Citation
A Study of the effects of diffusion and kinetics on the spatial distribution of products created by electron deposition /

Material Information

Title:
A Study of the effects of diffusion and kinetics on the spatial distribution of products created by electron deposition /
Creator:
Rio, Daniel Edward, 1951-
Publication Date:
Copyright Date:
1983
Language:
English
Physical Description:
ix, 185 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Chemicals ( jstor )
Elastic scattering ( jstor )
Electron energy ( jstor )
Electrons ( jstor )
Ions ( jstor )
Kinetics ( jstor )
Molecules ( jstor )
Physics ( jstor )
Radiation chemistry ( jstor )
Spectral energy distribution ( jstor )
Collisions (Nuclear physics) ( lcsh )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF
Nuclear Engineering Sciences Ph. D
Radiation chemistry ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1983.
Bibliography:
Bibliography: leaves 176-183.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Daniel Edward Rio.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
030550738 ( AlephBibNum )
11826643 ( OCLC )
ACQ0873 ( NOTIS )

Downloads

This item has the following downloads:


Full Text











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




Full Text
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 EN48B79UP_6N4JP4 INGEST_TIME 2017-07-17T20:36:44Z PACKAGE UF00098267_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES



PAGE 1

A STUDY OF THE EFFECTS OF DIFFUSION AND KINETICS Oil THE SPATIAL DISTRIBUTION OF PRODUCTS CREATED BY ELECTHON DEPOSITION BY DANIEL EDWAHD RIO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1983

PAGE 2

To my loving parents and to Chelsea for her first Ciiristaas

PAGE 3

ACKNOWLEDGEMENTS I would like to extend my qratitade to Dr. Genevieve Rosseier 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 viqilance and knowledqeable guidance. I also wish to thank the rest of my comiaittee members, Drs. Philip Achey, Lawrence Fitzqerald and James Keeslinq for the time they invested in reading this manuscript. They provided many helpful suggestions to improve it. Special mention and credit is due Dr. Robert Coldwell for his endless computer assistance. Paul SchippnicJc was very helpful in producing the contour graphs for the yields and calculating the diffusion coefficients, and 1 appreciate the many enlightening discussions I have had with him. I am also grateful to Dr. R. Singhal for his introduction to the computer methods associated with the Monte Carlo transport code. My parents Svelio and Jasmine Rio deserve exceptional recognition for ioining 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 goal would not have been reached. Xll

PAGE 4

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 iier 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 (NESDC) of the State University System of Florida. This manuscript was prepared at the NERDC using the UFTilESIS program. IV

PAGE 5

TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS ...,-.....--.-•»-••' n^ LIST OF TABLES vii ABSTRACT ..,,.....*..-..«'» ^ij-i CHAPTER I. INTRODUCTION ..,...,...-..* ^ *-•••• • 1 Overview .«....*-.--.. '' History and Motivation ..^.....^'••» 6 Description ........-....-••-•• 9 II. THE PRODUCTION Of INITIAL YIELDS 12 The Monte Carlo Method .,...*..**..'1-^ Basic Molecular Cross Sections 13 Elastic Cross Sections .,.,.*.*... 13 Inelastic Cross Sections . .... 20 Multiple Elastic Scatterinq .......... 29 The Yield Spectrum and Yields 44 III. DIfFDSION AND KINETICS ........*....'• o3 General 63 Kinetics ...,.....,....»••••» 66 Development of the Diffusion Kinetic System . 72 Computer Adaptation .,.._.... .....»90 IV. RESDLTS 93 Discussion ...*...••-' ^3 Computational Simulation .......*.. 93 Monte Carlo and Yields ^^5 Diffusion Kinetics ....».*...»• 115 Limitations .... ...... 122

PAGE 6

V. CONCLUSIONS 124 APPENDIX A. iSONTE CARLO PROGfiAM -....-..-<. 127 B. DIFFUSION KINETIC PBOGRAM ............ 133 BIBLIOGRAPHY 176 BIOGRAPHICAL SKETCH .................. 184 VI

PAGE 7

TABLE

PAGE 8

Abstract of Dissertation Presented to the Graduate School of the Dniversity of Plorida in Partial Fulfillment of the Bequicements for the Degree of Doctor of Philosophy A STDDY ON THE EFFECTS OF DIFFDSI08 AND KIWETICS ON THE SPATIAL DISTSIBOTION OF PBODOCTS PBODUCED BY ELECTRON DEPOSITION By Daniel Edward Eio December 1983 Chairman: Alex E. S. Green Hajor Department: Nuclear Enqineerinq Sciences We have updated our analytic representations of fundamental molecular cross sections for H2C vapor, drawinq 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 deqraded in H2O. 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 Vlll

PAGE 9

turn are used to describe tke 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 beaa of electrons. Additional results related to the yield spectruia such as eV/ion pair and deposition contours are also presentedThe results are intended to facilitate an understandinq of the effects of radiation in bioloqical systems. IX

PAGE 10

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 stady of radiation effects on bioloqical systems has in the past looked toward chemistry to supply the fundamental concepts necessary to understand this field. Radiation chemistry nas 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 from the study of macroscopic concepts suca as exposure, dose and LET (linear enerqy transfer) to the study of individual atomic and laolecular processes which require the formalism of quantum mechanics. Durinq the past few years the proqress in these fields has increased to the point where it is possible to beqin to understand the bioloqical effects of radiation in terms of individual atomic events and their aqqreqate behavior! This is due in large part to the development of high speed computers, experimental studies of basic atonic and

PAGE 11

2 molecular cross sections and picosscond studies of chemical kinotics. While the incident radiation may be qamma rays, protons or charged particles, ultimately it is electrons which dissipate much of the incident enerqy. Radiation deposition is split into two broad categories: (1) direct effects-in which the incident radiation interacts directly with the bioloqically sensitive sites and (2) indirect effects-in which the incidejit radiation interacts with the laaior component of the system, for example H which in turn forms free radicals which interact with the important biocheaicals (see fiq. 11) . Referinq 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 knowledge 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, ccllisional 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 C+, 0H+, H+, OH, H and H 0* (excited states). Part oi the radiate enerqy has oi course been absorbed directly by the critical

PAGE 12

DIRECT EFFECTS INDIRECT EFFECTS H m Biochemicals Ionized

PAGE 13

4 biochemicals. We move now from the physical statje to the chemical physical staqe. Furthermore, since the bulk of the dauiaqe is caused by indirect effects additional chamical and physical properties are required. A larqe percentaqe of the information needed in this study is drawn from literature in the combustion field (Kee et al, 19 83: Sestbrook, 198 1) since hydroqen-oxyqen flames form a maior area of study in this field and many of the chemical species present in this type of flame are also present in the irradiated H,^0 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, knowlodqe 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 biomoleculos. This information is found in the radiation chemistry literature (Scholes, et al. , 1969; i-'yers, 1974) but often does not take into account bioloqicai structure and function which will of course lead to modifications of the immediate chemical effect. For exaffi:ple, it is well known that DNA will repair itself if damaqed. Furthermore the ability to repair itself is also a

PAGE 14

5 function of the amcunt and or type of damaqe sustained by the molecule. While it is not possible at this time to study a realistic biological system iu this manner, several studies usinq this method have been applied in varyinq daqrees to study the radiolysis of H^O in both its liquid and qaseous state. The ma-jor proqrams 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 liaqee, 1930) and Battelle Northwest Laboratories (Wilson et al. , 1978). Much of the work by these qroups depends heavily on the pioneerinq work done on fundamental collisioaal cross sections of electrons with H 2O at the University of Florida (OF) (Green et al. , 1971; Olivero et al. , 1972; Kutcher and Green, 1976). Subsequently the thrust of the research at U7 has involved usinq similar cross section work to study the collisional history of electrons in various qases and representinq this information in analytic form (Green, Garvey and Jackman, 1977; Jackman, 1978; Green and Siuqhal, 1979 (GS) ; Sinqhal and Green, 1981; Sinqhal et al. , 1983). Kany of the techniques required for these calculations oriqinated in the field of atmospheric physics. With this thesis we brinq toqether the work on collisional cross sections and

PAGE 15

6 transport calculations at UF to iaply the initial yields of ionized, excited and dissociated H 2O in tiie qaseous state. These are then used as input, alonq with chemical kinetic rate constants and diffusion coefficients to calculate the spatial distribution of chemical species created by a iaonoenerqetic beam of electrons. Hi s to r Y and , Mo tiyati on The field of radiation chemistry originated in the 18th century with the study of chemical cJianqes produced by electrical discharqes in qases. Many of the reaction mechanisms to explain the experimental results studied in the iaterveninq years still remain opaque (Anderson, 1968) . The establishment of radiation chemistry as an exact science is basically due to the work of Lind (192 1, 1930) in the U.S.A. and Muud (1935) in Belqium. Much 01 this research used alpha particles as a radiation source and many of the quantitative measurements are still qood today. For example, it vjas observed that the radiolysis of pure water vapor produced almost no siqnificant decomposition unless chemically reactive scavengers were introduced 'into the system. The initial interpretation of the observed chemistry was centered on the assumption that the r adioloqically produced ions were surrounded by "clusters" of neutral molecules which would ultiiaately recoiubiae either with themselves, if they were of opposite charqe, or with

PAGE 16

7 free electrons, Dpon doinq so the enerqy released would be shared by the adjacent neutral molecules which would instigate further reactions. as early as 1909 Debierne (1909) proposed that free radicals were responsible for the observed chemistry. During the 1930 's two papers by Eyrinq, Hirshf elder and Taylor (1936a, 1936b) indicated that since the "W value" (average energy to form an ion pair) is much greater than the lowest ionization energy of a laolecuie that the excess energy must be channeled directly into excitation. In addition thay 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 tnat both ions and excited molecules produced directly by the incident radiation are important, Recent research by Anderson, Knight and Winter (1964,1966a, 1966b) has clarified the neutralization process in H O. These reactions produce free radicals which 2 continue to build up until their concentrations are sufficiently high tnat reverse cheaiical reactions compete with radical production and small steady state concentrations of H , O , and ii are observed. 2 2 2 2 The choice of using fl vapor instead of liguid as the 2 first test case for study reguires a coaraent. While the radiolysis of H liguid is obviously of more biological

PAGE 17

8 significance tlian H 2O vapor, physical processes in the liquid are not well known. For example, the question of defininq ionization in condensed matter (Douthat 1983) does not need to be addressed. In coaparison, the collisional cross sections for the qas phase are known to a Quch qreater deqree of accuracy. Often many of the liquid cross sections are extrapolated from qaseous models (Kutcher and Green, 1976; Hamin et al. , 1978, 1983) and questions of their validity have been raised by luoXuti (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 620 are comparable (Thoicas, 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 H2 C vapor represents a system in wnich the sensitivity to cross section inputs is most easily studied, comparisons witii experimental data are inherently simpler and the calculations accomplished for 'the vapor phase can be modified for tae liquid phase once the appropriate input data are obtained.

PAGE 18

9 £jgscri£jtioii The purposes of this dissertation are to 1. Describe multiple elastic scattering of low enerqy electrons in H^O vapor and characterize the spatial distribution of the resultant distribution of electrons. 2. Extend the work on initial yields due to incident electrons in H_0 vapor as reflected by the concept of a yield spectrum (see caapter II) 3. Use the above information to study the subsequent chemical reactions which occur durinq the diffusion of these chemical species in H 2O vapor. This will be developed in three maior seqments. In chapter II, using the earlier work by Kutcher and Green (1976) multiple elastic scattering distributions for electrons undergoing only elastic collisions in H _0 vapor are calculated. These results along with a compilation of cross sections based on empirical models developed by Green et al. (1971), Olivero et al. (1972) and Zaider et al. (1982) are used to calculate spatial yield spectra (Green and Sinqhal, 1979). These are then used to calculate spatial yields for all the important "initial products" and a spatial distribution for subexcit ation electrons. Analytic representations of all these quantities are givenIn chapter III the above calculated quantities are followed as they diffuse and react. This is accomplished by solving a

PAGE 19

10 system of nonlinear partial differential equations. This approach is based on the "classical" diffusion model as developed for the radiolysis of H^O which oriqinated with the radical theory of Weiss (1944) and expandinq spur kinetics as hypothesized by Lea (1946, 1947) and was ultimately formulated by Safisuel 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 Magee and Chatteriee (1930) and Chatteriee and Maqee (1980), Essentially, it consists of representinq the "initial" (in this instance those chemical species which are seen approximatly 10 (-13) seconds after the radiation impact) yields of radicals by a few estimated parameters and followinq their evolution in time, assuming spatial variation in only one dimension, radial in either spherical or cylindrical coordinates, depending on the L2T of the radiation. The calculations are then compared against experimental G-values(# pcoduced/100 eV) of, for example. In this study a maior step is taken to elucidate the mechanisms involved in this transition reqion between the physical and early cnemical physical stages. The problem will be solved in cylindrical coordinates, assuming cylindrical symmetry and in the steady state limit. Solving the temporal problem usiuq this technique would have been

PAGE 20

11 excessively expensive. The maior emphasis of this cofflputatioa is to establish initial spatial yields for the chemical species produced by the incident radiation and to tie the physics and chemistry toqether by taStinq into account the ion chemistry as well as radical recomaination. The calculations are performed usinq the IMSL finite element partial differential equation solver TWODEPEP (1981). The systeia 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.

PAGE 21

CHAPTER II THE PRODUCTION Of INITIAL YIELDS Tae_Mgn t e_Ca r lo„ Me th gd The Honte Carlo approach to solvinq the transport equation is a stochastic method of imitating the actual path of an electron as it is deqraded in enerqy by its interactions with the mediutn, Durinq the siaulation the moveaent 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 nature of the degradation process. Depending on the incident energy of the electron the number of incident electrons allowed to degrade is varied in order to build up a statistical profile of the actual solution of the transport equation. The statistical error resulting from the Monte Carlo calculation is discussed in the thesis by Jackiaan (1973) where a detailed description of an earlier version of the program used in this calculation may be found. A compact overview of this particular study may be found in Jackman and Green (1979), In brief, the information necessary to carry out this calculation is a detailed collection of atomic or molecular cross sections, the attributes of the incident electrons, in 12

PAGE 22

13 particular their energy and direction and the density of the mediuia 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 nay 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 tag delineating whether it is a primary or later generation electron. Extensive Eodif ica tions of the Monte Carlo proqraa have been jaade while applying it to K O. Those are documented in appendix A. Basic_Molecu,lar_Cross__Sectioas lis stic.jCross_ Sect ions The production of a spatial yield spectrum requires detailed comprehensive data on the elastic scattering of electrons from H 0. Unfortunately, such data are sparse or 2 nonexistent. To overcome this problem we rely on the extensive work done by Porter and Jump (1978), which was recently updated at Los Alamos National Labs (ZBN) (1962) for H 0, to characterize elastic cross sections with 2 convenient analytic functions.

PAGE 23

14 For electron scattering in ii ^C we use the standard Rutherford representation for differential elastic cross sections modified by a screeninq parameter for electron energies above 200 eV (Jackman, 1978). The functional form is d0(e.E) = Z^e^ 2.1 m V (i+2n-cos e) ^ where r, (E,Z),tiie screeninq parameter, is a modified formulation proposed by Grosswendt and Waibel (1973) and used by Zaider et al. (1982) for H2O. It is qiven oy n = n^k/[E(E/mc^+ 2)] n^ = a + plnE 2.2 2/3 k = kj^mz"^ where Z is the atomic number of the medium and e,p,v and theta are the charge, moiaentua, velocity and angular deflection of the electron. The parameters are listed in TaDle 2. 1. Below 200 eV the empirical function proposed 'by Porter and Jump (1978) dc(O.E)oc 1 + P(E) , dn 1 + 2\(E) cos e [1 + 26(E) + cos 6]" 2.3

PAGE 24

15 TABLE 2. 1 Differential and total elastic scattering parameters. Elastic Scattering ( < 200 eV ) PAX 7.51525 2.9612 -1.7013 •0.419122 -0.26375 -1.48284 0.72017 (-2) 0.4307 (-2) 0.6331 •0.4645 (-4) -0.26895 (-4) -0.10911 -0.10289 (-6) 0.583505(-7) 0.8358 (-2) -0.2388 (-3) -3.32517 0.10996 -0.45255 1-2) 0.58372 (-4) 0.58372 (-4) -0-24659 (-6) 0.24775 (-1) -0.296264 (-4) -0. 120655(-b) Eutlierford Scattering ( > 200 eV ) = 1.64 P = .0825 ^x = .000017 Total Inelastic Cross Section Eq = 1.798 Fj^ = 72.53 ?2 = 11.093 X = -.77 Gi = .4734 G2 = 19.9223 U = .00195 Ej^ .172 E2 = 13.93 V = 150.

PAGE 25

16 is used wiiece p , 5 and x ^^^ parameters fitted by Brenner (1982) to experimental data. These functional forms are given in Table 2. 2, The experimental data used in constructinq this model are those oy Seng (1975) (0.35-10 eV, 20-108 degrees), Traiiaar et al. (1973) (15-53 eV, 10-90 deqrees) , 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 along with experimental data are presented in Figure 2.1, For the total elastic cross section we again use a function suggested by Porter and Jump (1978) and fitted to experimental data by ZBK (1982) „x 2 F gI n(n+l)[V +E ] ^^^ ^^-^n^ "^ n where n=U/3 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) (U-36eV). A plot of the total elastic cross section is presented in Figure 2.2w

PAGE 26

17 TABLE 2.2 Differential elastic scattering functions. ill Pi-1^'"'' P(E)=e 6(E)=e « i-l ill'^i-l^ X(E)=:e .35eV
PAGE 27

EICRGY-0.60 -g^ 5J3m mTm 9o7oo lio.oo ito.oo iB>-:o IHE Tfl CO. 00 90.00 THEIR Ul.OO ISO. 00 60.00 90.00 THEIR lU.DO ISO. 00 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) .

PAGE 28

19

PAGE 29

20 Inelastic., Cross, Sect ions He beqin this section with a description of the differential scatterinq cross sections for ionization collisions. The theoretical work on the ionization continuum by Massey and Mohr (1933) and Kott and Massey (1965) provide a basic framework for the calculation of simple atouic and laolecular cross sections for ionizationGreen and Dutta (1967) have developed a practical method of determining the doubly differential cross sections d o/dS^dT (T is the enerqy of the secondary electron) for complex molecules, such as H 0, from sparse experimental data using 2 the formalism of generalized oscillator strengths (GOS) as described by Kott and Ilassey (1965). A specific analytic representation of the GOS for HO is given by Green et al. 2 (1971) and Olivero et al. (1972) with a correction to the optical oscillator strength part of the GOS being made by Kuccher and Gre^^n (1976). This representation of the angular dependence for ionization scattering is perhaps the most complete available at present; however because of its complexity it was felt that a less ambitious approach for this inelastic scatterinq type should bo made. Following the work of ZBW (1932) we will generate the scattering angles according to the prescription of Grosswendt and waibel (1978). The functions are given in Table 2.3,

PAGE 30

21 TABLE 2.3 Ionization scattering algorithms. Inelastic Scatterinq ( > 100 eV ) cos e^ = •l-T/E(l-E(l-T/E))/C cos 02 = /1-(1-T/E)/(1-T/C) T>200eV cos 92 = .707R T<200eV. 00362 = -1+2R T50eV = -1+2R T<50eV

PAGE 31

22 At low energies (<100 eV) these functions are rather arbitrary; fortunately at these enerqies 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)r^(E)/([T-T (E)] ^+r^(E)} ° 2.5 where E and T are the incident and secondary electron energy. The adiustable parameters r and Tq have been taken from Olivero et al. (1972). The function was then integrated over T to give the total cross section and the parameters Kw K„, J., and Jq were adjusted to reproduce experimental data. Four ionization cross sections were considered. The data for the 1D1+3A1 cross sections were taken from Kark and Egger (1976) and for the 1B2+2A1 cross sections from Schutten et ai. (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.

PAGE 32

23 TABLE 2.^ The total and secondary electron cross section. Ion. States Threshold K K A B 1B1 H20+ 12.620 2.590 173.1 3A1 H20+ 14.750 2.115 176-4 1B2 OH+,H+,0+ 18.510 2.069 116.2 2A1 H+,0+ 32.400 0.884 241-0 J

PAGE 33

24 ZUOT (axnoaxoui/ ^^) (3) '^

PAGE 34

25 > (U (axnoaioui/ uio) (3) ZUOT^

PAGE 35

26 q F. W.°'i "^3 W. ^ j v.. o.(E)= -J-L [l-(-=r^) ] C-^ ) + -^ 2.6 where tue resonance term is used only for the vibratioual 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 laodif icat ion for the vibrational states is due to ZBW (1982). Tae parameterization for all excitation states is described by Z3» (19^2) which draws on work by Olivero et al. (1972), Kutcher and Green (1976), and data by Beenakker et al. (1974), i^ohlaann and de Heer (1979), and Fuiita 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 Stolarski (1972) and Jackiaaii et al. (1977). Finally we calculate the total inelastic cross section by suraainq the aoove functions and above 20 eV we fit this cross section with the function ^o^ E P '^tOt^^> = ^1" t-'^W-^^ -^3 2.7 where F = 2.9b25, :if = 5.5343, °' = . 02239, and P=1. 7737. Below 20 eV we enter the total cross section discretely. The fit above 20 eV is virtually indistinquishable from the

PAGE 36

27 lABLS 2.5 Paraiaeters for the excited states of H20. Exc. States Threshold a P (A1) V2

PAGE 37

28 Vib .46 o (U rH O g \ E u t) vib 10 -16 10 •17 10 -IS 10 -19 10 -20 10 -1 Vib .899 10' 10 E(eV) Triplet 4.5 10' 10Figure 2.5 Total excitation cross sections versus incident energy.

PAGE 38

29 calculated value. Tte total elastic and inelastic cross sections are graphed in Fiq^ 2,2, Multiple Elastic___Scatterinq Below 50 eV the elastic cross section is at least twice as iarqe as the inelastic cross section. This ratio continues to increase as the electron enerqy decreases approachinq the threshold for the lowest excitational state of H2O. The electrons will then slowly approach thermal equilibrium with the H2O molecules. It would be uneconomical to follow these electrons collision by collision durinq this staqe of their degradation; therefore a multiple elastic scattering distribution (MESD) is used below 20 eV, MESDs have been used foe condensed histories calculations, for example, by Berqer (1963), Jackman (1978) and Sinqhal aud 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 uiean free pathlenqths and characterized the results using cumulative functions for the radial, lonqitudinal 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 pathlenqths, Usinq the computer code

PAGE 39

30 developed by Kutciier ve have essentially solved the one velocity Doltzmann equation 3f ~(R.Z.U.A4..S) + Q(.e)«7f (R.z.u.A(t>.s) 2.8 = [f (R.Z.U.A4..S) f (R.z.u' .A =ij> ~ <^ and s are defined in Fiq, A. 2 and oaeqa is a unit vector alonq the direction of the velocity^ In particular we have calculated the distribution functions for E (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 enerqies from .1 eV to 50 eV. The functional forms used in fittinq these distributions are presented in Table 2.5Eepresentative qraphs or the distributions are shown in Fiqs. 2.6-2. lU. The solutions qenerated by the Monte Carlo simulation are represented as discrete points. We see that the analytic functions represent these solutions quite accurately, altiiouqh for fewer than four nean free pathlecqths, the fits systematically undershoot or overshoot the qenerated solutions. This is of minor concerii, since these results are usually applied only to those cases in which the number of elastic collisions between each inelastic collision is much qreater than tour.

PAGE 40

31 TABLE 2.6 Cumulative distribution functions for elastic scatter inq. F(R.S) = 1-e"*^ V(s) = 2[1-Ae^^''\^ ] MS) = (-^) . i-^) F(z.s) = {l-e~^^[F(0)~^^''-l])-'' D -(s/s ) ^ v(s) = 1-A e V -(s/s ) F(0) = Fptl e "^ ] u(s) = (H + s^)/s'^ F(X.s) = (^ X)"X^ -s/s ir(s) = Le -s/s e ( s ) = Me -(s/s ) t(s) = 1-e X = (1-cos e)/2

PAGE 41

TABLE 2.7 Parameter values for radial distribution. 32 E(eV) .2 .5 B 5.24 5.37 4.74 Y 1.00 1. 10 .740 >728 .739 2.79 2.87 2.73 10.0 7.09 1.G9 20.0 6.88 1.11 50.0 9.00 1.07 W TToo" 1.95 2.06 1.0 5.94 1.00 .641 2.94 2-21 2.0 7.00 1.00 .596 3.09 2.28 5.0 6.37 1-02 .664 2.36 2.07 .623 3-05 2.07 .645 3.09 2.00 TToo" . 100 .100 w .734 .755 .755 -100 .790 .100 .786 .100 ,760 .63 3.73 2.32 -100 .100 . 100 -753 .731 .7 99

PAGE 42

33 TABLE 2.3 Parameter values for loiiqitudinal distributionE(eV) .2 -5 D H V V V '1 75 "173 4 ^7 50 ITT 5"50 795 7"~T ~ 5 o" 9.90 1.50 -.252 .600 1.00 1.80 9.20 1.40 .0085 .650 1.05 2-30 ^F B. F "F "F r435~~9722"7600' .435 11.5 . 640 .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 2.30 .435 16.1 .725 5.0 8.00 1-87 .400 .740 1-17 2-60 -440 12-3 -704 10.0 7-50 1.94 .451 .800 1.22 2.50 .428 10.2 .700 20.0 8.00 1.75 .500 .900 1.29 2.60 .435 9.10 .750 50.0 9.00 1.72 .475 .940 1.39 2.60 -425 14.4 .750

PAGE 43

34 TABLE 2.9 Parameter values for Polar Anqles distribution. E(eV) 1 .2 .5 '73 oo" .700 .580 L ""550" ,800 1. 10 c 'TasT >600 .840 M "IToo" 1.60 1.40 1 . oT 1.46 1.76 N T6 50' 1.07 1-25 1.0 2.0 5.0 . 523 .459 .500 1.28 1.26 1.25 1.13 1-13 2.58 1.27 1.50 1.10 2.84 1.08 1.30 1.30 1.98 1.23 10.0 20.0 5 0. .540 .770 1.40 1.20 .973 .550 1.10 . 315 .550 1.40 1.68 2.00 1.85 1.60 1.20 1.20 1. 19

PAGE 44

35 T3 • >, 0) en 42 U •T3 W 0) ^ -P -P 4-1 rO U CO &4J CD a 0) -H 4J iH e S-l o m C O -M M-l 4J Cl,4J to d o •iH s-l fo > o 13 •rH !-l 4J cn •H •H rO > •iH 4J to rH g u to o -H +J cn 4J to 4-1 > CO 0) CN a; • 4=1 4J >i Cn4J 0) tn cu u U) 4J G •H o p 0) in CN (U M -rH (H) J

PAGE 45

36 -a u2 >i

PAGE 46

37

PAGE 47

38 >

PAGE 48

39 fN

PAGE 49

13 U 0) -P +J rC • O W 0) 40

PAGE 50

41 P4 X = (l-COS9)/2 Figure 2.12 Cumulative polar angle distributions for multiply scattered electrons with energy .2eV for various pathlengths, S.

PAGE 51

4 2 1.0 8 6 r .4 2 _ .2 .4 .6 .8 X = (.i-cose)./2 Figure 2.13 Cumulative polar angle distributions for multiply scattered electrons with energy 2 . OeV for various pathlengths, S. 1.0

PAGE 52

43 1.0 .8 6 Cm 4 2 _ .4 .6 X (i-cose)/2 1.0 Figure 2.14 Cumulative polar angle distributions for multiply scattered electrons with energy 20eV for various pathlengths, S.

PAGE 53

44 Xkj§_lisi:^_S£ectrum_aiid_Yieids Tiie concept of a yield spectrum was introduced by Green, Garvey and Jackinan (1977) (GGJ) . This concept traces its evolution from the work on the equilibrium £lux of Spencer and Fano (1954)^ Peterson (1969) and Jura (197 1). The yield spectrum has properties whicii are much simpler than tie equilibrium electron distribution. Its shape can be described by a continuously varyinq function of incident and spectral electron enerqy and as demonstrated by GGJ (1977) this functional dependence is similar for many atoias and molecules. As introduced by GGJ (1977) the yield spectrum is defined by U(E.Eq) = N(E.E^)/AE = U^(E.E^)H(E^-E-Ejj^)+«(E^-E) 2.9 where N is the total number of inelastic collisions which occur in the enerqy interval A3 centered at E and E is the enerqy of the incident electron, II is the Heavisida function with E^ set at the lowest threshold value of all the inelastic interactions considered and * is the Dirac delta function which allows the second term to represent the source contribution to the yield spectrua. This definition has also been qeneralizcd to four dimensions (GS) . The functional form is

PAGE 54

45 U(E.p.X.E ) = N(E.p,z)/AE&sAz = U^(E.p.z.Eq)H(E^-E-Ejjj)+D(p.z.E^)«{E^-E) 2.10 where and N(E, p, z) is the total nuniher of inelastic collisions which occur iii the volume element As, z centered at p, z and the enerqy interval Ae centered at EAnalytic expressions for U and D are a U^(E.p.Z.E^) = I A.(E^)G.(p.z.E^Xi(E.E JD^^ DCp.Z.E^) = A4(E^)G4(p.Z.E^)D^^ 2.11 waere A. (E^) = {D^ j" r G. (p,z.E )pdpdz) Xi(E.E^) = Ci-^ X.E,e \^^i and D is the density of the mediua with units of qrams per o centimeter cubed. Note, by dcfininq the normalization constant A with D"^ explicitly displayed, we have o essentially taken all density dependence out of it since the inteqrated quantity is proportional to the inverse density cubed. The paraaeterization for li^O is qiven in Tables 2.10

PAGE 55

46 and 2.11Representative qcaphs of the two and four dimensional yield spectra are given 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 enerqy. These electrons undergo elastic collisions, but it is a well known result froia classical mechanics that they can exchange kinetic enerqy with the B JJ molecule, losing enerqy on the average, until they reach thermal equilibrium. Because the average enerqy lost to the H C molecule by tne electrons approaches zero as the electron approaches equilibrium, it is not practical to follow the electron to equilibrium iu the MC simulation. Therefore a cutoff or E = .125 eV was chosen; that is, once the electrons enerqy fell below this energy its position was recorded and it was no longer followed. A new guantity is defined, called tiie 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 Singhal et al. (1983) which represents the number of electrons with energy E after their last inelastic collision. The deposition spectrum can be analo^ou3ly represented by a function similar to that

PAGE 56

47 lABLE 2. 10 Parameters for two dimensional yield spectruia. U(E.E ) = I x: (E.E^) i = l ° Xi(E.E^) = Ci. \^e"''°'^i E. = E^/1000 Iv O ^i ^i ? 1 1 -0339 .8036 40.3 2 0.0 99-85 4,61 3 0.0 830.6 .074

PAGE 57

48 TABLE 2. 11 Parameters for four dimensional yield spectrum. G^(p.z.E^) = e -B .D /p \(> .(z-z . ) ^ ' 1 o ^ 1 oi p.(E^) = A./{E^-.B.)ln(E^-HC.) z^.(E) = (E^.B.)/A.ln-\E^.C.) 1 2 3 'T 2 3 4 A 21-i;3 (7) 20. 32(7) 13.30(7) 91.92(7) Il573l7r 619.6 (7) 34.07(7) 88.70 (7) B -75.90 -25.80 -5-224 87.39 "looTs 465.1 -52.95 355-6 C -92.63 -90-97 6. 005 'l0574 8331 8.82 -66.04 1 2 3 4 T7091' .5902 . 2835 .700

PAGE 58

49 > W 4-1 U (U Cu CO 10"' __ Deposxtxon Spectrum E„=1000eV Yield Spectrum Figure 2.15 E(eV) Two dimensional yield spectrum and electron deposition spectrum as function of spectral energy E at various incident energies, £„ . The points represent the distribution values obtained directly from the Monte Carlo Transport code.

PAGE 59

50 e > 0) >1 H
PAGE 60

51 10 19 10 18 u I m B I > 17 0)10 +> m CD \ u -P o 0) w Id •^ 10 ioi6 h 15 10 14 10 -1 10 10 E (eV) 10' 10Figure 2.17 Four dimensional yield spectrum for electron with incident energy 200eV versus spectral energy E at the spatial points (p, Z) .

PAGE 61

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.E^) = N(E^)A(E^)G(p.Z.E^) where the incident enerqy dependence is qiven by 2.12 GCp.z.Eq) = e 2.13 and the parameters are qiven in Table 2.12. Normalization factors A for yield and deposition spectra are qiveu in Table 2.13. Tue function is plotted for various isoelectronic densities in Piq. 2.18 and Fiq. 2.19. Havinq obtained the yield spectrum we can then calculate the specific state yields {#iflol. of H in a particular state per cubic centimeter due to all collisions for one electron) as a function of position by inteqratinq the yield spectrum multiplied by tae probability of a collision producinq a molecule in the i^^ state over E troai the threshold of that particular state to the incident electron enerqy. Analytically, the yields are E ° P^(E)U(E.p.z.^ )dE 2.14 i J.(p.z.E^) = I^° p.(E)U(E.p.z.^ ).

PAGE 62

53 TABLE 2. 12 Parameters for electron deposition spectrum. -PD /p^+i{Z-2^) ^ G(p.z.E^) = e ° ° P(E^) = A/(E^+B)ln(E +C) -(E /E )" 6(E^) = 1-Ae ° ^ ^o^^o^ = (E^+B)/A ln-^(E^+C) A=19.30(7) B=-22.77 C=-91.70 ^o A=102,8(7) B=213.7 C=-90.53 A-,3809(7) E=5n.9 N=a,98 Ene rqy (eV)

PAGE 63

54 TABLE 2. 13 Noriaalization constants, AIncident lield Spectrum Electron Enerqy (eV) 12 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(13) 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(15)

PAGE 64

55 OQ"QI m

PAGE 65

56 OQ-Q

PAGE 66

57 where J • is the yield for the i state, H ^ is the threshold for the i state and p^ is tne probability of interaction for this state, qiven by p^(E) = <^i(E)/o^Q^(E) 2.15 Pert oriainc,' the integration over all space as well qives 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. Reqardless, it is then possible to calculate W, the energy loss per ion pair produced. We have calculated a value of W(9V/ip) for H O which is comparable to experimental rasuits. The results are presented in caapter IV. A few representative isoyield contours are presented in Fiqs. 2.20 -2.23. Ihese will te 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 usinq the yielu spectrum by perioriuinq the followinq sum Dose{p.z.E^)a I W. J. (p.z.E^) 2.16 over all states.

PAGE 67

53 TABLE 2-14 States and associated products. states 7aT7v2 {^^) VI (Bi) V3 (Bl) VI (BI) V3 TRIPLET TRIPLET DIS^CONT DIS.CONT DIFFUSE BANDS H* LYHAN ALPHA H* BALtlER ALPHA OK* 3064A SERIES A SERIES B SERIES C SERIES D products

PAGE 68

59 E U CNl 00 I-l N 0) u P •H O M 0) IM 0) CA d o u «J u 0) I-l o o N u o + «3 O u p o 4-) c o o O U m o o CM 0) M P C7> Bu

PAGE 69

60 QQ-QI E O 00 P O M a c o u o r-t 0) c 0) TJ •H O c > 0) o o o o + w O a> CM :3 EC i-< «3 >-< > O P w o -o *-> ^ O U to o H CN! CM CJ u P

PAGE 70

61 CO H (M 0) U CP o u c o u o 0) c a •H V c > a o o CM u o X o CO l-l (0 > Ou'Qi uc u i-l o o c o >^ O U w o u p

PAGE 71

62 OO-Ql E U o U c o u V l-l 0) a d) -o o c > o o o u o •4-1 X o w 0) D I— I 03 > u 3 o C! o V u o (UiO) u o IM CO w • .-I >i 3 O Cr> 1— I pL, «*1 N N 0) U 3

PAGE 72

CHAPTER III DIFFDSION AND KINETICS General After calculatinq the yields of tiie various excited and ionized states of H2O/ which occur in less than 10 (-13) seconds, we are interested in studyinq 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 H2O2 which are created by the interaction of these initial products. In order to accomplish this we have developed a proqraai called DIFKT which can calculate time dependent and steady state spatial concentration profiles of all the initial species and their products. Details of the proqraui are addressed in chapter III. 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 (194b). Samuel and Kaqee (1953) extended this work, applyinq several mathematical approximations to laake the problea analytically tractable. Waqee (1955) considered the effects of dose rate for this aodfelThat is the case when the enerqy is absorbed at a 63

PAGE 73

en sufficiently aiyh rate that we cannot treat the problen as the sum of the effects of individual particles. Kuppermann (196 1) 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 ac.(r.t)=D.72c.(r.t)-Z C^(r . t)C^(r . t)k^^. 3.1 nm 3t C^Cr.O) = C^^Cr) where D., k. and Care respectively the diffusion i' nmi 1 th ^ . th constant of the i species, tne rats constants tor the i th species and the concentration or the x species as a function of time and space (Kuppermann, 1959, 1961 ; Kuppermann and Bolford, 1962a, 1962b) , The initial species present and their spatial concentration were considered somewhat arbitrary. Recently this approach has been extt^nded by Ilaqee and Chatterjee (1978,1980) and Chatteriee and [laqee (1980). They essentially start their calculations after ail ions and excited states have reacted and formed neutral qrouud state molecules. The approximations involved in these calculations are numerous; the most questionable are those iavolvinq the spatial distribution of the primary neutral species. They are adiusted to qive the best results in reproducing experimental data for differential and

PAGE 74

65 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 using Monte Carlo methods was initiated in the late 70 's at Oak Ridge National Labs (Hamm et al., 1978; Ritchie et al. , 1978) and Los Alamos National Labs (ZDW, 1982; Zaider and Brenner, 1983 and Brenner and Zaider, 1983). This approach was recently connected to the work of Magee and Caatterjee (1980) (Wrignt et al. , in press; Wriqht et al. 1983). Initial prediction of experimental G values are reasonable. While a tine dependent solution for one electron track as presented is very useful in predicting experimental results, a continuous beam (or at least pulsed) of electrons is normally encountered. The raaior part of the present analysis has concentrated on looking at a continuous beam of electrons. The transient behavior of this system has not been addressed in the diffusion kinetic calculation.

PAGE 75

66 Kinetics In solving a system which involves both diffusion and kinetics it is often instructive to solve the problem iqnorinq diffusion. In this section we present the chemical system which exists iiaiaediately after the incident electrons have deposited their energy. This system exists approximately 10 (-13) seconds after an electron is iuiected into the medium. Thus we see that the physical part of the deposition process finishes very quickly, A complete description of these events is presented iii chapter I. The system of nonlinear first order differential equations which describes the chemistry is ^^i^^^ = I k .C (t)C (t) , 7 i; ^ nmi n^ m^ ' 3.2 dt nm C. (O) = C . 1 ^ "^ 01 Table 3^1 lists the species considered and Tables 3.2 and 3-3 list the reactions. The system consists of electrons, ions and excited species of 1-12 as well as the dissociated products. The neutral chemistry reaction rates have been taken from a recent coicpilation by Westbrook (1931) . The ion neutral reaction rates are taken from Huntress (1973) and Ferguson (1973). The ion clustering reactions are taken from the work of Good et al. (1970). The ion recoabination rates froia Leu et al. (1973). The chemistry is fairly well known for H2 vapor and is

PAGE 76

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 112 0. In a recent study for liquid water these states were assumed to dissociate in the same tioe scale as the interaction of the H2O+ ion (Wriqht et al . , 1933) . For vapor we would expect a sliqhtly slower dissociation rate. However the information ia this area is hiqhly deficient (Inokuti, 1983). The system of differential equations is solved usinq the Gear method (Hindmarsh, 1974). The proqram is based on the routine GEAR (IM3L, 1981) and includes an adaptation of the kinetic subroutines from the diffusion kinetic proqram DIFKT (see chapter III) in order to set ap the kinetic equations. Usiiiq the results froia 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 214 and the thermal electron distribution. The reaction mechanism is presented in Tables 3.2 and 3.3. The results of tae calculation are presented in Fiq. 3.1 and discussed in chapter IV.

PAGE 77

68 TABLE 3. 1 Chemical species included in the diffusion kinetic system. Species Diffusion Coefficients {cm**2/sec) E

PAGE 78

69 lABLE 3. 2 Reaction rates at temperture equal to 400 K. Reaction Reaction Equation Number 1 H + 02 -> + OH 2 + OH -> H + 02 3 H2 + -> H + OH 4 H + OH -> H2 + 5 H20 + -> OH + OH 6 OH + OH -> H2C + 7 H20 + H -> H2 + OH 8 H2 + OK -> H20 + H 9 H2C2 + OH -> H20 + H02 10 H20 + H02 -> H202 + OH 1 1 H20 + H -> H + OH + M 12 H + OH + ;i -> H20 + a 13 H + 02 + M -> H02 + M 14 H02 + H -> H + 02 + M 15 H02 + -> OH + 02 16 OH + 02 -> H02 + 17 H02 + H -> OH + OH 18 OH + OH -> H02 + H 19 H02 + H -> H2 + 02 20 H2 + 02 -> H02 + H 21 H02 + OH -> H20 + 02 22 H20 + 02 -> H02 + OH Reaction Sate cm, moles, sec 1.244 ( 5) .6294 (13) .9954 ( 8) .5300 ( 9) 3.187 ( 3) .7910 (12) 7.729 ( 2) .3359 (11) . 1038 (13) .3432 (-4) .9265 (-41) .8812 (18) .5837 (16) , 1902 (-9) .1424 (14) .7544 (-17) .2299 (14) .1472 (-08) . 1041 (14) .1434 (-17) . 1424 (14) .2760 (-25)

PAGE 79

70 TABLE 3.3 Reaction rates at temperture equal to 400° K. Reaction reaction Equation UToleVslT lluDiber 23 24 25 26 27 H2C2 + 02 -> H02 + H02 .2000 (-9) H02 + H02 -> H202 + 02 ,2843 (13) H202 + tl -> OH + on + M .1651 (-7) OH + OH + M -> H202 + M ,5380 (18) H2C2 + li -> H02 + 112 .1514 (11) 28 HC2 + H2 -> H202 + H 43.84 29 + H + E -> OH + M I'OOO (16) 30 OU + il -> + H + M .4191 (-39) 31 02 < a -> + + H .7448 (-47) + + K -> 02 + :i .8743 (15) 32 33 H2 + 11 -> fi + H + M 7659 (-38) 34 H + K + M -> H2 + a .30 20 (16) 35 1120+ + n20 -> H30+ + OH .1130 (16) 36 0H+ + H20 -> H20+ + OK .3480 (15) 37 0H+ + n20 -> 030+ + 38 n+ + n20 -> H20+ + H 39 H30+ + H2C + M -> +H(n20)2 + M 40 +H(H20)2 + K20 + K -> +H(H20)3 + « i;l +H(H20)2 + S > H + H20 + H20 42 +H(H20)3 + K > H + 020 + H20 + H20 .2770 43 H20+ + H2 -> H30+ + H .3670 (15) 1^4 H30+ + 2-> H20 + H .7830 (18) .2839

PAGE 80

71 i, I J M i n i : 4 1, I i I I i r 4 i i Mi iii 4 11 1 )^ ! ,"n-^."" ' ?o-^-"" ' i"n-'-"'" Trf • REfiCTIQN TIME, SEC Figure 3.1 Results from kinetics program concentrations versus time. Based on system presented in Tables 3.2 and 3.3.

PAGE 81

72 Develq2aent_of_the_Dif fusiou_KiuGtic_SYS 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 qives rise to spatial distributions of H20+,CH+,0+ and H+ alonq with the associated dissociated products and electrons. Additionally, various excited states of H2O are produced. The theraal 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 followiiiq equation: 00 00 ]£ J„ J [J '^DS(p.z.E.E^)dE + 00 3.3 ^o 1 J p. (E)U(E./D.z.E )dE]pdpdx.^l i = l 1 ° The first term is the number of thermal electrons per incident electron, the second terra represents the total number of ions created and the third term accounts foe the source electron. Since the system of differential equation;

PAGE 82

73 includes an equatiou for the electrons, we need to consider what the ultimate fate of these source electrons will be. Dltimately, they will encounter a qround such as the c ontaiument vessel for the ^'2^' ^^ ^^ possible to hypothesize a reaction which eliminates these electrons, such as electron attachment to H2O (Hanrahan, 1983). Alternately we may simply iqnore these electrons in the deposition spectrum. For a 1000 eV electron this is the same as reiaovinq one electron out of thirty. At most this would slow up the ion chemistry sliqhtly. The physical system is described by the followinq set of equations: 3C ^(p.z.t) D^y ^ ^(p.z.t) 3t 3,4 -^ ^nmi^n^^'^-^^^m^^-^'^^ "^ Y.(p.z.t) n , m where the initial conditions are C^(p.z.O) = C^^(p.z) Yj^(p.z.t) = J^(p.z) Rate (tte /sec) In the steady state case, 'X:/ % is set equal to zero. The required boundary conditions in this case are that the concentrations equal zero at infinity and their derivatives, with respect to P, at P equal zero be equal to zero. The

PAGE 83

74 symbol Y. is used to denote tue yield of the i' species. The remaininq 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 Fiqs. 2.18-2.23. The diffusion coefficients for the molecules and atoms are based on qaseous diffusion theory as presented by Hirschfeider et al. (1954) and Monchick and Mason (1961) and experimental data compiled by Marrero and Mason (1972), The coefficients are calculated according to Chapinan-Enslcoq tiieory , implementinq the molecular paraiaeters given in the paper by Kee et al. (1933). The diffusion coefficient of the electron in {l20 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 laore tractable the variables have been transformed to dimensionless coordinates. The transformations are P'=p/ap^ .z'=z/bz^ . D.'=D./D^ ^and C.-=C./bC 35 2 ^ The transiorined equations and their associated initial and boundary conditions are

PAGE 84

75 3C' (p' .z' .t)=DW' ^C'^ (p' .2' .t) +YJ(p'.Z') 3t 3.6 ^^ '^;nni C;^(p'.z'.t)C;^(p'.z'.t) = nm ,'. .',y 2 2. a p b k° k . nmi D-, -, nmi where Y is qiven by i 2 2 a p YMp'.z') = —5 Y.(ap p.bz z) 3.7 -H,0-H,0 In the steady state case we make the additional t rail sf o riaa tiofl 2 -1 x= -tan p' 3.8 ir 2^ -1 , Y= —tan z' • IT which reduces the domain of the problem to finite dimensions. The equations can be restated in the followinq form: 3C ' ^ Ci ~ I k' .CL C'_ + Yi = 3.9 n.m ^ ^ \x.y)^i ^, „^'nmi^n "m "i :.' = Ci (x.y.t) Y| = Y^Cx.y.t)

PAGE 85

76 wnere and „2 4 2, irx. a , 2 ,irx .3 , ^x.y) = "1^°^ ^i~>-i5^ f^°^ ^-2^-^^ * ^^!^,^ ,^--^;,^^ 1^ . ^COS^ (^ ) |[COS 2( f A] irsin {irX/2) dX ir Z ay ^ ay 2 2,, 2 2 S = a p^ /b z^ o o The steady iitate solution is valid for restricted values of the yields. This is discussed in cliapter IV. Solutions for the steady state equations are presented in Fiqs. 3.2-3.14, The sif,iulation was made at a temperature of 400 °KThe density of the medium was set at 1.795 (-7) molecules per centimeter cubed and a, h 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 knowledqe acquired in previous calculations and reduce the computational cost.

PAGE 86

77 .OOOeV c p « N (0 g O c 10 -5 C o
PAGE 87

78 to a o N e M O 0) •H >. U o e o •H «J ro U «J C
PAGE 88

79 200eV 10 -12 *j c N •H rH (0 B u o c o CO c o e o V 10 -13 Figure 3.4 Concentration profiles (p = 0) of the H O"*" distribution obtained from the steadystate solution of the diffusion kinetics system. The solutions are normalized by Eq 3.5.

PAGE 89

80 w *-> c 3

PAGE 90

81 10 -11 10-12 10 z(cro) 10 Figure 3.6 Concentration profiles (p = 0) of the H O"^. H'^CH 0)„ and H'*"(H O) distributions oBtained fcom the steady state solution of the diffusion kinetics system. The solutions are normalized by Eq. 3.5.

PAGE 91

82 10 -2 CO C P N e u O c c o •H c V c o u 10 -3 I -10 z(cra) 10 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.

PAGE 92

to c •a -l c u c o o .1 lOOOeV 200eV 01 10 20 p(cra) 30 Figure 3.8 Concentration and yield rate profiles (z = of atomic hydrogen. Yields normalized to maximum value and concentrations normalized to (0, 0) point value. 0)

PAGE 93

84 00 c p •d N <0 g u o a c o •H 4^ (0 U 4J c <0 V G O u 1 -10 z(cro) 10 Figure 3.9 Concentration profiles (p = 0) of the OH distribution obtained from the steady state solution of the diffusion kinetics system. The solutions ace normalized by Eq. 3.5.

PAGE 94

85

PAGE 95

86 10 -5 to e u O a G o •H 4J 10 U 4J e a> u c o o 200eV 10 -6 10 z(cm) 10 Figure 3.11 Concentcation 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.

PAGE 96

87 10 -2 c: •O (I) N •H i-< (T3 e U o c c o u 4J c
PAGE 97

88 w 4-1 .1-1 c N ITJ e u o c c o f-i 4-> «8 U C o o c o u •10 z{cm) 10 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.

PAGE 98

89 (0 c p 0) N e u o c c o •H J c 0) u c: o o 10 z(cra) 10 Figure 3.14 Concentration profiles (p = 0) of the H O distribution obtained from the steady slale solution of the diffusion kinetics system. The solutions are normalized by Eq 3.5.

PAGE 99

90 Compu terAdaptation 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 Fig. 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 eguations 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 in 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

PAGE 100

INPUT 91 TWODEPEP Set up Equation System Boundary Conditions COEFF

PAGE 101

92 requested values of x and y. This subroutine can be easily modified to process the iuformation before it is stored on raaqnetic tape. The solution algorithm has been tested extensively by ir'iSL. In addition the procjram was tested aqainst a simple one diiuensional system similar to the system of interest. Grid dependent tests were run and it was found that a minimum of 130 finite elements was needed.

PAGE 102

CHAPTEB IV SESULTS JDiscussipn C om£ u t a t i on al_S i m u la t i on The results of the calculations can be divided into two broad cateqories. First, the calculation of initial yields directly resultiuq from the deposition of the incident electron energy in the medium. Second, the calculation of steady state concentrations resulting from the introduction of electrons into the medium at a constant rate. The system of proqraais which are used to calculate these quantities is also naturally divided into tuese two categories. These programs are listed in Fig, 4^1, The programs waich are used to calculate the initial yields are ELASTIC, T0TH2C, MESD, GETMESD, KC, GETMC, NLLSYS and YIELD. ELASTIC and T0TIi20 are service prograas which are used to generate cross sections for both elastic aiid inelastic collisions. In addition, they also produce graphs ox these cross sections. MESD and GETHESD are a pair of programs which follow the path of an electron which undergoes only elastic scattering. MESD generates a data base of the collisional history of the electron and GETMESD sorts this information and calculates cuiaulative 93

PAGE 103

94 Input cross sections Program MESD Monte Carlo Output •collision history collision hisi-oty ^ GET DAT (a) yield spectrum (b) deposition spectrum yield spectrum YIELD .yields yields — reactions KINETIC .concentration as function of time yields | kinetic reactions rates diffusion coefficients DIFKT concentration as function of space Figure 4.1 Overview of computation

PAGE 104

95 distributions for R, Z and X as a function of pathlenqtli and incident electron energy. These distributions are fit with analytic functions which are then inserted in the MC proqram. The MC proqram is the maior proqram in this seqment of the calculation. This program tracks electrons on a collision by collision basis as they undergo elastic and inelastic scattering. When the electrons reach the point at which the total elastic cross section is much greater than the total inelastic cross section the elastic distributions produced by MESD are used to calculate the movement of the electron between inelastic collisions. In addition, when the electron energy is less than the lowest excitation state being considered, these distributions are used to follow the electron as it approaches thermal eguilibriuia. GETMC uses tiie data generated by MC to calculate two and four dimensional yield spectra, collisional densities and theriaal electron distributions. The NLL3YS program is used to fit the yield spectra and thermal electron distribution. It consists of a nonlinear leasx square fitting algorithm which has been modified by the incorporation of a a normalization criterion which reguires that the integral of the function being used to fit the data over all space, integrate to unity. The laultiple dimensional integrals are numerically computed in spherical coordinates using Gauss-Laguerre guadrature in the radial coordinate and a simple Newton-Cotes composite formula in

PAGE 105

96 tiie theta coordinate. Finally, the YIELD program uses the analytic fits generated by NLLSYS and the cross section data from TOTH20 to calculate yields as described by Eg. 2.14. A variation of this program generates spatial data sets whicu are used to produce isoyield and isodose curves. The yields are then used in the diffusion kinetics calculation. The programs which are used to calculate the subseguent evolution of the system are KINETIC, DIFKT, and CONTOUR. KINETIC is a program which calculates at the time evolation of a set of first order nonlinear reaction rate eguations. This program is used to study various reaction mechanisms without diffusion. DIFKT is the maior program in this calculation. As input, it reguires the results from the YIELD program, the reaction mechanism studied in the KINETIC program and a set of diffusion coefficients in order to solve a system of second order nonlinear partial differential eguations which describes the physical chemical system set up by a steady beam of electrons incident in gaseous ii^C, i52^te_Carlg_and_Yieid3 The solutions to tue multiple elastic scattering problem presented in chapter II were obtained for electrons with energies from .2 eV to 50 eV. It is of interest to analyze theiie solutions in terms of the single scattering cross section at these energies and look at the energy dependence

PAGE 106

97 of the parameters used in representinq the multiple scattering distributions (Fiqs. 4,2-4.7). First, we should note that at all energies HO has a significant backscatterinq componentFor example, at 50 eV the scattering cross section is essentially split into two components with no scattering at 90 degrees. At lower energies (less than 1 eV) we see (Fig. 2.1) that the single scattering distribution is monotonically decreasing and the relative difference in magnitude between the forward scattering peak, and the backward scattering peak is approximately a factor of ten. At higher energies {15 eV to 50 eV) the probability for small angle scattering has increased as expected, bat the ratio of the areas under the curve from 90 to 180 degrees and to 90 degrees is increasing. Looking at the cumulative radial distributions, F(n) in Figs2.6-2.8, we can see that as the pathiength increases, the distribution widths increase. Furtherfiiore, as the electron energy increases the distributions spread out, the increased dispersion correlating witn the increase of the forward scattering aiaximufflAs the energy continues to increase, these distribution widths decrease because the ratio of backscattering tc forward scattering increases. These patterns are expressed by the parameter dependence on energy as depicted in Figs. 4-2 and 4.3The cumulative longitudinal distributions behave similarly. The laost apparent feature of the polar angle distributions is their

PAGE 107

98 relative uniformity for pathlenqths greater than four raeau free paths, (Fiqs. 2.12-2.14) at all energies. This feature simplifies the application of the HESD results in the Monte Carlo transport code since for pathlenqths greater than this it is not necessary to invert this function. In practice, the MESD results are not used above an energy v/here there is a 20r3 probability of having fewer than four elastic collisions between inelastic collisions. The energy dependence of the polar angle distribution is graphed in Fig. 4.4. The results of the Konte Carlo calculation are presented in condensed form by the analytic representations of the two and four dimensional yield spectra. Samples of these functions are presented in Fiqs. 2.15-2.17. The functional representation of this information originated in the papers by Green, Garvey and Jackman (1977) and Green and Sinqhal (1979) for the two and four dimensioiial yield spectra, respectively, and were used as recently as 1933 in a study of electron deposition in neon by Singhal, Rio, Schippnick and Green (1983). In this study the representations for the four dimensional spectra for H was better fitted with the 2 new functional forms presented in Tables 2.10 and 2.11. Two diaiensional yield spectrum are qraphed in Fig. 2.10. It is interesting to compare the spectra for HO with those presented for some other gases in a paper by Green, Jackman and Garvoy (1977). Some of these have been reproduced in Fig. 4.8.

PAGE 108

99 < 9 |B « •
PAGE 109

100 1

PAGE 110

101 \ \ • ^ B \ I ^ \ / \ / / \ / V / B • 9 CO \ «» \. B • \ B • B B \ \ fi

PAGE 111

102

PAGE 112

103 ^

PAGE 113

104 < s M cn w D C ui ^ OJ P OJ O 3 HCD O. rt fD fD rt W fD Hi M O CD i~{ O rt M M O O 3 3 in fD ft 3 c fD a I-! H^ ^ M Pi HCO rt^i cr rt O 3 < < 1 1

PAGE 114

105 kT

PAGE 115

106 By and larqe, ail the spectra show the same overall behavior except for a prominent difference in the low spectral enerqy behavior. The difference can be attributed to the structural variation of the scattering conponent, for example the difference between an atom and molecule. Por both H and CO the density of vibrational states at low 2 2 enerqy, as plotted in Fiq.2.2 for H qives rise to an area 2 of increased collisions at low enerqy. Diatomic molecules show less of this behavior in their yield spectra because they have only one low enerqy vibrational state whereas their spectra are comparable to those of atoms (see Fiq.4,8). In addition, a two dimensional spectrum for E 2 calculated in 1977 is shown. The low enerqy resonance behavior for the vibrational states was not known at that time, and by comparinq these spectra (Fiq.4.3) to those in Fiq. 2.15 it is obvious that their contribution to the spectral distribution is shown by the rapid increase in collision density at low enerqy (see Fiq. 2.2). The functional forms for the four dimensional yield spectra represent an improvement in the representation of the physical attributes of the deposition pattern of electrons in a qaseous medium. The functions used to model the two and four dimensional yield spectra retain their overall attributes as formulated by Green and Jinqhal (1979), That is, they are separable in spectral enerqy and spatial coordinates, the four dimensional yield spectra are

PAGE 116

107 constrained to inteqrate to the two dimensional spectra, and the spatial dependence is assumed to be distributed over three spectral energy regions. However, the current model has replaced the spatial functions with functions that are based on concentric ellipses (see Tables 2.10 and 2.11). Unlik:e the previous functions, these are continuous in the z coordinate. This shape is similar to that experimentally observed by Grun (1957). In addition the density dependence associated with the four dimensional yield spectrum has been corrected. In the previous representation the integrated yields were density dependent but, as can be easily demonstrated by looking at a proiection of a electron path on a surface which expands as one over the density, it is seen that while the spatial pattern may expand with decreased density the number of total collisions and in fact the spatial pattern remains the same. This concept has been incorporated into the spatial functions. Finally, as previously mentioned, the four dimensional yield spectra are constrained to inteqrate to the two dimensional yield spectra. This is accomplished by requiring that the constants A observe the normalization relationship expressed in Eg. 2.11. Unfortunately, the spatial integrals are not analytic and the integrations are performed numerically as described in Chapter IV. Furthermore, since the spatial parameters P, *, and z^ are dependent on incident energy, the normalization constants A are also dependent on the

PAGE 117

108 incident energy. These constants are tabulated in Table 2.13. One of the aost interesting results of using the concentric elliptic functions to model the spatial dependence of the yield and deposition spectra is that the incident energy dependence of the spatial parameters can be related to an analytic form similar to that used for the total cross section. That is, 3 can be thought of as being related to the total cross section since the collision density would fall off slower for a particle which had collisions spread far apart. Similarly the z offset increases with an increase in the collisional pathlength since it is proportional to one over the total cross section. The A parameter controls the eccentricity of the ellipses. In general it is seen to behave in a predictable manner. The yields for most of the states listed in Tables 2. 4 and 2-5 are calculated using Eg, 2.14 and presented in Figs. 4.9-4.13The G(#/eV) values are seen to become constant at high energy. The value of W ( eV/ip) for the ionization states versus incident electron energy is presented in fig. 4,14 and compared with experimental data. The variation of this guautity with incident energy has been verified by numerous experiments over the years and is accounted for by the properties of the yield spectrum {Green, Garvey and Jackman , 1 977) .

PAGE 118

109 > M (0 0) -p m +) CO u •H 0) rH > CTl 0) en (Ae/#)D

PAGE 119

110 en (U -P (tJ +) 01 o H 4-1 -H O QJ & CQ U O m >i Di U (0 (U iH > I o H en I o (A3/#)D 0)

PAGE 120

Ill ^

PAGE 121

-p m -p CO 112 > u •H M-l •H U 0) 04 o iw >i Cn QJ C (1) c o S-l +J o cu iH 0) -p (U Td •H u w m u > en CD iH rd > CN 0) en (A3/#)D h

PAGE 122

113 > W u (U m +» 01 o •H M-1 •H O 0) (0 O >1 0) CD o -p u 0) rH (1) -p 'd •H U a <-{ CO :3 01 U (U > 01 n CM I o CO I o (AS/#)D 0) •H

PAGE 123

114 Figure 4.14 v; (#ip/eV) vs. incident electron energy. Experimental results from Klots (1968) ( • and Berger and Seltzer (1964) (A).

PAGE 124

115 Looking at the yield contours presented in fiqs. 2.18-2.23 it is easy to see that the pattern for each species presented is characteristic of the physical parameters of the collisional cross section. For this reason we see that OH, which can be produced by states with lover energy thresholds than H 0+ can, has a more diffuse 2 pattern than H 0+. This is because the electron energy 2 diminishes as it moves away from the incident point. Also, as the incident electron energy increases the thermal electron distribution becomes more spherical. This is because the electrons, after as few as four collisions at low energy, lose the knowledge of their initial orientation (see Figs. 2.12-2.1^4). Pif f usion_Kinetics The results from the KINETIC and DIFKT programs provide us witii a unigue view into the physical-chemical mechanisms involved in the picosecond region. Refering to Fig. 3. 1 we can see that the ions which are initially produced by the incident electrons react very quickly and are neutralized essentially within a microsecond. The H O clustering 2 reactions enable the proton to remain stable for a relatively long time. This is because of the shielding it receives from the polar molecules. The electrons are finally seen to combine with these positively charged species until only the incident electrons remain. At the

PAGE 125

116 same time these species are beiaq depleted the concentrations of the radical H, 0, and OH are increasinq. The chemistry which involves the OH radical is completed in one second. In approximately two seconds the H 2O chemistry has essentially stabilized and we are left with H 2# 2, H 2O 2 and a small amount of HO2. However, we should note (Fiq. 3,1) that hydrogen has not yet reached steady state. DK calculations were performed for electrons with incident enerqies of 200 and 1000 eV. These tvo enerqies are representative of the types of deposition patterns which can be seen in the energy range from 100 eV to 10 keV. The 200 eV distribution patterns for all the primary yields are much tighter than those for 1 keV (see Fiqs, 2,18-2,23). As will be seen, this creates a dramatic difference in the steady state concentration distributions. We beqin the discussion of these calculations by analyzinq the results for the 1 keV electron. The primary species involved can be divided into three major types. The electrons, the posixive ions and the neutrals OH, H, C and H 9. As mentioned previously, the electrons are the only species considered which are not strictly conserved in the system. In the initial formulation of the problem they were introduced into the system by two methods in order to compensate for this imbalance. However, it was discovered that neither fix mentioned was necessary. The resultant steady state distribution for the electrons is presented in

PAGE 126

117 Fiq. 3.2. The distribution is presented in profile, as all results are, and shows that, as expected, near the oriqin the steady state distribution is dominated hy the yields. This region extends to approximately seven centimeters for the density at which the calculation was made and can be compared directly to the yield profile presented in Fiq, 3.3, In spite of the fast reaction rates associated with the electron their numerical abundance prevents them from beinq significantly depleted. Thus, those electrons which do not react with a corresponding ion quickly beqin to diffuse in the classical diffusion liait. This is demonstrated by the 1/r shape of the profile for distances greater then seven centimetersFurthermore, we see by lookinq at the profiles alonq both the p and z axes that the distribution is almost spherical except for a slight asya^raetric component in the forward z direction. This asymmetry (Fig. 3^2) is due to the preference of the incidant electrons to scatter in the forward direction. The H2O+ ion distribution (Fiq. 3.4) drops off very quickly. This is because the maior pathway for the destruction of this ion is its reaction with H2O and the reaction rate for this reaction is high. We see in Fig. 3.5 that, in fact, the ion reaction is so guick that it barely diffuses before reacting. The H2O+ distribution, unlike the electron distritjution, is fairly symmetric. The reaction of the H2O+ ion leads to the production of the H3O+ ion. Since

PAGE 127

118 the reaction which creates this ion is linear, due to the constancy of the H 2O backqround, the distribution for this ion is almost exactly like that of H 2O+ . This can also be attributed to the fast reaction rate of H3O+ with H2O, similar to its precusser fl 2O+ "^^^ remaininq reactions in this subset of ionic reactions, as presented in Table 3.2, continue the same pattern. That is, the ions form complexes with the water molecules and are neutralized by the electrons (Fiq. 3.6). The maior effect of these hydration reactions is to delay the production of hydroqen and thus shift the hydroqen concentration away from the oriqin. In addition, these reactions lead to decreased production of atomic hydroqen when compared with the direct reaction of the U 2O+ i°i^ with the electron. This is due to the exothermicity of the charqe neutralization reactions for the hiqher order hydrates which in turn leaves insufficient enerqy to dissociate a water molecule as compared with the neutralization reaction of the hydroniuia ion (Anderson, 1958) . Next, let us look, at the steady state distribution of the hydroqen atom. Unlike the H 2O+ ^°^ previously discussed, the maior source of these atoms is not solely due to the dissociation of the water molecule by the incident electron but also due to the neutralization process. We therefore see a rather extended reqion of constant concentration level for this species (see Fiq. 3.7) even thouqh the primary

PAGE 128

119 yield falls off much quicker (Fiq, 3.8) . The large diffusion coefficient of this species also contributes to this behavior. The hydrcqen molecule distribution follows the same pattern as that of atomic hydroqen. This is because the major pathway for production of this species is via the combination of atomic hydroqen. The OH radical exhibits a distribution pattern which shares some attributes with both the H2O+ ion and atom distribution. For instance, the OH distribution (see Fiq. 3-9) beqins to decrease almost immediately because OH is fairly reactive. However we can see that the distribution does not fall off as rapidly as that for the ions. Furthermore, by coraparinq the yield profile with the steady state distribution profile (Fiq. 3.10) we can see that the radical moves an appreciable distance before it reacts. The oxyqen atom distribution (see Fiq. 3.11) is similar to that of the OH radical. Essentially its fall ofx is slower than that of OH because Atomic oxyqen does not react as readily as does OH and taerefore its distribution is not as sharply peaked. The molecular oxyqen steady state distribution looks similar to that of molecular hydroqen except that the plateau reqion is not as flat. This is expected, to some extent, since O2 and H2 occupy somewhat of the same role in the cheaical mechanism. That is, they are botn relatively stable cnemical species. The final products of interest are H2O2 ^^'^ ^^02* ^^^^ hydroqen peroxide molecule is not

PAGE 129

120 produced in any appreciable amount. This is not unexpected since the kinetic model gives an indication tiat it is not stable enouqh to exist for any significant length of time. On the other hand, H2O2 appears relatively stable in the kinetic calculation and we see that its concentration distribution is significant. The peak value of this distribution is of course much smaller than that of either H2 or ©2. Eefering to Figs. 3.2, 3.4, 3.7 and 3.9 we can observe the critical differences in the steady state distribution patterns for the two incident electron energies studiedThe low energy incident electrons give rise to a much more compact region of positive ions which in turn deplete the thermal electron population. This inner core of positive charge, while present at higner energy, does not peak dramatically enough to deplete the thermal electrons. In general this pattern of high peak values with a fast fall off is seen for all the species created directly by the incident low energy electrons. The distribution patterns of the priaary species produced by the 1000 eV incident electron show a more gradual transition region to pure diffusion. While there is a significant difference in the distribution patterns of the priaary species for the two cases studied, those species which are essentially end products of the reaction mechanism are seen to have distributions which are similar in shape (see Fig3.12, 3. 13 and 3.14) .

PAGE 130

122 where we have an extended source. The resultant concentration profiles are essentially spherical and fail off gradually in comparison to the source term (yields). The electrous form an outer layer of charqe around the ions, which of course remain nearer the oriqin, because of their slower diffusion rate. The radicals, for example OH, have a profile which extends further out. Finally, the distribution profiles for H , 0^ and fi2°2 ' ^''^^^ '^^^ ^^'® m ost stable species, extend out the farthest. Limitations Let us discuss the iimitatious of the raatheaatical techniques used in this study to describe the effects of irradiating H vapor with electrons. Foremost is the yield spectrum which characterizes the immediate iiapact of the electron on the medium. The first maior consideration, especially in a gaseous medium, is that the density of the medium be sufficiently high that the radiation not diainish it to an appreciable extent. This is necessary because the yield spectrum is dependent on the physical properties of the medium. Since we have chosen to study the steady state case for the diffusion kinetic system it is also necessary to consider when this situation might break down. The system looked at assumes a medium which is essentially infinite compared to the spatial distribution of products. Thus we have not taken into account the finite boundari^is

PAGE 131

121 The results from the diffusion kinetic proqram DIFKT are presented in Fiqs. 3.2-3-14 for 200 and 1000 eV electrons incident at the rate of 10(12) per second. Cotnparinq the profiles of the steady state concentration distributions with the yield profiles, we see that the distributions of those species directly produced by the incident electron exhibit an appreciable relaxation of the system. Whereas the yields fall off dramatically with distance, the steady state concentrations fall off much more slowly. The electron, whose diffusion coefficient has been estimated at approximately five times that of H or H2 , travels the farthest distance before it beqins to be depleted by interactions with the positive ions. Molecular hydroqen, whicn is one of the major products of tae reaction mechanism, is seen to remain constant for a relatively lonq distance. An analoqous behaviour is also exhibited in the kinetic results presented in fiq. 3,1. This can he attributed to the relative stability of tnis molecule. Furthermore, we note (Fiq. 3.1) that li, is the last species to stabilize. Therefore wo would expect the molecular hydroqen distribution profile to extend further out than that of some of the other species, like H^ 0+ or OH, which react very quickly. The picture of the diffusion kinetic system that we see is essentially one of translatinq the time coordinate in the time dependent kinetic simulation (Fiq31) into space

PAGE 132

123 which would be associated with a containment vessel for the medium which of course could be bioloqical. Furthermore, the chemistry associated with the system (see Fiq, 3*1) implies that the ions are depleted in the time frame of 10 (-6) seconds. Therefore the rate of input of electrons must be greater than one over this number to avoid calculations with less than one electron* This may happen since we are using a continuum description in the mathematical formulation of the diffusion kinetics physical chemistry.

PAGE 133

CHAPTEE V CONCLUSIONS In the early fifties a qroup of scientists set in Hiqh Park, Illinois to discuss the physical mechanisms involved in the interaction of radiation with a bioloqical cell, la 1964 and aqain in 1974 in Airlie, Virginia the problems were addressed by an invited qroup of scientists froK the fields of bioloqy, chemistry and physics. Durinq the 1974 meetinq, the concept of following the radiation deposition mechanisa from initial principles was seriously discussed for the first time. One of the founders of this field, Robert Platzman (1974), said in a review article oa early time chemistry froia this meetinq that "at a meetinq of this type 20 years aqo and at another 10 years aqo, I encountered some of the same problems I have touched on here, and I am willinq to bet anyone that if I survive another 10 years, I will find that some of the complications I have stressed have survived also." In the last ten years the research in this area has literally exploded. The latest meetinq of participants in this interdisciplinary field was held at Gettysburq, Pennsylvania in 1932 in what was the fifteenth annual meetinq on "The Bioloqical Effects of lonizinq RadiationRadioloqical Physics". This meetinq led to a 124

PAGE 134

125 special "Workshop on the Interface Between Radiation Chemistry and Radiation Physics" held later iu the same year at Arqonne National Laboratory. As mentioned in the introductory chapter, the questions raised thirty years aqo are finally beinq answered. In this dissertation we have brouqht toqether information and techniques from many fields in order to qive one of the first complete descriptions of the physical and early time physical-chemical raechanisms involved in the radiation of bioloqical systems. Thus we have followed the deposition of enerqy by an electron as it produced initial yields and have tracked these species as they reacted and diffused. The sat of proqraas written to describe this system provides us with an apparatus to study various physical and caemicai effects in a very detailed manner. For example, the effect of varyinq the dose rate or enerqy of an electron beam can be seen from a new perspective. It is also possible to study the effects of addinq trace amounts of various chemicals to the system and lookinq for a unique mechanism which would modify the effects of the radiation. As a byproduct of these calculations we have computed yields for the various individual states and seen that they are similar to those observed in experiments. It follows that the ev/ion pair calculated is also quite close to tae observed value. Furthermore, we are able to construct isoyields or, of more importance to the practicinq medical

PAGE 135

126 physicist, isodose curves for a moaoeiierqetic beam of electrons. It is a straightforward procodure to qeneralize these computations for a spectral and spatially distributed beaiR of electrons.

PAGE 136

APPENDIX A MONTE CARLO PBOGRAtJ The proqraa used to qenerate the path of an electron inelastically scattering is MC. This program was originally written by Brinkman in 1970 and essentially rewritten by Jackman (1978). The program has since bean slightly modified by Singhal (Sinqhal et ai. 1980, Sinqhal et al . 1983). la the course of calculating the yield spectrum for H this program was again revised. Approxiuiately 70,< of 2 the program was rewritten; 30% of this was new, reflecting changes in both the logic of the program in order to decrease its computational time and to accommodate the necessity of calculating the collisional history of the electrons as they approach thermal equilibrium. There were some problems associated with this program prior to these modifications. During the research on Neon {Singhal et ai., 1983) the probabilities for inelastic collisions were incorrectly scaled. The allowed and forbiddfn state inelastic scattering probabilities were scaled only by a proportionality factor above 200 eV, whereas the forbidden probabilities fall off much faster than the allowed with increasing energy. Additional problem areas included the omission of scattering for the electron immediately 127

PAGE 137

123 followinq the MESD collision, cross correlations between MESD parameters because the same random number was used for generation of uncorrelated variables and finally the ratio of elastic to inelastic cross sections used in the MESD subroutine was incorrectly calculated since the elastic cross section changes for large pathlengths (see chapter II) . In order to clarify this problem with the MESD subroutine in tlie in the Honte Carlo transport calculation it is helpful to refer to Fig. K.2. Originally when an MESD calculation was applied between the origin (relatively speaking) and the next inelastic collision (the second set of axes) the inelastic scattering was ignored. The inelastic scattering algothrim introduced into the current program is diagramed in Fig. A. 3. The progra^a was completely annotated and the input was modified in the interest of siiaplif ication. In addition, a logical pathway was created which allowed us to follow the electrons as they elastically scatter iselow the lowest inelastic state considered. The electrons were then tracJced until they approached thermal eguiliJarium, Following these changes, the program was checked by running a few test electrons through and calculating different parts of the path with built in diagnostics aui coiaparing them with hand calculations. The diagram in Fig. A.I shows the new logic flow for the l\Z proqram. The program was found to be approximately 50 ;i faster than the older version.

PAGE 138

129 The companion proqrain to MC, GETMC was also modified. The changes reflect those made in the HC program in order to follow the low energy electrons. However, a few modifications were made in both the input and output of the program. The new output requires some discussion, since it enabled a longtime embedded problem with this code to be found. The GETMC code essentially calculates the two and four dimenional yield spectruas. In doing this it sums collisional events in energy and spatial bins and normalizes the results. Unfortunately, the information describing the number of collisions which occur in each bin is not displayed. In reworking this program it was deemed necessary to output this information in order to fit the yield spectrum data only where the number of collisional events was statistically meaningful. This uncovered a computer error which caused bins with no collisional events to have the number of collisional events displayed from the previous calculation. Finally, this code was modified to calculate the ion pairs per eV directly from the tabulated data instead of using the yield calculation as discussed in chapter IV. In closing this appendix, it should be made perfectly clear that due to the nature of the yield spactrua most of the errors mentioned will not have a significant effect on the prior results.

PAGE 139

130 Initial Energy Eq Position (p , Z,cos9 ,(})) Set E = E ^0 Calculate New Position, Energy V Is E<.125eV? V Calculate New Position, Energy YES NO Thermalized Electrons ( Is E<22eV? > SCAT Calculate elastic scatterinc angle 6 COLL Calculate : cross sections collision type energy loss MESD pathlength S=1500 Calculate : Ap , AZ, A({),cose AE A Elastic E-AE Type Collisio YES I n_?) \-*<] Excitation \ query COLL \ Is E<.198eV?, ) PETI Calculate secondary energy, T SCAT Calculate scattering angle NO MESD Calculate S, Ap,AZ, A(}) cos9 and AE Excitation E W * ^ype Co llision?/ I Ionization E = E W I NO -/is T^E ?y •YES

PAGE 140

131 (Xo''^^0'^'°^ M Figure A. 2 Coordinates for multiple elastic scattering calculation.

PAGE 141

132 dc (E ,T,e, j) primary $, = 2-!:R secondary i+ij = 7i(2R-l) high energy secondary E >_ lOOeV T > 200eV COSa. / ^ (1 + T/C) .707R E(1-T/'E) E < 100 eV cose, ,707 + .2929R Low er.ercv secondarv E < 100 eV T >^ 50 eV cose. T < 50 eV cose; .707R 1 + 2R Figure A. 3 Ionization scattering algorithms E is primary electron energy and T is secondary electron energy.

PAGE 142

APPENDIX B DIFFUSION KINETIC PROGEAl This appendix consist of the source code which interfaces with TWODEPEP, in the program DIFKT as presented in Fiq, 3.6. 133

PAGE 143

134

PAGE 144

135 n H m IT) o it r»— I— lO ' w n » W » m Vi « » 10 W » t/5 • OJ Eh O W E-t rn W 3 V) • a » ro CO » n in fc Q in <. O cruO »'— CNf^rJ-in k » Eh » ;3J3W m H » to • Eh H cr Eh CiJ M rs W CO to » CO in in EH tn D 3 W * » CO H EH f^ Eh EH EH . M cq Kl M ta t3 M CO CO to ^ ^ 33 !=) O H EH H WWW to to CO rn rn (V) 3 ra n k k « EH EH e^ www to to to (N CN CN 3 a S » « H FH EH H WWW to to to O 3 » k H «1 in in in o n a « » « ct d=Jta t=i 3 * k « rn 00 r^ 3 m 3 >• » * C^J CN r-4 a D » to to ".to ^ ::r EH ^ SOWS » k to • EH EH CN EH w w a w CO to * to cn n EH ro » rs w o * » to » H Eh «Eh W W 3 W to CO » to CN '.N >H CN n C3 in 3 k k t3 k fn tri k £H W W X frj CO CO in to T~ ^O *"" ca 3 on k k LT) » in in C3 in a =3 k r) k k c^ • ^ clE3 ^ O 3 «» 3 k k fo k f^ ro S "^ S C3 k O k k(N » r^ ;n 3 CN ZD a 03 IT) in k H EH W W to to in to 3 D k k iri Eh w w to to k k EH EH W W CO to n a k k EH H W W to CO CN CN a a k k M W CO CO m in k k EH EH W W to CO in in a a EH EH W W to to a a k k EH H W W to CO rn n a a k k EH EH W K to to CN (N a a k k w c^q CO to a a a a k k k k in in in in a a a a k k k » ^ ^ =r :3a a a a k k k k ro ro rn O a a a a k k k k iN rj CN CN a a a a to w EH CN en cjtn a »»'k a a a >H — — ^ "— k Q a Q o X CO CO CO to — ' U^ W fi-l w to a a aaaa oaaaaa k — >-i '— ^«-~-' ~Q k Q Q Q c:3 a CO X to to to to to W '"' W f "-''*• "H -^ CO fH k Q o a Q n X CO to CO to to "-* [1^ fc, &U (i^ fc( CO M H CO w H EH 2: r-cN(^^m t^(Nrn^in »— cNc^^rin snaaa aaaatz) aaaaa*** roro"nr^cnroj-:j;3-cr::j:j-LninininLnLn* »• * tuiWCufcjP-nU^rvi&LiWWWPtitiiCti&jCuWfcH* * »» ^H S^ ^H ^^ ^^ k k k k k X! X X X X 00000 tn Ctj Cn (in W 00000* EH fH EH EH PH * TCN n cr u^ *• a a a a a * CO o H Eh CD CJ M a Eh a k-5 fr. a < >H W « W a a h5 « M* •« *• » -h* * * * * )«*<** *

PAGE 145

136 (N

PAGE 146

137 n ^ k •< =r =r « k H EH M W m to in lo s <» Eh EH 1/1 (/i 3^ D n W W CO CO on (^ H H W W to to (N O-J a 3 m to « • r-) ro k « in W ^ to ^ CN in (N » 3 » uO * "" » 6H » =r !i] in » to k 5-1 ^ H WOW to k to in H m a w r) k t/1 k H n H Haw to k (/I H CM a in 3 k a k H k en ty X w 10 in to rra •— =) so k in k in a in t3 k o k ^ k 33 -^ O k a k ro k m n ^ 3 k o k(N k rs) a rv) a k a k T— k T— a •— D k a Q k Q to X -jO b-i — Ph CO cq W t/0 to in in k k H EH t/l to 3 D k k H H W M to to r^ rr, a 3 k k H H to to Csl CN) 3 3 k k H H to to to to in m 3 3 k ^ H EH M W to to 3 3 k k H H M W r^ to 3 3 k k H EH W W to to 'N CM 3 3 k >. H EH Kl W to t/1 3 3 3 3 k k k k in in in in 3 3 3 3 k k k k 3-acT 33 3 3 3 k k s k m r~i r^ r^ 3 3 3 3 k k k k CN -\J CN fN 3 3 3 3 3 3 3 3 Q Q O Q to l-O to CO fH fi-i (i^ fn CO cq -a) S M H rq CN CN £N -l tW JH >-l tH k k k k k X tx: X --^ X o o o o o CH (i^ Cm tv fi^ m=rin »— fNncrin 333 33333* 3J^ in ^o in uT in in »« [ij Csj fc( &< &-< Ck Sh tu Cn * H to o H H Q U W z: tH 3 J Cm 3 !« W < Q a a -4 3 ~i o fs; oa * * * * * * * * * * * * * cq * •««!«* * 03 **: if K > * EH * Jc * * Eh * * pq •*»• •»* CO -h!» n* •if * h«• •hro

PAGE 147

138 er) nt rn rn % » •> * rCN m ^ • * • » H H H H bq W w W »t-tNf^^ fc^rsirnjfc^fNfrjOfcT-CNrr):3EH »*»»H »»«»»H *»%»fc-( »»»% c/i »*»»cn »»»%[/! »fc%%c/i k»%% fctotnoiin fctnvjnt/5 ocnt/jnto ^cotnt/jto HiDunintnE-finLnmLnHLDininLnHtnLninin (/} k»%«v^ kkkkio ««»»cn «»«» nE-lHHfc-inHfcHE-tHf^HH&-"Ho-)HHHH * v^ m m m •cntot/ic/i ki/jc/into ^i/iw^v) (/} H%%%01 »*%»t/5 *»*»(/) *%** CNHfc-tHHCNE-ltHHEHrNb-tHtHHCNHHt-'EH »cn(/5io^ s m m vi in oiocot^io fctocomuj irj s»%%c/) «.%*»co *»»%i/j »»%* '-IrttHMt-i'-Hfc-tHH'-t-lHHH'-HHtHH fcwcnioto <*v) m m in '>m in tn vi ^iDinvjto s » •> k ks « « % «a C3 » ., « » »He^HH tt^ir^^fr* ^ fr* ir< fH iri % tri (r* fr^ i-l »r303C3 •c3r>C53 »ar3D3 »r300o O <. » * »Q * » «. %0 » » » -.O •. * % % *0000 fcOOOO fcOOOO »oooo ;3*«»»3»»-.*=r »»i.»:3*»*» »3»Do fcosrsa *Qooo *asQO n fc » H »f^ « *. % %o-i » 1 » orn =r r-cNfocr T-ojrncr OD&S Saoa DSSO 03a3.t»» WN\ WW WW WW* bb4b-iC>4l^b-iD>-iPbiI^eHC>H>^l!b(P^I^[<-iP<-iP^I^(^'t» «' rn ro ro m H » * » *-t s * % s X X X X o o o o (*< gt, ^l^ fM * O O O O * H H H H * 4» ^rs) O :3» -If D 3 O CI #

PAGE 148

139 o u^ t rST O « o o * 10

PAGE 149

140

PAGE 150

141 in on o o H PB O u « o Q V3 C3 o H en H O o U pa CO in u w to O Q U H at O 03 O o o * It CO 25 M K H CN I M in 4II II « o o II to >»4 Oi S5 fcH a w < Q 03 H 00 II SB u u u u t t*l w o u C3 H J u w u u u in a • ( O r~ t— »— ctq II II II II U U O O rt! 03 m Nl CO H ra tt o II to o II o « u 03 II » u H CO « II CO s \ CO w a II o o u u u u to < t-l to a o u CO to w u o 03 u u u < o o II (N tN IN f>4 U U O O < 03 03 C<3 O o * u •H^ m o IN u II < t5 O O O O in •«• * »u u — ' CQ 03 Q » * — 03 < kC * H H CN -q v4 U W W «J Q Q II II II « rII M u u u

PAGE 151

H 142

PAGE 152

< 143 O EH O U 04 O e-t » • P3 W >< S> H a to H Cn O ID CN in o o * tsi o a in * O t-t O in cj M P4 o EH H H pa < H en CN o EH O o . m «! (1^ in -t rtH tn ^ + E; t/j rS3 S) -H Q H a Oi H u > Z5 H O S3 "II P4 N ts3 W Pl< H a 3 « M H to + D3 P3 ^o w o tH "* II o + •pa II W ISI (S3 Z 53 VI cn II II CSl P3 » «: *-E-i ISI en o tn « O O H » «H fc, O H U «; <: o o P4 fci o in un (N m in in u u u o »U U u H H w o Q W 3 W o > >H ca w H CD H W H Q «: w ca H ?H H H O rCN o en O Q S5 25 »H Q H «: 2! w o « u o t Pm Oh pq O U CO o H EH ft, N H -^ o o < B5 O M H H to O U4 W « u o H ^ cq w CO A 03 I w in E-t II < M H CM en Q w « o o en A I II •-3 m in II II •» * 03 Q o o o o CN CN ^ 2. o « «: ««c u u u u u u u u u

PAGE 153

144 cr. II

PAGE 154

145 u

PAGE 155

146 < s » O U • KM »4 H Q

PAGE 156

147

PAGE 157

148

PAGE 158

149 Q U CO in o H « M w H Q W W M U Oi < O W H M w Q a PO EH W n c/1 KB W Q H H W U u c^ o w 3 tn CU CO to a-i •« M o ^ a tn w H tH 2^ o O u a w o >H O H-l I'M S M o H CM W M H H «< O Oi 35 » D H a a « s: • H II 03 fN W *. 3 H U ~Oi « — w aa o «a: en — s w Q ^^ * o o t • tn in H M W H \ U *• pa * •If cu w * P3 n « C/5 a, S3 53 00 a w 03 Oi fcH =3 CM z; M U H H P3 3 II CM to ^3 C4 «-— . H cn CO w a Q EH 25 » H a^o ^o o P5 r» T» H » cN m tQ Q >=«] K 5a ce w w o E-t H H H H « (K ,-» 38 1« m o ^ — . r»(N V5 O SB H ^«« (S w o CM ID O CO H H « OI «H CM to « 25 «»; m (-> (ji — w ra 2S . t — -ca Q w » » H — — ' M M (i-i &H a H M II • z; [*4 to Q pa na o CM C3 P3 H H CM P3 2; 31 fN ^ C3 -— CVi 25 (O 25 «* O N • 25 T~ fSJ » II • —. ^ • H irf Q — to — w )-M ix a Pm O — H CM 0* =** to < w U 03 S H cj: o H O C3 pa w^ 25 tti W M m 1*3 CO 25 u 03 o fe-i o CM PM H W 33 to U U < M fcH 33 CO 3 O CM 25 H H H K fc-t at H S i-l W Q O to m to W 25 53 H > w pa H hJ a H CM Q S3 -a: H W W H H 25 O H <»: N < H CO ^ oU Q "J trt pa o 25 EC O Q W P5 «a; W U c:) w «) CO u Q W 25 25 Q H -a: t-t I CQ n H f ui 25 II -=C CH M W Q z 6-1 W O « «: H >-l W C3 25 H 1-3 W C3 «: H M W P Q 23 « CQ C5 H O .m 1^1 03 n O on n EH Q < H W ^ V.— 03 II pa UUCJUUCJu u UCJuCJ uuuuuuuuuu uuuu uu

PAGE 159

150 * * * * * * * * * * « u H EH W ^5 O w u o H < O O CO 2: H 3: O ^A O ^H C3 O s u u + + * * * en H C3 H H o o U M a u H cd w X Q C3 A /\ I I + + « u * S S Q O « W * •• + + + +» (N * < U U O U U * * * A A A A A A I I I I I 1 * a a s # * + + * * < m cQ CQ PQ cq # + + + + Q 4. * «3; < rt rt << «3 »»• )»• to £3 CJ H H H w u <: Q >M (_) Ca 32 u U 04 03 en 2; o H H « » » * *•***•* * 4^ n o H o o M H U E-1 W Q IN O H O w M H O U ^ «• I fN tn Q cn R CQ V3 V) en « o 1— m z z I H I! II rCN f^ ^ • 3* II a on II w. 4. tn en CT tn — o Q Q Q en Z z: S Z Q » H % •• z M H M H 'en en en en t^ H H H H M n M E-t O U rsi en en o 25 H M en o 25 en uuuuuuuuuuuuuuuuuuu u u u u u u

PAGE 160

151 H Ck S5 H to

PAGE 161

152

PAGE 162

153 + in M H •.CN r»~ II »• cq + — n H « H H O! *-« 3t V3 ^-, H O -^ • H W I CN rr+ to to ato B3 H M aotMQy II fciOoo f^QMMaaH&uu CN n *

PAGE 163

154

PAGE 164

155 *

PAGE 165

156

PAGE 166

157

PAGE 167

158 H W Vi ir» D H PU to St a h H W VJ EH tn (N a H W =3 in o m o >H X be;

PAGE 168

159 »»*•»* . to a t-t w z CO U w o u o H CO H H U W w M (£( a I I VO CO 10 H W sa Z5 H O H o o « tri H H z: D O £ PM 05 •D o H H M •a: M H n. U t>-i «! H H Z 10 OS 03 W 05 H 6m U tH 05 u o H H O Ol ss H H =3 CM E-t o * * * * * * o o o (N O H O * M »*»* Of ^ 1*1 \M • WW u o to 1-5 M H fc-t O u u u u u u H U U o o o * * * * * * * *««*** * * P«^ o CO o M H M Zi • • • i < • in EH II W M CO »zs H M O 03 N. B H M n lit pm z; m A A I CO I I »w M U-) II U W W Z I H a< o w CO Z! »CO I I H r> I * w * UUUUCJOCJUUUUUUUU » *

PAGE 169

160 m rT-

PAGE 170

161 # « * # « * * 9 u o w H < > * * * * * I H O in ^ S \D D II . II --O ^^^ O C3 D U SS II II II II II rN m ain tn » O U «— — >-'^»O ;b II II II II II ^ II II II II H ^ r» « cri o E-i a r»— r" rCN O '—— — >»— O a H «" (N rn ^ in t>« »
PAGE 171

162 CN

PAGE 172

163

PAGE 173

164

PAGE 174

165

PAGE 175

166

PAGE 176

167 « * H 55 W ca :z a« w o o M U to o H U W * * « » in * * u H W H H rtj < 03 H EH P3 S3 Oi W O u 03 21 Oi O Oh U < pm pm o o 3 w H CQ H -< S M w pa u > ID ,-,^^,-»*-«.-».— .^-.--".^^(Nf^f^'^f^'^ T-T-»-T-T-CM(Nrs)D3~II II II II II II II II II II .O H * * ^ T(N| (T, 3O a 3 C3 U) ^ P> to £3 D C5 D a 3 W a •CN 00 :}• in ^ ^! a D a o a II II II II II o H --. ^^H :z; r-. IN rr) a' m U O O S 13 O C) u u u

PAGE 177

168 lii un w T— Q o CN 1^ 3" Ln 3 3 = 00 OOOOOOvOOOOO :3 II II II II II s: II II II UOOOOOOU3DO «• in C3in w O D O II II ^ =r ID tH
PAGE 178

169 H

PAGE 179

170

PAGE 180

171

PAGE 181

172 c

PAGE 182

173

PAGE 183

174

PAGE 184

o

PAGE 185

BIBLIOGRAPHY Alleu, A01961. The Radiation Chemist rjjf of Water and Ajjueous Solutions. Princeton, New Jersey: D. Van Nostrad Company, Inc. An bar, M. 1968. Fundamental Processes in Radiation Chemistry. ed, P. Ausloos, 651-680* New York: Interscience Pablishers, Wiley and Sons, Anderson, A. R. 1968. Fundamental Pro£«sse^s in Radiation Chemistry. edP. Ausloos, 281New York: Interscience Publishers, Wiley and Sons. Anderson, A. E. , B. Knight and J. A. Winter. 1964. Nature. 201: 1026. Anderson, A. R., B. Kniqht and J* A, Winter. 1966a, Nature. 209:199. Anderson, A. E., B. Kniqht and J. A. Winter. 1966b. Trau. LiO^iiY Soc. 62:359. Beenakker, C, I. M, , F, J. de, Heer, H. E. Krop and G. R. Mohlmann. 1974. J. Chera. Phjs. 6:445-454. Berqer, M. J. 1963. Methods in Com£Utational Physics. 1:135. New York: Academic Press. Berqer, M, J., and S. M. Seltzer. 1964, Tables of energy losses and ranges of electrons and positrons, in Studies in Penetration of Ch arg ed Particles in Matter, Publ. 2133, 205. Washinqtou, D. C. : Nat. Acad. Sci. Ees. Couric. Boag, J. W., and H. J. Hart. 1963. Nature. 197:45. Brenner, D, J. 1982. A Comgu tationally Conyenieat E^rameterization of Experiment a 1 Angular Distributions of Il9'1 Elkerg.1 Electrons Blast ically Scattered off Kater Va^or. Los Alamos, New Mexico: Los Alamos National LaoBrenner, D. J,, and M. Zaider. 1983, In proceedinqs. Workshop on the Interface between Radiatign Chemistry and E.adiatiou Physics, 58. Prepared for U. S. Dept. of Euerqy, Contract W-311 09-'ENG-3Arqonne, Illinois: Arqonne National Laboratory, 176

PAGE 186

177 Brinkmann, R. T. , and S. Trajmar. 1970* ABB* GeojEhx* 26:201. Bromberq, J. P. 1975. In proceedings. Ninth International Conference on the Physics ox Electronic and Atomic Collisions, (Seattle, Washington)", ed. J. S. Risley and R* Geballe, 98-111. Seattle, Hashinqton: Univ. of Washington Press. Bruche, E. 1929. Ann. Phisik. (GeriDany). 1:93-134. Chatteriee, A. , and J. L. Magee. 1980. J. Ph..Ys. Chem. 84:3537-3543. Debierne, M. 1909. Con£t. Rend. 148:703. Douthat, D. A. 1983. In proceedings Workshop on the Interfacebetween Radiation Chemistrx arid Ead.iation Phj^sics, 134. Prepared for U. S. Dept. of Energy, Contract W-3 1109-EKG-3. Argonne, Illinois: Argonne National Laboratory. Dyne, P. J., and J'd. Kennedy1958. Can. J. Ciiem. 36: 1518. Eyring, H. , J. 0. Hirschf elder and H. S. Taylor. 1936a, J. Chem. ?hY:s. 4:479. Eyring, H. , J. 0. tlirschfelder and H, S. Taylor. 1936x1. J. Chem. Ph^s. 4:570. Ferguson, E. E1973. Atom. Data Nucl. Data Tables 12:159. Flanders, D. A., and II. Fricke. 1958, J. Chem. Phys. 28:1126. Fujita, T., T. Iwai, K. Ogura, S. Watanabe and Y. Watanabe. 1977. J. Phis. Soc (Japan). 42:1296-1304. Ganguly, A. K. , and J. L. Magee. 1956. J. Chem. Phjfs. 21:1080. Good, A., D. A. Burden and P. Kerbarle. 1970. J. Chem. Phis. 52:212. Green, A. E. S. , and S. K. Dutta. 1967. J. Geqphjs. Res. 72:3933-3941. Green, A. E. S., E. H. Garvey and C. H. Jackman. 1977. Int. 1' Quantum Chem. S^mp. 11:97-103. Green, A. E. S. , C. H. Jackman and B. H. Garvey. 1977. J. Geophys. Res. 82:5104-

PAGE 187

178 Green A. E. S,, and J. H. Miller1974. Physical Mechanisms in Radiation Biolggj. ed. E. D. Cooper and R. W. Wood, 155-176. Conf. 721001. Washington: Tech. InfoCtr. , Of Cm Info. Services. Green, A. E. S., J. J, Olivero and fi. H. Staqat. 1971. Bioph ysical Aspects of fiadiation ^aiitj^. Vienna: Int. Atomic Energy Agency. Green, A. E. S. , and D. E. Bio. 1983. In proceedings. Workshop on the Interface betw een Sa dia t ion Che raistry p'.A.^ R^Mittion Physics, 65. Prepared for U. S. Dept. of Energy, Contract W-3 1109-ENG-3. Argonne, Illinois: Argonne National Laboratory. Green, A. E. S,, and T. Sawada. 1972. J. Atmos£ii. Terr. Phjs. 34:1719. Green, A. E. S. , and R. fi. Singhal, 1979. Geo^hys, Res, L ett ers. 6:62 5-628. Green, A. E. S. , and fi. S. Stolarski. 1972. J, Atmpspji, Terr. Phys. 34:1703-1717. Grosswendt, B, , and E. Waibel. 1978, Nucl. Instrura. M^th, 155: 145-156Grun, Von A. E. 1957. Z, Naturforschcj 12a: 89-95. Haam, S. N., R. H. Ritchie, J. E. Turner and H. A. Wright, 1983. In proceedings, iior|csh.O£ on the Interface between R§:^i3,tign Chemistry and Radiation £hysics, 3 2. Prepared for U. S. Dept. of Energy, Contract W-31 -1 09-SJ1G-3. Argonne, Illinois: Argonne National Laboratory. Hamm, RN., H. A. Wright, J. ETurner and fi. H. Ritchie. 1978. In proceedings, 6th Syjjgosium oa Microdosimetry (Brussels, Belgium), ed. J, Booz and H. G. Ebert, 179. Commission of the European Communities, EUH-6064-DE-2NER. London: Harwood Academic Pub, Hanralian, E. J. 1983. In proceedings. Workshop on the Iy:i§Ef3,ce between Radiat ion Chemistry an^ Radiation Physics, 132. Prepared for 0. S. Dept. of Energy, Contract W-3 1109-ENG-3Argonne, Illinois: Argonne National Laboratory, Hilgner, W. , J. Kessler and E. Steeb1969. Z. Physik (Germany) 221:324-332. fiindmarsh. A, C. 19 74. GEAR: ordinary differential eguation system solver. Report OCID 30001, Rev, 3, December, Oakland: Lawrence Livermore Laboratory.

PAGE 188

179 HirscLfelder, J. 0., C. F. Curtiss cind B. B. Bird1954. Kolecular Theorj of Gases and LiSQ?ids. New York; fJiley and Sons, Huat, J. S. 1976Advances in Radiation Chemistry, Ed, K. Burton and J. L. Maqee, 185. New York: Interscience Pub., Wiley and Sons. Huntress, W. T. 1973. Astrophy. JSujg^l. Ser. 33:495. IMSL Library Edition 8. 1981Houston, Texas: International Hathematical Subroutine Library, Inc, Inokuti, M1983. In proceedings, Sorkshoj on the Interface between Radiation Chemistri and Radiation PhYgics, 4, Prepared for U, S. Dept. of Energy, Contract W-31-109-SNG-3. Arqonne, Illinois: Arqonne National Laboratory, Jackman, C, U. 1978. Spatial and Energetic Aspects of Ei:^£tJ.09 Easrgi Deposition.. Ph. D. diss. , (Jniversity of Florida, Gainesville. Jackman, C. H. , R. H, Garvey and A. E. S. Green, 1977. J. Geo£his. Res. 82:5081-5090. Jackman, C. H. , and A. E, S, Green. 1979, Geofihis. Pes. 34:2715. Jaffe, G, 1913. Ann. Ph^sik (Germany) Ser, 4, 42:303, Jaffe, G, 1929, Ph^s, Zeit (Germany) 30:849, Jura, H. 1971. Models of the Interstellar Gas. Ph. D. diss.. Harvard Qniversity, Cambridge, Massachusets . Kee, E. J-, J. Warnatz and J. A. Miller. 1983. A Fortran Computer Code Package for the Evaluation of Gas-Phase Viscosities, Conductivities and Diffusion Coefficients, Sandia Report, Sand, 83-8209, Contract Dif:-AC047bD?00789, Albuguerque, New ^5exico and Livermore, California: Sandia National Laboratory. Klots, C. E. 1968, Energy deposition mechanisms, in Fundamental Processes in Radiation Chemistry, ed. P. Ausloos, 1. New York: Interscience Pub., Wiley and Sons. Kupperman, A. 1959. J. Chem, Educ, 36:279, Kupperman, A, 196 1. In The Chemical and Biological Action oj Radiations, ed, M. Haissinsky, 5:85-166, Mew York: Academic Press,

PAGE 189

180 Kupperman, A. 1974. Ph ysi cal Mechanisms in fiadiation Biologjr. ed. RD. Cooper aad E. 'A. Wood, 155-176Conf. 721001. Washinqton: TechInfo. Ctr., Ofc. InfoServices. Kuppenaan, A., and G. G. Belford. 1962a. J. Chem. Phis. 36:1412-1426Kupperraan, A., and G. G. Belford. 1962b. J. Chem. PhY§36: 1427-1441Kutcher, G. J., and A£. S. Green1976aJ. Applied Phys ics 4 7:2175. Kutcher, G. J, and AE. SGreen1976b, gad. Res. 67:408-425Lea, DE, 19 46. Actions of Radiation on Liviug Ceils. New York: Caaibridqe University Press. Lea, D. S. 1947. British JRadio.1. Suppl1:59. Leu, M. T., M. ABiondi and fi. Johnsen. 1973. Rhy^s^ Rev. 8:413. Lind, S. C. 1921. The Che mical Effects of Alpha Particles §:M ElgStronsNew York: The Chemical Cataioque CoLind, SC. 1938. The Chemical Effects jof Alfiha Particles and Electrons, 2d edNew Yor)^: The Cheiaical Cataioque Co. Maqee, J. L1955. J. Chem, Phjs. 52:528. Maqee, J. L. , and A. Chatteriee, 1978. J. Ph^s. ChejQ. 82:2219-2226. Maqee, J. L. , and A. Chatteriee. 1980. J. Phyi^. Chem. 84:3529-3536. Mark, TD-, and F. Sqqer1976Int. J. Mass S^ectrgm. Ion Phjs. 20:8 9-9 9Marrero, T, R. , and E. A, Mason1972JPhjrsChem. Ref. Data 1:3Massey, HS. H. , and C. B. 0. Mohr. 19 33. Prgc. Roy. Sgc. LoMon Al40:613-636Miller, JH. , and AE. SGreen. 1973. Pad. Res. 54:343-363-

PAGE 190

181 Miller, J. H, , and W. E. Wilson. 1983. In proceedings, Workshog on the Interface between Radiation Chemistrx and i^diation Physics, 73. Prepared for U. S. Dept, of Energy~ Contract H-311 09-ENG-3 Arqonne, Illinois: Argonne National Laboratory. . Mohlinann, G. R., and F. Jde Heer. 1979JChem. P.hjfs40: 157-162Monchick, L. , J. L. Maqee and A. H. Samuel. 1957. J. Chea. Zhis. 26:935-941Monchick, L, and EAMason19 61J. ChemPhy,s. 35:1676. Mott, NF-, and HS. HMassey1965. The Theory ofAtomic CollisionsLondon: Claren Press. Mund, W. 1935L 'Action Chimiiiue des Ray„ons AlEha en Phase Gazeu^e. Paris: Hermann et Cie. Myers, Jr., LS1974Physi cal H eg nanisms in Hadiation Biplogi. ed. E. D. Cooper and R, W. Wood, 185-206. Conf721001Washington: TechInfo. Ctr. , Ofc. Info. Services. Sishimura, fi. 1979. In proceedings, Hth Intexnatignal Conference on the Physics of Electronic and Atoaic Collisions, Kyoto, Japan, edK. Takayanagi and UCda, vol. 2, 314. Kyoto, Japan: The Society for Atomic Collision Research. Olivero, J. J., SW. Staqat and AES. Green1972JGeophys. Res. 17:4797-4811Paretzke, H. G., and «. JBerger1978. In proceedings, 6th SimEOsiuiE on Microdosiiaetri (Brussels, Belgium) , ed. JBooz and~HgT Ebert, 749-758, Commission of ti.e European Communities, EUR-6064-DE-EN-FELondon: Ilarwood Academic Pub. Peterson, L. R. 1969. Ph^s. Rev. 187:105. Platzman, R. L. 1955. Rad> Res. 2:1. Platziaan, R. L. 19 74Physical Mechanisms in Radiation Biology. ed. R. D, Cooper and R. a. Wood, 124Conf, 721001. Washington: TechInfoCtr., Ofc. Info. Services. Porter, H. S. , and F. W. Jump1978. Report CSC/TM-78/b 1 7. Prepared for Goddard Space Flight Ctr-, Washington, D. C: Computer Science Corp.

PAGE 191

182 Ritchie, H. H. , R» NHamm, J. E. Turner, and T. P, Turner. 1978In proceediuqs, _6th Symposiua on Jiicrodosiiuetrx (Brussels, Belqiuai) , ed. J. Dooz and HG. Ebert, 345-354, Commission of the European Comnunities, EUR-6064-DE-EN-FR. London: Harwood Academic Pub. Samuel, A. H. , and J, L, Maqee, 1953, J, Cheja* ELXS, 21:1080. Scholes, G. , R. L. Hilson and M. Ebert. 1969CheD;Comaun. 1:17-18Schutten, J., F. J. de Heer, H. R, Houstafa, A, J, H. Boer boom and J. Kisteraaker. 1966. J. ChemPnys. 44:3924-3928. Senq, G. 197 5. Das System (e-Hg^O) Differ en tielle S t r e u e X pe r i me n t bei Klexnsten Euergaen. Thesis, Kaiserslautern University. Sinqhal, R. P., and A. E. SGreen. 1981. J. of £eophYS. Res. 86 (A6) :4776-4780, Sinqhal, R. P., C. H, Jackman and A. E, S. Green. 1980, J. Oi Geo£hY.s. ges. 85 (A3) : 1 2461254 , Sinqhal, R. P., D. E. Rio, P, F. Schippnick and A. E. S. Green. 1983. Rad. Res. 95:32-44. Spencer, L. V., and V. Fano. 1954. Physical Review 93:1172-1181. Spinks, J. W., and S. J. Woods. 1976. An Introduction to Radiation Chemistry. New York: Hiley and Sons. Thomas, J, K. 1969. Advances in Radiation Cheaistry. Sd. tl. Burton and J. L. Maqee. 1;112. New York: Interscience Pub, Wiley and Sons. Traiman, S. , W, Williams and A. Kupperman, 1973. J. Chem. Phis. 58:2521-2531. Turner, J, E., H, G. Paretzke, R. N, Eiamm, H. A. Wriqht and R. HRitchie. 1983. In proceedinqs. Workshop on the Interface between Radiation Chemistry and Radiation Physics, 9 1. Prepared for Q. S. Dept. of Enerqy, Contract H-3 1109-EUG-3. Arqonne, Illinois: Arqonne National Laboratory. Trt'ODEPEP Edition 3. 1981. Houston, Texas: International Mathematical Subroutine Library, Inc.

PAGE 192

183 Venuqopolau, M* , and BA. Jones. 1963Chemistrjr of Dissociated Water Vapor and Related Syste ms. New York: John Hiley and Sons, Heiss, J, 194^1. Nature. 153:748. Restbrook, C. K. 1981. Combustion Science and TechQ.Pioay» 1:1. Uilson, H. E., L. H. Toburen and H. G. Paretzke. 1978. In proceedings, 5th Symposium on Microdosimetrj (Brussels, Belgium), ed. J. Booz and H. G, Ebert, 239. Comciission of the European Coiuaiunities, EDE-5064-DE-EN-FR. London: Harwood Academic Pub. h'riqht, H. A., J. E. Turner, EK. Hamia, H. H. Ritchie, J. L. Maqee and A. Chatterjee. in press. In proceedings, Eiailth Sytapogiuffl on Micrpdosimetrj (Juliph, i^est Germ any) . Wright, fi. A., J. £. Turner, E. N. Haam, BH. Ritchie, J. L. Maqee and A. Chatterjee. 1983. In proceedings, Wqckshop on the Interface between Eadiatioii Chemistry and Radiation Physics, 106. Prepared for U. S. Dept. of Energy, Contract W-31109-SNG-3. Argonne, Illinois: Arqonne National Laboratory. Zaider, M. , and D. J. Erenner. 1983. In proceedings. Wo rksho p on the I nterface between Radiation Chemistry and K^Mation Paysics, 113. Prepared for U. S. Dept. of Energy, Contract 'W-3 1-10 9-ENG-3. Argonne, Illinois: Arqonne National Laboratory. Zaider, H., D. J. Brenner and W. E. Wilson. 1982. The Applic ation of Tracjc Cal culations to Sadiobiolpgy I. ilonte Carlo Simulation of Proton Tra^SJSs. New Mexico: Los Alamos National Lab.

PAGE 193

BIOGfiAPHICAL SKETCH Daniel Edward Rio was born in Tampa, Florida, on Auqust 4, 1951. His parents, Evelio and Jasmine Eio, are native Floridians and his father graduated from the Colleqe of Engineering at the University of florida in 1952, He has one brother, Hichael, born seven years later. Developing an interest in aathematics and physics at a young age, he continued to study these fields in colleqe. He received his BA and HA degrees froia the Qniversity of South Florida, having aajored in mathematics and physics as an undergraduate, and in physics as a graduate. During his years at the University of South Florida, he held positions as tutor for the mathematics and physics departments, adjunct lecturer for the department of mathematics and graduate teaching assistant for the physics department. To aid in the support of his education, he also had many parttime "jobs. Daniel was married to Patti Duggins in the spring of 1976* Soon after, he entered the program of biomedical engineering at the University of Florida. At this time he accepted a position as research assistant with A. E. S. Green, thereafter producing a number of publications wnich added to iiis knowledge of various related fields and 184

PAGE 194

185 enhanced his education. While workinq on his deqree he became the father of two beautiful baby qirls, Loqan and Chelsea. He is currently employed at the National Institutes of Health in Bethesda, Maryland.

PAGE 195

I certify that I have read this study and that in ray 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.
PAGE 196

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 deqree of Doctor of Philosophy. Genevieve S. Eoessler Associate Professor of Nuclear Enqineerinq Sciences This dissertation was submitted to the Graduate faculty of the Colleqe of Enqineerinq and to the Graduate School, and was accepted as partial fulfillment of the requirements for the deqree of Doctor of Philosophy. December 1983 't-JL Kl ' O/^-^^ Dean, Colleqe of Enqineerinq Dean for Graduate Studies and Research