Group Title: study of the effects of diffusion and kinetics on the spatial distribution of products created by electron deposition /
Title: A Study of the effects of diffusion and kinetics on the spatial distribution of products created by electron deposition /
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Title: A Study of the effects of diffusion and kinetics on the spatial distribution of products created by electron deposition /
Physical Description: ix, 185 leaves : ill. ; 28 cm.
Language: English
Creator: Rio, Daniel Edward, 1951-
Publication Date: 1983
Copyright Date: 1983
 Subjects
Subject: Radiation chemistry   ( lcsh )
Collisions (Nuclear physics)   ( lcsh )
Nuclear Engineering Sciences Ph. D
Dissertations, Academic -- Nuclear Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1983.
Bibliography: Bibliography: leaves 176-183.
Statement of Responsibility: by Daniel Edward Rio.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00098267
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000483065
oclc - 11826643
notis - ACQ0873

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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




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