some effects on polymers of low-energy implanted...

Some effects on polymers of

low-energy implanted

positrons

FACULTY OF SCIENCES

Carlos Andres Palacio Gomez

Department of Subatomic and Radiation Physics

Ghent University

A thesis submitted for the degree of

Doctor in sciences: Physics

October 2008

mailto:[email protected]

http://ssf.ugent.be/linac/tmp_ssf/

http://www.ugent.be

This thesis was submitted to obtain the degree of ‘Doctor in Sciences: Physics’at the Ghent University. The public defense of this thesis was held on October 31,2008.

Examination committee:

Prof. Dr. Danny Segers Ghent University, promoterDr. Jeremie De Baerdemaeker Ghent University, copromoterProf. Dr. Charles Dauwe Ghent UniversityDr. Steven Van Petegem Paul Scherrer Institute (Villigen, Switzerland)Prof. Dr. Eddy de Grave Ghent UniversityProf. Dr. Roland Van Meirhaeghe Ghent UniversityProf. Dr. Dirk Ryckbosch Ghent University, chairman

This work was produced and published at:

Ghent UniversityDepartment Subatomic and Radiation PhysicsProeftuinstraat 86BE-9000 GentBelgium

Copyright c© 2008 Carlos Andres Palacio Gomez. All rights reserved.This document was prepared with LATEX 2ε.

I would like to dedicate this thesis to my charming and adorable wifeAlexandra and to my loving parents.

Acknowledgements

The acknowledgments go to all the people that I have had the oppor-tunity to meet and work with during all these years.

Without a preferential order, first of all I would like to thank threepersons who have been fundamental to the successful development andfor the proof reading of my Ph.D. thesis.

• I would like to thank my promoter Danny Segers who has givenme the opportunity to become a Ph.D. student in the NUMATgroup, for teaching me, for helping me at any time I needed andfor supporting me during all these years.

• My deepest gratitude also goes to Charles Dauwe for teaching me,for introducing me to the world of positrons, and at the same timefor giving me the right amount of criticism to bring my researchto a higher level. Charles, after all our discussions I must admitthat never before I had the opportunity to meet such an excellentresearcher as you are. I am glad to have known you.

• I am also entirely grateful to Jeremie De Baerdemaeker who hasspent days and nights helping me, teaching me, discussing withme any detail at any time, learning me to be critical even aboutobvious things, and giving me all his support and encouragementto get the better results and analysis obtained during my thesiswork. Jeremie, without you this thesis would have never been ac-complished and would have been considerably postponed. Thankyou so much.

Many special thanks to the reading committee of this thesis, for theirhelpful corrections and suggestions: Danny Segers, Jeremie De Baerde-maeker, Charles Dauwe and Steven Van Petegem.

Many thanks also to Eddy de Grave for giving me the opportunity towork half time in the Mossbauer group during the last year in parallelwith my Ph.D. thesis. Eddy, thank you for all your help inside andoutside the laboratory.

iii

I would like to express my gratitude to all my other colleagues ofthe NUMAT: Robert Vandenberghe, Bartel Van Waeyenberge, ArneVansteenkiste, Valdirene Gonzaga De Resende, Julieth Alexandra MejıaGomez (lee tu dedicatoria especial al final), Caroline Van Cromphaut,Khaled Mostafa, Nicolas Laforest, Abdurazak M. Alakrmi and ToonVan Alboom. Special thanks to Bartel Van Waeyenberge who was alsoteaching and helping me especially during the first years of my Ph.D.and to Khaled Mostafa for being one of my best friends in the labo-ratory, for his confidence, loyalty and transparency, and also for hissupport during all these years. Thanks Khaled!.

For the technical support I would like to thank the team of the ad-ministrative and technical staff: Philippe Van Auwegem, Roland DeSmet, Patrick Sennesael, George Wiewauters (still for me), ChristopheSchuerens, Bart Vancauteren, Daniella Lootens, Linda Schepens, BrigitteVerschelden and Rudi Verspille. Special thanks to Philip Van Auwegem,Linda Schepens, Roland De Smet and Rudi Verspille.

Furthermore, many thanks to all my other colleagues professors andgraduate students of our laboratory. Specially to Christine Iserentant,Luc Van Hoorebeke, Jelle Vlassenbroeck and Tim Van Cauteren.

Thanks to the Ghent University, the Fund for Scientific Research (Fondsvoor Wetenschappelijk Onderzoek (FWO)) and the Interuniversity At-traction Poles (IUAP/PAI) V/01–Network program of the Belgian Fed-eral Government, for their financial support.

Many special thanks to Jan Kuriplach, Steven Van Petegem and Niko-lay Djourelov for all your help and for your friendship.

My acknowledgements also go to the families of my professors and/orcolleagues for giving me their hospitality and for giving me some goodmemories. Specially to Carmen (Charles’s wife) and Nathalie (andkids) (Jeremie’s family).

Me gustarıa agradecer a todas las personas que he conocido y con lasque por una u otra razon he tenido la oportunidad de compartir muybuenos momentos. Por lo tanto, mis mas sinceros agradecimientosson para Yves, Clara, Julian, Carolina, Orlando, Valdirene, Milena,Luca, don Giovanny, Maria Isabel, Gladis, Elisa, Fredy, Wim, Johan,Geertrui, Julio (y esposa), Mauricio, Douglas, Liliana (y familia), Adri-ana, Patrick, Marco, Clarena (y familia), Felipe (y familia), Flavia yDiego. My acknowledgements also go to the several of my friends thatperhaps I have forgotten to include in this list.

I also would like to acknowledge to the staff of OBSG for allowing

iv

us (–me and my wife–) to live, during about 3 years, in their niceenvironment and for inviting us to participate in their several pleasantactivities. Definitely: “Our home away from home”. Special thanks toIsabelle Mrozowski and Marleen Van Stappen, both of you are reallyoutstanding. Here I would also like to mention to Piotr and Eliza.

Agradezco con todo mi corazon a todos mis familiares y amigos quesiempre estuvieron pendientes y en muchas ocasiones me preguntaronacerca de mis progresos con la tesis. Aunque no siempre estuve muypositivo, les agradezco por el interes y los animos que me brindaron.En especial agradezco a mis padres Jorge y Nury, a mis hermanosJorge y Paula (y a mi sobrinito Andres, que en el futuro entendera laimportancia de esta tesis), a mi cunado German, a mi suegra Marthay a mi cunado Jaider.

Finally, my deepest and unlimited words of gratitude go, of course, tomy wife Alexandra for her love, kindness and patience. Dear Alexan-dra, this thesis would not have existed without you. You have givenme the opportunity to let me work on it during many, many eveningsand weekends even in several occasions when probably it would havebeen very important for you that I would have given my 100 percentattention to you. I am a really lucky guy for ‘having’ you with me.Thank you very much for your time and for your encourage words.

Carlos Andres Palacio Gomez, October 2008

v

Contents

Acnowledgements iii

Table of contents vii

List of Figures xi

List of Tables xv

Nomenclature xvii

Introduction 1

PART I: INTRODUCTION TO POSITRON PHYSICS 5

1 Introduction to the positron annihilation spectroscopy 7

1.1 The positron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 Positron Annihilation . . . . . . . . . . . . . . . . . . . . . 8

1.1.2.1 Free positrons interaction in condensed matter . . 10

1.2 Positronium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 Positronium wave function . . . . . . . . . . . . . . . . . . 11

1.2.2 Annihilation selection rule and decay rates . . . . . . . . . 12

1.2.3 Positronium formation in molecular media . . . . . . . . . . 14

vii

CONTENTS

1.2.4 Positronium quenching . . . . . . . . . . . . . . . . . . . . . 17

2 Experimental techniques in positron annihilation 19

2.1 Positron sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Positron annihilation lifetime spectroscopy (PALS) . . . . . . . . . 21

2.2.1 The basic operating principle . . . . . . . . . . . . . . . . . 21

2.2.2 Lifetime Data Treatment . . . . . . . . . . . . . . . . . . . 23

2.2.3 Relation between the positronium lifetime and the free-vo-lume-hole size . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.3.1 The Tao-Eldrup model . . . . . . . . . . . . . . . 24

2.2.3.2 lifetimes in the free volumes of larger radii . . . . 26

2.3 Doppler shift or broadening of the annihilation radiation (DBAR) 27

2.3.1 The S- and W-parameters . . . . . . . . . . . . . . . . . . . 29

2.3.2 Coincidence DBAR (CDBAR) . . . . . . . . . . . . . . . . 31

2.4 Angular correlation of annihilation radiation (ACAR) . . . . . . . 33

2.4.1 Relation between the para-positronium momentum and thefree-volume-hole size . . . . . . . . . . . . . . . . . . . . . . 34

3 Interaction of positrons with solids and surfaces 37

3.1 Slow Positron beams . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 Introduction to positron moderation . . . . . . . . . . . . . 38

3.1.1.1 Positron re-emission . . . . . . . . . . . . . . . . . 39

3.1.2 Beam transport . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Positron beam interactions with solids and surfaces . . . . . . . . . 42

3.2.1 overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2 Positron backscattering . . . . . . . . . . . . . . . . . . . . 43

3.2.3 Positron implantation profile . . . . . . . . . . . . . . . . . 44

3.2.4 Positron diffusion . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.5 Epithermal positrons . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Experimental determination of the positronium fractions . . . . . . 50

3.4 Charging effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

viii

CONTENTS

PART II: EXPERIMENTAL DETAILS 55

4 Experimental set-up and Samples description 57

4.1 Variable energy positron beam . . . . . . . . . . . . . . . . . . . . 57

4.2 Doppler broadening . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Photon detection system . . . . . . . . . . . . . . . . . . . . 58

4.3 Polymer samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.1 Kaptonr . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.2 Thin polymer films . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.2.1 Spin Coating . . . . . . . . . . . . . . . . . . . . . 60

4.3.3 Free-standing nanometric polymer films . . . . . . . . . . . 61

PART III: RESULTS AND DISCUSSION 65

5 Parameterization of the median penetration depth of implantedpositrons in free-standing nanometric polymer films 67

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 Charging effects . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Analysis and results . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Determination of the positron diffusion length in polymers byanalysing the positronium emission 91

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.3 Analysis and results . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3.1 Position of the p-Ps contribution in annihilation spectra . . 103

6.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 111

A Basic derivation for the momentum measurements 113

A.1 Energy of the annihilation γ−rays . . . . . . . . . . . . . . . . . . 113

A.2 angular correlation of the two γ−rays decay . . . . . . . . . . . . . 114

ix

CONTENTS

B Positronium fraction from the Compton-to-peak ratio analysis ofthe annihilation spectrum 117

Bibliography 119

Publications 133

x

List of Figures

1.1 Schematic view on the terminal positron blob . . . . . . . . . . . . 16

2.1 Simplified decay scheme of the radioactive isotope 22Na . . . . . . 20

2.2 Schematic positron lifetime spectrometer . . . . . . . . . . . . . . . 22

2.3 Positron lifetime spectrum . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Graphical representation of the Tao-Eldrup model . . . . . . . . . 25

2.5 Graphical representation of the extended Tao-Eldrup model . . . . 27

2.6 The vector diagram of the momentum conservation in the 2γ-anni-hilation process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Schematic drawing of a typical Doppler broadening setup . . . . . 29

2.8 S- and W-parameters calculation . . . . . . . . . . . . . . . . . . . 30

2.9 Coincidence Doppler broadening spectrum . . . . . . . . . . . . . . 32

2.10 Schematic view of a 2D-ACAR setup . . . . . . . . . . . . . . . . . 34

2.11 Representation of the relation between the p-Ps momentum andfree-volume-hole size . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Comparison of the energy spectrum before and after the moderationprocess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Representation of the one dimensional potential representation fora thermalized positron near the surface of a metal . . . . . . . . . 41

3.3 Schematic representation of the possible positron interactions . . . 43

3.4 Examples of some Makhov profiles . . . . . . . . . . . . . . . . . . 46

xi

LIST OF FIGURES

3.5 Example of two extreme conditions of Ps formation . . . . . . . . . 51

3.6 Charging effects model . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Example of an imide group . . . . . . . . . . . . . . . . . . . . . . 59

4.2 General types of polyimides. . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Floating PMMA film . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Some of the free-standing nanometric PMMA films . . . . . . . . . 63

5.1 Scheme of the experimental DBAR setup . . . . . . . . . . . . . . 69

5.2 Charging test for polystyrene . . . . . . . . . . . . . . . . . . . . . 71

5.3 Compton-to-peak charging test for polystyrene . . . . . . . . . . . 71

5.4 Peak counts: extrapolation to high energy values . . . . . . . . . . 73

5.5 S-parameter for a 220 nm PMMA film in Ghent. Comparison whenthe chamber walls are internally cladded with Teflon with thosewithout clad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.6 Screenshot of the SIMION simulation of the trajectory of the trans-mitted positrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.7 S-W results obtained in Ghent for the different PMMA and PS filmsamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.8 Obtained S-parameters as a function of the implantation energy forthe PMMA and PS films . . . . . . . . . . . . . . . . . . . . . . . . 80

5.9 Transmission experiments performed at Washington State Univer-sity to some of the free-standing polymer samples . . . . . . . . . . 81

5.10 Graphical representation of the power-law z1/2(E) = αρE

n accord-ing to the data of the transmission experiments . . . . . . . . . . . 84

5.11 S and W-parameters of a Non-detached PMMA thin film . . . . . 86

5.12 Thicknesses of the PMMA samples obtained with the different val-ues for the parameters α and n compared with the experimentalthickness values at the extracted energy values E1/2 . . . . . . . . 88

5.13 Thicknesses of the PS samples obtained with the different values forthe parameters α and n compared with the experimental values ofthe thickness at the extracted energy values E1/2 . . . . . . . . . . 89

6.1 Experimental setup with the sample at 45 and perpendicular withrespect to the beam axis . . . . . . . . . . . . . . . . . . . . . . . . 94

xii

LIST OF FIGURES

6.2 Comparison of the peak statistics as a function of the positron im-plantation energy in Kapton for three successive measurements . . 95

6.3 Charging test for Kapton . . . . . . . . . . . . . . . . . . . . . . . 96

6.4 Annihilation peak obtained for the PMMA sample for an implantedpositron energy of 467 eV (a) at 45 and (b) perpendicular withrespect to the beam axis . . . . . . . . . . . . . . . . . . . . . . . . 99

6.5 Annihilation peak obtained for the Kapton sample (a) at 45 and(b) perpendicular with respect to the beam axis . . . . . . . . . . . 100

6.6 Schematic representation of the positronium emission from a samplesurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.7 Free p-Ps population as a function of the distance between the sam-ple and the annihilation position . . . . . . . . . . . . . . . . . . . 104

6.8 Ps emission from the of 310 nm-thick PMMA film . . . . . . . . . 105

6.9 Ps emission from the Kapton surface when the sample is at 45 withrespect to the beam axis . . . . . . . . . . . . . . . . . . . . . . . . 107

6.10 Ps emission from the Kapton surface when the sample is perpen-dicular with respect to the beam axis . . . . . . . . . . . . . . . . . 108

xiii

LIST OF FIGURES

xiv

List of Tables

4.1 Spin coating: preparation of the thin polymer films . . . . . . . . . 62

5.1 Extracted energy values E1/2 from the transmission experiments . 82

5.2 Comparison of the thicknesses (z1/2) of the thin polymer films ob-tained from the different values for the parameters α and n thatcharacterize the well-known power-law (z1/2 = α

ρEn) at the ex-

tracted energies E1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.1 Comparison of the values obtained from the fitting of the experi-mental intensities of the fly-away p-Ps, the fly-away o-Ps and thebulk p-Ps for the PMMA film by using the the different values forthe parameters α and n that characterize the well-known power-lawequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2 Comparison of the values obtained from the fitting of the exper-imental intensities (at 45 and perpendicular with respect to thebeam axis) of the fly-away p-Ps and the fly-away o-Ps for the Kap-ton sample by using the different values for the parameters α andn that characterize the well-known power-law equation . . . . . . . 109

xv

LIST OF TABLES

xvi

Nomenclature

Symbols

e+ Positron

e− electron

τ Positron mean lifetime

π ' 3.1416 . . .

D+ Positron diffusion coefficient

L+ Positron diffusion length

Acronyms

Ps Positronium

p-Ps para-Positronium

o-Ps ortho-Positronium

FWHM Full Width at Half Maximum

PALS Positron Annihilation Lifetime Spectroscopy

DBAR Doppler Broadening of Annihilation Radiation

CDBAR Coincidence Doppler Broadening of Annihilation Radiation

ACAR Angular Correlation of Annihilation Radiation

VEP Variable Energy Positrons

PMMA Poly(methylmethacrylate)

PS Polystyrene

xvii

Introduction

Background

The positron annihilation spectroscopy has shown to be a very effective and po-werful tool, which allows accurate analysis of a wide variety of materials. Positronscan penetrate into liquids and solids without damaging the material. The anni-hilation gamma rays give information about the structure of the material and itsinteractions with positrons.

Several important advances have been made in the last years, specially inpositron annihilation in metals, gasses, and in positronium chemistry. In addition,the commercial development of stable and fast electronic apparatus has encouragedseveral scientific groups to work in the field so that the rate of progress is rapidlygrowing up.

Aim

Experiments concerning positrons and monoenergetic positron beams have at-tracted the interest of several laboratories. The number of original papers wherethe wide application field of a positron beam is reported can be shocking for anonspecialist. Thus, this thesis is written with two purposes:

- To give an introduction to the field of positron annihilation spectroscopy.However, readers who are interested in going deeper into the subject shouldmake their second step with the help of specialized reviews and collectedpapers in the proceedings of recent topical meetings.

- To investigate some of the effects that have the low energy (E < 30 keV)

1

positrons when they are implanted on polymers. This thesis is focused es-pecially on two issues:

1. The median penetration depth of positrons as a function of the im-plantation energy z1/2(E), related to the positron implantation profileP (E, z) is assumed to be a power-law z1/2(E) = α

ρEn. Here ρ is the

sample density and the constants α = 4.0(±0.3) µg cm−2 keV−n andn = 1.60(±0.05) are the most frequently used empirical parameterswhich have been under some debate.A few years ago, specifically in the case of polymers, the values α =2.8(±0.2) µg cm−2 keV−n and n = 1.71(±0.05) have been suggested.These values were found by analyzing the ortho-positronium yield frompositron lifetime experiments at different implantation energies. How-ever, these experiments were performed on several non-detached (fromthe Si substrate) spin-coated polymers.From here arise the first motivation as it is expected that in a non-detached polymer the interaction at the interface with the substratewould have a higher contribution of annihilation of positrons in thepolymer than in the case of self-supporting films. Therefore, for thefirst time, positron annihilation experiments are performed on self-supporting nanometric polymer films.In addition, by performing transmission experiments, and with a previ-ous knowledge of the thickness of the samples, the values for the parame-ters α and n are obtained from (a) the measurements of the positron an-nihilation line-shape parameter, derived from Doppler broadening of an-nihilation radiation (DBAR) measurements, performed at the positronbeam in Ghent, together with (b) the peak rate measurements per-formed to some of the samples at the positron beam facility in Wash-ington. Finding the values for the parameters α and n by means ofthe line-shape parameter in the way as it is presented in this thesis isalso a novel method for the positron community. Therefore, as a sec-ond motivation, and by suggesting a novel method, these values for theparameters α and n are found and compared.Finally, all the values for the parameters α and n considered in this the-sis are used to calculate the thicknesses of the samples and to comparethem with the experimental ones.

2. The study of the positron motion is important for understanding theinteractions of positrons with matter. When bombarded by low-energypositrons, an interesting phenomenon that usually appears in somemetal oxides and in polymeric materials is the emission of positron-ium from the sample surface.

2

Nomenclature

In a longitudinal setup, with the γ−ray detector located behind thesample on the axis of the beam, it has been shown that the Ps emittedat the front side surface of the sample has a linear momentum mainlyaway from the detector. In that experiment, the detected photo-peakin DBAR measurements was approximated by a Gaussian distribution.The p-Ps contribution was detected as a narrow fly-away peak at thelow-energy side of the 511-keV-line (red-shift contribution).A motivation here is to prove that depending on specimen-detectorgeometry, the detected photo-peak from DBAR experiments (after thebackground subtraction) can also be affected by the contribution of thep-Ps emission at the high-energy side (blue-shift) or at the central partof the photo-peak. A blue-shifted peak has an advantage over the red-shifted because the Compton background contribution that appears atthe low-energy tail of the detected photo-peak can be avoided.Two different specimen-detector geometries are thus proposed: Thepolymer sample is located (1) at 45 and (2) perpendicular with respectto the positron beam axis. The detector is located beside the sampleposition, but perpendicular to the positron beam line.These experiments are performed in two different polymers: poly(methyl-methacrylate) (PMMA) and Kapton. This experiment is also interest-ing because Ps is formed only in PMMA and not in Kapton.The positronium (Ps) emission from the sample surface is studied byusing Doppler profile spectroscopy and Compton-to-peak ratio analysis.From the obtained results, and by using all the values of the parametersα and n discussed in the item 1., the thermal and or epithermal positrondiffusion length, the efficiency for the emission of Ps by picking up anelectron from the surface, and in addition for the case of PMMA, thebulk Ps fraction and the diffusion length of p-Ps and ortho-positronium(o-Ps) are obtained.

Although this work is not done with the pretext of solving all the questions, it isexpected, however, that the obtained results may contribute to the knowledge andmay open a new door for the investigation in the positrons field.

Scope

This thesis is divided in three parts: I. introduction (devoted to readers who haveno former familiarity with positron annihilation), II. experimental details and III.results and discussion.

3

• In Chapter 1 we give the reader a short introduction into positron physicsand positron annihilation spectroscopy.

• Chapter 2 describes the most conventional experimental techniques in positronannihilation.

• Chapter 3 gives an introduction about the low energy, or slow, positronbeams.

• In Chapter 4 the experimental setup used for this thesis as well as the de-scription (and in some cases the preparation) of the polymer samples aredescribed in detail.

• Chapters 5 and 6, respectively, deal with the interpretation, results, discus-sion and conclusions of the two issues described above.

4

PART I

INTRODUCTION TO

POSITRON PHYSICS

5

1Introduction to the positron

annihilation spectroscopy

A small background about the positron and positronium physics is given in thischapter.

1.1 The positron

1.1.1 Historical remarks

The prediction and subsequent discovery of the existence of the positron, e+, con-stitutes one of the big successes of the theory of relativistic quantum mechanicsand of the twentieth century physics. The first theory that was consistent withboth quantum mechanics and the special relativity was presented by the scientistPaul A.M. Dirac.

In special relativity the relationship between the total energy E ofa free particle with rest mass m0 is related to its linear momentump by:

E2 = p2c2 +m02c4 (1.1)

where c is the velocity of light. The two solutions for this equa-tion are E = c

√p2 +m2

0c2 and E = −c

√p2 +m2

0c2. Dirac stated that the ne-

gative energy solutions of the relativistically invariant wave equation had a realphysical significance leading to a fully occupied (in accordance with the Pauli ex-clusion principle) ‘sea’ of electron states with negative energies between −∞ and

7

1. INTRODUCTION TO THE POSITRON ANNIHILATIONSPECTROSCOPY

−m0c2 (with m0 the electron rest mass. – The Dirac sea is a theoretical model of

the vacuum as an infinite sea of particles possessing negative energy so that theanomalous negative-energy quantum states predicted by the Dirac equation forrelativistic electrons could be explained –). A ‘hole’ or ‘vacancy’ in this sea, how-ever, would appear itself as a positively charged particle with a positive rest mass,which, on the basis of uncalculated Coulomb energy corrections and the particlesthen known, Dirac assumed that the hole might be the proton [1, 2]. It was soonrealized by Hermann Weyl that this was not the case and that the theory actuallypredicted the existence of a new particle with the same rest mass and magneticmoment as an electron and equal but opposite charge, whereas the proton is over1800 times heavier. Dirac predicted therefore the existence of the positron.

The positron was discovered experimentally at a later date (1932) by C. D.Anderson when he was studying cosmic radiation with a cloud chamber [3–7]. Theexistence of the positron was likewise proved by Blackett and Occhialini (1933) [8]in the phenomena of “pair production” and proved by Curie [9] in radioactivedecay.

As the discovery of the positron in 1932 confirmed the theory of Dirac, he wasawarded the Nobel prize for Physics in 1933.

1.1.2 Positron Annihilation

In vacuum the positron is a perfect stable particle. However, our world is made ofmatter and not anti-matter. When anti-matter and matter meet, they annihilateconverting their masses into energy. In the theory of Dirac, this conversion of mat-ter to energy (i.e. the positron-electron annihilation) can be seen as the radiativede-excitation of the electron. This process can be described by quantum electrody-namics (QED) and may proceed by the creation of zero, one, two or three photonswithin the constraints of energy, momentum and spin conservation. Higher orderprocesses are also possible but they have never been observed for free positrons.The total annihilation cross-section σa is given by the sum of the single processescross-sections:

σa = σ0γ + σ1γ + σ2γ + σ3γ + ... (1.2)

The total spin S of the annihilating positron-electron pair can be either 0 or 1.In an unpolarized medium, the random orientation of spins leads to a statisticalweight of ws = 1

4 for the singlet state (S = 0) and wt = 34 for the triplet state

(S = 1).

Since a photon has spin 1, the conservation of spin will limit the annihilationof a positron-electron pair in the singlet S = 0 state to the emission of an even

8

1.1 The positron

number of photons and the triplet state to the emission of an odd number ofphotons. Due to the conservation of momentum, single photon and zero photonannihilation require a third and fourth body, respectively.

When the positron and the electron are in the singlet spin state (spins anti-parallel), the two-photon process is the most probable. The cross-section for thisprocess was derived by Dirac [2] to be:

σ2γ =4πr20γ + 1

(γ2 + 4γ + 1γ2 − 1

ln(γ +

√γ2 − 1

)+√γ2 − 1− γ + 3√

γ2 − 1

)(1.3)

with r0 = α~mc ≈ 2.8×10−15m the classical radius of the electron (or positron)(α is

given below in (1.6)), γ = 1/√

1− (v/c)2 and v the speed of the positron relativeto the stationary electron. Of most relevance for our discussion is annihilation atlow positron energies, where v c, (i.e. in the non-relativistic limit), so that theequation (1.3) is reduced to the familiar form:

σ2γ =4πr20cv

(1.4)

When the positron and the electron are in the triplet spin state (spins paral-lel), the lowest order process is the annihilation into three photons. Three-photonannihilation was first observed by Rich [10, 11]. The cross section for the three-photon annihilation in the approximation of low relative velocity of the two par-ticles (v c) was calculated in 1949 by Øre and Powell [12]. It can be written infunction of the two photon annihilation cross section (Eq. (1.4)) as:

σ3γ =4α9π

(π2 − 9)σ2γ (1.5)

with α the fine-structure constant:

α =e2

(4πε)~c≈ 1

137.036(1.6)

Due to the conservation laws, two other particles are required for the zero-photon process. These particles can be provided by two nucleons of a nucleus orby two electrons in an atom. This process is however very unlikely because of theunfavorable momentum transfer to two massive particles. The cross-section scaleswith Z8, with Z the atomic number of the atom involved. It has a maximum forEe+ = 500 keV at about 10−32 m2 for Z=80 [13].

The one-photon annihilation requires one extra particle and will emit a photonwith energy E + mec

2 − Eb where E the total energy of the positron according

9


to the equation (1.1) and Eb is the binding energy of the electron involved. Thisprocess is expected to occur mainly with inner shell electrons, e.g. the cross-section for single quantum annihilation with a K-shell electron is peaked aroundEe+ = 400 keV when the positron has sufficient energy to reach the deepestshells [14]. Experimentally the single quantum annihilation from the K–, L–, andM–shells has been observed for a number of materials by Palathingal et al. [15].Even for high Z materials, for low positron energies (Ee+ < 0.1 keV) this processis negligible compared to two- and three-photon annihilation.

When the positron and the electron are at rest, a characteristic radiation isemitted as a consequence of the annihilation. In the singlet state case (spinsantiparallel), the positron and the electron will annihilate into two anti-collinearphotons each carrying the rest mass energy of the electron (positron), i.e. 511keV. This was first observed by Klemperer [16]. In the triplet state case, thetotal energy of 1022 keV is distributed over three photons. They are emittedin a coplanar fashion with energy distributions up to 511 keV. This was verifiedexperimentally by Chang et al. [17] using high resolution gamma spectroscopy.

The annihilation rate λ (– the inverse of the positron lifetime τ –) of freepositrons with velocity v can easily be calculated from the cross-section:

λ =1τ

= σavne (1.7)

where ne is the electron density available for the annihilation process considered.

One can see that the two– and three–photon cross sections (equations (1.4)and (1.5) respectively) go to infinity for v going to zero. In contrast, notice thatthe annihilation rate stays finite and it is independent of the velocity v going tozero.

1.1.2.1 Free positrons interaction in condensed matter

Positrons rapidly loose their energy when injected into matter. The high energeticpositrons are believed to slow down to thermal energies in a very short time (1–10 ps) (– this rapidity has been experimentally proven in angular correlation ofannihilation radiation (ACAR) measurements [18], a brief information about theACAR technique can be found in section 2.4 on page 33–) compared to the meanlifetime of free positrons (which is typically 100–400 ps (for a review see [19])).This means that the mean time a positron spends at high energy is negligibleand therefore only the two– and three–photon annihilation should be taken intoaccount. The ratio of two– to three–photon annihilation can be calculated from

10

1.2 Positronium

the cross–sections:

wsσ2γ

wtσ3γ=

14σ2γ

34σ3γ

≈ 371, 2 (1.8)

This value was experimentally confirmed in metals by triple coincidence measure-ments by Basson [20].

In a system of non-interacting particles (i.e. neglecting the influence of thepositron on the electrons of the medium), the total annihilation rate (using theequations (1.2) and (1.7)) is given by:

λ = σavne

= (σ0γ + σ1γ + σ2γ + σ3γ)vne

≈ πr20cne

(1.9)

1.2 Positronium

In 1934 Mohorovicic [21] proposed the existence of a bound state of a positron andan electron which, he (incorrectly) suggested, might be responsible for unexplainedfeatures in the spectra emitted by some stars. However, Mohorovicic’s ideas onthe properties of this new atom were somewhat unconventional, and the name‘electrum’ which he gave to it did not become widespread. Later in 1945 Ruark [22]predicted it using quantum mechanics and named it ‘positronium’ (which is itspresent appellation), with the chemical symbol Ps.

Positronium itself was eventually discovered in 1951 by Deutsch [23–25] andits properties were investigated in a series of experiments based around positronannihilation in gases. Many of the techniques developed then are still in use today.

1.2.1 Positronium wave function

The spectroscopic differences between Ps and hydrogen (H) are due to the particle-antiparticle nature of Ps, which assures the equality of the positron and elec-tron masses and magnitudes of magnetic moments and the possibility of self-annihilation.

The non–relativistic quantum mechanics of the Ps atom is practically iden-tical to that of the hydrogen atom. The Schrodinger equations are the same,except for the magnitude of the masses of the positive particles. The reducedmass µ = m1m2

m1+m2of hydrogen is very close to the electron mass, and the one of

the positronium is exactly one half of it (µ = memp

me+mp= m2

e

2me= me

2 , where me and

11


mp are, respectively, the mass of the electron and the positron). When the centerof mass coordinates are eliminated, the one–body Schrodinger equation expressedin the internal coordinate r is[

− ~2

2µ∇2 − e2

(4πε)r

]ψ(r) = Eψ(r) (1.10)

The bound state energy eigenvalues of this equation are

En = − µ

2~2

(e2

4πε

)2 1n2

= −12µc2

α2

n2, n = 1, 2, 3, ...

(1.11)

where again α is the fine-structure constant. As the reduced mass is me

2 thegross values of the energy levels are decreased to half those found in the hydrogenatom, so that the binding energy of the ground state positronium (n = 1) isapproximately EB = −6.8 eV.

The spherical symmetric spatial wave function of the ground state in sphericalcoordinates is (as an example):

ψPs(r, θ, φ) =1√

π(2a0)3e−

r2a0 (1.12)

where a0 = 4πε~2

mee2 = ~mecα is the Bohr radius (e is the elementary charge). This

equation can be used to calculate the probability density of the Ps ground statewave function for r to be zero (i.e. at the origin):

|ψPs(0)|2 =1

π(2a0)3=m3

ec3α3

8π~3(1.13)

Positronium can exist in the two spin states, S=0, 1. The singlet state 1S0, inwhich the electron and positron spins are antiparallel, is termed para-positronium(p-Ps), whereas the triplet state 3S1 where the spins are parallel is termed ortho-positronium (o-Ps). The spin state has a significant influence on the energy levelstructure of the positronium, and also on its lifetime against self–annihilation.The hyperfine splitting of the Ps is characterized by an energy excess of the tripletstate over the singlet state [25]: ∆Ehfs = 7

12α4c2me+ ≈ 8.4× 10−4 eV.

1.2.2 Annihilation selection rule and decay rates

The first theoretical discussion of positronium is found in the work of Pirenne[26] who set the starting point for the many subsequent Ps studies concerning to

12

1.2 Positronium

the structure, means of formation and modes of decay. The selection rule thatmanages the e+ − e− annihilation process is fundamental to the understanding ofPs physics [27].

Energy and momentum conservation forbids the single-photon (1γ) annihila-tion of free Ps (– due to the need to conserve angular momentum and parity –).

The general selection rule for the annihilation of Ps from a state of orbitalangular momentum l and total spin S into n photons is given by:

(−1)l+S = (−1)n (1.14)

This follows from the n-photon states and charge conjugation properties ofPs: each photon contributes a factor of (−1), whereas in the Ps the electronand positron are interchanged, yielding a factor (−1)l+S, since they have oppositeintrinsic parity. For ground state positronium with l = 0, one concludes that theannihilation of the singlet (11S0) and triplet (13S1) spin states can only proceedby the emission of even and odd numbers of photons respectively. Thus, in theabsence of any perturbation the annihilation of p-Ps proceeds by the emissionof two, four, etc. gamma–rays; and the annihilation of o-Ps by the emission ofthree, five, etc. gamma–rays. In both cases the lowest order processes dominate,although the second order processes have been observed: the four–photon decayof p-Ps [28] and the five–photon decay of o-Ps [29].

It is expected from spin statistics that positronium will in general be formedwith a population ratio of ortho- to para- equal to 3:1, and in the absence ofany significant quenching (e.g. via the conversion of o-Ps to p-Ps considered insubsection 1.2.4 on page 17), most of the o-Ps which is formed will eventuallyannihilate in this state. Thus, the three–gamma–ray annihilation mode will bemuch more prolific for positronium than it is for free positron annihilation.

p-Ps has a lifetime of 125 ps and annihilates into two collinear 511 keV photons.o-Ps has a lifetime of 142 ns and annihilates into three photons with an energydistribution up to 511 keV.

The Equation (1.7) can be used to calculate the annihilation rate due to thenegligible effect of the Coulomb binding on the decay probability [12]:

λPs = σv |ψPs(0)|2 (1.15)

where |ψPs(0)|2 for the ground state is given by the Equation (1.13).

The decay rate for p-Ps in vacuum was first calculated to lowest order of per-turbation by Pirenne [26] and Wheeler [30] with the use of the Dirac’s cross sectionof the 2γ annihilation for positron-electron collisions at low energies (Eq. (1.4)).

13


The decay rate was found by multiplying this cross section by the flux of collid-ing particles taken as the relative velocity multiplied by the particle density atthe point of annihilation, i.e. the origin, taken as the square of the orbital wavefunction of p-Ps at the origin [31]:

λp−Ps =

4πcv r2

0︷︸︸︷σ2γ v

m3ec3α3

8π~3︷︸︸︷|ψPs(0)|2 =

4πcv

( α~mc )2︷︸︸︷r20 v

m3ec

3α3

8π~3

=12mec

2α5

~≈ 1

125 ps≈ 8 ns−1

(1.16)

In a substantially more involved calculation and by using the equation (1.5),the decay rate of o-Ps was determined to lowest order by Ore and Powell [12] as

λo−Ps =2

9π(π2 − 9)

mc2

~α6 ≈ 1

142 ns≈ 0.0072 ns−1 (1.17)

High order corrections to these equations can be found in literature [32, 33].Many experiments [34, 35] were performed to experimentally determine the o-Psdecay rate to compare it with the theoretical values found by QED. Reviews onthis topic can be found in references [13,36]. Only two experiments to accuratelydetermine the singlet lifetime have been reported. Theriot et al. [37] derivedthe singlet lifetime from the broadening of the radio-frequency resonance of thehyperfine splitting. Al-Ramadhan and Gidley [38] measured it by using the effectof singlet-triplet mixing in a static magnetic field (see subsection 1.2.4 on page 17).

1.2.3 Positronium formation in molecular media

In insulators, the Ps formation amounts to 20% to 70% from all positrons injectedinto the medium. This is higher than in metals and semiconductors because ofthe higher concentration of imperfections and impurities and the lower electrondensity. In metals, additionally, the high density of free electrons prevents thepositron to bind with a single electron and therefore positronium can only beformed at the surfaces (internal and external) [39,40].

• The Øre-gap modelAccording to the Øre-gap model [41], the positronium is formed by extractingan electron from a medium molecule in passing. There is a threshold of thepositron energy (Ee+) for forming Ps by this process, which is 6.8 eV lessthan the ionization energy of the molecule Ei (i.e. the energy necessary torelease the electron for the Ps formation). If the positron energy is greater

14

1.2 Positronium

than the ionization energy of the molecule, (Ee+ > Ei), then the resultingpositronium will have a kinetic energy greater than its own binding energyand hence it is a candidate for breakup in a subsequent collision. Thus thepositronium formation is most probable with the positron kinetic energy inthe range:

Ei − 6.8 eV < Ee+ < Ei (1.18)

thus, the positronium is formed when the positron energy during slowingdown lies within a gap where no other electronic energy transfer is possible.

• The Spur modelIn 1974 Mogensen [42, 43] suggested that positronium formation is a spurreaction process. The positron spur is a group of electrons, ions, radicalsand other excited species produced in the last ionization collisions duringthe slowing down of the positron (i.e. the terminal track of the positron,formed when it loses the last part of its kinetic energy). According to thismodel, positronium is formed mainly by the reaction between the positronand an excess electron in the spur. Thus, the positronium is formed when thepositron is thermalized and captures a thermalized electron in its own spur.The positronium yield can be quantitatively treated with reaction kineticsof positron spur reactions (Mogensen 1995 [44]).

Some believe that one of the two models (Øre-gap or spur) is right and the otheris wrong. Some believe that the two models are not inconsistent. In the begin-ning of the eighties, Eldrup et al. [45, 46] showed by making slow positron beamexperiments on ice that depending on the energy of the positron both processescan occur simultaneously.

• The Blob modelThis model, developed by Stepanov et al. [47, 48], is an extension of thespur reaction model. The distributions of excess electrons and positron wereapplied and then it was possible to explain the change of Ps formation prob-ability under an external electric field [49].

A summary of the most important properties of the blob is presented below(and illustrated in Figure 1.1). However, readers should refer to the originalpublications for complete details [47–49].

- Through ionizing collisions (the spur, cylindrical column in Figure 1.1),a positron of several hundred keV will lose most of its energy within10−11 s until its energy drops below the ionization threshold.

15


- In the final ionizing regime, with the positron energy Wbl ∼ 0.5 keVand the ionization threshold of several eV, n0 ≈ 30 overlapped ion–electron pairs are generated in the terminal blob. The terminal blobis a spherical micro-volume of “radius” abl ≈ 40 A which confines theend part of the positron trajectory, where ionization slowing down isthe most efficient (thermalization stage of subionizing positron is notincluded here).

- The subionizing positron further undergoes positron-phonon scatteringand may diffuse out of the blob, until it becomes thermalized in aspherical volume bigger than the blob volume, ap > abl.

- The intrablob electrons are tightly kept by electric fields of the positiveions. The positrons thermalized within the blob can not escape from it.On the contrary, the faster subionizing positrons can do it. Therefore,it becomes necessary to distinguish between the inside (e+in) and outside(e+out) blob positrons.

- Within the blob, the encounter of a thermalized positron with one ofthe thermalized intrablob electrons, followed by formation of weaklybound positron–electron pair is the first stage for the formation of Ps.

Figure 1.1: Schematic view on the terminal positron blob. Positronmotion is simulated as random walks with the energy dependent stepltr(W ). For more details, see Stepanov et al. [47].

16

1.2 Positronium

1.2.4 Positronium quenching

“Positronium quenching” is an effect wherein the mean lifetime of positronium isshortened by the interaction with matter (through different processes) or externalmagnetic fields. The term ‘quenching ’ is commonly used for o-Ps since its annihi-lation rate is 3 orders of magnitude lower than for p-Ps and therefore, the influenceis bigger on the o-Ps lifetime. The two most important processes are pick-off andconversion quenching.

In the pick-off process, the positron of the positronium (in matter) suffers2γ−annihilation in interaction with an electron from the surrounding moleculeshaving opposite (antiparallel) spin. The annihilation with such electrons reducesthe o-Ps lifetime typically from 142 ns to 1-5 ns. This process was first suggestedby Garwin in 1953 [50] and called ‘pick-off ’ quenching by Dresden [51].

Conversion (or exchange) quenching occurs when the parallel spin electron ofthe o-Ps exchanges with an atomic electron with anti-parallel spin to produce p-Ps [52], which then enables two-gamma annihilation before the reverse process canoccur. Because of the much higher annihilation rate of p-Ps, the annihilation isalmost immediate in the o-Ps time scale. Conversion quenching is clearly observedwhen o-Ps interacts with paramagnetic gases like NO [23,24] and O2 [53, 54].

A more detailed information on these and other processes that make possiblethe positronium quenching can be found in reference [55].

17


18

2Experimental techniques in

positron annihilation

The conventional experimental techniques frequently employed to study positronsare introduced in this chapter. Some of these techniques are used through thiswork, thus the principles behind them are briefly described as well as the me-thodology and illustrations of the apparatus for some of them. First we startwith a brief overview of the positron sources, the annihilation lifetime and finallythe momentum measurements (which include the Doppler Broadening (or Dopplershift) and the angular correlation of the annihilation radiation) are presented.

2.1 Positron sources

β+-emitting radioactive isotopes are used to obtain positrons in conventionalpositron measurements. A few well-known isotopes are 22Na (2.6 y), 58Co (288d), 68Ge (71 d) and 64Cu (12.8 h). The most used source material in positronresearch is the 22Na radioisotope. In addition to the half-life of 2.6 years andthe reasonable price of 22Na, an advantage is that the manufacture of laboratorysources is simple, due to the easy handling of the different sodium salts in aqueoussolution, such as sodium chloride or sodium acetate.

A simplified decay scheme of 22Na is shown in Figure 2.1. 22Na decays to theexcited state 22Ne* with a β+ branching ratio of around 90%. The ground state of22Ne is reached after 3.7 ps by emission of a γ−photon of 1274 keV. The positronemission is followed promptly by this photon and therefore, it can be used toregister the positron’s birth.

Because the β-decay reaction is a two particle decay, the positrons emitted by22Na exhibit a broad energy distribution extending from almost zero to 545 keV

19

2. EXPERIMENTAL TECHNIQUES IN POSITRONANNIHILATION

E = 1274 keVg

g

b+

Na22

Ne22

(2.6 y)

(3.7 ps)Ne*

22

Figure 2.1: Simplified decay scheme of the radioactive isotope22Na. 22Na decays to the excited state 22Ne*. This excitedstate has a half-life of 3.7 ps and de-excites to the ground stateof 22Ne by emitting a 1274 keV γ−photon.

and thus, can penetrate deep into a sample.

The sources are usually prepared by evaporating a solution of a 22Na salt on athin metal or polymer foil. The most common foil materials are Al, Ni, and Mylaror Kapton. In order to ensure the almost complete annihilation of positrons in thespecimen a “sandwich arrangement” is used, the foil source is placed between twoidentical samples. A minimum thickness of the samples is required to ensure thatthe essential fraction of positrons annihilates in the sample pair.

The implantation profile of high-energy positrons emitted from a radioactivesource into a solid can be described by an empirical law which first was foundfor electrons and later confirmed for positrons [56,57]. It states that the positronintensity I(z) decays as:

I(z) = I0e−α+z (2.1)

The mean implantation depth of the positrons is 1/αe+ and can be approximatedas

αe+ ≈ 17ρ [g/cm3]

E1.43max [MeV][cm−1] (2.2)

where Emax is the maximum energy of the emitted positrons and ρ the density ofthe solid. This approximation can be used for the determination of the minimalthickness of the samples.

20

2.2 Positron annihilation lifetime spectroscopy (PALS)

2.2 Positron annihilation lifetime spectroscopy -

(PALS)

After injected into matter, a positron will eventually annihilate with an electron.The lifetime of a positron is the mean time between the injection (“start”) andsubsequent annihilation (“stop”) of the positron. As the start signal is given atthe moment a positron is injected into the material, its origin depends on thepositron’s source. As stated previously in section 2.1 on page 19, in the case ofa radioactive source of 22Na, the start signal is given by the the detection of theγ−photon of 1274 keV that is emitted simultaneously with the positron. In a highenergy beam experiment, the positrons deposit a small amount of energy in a thinscintillator before being injected into the sample. As the positrons travel at thespeed of light the time difference between the scintillator signal and the injectioninto the sample is always equal and therefore, the scintillator signal can be used asstart signal. A last method is the use of a pulsed positron beam. In this case, thesignal is generated by the pulse electronics as the positrons are injected in fixedtime intervals [58].

In 1948 Debenedetti [59] build a setup to measure the time intervals betweenionizing events. In 1949 it was realized that the lifetime of a positron is an impor-tant property [60]. In 1951 Deutsch [23] investigated the lifetime of positrons inseveral gases, finding the definite proof for the existence of positronium. Bell andGraham [61] investigated more systematic the lifetime of positrons in solids andliquids.

2.2.1 The basic operating principle

The basic operating principle of all traditional positron lifetime systems is schemat-ically illustrated in Figure 2.2. A 22Na source is employed in the apparatus shown.The start signal is derived from the detection of the positron’s birth (i.e. theγ−photon of 1274 keV) and the stop signal from one of the annihilation photons(of 511 keV). Both the start and stop signals are registered using γ−ray scintilla-tion counters (SC) (see e.g. [62] for a general discussion). The scintillator signalsare converted into electronic signals by photomultipliers. These signals are later onprocessed by a pair of discriminators, and the simplest arrangement consists of twoconstant-fraction differential discriminators (CFDD), which combine good timingcharacteristics with the capacity to set upper and lower limits on the pulse heightaccepted by the instrument. Therefore, the higher energy (start) signal can easilybe selected. An appropriate delay is inserted in order to introduce a minimumfixed time between the start and stop signals. Then the signals are connected to a

21


time-to-amplitude converter (TAC). The TAC delivers a signal with an amplitudeproportional to the length of time between the start and stop signals. The outputof this module is recorded by a multichannel analyzer (MCA) of which the outputis stored in a computer. A lifetime spectrum is recorded frequently containing106−107 events, from which various lifetimes along with several other parameterscan be extracted.

g

Sample

e-

e+

22Na Source

511 keV

1274 keV

gg

511 keV

StopStart

Photomultiplier

tubeSC

CFDD

Photomultiplier

tubeSC

CFDD

delay

TAC

Start Stop

MCA

Figure 2.2: Schematic positron lifetime spectrometer. The lifetime ismeasured as the time difference between the appearance of the startand stop γ−photons. Key: SC, scintillator; CFDD, constant-fractiondifferential discriminator; TAC, time-to-amplitude converter; MCA,multichannel analyzer.

Another useful lifetime system, though less frequently encountered is the so-called β+−γ system. In this system the annihilation γ−ray still provides the stopsignal, but the start signal is derived via the energy deposited by the positrons asthey traverse a thin (typically 0.1 to 0.3 mm) scintillator. This method of startdetection has a high efficiency, usually around 50 %, which permits the use ofa relatively weak radioactive source, resulting in a superior signal-to-backgroundratio. This technique was first used by the pioneers Bell and Graham in 1953 [61].They used a stilbene scintillator in from of a 22Na source to deliver the startsignal for their delayed coincidence measurements. When used with a radioactivesource, the lowest energy positrons will annihilate in the scintillator itself and

22


give a contribution to the lifetime spectrum. When using a MeV mono-energeticpositron beam, almost no positrons will be stopped in the scintillator. The highbeam energy allows the use of a sufficient thick scintillator (2 to 5 mm), enhancinglight collection and signal amplitude. In this way a start detector with almost100% efficiency can be achieved. This will result in a virtually background freelifetime spectrum [63].

2.2.2 Lifetime Data Treatment

The positron lifetime spectrum describes the probability of an annihilation at timet. If positrons have several different states from which to annihilate (consider asystem with n independent positron states i), the lifetime spectrum is determinedby the solution of the differential equation

dni(t)dt

=∑

i

Iie−λit (2.3)

where λi is the decay rate constant associated with state i and Ii the correspondingintensities (

∑i Ii = 1). The lifetime components are defined as the reciprocal

values of the decay constants τi = λ−1i (see Equation (1.7) on page 10).

As an example, the lifetime spectrum of a standard polymethyl metacrylate(PMMA) sample is presented in Figure 2.3. The measured spectrum is the convo-lution of the ideal exponential spectrum presented in Equation (2.3) and the res-olution function of the system. This resolution function is usually approximatedby a Gaussian function. The typical Full Width at Half Maximum (FWHM) ofthe resolution function is around 200-250 ps (depending mainly on the sizes of thescintillators and the used energy windows).

In addition, a few percent of the positrons annihilate in the source materialproducing an additional component to the experimental spectrum. For this reason,usually several components can be reliably separated in the experimental lifetimespectra. The separation is normally performed by fitting the convoluted theoreticallifetime spectrum to the measured data. The effect of the source component canbe eliminated by measuring a defect free reference sample. On the other hand,when the resolution function is Gaussian, it does not affect the average positronlifetime defined as

τave =∑

i

Iiτi =∫ ∞

0

tP (t)dt (2.4)

The average positron lifetime (equal to the center of mass of the spectrum) is animportant quantity since it can be always determined even if the decompositionof the lifetime spectrum is difficult.

23


300 400 500 600 700 800 900 1000 1100 1200 1300 140010

1

102

103

104

105

Channel

Cou

nts

τ2=0.33 ns

τ3=1.8 ns

Figure 2.3: Experimental positron lifetime spectrum obtained froma poly(methyl-methacrylate) (PMMA) sample. After the decompo-sition of the spectrum, the obtained lifetime components τ2 and τ3are added as straight lines in the semi-logarithmic plot for illustra-tion. The τ1 component is not indicated as a straight line (τ1 = 0.125ns). The deviations from the straight line at higher times are dueto annihilations in the source and the background contribution. TheGaussian-like shape of the left part of the curve is mainly caused bythe resolution function.

2.2.3 Relation between the positronium lifetime and thefree-volume-hole size

2.2.3.1 The Tao-Eldrup model

As stated before in subsection 1.2.4 on page 17, in condensed matter the o-Pslifetime, τo-Ps (which is 142 ns in vacuum) is quenched to some nanoseconds asthe e+ of the o-Ps atom annihilates with an e− from the surrounding molecules(the so-called pick-off process).

When thermalised o-Ps annihilates from cavities, the probability for the occur-rence of the pick-off process is related to the electron density of the cavity wall.Therefore the o-Ps lifetime (τo-Ps) is related to the free-volume-hole (FVH) dimen-

24


sion. The measured τo-Ps can be related with the FVH size by a semi-empiricalequation known as the Tao-Eldrup model. It is derived from a simple quantum-mechanical model in which it is assumed that the Ps is confined in a sphericalvoid of radius R. In that model one assumes that the void represents a rectan-gular infinite potential well for Ps with a spatial overlap of the Ps wave functionwith molecules within a layer δR of the potential wall [64–66]:

τo-Ps = 0.5[1− R

R0+

12π

sin(2πRR0

)]−1

(2.5)

where τo-Ps is expressed in ns and R0 = R + δR in A (δR = 1.656 A is theempirical parameter that represents the o-Ps penetration depth into the wall ofthe hole wherein the o-Ps annihilates). A graphical representation is given inFigure 2.4.

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.651

2

3

4

5

6

7

8

9

Free−volume hole radius R (nm)

o−P

s lif

etim

e (n

s)

Figure 2.4: Graphical representation of the Tao-Eldrup model.

However, one has to be aware that in molecular crystals and polymers theseassumptions are not strictly fulfilled: the well is shallow, the potential is notrectangular, the voids are often not spherical. Thus, the Tao-Eldrup equationis only an approximation of real τo-Ps versus R relation (as emphasized by theseauthors). One can also apply other shapes, e.g. cuboids proposed by Jasinska etal. [67].

25


2.2.3.2 lifetimes in the free volumes of larger radii

The lifetime to pore radius relation is well approximated by the Tao-Eldrup modelfor R < 1 nm. Thus, in order to explain lifetimes in the free volumes of largerradii, some extensions to the model are required. There are a few approaches tothis problem resulting in different τo-Ps(R) dependencies each.

The Tao-Eldrup model’s simplicity is due to approximations like consideringPs as a particle without internal structure, use of a spherical potential well ofinfinite depth (broadened by δR = 0.166 nm) and taking into account only theground level of a particle in a rectangular potential well.

The simplest way to avoid these limitations is to discard some of the approx-imations. A modification proposed by Goworek et al. [68–70] takes into accountexcited levels of Ps in the potential well without changing any other assumptions.Such a simple extension substantially changes the Tao-Eldrup model curve forR > 1 nm at moderate temperatures. Moreover it explains a temperature depen-dence of the lifetime, which could not be done based on the original Tao-Eldrupmodel. The price paid for such a modification is that model equations are morecomplicated and have to be solved numerically.

The essential point in this model is the introduction of lifetime averaged overas many excited states as necessary; it is a second rank problem what geometryis most appropriate for the particular case. For better explanations related to themodel, refer to the original papers [68–70] or more recent papers [71,72].

For a visual idea however, the ortho-positronium lifetime curves versus free-volume-hole diameter at different temperatures are shown in Figure 2.5. It is seenthat great differences of lifetimes as a function of temperature can be observed inthe range of radii of several nm. For R → ∞ all lifetimes approach the vacuumvalue 142 ns. The shadow area corresponds to the original Tao-Eldrup model (seeSubsection 2.2.3).

To find simpler solutions of the problem, cubic geometry was proposed byGidley et al., [74]. A less realistic approximation of the potential well’s shapeallowed one to write τ in a form that uses only elementary functions. A newvalue of δR = 0.18 nm was suggested in order to fit the obtained curve to theTao-Eldrup model. Later it was found that the use of this δR value leads to agood agreement between the modified model and experimental data for R > 1 nm(Dull et al., [75]). A totally different approach to the problem of Ps annihilationin large free volumes was proposed by Ito et al., [76]. Instead of representing Psas a standing wave, it was considered as a Gaussian wave packet scattering inthe potential well, the same as in the TaoEldrup model. Unfortunately, empiricalparameters of this model were fitted to badly chosen experimental data (Dull et

26

2.3 Doppler shift or broadening of the annihilation radiation (DBAR)

Figure 2.5: Graphical representation of the extended Tao-Eldrupmodel proposed by Goworek et al. [68–70]. The figure represents theortho-positronium lifetime versus free-volume-hole diameter at differ-ent temperatures. The shadow area corresponds to the original Tao-Eldrup model. For R → ∞ all lifetimes approach the vacuum value142 ns. The figure is taken from reference [73].

al., [75]).

2.3 Doppler shift or broadening of the annihila-

tion radiation (DBAR)

In the frame of reference in which the center of mass of an electron-positron pair isat rest, the two annihilation photons arising from their annihilation in a spin singletstate each have an energy of 511 keV, and they are emitted in opposite directions,i.e. the angle between the directions of the two photons is 180. However, the centerof mass is not at rest in the laboratory frame of reference. When slowing downin matter, most positrons thermalize before annihilation. The linear momentum1

connected to the motion of the center of mass of the positron-electron pair will be,therefore, dominated by the electron motion. Thus, in the laboratory frame the

1For the remaining of this thesis ‘linear momentum’ is abbreviated by ‘momentum’

27


motion of the center of mass creates a Doppler shift in the γ−ray energies, andthe angle between the annihilation photons deviates from 180, depending on themomentum of the annihilating pair. This is shown in Figure 2.6.

PL

PP q

P1 , E1 1, g

P2 , E2 2, g

Figure 2.6: The vector diagram of the momentum conservation in the2γ-annihilation process. The momentum of the annihilation pair isdenoted by P , and the subscripts L and ⊥ refer to longitudinal andtransverse components, respectively.

The Doppler broadening of the annihilation line was first observed by DuMondet al. [77] when measuring the radiation from a 64Cu β+ source using a curvedcrystal spectrometer. The Doppler shift in the energy of the 511 keV annihilationline is given by (– for the derivation, see Appendix A, equation (A.1) –):

∆Eγ =cpL

2(2.6)

with pL the longitudinal momentum component, i.e. the projection of the momen-tum of the center of mass along the direction of emission of the gamma–ray. Fora typical electron energy of a few eV and a thermalized positron the Doppler shiftis of the order of 1.2 keV1. The shape of the 511 keV annihilation line is in factdue to the one-dimensional momentum distribution of the electron-positron pair

L(Eγ) ∝∫ ∞

∞

∫ ∞

∞ρ(px, py, pz)dpxdpy (2.7)

with pz = 2c (Eγ −m0c

2).

A typical Doppler broadening setup is shown in Figure 2.7. Since the energyshift is very small, except for the early experiments [77], the measurement ofDoppler profiles has only become possible by the development of Ge(Li) (Lithiumdrifted Germanium) gamma-ray detectors (–which have very high resolution–)[78, 79]. Nowadays High Purity Germanium (HPGe) detectors are used. Theamplification system following the detector is standard, usually consisting of apreamplifier and a spectroscopy amplifier; which allows the broadened annihilation

1In addition to the traditional quantification of the gamma-photon energy in keV, the fol-lowing units are sometimes used: atomic units (1 a.u. ≈ 3.73 keV/c) and millirad (1 keV ≈ 3.9mrad)

28


line to be examined in more detail. The γ−ray energy distribution can then bestored in a multichannel analyzer and processed in various ways depending uponthe details of the study.

Spectroscopy

amplifier

High Voltage

Power Supply

Multichannel

analyzer

HPGePreamplifier

Sample-Source

arrangement

Figure 2.7: Schematic drawing of a typical Doppler broadening setup.The signal is processed by the preamplifier and by the spectroscopyamplifier before being recorded in the multichannel analyzer.

This technique is mainly used in investigations of solid state, where, in mostcases the geometry of the experiment and the random nature of the directionof motion of the positron-electron pairs means that the angle θ (see Figure 2.6)has a continuous distribution and consequently, the 511 keV γ−line is Dopplerbroadened by an amount related to the momentum distribution of the annihilatingpair.

2.3.1 The S- and W-parameters

Extracting information from the whole shape of the annihilation peak is insuffi-cient due to the resolution and to the peak-to-background ratio of the Dopplerbroadening setup. Some deconvolution procedures have been developed but theirreliability is always limited. Doppler broadening spectra are, therefore, usuallycharacterized with the S- and W-parameters. These parameters were first intro-duced by MacKenzie et al. [80]. The Figure 2.8 illustrates both parameters. Theline-shape parameter (S-parameter) is calculated as the ratio of a central area ofthe 511 keV annihilation line to the total area. The wing parameter (W-parameter)is the ratio of the sum of the two wing areas to the total area. The choice of these

29


intervals is to some extend arbitrary. The highest sensitivity of the S-parameterfor changes in the line shape is usually obtained if the S-parameter is close to 0.5.

It should however be noticed that even after the background correction theDoppler curve can still have a small asymmetry. This is the reason why the W-parameter is sometimes calculated only using information in the high energy wingof the Doppler curve.

As

Inte

nsi

ty (a

rb.

units

)

E [keV]g

S-parameter

W-parameter

Aw1 Aw2

AT

Figure 2.8: Schematic view on how to calculate the S- and W-parameters. The areas indicate the summation regions for the cal-culation of the line-shape parameters

(S = AS

ATand W = AW1+AW2

AT

).

The S-parameter is a sensitive parameter to detect changes in low momen-tum contributions of the annihilation peak, which produce the smaller Dopplerbroadening of the annihilation line. This means that the annihilation with va-lence electrons gives the largest contribution to the S parameter. The formationof p−Ps also contributes almost entirely to the S-parameter because of its verysmall momentum. On the other hand, the W-parameter is sensitive to variationsin high momentum contributions; thus, mainly annihilations with core electrons.

As an example of the importance in using the S and W-parameters in a Dopplerspectrum we may consider a defect-free and some defect-rich samples of the samepure material. In a defect-free sample, the positron can be represented by aBloch state with its wave function overlapping the valence and core electrons. Inthe material with defects, the positron can be trapped in a defect and its wave

30


function is localized, and although its wave function still overlaps with the valenceand core electrons, the fraction changes: the larger the open volume, the less thewave function of the trapped positron will overlap with the core electrons, – whichhave the highest momentum –, and therefore, the more it overlaps with the valenceelectrons. Thus, the defect-rich sample should have the highest S- and the lowestW-parameter.

2.3.2 Coincidence DBAR (CDBAR)

A substantial problem that limits the capability of the conventional DBAR tech-nique is its relatively high background contribution. This is especially importantfor the measurements of the high-momentum contribution of the 2D annihilationpeak –mainly at the low-energy side of the photo-peak–.

In 1976 Lynn et al. added a NaI(Tl) detector opposite to the Ge detectorto reduce the background by detecting both photons in coincidence [81]. Thismethod is still used today (see e.g. [82–87]) due to the relative small additionalcost compared to a single detector setup. It is used to reduce the background atthe high energy side of the annihilation peak. In 1977 Lynn and co-workers [88,89]replaced the NaI by a second Ge detector to extract the energy information fromboth detectors. This is conventionally called coincidence-DBAR (CDBAR) ordouble-DBAR (DDBAR).

The coincident γ−quantum is almost anti-collinear to the primarily registeredannihilation photon and it serves to suppress the Compton background near the511 keV energy which arises from the 1274 keV γ−ray of 22Na. For comparisontwo photo-peaks, one recorded by means of the typical DBAR and the other oneby means of the CDBAR techniques are shown in Figure 2.9(a). Thus, from thisfigure it is clear that by using the coincidence Doppler broadening technique, thebackground reduction is highly improved. An important consequence from thedrastic background reduction is that this method allows the observation of thehigh momentum annihilations with core electrons and therefore the comparisonwith theoretical calculations is possible.

In a CDBAR experiment a two-dimensional spectrum is recorded where theaxes represent the energy scales of the respective detectors.

For illustration, an example of a two-dimensional coincidence Doppler broad-ening spectrum is presented in Figure 2.9(b). Considering the momentum andenergy conservation of the annihilation process, an increase in the energy of anannihilation γ in one detector according to the Doppler shift, is consequent witha simultaneous reduction of the energy of the γ recorded in the second detector.The sum of the energies of the two γ’s is E1 + E2 ≈ 1022 keV and therefore this

31


elliptical shape is explained.

(a) Comparison between a typicalphoto-peak obtained by a single detec-tor (open triangles) and a coincidencesetup (circles).

E1 (keV)

E2 (

keV

)

500 510 520 530

500

510

520

530

(b) Example of a two-dimensionalcoincidence Doppler broadening spec-trum.

Figure 2.9: Coincidence Doppler broadening spectrum

To obtain the corresponding one-dimensional spectrum, a projection onto theE1 + E2 = 1022 axis is made. The projection includes events in a window alongthe diagonal with 2m0c

2 − δ ≤ E1 + E2 ≤ 2m0c2 + δ, with δ the width of the

window (typically 1 keV up to 4 keV).

The horizontal and vertical bands extending from the central peak are producedby coincidences of a 511 keV photon with a background photon.

For many years this method was not really used by the scientific commu-nity. It was only in the early-nineties that it was ‘rediscovered’ [90–93]. Thenit was demonstrated that in some cases CDBAR can be used for the chemicalidentification of the atoms at the site of annihilation. Positron annihilation withlow-momentum valence or conduction electrons results in a small Doppler shift.Annihilations with core electrons result in a large Doppler shift, contributing tothe wings of the 511 keV annihilation line. The momentum distribution of thecore electrons is typical for each chemical element. Therefore, the shape of thecore contributions form some kind of fingerprint of the atoms from which theyoriginate.

A more detailed information on the coincidence Doppler broadening techniquecan be found in reference [94].

32

2.4 Angular correlation of annihilation radiation (ACAR)

2.4 Angular correlation of annihilation radiation

(ACAR)

The ACAR method was first studied by Beringer and Montgomery [95] in 1942 andwas further improved by DeBenedetti et al. [96] in 1949. This method involvesmeasuring θ, the small deviation from π radians in the angle between the twoannihilation γ−rays. As previously mentioned in section 2.3 on page 27, thisdeviation is a consequence of the center-of-mass momentum of the annihilatingelectron-positron pair.

The relationship between θ and the component of this momentum perpendicu-lar to the direction of one of the γ−rays, which can be taken to be the longitudinal-component, PL, as in Figure 2.6, is given by (– also for the derivation, see AppendixA; equation (A.8) –):

θ =PT

m0c(2.8)

The sample-to-detector distance amounts typically to several meters (– 5 m typi-cally –) so that γ−quanta from only a small solid angle are detected. Hence muchstronger sources compared with conventional PALS and DBAR techniques are re-quired. On the other hand, angular resolution can be adjusted in the range 0.2 to5 milliradians [97], which corresponds to the energy resolution of DBAR measure-ments in the range 0.05 to 1.3 keV. Thus this technique provides essentially thesame kind of information as DBAR, however, the momentum resolution is muchbetter.

An example of an angular correlation of annihilation radiation apparatus todetect such small angular deviations is shown in Figure 2.10. Apart from the digi-tization and storage electronics, basically it consists of a pair of two-dimensionalposition-sensitive gamma-ray detectors and a radioactive source, which is imme-diately adjacent to the sample being studied. The field of view of each detector isin addition limited by lead collimators.

In media in which there is not a preferred axis of symmetry (e.g. in gases andliquids in the absence of external fields) it is not necessary to use a two-dimensionalsystem (although in that case, the count rate will be lower). Instead, the one-dimensional technique can be used, in which the position-sensitive detectors arereplaced by two single detectors, each with a long collimator placed in front ofit, giving integration over one of the components of the momentum. One of thedetectors is fixed whereas the other one is scanned through the angle θ.

33


x

Detector 1

coincidence t

digitizationand storage

source

Detector 2

sample

gqx,y

ge+

Px

Py

Pz

x

Figure 2.10: Scheme of the apparatus for two-dimensional angular co-rrelation of annihilation radiation. The deviation angular, θ, of thetwo γ−quanta is recorded by position sensitive detectors in a coinci-dence setup and stored in a two-dimensional memory array.

2.4.1 Relation between the para-positronium momentumand the free-volume-hole size

The hole size can also be estimated from the electron-positron momentum distri-bution as observed via ACAR or DBAR (for a review, see refs [98,99]).

The detected photo-peak can (in first approximation) be fitted with a super-position of Gaussian distributions whose components arise from the different anni-hilation channels. The narrowest component is associated to the self-annihilationof para-positronium and the broader distributions to the free e+ and o-Ps pick-offannihilations [98–101]. The narrow component reflects the momentum of the p-Psannihilating from a free-volume-hole. According to the Heisenberg uncertaintyprinciple (∆x∆p ≥ ~

2 ), the momentum distribution is sensitive to the FVH size.The full width at half maximum of the narrow component may be related to thehole dimension via the same model used to derive Eq. (2.5) on page 25 [98,102]:

R = 16.6/θ 12− δR, (2.9)

where θ 12

is in mrad (1 keV = 3.913 mrad) and δR = 1.656 A is an empiricalparameter. A graphical representation is given in Figure 2.11. In this figure theeffects of the extended Tao-Eldrup (ETE) model discussed in 2.2.3.2 on page 26are not included. To my present knowledge no studies about these effects havebeen reported in literature.

34

2.4 Angular correlation of annihilation radiation (ACAR)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.652

2.5

3

3.5

4

4.5

5

Free−volume hole radius R (nm)

θ 1/2 (

mra

d)

Figure 2.11: Graphical representation of the relation between the p-Psmomentum and free-volume-hole size.

35


36

3Interaction of positrons with

solids and surfaces

In this chapter is given a brief summary about slow positron beam production,positron implantation into solids and detection.

3.1 Slow Positron beams

The conventional positron techniques with the “sandwich arrangement” are notsuitable for thin-layer materials or near-surface experiments because in such con-ventional cases the positrons are immediately implanted into the sample material.Mono-energetic positrons are therefore necessary in order to obtain a defined pe-netration depth. These positrons can easily be accelerated to well-defined implan-tation energies1. This is the base for the low-energy, or slow, positron beams, alsoreferred to as Variable Energy Positrons (VEP).

A detailed knowledge of the positron interaction with solids and surfaces is offundamental importance for the interpretation of the results. For example, Schultzand Lynn [103] have reviewed a variety of theoretical as well as experimental workperformed with positron beams. Thus, in this section we give a basic descriptionof the slow positron beam production through a discussion of the different inter-action processes as moderation, through implantation into solids; the subsequentthermalization and diffusion of the moderated positrons and also the emission intovacuum of those positrons which reach the surface before being annihilated; andtheir subsequent manipulation to form a beam.

1According to the energy of the positrons, the positron beams can be divided into twocategories: The high-energy positron beams with typical energies of 1-4 MeV and the low-energypositron beams ranging from 0-30 keV.

37

3. INTERACTION OF POSITRONS WITH SOLIDS ANDSURFACES

3.1.1 Introduction to positron moderation

The very first step in the production of a slow positron beam, namely the energymoderation of positrons coming from a high-energy source, is based on typicalprocesses of positron interaction with solids in the bulk and at the surface.

High energy positrons are produced either as a result of nuclear decay, usuallyof an artificially produced radioactive isotope (– as stated previously in section 2.1on page 19 –) or by pair production1 from a photon of sufficiently high energy.The positrons produced by these two ways have a very broad energy spectrum(see Figure 3.1) and a high mean energy, typically of the order of MeV.

To optimize the number of positrons that can be trapped in vacuum, mostexperiments require a beam of a narrow energy spread and a low average energy.Typically positrons with kinetic energy in the eV range are required.

The most desirable option would be to compress the entire beta distribution.This is done by stopping the positrons in a solid. Positrons implanted into a solidthermalize through collisions on a time scale of less than 10 ps. There the positronswill diffuse around in this environment with an average lifetime of ∼ 100 ps. Itwas realized by Madansky and Rasetti already in 1950 [104] that this time wouldbe sufficient for some of the thermal positrons to diffuse back to the surface wherethey could be re-emitted. Unfortunately, due to contamination of the surfacetheir search for slow positrons diffusing out of thin 64Cu activated foils failed; noslow positrons were observed, even though one slow positron per 103 fast β+ wasexpected.

This method of making low-energy positrons has later become known as ‘mo-deration’. Thus, moderation is the name given to the technique of producinga monochromatic positron beam from such a broad energy spectrum. In otherwords, moderation is done with the purpose of optimizing the number of positronsthat can be emitted from the moderator foil into the vacuum. Those positronsalso have a low average energy and a narrow energy spread (–typically positronswith kinetic energy in the eV range–).

Figure 3.1 shows a comparison of the energy spectrum of positrons emitted froma 22Na radioactive source to that of moderated positrons. From this figure, theadvantage of moderators becomes obvious: A positron beam of any desired energycan be obtained just by accelerating the near-zero energy moderated positrons.

1The phenomenon of pair production occurs when a high-energy photon interacts with anatomic nucleus, allowing it to produce an electron and a positron without violating conservationof momentum. This phenomenon, although interesting, is not used in this work, and thereforewill not be discussed here.

38


Posi

tro

n y

ield

Positron energy [eV]

Moderated positrons

Typical

energy spectrum

b+

10-2

10-3

10-4

10-5

10-6

10-7

10-8

10-9

10-1

100

101 10

210

310

410

510

6

Figure 3.1: Positron energy spectra before and after the moderationprocess, demonstrating the increased number of useful positrons forlow energy beam experiments. The curve is the normalized energydistribution for β+ particles emitted from a 22Na source. The full barshows the efficiency of moderation

3.1.1.1 Positron re-emission

The first report (not published in a journal article) of slow positron emission wasmade by Cherry in 1958 [105]. He reported that positrons with energies below 10eV were emitted from a 64Cu source, transmitted through mica, which was coatedwith a thin layer of chromium. The conversion efficiency was approximately oneslow positron per 107 fast positrons. In 1968 Groce et al. [106] observed moderatedpositrons with an energy of 1-2 eV from gold plated mica moderators. Later on,it was noted that the positrons emitted from these solids in general had meanenergies a few eV higher than expected for a thermal distribution. This meantthat the positrons gained energy leaving the solid, giving rise to the name ‘negativework function1 materials’. The theoretical explanation for this phenomenon (inmetals) was given by Tong in 1972 [107].

1Work function definition: is the minimum energy (–usually measured in electron volts–)needed to remove a positron from a solid to a point immediately outside the solid surface. Here“immediately” means that on the atomic scale the final positron position is far from the surfacebut still close to the solid on the macroscopic scale.

39


The work function, φ, for electrons and positrons has two contributions:

• One is the chemical potential of the particle (µ). For electrons this is theenergy required to lift an electron from the Fermi energy level into vacuum.For the positrons a similar ‘binding energy’ arises mainly from the sum ofthe repulsive interaction with the ion cores in the solid and the correlationpotential from the attractive interaction with the electrons of the solid.

• The second contribution to the work functions arise from the effect of thesolid surface dipole layer (D), which is repulsive for electrons and attractivefor positrons. At the surface the electron distribution tends to spill out intothe vacuum. Together with the ion cores inside the surface, these electronscreate a dipole layer at the surface. The effect of this layer on an electronis to decelerate it as it leaves the solid, making it harder to remove. Asthe positron has the opposite charge as the electron, the surface dipole hasthe reverse effect on positrons as it does on electrons. That is, positronsget accelerated by the surface layer which counters the binding effect of thechemical potential.

The work function, therefore, is the sum of the two terms: the chemical potential(experienced by the particle in the bulk), and the surface dipole potential. Thus,for positrons and electrons the relevant work functions can be written as [107,108]−φ± = µ± ±D.

For some materials the contribution from the dipole layer is larger than the onefrom the chemical potential, making the positron work function negative. This,finally allows mono-energetic positrons to be re-emitted into the vacuum from thesurfaces of these materials, or to be emitted into the interior of a large open-volumedefect such as a void. This is important because positron re-emission forms thebasis of moderation.

Figure 3.2 illustrates a one-dimensional representation of the single-particlepotential energy of a positron in the near-surface region for the case where φ+ isnegative, so that escape of the thermalized positron from the solid into the vacuumis energetically allowed.

Among the negative work function materials, tungsten (W(110)) has the high-est reported efficiency of ∼ 10−3 [109] (or later on ∼ 3× 10−3 [110]).

A moderator should in general be a single crystal solid. The only inconvenient isthat the defects tend to attract and trap positrons, preventing them from diffusingto the surface. To avoid this problem the moderator can be prepared by heating itto high temperatures (annealing) which removes vacancies and dislocations. Thewidth of the energy distribution of positrons emitted from a well-prepared singlecrystal is of the order of 0.5 eV (FWHM) or less. Popular alternatives to single

40


Figure 3.2: Representation of the one dimensional potential fora thermalized positron near the surface of a metal. The workfunction φ+ is a combination of the bulk chemical potential µ+

and the surface dipole layer D. The positron chemical potentialcontains a term Vcorr due to correlation with the conduction elec-trons and a term V0 due to the repulsive interaction with the ioncores, shown as black dots. (Figure extracted from Schultz andLynn [103]).

crystals are poly-crystalline tungsten foils or commercially available meshes. Forthe poly-crystalline tungsten foils and the meshes the moderation is in transmis-sion, whereas for the single crystals the moderation is in reflection mode. Howeverbeing poly-crystalline, the efficiency of these moderators is lower (∼ 10−4) and theenergy width larger (1-2 eV FWHM).

To summarize, the principle of energy moderation by interaction with a solid issimple:

1. An initial flux of fast positrons is implanted at some depth.2. The positrons rapidly reach thermal equilibrium.3. The thermalized positrons diffuse in the solid.4. A certain fraction of thermalized positrons reach the free surface where

they can be re-emitted in the vacuum.

41


3.1.2 Beam transport

Applying a magnetic field along the direction of transportation is the simplestway to guide a low-energy positron beam. If a positron in the beam has a velocitycomponent perpendicular to the beam direction it will be kept confined to thebeam by the cyclotron motion induced by the magnetic field. As a result thepositron will follow a spiraling trajectory along the magnetic field lines.

Since the positrons are so light and low in energy this guidance can be achievedby using even a modest magnetic field, which can be produced with solenoids orHelmholtz coils.

For some experiments however one may want to avoid having magnetic fieldspresent. An alternative is to use purely electrostatic guidance. The confinementof the beam in this case is not as good as in a magnetic beam which may causeloss of positrons during transport. However, the electrostatic beam optics are wellunderstood and can be simulated in advance to minimize any loss.

3.2 Positron beam interactions with solids and

surfaces

3.2.1 overview

When a slow positron beam hits the surface of a material, before being anni-hilated or ejected from the target surface, the positrons may undergo differentinteractions (for more complete details, see for example [111]). These interactionsare schematically represented in Figure 3.3.

1. The first possibility is that some of the incident positrons will scatter elasti-cally from the target and will form diffracted beams if the surface is a singlecrystal. The rest of the incident beam may enter the solid.

2. Several energy loss mechanisms will cause the positrons that penetrate thematerial to lose their kinetic energy; thus stopping and thermalizing them.

• Some of those thermal positrons may diffuse back to the surface wherethe positron can be trapped in a two-dimensional state or in a near-surface defect.In the case of a material having a negative work function, the positronmay be emitted from the surface as a positron, or after forming positro-nium at the surface, it may leave as positronium, –the formation of

42

3.2 Positron beam interactions with solids and surfaces

positronium at the surface is particularly interesting in this thesis andtherefore, it will be analyzed in more detail later on–.

• The thermalized positron that does not reach the surface may annihilatewith an electron of the material (free state) or may get localized duringdiffusion by a positron trap such as defects, where it will eventuallyannihilate. It may also de-trap from this trapped state, which occursin case of a shallow trap.

3. Scattering processes may cause that near-surface implanted positrons returnback to the surface before being thermalized. At the surface, these positronsmay annihilate inside the material (without even being fully thermalized) ormay be emitted directly from the surface as epithermal positrons or eitherform positronium at the surface and escape as epithermal ‘hot’ positronium.

Diffractedpositrons

Incidentpositronbeam

Non-thermal e+

Epithermal e+

Ps

Thermal e+

g

g

Annhihilation

Captureat defect

Slow e+

PsVacuum Solid

Figure 3.3: Schematic representation of the possible positron in-teractions. Part of the incident positrons may be backscatteredand may form diffracted beams if the surface is a single crystal.The thermalized positrons diffuse and (1) annihilate in a free state,(2) gets captured by a positron trap or (3) reach the surface againwhere it can be emitted as a positron or positronium. Also notthermalized positrons can return to the surface.

3.2.2 Positron backscattering

When a positron beam collides on a solid surface, some of the incident particleswill undergo a small number of large-angle collisions and leave the target with afraction of their original energy. These are termed ‘backscattered ’ positrons [112].

43


The backscattering probability depends on the incident energy of the positronsand on the sample material. Knowledge of positron backscattering probabilities isof practical use if positron annihilation is used to probe the surface and subsurfaceregions of solids. The positrons which backscatter from a sample can be annihilatedat solid surfaces – for example, chamber walls – in sight of the gamma detector,thereby disturbing the experimental data with unwanted contributions.

The first positron backscattering measurements were performed for the entirebeta spectrum of energies from a radioactive source (for example, 0 to 540 keVfor 22Na) [113, 114]. One attempt was made to select energy windows from thebeta spectrum using a magnetic spectrometer [115]. The first report of backscat-tering probabilities for truly mono-energetic positrons was for sub-3 keV positronsinteracting with Al foils [111], which formed part of a study of the penetration ofpositrons through thin metallic foils.

More recent reports of Monte-Carlo simulations compared with experimentalresults [116, 117]; showed that materials of higher atomic number cause a largerfraction of backscattered positrons. Since these fractions are not negligible, onehas to take into account that these positrons annihilate outside the sample, leadingto a contribution in the spectra recorded by the detector.

Besides the Monte-Carlo simulations, a full description of the experimentalfeatures and limitations, and a more detailed information on the positron andelectron backscattering from thick solids as a function of the incident and outgoingangles and energies can also be found in reference [117].

3.2.3 Positron implantation profile

As stated in the overview (subsection 3.2.1 on page 42), several elements are im-portant to study the parameters that influence the probability that a fast positron(injected into a solid) escapes into the vacuum after thermalization in the absorber:(a) the penetration depth, (b) the ability to move inside the host material and (c)the probability of being annihilated or trapped in a non-propagating state beforereaching the surface.

The penetration depth. Considering a number of positrons implanted into anabsorber, the penetration depth is the distance of the trajectory ‘endpoint’ ofthe positrons from the entrance surface (it is, therefore, a random variable). Thepenetration depth is characterized by P (E, z), a probability density function called“the implantation profile”1. P (E, z) describes the probability of a positron withthe initial energy E thermalizing at depth z in the material.

1also referred to as stopping profile.

44


Due to the difficulties in performing direct experimental measurements, re-searchers have often made use of computer simulations to gain information aboutparticle implantation. Monte Carlo simulations are in general, always reliable inpredictions of inelastic mean free path and stopping profiles of electrons. MonteCarlo codes can also give some ideas on the positron path in a material which canlater on be supported by experimental work.

Obviously, implantation profiles depend on the energy spectrum of the in-cident positron flux. In the monoenergetic case, extensive Monte Carlo simula-tions [118–122], were found to be well approximated by the well-known Makhovianimplantation profile1 [123]:

P (E, z) =mzm−1

zm0

exp[−(z

z0

)m], (3.1)

here the empirical shape parameter they find is m ≈ 1.9 and the parameter z0 isa function of incident positron energy, given by:

z = z0

(Γ(m+ 1m

))(3.2)

where z is the mean implantation depth. It has been proven that the relation(3.2) becomes simpler for certain profiles (discussed in detail in Sec. II.C.1 on page738 in ref. [103]), such as the exponential profile for which m = 1 and Γ(2) = 1,or the Gaussian derivative profile m = 2 and Γ( 3

2 ) =√

π2 .

For illustration, Figure 3.4 shows some Makhov profiles of mono-energeticpositrons implanted in Al and Si and for three different values of implantationenergies. A more detailed listing for some materials can be found in reference [124].

Traditionally it has been the mean implantation depth z that has been usedmore widely to characterize P (E, z). This is somewhat unfortunate because theparameter that is directly extracted from experiment is the median penetrationdepth z1/2

2.

From the Makhovian distribution z1/2 and z0 are related by

z1/2 = z0(ln 2)1m . (3.3)

1Named after the Makhov’s original electron implantation experiments [123].2However, by combining Eq. (3.2) together with the following Eq. (3.3) and for m = 2, the

values of the the mean implantation depth (z) and median penetration depth (z1/2) are verysimilar: z = 1.064 z1/2.

45


0 100 200 300 400 500 600 700 800 900 10000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Depth (nm)

Pos

itron

impl

anta

tion

prof

ile

Al Si

E = 2 keV

E = 5 keV

E = 8 keV

Figure 3.4: Calculated Makhov profiles of mono-energeticpositrons implanted in Al (dashed lines, ρ = 2.7 g cm−3) andSi (solid lines, ρ = 2.3 g cm−3) at 2 keV, 5 keV and 8 keV.The values of the parameters used are m = 2.0, n = 1.6 andα = 4.0 µg cm−2 keV−n.

The dependence of the mean implantation depth on energy is assumed to be apower law:

z(E) =α

ρEn, (3.4)

which was originally developed for electron stopping. In Equation (3.4) the pa-rameter ρ is the material density and α (with values often reported in units of µgcm−2 keV−n) and n are empirical parameters.

Mills and Wilson [111] measured the transmitted positron flux (1–6 keV)through thin wedge-shaped foils of Al and Cu supported on a thin carbon foil.From their experimental data they determined:

1. The implantation profile P (E, z) without presupposing any (Makhovian)functional relationship.

2. The the median penetration depth as a function of energy.

From the measurements of the median penetration depth as a function of energythey extracted values for α = 4.0± 0.3 µg cm−2 keV−n and n = 1.6± 0.05.

In literature different values for α (= 3.3 . . . 4) µg cm−2 keV−n and n (= 1 . . . 2)have been reported [125].

46


Thus, although the material and energy dependencies of the empirical param-eters α and n have been under some debate, the most frequently used values forall the materials in general are the values as determined by Mills and Wilson:α = 4.0 ± 0.3 µg cm−2 keV−n and n = 1.6 ± 0.05 (see for example the refer-ences [103,111,118,119]).

According to Vehannen et al. [125], these parameters were considered to bematerial independent.

3.2.4 Positron diffusion

If a positron is injected into a solid, the slowing down process will drop it closeto the bottom of the lowest energy band with a residual kinetic energy of theorder of kBT (where kB is the Boltzmann’s constant (1.38× 10−23J/K)). At theseenergies, the de Broglie wavelength λth (∼ 60 A at room temperature) is largerthan the interatomic distances in condensed matter (a few A). This means that athermalized positron in a solid actually is to be seen as a propagating wave (seethe references [103, 124, 126] and [127]). However, unless the temperature is verylow and the host medium a very perfect crystal, the number of collisions withphonons and imperfections during the positron lifetime τ can be so high, and atsuch large angles [40], that any directional correlation of the trajectory with theinitial motion is, in practice, lost.

In these conditions, positron migration over distances greater than de Brogliewavelength can be adequately described as a classical “random walk”. The deBroglie wavelength (λth) at temperature T is given by [128]:

λth =2π~√

3mpkBT≈ (62 A)

√300T

, (3.5)

where ~ is the reduced Planck’s constant and mp is the rest mass of the positron.

The semi-classical three-dimensional random walk theory of positrons [124] al-lows us, therefore, to calculate the displacement of thermalized positrons in theframework of the ‘diffusion approximation’, and to define the “diffusion coeffi-cient”. Scattering mechanisms determine the positron mean free path 〈l〉:

〈l〉 =3D+√3kBTm∗

, (3.6)

where the mean free path is a function of temperature T , D+ is the positrondiffusion coefficient and m∗ is the effective positron mass. The positron diffusion

47


coefficient is related to the mobility µ+ by the Einstein relation:

µ+ =eD+

kBT, (3.7)

with e the elementary charge (1.6× 10−19C).

The main process that determines the positron diffusion is the phonon scatte-ring: the positron momentum distribution broadens due to phonon absorption bythe low-energy positron. In materials such as semiconductors and a lot of metals,the scattering occurs by longitudinal acoustic phonons. This results in a temper-ature dependent diffusion coefficient D+ ∝ T−

12 . The temperature dependence of

positron diffusion in several metals and semiconductors have been widely reviewedby Schultz and Lynn [103].

In nonmetals, propagation of free positrons is not the only possibility; quasi-positronium can also contribute to transport [129], with a diffusion constant Dq−Ps

which is not necessarily much different from D+.

We do not need to discuss here the scattering process that determine Dq−Ps

or D+, but diffusion-coefficient-vs.-temperature curves are an important source ofinformation on solid excitations as probed by positrons [130].

In the framework of the diffusion approximation, it is a simple matter to solvethe problem of positron migration to a free surface. In the unidimensional case(flat boundary), given the stopping profile P (E, z) (Equation (3.1)), with z takennormal to the boundary, the probability of reaching the surface is proportionalto [131,132]:

F+(E) =∫ ∞

0

P (E, z) exp(− z

L+

)dz, (3.8)

where the subscript + stands for the positrons and L+ is the “diffusion length”.L+ is the distance that a thermal positron can travel before annihilated or capturedin a localized state [131].

Equation (3.8) tells us that F+(E) taken as a function of L−1+ , is nothing else

than the Laplace transform of P (E, z); thus the experimental determination ofF+(E), with a monoenergetic positron beam of variable energy, gives access toL+.

The relation between the diffusion length and the diffusion coefficient (D+) isgiven by

L+ =√D+τ∗, (3.9)

where τ∗ = (τ−1+κ)−1 is the mean time remaining in the propagating state whichis depopulated by annihilation at a rate τ−1 and by capture in a localized state at a

48


rate κ [131]. In kapton for example, τ∗ = 382 ps is the positron mean lifetime [133].

In addition to phonon scattering, the other processes that determine the positrondiffusion are the effect of the electron density enhancement in the vicinity of thepositive particle and the effect of the periodic lattice. Diffusion lengths are of theorder of 110–180 nm at 300K in the case of high-purity defect-free single-crystals,which is small compared to the implantation depth.

The Doppler-broadening VEP-beam experiments give depth-resolved and time-independent information. Thus, all processes to which thermal positrons are sub-jected, such as diffusion, drift, trapping at defects or free annihilation can be com-bined in a single one-dimensional equation. This steady-state positron diffusionequation can be written as:

D+d

dz2c(z)− d

dz

(µ+E(z)c(z)

)− λeff c(z) + I(E, z) = 0, (3.10)

with c(z) the steady-state positron density as a function of depth z, I(E, z) thenumber of positrons with an energy E implanted at a depth z, and E(z) a localelectric field.

In the diffusion equation, the positron trapping in defects is taken into accountand this leads to effective annihilation rate λeff:

λeff =1τb

+ κ(z) (3.11)

Next to the bulk annihilations, represented by τb, positrons are trapped in defectswith a rate κ(z), which is a function of the specific positron trapping rate fordefects νt and the defect concentration nt(z):

κ(z) = νtnt(z) (3.12)

In some materials, different specific positron trapping rates can be associated withtypical defect types like monovacancies and dislocations [134].

3.2.5 Epithermal positrons

The diffusion equation described in the previous subsection (Equation (3.10)) isbased completely on positrons that are thermalized. This is considered to be validfor higher positron implantation energies if inelastic scattering processes dominateelastic scattering and bulk trapping [135,136]. In the case when the implantationdepth is smaller than the thermalization length Ltherm(of the order of 10 to 20nm), one has to consider the epithermal positron effects. Therefore positrons

49


that are implanted with an energy lower than 1 keV have a high probability toend up in an epithermal state, so as a consequence, there is an enhancement ofthe Ps formation. This is the case specially for insulators and semiconductorswhere thermalization is slower due to inefficient stopping processes. An attemptfor calculating the number of epithermal positrons that return to the surface wasfirst given by Britton et al. [137] in 1988 and was further improved by Kong andLynn [135,136] who gave a more complete interpretation.

3.3 Experimental determination of the positro-

nium fractions

As explained in subsection 1.2.2 on page 12, Ps can be formed under certainconditions. While p-Ps annihilates into two anti-collinear 511 keV photons, o-Psdecays in vacuum via the emission of three photons. The Ps atom has a zerointrinsic momentum. Thus, p-Ps is observed as a narrow contribution to theDoppler-broadened annihilation peak obtained by a Ge-detector –which causeshigher S-parameter values–.

Due to the conservation laws the total energy of the three annihilation photonsof o-Ps is equal to 1022 keV. This results in a continuous energy distribution up to511 keV, which increases more or less linearly with increasing energy [12]. Thus, Psmay be detected by studying the energy spectrum of its annihilation photons [23].

With the use of a Ge-detector, this effect can be observed as an enhancementof the photon energy spectrum in the 0 to 511 keV region. In that way, thecomparison of the number of counts in the peak region P with a chosen fixed areaof the Compton region C (i.e. the region at energies lower than 511 keV) givesinformation about the amount of positronium.

The Ps fraction fPs can be deduced from the ratio R of counts accumulated indifferent regions of a spectrum [25,138,139]:

R =T − PP

, (3.13)

where T is the number of counts accumulated in the photon energy region below511 keV and P the number of counts in the peak area around the 511 keV annihi-lation line (–for a better clarity; see the derivation of this and a modified versionof the next expression (Equation (3.14)) in the Appendix B–).

To determine the fraction of 3γ o−Ps annihilation a calibration should be donewhere the Compton contribution is measured when no 3γ annihilation is presentand in the case where there is 100% of 3γ contribution. This is, for example,

50

3.3 Experimental determination of the positronium fractions

the case for Al(110) as shown in Figure 3.5 [140]. In the figure, both curves werenormalized to equal height of the 511-keV peak. It represents the complete spectrafor the two conditions (0% and 100% Ps formation).

Figure 3.5: Annihilation spectra measured at an Al(110) surface with aHPGe detector for situations representing 100% and 0% Ps [140]. Thefirst condition is realized by a low incident energy of 40 eV at a sampletemperature of 400 C, where all positrons are pushed back to the surfacefor Ps formation, the second one (no-Ps) by a high incident positronenergy of 15 keV.

Thus, for the Ps fraction (fPs) and with the use of Equation (3.13), the follow-ing expression is used:

fPs =

[1 +

P1

P0

(R1 −Rf )(Rf −R0)

]−1

(3.14)

where Tf and Pf (intrinsic in Rf ) are the “total” and “photopeak” HPGe countingrates for a given number of positrons annihilating per second and the subscripts 1and 0 refer to f = 1 (100% Ps) and f = 0 (no-Ps).

In samples where Ps formation is not possible in the bulk (e.g. Kapton), fPs

has to be correlated with Ps at the surface, which is proportional to the fraction of

51


positrons diffusing back to the surface. Thus, Equation (3.8) on page 48 is used.This is a measure for the fraction of positrons trapped in defects as a function ofthe incident energy, which can give information about the specific defect depthprofile.

With Equation (3.14), measurements of the Ps fraction as a function of thepositron implantation energy can therefore be fitted directly to obtain the positrondiffusion length (L+) and consequently the specific trapping rate.

3.4 Charging effects

In slow positron beam experiments (for positron implantation energies typicallybelow 2 keV), charging phenomena appear when the samples are insulators.

For polymers Coleman et al. [141] studied the time-dependent changes of: (a)the measured S-parameters at low (<few keV) incident energies, (b) the totalgamma count rate, and (c) the positronium formation fraction.

Their measured S(z1/2) and the corresponding total photo-peak data are shownin Figure 3.6. To translate E (in keV) into z1/2, they used the relation z1/2 =33E1.6 nm.

(a) (b)

Figure 3.6: (a) S vs z1/2 for an isolated polymer film. (b) Photo-peak countrate for the same run as in 3.6(a). One run performed per day: H day 1; Oday 2; day 3; day 4; N day 5. Figures taken from reference [141].

It is seen in Figure 3.6(b) that after 5 days, the photo-peak signal count rate at

52

3.4 Charging effects

the lowest incident energy decreases to about 25% of its original value. The photo-peak signal count rate is essentially unchanged for incident energies above about5 keV. The constancy of the count rate is consistent with the reproducibility inthe same energy range (high E) in the S-parameter values in Figure 3.6(a). As Edecreases (and hence z1/2) the results for S for the day 1 sample indicate a steadydecrease. The authors claim that this may indicate that positrons are forming lessPs in the first 10 – 50 nm below the surface. As the time elapses, the decreasestarts at progressively higher energies, and then S increases again at the lowest Evalues. To give an interpretation the authors proposed the following model:

a subsurface electric field builds up with time under positron bombardment, aselectrons are annihilated. The increasing field strength progressively sweeps anincreasing number of positrons from depths of 100–500 nm towards the lower Sregion just below the surface. At the lowest incident energies the positrons areswept through the surface, where they can form Ps which decays in the vacuum.Observation of vacuum Ps decay increases the observed S values.

This last observation is supported by the dramatic decrease in the photo-peakcount rate (Fig. 3.6(b)) with time at lower incident energies, which suggests thatpositrons are indeed being swept from the sample. After 100 h of exposure tothe positron beam over 70 % of positrons implanted at ∼1 keV are entering thevacuum. Note, however that for z1/2 = 300 nm (E ∼ 4 keV) the signal rate showsno significant decrease, whereas the effect on S is dramatic. This is consistent withthe model in which positrons are being swept to the near-surface region but notout of the sample.

In summary, the results may point to field drifting of positrons towards the sam-ple surface and they certainly have implications for the measurement of positron-related parameters (such as S) for insulating samples.

The authors conclude that these problems can be overcome by taking low-energy measurements immediately after exposing a sample to a positron beam(i.e. the first run), and that these results can be checked by repeating at a latertime and, if necessary, remeasuring after a short aeration of the system.

53


54

PART II

EXPERIMENTAL DETAILS

55

4Experimental set-up and Samples

description

A general description of the apparatus as well as a very short description of theused samples and some of their applications are given in this chapter.

4.1 Variable energy positron beam

For this thesis the measurements were mainly performed at the magneticallyguided positron beam in Ghent whose energy is variable between 0.1 and 30keV [58]. All the specifications about the design of the Ghent positron facilityare given in reference [142].

In one of the experiments, some of the samples were also measured at the slowpositron beam of the Washington State University (WSU) [143]. In this magneti-cally guided slow positron beam, monoenergetic positrons can be accelerated froma couple of eV up to an energy of 60 keV (–although in this thesis, the data areshown only up to 26 keV–) with a total peak rate of ∼ 2× 103 e+/s. The HPGedetector has a FWHM resolution of 1.6 keV at the 514 keV line of 85Sr.

4.2 Doppler broadening

The annihilation radiation is detected by a High Purity germanium (HPGe) de-tector. It is coupled to a digital signal processor (DSP) unit. The digital signalsof the DSP unit are further processed by an interface card and then transferredto the PC by a digital data acquisition card.

57

4. EXPERIMENTAL SET-UP AND SAMPLES DESCRIPTION

4.2.1 Photon detection system

The HPGe detector (Canberra) has a 25% efficiency1 and a resolution (FWHM)of 1.17 keV at the 514 keV line of 85Sr. The DSP unit (model 2060 from Can-berra) replaces the amplifier-ADC combination of a traditional high resolutionpulse processing chain [62]. In a conventional analog setup the energy signal isprocessed, shaped and filtered by a shaping amplifier before being digitized by anADC. In the setup with the DSP unit, the energy signal is digitized immediatelyafter the preamplifier and stored in the MCA. Such a system has the advantagethat the trade-off between throughput and resolution is more favorable comparedto the ordinary amplifier-ADC combination. Furthermore there is less chance fordrifts and instabilities leading to an improved system stability. The settings of aDSP can be saved. This leads to a higher flexibility (e.g. one can use differentsettings for different experiments) and a better reproducibility which is extremelyimportant for Doppler measurements. For more information see e.g. ref. [144].The DSP units have a conversion range of 16K. The gain was tuned so that thedetector has a calibration factor of 0.0332 keV/channel. The input count rate isabout 10 kHz. To obtain the best possible resolution a rise time of 5.6 µs and aflat top of 0.8 µs for the trapezoidal filter function was chosen. This is equivalentto a shaping time of 4 µs for a conventional shaping amplifier.

4.3 Polymer samples

4.3.1 Kaptonr

One of the selected materials for the experiment is Kaptonr Type HN2, a poly-imide film made by DuPont de Nemours [145]. Kapton is a material whose physi-cal, electrical, and mechanical properties remain constant over a wide temperaturerange (from 269C to 400C). The used Kapton sample has a thickness of 127 µmand a density (ρ) of 1.42 g cm−3.

Due to their properties, polyimides often replace glass and metals, such assteel, in many demanding industrial applications. They are used for the chassis insome cars as well as some parts under-the-hood because they can withstand theintense heat and corrosive lubricants, fuels, and coolants cars require. They arealso used in circuit boards, insulation, fibers for protective clothing, composites,

1Relative efficiency at the 1332 keV line of 60Co relative to a standard 3”-diameter, 3”-longThallium-doped Sodium Iodide (NaI(Tl)) scintillator.

2 A more complete description on the physical and chemical properties as well as the differentuses and applications is given in:http://www2.dupont.com/Kapton/en US/assets/downloads/pdf/summaryofprop.pdf

58

4.3 Polymer samples

and adhesives. They can also be used in the construction of many apparatus aswell as microwave cookware and food packaging because of their thermal stability,resistance to oils, greases, and fats, and their transparency to microwave radiation.

A polyimide is a polymer that contains an imide group, i.e. a group in amolecule (drawn in blue) that has a general structure which looks like the oneshown in Figure 4.1.

Figure 4.1: Example of an imide group. Thereare no special elements for R, R’ and R”. Theystand for any atom or group of atoms.

If the molecule shown above were to be polymerized1 the product would be apolyimide. There are two general types of polyimides. In one type (so-called linearpolyimides) the atoms of the imide group are part of a linear chain. The secondof these structures is a heterocyclic structure where the imide group is part of acyclic unit in the polymer chain (where R’ and R” in Figure 4.1 are two carbonatoms of an aromatic ring). Both types of polyimides are represented in Figure4.2.

(a) Linear polyimide (b) Aromatic heterocyclic polyimide

Figure 4.2: General types of polyimides.

Being one of the most commercial polyimides, DuPont’s Kapton belongs to thearomatic heterocyclic polyimides, like the one on the right (Figure 4.2(b)).

In particular, in positron annihilation spectroscopy, a 7.5 µm-thick Kaptonis commonly used as a positron-source-supporting foil because of its satisfactory

1Polymerization definition: Any process in which relatively small molecules, called monomers,combine chemically to produce a very large chainlike or network molecule (i.e. a polymer).

59


resistance to radiation and also because it has the property of the complete inhibi-tion of the positronium in the bulk and therefore avoids difficulties at the momentof analysing the experimental data. Therefore, it is an important and interestingmaterial for research purposes.

4.3.2 Thin polymer films

Thin films are thin material layers ranging from fractions of a nanometer to severalmicrometers in thickness.

From a physical point of view of the properties of the system, one can defineif a film is thin or not. Thin films have a behavior that differs from “bulk” be-havior, with different structural attributes and/or different dynamics that resultin important differences in properties. This difference arises from the fundamen-tal difference between the environment of molecules (or chain segments) at theinterface with another phase and the environment of these molecules in bulk.

On the material itself, there is a big dependence on how is the influence betweeninterfaces into the material, but also on external parameters such as temperature.When this influence expands over a significant proportion of a film, it can bedefined as thin.

In the case of polymers, the thicknesses of the “thin” films can be up to mi-crometers, because of the large size of the polymer chains. There are many knowntechniques to prepare thin polymer films. In this thesis, we will only describespin-coating from a solution.

4.3.2.1 Spin Coating

In the process of spin coating, a polymer solution is deposited on a substrate, andthe substrate is then quickly accelerated to the desired spinning velocity (normally1000 - 4000 rpm) during a certain time. Due to the action of the centrifugal force,the liquid flows radially, and the excess is ejected off the edge of the substrate.The film continues to thin slowly until it turns solid-like due to a dramatic rise inviscosity caused by solvent evaporation or until pressure effects cause the film toreach an equilibrium thickness. The final thinning of the film is only due to solventevaporation [146]. The final thickness of the deposited polymer film depends onseveral parameters [147,148] among which are the initial viscosity of the polymersolution, the spinning speed, and the concentration of the solution.

60

4.3 Polymer samples

4.3.3 Free-standing nanometric polymer films

In this thesis, free-standing nanometric polymer films have been prepared. This isthe first time that the positron annihilation spectroscopy technique is performedin self-supporting nanometric polymer films with the purposes discussed in thenext chapters.

The chosen materials for the free-standing films are:

1. A standard poly(methyl-methacrylate) (PMMA) resist (used in lithography)with low molecular weight in a Spin Bowl Compatible solvent system (SBC)5% (purchased from Brewer science).

2. Polystyrene (PS) (Acros Organics ref No. 17889; average molecular weight(M.W.) 240,000 (SEC); Tg = 100oC (DSC 10C/min)) which was dissolvedin toluene to concentrations of 30, 50 and 70 mg/mL.

The polymer films were prepared by spin-coating on SiO2 wafers of 2 inchesdiameter. As the concentration of the PMMA solution was always the same, thecoatings were performed by varying the spinning velocity from 500 to 4000 rpmand with the spinning time of 30 seconds. Immediately after spin-coating, thePMMA samples were dried at 110C during ∼10 seconds. In the case of PS thesamples were prepared by changing the spinning velocities (from 800 to 3000 rpm)and also the concentrations, but the samples were not dried.

Before detaching the polymer films from the substrate their thickness was mea-sured with a surface profilometer (Talystep).

Table 4.1 collects all the information about the spin-coating of the samples andalso lists the measured thicknesses.

The preparation of each of the self-supporting films is described as follows: Afterthe spin coating procedure, it was immersed in a distilled water bath for about 24h.Later on, the film was easily detached from the silicon wafer by pulling carefullyfrom the borders of the polymer1.

The polymers used are very hydrophobic, while SiO2 is a very hydrophilicsurface. Thus, the polymer films are easily detached from silicon wafers whenimmersed in a water bath because the water wets SiO2 much better than thepolymer and the interaction between the polymer and SiO2 becomes weaker [149–152]. A picture of one of the floating films is shown in Figure 4.3.

1According to literature [149], the samples should detach by themselves when immersed inwater. However, only the thickest PMMA film (listed in Table 4.1) was detached by itself after24h.

61


Table 4.1: Spin coating: preparation of the thin polymer films.a,b.

Polymer Material

Spin Measuredd

Concentrationc velocityz thickness

(mg/mL) (rpm) (nm)

PMMA SBC 5% 4000 220± 10

PMMA SBC 5% 2000 310± 10

PMMA SBC 5% 3000 400± 20

PMMA SBC 5% 1000 480± 10

PMMA SBC 5% 500 1700± 50

PS 30 3000 210± 10

PS 50 3000 340± 10

PS 50 1500 460± 10

PS 70 3000 650± 20

PS 50 800 670± 20

PS 50 1500 960± 20

aImmediately after the spin coating procedure, the PMMA samples were dried at110C during ∼10 seconds. The PS samples were not dried.bThe spinning time for all the samples was 30 seconds.cThe PMMA resist comes in a Spin Bowl Compatible solvent system (SBC) 5%.The PS was dissolved in Toluene.dThe thickness of each polymer film was measured with a surface profilometer(Talystep).zThe number ‘rpm’ is nominal. One sets certain spin velocity, but if the motor ischanged (as it was the case), the real spin velocity is no longer the same as it waspreviously.

Figure 4.3: Floating PMMA film.

62

4.3 Polymer samples

Eventually, the floating film was picked up by an aluminum holder with a 3cm diameter hole in its center and it was finally dried in a furnace at about 90Cduring 15 min.

For illustration, some of the spin-coated PMMA films are shown in Figure 4.4.

Figure 4.4: Some of the free-standing nanometric PMMA films.

63


64

PART III

RESULTS AND DISCUSSION

65

5Parameterization of the median

penetration depth of implanted

positrons in free-standing

nanometric polymer films

5.1 Introduction

A detailed knowledge of the positron interaction with solids and surfaces is offundamental importance for the interpretation of the results.

It has already been noted (–in the subsection 3.2.3 on page 44–) that the medianpenetration depth as a function of the implantation energy, z1/2(E), related to thewell-known Makhov distribution, P (E, z) (Equation (3.1)), can be parameterizedby means of the power-law z1/2(E) = α

ρEn, where ρ is the target density and

the constants α = 4.0(±0.3) µg cm−2 keV−n and n = 1.60(±0.05) are the mostfrequently used empirical parameters.

In the case of polymers however, by analyzing the ortho-positronium yield frompositron lifetime experiments at different implantation energies, Algers et al. [153]have found the values n = 1.71(±0.05) and α = 2.81(±0.20) µg cm−2 keV−n.In their experiment, the positron lifetime experiments were performed on severalspin-coated polymers on Si substrates. However, these polymers were not detachedfrom the substrates.

It is expected that in a non-detached polymer (from the substrate) the in-teraction at the interface with the substrate would have a higher contribution ofannihilation of positrons in the polymer than in the case of self-supporting films.

67

5. PARAMETERIZATION OF THE MEDIAN PENETRATIONDEPTH OF IMPLANTED POSITRONS IN FREE-STANDINGNANOMETRIC POLYMER FILMS

Therefore, for the first time, positron transmission experiments have been per-formed on the free-standing poly(methyl-methacrylate) and polystyrene films ofnanometric thicknesses described in the subsection 4.3.3 on page 61 for incidentenergies in the range 0.1–21 keV. These results are reported in this chapter.

One of the motivations in this chapter is to parameterize z1/2(E) from thesepositron transmission experiments, and so, we are able to compare our results withthe ones of Algers et al. and with the most commonly used parameters.

The calculation of positron implantation profiles is necessary for the analysis ofDoppler broadening of annihilation radiation (DBAR) experiments. The literatureis plenty of reports on experiments where the Doppler broadened line S-parameteris investigated in function of the energy of the implanted positrons by means of aVariable Energy Positron beam (VEP).

A few years ago, Coleman et al. [154] have suggested the idea of parameteriz-ing z1/2(E) from the S-parameter derived from DBAR measurements. They useda technique based on the difference between positron annihilation lineshape pa-rameters in two bi-layered materials that appear similar to the incident positrons(aluminium-glass and gold-tungsten)1.

Although the S-parameter is also employed, as a second motivation for thischapter, the bi-layer method is avoided. From the measurements of the S-parameteras a function of the positron implantation energy it is possible to obtain the valuesfor the parameters n and α that characterize the median penetration depth. Theanalysis of the measured data shown in this chapter is a novel way, to parameterizez1/2(E). It is also worthwhile to emphasize here that this is the first time thatthis type of measurements with such a purpose is performed in polymers whichadditionally are free-standing nanometric films.

In addition, at the positron facility of the Washington State University andfor some of the samples, the same approach of Mills and Wilson of measuring thetransmission coefficient of positrons through thin films of known thickness [111] isfollowed. The inspection of the peak count rate in these transmission experimentsare compared with the values obtained for the S-parameter experiments performedin Ghent. These data are also used for the parameterization of z1/2(E).

Finally, with all the values for the parameters α and n the theoretical thick-nesses of the samples are calculated and compared with the experimental ones.

1For a better clarity on this experiment, readers should refer to the original paper [154].

68

5.2 Experimental

5.2 Experimental

In addition to the experimental considerations of Chapter 4, there are other ex-perimental facts described in this section.

- The the densities (ρ) of the samples were not measured. However, it isexpected that there is not a significant change with respect to the samplesused by Algers et al. [153]. Thus, for allowing the comparison with theirdata, their density values were used. They were considered to be 1.197 gcm−3 and 1.040 g cm−3 for PMMA and PS respectively.

- The experiments where the S-parameter is obtained as a function of thepositron implantation energy were performed at the variable energy positronbeam in Ghent [58].

- All the DBAR spectra were collected for several implantation energies from0.1 to 21 keV. In these DBAR experiments two of the PMMA samples (310nm and 480 nm in Table 4.1 on page 62) were measured at each positronimplantation energy for every 30 minutes, the other films were measured ateach positron implantation energy for every 10 minutes.

- The measurements were performed with the sample mounted in the vacuumchamber in front of and perpendicular to the beam line and the Ge-detectorbeside, at the sample position, but perpendicular with respect to the positronbeam axis. A scheme is shown in Figure 5.1.

HPGedetector

Positron beam

Sample

Detector axis

Beam axis

Chamber walls

Figure 5.1: Scheme of the experimental DBAR setup.

- The absorption experiments where the count rate is taken into account weredone in only some of the samples (see Table 5.1 on page 82). They were

69


performed at the slow positron beam of the Washington State University(WSU). In this experiment, the monoenergetic positrons were acceleratedfrom ∼ 0.06 keV up to an energy of 60 keV (–although in this thesis, thedata are shown only up to 26 keV–).

5.2.1 Charging effects

Special care was taken to minimize the effect of the charge of the sample whichcan influence the measurements when the samples are insulators

Figure 5.2 shows the peak counts as a function of time for the 460 nm-thick PSfilm (see Table 4.1 on page 62). All the data points were collected every 10 minat 0.1 keV. It becomes obvious from the figure that as the time elapses the peakcounts decreases until certain saturation level. An explanation can be given withthe model proposed by Coleman et al. [141] (see the overview in the section 3.4on page 52). Thus, this may be interpreted as a subsurface electric field builds upwith time under positron bombardment, as electrons are annihilated. Positronswith low energy are swept through the surface, where they can form Ps whichdecays in vacuum.

The fitting curve in Figure 5.2, which represents a charging time constantτcharge is an exponential decay y(t) = y0 +A exp(−t/τcharge). Here y(t) representsthe peak counts as a function of time. Fitting this equation to the data of Figure5.2 resulted in a charging time constant of τcharge = 3.08 hours for polystyrene.

Figure 5.3 shows the compton-to-peak ratio as a function of time for E = 0.1keV for the 460 nm-thick PS film (from the same experiment as for the peakcounts). The significant increase in positronium formation is consistent with theinterpretation given above and therefore, serves as a support for it.

Thus, to reduce the influence of the charging effects on the results, the data inthe following experiments for this thesis are taken only from the first run (i.e. freshsample, see Coleman et al. [141] and Ito et al. [155]) so the charging effects are arethe lowest possible. Also the measuring time for each data point was relativelyshort (10 minutes) in comparison to the charging time constant. A complete runfor different implantation energies from 0.1 to 21 keV consists of 29 data files i.e.∼ 4.8 hours. However, after the first hour the positron implantation energy isalready 1.5 keV where the charging effect does not longer affect the measurements(as will be discussed in the next chapter).

70

5.2 Experimental

0 100 200 300 400 500 600 700 800 76000

78000

80000

82000

84000

86000

88000

90000

Pea

k co

unts

Time (min) (10 min each point)

PS

Figure 5.2: Charging test for the 460 nm-thick polystyrenefilm at 0.1 keV. The fitting curve is an exponential decayy(t) = y0 + A exp(−t/τcharge). This fitting curve representsthe charging time constant which resulted to be τcharge =184.8 min/60 min = 3.08 hours for polystyrene.

0 100 200 300 400 500 600 700 800 2.78

2.80

2.82

2.84

2.86

2.88

2.90

2.92

2.94

2.96

2.98

3.00

3.02

3.04

C/P


PS

Figure 5.3: Compton-to-peak charging test for the 460 nm-thickpolystyrene film at 0.1 keV.

71


5.3 Analysis and results

All the implanted and transmitted positrons contribute to the S-parameter directlyobtained from the experimental data. The S-parameter was taken to be equal tothe integral of P (E, z) times S(z) from z = 0 to z = d (d = polymer thickness):

S=∫ z=d

z=0P (E, z)S(z)dz, (5.1)

where S(z) represents the contribution to the S-parameter from a positron at acertain depth z.

At the specific energy E1/2 in keV, 50% of the implanted positrons annihilatein the polymer and 50% are transmitted. There the S-parameter is defined as S =S1/2. At E1/2, the median penetration depth (z1/2)1 is thus equal to the polymerfilm thickness d in nm. By plotting the film thickness d versus E1/2, the shape ofz1/2(E) was obtained and parameterized by means of the well-known power-lawequation z1/2(E) = α

ρEn.

It is expected that once the positrons have the enough energy to be transmitted,no peak counts should be detected (these positrons should annihilate far from thesample position). In that case, these transmitted positrons do not contribute tothe S-parameter. In our case, as displayed in Figure 5.4, the extrapolation to highenergy values (50 keV) gives as a result that about 4.5% of the maximum of themeasured curve (y = 275.8) is still detected. This means that, as a consequence,these high energy transmitted positrons give a contribution to the S-parameter.

It might be argued that the background is the responsible for this effect. Thus,in order to determine the background, a measure was performed in exactly thesame conditions, at different positron energies, with a dummy sample holder butwithout a sample. The measured background resulted not to be the responsiblefor the detected photo-peaks at high energies (it is displayed also in Figure 5.4).

1For a better clarity, see the definition of z1/2 on page 45.

72


0 5 10 15 20 25 30 35 40 45 50 55

0.0

30.0

60.0

90.0

120.0

150.0

180.0

210.0

240.0

270.0

300.0

y=275.8

y=12.5

Pea

k co

unts

(C

PS

)

Positron Energy (keV)

PMMA Background

Figure 5.4: Experiment performed in Ghent. Peak counts: extrap-olation to high energy values. The extrapolation to high energyvalues gives as a result that about 4.5% of the maximum of themeasured curve (y = 275.8) is still detected. From the figure, itbecomes obvious that the measured background is not responsiblefor this effect.

73


Thus, with the purpose to find out an explanation for this phenomenon, a diffe-rent transmission experiment with one of the polymer films was performed. Theexperiment consisted in measuring the S-parameter with teflon sheets everywhereat the vacuum chamber walls were the sample is located. Thus, the ‘new’ walls ofthe beam around the sample were internally “cladded” with Teflon.

The measurements of the S-parameter can therefore be compared with thosewithout clad. It was done because the material type of the chamber walls isstainless steel and the characteristic S-parameter in Teflon is higher than in thestainless steel.

The experiment is shown in Figure 5.5. When the walls were cladded withTeflon, the constant level observed for the S-parameter at high energies (E >∼ 8keV) corresponds to the bulk S-parameter value obtained for a Teflon sample.Without the Teflon clad, the S-parameter of the films at high energies is compa-rable (in good approximation) to the S-parameter value obtained for a standardnon-magnetic sample of stainless steel. This result suggests that the main contri-bution of the transmitted positrons comes from their annihilation in the chamberwalls.

74


0 2 4 6 8 10 12 14 16 18 20 22

0.480

0.495

0.510

0.525

0.540

0.555

PMMA 220nm normal setup PMMA 220nm + teflon clad inside non magnetic Stainless Steel sample Teflon sample

S-p

aram

eter

Positron Implantation Energy (keV)

Figure 5.5: Obtained S-parameter for a 220 nm PMMA film in Ghent.Comparison when the chamber walls are internally cladded with Teflon (N)with those without clad (). In addition, the same Teflon material usedfor the clad has been measured (4) and a standard non-magnetic stainlesssteel sample was also measured (). This stainless steel sample was not ofthe same material than the chamber walls but gives an indication that theannihilation of positrons at high energies comes from the stainless steel.All the points were measured during 20 minutes except the ones of thestainless steel sample (10 min). The vertical dashed line highlights theenergy E1/2 at which S = S1/2.

75


A simulation was also performed with the SIMION 7.0 program [156] in orderto have an idea about the trajectory of the high energy (transmitted) positrons(E >∼ 8 keV). SIMION is a ion optics software program which is widely used tosimulate electron and positron trajectories under the presence of electric and/ormagnetic fields. The simulation consisted in:

• An homogeneous magnetic field of about 65 gauss, which is about the sameas the magnetic field given by the Helmholtz coils at the positron beam line.

• The vacuum chamber were the sample is located.

• 60 positrons with energies varying between 8 keV and 21 keV. This is becausewe were interested in the behavior of the high energy positrons.

• These positrons ejecting from the sample at the same point but at anglesvarying from 13 to 25 degrees. This is because according to the simulation,the positrons that are ejected at angles lower than 13 degrees are efficientlyguided by the homogeneous magnetic field away from the sample so thatthese annihilations can not be detected by the HPGe detector. On the con-trary, positrons ejected with angles higher than 25 degrees always annihilateat the chamber walls close to the sample. Therefore, these annihilations arealways detected by the HPGe detector.

From the simulation, the positrons with energies higher than 8 keV do not followthe homogeneous magnetic field and annihilate in the chamber walls from approxi-mately 5 to 10 cm from the sample position (depending on their ejection angle andtowards the direction of the beam axis). These annihilations might be detectedby the detector which has 4.89 cm of diameter and is centered with respect to thesample position.

The simulation also shows that if the chamber would be 3 times its diameter(– it is 5 cm diameter –), the positrons would annihilate farther from the sampleand therefore, they would not be detected due to the lead shield. Unfortunately,it is not possible to construct a vacuum chamber with such a dimensions at thepositron beam facility in Ghent.

Another possibility to efficiently guide the transmitted positrons away fromthe sample would have been by increasing the homogeneous magnetic field. Fromthe simulation ∼ 120 Gauss would have been enough. However, to generate thesefield the current in the helmholtz coils has to be increased. The current wouldhave been that high for the helmholtz coils that they would have burned out.

The most important conclusion from the simulation is, therefore, that thehomogeneous magnetic field can not efficiently guide the positrons (with energieshigher than 8 keV) away from the neighborhood of the detector, and consequently

76


the positrons annihilate in the chamber walls. This result hence, could supportthe previous hypothesis. A screenshot of the simulation is shown in Figure 5.6.

y

x

Magnetic field

Ejected positrons

Sample positionz

Figure 5.6: SIMION simulation of the trajectory of the ejected(transmitted) positrons in a homogeneous magnetic field of about65 Gauss. In the simulation 60 positrons with energies varyingbetween 8 keV and 21 keV are ejected from the sample at thesame point but with angles also varying from 13 to 25 degrees.The chamber tube is 5 cm in diameter and has a length of 28 cm.

Figure 5.7 shows the S-parameter as a function of the W-parameter for the dif-ferent PMMA and PS film samples. In the figures, when increasing the energy,an evolution on a fairly straight line is seen from a surface value A to a clusteringpoint1 B representing the bulk value of the polymer film. Then, as the positronincident energy is increased the data points move from the cluster point B to C(slightly below the straight line segment that connects the cluster points B and C).This means that for higher implantation energies the S-W plot evolves from thebulk polymer value B to a cluster point C, which from our previous hypotheses,represents the bulk value of the stainless steel (or a combination of stainless steeland polymer). The evolution from B to C in Figure 5.7 indicates that in all themeasurements, for the PMMA as well as for the PS, positrons only annihilate inthe film and in the stainless steel.

1A cluster point is a point in the S-W representation where it is clear that for differentimplantation energies the positrons annihilate in an identical state.

77


0.05 0.06 0.07 0.08 0.09 0.10 0.47

0.48

0.49

0.50

0.51

0.52

0.53

0.54

0.55

0.56

C

B

A

PMMA

220 nm 310 nm 400 nm 480 nm 1700 nm

S-p

aram

eter

W-parameter

(a) S-W representation for PMMA

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11

0.46

0.48

0.50

0.52

0.54

0.56

0.58

0.60

C

B

A

PS

210 nm 340 nm 460 nm 650 nm 670 nm 960 nm

S-p

aram

eter

W-parameter

(b) S-W representation for PS

Figure 5.7: S-W results obtained in Ghent for the different (a)poly(methyl-methacrylate) (PMMA) and (b) polystyrene (PS) film sam-ples. When increasing the positron implantation energy an evolution ona fairly straight line is seen from a surface value A to a clustering point Brepresenting the bulk value of the polymer film. Then, for higher implan-tation energies the S-W plot evolves from the bulk polymer value B to acluster point C which represents the bulk value of the stainless steel (ora combination of stainless steel and PMMA). The evolution from B to Cindicates that in all the measurements, for the PMMA as well as for thePS, positrons only annihilate in the film and in the stainless steel.

78


Figures 5.8(a) and 5.8(b) display the obtained S-parameters as a function ofthe implantation energy for each film. In the figures, the upper and lower horizon-tal lines correspond respectively to the bulk polymer value and the bulk value ofthe stainless steel (or a combination of stainless steel and polymer). The smoothlines in the region between the upper and lower horizontal lines correspond to apolynomial fitting grade 3. The fitting was performed with the purpose of extract-ing the energy values at the point where the data intercept with the line/value S= S1/2. Thus, the extracted energy values were obtained from the intercept (withS1/2) of the polynomial fitting and not from the intercept of the lines that connectthe data points. The extracted energy values are listed in Table 5.1.

With the measured thicknesses and the obtained energies, the parameters αand n that parameterize z1/2(E) can, therefore, be found (z1/2(E) = α

ρEn). In

Figure 5.5 on page 75, the vertical dashed line highlights the energy E1/2 at whichS = S1/2. This is plotted to show that for both cases, i.e. with and without theinternal Teflon clad, the same energy value can be obtained (E = 4.75±0.12 keV).

Figures 5.9(a) and 5.9(b) show the resulting transmission experiments per-formed at WSU. For comparison, the S-parameter of the same samples measuredin Ghent is also displayed. By inspecting the Figures, one can see that the twoexperiments more or less overlap. Thus, by using the same procedure that wasperformed for the S-parameters in Figure 5.8 the energies E1/2 can be obtained.These values are also listed in Table 5.1.

79


0 2 4 6 8 10 12 14 16 18 200.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

Positron energy (keV)

S−

para

met

er

220 nm310 nm400 nm480 nm1700 nm

S = S1/2

PMMA

(a)

0 2 4 6 8 10 12 14 16 18 200.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

Positron energy (keV)

S p

aram

eter

210 nm340 nm460 nm650 nm960 nm

S = S1/2

PS

(b)

Figure 5.8: Experiment performed in Ghent. The figure shows the ob-tained S-parameters as a function of the implantation energy for the poly-mer films of (a) poly(methyl-methacrylate) (PMMA) and (b) polystyrene(PS). The upper and lower horizontal lines correspond respectively to thebulk polymer value and the bulk value of the stainless steel (or a combi-nation of stainless steel and polymer). The line S = S1/2 yields the energyat which 50% of the positrons have stopped in the films. The smooth linesin the region between the upper and lower horizontal lines correspond toa polynomial fitting grade 3. The energy values in Table 5.1 are thusextracted from the intercept of the polynomial fitting with S = S1/2.

80


0 2 4 6 8 10 12 14 16 18 20 22 24 26

0.0

0.2

0.4

0.6

0.8

1.0 N

orm

alis

ed P

eak

rate

(10

3 /s)

Positron implantation energy (keV)

PMMA 400 nm WSU PMMA 480 nm WSU

0.48

0.49

0.50

0.51

0.52

0.53

0.54

0.55

0.56

S-p

aram

eter

S-parameter PMMA 400 nm S-parameter PMMA 480 nm

(a)

0 2 4 6 8 10 12 14 16 18 20 22 24 26

0.0

0.2

0.4

0.6

0.8

1.0

PS 670 nm WSU

Nor

mal

ised

Pea

k ra

te (

10 3 /s

)


0 2 4 6 8 10 12 14 16 18 20 22 24 26

0.46

0.48

0.50

0.52

0.54

0.56

0.58

0.60 S-parameter PS 670 nm Ghent

S-p

aram

eter

(b)

Figure 5.9: Transmission experiments performed at WashingtonState University (WSU) to some of the free-standing polymer sam-ples (filled squares). For comparison the resulting S-parametersobtained in Ghent for the same samples is also displayed (emptytriangles). One can see that the two experiments more or lessoverlap. This agreement gives support and a high degree of con-fidence that the findings are truthful.

81


Table 5.1: Extracted energy values E1/2 defined as the energy at which the medianpenetration depth z1/2(E) = polymer film thickness. At that point 50% of thepositrons have stopped in the films. The samples marked with star (?) were alsomeasured at Washington State University (WSU). The extracted energies fromthese transmission experiments are also listed.

Experimental E1/2 from S at E1/2 from

Polymer Material Thickness Ghent transmission at WSU

(nm) (keV) (keV)

PMMA 220± 10 4.75± 0.12

PMMA 310± 10 6.08± 0.09

PMMA (?) 400± 20 6.52± 0.17 6.49± 0.08

PMMA (?) 480± 10 7.17± 0.06 6.96± 0.11

PMMA 1700± 50 13.73± 0.10

PS 210± 10 3.96± 0.03

PS 340± 10 5.39± 0.13

PS 460± 10 6.93± 0.10

PS 650± 20 7.79± 0.17

PS (?) 670± 20 8.33± 0.17 7.93± 0.07

PS 960± 20 9.45± 0.13

82


The values for the parameters α and n can be obtained as a result from a linearregression of our data:

z1/2 =α

ρEn ⇒

Y︷︸︸︷log(z1/2 ρ) =

a︷︸︸︷log(α) +

b X︷︸︸︷n log(E),

so it becomes linearized (y = a + bx). In that way the parameter n = b and theparameter α is:

log(α) = a ⇒ α = 10a.

Thus, in order to obtain the values for the parameters α and n, the data points thatare listed in Table 5.1 on page 82 were plotted in the Figure 5.10. The error barsof the data (PMMA (), PS (N) and transmission WSU (•)) were also taken intoaccount. One has to be careful as the error bars become asymmetric in logarithms.Imagine the data 1 ± 1. When taking logarithm log(1) = 0. The upper limit islog(2) = 0.3. The lower limit log(0) = −∞. Thus, If we consider a certain datavalue V with an upper value V + ∆V and a lower value V −∆V ; the upper errorbars = log(V + ∆V )− log(V ) and the lower error bars = log(V )− log(V −∆V ).

In Figure 5.10, the line that fits our data (solid line) is the result from a linearregression log(z1/2 ρ) = 1.899 log(E) + 0.1340. For these fitting line, the errors ofthe data on both axis were taken into account. With ρ the density, the resultingfitting parameters were n = 1.90(±0.04) and α = 1.36(±0.04) µg cm−2 keV−n.

In the same figure, the data represented by open squares () and triangles(4) were taken, respectively, from the PMMA and PS films measured by Algerset al. [153]. The corresponding fitting line (dashed line) was created by usingthe resulting values for the parameters α and n reported by Algers et al. n =1.71(±0.05) and α = 2.81(±0.20) µg cm−2 keV−n. The solid and dashed fittinglines in Figure 5.10 are clearly different. As suggested in Subsection 3.2.3 on page44 the differences does not arise from the shape of the profile. A probable reasonis that in the experiment of Algers et al., the spin-coated films were not detachedfrom the silicon substrate and subsequently the interaction at the interface withthe substrate would contribute to more annihilation of positrons in the polymerthan the expected in the self-supporting films.

In addition, the dotted line in Figure 5.10 was created with the most com-monly used values for the parameters α and n in the Makhovian equation (n =1.60(±0.05) and α = 4.0(±0.3) µg cm−2 keV−n). The differences with respect tothe dashed and solid lines are obvious. This suggest that the ‘standard’ values forthe parameters α and n are far from being the most appropriate for the analysisof polymers.

83


0.6 0.7 0.8 0.9 1.0 1.1 1.2

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

PMMA thin films

PS thin films

Fit

PMMA Algers

PS Algers

Fit Algers

Common values

Transmission WSU

log(z

1/2 ρ)

log(Energy)

Figure 5.10: Graphical representation of the power-law z1/2(E) =αρE

n according to the data of the transmission experiments. The re-sults are compared with the results of Algers et al. [153] and with themost commonly used values. The line that fits our data (filled squares(), triangles (N) and transmission WSU (•)) is the result from a lin-ear regression: log(z1/2ρ) = 1.899 log(E) + 0.134; χ2 = 3.26. Fromhere that n = 1.90 ± 0.04 and α = 1.36 ± 0.04 µg cm−2 keV−n. Thedata represented by open squares () and triangles (4) were taken,respectively, from the PMMA and PS films measured by Algers etal. [153]. The corresponding fitting line (dashed line) was created byusing the resulting values for the parameters α and n reported by Al-gers et al. The dotted line is created with the most commonly usedvalues for the Makhovian equation.To clarify the legend in the figure, the circles (•) are the data corre-sponding to the transmission experiments performed in WashingtonState University (WSU).

84


It is also worthwhile to emphasize that for comparison a non-detached (from theSi substrate) PMMA sample has also been measured. The obtained experimentalthickness was (220± 10) nm. The resulting S, S-W and W-parameters are shownin Figure 5.11. The S-parameter has been analyzed with the well-known VEPFITprogram [157]. The steps followed in such an analysis were:

1. All the parameters corresponding to the Si substrate were fixed.2. The only free parameters to be fitted by the program were (a) the thickness

for the PMMA film and for the interface and (b) the S-parameter of thefilm.

3. For comparison, the fitting was performed by employing the different val-ues for the parameters α and n: the values proposed in Ghent-WSU, thevalues proposed by Algers et al., and the most commonly used values.

Thus, the calculated thickness with the different values for the parameters αand n was:

(132± 6) nm for Ghent-WSU.(227± 9) nm for Algers et al.(292± 12) nm for the most common used values.

Hence, when compared with the experimental thickness, these results suggestthat the values proposed by Algers et al. are the ones that best fit our data.

In Table 5.2 are listed what would be the different thicknesses of the samples.They were obtained by employing in the power-law (z1/2 = α

ρEn):

1. the different values for the parameters α and n,2. the extracted energy values E1/2 that are listed in Table 5.1 on page 82,3. the densities ρ = 1.197 g cm−3 for PMMA ρ = 1.040 g cm−3 for PS.

For comparison the experimental values of the thickness are also listed in thistable.

For a visual interpretation, these different thicknesses of the samples are alsoshown in Figure 5.12 for PMMA and in Figure 5.13 for PS.

85


-2

0 2

4 6

8 10

12

14 1

6 18

20

22 2

4 26

28

0.54

5

0.55

0

0.55

5

0.56

0

0.04

0 0.

045

0.05

0

0.54

5

0.55

0

0.55

5

0.56

0

0.04

0 0.

045

0.05

0

-2 0 2 4 6 8

10

12

14

16

18

20

22

24

26

28

Non

-det

ache

d P

MM

A th

in fi

lm

S P

ositr

on Im

plan

tatio

n E

nerg

y (k

eV)

S

W


W

Fig

ure

5.11

:S,

S-W

and

W-p

aram

eter

sof

aN

on-d

etac

hed

(fro

mth

eSi

O2

subs

trat

e)22

0nm

PM

MA

thin

film

.

86


Tab

le5.

2:C

ompa

riso

nof

the

thic

knes

ses

(z1/2)

ofth

eth

inpo

lym

erfil

ms

obta

ined

from

the

diffe

rent

valu

esfo

rth

epa

ram

eter

sα

andn

that

char

acte

rize

the

wel

l-kn

own

pow

er-l

aw(z

1/2

=α ρE

n)

atth

eex

trac

ted

ener

giesE

1/2.

The

used

valu

esof

the

para

met

ers

are:

Ghe

nt-W

SU,n

=1.

90(±

0.04

)an

dα

=1.

36(±

0.04

)µ

gcm

−2

keV−

n;

Alg

ers

etal

.n

=1.

71(±

0.05

)an

dα

=2.

81(±

0.20

)µ

gcm

−2

keV−

nan

dC

omm

onva

luesn

=1.

60(±

0.05

)an

dα

=4.

0(±

0.3)

µg

cm−

2ke

V−

n.

The

extr

acte

den

ergy

valu

esfr

omth

eW

ashi

ngto

nSt

ate

Uni

vers

ity

(WSU

)ex

peri

men

tsan

dth

eir

corr

espo

ndin

gth

ickn

ess

calc

ulat

ion

are

liste

dat

the

end

ofth

eta

ble

and

mar

ked

wit

hw

ith

ast

ar(?

).

Exper

imen

tal

Thic

knes

sT

hic

knes

sT

hic

knes

s

Poly

mer

Mate

rial

Obta

ined

E1/2

Thic

knes

sG

hen

t-W

SU

Alg

ers

etal.

Com

mon

(keV

)(n

m)

(nm

)(n

m)

valu

es(n

m)

PM

MA

4.7

5±

0.1

2220±

10

219±

31

337±

65

404±

78

PM

MA

6.0

8±

0.0

9310±

10

351±

45

514±

96

600±

113

PM

MA

6.5

2±

0.1

7400±

20

400±

62

579±

121

671±

141

PM

MA

7.1

7±

0.0

6480±

10

480±

60

682±

125

781±

146

PM

MA

13.7

3±

0.1

01700±

50

1648±

244

2070±

444

2209±

481

PS

3.9

6±

0.0

3210±

10

179±

18

284±

43

348±

54

PS

5.3

9±

0.1

3340±

10

321±

46

482±

95

570±

113

PS

6.9

3±

0.1

0460±

10

517±

69

740±

143

852±

166

PS

7.7

9±

0.1

7650±

20

646±

99

904±

191

1027±

218

PS

8.3

3±

0.1

7670±

20

734±

112

1014±

215

1143±

244

PS

9.4

5±

0.1

3960±

20

933±

136

1258±

260

1399±

293

PM

MA

(?)

6.4

9±

0.0

8400±

20

397±

51

575±

107

666±

125

PM

MA

(?)

6.9

6±

0.1

1480±

10

453±

62

648±

126

745±

147

PS

(?)

7.9

3±

0.0

7670±

20

669±

86

932±

177

1056±

204

87


100

200

300

400

500

600

700

800

T

hick

ness

(nm

)

300

400

500

600

700

800

900

Thi

ckne

ss (

nm)

400

500

600

700

800

900

1000

Thi

ckne

ss (

nm)

1400

1600

1800

2000

2200

2400

2600

2800

Thi

ckne

ss (

nm)

400

500

600

700

800

Energy data From WSU

Thi

ckne

ss (

nm)

300

450

600

750

900

1050

PMMA


T

hick

ness

(nm

)

Figure 5.12: Thicknesses of the PMMA samples obtained with the different valuesfor the parameters α and n compared with the experimental thickness values atthe extracted energy values E1/2 (from z1/2(E) = α

ρEn).

The order of the data points, respectively, from left to right are: (1) experimentalthickness; (2) Ghent-WSU, n = 1.90(±0.04) and α = 1.36(±0.04) µg cm−2 keV−n;(3) Algers et al. n = 1.71(±0.05) and α = 2.81(±0.20) µg cm−2 keV−n and (4)Common values n = 1.60(±0.05) and α = 4.0(±0.3) µg cm−2 keV−n.

88


200

300

400

500

600

700

Thi

ckne

ss (

nm)

400

500

600

700

800

900

1000

1100

Thi

ckne

ss (

nm)

600

750

900

1050

1200

Thi

ckne

ss (

nm)

600

800

1000

1200

1400

Thi

ckne

ss (

nm)

750

900

1050

1200

1350

1500

1650

1800

Thi

ckne

ss (

nm)

600

750

900

1050

1200

1350

PS


Thi

ckne

ss (

nm)

Figure 5.13: Same caption that in the previous Figure 5.12 but in this case for thePS films.

89


5.4 Summary

Positron transmission experiments have been performed on free-standing polystyreneand poly(methyl-methacrylate) films of nanometric thicknesses.

The constants that parameterize the median penetration depth z1/2(E) = αρE

n

have been found to be n = 1.90±0.04 and α = 1.36±0.04 µg cm−2 keV−n. Thesevalues were successfully determined with the previous knowledge of the thicknessof the samples and with the data obtained from the transmission measurementsperformed in two different positron facilities: (1) with the S-parameter as a func-tion of the positron implantation energy and (2) with the peak rate obtained insome of the samples.

As the results obtained from the experiments performed in Washington StateUniversity resulted in a high agreement with the results obtained in Ghent, thisgives support and a high degree of confidence that the procedures and respectivefindings are truthful.

The results suggest that special care has to be taken into account in selectingcorrect values of the parameters when analyzing the experimental data as standardvalues might lead to wrong results.

Our results seem to indicate that the parameters proposed by Algers et al. arenot valid in the case of self supporting films. However, when analyzing a non-detached sample from the Si substrate, their values are the ones that best fit ourdata.

90

6Determination of the positron

diffusion length in polymers by

analysing the positronium

emission

In this chapter are shown the results concerning some positron beam experimentsperformed on the self-supporting poly(methylmethacrylate) (PMMA) film of 310nm-thick (see Table 4.1 on page 62) and on the Kaptonr samples, both describedin Chapter 4.

The positronium (Ps) emission from the PMMA and Kapton surfaces is stud-ied as a function of the positron implantation energy by using Doppler profilespectroscopy and Compton-to-peak ratio analysis.

This experiment is also interesting because Ps is formed only in PMMA andnot in Kapton.

6.1 Introduction

Many solids are known to emit positronium when bombarded by low-energy po-sitrons. The study of the positron motion is important for understanding theinteractions of positrons with matter. The Ps emission is a phenomenon partic-ularly interesting in some metal oxides and in polymeric materials. An overviewabout the several mechanisms at the basis of Ps emission in insulators can be foundin Reference [158]. The mechanisms of Ps formation at the materials surface havebeen described by the following processes:

91

6. DETERMINATION OF THE POSITRON DIFFUSION LENGTHIN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION

a. Implanted positrons can get trapped into a surface state that can be subse-quently thermally activated into Ps emission [159].

b. Implanted positrons can reach the surface, capture an electron at the surfaceand thus emerge as a Ps rather than free positron [40].

c. Ps can be formed in the bulk of the material and can diffuse back to thesurface where it is emitted [160].

The Doppler broadening of annihilation radiation (DBAR) technique (describedin section 2.3 on page 27) is one of the methods that may be used to study theemission of para-positronium (p-Ps) (see for example [46]).

DBAR measurements performed in a longitudinal setup at the Variable EnergyPositron beam (VEP) in Ghent have been described in reference [161]. In thesesetup the γ−ray detector is located behind the sample on the axis of the beam.The authors concluded that the Ps emitted at the front side surface of the samplehas a linear momentum mainly away from the detector. This causes a red shift ofthe p-Ps contribution in the annihilation spectrum. A ‘red-shifted’ contributionmeans that, by approximating the detected photo-peak by a Gaussian distribution,the p-Ps contribution is detected as a narrow fly-away peak at the low-energy sideof the 511-keV-line [161].

In this chapter it is proven that the detected photo-peak from DBAR experi-ments (after the background subtraction) can also be affected by the contributionof the p-Ps emission at the high-energy side (blue-shift) or at the central part ofthe photo-peak. A blue-shifted peak has an advantage over the red-shifted becausethe Compton background contribution that appears at the low-energy tail of thedetected photo-peak can be avoided.

Two different specimen-detector geometries are thus proposed: The polymersample is located (1) at 45 and (2) perpendicular with respect to the positronbeam axis. The detector is located beside the sample position, but perpendicularto the positron beam line.

For the PMMA sample, the following parameters are calculated and discussed inthis chapter:

1. The bulk Ps fraction.2. The efficiency for the emission of Ps by picking up an electron from the

surface.3. The diffusion lengths of positrons (thermal (and epithermal)1), p-Ps and

ortho-positronium (o-Ps).

These parameters were obtained from:

92

6.2 Experimental

a. The analysis of the fly-away p-Ps.b. The bulk p-Ps.c. The fly-away ortho-positronium o-Ps observed in the Compton-to-peak

ratio analysis.

In the case of the Kapton sample the thermal (and epithermal) positron diffusionlength and the efficiency for the emission of Ps by picking up an electron from thesurface were obtained.

6.2 Experimental

This section describes the experimental parameters that were not already given inChapter 4.

• All the experiments were performed at the variable energy positron beam inGhent [58].

• Two different specimen-detector geometries were performed for these DBARmeasurements. They are schematically represented in Figure 6.1:

1. In the first case the polymer sample was located at 45 with respectto the positron beam axis. The HPGe-detector was located beside thesample position, but perpendicular to the positron beam line (Figure6.1(a)).

2. In the second case, the polymer sample was located perpendicular withrespect to the positron beam axis. The HPGe-detector was also lo-cated beside the sample position and perpendicular with respect to thepositron beam axis. The scheme is shown in Figure 6.1(b).

93


HPGedetector

Positron beam

Sample

Detector axis

SampleOrientationAngle

45°Beam axis

(a)

HPGedetector

Positron beamSample

Detector axis

Beam axis

(b)

Figure 6.1: Experimental setup with the sample at (a) 45 and (b) per-pendicular with respect to the beam axis.

⇒ For the 310 nm-thick PMMA sample:

- The DBAR spectra were collected for several implantation energiesfrom 0.1 to to 1.2 keV. The measurements were recorded at eachpositron implantation energy for every 30 minutes.

⇒ For the Kapton sample:

- The DBAR measurements were recorded every 30 minutes mini-mum three times.

- Different implantation energies were used depending on the specimen-detector geometries:a. From 0.1 to 0.65 keV for the measurements with the sample at

45 with respect to the beam axis (Fig. 6.1(a)).b. From 0.1 to 0.75 keV for the measurements with the sample

perpendicular to the beam axis (Fig. 6.1(b)).- An explanation of the charging-up of a polymer sample has been

given by Coleman et al. [141] (see a review in section 3.4 on page 52).

In this Chapter, special care was taken to study and minimize theeffect of charging of the sample. Three successive measurementswere done for testing the reproducibility and hence the effect ofcharging:a. The sample was measured in perpendicular geometry.b. The measurements were recorded every 30 minutes.c. In the first run, the positron implantation energy was increased

from 0.1 to 1.1 keV at intervals of 0.1 keV. Then, for the second

94

6.2 Experimental

run, the positron implantation energy was decreased from 1.1keV to 0.1 keV at the same intervals. Finally for the third run,the positron implantation energy was increased again in thesame way as for the first run.

The peak statistics as a function of the positron implantation en-ergy of such an experiment is shown in Figure 6.2. From the figure,it follows that the second and the third runs were nearly equal.They, in addition, differ from the first run only below ∼0.3 keV.

0 200 400 600 800 1000 10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

Pea

k S

tatis

tics

(x 1

0 4 )

Positron implantation energy (eV)

1st 2nd 3rd

Figure 6.2: Comparison of the peak statistics as a function ofthe positron implantation energy in Kapton for three successivemeasurements. It was performed to test the reproducibility andthus the effect of charging. The positron implantation energy wasincreased during the first and third runs and was decreased in inthe second run. The setup used for this experiment was with theKapton perpendicular with respect to the beam axis (Fig. 6.1(b)).

Figure 6.3 shows the peak statistics as a function of time for aKapton sample. The experiment was performed with the sampleat 45 with respect to the beam axis. All the data points werecollected every 30 min at 0.1 keV. The fitting curve in the figure,

95


which represents a charging time constant τcharge is an exponentialdecay y(t) = y0 +A exp(−t/τcharge). Here y(t) represents the peakcounts as a function of time. Fitting this equation to the data ofFigure 6.3 resulted in a charging time constant of τcharge = 7.26hours for Kapton.In the following experiments the data are taken only from the firstrun so the charging effects are are the lowest possible (see Colemanet al. [141] and Ito et al. [155]). In addition, the influence of thecharging-up of the sample is only given below ∼300 eV and themeasuring time for each data point is relatively short (30 min) incomparison to the charging time constant at these low energies. Acomplete run for different implantation energies from 0.1 to 0.75keV consists of 7 data files i.e. ∼ 3.5 hours, which is not even thecharging time constant. In addition, after 2 hours the sample isalready above 0.3 keV where the charging effect does not longeraffect the measurements (see Figure 6.2).

0 500 1000 1500 2000 2500 230000

240000

250000

260000

270000

280000

290000

300000

310000

Pea

k S

tatis

tics


Kapton

Figure 6.3: Charging test for Kapton. The setup used for thisexperiment was with the Kapton at 45 with respect to the beamaxis (Fig. 6.1(a)). The data points correspond to the peak statis-tics as as a function of time for implanted positrons at 0.1 keV. Thefitting curve is an exponential decay y(t) = y0 +A exp(−t/τcharge)that represents the charging time constant. It resulted to beτcharge = 7.26 hours for Kapton.

96


• Time-of-flight (TOF) spectroscopy which uses a specialized setup [160] isthe basic method to investigate Ps emission in a direct way. In those exper-iments a detailed description of the energy distribution of the emitted p-Pscan be obtained. Since we are only interested in the fraction and energy shiftof the emitted p-Ps from the sample surface, the detected photo-peak (byonly one Ge-detector) can (in first approximation) be fitted with a superpo-sition of Gaussian distributions whose components arise from the differentannihilation channels [44,101].

As Ps is only present in PMMA and not in Kapton, all the DBAR spectrawere independently analyzed with a sum of four Gaussians for the PMMAfilm (see Fig. 6.4) and three Gaussians for the Kapton (Fig. 6.5). For theanalysis the DBAN program was used [162].

As the detected photo-peak has a big contribution from the Compton back-ground at the low-energy side, it is necessary to subtract it and thus, a moreor less symmetric peak can be obtained.

In the DBAN program, the stepwise background is subtracted by using: (1)the erfc function, (2) estimating a flat level before the 511 keV annihilationpeak and (3) a line with negative slope from the right of the peak. Theresolution function is convoluted and then a minimization by least squaremethod is performed1.

Thus, once the stepwise background is subtracted, in the DBAN programthe 511-keV-line can be fitted with a sum of Gaussians (up to four). Thisis convoluted with the resolution function of the detector, which is alsotaken as a Gaussian, so the convolution results in a Gaussian. The observed(FWHMfit) values for the Gaussians are finally obtained.


In Figures 6.4 and 6.5 all the contributions of annihilation of positrons with lowand high-momentum electrons are represented by Gaussians:

- In both cases (PMMA and Kapton) for the fitting of the detected photo-peakthe fixed parameters were:

a. The peak-shift of the component 1. It was centered with respect to the511-keV-line (i.e., 0 keV in the figures).

1For clarity about the code and functionality of the DBAN program, please contact his authorNikolay Djourelov at [email protected]

97

mailto:[email protected]


b. A full width at half maximum (FWHM) of 1.1 keV for the component3.

- The two main contributions (labeled with 1 and 2) have about the samecentroid and they describe fairly well the low and the high-momentum con-tribution of the annihilation of positrons in the bulk and/or on the surface1.

- The third Gaussian contribution represents the emitted p-Ps which is:

a. shifted towards high energy with respect to the 511-keV-line (–causing astrong asymmetry in the annihilation peak–) when the sample is at 45

with respect to the beam axis (displayed in Figures 6.4(a) and 6.5(a))).The linear momentum of the Ps emitted at the surface of the sample isoriented mainly towards the detector. This allows the detection of thep-Ps emission as a narrow fly-away peak at the high-energy side of the511-keV-line (blue-shift).

b. centered with respect to the 511-keV-line (Figures 6.4(b) and 6.5(b))when the sample is perpendicular with respect to the beam axis.

In DBAR experiments, the low-energy tail of the detected photo-peak hasa big contribution of the Compton background which arises from the 1274keV γ−ray of 22Na. Therefore, a blue-shifted peak has an advantage over thered-shifted because these Compton background contribution can be avoided.

- In the case of PMMA, the fourth contribution is centered with respect tothe 511-keV-line and has a narrow FWHM. It is identified as annihilation ofp-Ps in the bulk. This only applies to the PMMA because Ps is not formedin bulk of Kapton, but it does in PMMA.

1For the importance of the low and high momentum contribution of the annihilation ofpositrons see e.g. the final paragraph of subsection 2.3.1 on page 29.

98


−4 −3 −2 −1 0 1 2 3 440

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Pe

ak

Co

un

ts

Energy Shift(keV)

1

234

(a)

−4 −3 −2 −1 0 1 2 3 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Pe

ak

Co

un

ts

Energy Shift(keV)

1

2

34

(b)

Figure 6.4: Annihilation peak obtained for the PMMA sample for animplanted positron energy of 467 eV.(a) The sample is at 45 with respect to the beam axis. The fitting isdone with (1) a low momentum, (2) high momentum (dashed line), (3) acontribution from the emitted p-Ps which is blue-shifted with respect tothe 511-keV-line and (4) annihilation of p-Ps in the bulk (dotted line).(b) The sample is perpendicular with respect to the beam axis. The fittingis done with (1) a low momentum, (2) high momentum (dashed line), (3)a centered contribution from the emitted p-Ps with respect to the 511-keV-line and (4) annihilation of p-Ps in the bulk (dotted line).

99


−5 −4 −3 −2 −1 0 1 2 3 4 550

2

4

6

8

10

12

14x 10

4

Pe

ak

cou

nts

Peak shift (keV)

1

23

(a)

−5 −4 −3 −2 −1 0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Pe

ak

cou

nts

Peak shift (keV)

1

23

(b)

Figure 6.5: Annihilation peak obtained for the Kapton sample (a) at 45

and (b) perpendicular with respect to the positron beam axis for an im-planted positron energy of 103 eV. The fitting is done with (1) a lowmomentum, (2) high momentum (dashed line) and (3) with respect tothe 511-keV-line, (a) a blue-shifted or (b) centered contribution from theemitted p-Ps.

100


Both, positron and Ps can diffuse back to the surface. First we will considerthe positrons and then Ps.

If Ps is formed in the bulk of the material, we may assume that the initialdistribution of Ps is equal to the positron implantation profile (see subsection 3.2.3on page 44). We call fPs the fraction of positrons that form Ps. The remainingfraction (1- fPs) of positrons can diffuse back to the surface and may emerge aso-Ps or p-Ps by picking up an electron from the surface. The fraction that capturessuch an electron is fpu. The situation is schematically represented in Figure 6.6.

Ps

Vacuum Sample

e+

e+

F ( )p-Ps

E

F ( )o-Ps

E

(3/4)fPs

(1/4)fPs

F ( )=[1-b -Ps

E fp( )]E

1-fPs

fPs

F ( )+

E

fb

fpu

fs

Ps

Figure 6.6: Schematic representation of the positronium emissionfrom a sample surface.

The bulk Ps diffuses and is either trapped in free-volume sites or reaches thesurface whereupon it is ejected. As stated in Ref. [161], due to the self-annihilationin these free-volume sites, only a fraction fb of p-Ps is observed as a narrow centralcontribution in the Doppler line-shape.

Given the implantation profile P (E, z) = mzm−1

zm0

exp[−(

zz0

)m], (with z0 =

z(Γ(

m+1m

))−1), the probability of reaching the surface is proportional to [131,132]:

Fj(E) =∫ ∞

0

P (E, z) exp(− z

Lj

)dz, (6.1)

101


where the subscript j stands for the positrons (+), p-Ps and o-Ps. The respective“diffusion lengths” are L+, Lp-Ps and Lo-Ps

1.

The intensities given by:

• The fly-away p-Ps that is observed in the blue-shifted contribution.

• The fly-away o-Ps that is observed in the Compton-to-peak ratio analysis.

• The bulk p-Ps which is detected as a supplementary central narrow contri-bution in the detected photo-peak (in the PMMA case).

can be described by the following set of equations:

Iep-Ps =

(14

)[fF+(E) + fPsfeFp-Ps(E)] , (6.2a)

Ieo-Ps =

(34

)[fF+(E) + fPsfeFo-Ps(E)] , (6.2b)

Ibp-Ps =

(14

)fbfPs [1− feFp-Ps(E)] , (6.2c)

where f = fpu(1− fPs) and fe represents the emission efficiency. Figure 6.6 isalso a graphic representation of these set of equations.

In the case of a Kapton sample no positronium is formed in the bulk. Equations(6.2) become much simpler because then fPs = 0. Thus, only a fly-away p-Ps peakwill be observed. In this case, the equations are reduced to:

Iep-Ps = (1/4)F+(E)fpu, and Ie

o-Ps = (3/4)F+(E)fpu (6.3)

The experimental fraction of emitted o-Ps at implantation energy E is obtainedfrom a Compton-to-peak ratio analysis of the annihilation spectrum2 [138]:

Iexpo-Ps = α

[1 +

P (0)P (∞)

R(0)−R(E)R(E)−R(∞)

]−1

(6.4)

where P is the number of counts accumulated in the region centered around the511-keV annihilation line and R = C/P is the ratio of the number of counts in a

1For a brief description of the diffusion length refer to the subsection 3.2.4 on page 47 (thediffusion length is defined after Equation (3.8)).

2see also Section 3.3 on page 50 and Appendix B on page 117.

102


chosen fixed area of the Compton region C to the peak counts P . P (0) and R(0)are the values extrapolated to zero implantation energy and P (∞) and R(∞) arethe asymptotic values for high implantation energy, i.e. in the bulk of the material.Equation (6.4) is applied only if P (∞) and R(∞) correspond to a situation whereno o-Ps is detected by three-quantum annihilation.

6.3.1 Position of the p-Ps contribution in annihilation spec-tra

When p-Ps is emitted out of the sample in the vacuum, its distance tothe detector increases or decreases, depending on the detector-sampleposition, before the annihilation takes place. This distance influencesthe detection efficiency and thus the intensity of the p-Ps contributionto the peak.

The distance between the sample and the annihilation position can beestimated using the law for a statistical decay process:

N(t) = N0e− t

τp-Ps (6.5)

with N(t) the number of particles that are not annihilated at time t andN(0) the number of particles at time zero. τp-Ps is the mean lifetime ofp-Ps in vacuum, which is 124 ps. The time dependence can be replacedby a place dependence x = vp-Pst, with x the distance from the sampleto the annihilation position and vp-Ps the p-Ps velocity. This leads to1:

N(x) = N0e− x

vp-Psτp-Ps = N0e− xr

Ep-Psme

τp-Ps (6.6)

If one takes the representative numerical example of p-Ps emitted withan energy of Ep-Ps = 1 eV, then the velocity is vp-Ps = 4.2× 105 m/s.Figure 6.7 displays N/N0 as a function of x. For this emission energy, apercentage of 99.9% of the p-Ps annihilates within the distance of 0.36mm. For positrons with emission energies up to 5 eV this distance is0.7 mm. This means a maximum decrease of the detection efficiency of1% for the longest living Ps. Thus one can consider that the influenceon the detection efficiency for this distance is negligible.

For o-Ps the same equation can be used, but with a mean lifetime of142 ns. Since its lifetime is approximately 1000 times larger than the

1By simple calculations, one can determine the energy and velocity of the emitted p-Ps. Seefor example the reference [163]

103


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance from the sample (mm)

N/N

0

1 eV5 eV

Figure 6.7: Free p-Ps population as a function of the distancebetween the sample and the annihilation position. The p-Ps isemitted with an energy of 1 eV and 5 eV.

p-Ps lifetime, most of o-Ps will annihilate within a distance of a fewcentimeters. For this range the detection efficiency strongly dependson the emission energy.

As fly-away o-Ps may annihilate at several cm in front of the specimen [130](in contrast to the p-Ps fly-away), the the solid angle for the detection of thecorresponding three-quantum annihilation is increased by a factor ≈ 2.26 withrespect to all two-quantum annihilation1. Such an effect is taken into account byadjusting the value of the proportionality constant α in Equation 6.4 in the waythat Io-Ps(0)/Ip-Ps(0) = 3.

With the sample at 45 (Figure 6.1(a)), the experimental data for the intensitiesof the fly-away p-Ps, the fly-away o-Ps and the bulk p-Ps for the PMMA film areshown in Fig. 6.8. The solid lines represent the simultaneous fitting of Eqs. (6.2) todetermine the positron diffusion length L+ (thermal (and epithermal)), the Lp-Ps

and the Lo-Ps. The fitting shown in the figure was performed using the values forthe parameters α and n in the Makhovian equation found in the previous Chapter(α = 1.36(±0.04) µg cm−2 keV−n and n = 1.90(±0.04)). It can be seen thatthe intensity of the bulk p-Ps increases as function of the positron implantationenergy mainly at the expenses of the intensity of the emitted p-Ps (blue-shiftedcontribution).

1The solid angle is given by Ω = π r2

d2+r2 , where r = 2.44 cm is the detector radius and

d is the distance of annihilation (with respect to the detector axis). Thus, with d = 8 cm,Ω1 = 0.27 sr and with d = 5 cm, Ω2 = 0.61 sr. The ratio is therefore, Ω2

Ω1= 2.26.

104


0 0.2 0.4 0.6 0.8 1 1.2−2

0

2

4

6

8

10

12

14

16

18


Pos

itron

ium

inte

nsity

(%

)

Figure 6.8: Ps emission from the of 310 nm-thick PMMA film.The figure shows the intensity of the p-Ps formed in the bulk(×), the intensity of the p-Ps emitted from the surface (•) andthe intensity of the emitted o-Ps (4). The solid lines representthe fit of Eqs. (6.2) to determine the positron diffusion lengths(thermal (and epithermal)) L+, Lp-Ps and Lo-Ps. The fitting wasperformed using the values for the parameters found in the pre-vious Chapter for the Makhovian equation (n = 1.90(±0.04) andα = 1.36(±0.04) µg cm−2 keV−n).

105


For comparison, a fitting was done by using the most frequently used valuesof the power-law equation on the Makhov distribution (n = 1.60(±0.05) and α =4.0(±0.3) µg cm−2 keV−n) and by using the values for the parameters as proposedby Algers et al. (n = 1.71(±0.05) and α = 2.81(±0.20) µg cm−2 keV−n). All theresulting values are summarized in Table 6.1.

Table 6.1: Comparison of the values obtained from the fitting of the experimentalintensities of the fly-away p-Ps, the fly-away o-Ps and the bulk p-Ps for the PMMAfilm by using the the different values for the parameters α and n that characterizethe well-known power-law equation (z1/2(E) = α

ρEn). The used parameters are:

Common values n = 1.60(±0.05) and α = 4.0(±0.3) µg cm−2 keV−n; Algerset al. n = 1.71(±0.05) and α = 2.81(±0.20) µg cm−2 keV−n; and Ghent-WSU,n = 1.90(±0.04) and α = 1.36(±0.04) µg cm−2 keV−n.

L+ (nm) Lp-Ps (nm) Lo-Ps (nm) fPs fpu

Common values 5.18± 0.20 12.68± 0.09 8.67± 0.08 40.07± 0.03 0.47± 0.08

Algers et al. 3.46± 0.01 8.38± 0.09 5.82± 0.03 39.87± 0.03 0.47± 0.08

Ghent-WSU 1.78± 0.22 3.63± 0.10 2.66± 0.09 39.24± 0.03 0.47± 0.08

From Table 6.1 it is clear that the values for the diffusion lengths are dependenton the values of α and n used to describe the implantation profile. The values forthe total bulk Ps formation (fPs) and the fraction of surface positrons that pick upan electron (fpu) do not depend on the different values for α and n used to describethe implantation profile. From this it can be concluded that for determining thediffusion lengths, special care has to be taken into account for describing theimplantation profile.

The experimental values for Lp-Ps and Lo-Ps are within the experimental errornot the same but they are comparable. This can be explained within a model asproposed by Van Petegem et al. [161]:

The diffusion length of positrons is related with the diffusion coefficient (D+)by L+ =

√D+τ∗ (see equation 3.9 on page 48), where τ∗ = (λ+ κ)−1. Here λ is

the decay constant of the free particle and κ is the trapping rate of the particleinto some sink. As in the bulk all positronium is efficiently trapped into the freevolume sites, κ λ for both, o-Ps and p-Ps. Therefore, Lo-Ps ≈ Lp-Ps ≈

√DPs/κ.

However, one has to be aware that this experiments are performed at verylow implantation energies (from 0.1 to to 1.2 keV) and epithermal positrons mayreach the surface, in which case it is the diffusion of a positron that is not inthermal equilibrium. This may enhance the emission for very low energies, and

106


thus shorten the apparent diffusion length L+.

It is worthwhile to emphasize here that when the PMMA sample is perpendi-cular to the beam axis (Figure 6.1(b)), when increasing the positron implantationenergy, the intensity of the emitted p-Ps (i.e., the component 3 in Figure 6.4(b))can not be clearly distinguished from the intensity of the bulk p-Ps (component 4in the same figure) because they are at the same position. Thus, this fitting wasnot performed.

Figures 6.9 and 6.10 show the intensities of the fly away p-Ps and the fly awayo-Ps as function of the positron implantation energy for the Kapton sample as wellas the results from the simultaneous fitting of the Equations (6.3).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

2

4

6

8

10

12


Pos

itron

ium

inte

nsity

(%

)

Figure 6.9: Ps emission from the Kapton surface when the sampleis at 45 with respect to the beam axis as a function of the implan-tation energy. In the figure are shown the intensity of the p-Ps(×), the intensity of the o-Ps (•) and the fit (solid lines) of Eqs.(6.3) to determine the positron diffusion length (thermal (and ep-ithermal)) L+. The fitting in the figure was performed using thevalues for the parameters α and n in the Makhovian equationfound by Ghent-WSU in the previous Chapter (n = 1.90(±0.04)and α = 1.36(±0.04) µg cm−2 keV−n).

From both figures, one can see that the intensities decrease smoothly withincreasing the implantation energy and at about 850 eV the intensity of the p-Pspeak (i.e. the component 3 in Fig. 6.5) is too low to be clearly distinguished from

107


the broad central Gaussian (component 2 in Fig. 6.5).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

2

4

6

8

10

12

14

16

18

20


Pos

itron

ium

inte

nsity

(%

)

Figure 6.10: Ps emission from the Kapton surface when the sam-ple is perpendicular with respect to the beam axis as a functionof the implantation energy. In the figure are shown the intensityof the p-Ps (×), o-Ps (•) and the fit (solid lines) of Eqs. (6.3)to determine the positron diffusion length (thermal (and epither-mal)) L+. The fitting in the figure was performed using the val-ues for the parameters α and n in the Makhovian equation foundby Ghent-WSU in the previous Chapter (n = 1.90(±0.04) andα = 1.36(±0.04) µg cm−2 keV−n).

From the fitting procedure, the obtained diffusion lengths (L+) for thermal(and epithermal) positrons, their respective diffusion coefficient D+ (see Equation(3.9) on page 48) and the fraction of positrons that capture an electron from thesurface and are emitted as Ps (fpu) are listed in Table 6.2.

108


Table 6.2: Comparison of the values obtained from the fitting of the experimentalintensities (at 45 and perpendicular with respect to the beam axis) of the fly-away p-Ps and the fly-away o-Ps for the Kapton sample by using the differentvalues for the parameters α and n that characterize the well-known power-lawequation (z1/2(E) = α

ρEn) to determine the positron diffusion length (thermal

(and epithermal)) L+, their respective diffusion coefficient D+ (see Equation (3.9)on page 48) and the fraction of positrons that capture an electron from the surfaceand are emitted as Ps (fpu). The used parameters are: Common values n =1.60(±0.05) and α = 4.0(±0.3) µg cm−2 keV−n; Algers et al. n = 1.71(±0.05)and α = 2.81(±0.20) µg cm−2 keV−n; and Ghent-WSU, n = 1.90(±0.04) andα = 1.36(±0.04) µg cm−2 keV−n. For the calculation of the diffusion coefficientD+, τ∗ was considered to be 382 ps [133].

L+ (nm) D+ (×10−4 cm2/s) fpu

Sample at 45

Common values 8.22± 0.41 17.69± 1.76 0.158± 0.003

Algers et al. 5.51± 0.28 7.95± 0.81 0.155± 0.003

Ghent-WSU 2.41± 0.12 1.52± 0.15 0.151± 0.003

Sample perpendicular

Common values 8.17± 1.06 17.47± 4.53 0.250± 0.012

Algers et al. 5.43± 0.71 7.72± 2.02 0.247± 0.012

Ghent-WSU 2.33± 0.31 1.42± 0.38 0.241± 0.011

The differences of the values used for the parameters α and n for the dataanalysis are clear when inspecting the Table 6.2. In the table, one can see that thevalues for the parameters α and n have a big influence on L+ (or D+) dependingon the chosen model.

The only experiment found in the literature with the purpose of investigatingthe positron mobility in Kapton was done by Brusa et al. [164]. They have analyzedthe drift of positrons under the action of an electric field. These experiments wereperformed in the energy range between 0.85 keV and 2.5 keV. Our experimentswere performed with positrons with an energy below 0.85 keV where positroniumemission is possible.

This positron mobility experiment in Kapton resulted in a very small diffusioncoefficient: D+ = 2.5× 10−5 cm2/s with an error of 0.1× 10−5 – 3× 10−5 cm2/s.This value is lower and differs in 1 order of magnitude in comparison to our valueslisted in Table 6.2.

Both experiments (Ghent and Brusa’s) have different strengths and weaknessesthat can be summarized as follows:

109


1. Experiments performed in Ghent: The positron mobility is obtained bymeans of a non-conventional method which is the positronium emission.However, in Table 6.2, notice that once the values for the parameters αand n that describe the implantation profile have been chosen, the analy-sis of two independent experiments (sample at 45 and perpendicular withrespect to the beam axis) produce the same results for L+ (within the er-rors) which gives us an indication of the veracity of the followed procedure.Nevertheless:

a. our values can also be influenced by epithermal positrons (at least atthe very low energy values E <∼300 eV).

b. If the model proposed by Coleman et al. [141] is valid (see a reviewin Section 3.4), there might be an electric field drifting the positronstowards the sample surface. This means that the positron mobility µ+

is affected by this electric field. As a consequence, the measured valueof the positron diffusion coefficient can be influenced by the chargingeffects.

2. Experiments performed by Brusa et al.: The positron mobility is obtained ina standard manner. By employing high implantation energies they measurethermalized positrons and they can also avoid the influence of the chargingeffects. However:

a. The authors measured the same Kapton sample type, from the samemanufacturer than for the experiments in Ghent. However, they used asample density of (ρ = 1.7 g cm−3). The experimental value measuredin Ghent for the sample density was 1.4 g cm−3 which is the same asthe value given by the manufacturer (1.42 g cm−3).

b. The sample was coated on both sides with 0.02 µm-thick aluminium.As it has been proven in the previous Chapter, as the values for theparameters α and n differ for metals and polymers the positron implan-tation profiles should be different for a bi-layered material that containsmetal-polymer.

c. They used the values for the parameters α and n that best fitted theirdata. The obtained values were α = 5.4 µg cm−2 keV−n and n = 1.7(without an error estimation). The parameter α differs from the valuesfound for polymers by Algers et al. and in Ghent-WSU (see the previouschapter). The value found for α by Brusa et al. is even too high incomparison with the standard values.

d. The estimation of the experimental error is not clear.

Thus, studying the positronium emission by means of DBAR is a new tech-nique that may give clear and truthful results without the need of a complicated

110

6.4 Summary and Conclusions

setup.More research should be done in order to further clarify the effects of charg-ing and epithermal positrons. However, it appears as a promising technique thatin principle might be accessible for many positron laboratories.

6.4 Summary and Conclusions

The emission of Positronium as a function of the positron implantation energyfrom a 310 nm-thick PMMA film and from a Kapton surface has been studied bymeans of Doppler broadening, blue-shift spectroscopy and Compton-to-peak ratioanalysis.

In the analysis the detected photo-peak has been approximated by a sum ofGaussians.

The narrow component attributed to the p-Ps emission is centered with respectto the 511-keV-line when the sample and the detector are perpendicular to thepositron beam axis. However, it is shifted towards high energy when the sampleis at 45 with respect to the beam axis. This fact has been explained as thelinear momentum of the Ps emitted at the surface of the sample is oriented mainlytowards the detector causing a blue shift of the p-Ps.

From a detailed analysis of the experimental results, the thermal (and epither-mal) positron diffusion lengths, the fraction of positrons that pick up an electronfrom the surface and form positronium, and in the case of the PMMA also thefraction fPs of positronium formed in the bulk can be obtained.

In the case of Kapton, the analysis of two independent experiments suggestthat the positron diffusion coefficient might not be as low as the one proposed byBrusa et al. [164].

Special care has to be taken into account in selecting the correct values forthe parameters α and n that describe the implantation profile when analyzing theexperimental data as standard values might lead to wrong results.

The experiments performed on the Kapton sample also suggest that once thevalues for the parameters α and n that describe the implantation profile havebeen chosen, the analysis for two independent experiments (sample at 45 andperpendicular) produce the same results for L+ (within the errors) which gives usan indication of the veracity of the followed procedure.

Both experiments (Ghent and Brusa’s) have different strengths and weaknesses,however the new technique may give clear and truthful results without the needof a complicated setup. This, therefore, opens a new field of possible experimentsthat might be accessible for many positron laboratories.

111


112

ABasic derivation for the

momentum measurements

In this appendix is presented the derivation necessary to obtain the energy of theannihilation γ−rays and also the angular correlation of the two γ−ray decay, whichare used in Subsection 2.3 and in subsequent subsections.

A.1 Energy of the annihilation γ−rays

The kinetic energy of the positron-electron pair before annihilation causes a Dopplershift of the two γ−ray energies (see Fig. 2.6). the doppler shift is ∆ν

ν = νL

c (withthe longitudinal center-of-mass velocity νL = PL

2me). For energy E = mec

2 thedevelopment is as follows:

As the energy of a photon is proportional to its frequency

∆EE

∝ ∆νν

⇒ ∆E = ±(∆νν

)E

⇒ ∆E = ±(νL

c

)E, but νL =

PL

2me, then:

∆E = ±

(PL

2mec

)E, also E = mec

2, so that

∆E = ±

(PL

2mec

)HHmec

2

113

A. BASIC DERIVATION FOR THE MOMENTUMMEASUREMENTS

∴ ⇒ ∆E = ±PL

2c (A.1)

A.2 angular correlation of the two γ−rays decay

In the center-of-mass system the energy of each annihilation γ−ray is exactlymec

2 = 511 keV, and the emission angle between the directions of the two photonsis 180. In the laboratory system, this is true only if the positron-electron pairhas no kinetic energy. When slowing down in matter, most positrons thermalizebefore annihilation and the momentum of the center of mass motion of an positron-electron pair is not along the line of emission of the two photons (see Fig. 2.6).

For this development, it is necessary to use relativistic kinematics. The relevantformulae will be quoted without derivation1. In particular, the total energy, E, ofa particle having a rest mass, m0 and moving with a velocity v is given by

E =mc2(

1− v2

c2

) 12

(A.2)

It is also known in special relativity that the energy and momentum are relatedby:

E2 = p2c2 +m20c

4 (A.3)

Since the velocity of a photon is c and its energy E = hν = hcλ is finite, we see

from (A.2) that we must take the mass of a photon to be zero, in which case weobserve from (A.3) that the magnitude of its momentum is

p =E

c=h

λ(A.4)

Now, we start with the derivation. For clarity, see Fig. 2.6. From the energyconservation:

E1 + E2 = 2m0c2, but (A.5)

Pc = E1 + E2, so we have:

PLc = E1 − E2

≈1︷︸︸︷cos(θ) ⇒ PLc = E1 − E2, and therefore:

1For a discussion of the theory of special relativity, see for example the text by Taylor, E. F.and Wheeler, J.A. (1966). Spacetime physics. Freeman, San Francisco.

114

A.2 angular correlation of the two γ−rays decay

E1 = E2 + PLc, and also: E2 = E1 − PLc. (A.6)

Now replacing Eq. (A.6) in (A.5):

2E2 + PLc = 2m0c2 ⇒ E2 = m0c

2 − PLc

2, and also: (A.7)

E1 = m0c2 +

PLc

2.

Now we make E = pc (from Eq. (A.4)):

⇒ P2 = m0c−PL

2, and also: P1 = m0c+

PL

2.

On the other hand, by looking at the transverse components:

PT c = E2

≈θ︷︸︸︷sin(θ) +HHE10 ⇒ PT c = θE2, and replacing (A.7):

PT c = θ(m0c

2 − PLc

2

), but

PLc

2 m0c

2,

⇒ PLc

2≈ 0, so we have: PTc = θm0c

2

∴ ⇒ θ =PT

m0c(A.8)

115

A. BASIC DERIVATION FOR THE MOMENTUMMEASUREMENTS

116

BPositronium fraction from the

Compton-to-peak ratio analysis

of the annihilation spectrum

Ps may be detected by studying the energy spectrum of its annihilation photonsbecause the 3S1 state decays into 3γ’s with a continuous energy distribution in therange 0 to mc2 = 511 keV, whereas positrons which annihilate with an electron ofa solid decay principally by 1S0 overlap via 2γ’s with energy mc2.

Assume that a fraction f of the positrons which annihilate in the target regionform Ps. Let Tf and Pf be the “total” and “photopeak” detector counting ratesfor a given number of positrons annihilating per second. It is evident that

Tf = fTk + (fk− f)T0 (B.1) , and Pf = fPk + (fk−f)P0 (B.2)

where the subscripts 1 and 0 refer to f = 1 and f = 0 respectively. To eliminateany dependence on the positron beam strength, we can form the ratio R = T−P

P(well-known as the Compton to peak ratio), so we have:

Rf =Tf − Pf

Pf, now replacing (B.1) and (B.2):

RfPf = fTk + (fk − f)T0 − fPk − (fk − f)P0

substituting T = RP + P (from the ratio R):

RfPf = f(Pk +RkPk) + (fk − f)(P0 +R0P0)− fPk − (fk − f)P0

RfPf = fPk + fRkPk +HHHfkP0 −HHfP0 + fkR0P0 − fR0P0 −fPk −HHHfkP0 +HHfP0

RfPf = fRkPk + fkR0P0 − fR0P0

117

B. POSITRONIUM FRACTION FROM THE COMPTON-TO-PEAKRATIO ANALYSIS OF THE ANNIHILATION SPECTRUM

Replacing (B.2):

Rf (fPk + (fk − f)P0) = fRkPk + fkR0P0 − fR0P0

fPkRf + fkP0Rf︸︷︷︸−fP0Rf = fRkPk + fkR0P0︸︷︷︸−fR0P0

⇒ fkP0Rf − fkR0P0 = fP0Rf − fPkRf + fRkPk − fR0P0

fk(P0Rf −R0P0) = f(P0Rf − PkRf +RkPk −R0P0)

f =fk(P0Rf −R0P0)

P0Rf︸︷︷︸−PkRf +RkPk −R0P0︸︷︷︸f =

fk(P0Rf −R0P0)P0(Rf −R0) + Pk(Rk −Rf )

f = fk

[P0(Rf −R0) + Pk(Rk −Rf )

(P0Rf −R0P0)

]−1

f = fk

[P0(Rf −R0) + Pk(Rk −Rf )

P0(Rf −R0)

]−1

∴ ⇒ f = fk

[1 +

Pk

P0

(Rk −Rf )(Rf −R0)

]−1

(B.3)

The equation (B.3) was first introduced by Mills (A. P. Mills, Jr., Phys. Rev.Lett. 41 (1978) 1828-1831) who showed that the 3 γ annihilation can be de-termined using the energy spectrum and calibration values for the counts in the511-keV peak (P) and R for two samples with 0 and 100% yield of 3γ annihilation.The formula is easily modified if the second calibration sample is with a knownnonzero 3γ annihilation yield (fk) (e.g. in Kapton). In Kapton Ps is not formedin the bulk.o−Ps which escapes from the sample will annihilate in the vacuum via 3γ anni-hilation, while the e+ annihilation is mostly via 2γ annihilation (the 3γ positronannihilation cross section is only is 1/372 of the one of 2γ positron annihilationand can be neglected). If there is Ps formation there will be 3γ o−Ps annihilationwith a fraction dependent on the o−Ps lifetime. The relative ratio of the 3γ/2γannihilation is known as Compton-to-peak ratio (sometimes this parameter is alsodescribed by its inverse known as peak-to-valley ratio).

118

Bibliography

[1] P. A. M. Dirac. A theory of electrons and protons. Proc. Roy. Soc. Lond.A, 126:360, 1930.

[2] P. A. M. Dirac. On the annihilation of electrons and protons. Proc. Camb.Phil. Soc., 26:361, 1930.

[3] C. D. Anderson. The apparent existence of easily deflectable positives. Sci-ence, 76:238, 1932.

[4] C. D. Anderson. Energies of cosmic-ray particles. Phys. Rev., 41:405, 1932.

[5] C. D. Anderson. Cosmic-ray positive and negative electrons. Phys. Rev.,44:406, 1933.

[6] C. D. Anderson. The positive electron. Phys. Rev., 43:491, 1933.

[7] C. D. Anderson and Neddermeyer S.H. Positrons from gamma-rays. Phys.Rev., 43:1034, 1933.

[8] P.M.S Blackett and G.P.S. Occhialini. Some photographs of the tracks ofpenetrating radiation. Proc. Roy. Soc. Lond. A, 139:699, 1933.

[9] I. Curie and F. Joliot. Electrons de materialisation et de transmutation. J.Phys., 4:494, 1933.

[10] J.A. Rich. Experimental evidence for the three-photon annihilation of anelectron-positron pair. PhD thesis, Yale University, 1950.

[11] J.A. Rich. Experimental evidence for the three-photon annihilation of anelectron-positron pair. Phys. Rev., 81:140, 1951.

[12] A. Øre and J.L. Powell. Three-photon annihilation of an electron-positronpair. Phys. Rev., 75(11):1696, 1949.

119

BIBLIOGRAPHY

[13] M. Charlton. Positron physics. In Cambridge monographs on atomic, molec-ular and chemical physics, volume 173, Cambridge, 2001. Cambridge Uni-versity Press.

[14] W.R. Johnson, D.J. Buss, and CO. Carroll. Single-quantum annihilation ofpositrons. Phys. Rev., 135, 1964.

[15] J.C. Palathingal, P. Asoka-Kumar, G. Lynn, and X.Y. Wu. Nuclear-charge and positron-energy dependence of the single-quantum annihilationof positrons. 1995.

[16] O. Klempere. On the annihilation radiation of the positron. Proc. Camb.Phil. Soc., 30:347, 1934.

[17] T. Chang, H. Tang, and L. Yaoqing. The gamma ray energy spectrumin orthopositronium 3 gamma decay. In P.C. Jain, P.C. Singru, and K.P.Gopinathan, editors, Positron Annihilation, page 212, Singapore, 1985.World Scientific.

[18] P. Kubica and A. T. Stewart. Thermalization of positrons and positronium.Phys. Rev. Lett., 34:852, 1975.

[19] B. Bergersen and E. Pajanne. Motion of positrons in metals. Appl. Phys.,4:25, 1974.

[20] J.K. Basson. Direct quantitative observation of the three-photon annihilationof a positron-negatron pair. Phys. Rev., 96:691, 1954.

[21] S. Mohorivicic. Moglichkeit neuer elemente und ihre bedeutung fur die as-trophysic. Astron. Nachr., 253:93, 1934.

[22] J.M. Ruark. Positronium. Phys. Rev., 68:278, 1945.

[23] M. Deutsch. Evidence for the formation of positronium in gases. Phys. Rev.,82:455, 1951.

[24] M. Deutsch. Three-quantum decay of positronium. Phys. Rev., 83:866, 1951.

[25] M. Deutsch and E. Dulit. Short range interaction of electrons and finestructure of positronium. Phys. Rev., 84:601, 1951.

[26] J. Pirenne. Le champ propre et l’interaction des particules de dirac suivantl’´lectrodynamique quantique. Arch. Sci. Phys. Nat., 28:233, 1946.

[27] C.N. Yang. Selection rules for the dematerialization of a particle into twophotons. Phys. Rev., 77:242, 1950.

120

BIBLIOGRAPHY

[28] S. Adachi, M. Chiba, T. Hirose, S. Nagyama, Y. Nakamitsu, T. Sato, andT. Yamada. Measurement of e+e− annihilation at rest into four γ rays.Phys. Rev. A, 65:2634, 1990.

[29] T. Matsumoto, M. Chiba, R. Hamatsu, T. Hirose, J. Yang, and J. Yu. Mea-surement of five-photon decay in orthopositronium. Phys. Rev. A, 54:1947,1996.

[30] J.A. Wheeler. Ann. NY Acad. Sci., 48:219, 1946.

[31] Michael D. Harpen. Positronium: Review of symmetry, conserved quantitiesand decay for the radiological physicist. Med. Phys., 31:57, 2004.

[32] G.S. Adkins. Radiative corrections to positronium decay. Ann. Phys., 146:78,1983.

[33] G.S. Adkins. Analytic evaluation of an o(α) vertex correction to the decayrate of ortho-positronium. Phys. Rev. A, 27:530, 1983.

[34] J.S. Nico, D.W. Gidley, A. Rich, and P.W. Zitzewitz. Precision-measurementof the orthopositronium decay-rate using the vacuum technique. Phys. Rev.Lett., 65:1344, 1990.

[35] S. Asai, S. Orito, and N. Shinohara. New measurement of the orthopositro-nium decay-rate. Phys. Lett. B, 357:475, 1995.

[36] J.A. Rich. Recent experimental advances in positronium research. Phys.Mod. Phys., 53:127, 1981.

[37] E.D. Theriot Jr., R.H. Beers, V.W. Hughes, and K.O.H Ziock. Precisionredetermination of the fine-structure interval of the ground state of positro-nium and a direct measurement of the decay rate of parapositronium. Phys.Rev. A, 2:707, 1970.

[38] A. H. Al-Ramadhan and D. W. Gidley. New precision-measurement of thedecay-rate of singlet positronium. Phys. Rev. Lett., 72:1632, 1994.

[39] I.J. Rosenberg, A.H. Weiss, and K.F. Canter. Positronium emission frommetal-surfaces. J. Vac. Sci. Technol., 17:253, 1980.

[40] R.M. Nieminen and J. Oliva. Theory of positronium formation and positronemission at metal surfaces. Phys. Rev. B, 22(5):2226, 1980.

[41] A. Øre. Annihilation of positrons in gases. Naturvitenskapelig Rekke, 9:1,1949.

121

BIBLIOGRAPHY

[42] O.E. Mogensen. Spur reaction model of positronium formation. J. Chem.Phys., 60:998, 1974.

[43] O.E. Mogensen. Positronium formation in condensed matter an high-densitygases. In P.G. Coleman, S.C. Sharma, and L.M. Diana, editors, PositronAnnihilation, page 763. North-Holland, 1982.

[44] O.E. Mogensen. Positron annihilation in chemistry. Berlin, Heidelberg, 1995.Springer-Verlag.

[45] M. Eldrup, A. Vehanen, P.J. Schultz, and K.G. Lynn. Positronium formationand diffusion in a molecular-solid studied with variable-energy positrons.Phys. Rev. Lett., 51:2007, 1983.

[46] M. Eldrup, A. Vehanen, P.J. Schultz, and K.G. Lynn. Positronium forma-tion and diffusion in crystalline and amorphous ice using a variable-energypositron beam. Phys. Rev. B, 32(11):7048, 1983.

[47] S.V. Stepanov and V.M. Byakov. Electric field effect on positronium forma-tion in liquids. J. Chem. Phys., 116 (14):6178, 2002.

[48] S.V. Stepanov and V.M. Byakov. Physical and radiation chemistry of thepositron and positronium. In Y.C. Jean, P.E. Mallone, and D.M. Schrader,editors, Principles and Applications of Positron and Positronium Chemistry,pages 117–149, Singapore, 2003. World Scientific.

[49] S.V. Stepanov, V.M. Byakov, and Y. Kobayashi. Positronium formation inmolecular media: The effect of the external electric field. Phys. Rev. B,72:054205, 2005.

[50] R.L. Garwin. Thermalization of positrons in metals. Phys. Rev., 91:1571,1953.

[51] M. Dresden. Speculations on the behavior of positrons in superconductors.Phys. Rev., 93:1413, 1954.

[52] R.A. Ferrell. Ortho-parapositronium quenching by paramagnetic moleculesand ions. Phys. Rev., 110:1589, 1958.

[53] M. Heinberg and L.A. Page. Annihilation of positrons in gases. Phys. Rev.,107:1589, 1957.

[54] M. Kakimoto, T. Hyodo, and T.B. Chang. Conversion of ortho-positroniumin low-density oxygen gas. J. Phys. B, 23:589, 1990.

122

BIBLIOGRAPHY

[55] Bartel Van Waeyenberge. The interaction of Positronium with surfaces:A study with the Age-momentum Correlation Technique. Doctoral Thesis,Ghent University, 2002.

[56] W. Brandt and R. Paulin. Positron implantation-profile effects in solids.Phys. Rev. B, 15:2511, 1977.

[57] R. Paulin. Positron and positronium dynamics in solids. In R.R. Hasigutiand K. Fujiwara, editors, Positron Annihilation, page 601, Tokyo, 1979.Japan Institute Met.

[58] J. De Baerdemaeker and C. Dauwe. Development and application of theGhent pulsed positron beam. Appl. Surf. Sci., 194, 2002.

[59] S. DeBenedetti, F.K. McGowan, and J.E. Francis Jr. Self-delayed conci-dences with scintillation counters. Phys. Rev., 74:1404, 1948.

[60] I. Pomeranckuk. Lifetime of slow positrons. Zh. Ekspt. Teor. Fir., 19:183,1949.

[61] R.E. Bell and R.L. Graham. Time distribution of positrons annihilation inliquids and solids. Phys. Rev., 90:644, 1953.

[62] G.F. Knoll. Radiation detection and measurement. John Wiley & Sons,1979.

[63] H. Stoll, P. Castellaz, S. Coch, J. Major, H. Schneider, A. Seeger, andA. Siegle. Age-momentum-correlation (amoc) experiments by means of anmev positron beam. Mater. Sci. Forum, 255, 1997.

[64] S.J. Tao. Positronium annihilation in molecular substances. J. Chem. Phys.,56:5499, 1972.

[65] M. Eldrup, D. Lightbody, and J. N. Sherwood. The temperature dependenceof positron lifetimes in solid pivalic acid. J. Chem. Phys., 63:51, 1981.

[66] H. Nakanishi, S.W. Wang, and Y. C. Yean. Positron annihilation studies influids. World Scientific, Singapore, 1988.

[67] B. Jasinska, A.E. Koziol, and T. Goworek. Ortho-positronium lifetimesin nonspherical voids. Journal of Radioanalytical and Nuclear Chemistry,210(2):617, 1996.

[68] T. Goworek, K. Ciesielski, Jasinska B., and J. Wawryszczuk. Positroniumin large voids. silicagel. Chemical Physics Letters, 272:91, 1997.

123

BIBLIOGRAPHY

[69] T. Goworek, K. Ciesielski, Jasinska B., and J. Wawryszczuk. Lifetimes ofo-Ps in the pores of silica gel. Materials Science Forum (ICPA 11), 255-257:296, 1997.

[70] T. Goworek, K. Ciesielski, Jasinska B., and J. Wawryszczuk. Positroniumstates in the pores of silica gel. Chemical Physics, 230:305, 1998.

[71] T. Goworek, K. Ciesielski, Jasinska B., and J. Wawryszczuk. Mesoporecharacterization by PALS. Radiation Physics and Chemistry, 68:331, 2003.

[72] M. Sniegocka, B. Jasinska, J. Wawryszczuk, Zaleski R., A. Dery lo-Marczewska, and I. Skrzypek. Testing the extended Tao-Eldrup model. silicagels produced with polymer template. Acta Phys. Pol. A, 107:868, 2005.

[73] D.W. Gidley, H-G Peng, and R.S. Vallery. Positron annihilation as a methodto characterize porous materials. Annu. Rev. Mater. Res, 36:49, 2006.

[74] D.W. Gidley, W.E. Frieze, T.L. Dull, A.F. Yee, E.T. Ryan, and H.-M. Ho.Positronium annihilation in mesoporous thin films. Phys. Rev. B, 60(8),1999.

[75] T.L. Dull, W.E. Frieze, D.W. Gidley, J.N. Sun, and A.F. Yee. Determina-tion of pore size in mesoporous thin films from the annihilation lifetime ofpositronium. J. Phys. Chem. B, 105(20):4657, 2001.

[76] K. Ito, H. Nakanishi, and Y. Ujihira. Extension of the equation for theannihilation lifetime of ortho-positronium at a cavity larger than 1 nm inradius. J. Phys. Chem. B, 103(21):4555, 1999.

[77] J.M.W. Dumond, D. A. Lind, and B.B. Watson. Precision measurementof the wavelength and spectral profile of the annihilation radiation from64Cu with the two-meter focusing curved crystal spectrometer. Phys. Rev.,75:1226, 1949.

[78] G. Murray. Doppler broadening of annihilation radiation. Phys. Lett. B,24:268, 1967.

[79] F.H.H. Hsu and C.S. Wu. Correlation between decay lifetime and angulardistribution of positron annihilation in the plastic scintillator NATON. Phys.Rev. Lett., 18:889, 1967.

[80] I.K. MacKenzie, J.A. Eady, and A.A. Gingerich. The interaction betweenpositrons and dislocations in copper and in an aluminum alloy. Phys. Lett.A, 33:279, 1970.

124

BIBLIOGRAPHY

[81] K.G. Lynn and A.N. Goland. Observation of high momentum tails ofpositron-annihilation lineshapes. Sol. Stat. Comm., 18:1549, 1976.

[82] S. Szpala, P. Asoka-Kumar, B. Nielsen, J.P. Peng, S. Hayakawa, and K.G.Lynn. Defect identification using the core-electron contribution in Doppler-broadening spectroscopy of positron-annihilation radiation. Phys. Rev. B,54:4722, 1996.

[83] U. Myler, R.D. Goldberg, A.P. Knights, D.W. Lawther, and P.J. Simpson.Chemical information in positron annihilation spectra. Appl. Phys. Lett.,69:3333, 1996.

[84] J. Dryzek and C.A. Quarles. A spectrometer for the measurement of theDoppler broadening of the annihilation line with efficient reduction of back-ground. Nucl. Instr. & Meth. A, 378:337, 1996.

[85] U. Myler and P.J. Simpson. Survey of elemental specificity in positron an-nihilation peak shapes. Phys. Rev. B, 56:14303, 1997.

[86] H. Kauppinen, L. Baroux, K. Saarinen, C. Corbel, and Hautojarvi. Identi-fication of cadnium vacancy complexes in CdTe(In), CdTe(Cl) and CdTe(I)by positron annihilation with core electrons. J. Phys.: Condens. Matter,9:5495, 1997.

[87] P.M.G. Nambissan and P. Sen. Probing the site-specific annihilation ofpositrons in a nanocomposite medium by 2-detector Doppler broadeningmeasurements. Phys. Lett. A, 272:412, 2000.

[88] K.G. Lynn, J.E. Dickman, W.L. Brown, R.A. Boie, L.C. Feldman, J.D.Gabbe, M.F. Robbins, E Bonderup, and J. Golovchenko. Positron-annihilation momentum profiles in aluminum: core contribution and theindependent-particle model. Phys. Rev. Lett., 38:241, 1977.

[89] J.R. MacDonald, K.G. Lynn, R.A. Boie, and M.F. Robbins. A two-dimensional Doppler broadened technique in positron annihilation. Nucl.Instr. & Meth., 153:189, 1978.

[90] D.T. Britton, W. Junker, and P. Sperr. A high resolution Doppler-broadening spectrometer. Mat. Sci. For., 105-110:1845, 1992.

[91] M. Alatalo, H. Kauppinen, K. Saarinen, M.J. Puska, J. Makinen, P. Hau-tojarvi, and R.M. Nieminen. Identification of vacancy defects in compoundsemiconductors by core-electron annihilation application to InP. Phys. Rev.B, 51:4176, 1995.

125

BIBLIOGRAPHY

[92] P. Asoka-Kumar, M. Alatalo, V.J. Ghosh, A.C. Kruseman, B. Nielsen, andK.G. Lynn. Increased elemental specificity of positron annihilation spectra.Phys. Rev. Lett., 77:2097, 1996.

[93] P. Asoka-Kumar. Chemical identity of atoms using core electron annihila-tions. Mat. Sci. For., 255-257:166, 1997.

[94] Steven Van Petegem. Positron annihilation study of nanocrystalline mate-rials: a synergy of theory, experiment and computer simulations. DoctoralThesis, Ghent University, 2003.

[95] R. Beringer and C.G. Montgomery. Angular distribution of positron anni-hilation radiation. Phys. Rev., 61:222, 1942.

[96] S. DeBenedetti, C.E. Cowan, and W.R. Konneker. Angular distribution ofannihilation radiation. Phys. Rev., 76:440, 1949.

[97] R Krause-Rehberg and H.S. Leipner. Positron annihilation in semiconduc-tors. Springer Verlag, Heidelberg, 1999.

[98] H. Nakanishi and Y. C. Yean. Positrons and positronium in liquids. In D. M.Schrader and Y. C. Yean, editors, Positron and Positronium Chemistry,Studies in Physical and Theoretical Chemistry No.57, page 159. Elsevier-Amsterdam, 1988.

[99] O.E. Mogensen. Positron Annihilation in Chemistry. Springer-Verlag, 1995.

[100] G. Dlubek, H. M. Fretwell, and M. A. Alam. Positron/positronium annihila-tion as probe for the chemical environment of free volume holes in polymers.Macromolecules, 33:187, 2000.

[101] G. Dlubek and M. A. Alam. Studies of positron lifetime and Doppler-broadened annihilation radiation of polypropylene-polystyrene alloys. Poly-mer, 43:4025, 2002.

[102] C.V. Briscoe, S.-I. Choi, and A.T. Stewart. Zero-point bubbles in liquids.Phys. Rev. Lett., 20:493, 1968.

[103] P.J. Schultz and K.G. Lynn. Interaction of positron beams with surfaces,thin films and interfaces. Rev. Mod. Phys., 60:701, 1988.

[104] L. Madansky and F. Rasetti. An attempt to detect thermal energy positrons.Phys. Rev., 79:397, 1950.

[105] W. Cherry. PhD thesis, Princeton University, N. J., 1958.

126

BIBLIOGRAPHY

[106] D.E. Groce, D.G. Costello, J.Wm. McGowan, and D.F. Herring. Time-of-flight observation of low-energy positrons. Bulletin of the american physicalsociety, 13, 1972.

[107] B.Y. Tong. Negative work function of thermal positrons in metals. Phys.Rev. B, 5:1436, 1972.

[108] C.H. Hodges and M.J. Stott. Work functions for positrons in metals. Phys.Rev. B, 7:73, 1973.

[109] J.F. Dale, L. D. Hulett, and S. Pendyala. Low energy positrons from metalsurfaces. Surface and Interface Analysis, 2(6):199, 1980.

[110] A. Vehanen, K.G. Lynn, P.J. Schultz, and M. Eldrup. Improved slow-positron yield using a single-crystal tungsten moderator. Appl. Phys. A,32(3):163, 1983.

[111] A.P. Mills Jr. Positron and positronium interactions at surfaces. In P.G.Coleman, S.C. Sharma, and L.M. Diana, editors, Positron Annihilation, page121. North-Holland, 1982.

[112] A.H. Weiss and P.G. Coleman. Surface science with positrons. In P. Coleman,editor, Positron beams and their applications, page 129, Singapore, 2000.World scientific Publ. Co.

[113] I.K. MacKenzie, C. W. Shulte, T. Jackman, and J. L. Campbell. Positrontransmission and scattering measurements using superposition of annihila-tion line shapes: Backscatter coefficients. Phys. Rev. A, 7:135, 1973.

[114] V.A. Kuzminikh and S.A. Vorobiev. Backscattering of beta-particles fromthick targets. Nucl. Instr. & Meth., 167(3):483, 1979.

[115] P.U. Arifov, A.R. Grupper, and H. Alimkulov. Coefficients of positron massabsorption and backscattering. In P.G. Coleman, S.C. Sharma, and L.M.Diana, editors, Positron Annihilation, page 699. North-Holland, 1982.

[116] J.S. Makinen, S. Palko, J. Martikainen, and P. Hautojarvi. Positronbackscattering probabilities from solid surfaces at 2-30 kev. J. Phys. Cond.Mat., 4:L503, 1992.

[117] G.R. Massoumi, W.N. Lennard, Peter J. Schultz, A.B. Walker, and Kjeld O.Jensen. Electron and positron backscattering in the medium-energy range.Phys. Rev. B, 47:11007, 1993.

[118] S. Valkealahti and R.M. Nieminen. Monte-carlo calculations of kev electronand positron slowing down in solids. Appl. Phys. A, 32:95, 1983.

127

BIBLIOGRAPHY

[119] S. Valkealahti and R.M. Nieminen. Monte-carlo calculations of kev electronand positron slowing down in solids. ii. Appl. Phys. A, 35:51, 1984.

[120] E. Soininen, J. Makinen, D. Beyer, and P. Hautojarvi. High-temperaturepositron diffusion in Si, GaAs and Ge. Phys. Rev. B, 46:12394, 1992.

[121] K.O. Jensen and A.B. Walker. Monte-carlo simulation of the transport offast electrons and positrons in solids. Surf. Sci., 292:83, 1993.

[122] K.A. Ritley, K.G. Lynn, V.J. Ghosh, D.O. Welch, and M. McKeown. Low-energy contributions to positron implantation. J. Appl. Phys., 74:3479, 1993.

[123] A.F. Makhov. Penetration of electrons into solids, l– the intensity of anelectron beam, transverse path of electrons. Sov. Phys. Solid State, 2:1934,1961.

[124] M.J. Puska and R.M. Nieminen. Theory of positrons in solids and on solidsurfaces. Rev. Mod. Phys., 66:841, 1994.

[125] A. Vehanen, K Saarinen, P. Hautojarvi, and H. Huomo. Profiling multilayerstructures with monoenergetic positrons. Phys. Rev. B, 35(10):4606, 1987.

[126] R.M. Nieminen, J. Laakonen, P. Hautojarvi, and A. Vehanen. Temperaturedependence of positron trapping at voids in metals. Phys. Rev. B, 19:1397,1979.

[127] W. Brandt and N.R. Arista. Diffusion heating and cooling of positrons inconstrained media. Phys. Rev. A, 19:2317, 1979.

[128] P.J. Schultz and C.L. Snead. Positron spectroscopy for materials character-ization. Metallurgical Transactions A, 21(5):1121, 1990.

[129] A. Dupasquier. Positron Solid-State Physics, page 510. Proceedings of the in-ternational school of physics Enrico Fermi Course LXXXIII. North-Holland,Amsterdam, 1983.

[130] H.H. Jorch, K.G. Lynn, and T. McMullen. Positron diffusion in germanium.Phys. Rev. B, 30(1):93, 1984.

[131] A. Dupasquier and A. Zecca. Atomic and solid-state physics experimentswith slow-positron beams. Riv. Nuovo Cimento, 8(12):1, 1985.

[132] W Swiatkowski. Some remarks on positron/positronium diffusion models.Nucleonika, 48(3):141, 2003.

128

BIBLIOGRAPHY

[133] I. MacKenzie. Positron Solid State Physics, page 196. Proceedings of the in-ternational school of physics Enrico Fermi Course LXXXIII. North-Holland,Amsterdam, 1983.

[134] R.M. Nieminen and M. Manninen. Positrons in solids. In P. Hautojarvi,editor, Topics in Current Physics, volume No. 12, page 145, Berlin, 1979.Springer-Verlag.

[135] Y. Kong and K. G. Lynn. Transport model of thermal and epithermalpositrons in solids. i. Phys. Rev. B, 41:6179, 1990.

[136] Y. Kong and K. G. Lynn. Transport model of thermal and epithermalpositrons in solids. ii. Phys. Rev. B, 41:6185, 1990.

[137] D.T. Britton, P. C. Rice-Evans, and J.H. Evans. Epithermal effects inpositron depth profiling measurements. Philosophical Magazine Letters,57:3:165, 1988.

[138] A.P. Mills Jr. Positronium formation at surfaces. Phys. Rev. Lett., 41:1828,1978.

[139] K.G. Lynn and D.O. Welch. Slow positrons in metal single crystals. i. positro-nium formation at Ag(100), Ag(111), and Cu(111) surfaces. Phys. Rev. B,22:99, 1980.

[140] J. Lahtinen, A. Vehanen, H. Huomo, J. Makinen, P. Huttunen, K. Rytsola,M. Bentzon, and P. Hautojarvi. High-intensity variable-energy positronbeam for surface and near-surface studies. Nucl. Instr. & Meth. B, 17:73,1986.

[141] P. G. Coleman, S. Kuna, and R. Grynszpan. Slow positron implantationspectroscopy of insulators: Charging effects. Materials Science Forum (ICPA11), 255-257:668, 1997.

[142] Jeremie De Baerdemaeker. Defect characterization of sputtered, nitrided andion implanted materials using the new Ghent slow positron facility. DoctoralThesis, Ghent University, 2004.

[143] K.G. Lynn, B. Nielsen, and J. H. Quateman. Development and use of a thin-film transmission positron moderator. Appl. Phys. Lett., 47(3):239, 1985.

[144] Canberra: http://www.canberra.com.

[145] Dupont Electronics: http://www2.dupont.com.

129

BIBLIOGRAPHY

[146] D. E. Bornside, C. W. Macosko, and L. E. Scriven. On the modeling of spincoating. J. Imaging. Technol., 13:122, 1987.

[147] D. Meyerhofer. Characteristics of resist films produced by spinning. J. Appl.Phys., 49:3993, 1978.

[148] J. D. LeRoux and D. R. Paul. Preparation of composite membranes by aspin coating process. J. Membrane Sci., 74:233, 1992.

[149] L. B. R. Castro, A. T. Almeida, and D. F. S. Petri. The effect of water orsalt solution on thin hydrophobic films. Langmuir, 20:7610, 2004.

[150] D. Freitas Siqueira, D. W. Schubert, V. Erb, M. Stamm, and J. P. Amato. Interface thickness of the incompatible polymer system PS/PnBMA asmeasured by neutron reflectometry and ellipsometry. Colloid. Polymer Sci.,273:1041, 1995.

[151] D. F. S. Petri, G. Wenz, P. Schunk, and T. Schimmel. An improved methodfor the assembly of amino-terminated monolayers on SiO2 and the vapordeposition of gold layers. Langmuir, 15:4520, 1999.

[152] A. T. Almeida, M. C. Salvadori, and D. F. S. Petri. Enolase adsorption ontohydrophobic and hydrophilic solid substrates. Langmuir, 18:6914, 2002.

[153] J. Algers, P. Sperr, W. Egger, G. Kogel, and F. H. J. Maurer. Medianimplantation depth and implantation profile of 3-18 keV positrons in amor-phous polymers. Phys. Rev. B, 67:125404, 2003.

[154] P. G. Coleman, J. A. Baker, and N. B. Chilton. Experimental studies ofpositron stopping in matter: the binary sample method. J. Phys.: Condens.Matter, 5:8, 1993.

[155] Y. Ito, H. Abe, and Y. Tabata. Positron spur structure emulated by variableenergy slow positron beam. Journal de Physique IV, 3(Colloque 4) supl. JPII n9:131, 1993.

[156] D.A. David, EG & G Idaho. Simion 3d version 7.0 user’s manual. IdahoNational Engineering and Environmental Laboratory, Idaho Falls, Rev.5,2000.

[157] A. vanVeen, H. Schut, J. deVries, R.A. Hakvoort, and M.R. Ijpma. Analysisof positron profiling data by means of “VEPFIT”. In P.J. Schultz, G.R.Massoumi, and P.J. Simpson, editors, Positron Beams for Solids and Sur-faces, AIP Conf. Proc 218, pages 171–196. American Intitute of Physics,New York, 1990.

130

BIBLIOGRAPHY

[158] C. Dauwe, T. Van Hoecke, and D. Segers. Positronium physics in fine pow-ders of insulating oxides. In K. Tomala and E. Gorlich, editors, CondensedMatter Studies by Nuclear Methods, Proceedings of XXX Zakopane School ofPhysics, page 275, Krakow, 1995. Institute of Physics, Jagielloian Universityand H. Niewodniczanski Institute of Nuclear Physics.

[159] P. Sferlazzo, S. Berko, and K.F. Canter. Experimental support for ph-ysisorbed positronium at the surface of quartz. Phys. Rev. B, 32(9):6067,1985.

[160] P. Sferlazzo, S. Berko, and K.F. Canter. Time-of-flight spectroscopy ofpositronium emission from quartz and magnesium oxide. Phys. Rev. B,35(10):5315, 1987.

[161] S. Van Petegem, C. Dauwe, T. Van Hoecke, J. De Baerdemaeker, andD. Segers. Diffusion length of positrons and positronium investigated us-ing a positron beam with longitudinal geometry. Phys. Rev. B, 70:115410,2004.

[162] DBAN program is written in Matlab by Nikolay Djourelov and it is availableon request ([email protected]).

[163] Toon Van Hoecke. Construction and application of a source-based slowpositron beam. Doctoral Thesis, Ghent University, 2003.

[164] R.S. Brusa, A. Dupasquier, E. Galvanetto, and A. Zecca. Experimentalstudy of positron motion in Kapton. Appl. Phys. A, 54:233, 1992.

[165] A.P. Mills Jr. and Murray C.A. Diffusion of positrons to surfaces. Appl.Phys., 21:323, 1980.

131

BIBLIOGRAPHY

132

Publications

List of publications as student of Ghent University

1. C. A. Palacio, J. De Baerdemaeker, M. H. Weber, D. Segers, K. G. Lynn,K.M. Mostafa and C. Dauwe. “Parameterization of the median penetrationdepth of implanted positrons in free-standing polymer films”. Ready to besubmitted.

2. Khaled M. Mostafa, J. De Baerdemaeker, C. A. Palacio, D. Segers and Y.Houbaert. “Effect of annealing on deformed iron”. Ready to be presented atthe XVth International Conference on Positron Annihilation. Saha Instituteof Nuclear Physics. Kolkata, India. January 18 - 23, 2009.

3. Dirk I. Uhlenhaut, Florian H. Dalla Torre, Alberto Castellero, Carlos A.Palacio Gomez, Nikolay Djourelov, Gunter Krauss, Bernd Schmitt, BrucePatterson and Jorg F. Loffler.“Structural analysis of rapidly solidified Mg−Cu−Y glasses during room-temperature embrittlement”. Submitted to PHIL-OSOPHICAL MAGAZINE (31/07/2008). (A1 journal).

4. C. A. Palacio, J. De Baerdemaeker, D. Segers, K.M. Mostafa, D. Van Thourhout,and C. Dauwe. “Positron implantation and transmission experiments onfree-standing nanometric polymer films”. Accepted for publication in MA-TERIALS SCIENCE FORUM in the conference proceedings of the 9th In-ternational Workshop on Positron and Positronium Chemistry. May 11−15,2008 Wuhan University, China. (C1 journal).

5. C. A. Palacio, J. De Baerdemaeker, D. Van Thourhout and C. Dauwe. “Emis-sion of positronium in a nanometric PMMA film”. APPLIED SURFACESCIENCE, Vol255, p. 197 (2008). (A1 journal).doi:10.1016/j.apsusc.2008.05.235.

133

http://dox.doi.org/10.1016/j.apsusc.2008.05.235

BIBLIOGRAPHY

6. C. A. Palacio, J. De Baerdemaeker, and C. Dauwe. “Determination of thepositron diffusion length in Kapton by analysing the positronium emission”.APPLIED SURFACE SCIENCE, Vol255, p. 213 (2008). (A1 journal).doi:10.1016/j.apsusc.2008.05.234.

7. N. Djourelov, C. A. Palacio, J. De Baerdemaeker, C. Bas, N. Charvin, K.Delendik, G. Drobychev, D. Sillou, O. Voitik and S. Gninenko. “A studyof positronium formation in anodic alumina”. JOURNAL OF PHYSICS:CONDENSED MATTER, Vol20, p. 095206 (2008). (A1 journal).doi:10.1088/0953-8984/20/9/095206.

8. C. A. Palacio, N. Djourelov, J. Kuriplach, C. Dauwe, N. Laforest, and D.Segers. “Doppler broadening of positron annihilation radiation as a probefor the anisotropy of free-volume-holes in polymers”. PHYSICA STATUSSOLIDI (C) 4, No 10, 3755-3758 (2007). (C1 journal).doi:10.1002/pssc.200675781.

9. N. Djourelov, C. Dauwe, C. A. Palacio, N. Laforest and C. Bas. “Positronstates in polypropylene and polystyrene at low temperature”. PHYSICA STA-TUS SOLIDI (C) 4, No 10, 3743-3746 (2007). (C1 journal).doi:10.1002/pssc.200675779.

10. N. Djourelov, C. Dauwe, C. A. Palacio, N. Laforest and C. Bas. “On theconsistency between positron annihilation lifetime and Doppler broadeningresults in polypropylene”. PHYSICA STATUS SOLIDI (C) 4, No 10, 3710-3713 (2007). (C1 journal). doi:10.1002/pssc.200675732.

11. C. Dauwe, C. Bas, C. A. Palacio. “Formation of positronium: Multi-exponentialsversus blob model”. RADIATION PHYSICS AND CHEMISTRY 76, p. 280-284 (2007). (A1 journal). doi:10.1016/j.radphyschem.2006.03.051.

Awards

Honorable mention in the student poster competition. Award received during the14th International Conference on Positron Annihilation (ICPA-14). It was held atMcMaster University, Hamilton, Ontario, Canada, July 23− 28 / 2006.

134

http://dox.doi.org/10.1016/j.apsusc.2008.05.234

http://dox.doi.org/10.1088/0953-8984/20/9/095206

http://dox.doi.org/10.1002/pssc.200675781



http://dox.doi.org/10.1016/j.radphyschem.2006.03.051

some effects on polymers of low-energy implanted...

Documents