Particle Size Determination for

Alpha-Emitters Using CR-39

Gyorgy Hegyi

Department of Medical Physics

McGill University, Montréal

Wy 1999

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FULFILLMENT OF THE REQUEEMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN MEDICAL RADIATION PHYSICS

O Gyorgy Hegyi 1999

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Abstract

This project is to develop methods to retrospectively

determine the size of alpha-emitting particles that have been

coliected on personal air sarnplers. The alpha radiation from such

particles produces a cluster of tracks on the surface of an etched

nuclear track detector, CR-39. The number of tracks in a cluster, as

well the diameter of the cluster, are dependent on several factors:

the diameter of the hot particle, the distance between the particle

and the CR-39 surface, the composition of the particle, and the

aip ha-particle energy.

The dependence of the alpha-ernitting particle size and the

number of registered tracks were revealed, and produced

predictions of the track density distribution observed on the CR-39

plastic. There is a good f i t between the simulation of track density

observed on the CR-39 and the tracks arising from uranium oxide

and plutonium oxide particles.

Résumé

Ce projet consiste à ddévelopper des méthodes pour déterminer le

diamètre de particules radioactives alpha émethices, recuueillies par un

échantillomeur d'air personnel. La radiation alpha produit des groupes de

traces sur la surface d'un détecteur de traces nucléares gravées, CR-39. Le

nombre de traces dans un groupe et le diamètre de ce groupe dépendent de

plusieurs facteurs: le diamètre de la particule active, la distance entre la

particule et la surface du CR-39, la composition de la particule et l'énergie de

la particule alpha.

La dépendence du diamètre de la particule active alpha émetrice au

nombre de traces produites a été démontrée. Il est aussi possible de prédire la

distribution de densité de traces observée sur le plastique CR-39. ll y a un bon

accord entre la simulations des densités de traces observées sur le CR-39 et les

traces produites par les particles d'o xide d'uranium et d'oxide de plutonium.

Acknow Zedgments

I would like to thank my supervisor, Dr. Richard Richardson, for

his valuable advice and suggestions. Also I am thankful for his

patience in reading the manuscript over and over again, and putting it

on its feet.

1 would like to thank Dr. Ervin B. Podgorsak for providing me

financial assistance.

Also Stan Frost, Engin Ozberkand Hailing Liu of the Comeco

Corporation for supplying the particles of natural uranium used for

validating the technical and analytical methods.

In addition, 1 would like to thank Patrick Wilson for kindly

supplying the composition and density of plutonium oxide and mixed

oxides used in the AECL fuel developing sites.

1 would also like to thank Bruce Heinmiller for checking my

specific activity calculations.

Also, 1 would like to express my gratitude to François DeBlois for

helping me translate the abstract into French.

Finaliy, 1 would like to thank Sheliy Starling for the experimental

data suppfied for the validation of the theoreticai results.

List of figures

Figure 1.1

Figure 2.1

Figure 2.2

Figwe 3.1

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

The regional deposition fractions in habitually oral-breathing workers. Figure courtesy Robert Coms [1996].. ................ 1 1

.......................... Track etch rate variation uersus range.. -15

T ~ E measurable parameters of a track etch cone. ... ......... -16

Alpha particle range in CR-39, calculated with hiedlander,

[1964] approximation CO mpared with alp ha-parti.de ranges

........................................... supplied b y Fews, [1982J.. .19

Schematical representation of an alp haemitting point

........................... source, P above a CR-39 track detector. 22

Schematic representation for the experimental set up (not to

scale) ...................... ... .............................................. -24

Represenfation of the 3 dimensional integration of

registratio n e ffectiveness of infinitel y small sources

throughout the hot particle (not to scale).. ....................... .26

Restriction schemes for the registration effectiveness

integral evaluation ..................................................... .2 9

Figure 4.5 Flow chart for the evaluation of the solid angle R(q,<p) as

................. integrand for the Gauss quadrature scheme.. .3 1

Figure 4.6 Representation of the path of an alpha particle in Cartesian

............................................. coordinates (not to scale). -3 2

.................. Figure 4.7 Flow chart forfinding the limiting angle 8 iim.. -36

Figure 4.8 Case Ob function G(u) with parameters: p i = 1 1.46 g cm-? p2

= 0.001226g cm-3, p, = 1.3259 cm-3, Rc = 32.0 Pm, r = 6.0

pm, h = 30.0 ym, q = -0.5pm, <p = 30") 4 = I O 0 ................. 37

Figure 4.9 Case I a Function G(u), with parameters: pi = 1 1.46 g cm-3,

p2 = 1.226~ 1 0 3 g cm-3, p, = 1.325 g cm-3, R = 32.0 pm, r =

6.0pm, h=30.O ym, q = 3 . 5 p m , <p = IO0, 4 = I O 0 ............. 38

Figure 4.10 Case 1 a and 1 b for a point source track cluster (not to scale)

Figure 4.1 1 Case 1 a Point source track cluster for conditions shown in

Figure 4.10, with coordinates of point source P r + = 2.985

..... pm, yp=0.526ym, zp = 1.75pm, r=5pm, h = 3 0 p m 40

Figure 4.12 Case 1 b Fùnction G(u) with parameters: p i = 2 0. O g cm-3, pz

=l.Og cm-3, pc=1.5gcm-3, Rc=40.0pm, r=6.0pm, h

......................... = 30.0pm, q = 6.0pm, <p = 90°, + = 120° 40

Figure 4.13 Case 2 Function G(u) with parameters: p l = 1 1.46 g cm-3, p2

= 1.226x103gcm-3, p c = 1.325gcm-3, Rc=32.0pm, r=6.0

.................. pm, h = 30.0 ym, q = 3.3 pm, <p = 140°,4 = 80° 41

Figure 4.14 Case 2 Point source track cluster resulted in conditions

shown in Figure 4.4, coordinates of point source P: x, =

3.953 pm, yp = 0.697 pm, zp = -2.81 1 pm, r = Sprn, h = 30

................................................ ......................... pm. .. -42

Figure 5.1 Representation of geometncal set-up for the detemination of

the track distribution fitnction.. ........................................ -45

Figure 5 -2 Typical probability density for 239PuO2 particle, d = 1 pm, p =

1 1 -43 g cm-3, for different particle - detector separation ..... -48

Figure 6.1 The variation of regiçtration effectiveness of alpha-parficles

emitted by 2 3 9 P ~ 0 2 of p = 11 -45 g cm-3> versus hot particle -

detector separution.. .. ................................................... .53

Figure 6.2 Three nQtU02 particles and the respective CR39 track

clusters, each to the lower lefi of their respective hot

p article. ........................................................................ -5 4

Figure 6.3 Dependence of registered number of tracks on particle

composition and size .................................................... -58

Figure 6.4 Registration effectiveness for nat U02 particles exposed with a

400 prn air gap ............................................................. -59

Figure 6.5 Cornparison of predicted and measured diameters of natU02

particles .................... .... ........................................... .6 1

Figure 7.1 Calculated radial probability density for a U 0 2 particle, ( d m e ,

= 64 pm) with a 400 p m spacer compared with radial

probability density resulted from experimerztal determination

-65 ....................................................................................

Figure 7.2 Predicted (theoretical) track distribution for a U 0 2 pa?ficle,

(dm, = 64 prn) compared with experimentally detemined

counts ........................................

Figure 7.3 Calculated radialprobability density for a U02 particle, (dmeas

= 76pm) with a 400 prn spacer compared with radial

probability density resulted Rom ewpenmental

.............................................................. d etennination.. .66

Figure 7.4 Predicted (theoretical) track distribution for a U02 particle,

(dmem = 76 pn) compared with experimentally detenined

counts.. ......................................................................... -66

Figure Al Geornetrical parameters used in the proue of the solid angle

formula. .................................... .. ................................ -74

List of Tables

Table 4 . 1

Table 6 . 1

Table 6.2

Table 6.3

Table 6.4

Surnmary of thejive cases where findion G(u) = O... ... ....... 35

Registration effediveness for a 1 pm diameterparticle as a

function of cutoff angle, OcumE and minimum dip depth L . 5 1

Registration eflectiwness for a 1 0 pm diameterparticle as a

function of cutoff angle, & m ~ and minimum dip depth &52

Erperimental data used for calibmtion calculations.. . . . . . . . . . .60

Best fit equation for the predicted diameter from input of track

cluster data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

Table A l .a Specific ah'vities for aged Ar (1 983). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75

Table A 1 .b Specific actiuities for mtural uranium.. . . . . . . . .. . . . .. . . . . . . . . . .. . .,.75

Table A1.c Specific activities for low enriched uranium ...................... 76

Table A l .d Specifc activities for depleted uranium.. .. . .. ... . .. . . . ... ... .. .. ..76

Table A 1 .e

Table A 1. f

Table B1.a

Table B1.b

Table B2.a

Table B2.b

Table C1.a

Table C l .b

Table D 1

Table E. 1

Table E.2

Specifc acriuities for mixed oxide MOX of aged plutonium

with mtural uranium.* ..-............... .,,,. ............................ -77

Specific activities for mixed oxide MOX of aged plutonium

................................................. with depleted uranium.. -77

Exposure tirnes for U02 for 10 tracks with 400 pm air gap..78

E-posure times for U02 for 1 0 tracks with no air gap.. ....... -79

Exposure times for 10 tracks from 15 years aged Pu02

partrocles with 400 pm air gap.. ........................................ -80

Exposure times for 10 tracks from 1 5 years aged Pu02

particles with no air gap.. ............................................... .8 1

DiçtTibution of trackper days for ............................. ..82

DzDzstntnbution of trackper days foragePu. ..................... .. ..... 83

The alpha particle energy and its range in Cr-39 for al1 alpha

ernitting radioadive isotopes mnsidered in this thesis.. ..... .84

Cornparison betweenpredided and experimentally

detennined truck counts for sample one shown in Figures

7.1 and 7.2..-• ............................................................... 85

Cornparison between predicted a d expenmentally

determined track counts for sample one shown in Figures

7.1 and 7.2 ................................................................... 86

Table of Contents Chapter 1 3

Introduction 3

1.1. Radioactive isotopes present in nuclear reactor fuels used in

the nuclear power indus- 5

1.2. Radioprotection for inhaled radioactive particles 6

1.3. The effects of size on the aerodpnamics of airborne particles

1.4. The effects radioactive particles on the human respiratory

tract 11

Chapter 2 13

Characteristics of plastic nuclear truck detector 13

2.1. Solid state nuclear track detectors 13

2.2, Priaciples of track formation on the CR-39 nuclear track

detector 14

Chapter 3 18

Theoretical considerations related to truck formation 1 8

3.1. Alpha-particle interaction wïth matter 18

Chapter 4 21

Computatïon of registration effectiveness of the CR-39 detector 21

4.1. Registration effectiveness for a point source 21

4.2. Registration effectiveness of hot particles 25

4.2.1. Domain restriction schemes in the registration effectiveness integral 27

4.2.2. Determination of the soiid angle R as integrand of the registration

effectiveness integral 29

4.3. Evaluation algorithm for the generic Lùniting angle - 32

4.4. Methods developed to fmd the limiting angle 8u, 35

4.4.1.Description of cases Oa and O b (Table 4.1) with no solution of e ~ , - 37 4.4.2.Descnption of cases la and 1 b (Table 4.1) with one solution of - 38

4.4.3.Description of case 2 (Table 4.1) with two solutions of 61, 41

Chapter 5 43

Distribution of tracks in a cluster 43

5-1. Introduction 43

5.2. Radiai track formation probability density, D(o) fkom a single

disintegration event 44

5.3. Raaial track density, Db(o) from multiple disintegration

evemt during the exposure t h e period T 48

Chapter 6 50

Validation of the sizing method 50

Introduction

6.2. Dependence on the minimum dip depth L and the cutoff

6.3. Dependence on the air gap thickness 52

6.4, Experimental procedure 53

6.5. Errposure time evaluation 55

6.6. Determination of the computed particle diameter 56

6 -6.1 .Registration effectiveness of the mixed isotope particles 56

6.6.2.Verifcation of particle sizing method with naturai uranium oxîde

particles 57

6.7. Implementation of the size determination method into

routine sizing 61

Chapter 7 63

Validation of the sizing method for overlapping track clusters 63

Chapter 8 67

Discussion and conclusion 67

Appendix 1 71

Appendix A 75

Appendix B 78

Appendix C 82

Appendix D 84

Bibliograp hy 87

Chapter 1

Introduction

The widespread use of nuclear energy in the world raises

problems of radioprotection for personnel working at nuclear power

stations as well as nuclear fuel development sites. The greatest hazard

is related to inhaled radioactive particles [Richardson, 19961. Alpha-

ernitters are among the most harrnful. After deposition in the human

body, hot particles produce a high local dose even though the average

dose may be low. Also the level of hazard is still debated [Charles et al.,

19961. This can lead to the risk of cellular or subcellular damage, which

could lead to radiation-induced cancers. Inhaling hot particles is

undesirable, but once this has occurred, or the potential exists for the

occurrence, the evaluation of the acquired dose is essential. Interna1

dosimetry assesses the absorbed dose to the sensitive or target cells or

organs of inhaled hot particles, Le. the lung, stomach or other interna1

organs, where the inhaled radioactive materials may have migrated. To

evaluate the dose absorbed in the lungs from inhaled alpha-emitting hot

particles, the size of the particles must be evduated. A few attempts to

evaluate alpha-ernittuig particle size have been made in the past. Moss

et al. as early as 1961 made a simplifed size determinstion, u s h g track

counts which did not take into account the self absorption of alpha-

particles inside the hot particle. Some groups employed emulsion-based

alpha-particle detection [Sanders, 19791, and size determination by

reconstruction of track geometry data, combined with empirical fittulg

procedures [Akopova et al., 19931. The CR-39 solid state nuclear

particle track detector was preferred by many authors as the most

practical approach for taking autoradiographs of alpha-emitting hot

particles. Researchers at the University of Bristol used an automated,

optical image analysis system to measure the charactenstic dimensions

of individual registered tracks to ascertain the track length and

direction [Fews, 19861. This work was continued by Bondarenko et al.,

[1995, 19961 to derive the alpha-ernitting particle size starting from the

geometrical characteristics of the tracks registered on the CR-39

nuclear track detector. Later by backprojecting the geometrical tracks of

the alpha-particles, the hot particle size is determined by statistical

arialysis of the emerging points. Other groups had determined the Pu02

particle activity and size by registering the alpha-tracks using a

charged-particle imaging video monitor [Iida et al., 19901. The radial

distribution of dose from beta and gamma particles detected by laser

heated TLDs (Thermo Luminescent Diode) was calculated with Monte

Car10 simulation by Setzkorn et al., [1997] and measurements with an

extrapolation chamber, related to the topic, was performed by Leroux et

al., [1996]. The discrepancy between calculations and measurements

were quite large, about a factor of 2. Kushin et al., [1993, 19971 made

alpha and beta autoradiographic rneasurements of the biological tissues

collected from the surroundings of the Chernobil NPPl . Their activity

1 Nuclear power plant

assessment does not take into account the self-absorption of the alpha-

particles inside the hot particles, circumventing this by assuming a

quadratic relationship between particle size and activity.

The present work determiries the particle size based only on the

n-ber of tracks detected in a cluster, created by a hot particle on the

CR-39 solid state nuclear track detector and the exposure time. The

mathematical model developed here gives the relationship between the

activity of alpha-emitting particle and the nurnber and distribution of

tracks created on the surface of the track detector.

1.1. Radioactive isotopes present in nuclear reactor fuels

used in the nuclear power industry

Professionally exposed personnel can be exposed to alpha-

emitting hot particles, especially by inhalation, at nuclear plants or

mines where uranium or plutonium is handled. The present work has

been fmanced by COG (Candu' Owner Group). The methods developed

would be useful in assessing particle size a t decommissioning sites and

also the Recycled Fuel Fabrication Laboratory (RFFL) at the Chalk River

Laboratories (CE) , which, for many years, has produced mixed oxide

(MOX) fuels for the Canadian experimental fuels program. Mixed

plutonium and uranium oxides are available for use as reactor fuels,

due to weapon-grade plutonium being stockpiled from dismantled

nuclear weapons [Bairot et al., 19951. I t is assumed that for fuel

purposes, the plutonium has aged from the time it was frst produced,

Le. at least 15 years old, with a significant arnount of in-bred

americium-241 present in its composition (see Table Ala) . The main

purpose of d n g plutonium oxïdes with uranium oxides is to bring the

spontaneous fission level of the fuel to a higher level. This can be

achieved using up to 0.5% plutonium by mass in mixtures with natural

uranium natU or depleted uranium d e p u (see Tables A 1. e-f) .

The alpha-emitters to which AECL2 personnel are most likely

exposed to are the following:

- natural uranium oxide,

- depleted uranium oxide,

- low enriched uranium oxide,

- plutonium oxide, in the AECL practice it is aged more than 15

years as a consequence of reusing weapon grade Pu oxide for

fuel purposes,

- mixed oxides of the above.

MOX particles were not separately studied for their alpha-

emitting properties as this is entirely determined by the 0.5% Pu it

contains.

1.2. Radioprotection for inhaled radioactive particles

One objective of air sampling is to obtain information on the

nature and magnitude of the potential health hazard resulting from the

inhalation of airborne particles. As the sensitivity of in vivo and

bioassay monitoring for actinides at radiation protection levels is

technically extremely demanding, the introduction of a personal air

sampling (PAS) program is a step towards improving radiation

' Canadian deuterium uraniuma ' Atomic Energy of Canada Limited

protection for employees working with plutonium and other types of

nuclear fuel. Personal air samplers use a compact, light-weight pump

worn on the person, with the ffiter located on the lapel, with the result

that the air-sarnple is drawn-in near the breathing zone. Studies show

that even these samples can be in error by a factor of 5 but on the

average are representative of the exposure the person receives. The

persona1 air sarnpler used at AECL [Johnson et al., 19891 and [Kalos et

al., 19851 consists of a commercially available regulated-flow pump

(DuPont Mode1 P-2500) at 2L/min, attached to a workerJs belt and

connected to a CRL-designed filter head with a flexible hose. Following

use, the fdter heads are tested for gross alpha-contamination and are

then sent to the CRL Bioassay Laboratory. There they are disassembled,

the fdters are removed for analyses, the heads are reassembled with

new fdters, leak tested, and then returned to their original destination.

After a delay of at least 40 hours to allow for the decay of radon and

thoron progeny, the used fdters are f rs t screened in an automatic

alpha-counter. AECL maintains a filter analysis service, at Ch& River

Laboratones, for identification and quantification of alpha-

contaminants [Linauskas, 19951. The service utilises 2n alpha-

spectrometry systems, based on surface barrier detectors, to determine

alpha-contamination on filter papers used in PAS and Continuous Air

Monitors (CAM).

The personal air samplers are sensitive enough to detect 1/ 10 of

the annual limit intake during any monitoring interval. Because the

personal air sarnpler does not measure retained activiq or uptake, we

have to assume that activiw measured by this means is a good

approximation of the actual intake.

1.3. The effects of size on the aerodynamics of airborne

particles

Since the hot particles considered in our project are coLlected

from air, it is important to state some basic assumptions about their

behaviour as airbome particles. An aerosol is the suspension of small

liquid or solid particles in a gas. They can be stable in position for a

period between a few seconds and more than a year. In 1961 the US.

Atornic Energy Commission, Office of Health and Safety (Los Alamos)

defmed respirabie dust as the portion of inhaled insoluble dust that

reaches the nonciliated region of the lung.

Solid aerosol particles usually have cornplex shapes, but for the

developrnent of our theory it was assumed that the particles were

spherical. It is convenient to describe more complex shapes found in

practice by a single diameter and have the additional flow resistance or

drag represented by a factor. This dynamic shape factor, X , is the ratio

of the drag force of the particle in question (particle diameter d,) to that

of a sphere of equivalent volume (volume-equivalent diameter dm). The

relationship between the volume-equivalent diarneter dev and the

aerodynamic diameter da can be deduced from [Wileke et al, 19931 :

where pp and po are the densities of the particle respective the medium

and C(da) and C(dev) are the Cunningham slip factors whose value

depends on the chosen diameter, and x is the shape factor. The shape

factor is always equal to or greater than one. Compact shapes typicdy

have values between one (a sphere) and two, while more extreme

shapes, such as fibres and high-volume aggregates, may have larger

values. Willeke et al. give 1.28 for U02. Shape factors are useful for

converting a readily measurable equivalent diameter to one that

depends on particle behaviour, such as aerodynamic diameter or

diffusion-equivalent diameter. Some particles have relatively regular

shapes with volumes that can be calculated or compact shapes that can

be rneasured with a microscope to determine a volume-equivalent

diameter. For such particles, the shape factor is, from Equation ( 1.1) ,

The rnost commonly used equivdent diarneter is the projected

area diameter dp. It is defmed as the diameter of the circle that has the

same projected area as the particle silhouette [Hinds, 19821. It has the

advantage of providing a unique size for a given silhouette regardless of

its orientation. The measurement of dpa for a single particle of irregular

shape cm be determined using image andysis software.

We can also introduce the volume shape factor av which relates

the volume of the particle vp to one of the silhouette diameters described

above. It is defmed for a projected area diameter by

For regular geometric shapes, av can be calculated; for irregular

shapes it must be detemiined empirically by a combination of two or

more measurement methods. The equivdent volume diameter dev is

related to the volume shape factor av by

Except for fibres and platelets, d, and d p do not differ by more

than a factor of 2.

In the absence of specifc information about the physical

characteristics of the characteristics of aerosol to which a subject is

exposed, the ICRP 66, [1994] recommends a default value for the

particle shape factor, x = 1 .S.

In this work the effect of a non-spherical shape on the self-

absorption of the alpha-particles has been neglected, which may provide

a significant contribution to the overall errors involved in the evaluation

of the hot particle diarneter.

Particles in an aerosol sample u s u d y corne in a spectrum of

sizes, and the deposition of activity is averaged over this size

distribution. Most aerosols are described by log-normal distribution.

Frequently the size distribution is completely specified by the median

value and the geornetric standard deviation. The activity median

diarneters are particularly important in radiation protection. The activiq

median aerodynamic diarneter AMAD is that diameter for which 50% of

the total activity is in all of the particles of diameters larger than the

AMAD.

1.4. The effects radioactive particles on the human

respiratory tract

The effects of the radioactive particles on the human respiratory

tract had been described by the ICRP Publication 66. The ICRP 66

respiratory-tract model uses a combination of morphometric, empirical,

and mathematical modeiing. The objective is to descnbe the deposition,

clearance, and dosimetry of radioactive particles. The model itself may

be thought of as a series of linearly-coupled filters whose efficiencies

depend parametrically upon the particle's charactenstics and the

subject's characteristics. The respiratory tract is an important pathway

by which radioactive materials enter the body. A complete dosimetric

estimate requires not only the new lung model but a metabolic model of

the gut and body organs. The dimensions of the airways influence the

Figure 1.1 The regional deposition fractions in habitually oral-breathing

workers. Figure courtesy Robert Corns [1996]

dose received by the radiosensitive tissues by chanmg the deposition of

particles. Quantities such as air pressure and volume flow rates in the

lung depend upon the airway dimensions and affect the deposition of

radioactive material. The deposition, and consequently the dose to the

respiratory tract, is affected by the size of the particles. The effect of the

size of the particles on the respiratory track has been computed by

some dedicated codes such as GENMOD [Richardson et al., 19981 and

LUDEP [Birchall et al., 199 11. Figure 1.1 compares cdculations, using

GENMOD, of the fraction of intake activity deposited in various lung

regions by Corns, [1996] with data from LUDEP displayed by markers

V, 9, M, a, and ).

Chapter 2

Characteristics of pzastic

nuclear track detector

2.1. Solid state nuclear track detectors

The traverse of a charged particle through an isotropic,

homogeneous dielectric produces an axial degradation of the material

on an atornic scale. This damage is called latent tracks. Their diameter

is about 10-3 pm and their length is strictly related to the incoming

alpha-particle energy. For this study, the latent tracks of alpha-particles

in CR-39 will be considered to be straight-line segments.

To be observable using optical microscopy, the latent tracks must

be enlarged. To achieve this, a chernical etching is performed with a

concentrated, hot sodium hydroxide solution (7N and 70°C). Detectors

based on the etching out of the latent tracks are Solid State Nuclear

Track Detectors (SSNTD). The SSNTD we used was the CR-39 polymer,

which is an optically transparent, amorphous, thermoset plastic with a

high degree of isotropy. The principal reasons for using it were its

unique sensitivity and uniforrn response. Track detecting plastics

possess many advantages over electronic detectors. In particular spatial

resolution of a few microns is available simultaneously with the

potential for high-resolution spectroscopie measurements. O the r

advantages are cheapness, ease of use, abiliw to provide a permanent

record of events, and the ability to operate in adverse environments. For

these reasons we decided to design our measurement protocol based on

CR-39. The plastic detector used for this project is manufactured at

Bristol University under the TASTRAKO brand name. TASTRAKQ is a

highly sensitive track recorder for alpha-particles. The response

changes with time which is charactenstic of this class of track detector.

The majority of aging of the CR-39 occurs withïn three weeks of

manufacture, where after storage at -20°C maintains the response

within narrow limits [Henshaw, 19891.

2.2. Principles of track formation on the CR-39 nuclear

track detector

When the charged particle traverses the CR-39 it is delineated by

a trail of chernical darnage. The subsequent immersion of the plastic in

a suitable etchant such as NaOH results in bulk etching of the material

at a characteristic rate, VB and preferential etchïng of the material at a

characteristic rate, VT along the axis. The pit so formed, when enlarged

to a size that is easily visible under a microscope, can be measured to

find the track etch rate ratio VT/VB and hence the ionization of the

particle. In terms of particle nuclear charge (2) and relativistic velociw

(p) it records particles in the range 6 < Z/P 100. Natural alpha-

particles therefore lie towards the middle of the response cu rve , the

relationship between ionisation and VT/VB, which means that all

species of particles can be recorded at full energy and a wide span of

acceptance angles.

The VT range curve for the CR-39 utilized in this work has been

determined, by other workers, in the manner described by Hatzielekou

et al., [1988]. This is plotted in Figure 2.1 as the variation of track etch

rate versus range, a relation which is found to scale dong the ordinate

by a simple factor with changes in the plastic sensitivity. This enables

the response to be expressed as the track etch rate at an arbitrary value

of particle range.

Figure 2.1 Truck etch rate vanation versus alpha-particle range in CR39

The track that results after the etching process is visible by use of

an ordinary microscope, and its elements are displayed in Figure 2.2.

The track depth 2, has a minimum Zm, and the dip angle 81, a cut-off

Oc, such that only tracks with both a depth and dip angle greater than

In Figure 2.2 appears the angle 6 which is related to 8 by the relation Zi = z/2 - 8

these values will be registered and observed. & is dependent on the

etching conditions, which if are kept constant (as those mentioned in

Section 2.1) will render & almost constant. The value of Zm, chosen

for theoretical purposes is discussed in Section 6.2. The thickness of

the etched out CR-39 layer is A. The shortest and longest diameter of

the track in the final surface plane is Mi and Mj respectively, and rn is

the diameter of the tip of the track. The parameter L is the length of

track present in the detector plastic, from its pre-etch surface to the

track end. Tracks, which will have dimensions less than mentioned

above, WU leave observable trace on the frnal surface. We will cal1 valid

tracks ail tracks which will have both Z and 8 greater than the

minimum or cut-off values.

Figure 2.2 The meamrable parameters of n track etch cone

The resolution of the CR-39 nuclear detector is dependent on the

etching conditions and the alpha-particle range in CR-39 [Jeffs, 19681.

The maximum number of tracks registered for a given particle, cari be

obtained by direct exposure of the hot particle on the surface of the

nuclear detector, but this will lead to a poor track readability due to

overlapping tracks. A reasonable compromise is the use of a 400 pn

spacer between the hot particle and the CR-39 surface, which will

produce a reasonable spread of the registered tracks. The loss of

registration effectiveness due to the 400 pm air gap is s m d compared

with the gain due to the greater readability as shown in Chapter 6.

Chapter 3

Theoretica Z considerations

related to track formation

3.1. Alpha-particle interaction with matter

Ail calculations related to alpha-particle interaction with matter

are based on the range of alpha-particles in different media. The range

in solids of known composition can be computed approximately by the

following relationships given by Friedland et al., [ 19641 :

where

Rz is the range,

Ra is the range,

R, is the range,

mg cm-2, in the element of atomic number

mg cm-2, in air

E is the initial

M is the mass

mg cm-2, in the solid containhg the atoms A, B,

... Z in the relative proportions by weight wa, wb, WC ... WZ.

particle energy in MeV, and

number of the particle (for alpha-particles, M = 4).

- Friediander et ai. : L i Fews

- ----- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 6 7 8

Alpha-particle enetgy (MeV)

Figure 3.1 Alpha-particle range in CR-39, calculated with Friedlander,

[1964] approximation cornpared with alpha-particle ranges

supplied by Fews, [1982]

The calculated values for alpha-particle range in CR-39 by

Equation (3.1) are very similar to the values given by Fews, [1982] as

shown in Figure 3.1 for z = 9, which is slightly higher than the

calculated vaiue of 7.12, based on the approxhate chernical formula of

the CR-39 found also in Fews, [1982] thesis.

The discrepancy is mainly due to the polymerized character of the

CR-39 plastic whose exact chernical formula is not hown .

As the emitted alpha-particle passes through different materials

dong its range, from its source in the hot particle to the end of its range

in CR-39, we consider that its energy loss is only due to mass

attenuation. We ignored any other types of interactions or boundary

effects. The slowing down of the alpha-particle is proportional to the

densi@ of the medium. This hypothesis had been sustained by earlier

calculations made by Howarth, [1965], Charlton et al., [1962] and Fews,

[1982]. If we know the range (R) of the alpha-particle in one medium,

particularly in CR-39, we can calculate the range in other media using

this simple relationship:

where p is the density and subscript m refers to a particular medium,

while c refers to CR-39. This approximation enables us to find the range

of the alpha-particles in any media having the range of them in Cr-39

for every energy. Table Dl in Appendix D gives the emitted alpha-

particle energy and range in CR-39.

Chapter 4

Computation of registration

effectiveness of the CR-39

detector

4.1. Registration effectiveness for a point source

The conversion of the number of tracks in a track cluster to hot

particle size requires the calculation of registration effectiveness or

e fficiency .

The effectiveness of registration of detection of alpha-particles is

defined as the ratio of the nurnber of alpha-particles registered by the

detector in a certain time interval, to the nurnber of the alpha-particles

emitted, iri the same interval. The registration effectiveness of the CR-39

plastic can be defmed as:

where Nm is the number of valid alpha-particles tracks, A (Bq) is the

activity of the hot particle, and T (d) is the exposure time.

Equation (4.1) enables us to fmd the activity of a hot particle if we

know the registration effectiveness and the number of tracks in the

etched-out CR-39. The registration effectiveness can be considered to

equal the probability of the alpha-particle producing a valid track on the

detector.

Figure 4.1 Schematic representation of an alpha-emitting point source, P

above a CR-39 track detector

Consider a small alpha-ernitting point source P as a

dimensionless point. If the alpha-particles pass in a straight line

through a homogeneous and isotropie medium, the cluster of tracks

present on the CR-39 is contained within a circle as illustrated above.

At a certain limiting angle, 0- the equivalent path through the medium

and the pre-etched CR-39 surface wiU equal the range of the alpha-

particle in CR-39; beyond Biim no alpha-particle will register on the

detector. The solid angle is bound by a cone as shown in Figure 4.1,

and is given by:

The registration effectiveness of the detector will be

2n(l- cos 8,) 1 - cos 0, m, = - - 47r 2

Consider a sphencal hot particle of radius r, with homogeneous

distribution of alpha-emitting radionuclides, held at height h (pm) above

the CR-39 pre-etched surface (Figure 4.2). An alpha-particle emitted

from a point P inside the hot particle will have range R and pass

through the particle material of density pl (g cm-31, and air of densisr pz

(g c m 9 A s a result it leaves a track in the pre-etched surface of length

NM, which is long enough to produce a valid track after the etching

process.

In order to make a distinguishable track on the post-etched

surface of the CR-39, the alpha-particle must enter the pre-etched

surface of the detector and produce a latent track to a vertical depth of

Zm, and an incident angle less than an incident cut off angle, Bcutotr. The

incident cutoff angle €lcuton is related to the cutoff angle &utoff defmed in

section 2.2, Figure 2.2 as Bcutoa = 4 2 - Gcuton. The influence of these two

parameters on the registration effectiveness will be discussed in Section

6.2. The track cluster formation will be limited by two mutudy

-- - p-

independent angles. The Lunithg angle O b , as described

Figure 4.2 Schematic representation for the experirnental se t up (not to

scale)

above, will limit the track cluster dimensions dependent on alpha-

particle energy and conversely the incident cutoff angle B o r will limit

the registration effectiveness of the CR-39 nuclear track detector. If the

incident angle is too large, 8 > Bcutoff, the registered track will be too

shallow and will be etched out and, hence, will not give a readable, valid

track.

The sum of the alpha-particle track lengths through the three

media can be considered to be equivalent to the alpha-particle range Rc

(pm) in the CR-39 [Fews, 19821:

where PQ is the alpha-particle path length in the hot particle, QM is the

path length in air, MN the path length in the CR-39 and kl and k2 are

range ratio correction factors calculated using stopping power data of

Northcliffe et al., [1970]. The values of ki and k2 are here considered to

be 1 / 1.2 and 1.2 respectively, based on assuming the particle material

is uranium oxide [Richardson, Pnvate communication].

4.2. Registration effectiveness of hot particles

In the case of a f~te-sized particle of alpha-emitting

radionuclides the registration effectiveness q is the ratio of the SM of

the solid angles for infmitely small sources of volume dx dy clz within

the hot particle, which produce valid tracks to 4n, the solid angle of the

three dimensional space

where the dornain of both triple integrals is the volume V, of the hot

particle. Sirice we are assurning a spherical particle, Equation (4.5) c m

be written as:

Due to the syrnmetry of the sphere, it is easier to perform

calculations and defme elementaq voxels in spherical coordinates (see

Figure 4.3), therefore transforming the Cartesian coordinates into

spherical coordinates :

Hot particle

Post-etched su rtàce

Point Source crack cIuster fiom the point source P

Figure 4.3 Representation of the 3 dimensional integration of

registration effectiveness of ULfinitely srna22 sources

throug hout the hot particle (not to scale)

Furthemore, if we take into account the symmetry of the hot

particle and the nuclear track detector interaction, which is axially

symmetrical around the Z axis, we can write the Equation (4.7) as:

where the radial distance q- and <pmax are the restricted domain

boundary values.

4.2.1. Domain restriction schemes in the registration

effectiveness integrai

Domain restriction in an integral evaluation scheme means

reducing the initial domain to a smaller one without altering its value in

order to eliminate noncontributing points and speed up cornputhg

process. If the particle is big enough, there are many points inside the

particle whose alpha-particle emission will be absorbed before they

reach the detector surface. Elixninating them is a worthwhile operation,

and it is done before every registration effectiveness evaluation process.

We have to fhd new limiting values for q and <p which initially ruris from

q = O to q = r and from cp = O to <p = n. The problem is to contain the

active part of the hot particle (the one which c m send recordable alpha-

particles to the detector, Le. which are not absorbed on their way to the

detector) in an easily defmable geometric shape in q and cp coordinates

in the zOx plane. In Figure 4.4.1, we cari see that the active part is

CO'C'O". So in the zOx plane, the cuve CC' is the borderhe between

the active part and the rest of the hot particle. The closest regulate

geometrical figure to the active part of the hot particle is the annulus

AA'B'B .

The value of qm, was detennined by computational analysis

cornmencing in O", at <p = 0, and q = r, the sphere periphery, (Figure 4.4

1) then approaching the sphere center O in steps of r/20. The limiting

value of q at 0', when the alpha-particle ernitted toward the surface of

the CR-39 does not make a valid track, was chosen as qm, in the

integration scheme.

The upper Limiting value of the angle <p was determined setting q =

r and moving on the periphery of the sphere in steps of n/100, from

point O" at angle <p = O to point B' at angle qm, where from the alpha-

particle emitted toward the surface of the CR-39 can not make a valid

track (Figure 4.4 1).

The consequence of the restriction scheme is that the integration

domain will become equal to A A'B'B (Figure 4.4 1), which will enable us

to Save computer runtime and use less quadrature points in the

integration scheme.

The only situation when the above-mentioned restricted domain

algorithm is not applied is when q- is O, which means no restriction

on the initial domain for q. This situation can occur when we have

points in the upper side of the sphere which emanate alpha-particles

capable of being registered. In this situation, is automatically taken

as K , without any attempt to be evaluated, since Tm, c n would have the

consequence of scooping out a cone shaped volume from the initial

sphere (COC' from Figure 4.4 II), which can give valid tracks on the

detector. In this case the only "invisible" part of the hot particle is the

one lùnited by CC' and the upper surface of the sphere, which will be

included in the integration domain in spite of its ni1 contribution to the

value of the integral.

The double integral in Equation (4.8) had been evaluated using a

double Gauss-Legendre quadrature method using 10 quadrature points

[Abramowitz. et al., 19701. Increasing the nurnber of quadrature points

to 12 would change the accuracy by less than 0.1%, but signifcantly

increase the run time by 50%.

Figure 4.4 Restriction schemes for the registration effectiveness integral

evaluation, shown in the zOxplane

To assess the registration effectiveness q for typical input data: p i

= 11.46 g cm-3, p2 = 0.001226 g cm-3, pc = 1.325 g cm-3, Rc = 32 Pm, r =

9 Fm, h = 400 pm the runtirne was about 2 min on a Pentiumm 120 PC

running Windows 95@ and using Microsoft Visual C++ 4.0@.

4.2.2. Determination of the solid angle C2 as integrand of the

registration effectiveness integral

The mathematical definition of the solid angle C l is a surface

integral defined on the point source generated track cluster. The shape

of the track cluster is generally not a regulate geometrical figure, thus

setting up an elementary area dS is not a trivial task and can be done

in quite a few ways. This thesis adopted the following algorithm: the

track cluster had been divided into circular sectors with the central

point the projection of the emission point P to the post etched surface,

Pl. The sector angle pst, was taken as a constant with two possible

values: if P' feu inside the track cluster PaeP = IO0 (see Figure 4.10),

otherwise Pstep = la. AS a consequence, the angle between the emission

plane and the XOZ plane B had its current value P = j*Pstep, where the

index "j" runs from O to 25(. The next step to defme the elementary area

dS is to divide the circular sector into partial annuli I'IWJ"J', (see Figure

4.3) called voxel, of equal thickness I'I", which is equal to the sector

length P1A/400, if Pr is inside the track cluster or BA/400 if Pl outside.

The current position of the partial annuli will be defined by the index i,

where PI' = i* 1'1" and i is running from O to nvoxel = 400. The position of

the points A and B is determined by the limiting angle Bÿmi and

respective 8 ~ .

The evaluation of the total solid angle Q(q,<p), Equation (4.9), (See

flow chart in Figure 4.5) is based on its additive propem. First, the

solid angle on a circular sector is evaluated and later added to the

whole cluster sector by sector.

The elementaiy solid angle, Ri j(q,<p), defmed by the alpha-particle

emitting point P and elementary area I'I"JWJ' Rij(q,g), was found as it is

in Equation (4.10) (see proof in Annex 1): The choice of nvoxel = 400

and Baep = 10° (pstep = 1°) assures a 4 digit accuracy for the solid angle,

Q(q9rp) *

* t ~ e ~ 2 . j * 1 tane,2,j-tane,~,j P ,tep [tanûhlPj + (i + -) * nvoxel 2 nvoxel

Set initial alp ha-em itting plane p =O and $,,, =1P

L I

I + no

Get the lirniting angle

Get next lirniting

Add up solid angle for circular sector area between O,,, and

t ( P =P +P,,, Add up the solid angles

Figure 4.5 Flow chart for the evaluation of the solid angle R(q,<p) as

integrand for the Gauss quadrature scherne

4.3. Evaluation algorithm for the generic limiting angle 8,

Emission plane

Hot particle

Point source track cluster

Figure 4.6 Representation of the path of an alpha-particle in Cartesian

coordinates (not to scale)

The evaluation of the solid angle in the integral of Equation (4.8)

is conditioned by the determination of the generic lirniting angle, 0 b , for

which particular cases Orni and 0- were described earlier. For

practical reasons Cartesian coordinates have been chosen for the hot

particle, with the ongin defmed at the center of the hot particle (Figure

4.6.). The coordinates of the alpha-emitting point source P are x,, yp, zp.

The lirniting condition defmed by Equation (4.4) will becorne:

k A h-z, R, =A--+( - 1)- k * ~ , + Zm, Pc COS 0, p, C O S ~ ,

where = PQ (Figure 4.2) is the distance traveled by the aipha-particle

inside the hot particle. The solution of this equation gives the limitirig

value of 8. Rearrangement of Equation (4.1 1) gives:

where

and

The equation for the trajectory in terms of x, y, z of the alpha-

particle emerging from a point inside a particle can be given by:

X-X, Y - Y p z -zp - --- - -1

cosJ3sinû-sinBsin0 cos8

where angle P had been defmed in Section 4.2.2 and is shown in Figure

4.6.

Using a spherical model, the equation for the boundary surface of

the hot particle is:

x2 +y2 +z2 = r 2 (4- 16)

where r is the radius of the spherical particle. Combining Equations

(4.15) and (4.16) we can find:

(x, + h sin 8 cos P ) ~ + (yp + h sin 8 sin P ) ~ -

which after rearrangernents wifl give:

where:

Finally, substituthg A from Equation (4.12), Equation (4.18) will

give an equation in terms of the limiting angle 0rim as:

C C (e - )2 +2(e- )((x, COS p + y p sin~)J- + zp cos 0,) - f = O

COS 0, COS 0,

4.4. Methods developed to find the limiting angle 8,

After substituting u = cos9 in Equation (4.20), and after several

raising to square, it becomes a 6th degree polynomial equation in u. For

simplicity, the function on the left-hand side of Equation (4.20) will be

called G(u). Raising to square an equation will lead to the introduction

of strange solutions into the original set, which u s u d y has no physical

~ i g ~ c a n c e . The limïting angle 0~ is a physicaiiy acceptable solution of

Equation (4.20) for G(u) = O. For the physically meaningless solutions

will use the notation 8 1, 92. ..

AU analyses of the solutions of Equation (4.20) were made using

Mathematica 3.0TM [Wolfram, 19961, which provides fast, accurate

solutions and graphical representation of the behavior of the function

G(u) .-

Where O < u -= 1, Equation (4.20) can have O, 1, 2 or 3 distinct

solutions, and to fmd and sort out the physically meaningful solutions

of G(u) = O is a nontrivial and time-consuming operation. Table 4.1

Table 4.1 Summary of the five cases where function G(u) = O

Case G(1) Occurrences Phy sically Valid Relevant where acceptable intervals for 8 figure

G(u) = O solutions for G(0) Oa > O O none none

Ob < O 81 none none 4.8

la > O &in~l , 0s one 2 8 > O 4.9-4.11

l b > O em1 one 8 b l 2 8 2 0 4.12

2 < O €lm, 8 ~ , 03 two 2 8 2 4.13, 4.14

summarizes the five different cases (Oa, Ob, la, lb, 2) to evaluate d

cases of 8 b for emission from point P (see sections 4.4.1., 4.4.2. and

4.4.3. for detailed description of the cases). Figure 4.7 shows the flow

chart of the structure of the computational methods developed for the

five different cases for evaluating €lm.

Find G'(u)

End u, - the zero of G(u)

f Case ûa O,,, = O

\

f Case Ob O,,, = O

i

End u ' the zero of GT(u)

Yes

f 3 f v \

Case ûa Yes Case 2 O,, = O cos (QIi,) =u,

\ J L 1

End u2 the zero of G(u) between

u 'and 1

COS (Olim) =u* cos (eiim) =uz L LrLJ Figure 4.7 Flow chart for the procedure offlnding the limiting angle Oiim

The computational-solution-finding process starts with testing

G(1) sign (see flow chart Figure 4.7). This test leads us to cases Oa, la,

and lb if G(1) > O, or cases Ob and 2 if G(1) 4 O (see Table 4.1). Once the

solution is identified as acceptable and the interval of its occwence

deiimited, an interval halving subroutine determines the root of the

function with the required accuracy. The present code determines the

root of function G(u) to an accwacy of 10-9.

4.4.1. Description of cases Oa and Ob (Table 4.1) with no solution

of &im

Since G(u) always has a value of +m at G(O), and if G(1) > 0, the

positive branch of Figure 4.7, there can be no or zero solutions if the

Figure 4.8 Case Ob finction Glu) with parameters: p l = 1 1-46 g cm-3, pz

=0.001226gcm-3, pC=1.325gcm-3, Rc=32.0pm, r=6.0

pm, h = 30.0 Pm, q = -0.5 pm, <p = 30°, @ = 10"

function does not intersect the X axis in the 0-1 intemal, case O a Table

4.1. On the negative branch, there is a zero of the function G(u), 81 case

Ob, Figure 4.8. Case Ob is where G(u) is corresponding to a large

emission angle, 81 almost pardel to the CR-39 surface, where the range

of the alpha-particle is exceeded.

4.4.2. Description of cases la and 1b (Table 4.1) with one solution

of &im

Case l a is where there is one solution, on the G(1) > O branch of

Figure 4.7, if the function G(u) has a minimum in a position where the

value of the function G(u) is negative (see Figure 4.9).

Figure 4.9 Case 1 a hcnction G(u), with parameters: p 1 = 1 1.46 g cm-3, p;.

= 1.226x103g cm-3, p, = 1.325g cm-3, Rc = 32.0 Fm, r = 6.0

Pm, h = 3 0 . 0 p m , q = 3 . 5 p m , q = 10°,4= 10"

The right-hand side zero is accepted as a physically meaningful solution

for 8ii, and is the particular case 8hl. For 8, values will range between

O and e h i as showri for Case la. 02 is rejected as a physically

meaningful solution for the same reason previously described for case

Ob. The event cluster of all registered tracks is defmed by Bÿmi for P run~ ing from O to 360°, as shown in Figure 4.11. The outer closed

curve in Figure 4.11 shows a track cluster created from a point P inside

the hot particle, whose projection on to the detector surface is within

the event cluster. The inner circle represents the hot particle of

diameter 10 Pm.

For case lb the ssme track cluster pattern will be generated,

Figure 4.12, which cari be considered as a limiting case of la, where we

have two superposed zeros of the function G(u).

Hot particle

Single event track cluster h m the point source P

Post-etched su dace

Figure 4.10 Case 1 a and 1 b for a point source track cluster (not to scale)

Figure 4.11 Case l a Point source truck cluster for conditions shown in

Figure 4.10, with coordinates of point source P: xp = 2.985

Pm, yp = 0.526 Fm, zp = 1.75 Pm, r = 5 Pm, h = 3 0 prn

Figure 4.12 Case 1 b Function G@) with parameters: p r = 1 0.0 g cm-3, p î

= l . O g ~ r n - ~ , pc = l . 5 g cm-3, Rc= 40.0pm, r = 6.0 p m , h =

30.0 Pm, 4 = 6.0 Pm, cp = 90°, @ = 120"

4.4.3. Description of case 2 (Table 4.1) with two solutions of 8um

In case 2, when there are 3 zeros of the function G(u) (Figure

4-13), the valid solutions are taken as those which border the positive

part of G(u). So in this case, there are two physically meaningful

solutions: and 8 ~ . The value 83 corresponds to alpha-particle

emission approximately paraIlel relative to the detector surface, so the

emission can not lead to the registration of a valid track. Case 2

generates track clusters as shown in Figures 4.3 and 4.14. 0 = O

corresponds to vertical direction, perpendicular to the detector surface

at P' (Figure 4.3), while alpha-particles emerging at will produce

tracks at point B ',

Figure 4.13 Case 2 Fùnction G(uJ with parametersr p 1 = 1 1.46 g cm-3, pz =

1 .226x lWgcm-3 , p c = 1.325gcm-3, R=32 .0pm, r = 6 . 0

Pm, h=30.0pm, q = 3.3 Pm, <p = 140°, 4 = 80"

similarly those at 9~ at A'- In this case, the detector will build up a

track cluster away from the perpendicular traced from the ernitting

point to the detector. The track cluster resulted from the set up shown

in Figure 4.3 is represented in polar coordinates in Figure 4.14. The

central circle is the hot particle projection on the XY plane at Z = h - &, the z coordinate of pre-etched surface. The off set closed cunre

represents the hot particle of diameter 10 Fm.

Figure 4.14 Case 2 Point source track cluster resulted in conditions

shown in Figure 4.4, coordinates of point source P: xD = 3.953

pn, y, = 0.697pm, zp = -2.811 F m , r = Spm, h = 30 p m

Chapter 5

Distribution of tracks in a cluster

5,1, Introduction

In this chapter, a method is described which calculates the track

density distribution dong the radius of a track cluster. This method is

independent but complirnentary to the theory developed in Chapter 4 to

calculate registration effectiveness.

Where there is a range of particle sizes being exposed to CR-39,

an exposure t ime is chosen to ensure a statisticaily adequate number of

tracks for the smaller particles. This results in the larger particles

producing clusters with their centers having overlappÏng tracks. The

presence of overlapping tracks is more likely to occur for particles in

direct contact with CR-39, as the use of an air-gap separator results in

lower track densities in the central portion of the cluster. If no account

is taken of the overlapping tracks, an inaccuracy will be introduced in

the total track count for the cluster, and hence in the assessment of the

particle size. In cases where overlapping tracks are present, it is

desirable to calculate the particle size based on the number of tracks in

annuli surroundhg the cluster center.

5.2. Radial track formation probability density, D(o) from a

single disintegration event

In the followirig, we denve a mathematical expression for the

track formation probability density function D(o). We consider the

number of valid tracks on an infinitesimal area dS at coordinate x, y, on

the post-etched surface. I t is assumed that in the case of a single

disintegration in the elementary volume dV, the probability of making a

viable track on an elementary surface dS is proportional to a) the

probabil@ of an alpha-particle being emitted frorn elementary volume

dV, around P towards area dS, and b) the probabiliv p of a valid track

being produced on the post-etched surface.

The combined probability of getting a valid track d N on

elementary area dS, at coordinates x, y on the post-etched CR-39 from a

single alpha-emission from eiementary volume dV is:

where C is the normalization factor; 1 is the alpha-particle path length;

p(x, y,q,cp,$) is the registration factor which has the value: 1 if the right

hand side of Equation (4.4) is 5 Rc and O otherwise.

Y

Hot particle

a particIe trajecco ry

Figure 5.1 Representation of geometrical set-up for the detemination of

the track distributionfunction

The overall probabiliw dN(x,y) of a valid track being located on dS

from a single alpha-disintegration onginating from within the hot

particle of volume V, assuming homogeneous distribution of alpha-

ernitting nuclides inside the hot particle, will be:

Substituting Equation (5.1) into Equation (5.21, including 4n: into

the normalization constant, and expressing dV in spherical coordinates,

Equation (5.2) gives:

where we can determine C by:

If we def ie the density of probability of getting a track on the

detector surface from a single alpha-disintegration:

which when substituting Equation (5.3) into Equation (5.5) gives the

firial form of the densiw of probability of track formation function as:

Normalized to one, D(x,y) (pm-2) will give the probability of having

a valid track on a unit area at coordinate x, y of the post-etched CR-39

surface, starting from a single alpha-emission. D (x, y) behaves much as

a normal distribution function except it vanishes at a finite distance

from the Z axes. Since there is an axial symrnetry around the Z axis, for

a spherical hot particle, the radial probability density D(o) can be taken:

Another consequence of the axial syrnmetty of the hot particle

detector interaction is that the integral in Equation (5.4) can be

perforrned as a one dimensional integral, depending only on the radial

distance, w, limited to %ax where D(omax) = O. After perfonning a

coordinate transformation into polar coordinates, (x, y) becomes ( o , ~ ) ,

and performing the integration of yr from = O to 2rr, Equation (5.4) c m

be expressed as:

In practice, the D(x,y) (x = a, y = O) in Equation (5.6) had been

determined in the following way: as a fïrst step a set of raw D(w) is

determined from Equation (5.6), with coi = 20 pm steps, which will

contain the unknown C in its values; the next step is to evaluate C from

Equation (5.8); the last step is to nomalize the set of raw probability

density function D(m) to one. The path of the alpha-particle inside the

emitting particle was deterrnined with the help of MathematicaTM

software [WoIfrarn, 19961 and implemented as a formula in the

probability density written in "C" computer code.

A srpical probability density function is displayed in Figure 5.2,

Pu02 at different particle-detector separations.

To perform the Equation (5.8) integral as step two, the raw D(m)

values were curve fitted with cubic splines. It was necessary to perform

a good curve fitting in order to get a fair integration convergence.

Previous attempts with polynomial curve fitting gave modest results,

but cubic splines could reach &-figure accuracy, when 96 quadrature

points were used in the Gauss integration scheme [Abrarnowitz et al.,

l 9 ï O I .

h = 200 micron

- - - - h = 4 0 0 m i c r o n 0-

O 500 1 O00 1500 2000 2500

Radial distance o (pm)

Figure 5.2 Typical probability density for 239PuO2 particle, d = 1 pm, p =

1 1.43 g cm-3, for different particle - detector separation h

5.3. Radial track density, D,(o) from multiple

disintegration event during the exposure time period T

I n order to estimate the number of registered tracks within an

annulus of radii u>i and q, track density distribution is required dong

the cluster radius. (See Figure 5.1)

For a particular hot particle exposed to CR-39, consider an

annulus present between radii ai and q, the number Nij of registered

tracks will be:

where NP (#tracks x pm-2) is the total number of registered tracks in the

track cluster (including overlapping tracks) and i and j are designating

the two annulus radii, in which the track counting is perfonned.

Nomaked to the total number of tracks in the track cluster

registered on the detector Dt&)

will give the track density distribution on the detector. In practice, if we

can d e t e d e experimentally the number of tracks in an annulus

defmed between radii ai and q, Nu Equation (5.9) will enable us to fmd

the total number of registered tracks in the whole cluster. Finding the

registration efficiency with Equation (4.8), we can determine the total

number of disintegrations in the hot particle, and thus, the size of it.

A s an observation, we can remark that both algorithrns used in

chapters 4 and 5 were dealing with a five fold integral which, as a result

of the axial sy-rnmetry of the particle-detector interaction, breaks d o m

to a four fold integration.

Chapter 6

Validation of the sizing method

6.1. Introduction

Validation consists of demonstrating that the mode1 proposed is

an adequate representation of the real environment. Validation of the

technical and analyticd methods developed was obtained by analyzing

the CR-39 track clusters resulting from the exposure to particles of

natural uranium. Natura l uranium particles were used as they were

more easily obtained than plutonium and less of a hazard for a given

size,

6.2. Dependence on the minimum dip depth Z,, and the

cutoff angle e,,,,

This section addresses parameters that affect the theoretical

evaluation of the particle size using CR-39. The registration

effectiveness for producing valid tracks is dependent on the minimum

dip depth &, that an alpha-particle has to enter the CR-39 surface.

The registration effectiveness is also dependent on the maximum

incident angle at which an alpha-particle, emitted from a point source,

cari produce a valid track.

Studies made in the literature related to the topic by Fews, [1982]

and Richardson, [ 19921, reveals that the registration effectiveness

significantly varied with parameters Zmin and B,toff. In case of 1 pm

diameter particle, where self-absorption is not a signiflcant factor,

Wtually all alpha-particles heading toward the detector will have

enough energy to produce a valid track. Therefore the registration effec-

Table 6.1 Registration effectiveness for a "9AC02 particle, with 1

diameter and 400 p m particle-detector separation, as

jùnction of cutoff angle e m t O ~ and minimum dip depth &in

tiveness q does not Vary with & (Table 6.1). There is a relatively large

change in q with ûcutoff as this radically alters the solid angle where valid

tracks can occur.

For larger particle sizes, e.g. 10 pm the registration effectiveness

is dependent on both parameters (Table 6.2). In this project, considering

the etching conditions, Zm, is assumed to be independent of the

incident angle 9 and was assigned the value Zmin = 2.4 Fm [Richardson,

Private communication].

The incident cutoff angle was assigned the value Bcuton = 16.5' on

the basis of the bulk etch rate (pm h-1) and the track etch rate (pm h-l),

charactenstic of the TASTRAK plastic [Meyer et al., 19971.

Table 6.2 Registrution effectiveness for a 239PUO2 particle, with 10 pm

diameter and 400 pm particle-detector separation, as a

function of cutoff angle Oaitoffand minimum dip depth &in

6.3. Dependence on the air gap thickness

In order to reduce the number of overlappïng tracks in the center

of a track cluster, an air gap can be introduced between the particles

and the CR-39 using a plastic spacer. Figure 6.1 shows no variation in

the registration effectiveness for small particles of 0.1 and 1 pm where

the air gap is up to 5000 pm. The largest variation in q for relatively

s m d airgaps is for particles around 10 prn diarneter; however, for an

airgap of O to 400 pm the change in q is s t i l l ~ 2.5%

O 5000 IO000 15000 20000 25000 30000

Particle - detector separation (pn)

Figure 6.1 The variation of registration effectiveness of alpha-particles

emitted by 239fiO2 of p = 11.46 g cm-3 versus hot particle -

detector separation

6.4. Experimental procedure

Natura l uranium particles of three size ranges (de 1 Fm, 10 Fm >

d > 1 pm, d> 10 pm) were deposited on the surface of a substrate called

BIORETM. The uranium particles were placed either directly in contact

with the CR-39 surface or with a spacer producing an air gap of 100,

200, 400 pm. The particle-bonding substrate (attached to a blank CR-

39) and the exposed CR-39 were held in a specially developed holder

during the exposure, which allowed them to reposition after etchhg in

the same position (x = +5 ym, y = k5 pm). The CR-39 was etched after

exposure in 6.25N NaOH solution at 75OC for 6h. After etching, the

plastic was rinsed in 2% acetic acid and then in distilled water. The

length of the exposure was dependent on particle size. Tables BLa,

B 1 .b, B2.a and B2.b in Appendix B gives the exposure times required

for particles of plutonium and natural uranium oxides of different sizes.

Figure 6.2 Three n a t m particles and their respective CR-39 track

clusters, each to the lower ieff of their respective hot particle

After etching, the track clusters were analyzed by an automated

image analysis system based on an OlyrnpusTM microscope and a PriorTM

motorized microscope stage (see Figure 6.2). The particle-bonding

substrate and the etched CR-39 were placed under the microscope in a

position for exposure employing a specidy designed holder. Image

analysis of the X-Y coordinates of the individual alpha-tracks was

carried out using the image analysis software ImageProPlusTM. These

methods are more fully described by Richardson et al., [1999]. These

and track cluster data were provided for analysis by S. Starling.

6.5. Exposure time evaluation

It is important to choose exposure tirnes of hot particles to CR-39

necessary for the cornparison of theoretical predictions and the actual

track count. The choice of the exposure time is important, as

overexposure will produce track clusters that have an over abundance

of overlapping tracks, while underexposure will introduce statistical

incertitude. The exposure time for 10 valid tracks was calculated for

different fuel oxides ~a tU02, d W J 0 2 , 1euUOp , agedPU02 using Equation

(4.1). The specific activity of each fuel matenal is given in Tables A1.a-f

in Appendix A.

The exposure tirnes for natU02 for 10 tracks (with 400 pm air gap) ,

Tables B 1 .a and B 1 .b, were u p to about 6.7 years, 6.7 days and 42

minutes for particles of 1, 10 and 150 Pm diameter, respectively. There

was 4 +16% difference in the exposure times for d e p u 0 2 compared with

leuUO2, the largest difference in the set. The difference in exposure tirnes

for fuel particles with a 400 pm and no air gap was dependent on the

particle size. The difference of exposure times for various types of

uranium particles of 1, 10 and 150 pm were about 1%, 4% and 17%

respectively. The exposure times for aged plutonium oxides for 10

tracks (with 400 pm air gap) were 2 1 years, 7.68 days, 1 1 minutes and

1.2 seconds for particles of 0.01, 0.1, 1 and 10 pm diameter,

respectively. There was no difference be tween the exposure tirnes for

agedPu02 particles with diameters less than 3 Pm, exposed at 400 pm

and no air gap. For agedPuO2 particles with diameters of 10 pm (Tables

B2 .a and B2. b) the difference in exposure times was about 2%.

6.6. Determination of the computed particle diameter

6.6.1. Registration effectiveness of the mined isotope particles

AU calculations performed, based on the theoretical

considerations made in Chapters 4 and 5, assume a homogeneous hot

particle containhg one alpha-emitting substance. In practice, these

radioactive substances are mixtures of different isotopes, each one of

which has different spec5c activities and alpha-particle energies. The

registration effectiveness of the component radionuclides of uranium

oxide fuels are given in Tables B1.a and B1.b for given particle

diameters ranging from 1 pm to 150 Fm. A similar tabulation of data for

plutonium oxide fuels is shown in Tables B2.a and B2.b. Both the

registration effectiveness and the distribution fûnction for a specific

mixture of isotopes will be the surn of its parts taking into account the

fractional radionuclide abundance and specific activity.

where Fi is the fraction of the total activity of the given nuclide in the

oxide mixture, and i is running through the number of nuclides in the

mixture.

Even though the percentage specific activities of the various

radionuclide cornponents of the various rnixed oxïde fuels is simiZar to

those of pure plutonium oxides, the exposure time of mked oxides is

increased related to the later ones. Mïxïng plutonium oxides with

uranium oxides will result in a total take-over by the plutonium oxides

from the point of view of specific activity and has the effect of

downgrading of the plutonium regardless of how small its share in the

mixture is.

6.6.2. Verification of particle sizing method with natural uranium

oxide particles

The predicted number of tracks Nt in a track cluster per day of

exposure for a particle of diameter d is given by:

where p is the particle density, d is the particle diameter, SA is the

specific activity of the isotope mixture, and Texp is the exposure tirne.

Figure 6.3 shows a very strong dependency of the registered

tracks with particle size. For example, for aged plutonium oxide, there is

a h e a r relationship for the Log/log graph of tracks per day against

particle size for a diameter range of from 0.01 Fm to 4 pm when self-

absorption starts to become ~ i g ~ c a n t . If the composition of the hot

particle is known apriori, the particle diameter can be experirnentally

determined from the number of tracks present, i.e.:

Cornparison of predicted particle diameter with measured data

has been performed for natural uranium oxide, provided by the Comeco

Corporation. The experimental set-up and the method used are

described in Section 6.4, and the data collected is summarized in TabIe

6.3.

W."-""*

Particle diame ter (p m)

Figure 6.3 Dependence of registered number of

composition and size

The results obtained through the calibration process are shown in

Figures 6.4 and 6.5. There is a good agreement between the

experimental and theoretical relationship between the registration

effectiveness and particle diameter [Richardson et al. 19991. The

greatest difference occurs in the rapid change in registration

effectiveness that occurs at 20 pm diameter particles, where self-

absorption becomes important, and which is in good agreement with

values found by Terry, [1995]. The best fit straight Iine of Figure 6.5 is

the trend line, which was forced to pass through the origin, and has a

slope discrepancy from the line of identity of 14%. The correlation factor

obtairied under these circumstances was fomd R = 0.94. However

using the method described in reference [Moroney, 19771, the

correlation factor obtained for the measured data was R = 0.95, but in

this case the intercept was found y = 8.3 Fm, and the standard error of

the estimate was found 16%. The individual error occurruig during the

experimental track counting was considered Poissonian [Repin et al.,

preprin t

expected

19983; also, however some systematic errors could be also be

to be present.

+ hperimentd determination

#O 60 80 100

Particle àiameter (pm)

Figure 6.4 Registration effectiveness for mfU02 particles exposed with a

400 prn air gap

Table 6.3 E-penmental data used for calibration calculations

Track Exposure Track Measured Predicted Measured Predicted count time count registration registration diameter diameter

(4 per day effectiveness effectiveness (pm) ( ~ m )

Figure 6.5

Predicted diameter ( p m)

Cornparison of predicted and measured diameters of nat U02

w articles

6.7. Implementation of the size determination method

into routine sizing

Under routine working conditions,

determine the particle sizes in reading the

using Figure 6.3. In order to facilitate

it is rather impractical to

actual value from the graph,

particle size estimations, a

curvefit to the precomputed number of tracks per hour was carried out

and implemented into a spreadsheet software. In this thesis, the

curvefitthg was performed ushg TableCurveT" 2D version 4. The

relationship between particle diameter and the number of

day of exposure was determined for various uranium and

oxide fuels, and are given in the form:

tracks per

plutonium

The best fit correlation coefficients were all very high, 0.998 4 r2 c

0.999. Table 6.4 gives the cuvefit parameters for the most comrnon

fuels encountered in Canadian nuclear environments. Also taken into

account is the porosity of the hot particle which would result in a

reduction of its density: nominal density reductions by 0.75 and 0.5.

Table 6.4 Best fit equation for the predicted diameter from input of

track cluster data

Fitted range

z~f particle diameter

Density Best fit equations dpred = a + b*(N/T)C

-

( P d

1 - 150

1 - 150

Chapter 7

Validation of the sizing method

for overlapping track clusters

When exposing a PAS fdter with hot particles of different sizes to

CR-39 it is possible that the larger particles will have overlapping tracks

in the center of the track cluster. The method developed for these cases

consists in excluding a circular central region and counting the number

of tracks in the remaining part of the track cluster. The radial track

probability distribution D(o) and track density functions D&) are

described in sections 5.2 and 5.3.

The va3idation of the track distribution firnctions was carried out

using large track clusters with no overlapping tracks from the

experimental methods using natural uranium particles described in

Section 6.4. Two examples are shown here based on track clusters that

were created by UO:! particles exposed to CR-39 for 22.66 days both

with 400 pm air gap. The radial probability density and track density

are shown in Figures 7.1 and 7.2 for a 76 pm diameter particle. The

total number of tracks counted was 267 which results in a theoretical

particle diameter of 15 1 f 23 Pm. The correlation factor found in this

case using the method descnbed by Moroney, [1977], was R = 0.80 and

a standard error of the estirnate was S = 2.4%. Simitarly, Figures 7.3

and 7.4 are for a UO2 particle of 64 pm measured diameter that created

a cluster of 182 tracks and which corresponds to a theoretical particle

diameter of 105 t 18 ym. For this case, the correlation factor found was

R = 0-9 1, and the standard error of the estimate was 2.3%.

Both distribution functions, radial probabili~ density D(w) and

radial track counts Nij Equation (5.9) were computed for steps of 20 pm

radial distance. Performing the numerical integration on different

particle diameters ranging from 1 pm to 150 p m for uranium oxides and

from 0.05 Fm to 100 pm for plutonium oxides respectively, in intervals

of 100 pm in the radial distance direction, one can determine the

number of tracks in annulus shaped domains inside the track cluster.

The numerical integration was performed with the Gauss quadrature

method [Abramowitz et al., 19701. As the Gauss quadrature method is

notoriously bad behaving on interpolated functions, it was necessary to

use the spline interpolation scheme of the track per hour distribution

points in order to achieve an integral convergence with the constancy of

at least six significant figures. Results are shown in Tables C1.a and

C1.b. Figures 7.1 and 7.3 are the probability density cornparisons

between the predicted density and the experimentally deterrnined one.

Figures 7.2 and 7.4 give the radial track count cornparison between the

predicted (computed) track distribution and the experimentally

determined ones. The data on which these graphs are based is shown in

Appendix E: Tables E. 1 and E. 2.

Figure 7.1 Calculated radial probability dençity for a natU02 particle,

(dm, = 64 pm) with a 400 pm spacer compared with radial

pro bability density resulting from expen'mental detennination

-LI- -- ---- -. - - - . - A . - - Pred icted

CI---------- track count -

track count 1--- - --

Figure 7 .2 Predicted (theoretical) track distribution for a Mt U02 particle,

(dm=- = 64 pm) compared with experimentally determined

counts

Experirn enta1 . . -_ determ ination - -

O 200 400 600 800 ~ 0 0 0 1200 1400

Radial distance (mm)

Figure 7.3 Calculated radial probability density for a natU02 particle,

(dme, = 76pm) with a 400 p m spacer compared with radial

probability density resulting frorn experimental detemination

1.g ------ Pred icted - - - tnck counr

-. ---.-. m Exp erim ental

Track count - -- -

Figure 7.4 Aedicted (theoretical) track distribution for a U02 pahcle,

(dme, = 76pm) compared with expenmentally determined

counts

Chapter 8

Discussion and

Theoretical and experimental methods have been developed for

estimating the size of alpha-ernittuig particles. The autoradiographs of

alpha-emitting particles on the CR-39 nuclear detector can be O btaïned

easily with a minimum of equipment making this approach a very

attractive procedure. These techniques provide a flexible way of

determining the hot particle diameter starting with the track cluster

registered by a solid state nuclear detector during an autoradiograph

process. Although counting the number of tracks in a cluster is not a

trivial problem, an image analysis system and dedicated software codd

be coded to identify and count track clusters and individual tracks. The

method elaborated in this thesis can also deal with overexposed

autoradiographs of large particles which produce a large number of

overlapping tracks. This problem is handled by counting up the tracks

only from penpheral regions of the track cluster where individual tracks

are easily identifed and by cornparing this data with the theoretical

track distribution on the CR-39 nuclear detector.

The core part of the particle size determination method is the

computation of the registration effectiveness and the track density

distribution. Both these features use two dimensional Gauss-Legendre

integration schemes. This mathematical method allows the required

accuracy in the evaluation of registration effectiveness to be achieved in

a simple manner so that computation is manageable with simple

desktop computing resources using a PentiumB processor without

special memory or hard drive space requirements.

The computation of the registration effectiveness accounts for

self-absorption of the alpha-particle inside the emitting particle. Self-

absorption is significant in high density particles of uranium or

plutonium when the diameter is beyond 3 pm for uranium oxides and 5

pm for plutonium oxides.

The various predicted particle sizes of natural uranium oxide

particles were in a reasonable agreement with the measured values

obtained using an optical microscope. However, there was data outside

the error margins due to the Poisson distribution error arising from the

number of alpha-particle tracks counted. Discrepancy between the

predicted particle and the measured data is caused by several factors:

- Individual measwements of particle size were determined by the

CR-39 method and had a signifïcant scatter, although in practice rnany

particles would be sized and the median particle diameter would be

assessed. I n order to achieve more constant and reproducible

measurements, further research may be needed to irnplement the

method of hot particle size deterrnination before it is suitable for routine

use.

- As regarding the theoretical approach, improvements cari be

achieved by refmement of the theoretical parameters used, for example,

a better choice of the minimum dip depth &, the cut-off angle Oc, and

the hot particle density p. The uranium oxide and the plutonium oxide

densities are known. However, the porosity of the particle is not known,

which can reduce the densiw to as much as one third of the compact

particle density. This has to be studied as a site dependent factor. This

project has made allowance for porosity and its effect on the registration

effectiveness by considering different fractions of the nominal values.

- Deviations from the sphencal mode1 should be considered. In

this project, hot particles were considered to be spherical in shape. As

the size of the particle becomes larger, the shape factor and

corresponding effect on the track registration effectiveness will be

increasingly important, shce it is only the part of the hot particle which

faces the detector which will be able to register tracks. Theoretically, it

may be possible to estimate the registration effectiveness for different

ovoid shapes, but the practicality of making use of this in estimating

particle size would be limited.

The minimum particle diameter that the CR-39 method developed

would be useful in measuring is mainly determined by the length of the

exposure time. The exposure tirne of a given particle size is detemiined

by the specific activiw of the hot particle and the time needed to render

a good exposure (Le. a minimum of a significant nurnber of tracks at the

one extreme, and a track cluster with few overlapping tracks at the

other extreme). As can be seen in the exposure time tables given in the

Appendix, for plutonium oxide particles, and also the particles of mked

oxide, only particles with diameters greater than 0.06 pm give a

significant nurnber of tracks (10 or more) for an exposure time of 5

days. For particles smaller than the above mentioned, the exposure

time goes up dramatically; for example, a part..de of 0 .O 1 pm requires

an exposure of over a thousand days. Uranium oxide particles, which

have a much lower specific activity than particles with a plutonium

content, consequently need to be larger in size to be analyzed by the

methods described. The smallest uraniurn oxide particle, which can be

sized after an exposure of 5 days, is between 2 pm and 15 pm in

diameter, dependhg on the level of enrichment, and considering a

minimum of 10 registered tracks for acceptable confidence.

Appendix 1

The mathematical proof of

elementary solid angle formula

The defmition of the elementary, discrete solid angle defined on

the e l e m e n t q surface area (voxel) dSij of the ith annulus in the jth

circdar sector, from an exterior point P, is:

In our geometrical set up, the elementary, discrete solid angle Clij

will become:

express the

dSLjPP1 a - . = "' (PM' )

solid angle in terms of known quantities,

some additional geometrical elements:

we need

The defmition of the length of the circular arc I'J' is:

The area of the circular sector P'I'J' is:

In terms of the above defined quantities the elementary area dSij

will give:

Segments P'I" and P'I' cari be expressed in the following form:

P' 1" = Pt 11+1' 1" and PI' = PP' + i * 1' I"

So the elementary area of the ith voxel dSij gives:

The distance from the emittuig point to the elementary area dSij

can be expressed as:

PM' = ,/(PI II+=)* + P P ' ~ 2

Using the above formula for PM' the elementary solid angle nij will give:

1 [PP1*tan9,,, + (i + -) * 1'1" ] * I'I"*P,, * PP'

= 2 3

l * ' , ' {[PP'*tanû,,,j + (i + -) I I l2 + (PP')2 )2 2

Using the defmition of 1'1":

1' 1" = PP1*(tanBI,, - tane,,) nvoxel

and the abreviation t:

1 tme,,j - taneh1.j t = ta~~û,,,~ + (i + -) *

2 nvoxel

the elementary, discrete solid angle gives:

Pstep

= nvoxel 3

After doing the simplifications, d l get the final

elementary, discrete solid angle nij, which will depend

form of

only on

the

the

limiting angles 8mi and 9 ~ , the stepping angle Pstep and the position of

the voxel defmed by the running indexes i and j:

ta~~û, , ,~ - taMhlVj , - * -- 1 + (i + -) * - --- --

~ , t e p [ 2 nvoxel -- I 2 - nvoxel 0i.j - 1 taneh2.j -tan',~,j 2 ([tanO,l,j + (i + -) * -1 + I ) " ~

2 nvoxel

Hot particle

Cluster from point source

Posterched surface

Figure Al Geometncul parameters used in the proof of the solid

angle fomula

Appendix A

Specific activities (SA) of most frequently encountered aipha-emitting oxide fiels

Table A 1 .a Specific activities for agedPu1

Half iife Radionuclide Mass Specific . Specific Percentage Radio T1/2 Specinc composition activity activity of total

nuclide activity as a component a g c * P ~ 0 2 specific (Y) lS-l ~ q 2 (%) (s-l gl) ( - 1 ) activiîy (%)

238Pu 8.773+0 1 6.34E-t- L 1 0.20 1.273+09 1.12E+09 16-4

Table A 1 . b Specific activities for n ~ t U 3

Radio Half life Radionuclide Mass Specific Specific Total nuclid T1j2 specifk activity composition activity activïty specific

e @) (Bq g-9 ph) as a NATU02 activity component

q gl) (Bq gL1 ("A)

234U 2.463+05 2-30E+08 0.0055 1.273+04 1.1 1E+04 49 -4

* 3 W 7.043+08 8.00E+04 0.7200 5.763+02 5.073+02 2 -2

2 3 W 2.343+07 2,39E+06 0.0000 0.00E-t-00 0.003+00 0-0

2 3 W 4.473+09 1.243+04 99.2745 1.233+04 1 .09E+04 48.3

Total 100.00 2.563+04 2,25E+04 100

1 Recycled fiorn 1983 weapon grade Pu ' Plutonium isotopes are not only alpha-emitters, and here only the alpha-ernitting component of their activity was considered; so to enhance that the units had been taken (s-' g-') 3 Natural uranium

Table Al .c Specific actiuities for f e ~ ( I i

Half Iife Radionuclide Mass Spe&c Specific Percentage Radio TL/Z specific composition activïty activity of total

nuclide activiîy as a "*TU02 specific (Y) (Bq g') (%) component (Bq g-1) activity (?A)

q

*34U 2.46E+05 2.30E+08 O. 15 3.45334-05 3.04E+O5 92.4

Total 100 3-743+05 3.29E+05 100

Table A 1 .d Specifc activities for dep U2

Radio Half M e Radionuclide Mass Specific Specific Percentage nucl..de TL,^ specific composition activiîy activity of total

activity as a component natU02 specifÏc -

(Y) (Bq gr) (5%) (Bq g-1) q F;') activity (%) 734U 2.46E+05 2.30Et08 0.003 6 -9 1 E+03 6.08E+03 35.2

Total 100 1.96E+04 1.73Ei-04 100

' Low enriched uranium ' Depleted uranium

Table Al .e Specifc activities for mked oxide of a@pU with natU

Half life SA M a s s SA SA 0 2 Percentage 7'112 composition of total

Total 100 3-873+07 3.42E+07 100

Table A 1. f Specific activities for mixed oxide of a@Pu with dep U

MOX depU 99.7% agedm;i 0 -3% Half Tife SA M a s s SA SA 0 2 Percentage

T1/2 composition of total (Y) (s- 1 .g L) (%) (s-~ g ~ ) (s-I gq SA(%)

234U 2.46E+05 2.30E+08 0.003 6.89E+OS 6.06E+05 0.030

Total 100 2.323+09 2.05E+09 100

' Mixed oxides

Appendix B

Exposure-time tables for the ficek

considered in this studg

Table B 1 .a Exposure times for U02 for 1 O tracks with 400 prn air gap

Particle diameter

(w)

Registration effectiveness Exposure tirne for 10 tracks

(days) naru l e u u dcpU

2434 2434 2434

304.8 306.2 305.3

96.03 92.60 97.51

47.37 44-22 48.66

28.75 26.58 29.63

19.39 17.81 20.04 13.99 12.81 14.48

10.63 9.705 11.01 8.353 7.629 8.649 6.730 6.139 6.972 1.664 1.514 1.725 0.736 0.670 0.764 0.416 0.378 0.432 0.266 0.243 0.275 0.184 0.167 0.191 0.135 0.123 0.140 0.103 0.093 0.107 0.081 0.074 0.084 0.066 0.060 0.069 0.054 0.049 0.057 0.046 0.042 0.047 0.039 0.036 0.040 0.034 0.031 0.035 0.029 0.027 0.030

79

Table B 1 .b Exposure times for U02 for 1 0 tracks with no air gap

Registration effectiveness Exposure t h e for 10 tracks (days)

-tu lcuU d'pu 2434 2434

80

Table B2.a Exposure times for 1 0 tracks from 1 5 years aged A r 0 2 partzrtzcIes with 400 pm air gap

~ s s t r a t i o n effectiveness Exposure t h e

Table B2.b Exposure times for IO tracks from 15 years a g e d Pu02 particles with no air gap

Registration effectiveness Exposure time

(days)

7.6830E+03

9.6038E+02

2.8456E+02

1.2005E+02

6.1464E+0 1

3.5570E+0 1

2.2399E+O 1

1.5006E+O 1

1.05393+0 1

7.6830E+00

9.6038E-0 1

2.84563-0 1

2.2005E-0 1

6.1464E-02

3.5570E-02

2.2399E-02

1 -5006E-02

1.05393-O2

7.6830E-03

9.6038E-04

2.84563-04

2.2040E-04

6-45733-05

4.1274E-05

2.8827E-05

2.15613-05

1.6669E-05

1.3335E-05

3.2082E-06

1 -4232E-06

7.9736E-07

5.1027E-07

3.5394E-07

2.6 145E-07

1.9855E-07

1.5703E-07

1.2741E-07

5.69903-08

Appendix C

Truck distribution data, registered on CR-39, for the aïpha-emitting isotopes studied in this project

Table C 1 .a Distribution of track per days for nat UO2

83

Table C 1. b Distribution of track per days for agedPu02

Table D 1 The ahha particle energy and its range R CR39 for al1 alpha

emitting radioactive isotopes considered in thk thesis.

Nuclide Alpha-particle energy Range in CR-391 Range in CR-392

(MeV) (P-4 bm) 2 4 1 ~ f n 5.48 34.77 35.0

238% 5.50 34.97 35.0

2 3 9 ~ 5.15 31.52 32.0

24om;l 5. 17 31.71 32.0

24 1 p u 3 - - -

2 4 2 ~ 4.90 29.14 29.0

2 3 3 ~ 4.9 1 29.14 28.0

2 3 4 ~ 4.86 27.97 27.8

2 3 5 ~ 4.40 24.62 24.0

2 3 8 ~ 4.20 22.85 22.8

I Calculated with equation (3.1) using z = 9 ' Values taken fkom tables published in Fews, 19821 Beta emitter

Appendix E

Table E. 1 Cornparison between predicted and experimentally

detemined track counts for sample one shown in Figures

7 . 1 and 7.2

Track cluster Predicted Experirnentd diameter track count track count

Table E. 2 Cornparison between predicted and experïmentally

detenined track counts for sample one shown in Figures

7.1 and 7.2

Track cluster Predicted Experimental diameter track count track count

(prn) 50 2 2 100 6 3 150 9 10 200 11 9 250 12 12 300 12 8 350 12 11 400 12 8 450 11 7 500 10 13 550 9 14 600 9 9 650 8 9 700 7 4 750 7 10 800 6 7 850 6 6 900 5 7 950 5 8 1000 4 7 1050 4 5 1100 4 7 i 150 3 2 1200 3 1 1250 3 3 1300 2 O 1350 1 O 1400 1 O 1450 O O

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