382 (15) .1020(18) .1420(16)
> CN
300
00
0 C0
-i 0 -0
0
UN
S HOLn H
HHO
0 *r
So e
4-Q)
0 0
C) :C -
0 U
0
U 0 0
aO
r 0 n
Ou
O
S--4
-a 0 ^-
0 0
4
C.
CO
1-1
0'99
Cl)
0
04-
0-I
S 4JCN
U
(1) Q
00
oQ
Hk
o
*-)
4 -)
0
4.1 P:
Q1)
00
S4j
0 Q)
S*-H 4-1
0
04
Q) r
0 0
>4
S-H
to
0 Ou
U Q)
Q)
r-i 4-
Q)
N0
*rl 0
-4 0
f(4
where J i is the yield for the i state, W i is the threshold
th
for the i state and pi is the probability of interaction
for this state, given by
pi(E) = ai(E)/otot(E) 2.15
Performing the integration over all space as well gives the
number of collisions of a particular kind which occur for
each electron as it is degraded. Of course in this case it
is simply easier to start with the two dimensional yield
spectrum. Regardless, it is then possible to calculate W,
the energy loss per ion pair produced. We have calculated a
value of W(eV/ip) for H O which is comparable to
2
experimental results. The results are presented in chapter
IV. A few representative isoyield contours are presented in
Fiqs. 2.20 -2.23, These will be used as input into the
diffusion kinetic calculations which follow. In these
calculations more than one state has contributed to the
production of a specific species. The states and their
products are presented in Table 2.14, It is also possible
to calculate isodose curves using the yield spectrum by
performing the following sum
n
Dose(pz.E )a I W. J. (p.Z.E ) 2.16
0 j=1 0
over all states.
TABLE 2.14
States and associated products.
states products
------------------ ------^---------------
(A) V2 H20*
(Al)VI(B1)V3 H20*
(B1)V1 (B1)V3 20*
TRIPLET H20*
TRIPLET H20*
DIS.CONT li, OH
DIS.CONT H2, 0
DIFFUSE BANDS H20+, e-
II* LYMAN ALPHA H, OH1
H* BALMER ALPHA H, OH
OH* 3064A H, OH
SERIES A H20+, e-
SERIES B H20+, e-
SERIES C 120+, e-
SERIES D H20+, e-
1B1 H20+, e-
3A1 i20+, e-
1B2 OH+,H/ H+,OH/ 0+,H2
2A1 H+,OH/ 0+,H2
~ a)
0
o -1
/C.a
.r4
0
0
c,
V
,1 -l-
0
o
\ \f
ON
L1
n o
o4-
0) 0
>
i0
_____ ------------------------ i ------- 17
6C0
\ \\\ ~I
60
CO
co
N
.r-
CM
CF
0
0
c,
U
c
w >
0
0
) 0
'-4
0
.,. o
o 0
C o
0U
Cn3
3 w14-
C0'9
(Uro) -
61
o
C
0
/ *
r LI
0
SOc,
0,
m 0
rO
0
0
S0,
N
\--
\ \ I 2 3 Ia
\ **.. ~ -i *-*
V ^,
^ ^ *r-1
^ ^~~-^ >-
___________ ____________ _-_ ^a
UL-UI UU ^ *- ^ ^ ^^ Jl
62
0
0
0
a
C
0
0 0
o
w,
0
o
O
L,
W(
0 C-
C3
*a
'-4
/ N
(uD ^ r
\ \. 5 ^
\~~~~r \ o0(l.
\ \ \ <"
\ \ \ ^*
\ \ \ 0
\~~ \ .^
\ \ \ 0 -
\ \ V '-'
\ \ V
\ \ -^ -1 *'-
\~~~ \ r(
\ \ ^~-. '"
\ -^^a i-
\~~V ^^- T O
\~~~S w^'-- 't
\ ~- __
__
OO'OI~ OQ' GQ' 0'i 0^ CO
u 3L
CHAPTER III
DIFFUSION AND KINETICS
General
After calculating the yields of the various excited and
ionized states of H20, which occur in less than 10 (-13)
seconds, we are interested in studying the effects of
diffusion and kinetics on these chemicals. Usinq standard
diffusion theory it is possible to calculate the steady
state distribution of the initial species and those products
like H202 which are created by the interaction of these
initial products. In order to accomplish this we have
developed a program called DIFKT which can calculate time
dependent and steady state spatial concentration profiles of
all the initial species and their products. Details of the
program are addressed in chapter II.
The use of a diffusion kinetic (DK) model to explain the
dependence of final products on the initial spatial
distribution of radiation products was first formulated by
Jaffe (1913,1929) and Lea (1946). Samuel and Maqee (1953)
extended this work, applying several mathematical
arproximations to make the problem analytically tractable.
Maqee (1955) considered the effects of dose rate for this
Molel. That is the case when the energy is absorbed at a
64
sufficiently high rate that we cannot treat the problem as
the sum of the effects of individual particles. Kuppermann
(1961) treating the problem in the absence of dose rate
effects removed the mathematical restrictions imposed in the
previous work. He solved a system of diffusion kinetic
equations of the form
8Ci(r.t)=DiV 2Ci (r.t)-J Cn(r.t)C(r.t)k mni3.1
at~ nm
Ci(r.o) = Co (r)
where D., knmi and C. are respectively the diffusion
constant of the ith species, the rate constants for the it
th
species and the concentration of the i species as a
function of time and space (Kuppermann, 1959, 1961 :
Kuppermann and Belford, 1962a, 1962b) The initial species
present and their spatial concentration were considered
somewhat arbitrary. Recently this approach has been
extended by aaqee and Chatterjee (1978,1980) and Chatterjee
and Maqee (1980). They essentially start their calculations
after all ions and excited states have reacted and formed
neutral around state molecules. The approximations involved
in these calculations are numerous; the most questionable
are those involving the spatial distribution of the primary
neutral species. They are adjusted to qive the best results
in reproducing experimental data for differential and
integral G(ip/eV) values. There are no guarantees that
their choices for the primary distributions represent the
correct input for the problem, though the results presented
for a Fricke dosimeter system compare well with experiments.
In general this approach does not address the early time
chemistry.
A more basic approach to the problem of calculating the
initial species present usinq Monte Carlo methods was
initiated in the late 70's at Oak Ridqe National Labs (Hamm
et al., 1978; Ritchie et al., 1978) and Los Alamos National
Labs (ZBW, 1982; Zaider and Brenner, 1983 and Brenner and
Zaider, 1983). This approach was recently connected to the
work of Maqee and Chatterjee (1980) (Wriqht et al., in
press; Wriqht et al. 1983). Initial prediction of
experimental G values are reasonable.
While a time depernden-t solution for one electron track as
pre3nteid is very useful in predictinqi experimental results,
a continuous beam (or at least pulsed) of electrons is
normally encountered. The major part of the present
analysis jhas cor.centr. at.-d on lookiq, at a continuous beam of
electrons. The transieiit behavior of this systei-i has not
ben addressed in the diffusion kinetic calculation.
Kinetics
In solvinL- a system which involves both diffusion and
kinetics it is often instructive to solve the problem
iqnorin'j diffusion. In this section we present the chemical
system whizc exists im-aediately after the incident electrons
have deposited their energy. This system exists
approxi Imately 10 (-13) seconds after an electron is injected
into the medium. Thus we se that the physical part of the
deposition process finishes very quickly. A complete
description of these events is presented in chapter I.
The s yste of nonlinear first order differential
equations wilica d-scriLes the cheiaistry is
dCi (t) C (t)C (t)
= k nmi n(t) m t3.2
dt nm
Ci(0) = Co
1 01
T.ole 3. 1 lists tihe- spec.~s coiisiJreed and Tables 3.2 and
3.3 list the reactions. The system consists of
electrons, Lons and excited species of H.20 as well as the
dis:ociated products. The neutral chemistLy reaction rates
ha ve .ea&: taren froo a recei.t compilation by Westhro>k
(19 1) Th. ion neutral reaction rates are taken from
Hunrtress (1973) and Ferquson (1973). The ion clustering
reactions are taker from the work of Good et al. (1970).
The ion reco;biiiatior, rates froma Leu et al. (1973). The
chemistry is fairly well kncwn for 'l 0 vapor and is
67
delineated in the books by Allen (1961), Spinks and Woods
(1976), Venuqopolan and Jones (1968) and articles by Anbar
(1968), Thomas (1969) and Hunt (1976). The most serious
deficiency is the lack of information on the fate of the
excited states of H20. In a recent study for liquid water
these states were assumed to dissociate in the same tine
scale as the interaction of the H20+ ion (Wriqht et al.,
1983). For vapor we would expect a slightly slower
dissociation rate. However the information in this area is
highly deficient (Inokuti, 1983).
The system of differential equations is solved using the
Gear method (Hindmarsh, 1974). The program is based on the
routine GEAR (IMSL, 1981) and includes an adaptation of the
kinetic subroutines from the diffusion kinetic program DIFKT
(see chapter III) in order to set up the kinetic equations.
Usinq the results from the calculation of yields to set the
relative concentrations (see Table 3.1) of the species
initially produced by the incident electron, it is possible
to study the time development of these concentrations. The
relative concentrations are based on the distribution of
products presented in Table 2.14 and the thermal electron
distribution. The reaction mechanism is presented in Tables
3.2 and 3.3. The results of the calculation are presented
i. Fiq, 3.1 and discussed in chapter IV.
TABLE 3.1
Chemical species included in the diffusion kinetic system.
Species Diffusion Initial Concentrations
Coefficients Kinetic System
(cm**2/sec) (moles/cm**3)
E
H2
02
H202
H02
H+(H120)3
1120 +
H+ (H120)2
H30+
OH+
5.05
.8644
.2305
.2327
.2339
.3695
.3754
1.292
.070
.250
1.290
.110
.245
.370
H20 .250
9.900 (-08)
2.648 (-12)
0.0
0.0
0.0
1.395 (-12)
6,044 (-12)
6.215 (-12)
1.572 (-11)
1.101 (-11)
1.091 (-12)
1.581 (-12)
0.0
0.0
0.0
TABLE 3.2
Reaction rates at temperature equal to 4000 K.
Reaction Reaction Equation Reaction Rate
Number cm,moles,sec
1 H + 02 -> 0 + OH 1.244 ( 5)
2 0 + OH -> H + 02 .6294 (13)
3 H2 + 0 -> H + OH .9954 ( 8)
4 H + OH -> H2 + 0 .5300 ( 9)
5 H20 + O -> OH + OH 3.187 ( 3)
6 OH + OH -> H20 + 0 .7910 (12)
7 H20 + H -> H2 + OH 7.729 ( 2)
8 H2 + OH -> 120 + H .3359 (11)
9 H202 + OH -> H20 + H02 .1038 (13)
10 H20 + 0H2 -> H202 + OH .3432 (-4)
11 H20 + M -> H + OH + M .9265 (-41)
12 H + OH + M -> H20 + M .8812 (18)
13 H + 02 + M -> HO2 + M .5837 (16)
14 H102 + i -> H + 02 + M .1902 (-9)
15 HO2 + 0 -> OH + 02 .1424 (14)
16 OH + 02 -> HO2 + 0 .7544 (-17)
17 HO2 + H -> OH + OH .2299 (14)
18 OH + OH -> H02 + H .1472 (-08)
19 H02 + H -> H2 + 02 1041 (14)
20 H2 + 02 -> HO2 + H .1434 (-17)
21 1J.O + Oil -> H20 + 02 .1424 (14)
22 1120 + 02 -> -102 + OH .27E0 (-25)
TABLE 3.3
Reaction rates at temperature equal to 4000 K
Reaction Reaction Equation Reaction Rate
Lumber c, moles,sec
23 H202 + 02 -> HO2 + 1102 .2000 (-9)
24 H02 + H02 -> H202 + 02 .2843 (13)
25 H202 + H -> OH + Oil + M .1651 (-7)
26 OH + OII + M -> H202 + M ,5380 (18)
27 H202 + I -> H02 + H12 .1514 (11)
28 H02 + H2 -> H202 + H 43.84
29 O + H + -> Oi + M 1.000 (16)
30 OH + t -> 0 + H + 1 .4191 (-39)
31 02 + a -> 0 + O + H .7448 (-47)
32 O + 0 + M -> 02 + n .8743 (15)
33 H2 + I -> H + H + M .7659 (-38)
34 H + H + M -> H2 + M .3020 (16)
35 1120+ + 1120 -> H30+ + OH .1130 (16)
36 OH+ + H20 -> H20+ + OH .3480 (15)
37 OH+ + H120 -> [30+ + 0 .2839 (15)
38 H+ + H20 -> H20+ + H .4939 (16)
39 H30+ + H20 + M -> +H(H20)2 + M .1230 (22)
40 +H(H20)2 + H20 + H -> +11(H20)3 + M .8339 (21)
41 +H(H20)2 + E- -> H + H20 + H20 .1630 (19)
42 +H(H20)3 + E- -> H + 120 + H20 + H20 .2770 (19)
43 1120+ + H2 -> I30+ + H .3670 (15)
44 H30+ + E- -> H20 + H .7830 (18)
dH
i OH
0
2
? OH
o H o2o
SH O
1, H ) \I
S+(H2H 0)
HO2
I
S4 \ / 4
UL L u1
IO"L'" ]O" "" VI" \" \ \" 10" "
F.E-,[ lO ll TIM1E. 5EC.
Figure 3.1 Results from kinetics program concentrations
v.ersus time. Based on system presented in
Tables 3.2 and 3.3.
Development of the Diffusion Kinetic System
The DK system as formulated in this study is described by
a system of n nonlinear second order partial differentia].
equations (Green and Rio, 1983). The yields discussed in
the previous chapter are converted to yield rates by
multiplying them by a rate term (#elec./s). As previously
stated, the incident electron gives rise to spatial
distributions of H20+,OH+,O+ and H+ along with the
associated dissociated products and electrons.
Additionally, various excited states of H20 are produced.
The thermal electrons consist of the source electrons
plus those electrons which are produced in the ionizing
process and are degraded below .125 eV. The source
electrons, of course, have no companion ion as exhibited by
the following equation:
C m E
I J [ I DS(p,z.E,E )dE +
o o 3.3
4 E
I P (E)U(E.p,z.E )dE]pdpdx=l
i=1 i
The first term is the number of thermal electrons per
incident electron, the second term represents the total
number of ions created and the third term accounts for the
source electron. Since the system of differential equations
73
includes an equation for the electrons, we ueed to consider
what the ultimate fate of these source electrons will be.
Ultimately, they will encounter a ground such as the
containment vessel for the H20. It is possible to
hypothesize a reaction which eliminates these electrons,
such as electron attachment to H20 (Hanrahan, 1983).
Alternately we may simply iquore these electrons in the
deposition spectrum. For a 1000 eV electron this is the
same as removing one electron out of thirty. At most this
would slow up the ion chemistry slightly.
The physical system is described by the following set of
equations:
aci(p.z.t) = D V 2C (p.z.t)
1 1 1
at 3.4
-C k miCn(Pzt)C (P,2t) + Y i(p.zt)
n ,m
where the initial conditions are
Ci(p.z.O) = C o(p.Z)
Yi(p.z.t) = Ji(p.z) Rate (te-/sec)
I:L the steady state case, 'C/ I is set equal to zero. The
rctquircJ bouinary conditionlL. in this case are that the
concentratio:;s equal zeCLo at iL;finitv and their derivative's,
with respect to P, at P equal zero be equal to zero. The
th
symbol Y. is used to denote the yield of the i species.
1
The remaining symbols have been previously defined. The
species involved are listed in Table 3.1, the reactions
considered are listed in Table 3.2, the diffusion
coefficients are presented in Table 3.1 and representative
yields are presented in Figs. 2.18-2.23. The diffusion
coefficients for the molecules and atoms are based on
gaseous diffusion theory as presented by Hirschfelder et al.
(1954) and Monchick and Mason (1961) and experimental data
compiled by Marrero and Mason (1972), The coefficients are
calculated according to Chapman-Enskoq theory, implementing
the molecular parameters given in the paper by Kee et al.
(1983). The diffusion coefficient of the electron in H120 is
based on the total elastic cross section as presented in
chapter II. The complete set of parameters needed to
represent the yields in the diffusion kinetics program are
presented in Tables 2.11, 2.12 and 2.13.
In order to make the problem more tractable the variables
have been transformed to dimensionless coordinates. The
transformations are
p'=p/apo ,z=z/bz D =D /D0 and C '=C./ 35
oo 1 HO = C HO 3.5
2 2
The transformed equations and their associated initial and
boundary conditions are
aC' (p ,z' .t)=D'.V 2C'i (p'z t) + Y' (p' z')
1 1 1 1
at 3.6
+I k' C'(p'.z'.t)C'(p'.z'.t) = 0
mni n m
nm
2 2 2 2
V' = a2p V
o
2 2
ap0 b
HO
k' = .
nmi DH20 nmi
2
where Y. is given by
I
2 2
a p
Yl(p'.z') = 2 i(ap op.bz oz) 3.7
bC D
H20 H20
In the steady state case we make the additional
tran sforma tion
2 -1
x= -tan p1 3.8
2 -13.8
y= -tan z'
which reduces the domain of the problem to finite
dimensions. The equations can be restated in the following
form:
ac.
S + C k' .C' C' + Y'. = 0 3.9
at (x.y) 1 m nmi n m In
n~m
C! = C!(x,y.t) Y! = Y!(x.y.t)
1 1 1 1
where
2 4 2(ix 2 rX .a
) = -c ( ) [cos ( )] +
(x,y) 2 2 ax 2 aX
2cos (rx/2) a 4S 2 cos ) 2 y 8
rsin (vX/2) ax i 2 ay 2 ay
and
S = a2p /b z
O O
The steady state solution is valid for restricted values
of the yields. This is discussed in chapter IV. Solutions
for the steady state equations are presented in Fiqs.
3.2-3.14, The simulation was made at a temperature of
400 K. The density of the medium was set at 1.795(-7)
molecules per centimeter cubed and a, b and p were set to
1, 1, and 1.45 respectively. The computational time was
approximately two hours and involved ten iterations.
Converqence for ten iterations was nominal, however the cost
involved in extending these calculations was prohibitive.
In the future, modifications to the DIFKT proq'ram will
enable it to make use of the knowledge acquired in previous
calculations and reduce the computational cost.
77
OOOeV
10-5
0
o
1000eV
O -
10-6
CI
^ 10-5
z(cm)
Figure 3.2 Concentration profiles (p = 0) of the
electron distribution obtained from the
steady state solution of the diffusion
kinetics system. The solutions are
normalized by Eq. 3.5.
normalized by Eq. 3.5.
1.0
10 20 30
p(cm)
Figure 3.3
Concentration and yield rate profiles (z = 0)
(dashed line) of the electron. Yields
normalized to maximum value and
concentrations normalized to (0, 0) point
value.
10-12
0
S 10-13 000eV
C)
-10 0 10
z(cm)
Figure 3.4 Concentration profiles (p = 0) of the
H 0+ distribution obtained from the steady
siate solution of the diffusion kinetics
system. The solutions are normalized by Eq.
3.5.
80
a 1.0
S-\
>1 I \
0 \
0 .1
o 1000eV
\ 1000V
c 200eV
o
O \eV
.01
10 20 30
p(cm)
Figure 3.5 Concentration and yield rate profiles (z = 0)
(dashed line) of H20 .Yields normalized to
maximum value and concentrations normalized
to (0. 0) point value.
10-11
+(H
10-12
-10 0 10
z(cm)
Figure 3.6 Concentration profiles (p = 0) of the
H O+ H+(H 0)2 and H+(H20) distributions
o tainted fom the steady state solution of
the diffusion kinetics system. The solutions
are normalized by Eq. 3.5.
2200eV
N
S1000eV
0
S10-3
C,
0
-10 0 10
z(cm)
Figure 3.7 Concentration profiles (p = 0) of the
atomic hydrogen distribution obtained from
the steady state solution of the diffusion
kinetics system. The solutions are
normalized by Eq. 3.5.
83
0= 1
M
N
"0000eV
S200eV200eV
.f-
S\ \ Yields
.01
10 20 30
p(cm)
Figure 3.8 Concentration and yield rate profiles (z 0)
of atomic hydrogen. Yields normalized to
maximum value and concentrations normalized
to (0, O) point value.
84
10-E
S1000eV
0
N
S200eV
o 10-6
z(cm)
Figure 3.9 Concentration profiles (p = O) of the OH
distribution obtained from the steady state
solution of the diffusion kinetics system.
The solutions are normalized by Eq. 3.5.
0
U
-10 0 10
z(cm)
Figure 3.9 Concentration profiles (p = 0) of the OH
distribution obtained from the steady state
solution of the diffusion kinetics system.
The solutions are normalized by Eq. 3.5.
1.0
o
a)
S1000eV
0 \
u \ 00eV
c 1
o \20eeV
r \ \ 1000e-
0 \200eV
.01
10 20 30
p(Cm)
Figure 3.10 Concentration and yield rate profiles (z = 0)
(dashed line) of OH. Yields normalized to
maximum value and concentrations normalized
to (0, 0) point value.
W 200eV
0
Zo1000ev
0
-10 0 10
z(cm)
Figure 3.11 Concentration profiles (p = 0) of the
atomic oxygen distribution obtained from the
steady state solution of the diffusion
kinetics system. The solutions are
normalized by Eq. 3.5.
10-2
200eV
o ^ 1000eV
0
l 10-3
o
0
-10 0 10
z(cm)
Figure 3.12 Concentration profiles (p = 0) of the
molecular hydrogen distribution obtained from
the steady state solution of the diffusion
kinetics system. The solutions are
normalized by Eq. 3.5.
88
200eV
1000
0
o0
o 10-
0
U
-10 0 10
z(cm)
Figure 3.13 Concentration profiles (p = 0) of the
molecular oxygen distribution obtained from
the steady state solution of the diffusion
kinetics system. The solutions are
normalized by Eq. 3.5.
89
00eV
000eV
N
. 10-8
0
0
I S
1 10-9
z(cm)
Figure 3.14 Concentration profiles (p = O) of the
H O distribution obtained from the steady
sia~e solution of the diffusion kinetics
system. The solutions are normalized by Eq.
3.5.
3.5.
Compu ter Adaptation
The solution of the system of equations described in
chapter III is based on a finite element approach. The
basic program consist of the IMSL program TWODEPEP (1981).
Extensive input and output routines have been written to
apply this program to the current problem. A flow diagram
is presented in Fiq. 3.15. The program is listed in
appendix B and the main input routines can be tailored to
the time dependent or steady state case.
The program consists of five main input subroutines.
1. COEFF reads in scaling parameters, diffusion
constants and miscellaneous numbers.
2. REACIN reads in the species, reaction rate and yield
parameters and sets up the associated matrices which
the computational subroutines will access.
3. MAIN consist of a number of general equations which
are tailored to the system of interest.
4. FS is the main computational subroutine which
calculates the production and destruction terms in
the differential system. It was also adapted to the
Kinetic program described inl chapter III.
5. FSD calculates the partial derivatives of all
nonlinear terms with respect to the concentrations.
Output is taken care of by subroutine OUTPUT which produces
a grid of concentration values for all species at the
INPUT
PROCESS
TWODEPEP
Nonlinear Partial
Differential Equation
Solver
-T~
FC
Calculate
Coefficients
of 2nd order
derivatives
FSD
Calculate
partial derivatives
of kinetic terms
I
OUTPUT
&
Figure 3.15 Flow diagram for diffusion kinetic code.
TWODEPEP
Set up Equation
System
Boundary Conditions
COEFF
Input:
Diffusion coefficents
Normalization Constants
Yield Information
REACIN
Input: Reactions
Preprocess:
Set up kinetic terms
and their partial
derivatives
FS
Calculate kinetic
terms and yields
and coefficients of
1st order derivatives
__
__ |