Download Some effects on polymers of low-energy implanted positrons

Some effects on polymers of
low-energy implanted
positrons
FACULTY OF SCIENCES
Carlos Andrés Palacio Gómez
Department of Subatomic and Radiation Physics
Ghent University
A thesis submitted for the degree of
Doctor in sciences: Physics
October 2008
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
Dr. Jérémie De Baerdemaeker
Prof. Dr. Charles Dauwe
Dr. Steven Van Petegem
Prof. Dr. Eddy de Grave
Prof. Dr. Roland Van Meirhaeghe
Prof. Dr. Dirk Ryckbosch
Ghent University, promoter
Ghent University, copromoter
Ghent University
Paul Scherrer Institute (Villigen, Switzerland)
Ghent University
Ghent University
Ghent University, chairman
This work was produced and published at:
Ghent University
Department Subatomic and Radiation Physics
Proeftuinstraat 86
BE-9000 Gent
Belgium
c 2008 Carlos Andrés Palacio Gómez. All rights reserved.
Copyright This document was prepared with LATEX 2ε .
I would like to dedicate this thesis to my charming and adorable wife
Alexandra and to my loving parents.
ii
Acknowledgements
The acknowledgments go to all the people that I have had the opportunity to meet and work with during all these years.
Without a preferential order, first of all I would like to thank three
persons who have been fundamental to the successful development and
for the proof reading of my Ph.D. thesis.
• I would like to thank my promoter Danny Segers who has given
me the opportunity to become a Ph.D. student in the NUMAT
group, for teaching me, for helping me at any time I needed and
for 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 time
for giving me the right amount of criticism to bring my research
to a higher level. Charles, after all our discussions I must admit
that never before I had the opportunity to meet such an excellent
researcher as you are. I am glad to have known you.
• I am also entirely grateful to Jérémie De Baerdemaeker who has
spent days and nights helping me, teaching me, discussing with
me any detail at any time, learning me to be critical even about
obvious things, and giving me all his support and encouragement
to get the better results and analysis obtained during my thesis
work. Jérémie, without you this thesis would have never been accomplished and would have been considerably postponed. Thank
you so much.
Many special thanks to the reading committee of this thesis, for their
helpful corrections and suggestions: Danny Segers, Jérémie De Baerdemaeker, Charles Dauwe and Steven Van Petegem.
Many thanks also to Eddy de Grave for giving me the opportunity to
work half time in the Mössbauer group during the last year in parallel
with my Ph.D. thesis. Eddy, thank you for all your help inside and
outside the laboratory.
iii
I would like to express my gratitude to all my other colleagues of
the NUMAT: Robert Vandenberghe, Bartel Van Waeyenberge, Arne
Vansteenkiste, Valdirene Gonzaga De Resende, Julieth Alexandra Mejı́a
Gómez (lee tu dedicatoria especial al final), Caroline Van Cromphaut,
Khaled Mostafa, Nicolas Laforest, Abdurazak M. Alakrmi and Toon
Van Alboom. Special thanks to Bartel Van Waeyenberge who was also
teaching 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 laboratory, for his confidence, loyalty and transparency, and also for his
support during all these years. Thanks Khaled!.
For the technical support I would like to thank the team of the administrative and technical staff: Philippe Van Auwegem, Roland De
Smet, Patrick Sennesael, George Wiewauters (still for me), Christophe
Schuerens, Bart Vancauteren, Daniëlla Lootens, Linda Schepens, Brigitte
Verschelden 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 and
graduate 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 (Fonds
voor Wetenschappelijk Onderzoek (FWO)) and the Interuniversity Attraction Poles (IUAP/PAI) V/01–Network program of the Belgian Federal Government, for their financial support.
Many special thanks to Jan Kuriplach, Steven Van Petegem and Nikolay Djourelov for all your help and for your friendship.
My acknowledgements also go to the families of my professors and/or
colleagues for giving me their hospitality and for giving me some good
memories. Specially to Carmen (Charles’s wife) and Nathalie (and
kids) (Jérémie’s family).
Me gustarı́a agradecer a todas las personas que he conocido y con las
que por una u otra razón he tenido la oportunidad de compartir muy
buenos momentos. Por lo tanto, mis mas sinceros agradecimientos
son para Yves, Clara, Julián, Carolina, Orlando, Valdirene, Milena,
Luca, don Giovanny, Maria Isabel, Gladis, Elisa, Fredy, Wim, Johan,
Geertrui, Julio (y esposa), Mauricio, Douglas, Liliana (y familia), Adriana, Patrick, Marco, Clarena (y familia), Felipe (y familia), Flavia y
Diego. My acknowledgements also go to the several of my friends that
perhaps 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 nice
environment and for inviting us to participate in their several pleasant
activities. Definitely: “Our home away from home”. Special thanks to
Isabelle Mrozowski and Marleen Van Stappen, both of you are really
outstanding. Here I would also like to mention to Piotr and Eliza.
Agradezco con todo mi corazón a todos mis familiares y amigos que
siempre estuvieron pendientes y en muchas ocasiones me preguntaron
acerca de mis progresos con la tésis. Aunque no siempre estuve muy
positivo, les agradezco por el interés y los ánimos que me brindaron.
En especial agradezco a mis padres Jorge y Nury, a mis hermanos
Jorge y Paula (y a mi sobrinito Andrés, que en el futuro entenderá la
importancia de esta tésis), a mi cuñado Germán, a mi suegra Martha
y a mi cuñado Jaider.
Finally, my deepest and unlimited words of gratitude go, of course, to
my wife Alexandra for her love, kindness and patience. Dear Alexandra, this thesis would not have existed without you. You have given
me the opportunity to let me work on it during many, many evenings
and weekends even in several occasions when probably it would have
been very important for you that I would have given my 100 percent
attention 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 Andrés Palacio Gómez, October 2008
v
vi
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
1.2
Free positrons interaction in condensed matter . .
10
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 . . . . . . . . . . . . . . . . . . . . .
2 Experimental techniques in positron annihilation
17
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-volume-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
2.4
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
Angular correlation of annihilation radiation (ACAR) . . . . . . .
33
2.4.1
Relation between the para-positronium momentum and the
free-volume-hole size . . . . . . . . . . . . . . . . . . . . . .
3 Interaction of positrons with solids and surfaces
3.1
34
37
Slow Positron beams . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.1.1
Introduction to positron moderation . . . . . . . . . . . . .
38
3.1.1.1
Positron re-emission . . . . . . . . . . . . . . . . .
39
Beam transport . . . . . . . . . . . . . . . . . . . . . . . . .
42
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
3.1.2
3.2
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
Photon detection system . . . . . . . . . . . . . . . . . . . .
58
Polymer samples . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
4.2.1
4.3
r
4.3.1
Kapton
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
4.3.2
Thin polymer films . . . . . . . . . . . . . . . . . . . . . . .
60
4.3.2.1
Spin Coating . . . . . . . . . . . . . . . . . . . . .
60
Free-standing nanometric polymer films . . . . . . . . . . .
61
4.3.3
PART III: RESULTS AND DISCUSSION
65
5 Parameterization of the median penetration depth of implanted
positrons 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 by
analysing the positronium emission
91
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
6.2
Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.3
Analysis and results . . . . . . . . . . . . . . . . . . . . . . . . . .
97
6.3.1
6.4
Position of the p-Ps contribution in annihilation spectra . . 103
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 of
the annihilation spectrum
117
Bibliography
119
Publications
133
x
List of Figures
16
1.1
Schematic view on the terminal positron blob . . . . . . . . . . . .
2.1
Simplified decay scheme of the radioactive isotope
Na . . . . . .
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-annihilation 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 and
free-volume-hole size . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.1
22
Comparison of the energy spectrum before and after the moderation
process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Representation of the one dimensional potential representation for
a 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
3.2
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 when
the chamber walls are internally cladded with Teflon with those
without clad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Screenshot of the SIMION simulation of the trajectory of the transmitted positrons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
S-W results obtained in Ghent for the different PMMA and PS film
samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
Obtained S-parameters as a function of the implantation energy for
the PMMA and PS films . . . . . . . . . . . . . . . . . . . . . . . .
80
Transmission experiments performed at Washington State University to some of the free-standing polymer samples . . . . . . . . . .
81
5.6
5.7
5.8
5.9
α n
ρE
5.10 Graphical representation of the power-law z1/2 (E) =
according 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 values for the parameters α and n compared with the experimental
thickness values at the extracted energy values E1/2 . . . . . . . .
88
5.13 Thicknesses of the PS samples obtained with the different values for
the parameters α and n compared with the experimental values of
the thickness at the extracted energy values E1/2 . . . . . . . . . .
89
Experimental setup with the sample at 45◦ and perpendicular with
respect to the beam axis . . . . . . . . . . . . . . . . . . . . . . . .
94
6.1
xii
LIST OF FIGURES
6.2
Comparison of the peak statistics as a function of the positron implantation 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 implanted
positron energy of 467 eV (a) at 45◦ and (b) perpendicular with
respect 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 sample
surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.7
Free p-Ps population as a function of the distance between the sample 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◦ with
respect to the beam axis . . . . . . . . . . . . . . . . . . . . . . . . 107
6.10 Ps emission from the Kapton surface when the sample is perpendicular 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 obtained from the different values for the parameters α and n that
characterize the well-known power-law (z1/2 = αρ E n ) at the extracted energies E1/2 . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.1
Comparison of the values obtained from the fitting of the experimental intensities of the fly-away p-Ps, the fly-away o-Ps and the
bulk p-Ps for the PMMA film by using the the different values for
the parameters α and n that characterize the well-known power-law
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2
Comparison of the values obtained from the fitting of the experimental intensities (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 different values for the parameters α and
n 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
xviii
Introduction
Background
The positron annihilation spectroscopy has shown to be a very effective and powerful tool, which allows accurate analysis of a wide variety of materials. Positrons
can penetrate into liquids and solids without damaging the material. The annihilation gamma rays give information about the structure of the material and its
interactions with positrons.
Several important advances have been made in the last years, specially in
positron annihilation in metals, gasses, and in positronium chemistry. In addition,
the commercial development of stable and fast electronic apparatus has encouraged
several scientific groups to work in the field so that the rate of progress is rapidly
growing up.
Aim
Experiments concerning positrons and monoenergetic positron beams have attracted the interest of several laboratories. The number of original papers where
the wide application field of a positron beam is reported can be shocking for a
nonspecialist. 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 should
make their second step with the help of specialized reviews and collected
papers 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 especially on two issues:
1. The median penetration depth of positrons as a function of the implantation energy z1/2 (E), related to the positron implantation profile
P (E, z) is assumed to be a power-law z1/2 (E) = αρ E n . Here ρ is the
sample density and the constants α = 4.0(±0.3) µg cm−2 keV−n and
n = 1.60(±0.05) are the most frequently used empirical parameters
which 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 from
positron lifetime experiments at different implantation energies. However, these experiments were performed on several non-detached (from
the Si substrate) spin-coated polymers.
From here arise the first motivation as it is expected that in a nondetached polymer the interaction at the interface with the substrate
would have a higher contribution of annihilation of positrons in the
polymer than in the case of self-supporting films. Therefore, for the
first time, positron annihilation experiments are performed on selfsupporting nanometric polymer films.
In addition, by performing transmission experiments, and with a previous knowledge of the thickness of the samples, the values for the parameters α and n are obtained from (a) the measurements of the positron annihilation line-shape parameter, derived from Doppler broadening of annihilation radiation (DBAR) measurements, performed at the positron
beam in Ghent, together with (b) the peak rate measurements performed to some of the samples at the positron beam facility in Washington. Finding the values for the parameters α and n by means of
the line-shape parameter in the way as it is presented in this thesis is
also a novel method for the positron community. Therefore, as a second motivation, and by suggesting a novel method, these values for the
parameters α and n are found and compared.
Finally, all the values for the parameters α and n considered in this thesis are used to calculate the thicknesses of the samples and to compare
them with the experimental ones.
2. The study of the positron motion is important for understanding the
interactions of positrons with matter. When bombarded by low-energy
positrons, an interesting phenomenon that usually appears in some
metal oxides and in polymeric materials is the emission of positronium from the sample surface.
2
Nomenclature
In a longitudinal setup, with the γ−ray detector located behind the
sample on the axis of the beam, it has been shown that the Ps emitted
at the front side surface of the sample has a linear momentum mainly
away from the detector. In that experiment, the detected photo-peak
in DBAR measurements was approximated by a Gaussian distribution.
The p-Ps contribution was detected as a narrow fly-away peak at the
low-energy side of the 511-keV-line (red-shift contribution).
A motivation here is to prove that depending on specimen-detector
geometry, the detected photo-peak from DBAR experiments (after the
background subtraction) can also be affected by the contribution of the
p-Ps emission at the high-energy side (blue-shift) or at the central part
of the photo-peak. A blue-shifted peak has an advantage over the redshifted because the Compton background contribution that appears at
the low-energy tail of the detected photo-peak can be avoided.
Two different specimen-detector geometries are thus proposed: The
polymer sample is located (1) at 45◦ and (2) perpendicular with respect
to the positron beam axis. The detector is located beside the sample
position, but perpendicular to the positron beam line.
These experiments are performed in two different polymers: poly(methylmethacrylate) (PMMA) and Kapton. This experiment is also interesting because Ps is formed only in PMMA and not in Kapton.
The positronium (Ps) emission from the sample surface is studied by
using Doppler profile spectroscopy and Compton-to-peak ratio analysis.
From the obtained results, and by using all the values of the parameters
α and n discussed in the item 1., the thermal and or epithermal positron
diffusion length, the efficiency for the emission of Ps by picking up an
electron from the surface, and in addition for the case of PMMA, the
bulk 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 is
expected, however, that the obtained results may contribute to the knowledge and
may 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 have
no 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 physics
and positron annihilation spectroscopy.
• Chapter 2 describes the most conventional experimental techniques in positron
annihilation.
• Chapter 3 gives an introduction about the low energy, or slow, positron
beams.
• In Chapter 4 the experimental setup used for this thesis as well as the description (and in some cases the preparation) of the polymer samples are
described in detail.
• Chapters 5 and 6, respectively, deal with the interpretation, results, discussion and conclusions of the two issues described above.
4
PART I
INTRODUCTION TO
POSITRON PHYSICS
5
6
1
Introduction to the positron
annihilation spectroscopy
A small background about the positron and positronium physics is given in this
chapter.
1.1
1.1.1
The positron
Historical remarks
The prediction and subsequent discovery of the existence of the positron, e+ , constitutes one of the big successes of the theory of relativistic quantum mechanics
and of the twentieth century physics. The first theory that was consistent with
both quantum mechanics and the special relativity was presented by the scientist
Paul A.M. Dirac.
In special relativity the relationship between the total energy E of
a free particle with rest mass m0 is related to its linear momentum
p by:
E 2 = p2 c2 + m0 2 c4
(1.1)
of light. The two solutions for this equap where c is the velocity p
tion are E = c p2 + m20 c2 and E = −c p2 + m20 c2 . Dirac stated that the negative energy solutions of the relativistically invariant wave equation had a real
physical significance leading to a fully occupied (in accordance with the Pauli exclusion principle) ‘sea’ of electron states with negative energies between −∞ and
7
1. INTRODUCTION TO THE POSITRON ANNIHILATION
SPECTROSCOPY
−m0 c2 (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 the
anomalous negative-energy quantum states predicted by the Dirac equation for
relativistic electrons could be explained –). A ‘hole’ or ‘vacancy’ in this sea, however, would appear itself as a positively charged particle with a positive rest mass,
which, on the basis of uncalculated Coulomb energy corrections and the particles
then known, Dirac assumed that the hole might be the proton [1, 2]. It was soon
realized by Hermann Weyl that this was not the case and that the theory actually
predicted the existence of a new particle with the same rest mass and magnetic
moment as an electron and equal but opposite charge, whereas the proton is over
1800 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]. The
existence of the positron was likewise proved by Blackett and Occhialini (1933) [8]
in the phenomena of “pair production” and proved by Curie [9] in radioactive
decay.
As the discovery of the positron in 1932 confirmed the theory of Dirac, he was
awarded 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 of
matter and not anti-matter. When anti-matter and matter meet, they annihilate
converting their masses into energy. In the theory of Dirac, this conversion of matter to energy (i.e. the positron-electron annihilation) can be seen as the radiative
de-excitation of the electron. This process can be described by quantum electrodynamics (QED) and may proceed by the creation of zero, one, two or three photons
within the constraints of energy, momentum and spin conservation. Higher order
processes 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 processes
cross-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 statistical
weight of ws = 14 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 annihilation
of 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 of
photons. Due to the conservation of momentum, single photon and zero photon
annihilation require a third and fourth body, respectively.
When the positron and the electron are in the singlet spin state (spins antiparallel), the two-photon process is the most probable. The cross-section for this
process was derived by Dirac [2] to be:
!
p
p
γ+3
4πr02 γ 2 + 4γ + 1 2
2
(1.3)
ln γ + γ − 1 + γ − 1 − p
σ2γ =
γ+1
γ2 − 1
γ2 − 1
α~
≈ 2.8 × 10−15 m the classical radius of the electron (or positron)(α is
with r0 = mc
p
given below in (1.6)), γ = 1/ 1 − (v/c)2 and v the speed of the positron relative
to the stationary electron. Of most relevance for our discussion is annihilation at
low positron energies, where v c, (i.e. in the non-relativistic limit), so that the
equation (1.3) is reduced to the familiar form:
σ2γ =
4πr02 c
v
(1.4)
When the positron and the electron are in the triplet spin state (spins parallel), the lowest order process is the annihilation into three photons. Three-photon
annihilation was first observed by Rich [10, 11]. The cross section for the threephoton annihilation in the approximation of low relative velocity of the two particles (v c) was calculated in 1949 by Øre and Powell [12]. It can be written in
function of the two photon annihilation cross section (Eq. (1.4)) as:
σ3γ =
4α 2
(π − 9)σ2γ
9π
(1.5)
with α the fine-structure constant:
α=
e2
1
≈
(4π)~c
137.036
(1.6)
Due to the conservation laws, two other particles are required for the zerophoton process. These particles can be provided by two nucleons of a nucleus or
by two electrons in an atom. This process is however very unlikely because of the
unfavorable momentum transfer to two massive particles. The cross-section scales
with Z 8 , with Z the atomic number of the atom involved. It has a maximum for
Ee+ = 500 keV at about 10−32 m2 for Z=80 [13].
The one-photon annihilation requires one extra particle and will emit a photon
with energy E + me c2 − Eb where E the total energy of the positron according
9
1. INTRODUCTION TO THE POSITRON ANNIHILATION
SPECTROSCOPY
to the equation (1.1) and Eb is the binding energy of the electron involved. This
process is expected to occur mainly with inner shell electrons, e.g. the crosssection for single quantum annihilation with a K-shell electron is peaked around
Ee+ = 400 keV when the positron has sufficient energy to reach the deepest
shells [14]. Experimentally the single quantum annihilation from the K–, L–, and
M–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 process
is negligible compared to two- and three-photon annihilation.
When the positron and the electron are at rest, a characteristic radiation is
emitted as a consequence of the annihilation. In the singlet state case (spins
antiparallel), the positron and the electron will annihilate into two anti-collinear
photons each carrying the rest mass energy of the electron (positron), i.e. 511
keV. This was first observed by Klemperer [16]. In the triplet state case, the
total energy of 1022 keV is distributed over three photons. They are emitted
in a coplanar fashion with energy distributions up to 511 keV. This was verified
experimentally by Chang et al. [17] using high resolution gamma spectroscopy.
The annihilation rate λ (– the inverse of the positron lifetime τ –) of free
positrons with velocity v can easily be calculated from the cross-section:
λ=
1
= σa vne
τ
(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 that
the annihilation rate stays finite and it is independent of the velocity v going to
zero.
1.1.2.1
Free positrons interaction in condensed matter
Positrons rapidly loose their energy when injected into matter. The high energetic
positrons 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 of
annihilation radiation (ACAR) measurements [18], a brief information about the
ACAR technique can be found in section 2.4 on page 33–) compared to the mean
lifetime 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 negligible
and therefore only the two– and three–photon annihilation should be taken into
account. The ratio of two– to three–photon annihilation can be calculated from
10
1.2 Positronium
the cross–sections:
ws σ2γ
=
wt σ3γ
1
4 σ2γ
3
4 σ3γ
≈ 371, 2
(1.8)
This value was experimentally confirmed in metals by triple coincidence measurements by Basson [20].
In a system of non-interacting particles (i.e. neglecting the influence of the
positron on the electrons of the medium), the total annihilation rate (using the
equations (1.2) and (1.7)) is given by:
λ = σa vne
= (σ0γ + σ1γ + σ2γ + σ3γ )vne
≈
1.2
(1.9)
πr02 cne
Positronium
In 1934 Mohorovic̆ić [21] proposed the existence of a bound state of a positron and
an electron which, he (incorrectly) suggested, might be responsible for unexplained
features in the spectra emitted by some stars. However, Mohorovic̆ić’s ideas on
the 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 its
present appellation), with the chemical symbol Ps.
Positronium itself was eventually discovered in 1951 by Deutsch [23–25] and
its properties were investigated in a series of experiments based around positron
annihilation 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 particleantiparticle nature of Ps, which assures the equality of the positron and electron masses and magnitudes of magnetic moments and the possibility of selfannihilation.
The non–relativistic quantum mechanics of the Ps atom is practically identical to that of the hydrogen atom. The Schrödinger equations are the same,
except for the magnitude of the masses of the positive particles. The reduced
m2
of hydrogen is very close to the electron mass, and the one of
mass µ = mm11+m
2
the positronium is exactly one half of it (µ =
11
me mp
me +mp
=
m2e
2me
=
me
2 ,
where me and
1. INTRODUCTION TO THE POSITRON ANNIHILATION
SPECTROSCOPY
mp are, respectively, the mass of the electron and the positron). When the center
of mass coordinates are eliminated, the one–body Schrödinger equation expressed
in the internal coordinate r is
2
~ 2
e2
ψ(r) = Eψ(r)
(1.10)
− ∇ −
2µ
(4π)r
The bound state energy eigenvalues of this equation are
2 2
e
µ
1
En = − 2
2~
4π
n2
(1.11)
1 2 α2
= − µc 2 , n = 1, 2, 3, ...
2
n
where again α is the fine-structure constant. As the reduced mass is m2e the
gross values of the energy levels are decreased to half those found in the hydrogen
atom, so that the binding energy of the ground state positronium (n = 1) is
approximately EB = −6.8 eV.
The spherical symmetric spatial wave function of the ground state in spherical
coordinates is (as an example):
1
ψPs (r, θ, φ) = p
π(2a0
r
)3
e− 2a0
(1.12)
2
~
where a0 = 4π~
me e2 = me cα is the Bohr radius (e is the elementary charge). This
equation can be used to calculate the probability density of the Ps ground state
wave function for r to be zero (i.e. at the origin):
2
|ψPs (0)| =
1
m3 c3 α3
= e 3
3
π(2a0 )
8π~
(1.13)
Positronium can exist in the two spin states, S=0, 1. The singlet state 1 S0 , in
which the electron and positron spins are antiparallel, is termed para-positronium
(p-Ps), whereas the triplet state 3 S1 where the spins are parallel is termed orthopositronium (o-Ps). The spin state has a significant influence on the energy level
structure 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 triplet
7 4 2
state over the singlet state [25]: ∆Ehf s = 12
α c me+ ≈ 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 that
manages the e+ − e− annihilation process is fundamental to the understanding of
Ps physics [27].
Energy and momentum conservation forbids the single-photon (1γ) annihilation 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 orbital
angular 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 of
Ps: each photon contributes a factor of (−1), whereas in the Ps the electron
and positron are interchanged, yielding a factor (−1)l+S , since they have opposite
intrinsic parity. For ground state positronium with l = 0, one concludes that the
annihilation of the singlet (11 S0 ) and triplet (13 S1 ) spin states can only proceed
by the emission of even and odd numbers of photons respectively. Thus, in the
absence of any perturbation the annihilation of p-Ps proceeds by the emission
of two, four, etc. gamma–rays; and the annihilation of o-Ps by the emission of
three, five, etc. gamma–rays. In both cases the lowest order processes dominate,
although the second order processes have been observed: the four–photon decay
of p-Ps [28] and the five–photon decay of o-Ps [29].
It is expected from spin statistics that positronium will in general be formed
with a population ratio of ortho- to para- equal to 3:1, and in the absence of
any significant quenching (e.g. via the conversion of o-Ps to p-Ps considered in
subsection 1.2.4 on page 17), most of the o-Ps which is formed will eventually
annihilate in this state. Thus, the three–gamma–ray annihilation mode will be
much 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 energy
distribution up to 511 keV.
The Equation (1.7) can be used to calculate the annihilation rate due to the
negligible effect of the Coulomb binding on the decay probability [12]:
2
λPs = σv |ψPs (0)|
(1.15)
2
where |ψPs (0)| 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 perturbation by Pirenne [26] and Wheeler [30] with the use of the Dirac’s cross section
of the 2γ annihilation for positron-electron collisions at low energies (Eq. (1.4)).
13
1. INTRODUCTION TO THE POSITRON ANNIHILATION
SPECTROSCOPY
The decay rate was found by multiplying this cross section by the flux of colliding particles taken as the relative velocity multiplied by the particle density at
the point of annihilation, i.e. the origin, taken as the square of the orbital wave
function of p-Ps at the origin [31]:
λp−Ps
3 3
m3
ec α
8π~3
2
α~
( mc
)
z }| {
z}|{
z}|{
4πc 2
m3 c3 α3
2
r0 v e 3
= σ2γ v |ψPs (0)| =
v
8π~
1 me c2 α5
1
=
≈
≈ 8 ns−1
2
~
125 ps
4πc 2
v r0
(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 2
mc2 6
1
(π − 9)
α ≈
≈ 0.0072 ns−1
9π
~
142 ns
(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-Ps
decay rate to compare it with the theoretical values found by QED. Reviews on
this topic can be found in references [13, 36]. Only two experiments to accurately
determine the singlet lifetime have been reported. Theriot et al. [37] derived
the singlet lifetime from the broadening of the radio-frequency resonance of the
hyperfine splitting. Al-Ramadhan and Gidley [38] measured it by using the effect
of 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 injected
into the medium. This is higher than in metals and semiconductors because of
the higher concentration of imperfections and impurities and the lower electron
density. In metals, additionally, the high density of free electrons prevents the
positron to bind with a single electron and therefore positronium can only be
formed at the surfaces (internal and external) [39, 40].
• The Øre-gap model
According to the Øre-gap model [41], the positronium is formed by extracting
an electron from a medium molecule in passing. There is a threshold of the
positron energy (Ee+ ) for forming Ps by this process, which is 6.8 eV less
than the ionization energy of the molecule Ei (i.e. the energy necessary to
release 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 resulting
positronium will have a kinetic energy greater than its own binding energy
and hence it is a candidate for breakup in a subsequent collision. Thus the
positronium formation is most probable with the positron kinetic energy in
the range:
Ei − 6.8 eV < Ee+ < Ei
(1.18)
thus, the positronium is formed when the positron energy during slowing
down lies within a gap where no other electronic energy transfer is possible.
• The Spur model
In 1974 Mogensen [42, 43] suggested that positronium formation is a spur
reaction process. The positron spur is a group of electrons, ions, radicals
and other excited species produced in the last ionization collisions during
the 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 this
model, positronium is formed mainly by the reaction between the positron
and an excess electron in the spur. Thus, the positronium is formed when the
positron is thermalized and captures a thermalized electron in its own spur.
The positronium yield can be quantitatively treated with reaction kinetics
of positron spur reactions (Mogensen 1995 [44]).
Some believe that one of the two models (Øre-gap or spur) is right and the other
is wrong. Some believe that the two models are not inconsistent. In the beginning of the eighties, Eldrup et al. [45, 46] showed by making slow positron beam
experiments on ice that depending on the energy of the positron both processes
can occur simultaneously.
• The Blob model
This model, developed by Stepanov et al. [47, 48], is an extension of the
spur reaction model. The distributions of excess electrons and positron were
applied and then it was possible to explain the change of Ps formation probability 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 original
publications 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 within
10−11 s until its energy drops below the ionization threshold.
15
1. INTRODUCTION TO THE POSITRON ANNIHILATION
SPECTROSCOPY
- In the final ionizing regime, with the positron energy Wbl ∼ 0.5 keV
and the ionization threshold of several eV, n0 ≈ 30 overlapped ion–
electron pairs are generated in the terminal blob. The terminal blob
is a spherical micro-volume of “radius” abl ≈ 40 Å which confines the
end part of the positron trajectory, where ionization slowing down is
the most efficient (thermalization stage of subionizing positron is not
included here).
- The subionizing positron further undergoes positron-phonon scattering
and may diffuse out of the blob, until it becomes thermalized in a
spherical volume bigger than the blob volume, ap > abl .
- The intrablob electrons are tightly kept by electric fields of the positive
ions. 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 of
the thermalized intrablob electrons, followed by formation of weakly
bound positron–electron pair is the first stage for the formation of Ps.
Figure 1.1: Schematic view on the terminal positron blob. Positron
motion is simulated as random walks with the energy dependent step
ltr (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 is
shortened by the interaction with matter (through different processes) or external
magnetic fields. The term ‘quenching’ is commonly used for o-Ps since its annihilation rate is 3 orders of magnitude lower than for p-Ps and therefore, the influence
is bigger on the o-Ps lifetime. The two most important processes are pick-off and
conversion quenching.
In the pick-off process, the positron of the positronium (in matter) suffers
2γ−annihilation in interaction with an electron from the surrounding molecules
having opposite (antiparallel) spin. The annihilation with such electrons reduces
the o-Ps lifetime typically from 142 ns to 1-5 ns. This process was first suggested
by Garwin in 1953 [50] and called ‘pick-off ’ quenching by Dresden [51].
Conversion (or exchange) quenching occurs when the parallel spin electron of
the o-Ps exchanges with an atomic electron with anti-parallel spin to produce pPs [52], which then enables two-gamma annihilation before the reverse process can
occur. Because of the much higher annihilation rate of p-Ps, the annihilation is
almost immediate in the o-Ps time scale. Conversion quenching is clearly observed
when 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 possible
the positronium quenching can be found in reference [55].
17
1. INTRODUCTION TO THE POSITRON ANNIHILATION
SPECTROSCOPY
18
2
Experimental techniques in
positron annihilation
The conventional experimental techniques frequently employed to study positrons
are introduced in this chapter. Some of these techniques are used through this
work, thus the principles behind them are briefly described as well as the methodology and illustrations of the apparatus for some of them. First we start
with a brief overview of the positron sources, the annihilation lifetime and finally
the momentum measurements (which include the Doppler Broadening (or Doppler
shift) and the angular correlation of the annihilation radiation) are presented.
2.1
Positron sources
β + -emitting radioactive isotopes are used to obtain positrons in conventional
positron measurements. A few well-known isotopes are 22 Na (2.6 y), 58 Co (288
d), 68 Ge (71 d) and 64 Cu (12.8 h). The most used source material in positron
research is the 22 Na radioisotope. In addition to the half-life of 2.6 years and
the reasonable price of 22 Na, an advantage is that the manufacture of laboratory
sources is simple, due to the easy handling of the different sodium salts in aqueous
solution, such as sodium chloride or sodium acetate.
A simplified decay scheme of 22 Na is shown in Figure 2.1. 22 Na decays to the
excited state 22 Ne* with a β + branching ratio of around 90%. The ground state of
22
Ne is reached after 3.7 ps by emission of a γ−photon of 1274 keV. The positron
emission is followed promptly by this photon and therefore, it can be used to
register the positron’s birth.
22
Because the β-decay reaction is a two particle decay, the positrons emitted by
Na exhibit a broad energy distribution extending from almost zero to 545 keV
19
2. EXPERIMENTAL TECHNIQUES IN POSITRON
ANNIHILATION
(2.6 y)
22
Na
+
b
(3.7 ps)
22
Ne*
Eg = 1274 keV
22
Ne
g
Figure 2.1: Simplified decay scheme of the radioactive isotope
22
Na. 22 Na decays to the excited state 22 Ne*. This excited
state has a half-life of 3.7 ps and de-excites to the ground state
of 22 Ne 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 22 Na salt on a
thin metal or polymer foil. The most common foil materials are Al, Ni, and Mylar
or Kapton. In order to ensure the almost complete annihilation of positrons in the
specimen a “sandwich arrangement” is used, the foil source is placed between two
identical samples. A minimum thickness of the samples is required to ensure that
the essential fraction of positrons annihilates in the sample pair.
The implantation profile of high-energy positrons emitted from a radioactive
source into a solid can be described by an empirical law which first was found
for electrons and later confirmed for positrons [56, 57]. It states that the positron
intensity I(z) decays as:
I(z) = I0 e−α+ z
(2.1)
The mean implantation depth of the positrons is 1/αe+ and can be approximated
as
αe+ ≈ 17
ρ [g/cm3 ]
−1
]
1.43 [MeV] [cm
Emax
(2.2)
where Emax is the maximum energy of the emitted positrons and ρ the density of
the solid. This approximation can be used for the determination of the minimal
thickness 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”) and
subsequent annihilation (“stop”) of the positron. As the start signal is given at
the moment a positron is injected into the material, its origin depends on the
positron’s source. As stated previously in section 2.1 on page 19, in the case of
a radioactive source of 22 Na, the start signal is given by the the detection of the
γ−photon of 1274 keV that is emitted simultaneously with the positron. In a high
energy beam experiment, the positrons deposit a small amount of energy in a thin
scintillator before being injected into the sample. As the positrons travel at the
speed of light the time difference between the scintillator signal and the injection
into the sample is always equal and therefore, the scintillator signal can be used as
start signal. A last method is the use of a pulsed positron beam. In this case, the
signal is generated by the pulse electronics as the positrons are injected in fixed
time intervals [58].
In 1948 Debenedetti [59] build a setup to measure the time intervals between
ionizing events. In 1949 it was realized that the lifetime of a positron is an important property [60]. In 1951 Deutsch [23] investigated the lifetime of positrons in
several gases, finding the definite proof for the existence of positronium. Bell and
Graham [61] investigated more systematic the lifetime of positrons in solids and
liquids.
2.2.1
The basic operating principle
The basic operating principle of all traditional positron lifetime systems is schematically illustrated in Figure 2.2. A 22 Na source is employed in the apparatus shown.
The start signal is derived from the detection of the positron’s birth (i.e. the
γ−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 scintillation counters (SC) (see e.g. [62] for a general discussion). The scintillator signals
are converted into electronic signals by photomultipliers. These signals are later on
processed by a pair of discriminators, and the simplest arrangement consists of two
constant-fraction differential discriminators (CFDD), which combine good timing
characteristics with the capacity to set upper and lower limits on the pulse height
accepted by the instrument. Therefore, the higher energy (start) signal can easily
be selected. An appropriate delay is inserted in order to introduce a minimum
fixed time between the start and stop signals. Then the signals are connected to a
21
2. EXPERIMENTAL TECHNIQUES IN POSITRON
ANNIHILATION
time-to-amplitude converter (TAC). The TAC delivers a signal with an amplitude
proportional to the length of time between the start and stop signals. The output
of this module is recorded by a multichannel analyzer (MCA) of which the output
is stored in a computer. A lifetime spectrum is recorded frequently containing
106 − 107 events, from which various lifetimes along with several other parameters
can be extracted.
22
Na Source
1274 keV
Photomultiplier
tube
Sample
g
SC Start
+
e
-
e
g
g
Stop SC
511 keV
Photomultiplier
tube
511 keV
delay
CFDD
Start
Stop
CFDD
TAC
MCA
Figure 2.2: Schematic positron lifetime spectrometer. The lifetime is
measured as the time difference between the appearance of the start
and stop γ−photons. Key: SC, scintillator; CFDD, constant-fraction
differential discriminator; TAC, time-to-amplitude converter; MCA,
multichannel analyzer.
Another useful lifetime system, though less frequently encountered is the socalled β + − γ system. In this system the annihilation γ−ray still provides the stop
signal, but the start signal is derived via the energy deposited by the positrons as
they traverse a thin (typically 0.1 to 0.3 mm) scintillator. This method of start
detection has a high efficiency, usually around 50 %, which permits the use of
a relatively weak radioactive source, resulting in a superior signal-to-background
ratio. This technique was first used by the pioneers Bell and Graham in 1953 [61].
They used a stilbene scintillator in from of a 22 Na source to deliver the start
signal for their delayed coincidence measurements. When used with a radioactive
source, the lowest energy positrons will annihilate in the scintillator itself and
22
2.2 Positron annihilation lifetime spectroscopy (PALS)
give a contribution to the lifetime spectrum. When using a MeV mono-energetic
positron beam, almost no positrons will be stopped in the scintillator. The high
beam energy allows the use of a sufficient thick scintillator (2 to 5 mm), enhancing
light collection and signal amplitude. In this way a start detector with almost
100% efficiency can be achieved. This will result in a virtually background free
lifetime spectrum [63].
2.2.2
Lifetime Data Treatment
The positron lifetime spectrum describes the probability of an annihilation at time
t. If positrons have several different states from which to annihilate (consider a
system with n independent positron states i), the lifetime spectrum is determined
by the solution of the differential equation
dni (t) X −λi t
=
Ii e
(2.3)
dt
i
where λi is the
P decay rate constant associated with state i and Ii the corresponding
intensities ( i Ii = 1). The lifetime components are defined as the reciprocal
values of the decay constants τi = λ−1
i (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 convolution of the ideal exponential spectrum presented in Equation (2.3) and the resolution function of the system. This resolution function is usually approximated
by a Gaussian function. The typical Full Width at Half Maximum (FWHM) of
the resolution function is around 200-250 ps (depending mainly on the sizes of the
scintillators and the used energy windows).
In addition, a few percent of the positrons annihilate in the source material
producing an additional component to the experimental spectrum. For this reason,
usually several components can be reliably separated in the experimental lifetime
spectra. The separation is normally performed by fitting the convoluted theoretical
lifetime spectrum to the measured data. The effect of the source component can
be eliminated by measuring a defect free reference sample. On the other hand,
when the resolution function is Gaussian, it does not affect the average positron
lifetime defined as
Z ∞
X
τave =
Ii τi =
tP (t)dt
(2.4)
i
0
The average positron lifetime (equal to the center of mass of the spectrum) is an
important quantity since it can be always determined even if the decomposition
of the lifetime spectrum is difficult.
23
2. EXPERIMENTAL TECHNIQUES IN POSITRON
ANNIHILATION
5
10
τ =0.33 ns
2
4
Counts
10
τ =1.8 ns
3
10
3
2
10
1
10
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
Channel
Figure 2.3: Experimental positron lifetime spectrum obtained from
a poly(methyl-methacrylate) (PMMA) sample. After the decomposition of the spectrum, the obtained lifetime components τ2 and τ3
are added as straight lines in the semi-logarithmic plot for illustration. The τ1 component is not indicated as a straight line (τ1 = 0.125
ns). The deviations from the straight line at higher times are due
to annihilations in the source and the background contribution. The
Gaussian-like shape of the left part of the curve is mainly caused by
the resolution function.
2.2.3
Relation between the positronium lifetime and the
free-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-Ps
lifetime, τo-Ps (which is 142 ns in vacuum) is quenched to some nanoseconds as
the 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 occurrence 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
2.2 Positron annihilation lifetime spectroscopy (PALS)
sion. The measured τo-Ps can be related with the FVH size by a semi-empirical
equation known as the Tao-Eldrup model. It is derived from a simple quantummechanical model in which it is assumed that the Ps is confined in a spherical
void of radius R. In that model one assumes that the void represents a rectangular infinite potential well for Ps with a spatial overlap of the Ps wave function
with molecules within a layer δR of the potential wall [64–66]:
2πR i−1
h
1
R
+
sin
τo-Ps = 0.5 1 −
R0
2π
R0
(2.5)
where τo-Ps is expressed in ns and R0 = R + δR in Å (δR = 1.656 Å is the
empirical parameter that represents the o-Ps penetration depth into the wall of
the hole wherein the o-Ps annihilates). A graphical representation is given in
Figure 2.4.
9
8
o−Ps lifetime (ns)
7
6
5
4
3
2
1
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Free−volume hole radius R (nm)
Figure 2.4: Graphical representation of the Tao-Eldrup model.
However, one has to be aware that in molecular crystals and polymers these
assumptions are not strictly fulfilled: the well is shallow, the potential is not
rectangular, the voids are often not spherical. Thus, the Tao-Eldrup equation
is only an approximation of real τo-Ps versus R relation (as emphasized by these
authors). One can also apply other shapes, e.g. cuboids proposed by Jasińska et
al. [67].
25
2. EXPERIMENTAL TECHNIQUES IN POSITRON
ANNIHILATION
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 model
for R < 1 nm. Thus, in order to explain lifetimes in the free volumes of larger
radii, some extensions to the model are required. There are a few approaches to
this problem resulting in different τo-Ps (R) dependencies each.
The Tao-Eldrup model’s simplicity is due to approximations like considering
Ps as a particle without internal structure, use of a spherical potential well of
infinite depth (broadened by δR = 0.166 nm) and taking into account only the
ground level of a particle in a rectangular potential well.
The simplest way to avoid these limitations is to discard some of the approximations. A modification proposed by Goworek et al. [68–70] takes into account
excited levels of Ps in the potential well without changing any other assumptions.
Such a simple extension substantially changes the Tao-Eldrup model curve for
R > 1 nm at moderate temperatures. Moreover it explains a temperature dependence of the lifetime, which could not be done based on the original Tao-Eldrup
model. The price paid for such a modification is that model equations are more
complicated and have to be solved numerically.
The essential point in this model is the introduction of lifetime averaged over
as many excited states as necessary; it is a second rank problem what geometry
is most appropriate for the particular case. For better explanations related to the
model, refer to the original papers [68–70] or more recent papers [71, 72].
For a visual idea however, the ortho-positronium lifetime curves versus freevolume-hole diameter at different temperatures are shown in Figure 2.5. It is seen
that great differences of lifetimes as a function of temperature can be observed in
the range of radii of several nm. For R → ∞ all lifetimes approach the vacuum
value 142 ns. The shadow area corresponds to the original Tao-Eldrup model (see
Subsection 2.2.3).
To find simpler solutions of the problem, cubic geometry was proposed by
Gidley et al., [74]. A less realistic approximation of the potential well’s shape
allowed one to write τ in a form that uses only elementary functions. A new
value of δR = 0.18 nm was suggested in order to fit the obtained curve to the
Tao-Eldrup model. Later it was found that the use of this δR value leads to a
good 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 annihilation
in large free volumes was proposed by Ito et al., [76]. Instead of representing Ps
as a standing wave, it was considered as a Gaussian wave packet scattering in
the potential well, the same as in the TaoEldrup model. Unfortunately, empirical
parameters 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-Eldrup
model proposed by Goworek et al. [68–70]. The figure represents the
ortho-positronium lifetime versus free-volume-hole diameter at different temperatures. The shadow area corresponds to the original TaoEldrup model. For R → ∞ all lifetimes approach the vacuum value
142 ns. The figure is taken from reference [73].
al., [75]).
2.3
Doppler shift or broadening of the annihilation radiation (DBAR)
In the frame of reference in which the center of mass of an electron-positron pair is
at rest, the two annihilation photons arising from their annihilation in a spin singlet
state 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 center
of mass is not at rest in the laboratory frame of reference. When slowing down
in 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
1 For
the remaining of this thesis ‘linear momentum’ is abbreviated by ‘momentum’
27
2. EXPERIMENTAL TECHNIQUES IN POSITRON
ANNIHILATION
motion of the center of mass creates a Doppler shift in the γ−ray energies, and
the angle between the annihilation photons deviates from 180◦ , depending on the
momentum of the annihilating pair. This is shown in Figure 2.6.
PL
P
P2 , E2 , g2
P
q
P1 , E1 , g1
Figure 2.6: The vector diagram of the momentum conservation in the
2γ-annihilation process. The momentum of the annihilation pair is
denoted by P, and the subscripts L and ⊥ refer to longitudinal and
transverse components, respectively.
The Doppler broadening of the annihilation line was first observed by DuMond
et al. [77] when measuring the radiation from a 64 Cu β + source using a curved
crystal spectrometer. The Doppler shift in the energy of the 511 keV annihilation
line 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 momentum of the center of mass along the direction of emission of the gamma–ray. For
a typical electron energy of a few eV and a thermalized positron the Doppler shift
is of the order of 1.2 keV1 . The shape of the 511 keV annihilation line is in fact
due to the one-dimensional momentum distribution of the electron-positron pair
Z ∞Z ∞
L(Eγ ) ∝
ρ(px , py , pz )dpx dpy
(2.7)
∞
∞
with pz = 2c (Eγ − m0 c2 ).
A typical Doppler broadening setup is shown in Figure 2.7. Since the energy
shift is very small, except for the early experiments [77], the measurement of
Doppler profiles has only become possible by the development of Ge(Li) (Lithium
drifted Germanium) gamma-ray detectors (–which have very high resolution–)
[78, 79]. Nowadays High Purity Germanium (HPGe) detectors are used. The
amplification system following the detector is standard, usually consisting of a
preamplifier and a spectroscopy amplifier; which allows the broadened annihilation
1 In addition to the traditional quantification of the gamma-photon energy in keV, the following units are sometimes used: atomic units (1 a.u. ≈ 3.73 keV/c) and millirad (1 keV ≈ 3.9
mrad)
28
2.3 Doppler shift or broadening of the annihilation radiation (DBAR)
line to be examined in more detail. The γ−ray energy distribution can then be
stored in a multichannel analyzer and processed in various ways depending upon
the details of the study.
Sample-Source
arrangement
High Voltage
Power Supply
Preamplifier
HPGe
Spectroscopy
amplifier
Multichannel
analyzer
Figure 2.7: Schematic drawing of a typical Doppler broadening setup.
The signal is processed by the preamplifier and by the spectroscopy
amplifier before being recorded in the multichannel analyzer.
This technique is mainly used in investigations of solid state, where, in most
cases the geometry of the experiment and the random nature of the direction
of 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 Doppler
broadened by an amount related to the momentum distribution of the annihilating
pair.
2.3.1
The S- and W-parameters
Extracting information from the whole shape of the annihilation peak is insufficient due to the resolution and to the peak-to-background ratio of the Doppler
broadening setup. Some deconvolution procedures have been developed but their
reliability is always limited. Doppler broadening spectra are, therefore, usually
characterized with the S- and W-parameters. These parameters were first introduced by MacKenzie et al. [80]. The Figure 2.8 illustrates both parameters. The
line-shape parameter (S-parameter ) is calculated as the ratio of a central area of
the 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
2. EXPERIMENTAL TECHNIQUES IN POSITRON
ANNIHILATION
intervals is to some extend arbitrary. The highest sensitivity of the S-parameter
for 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 the
Doppler curve can still have a small asymmetry. This is the reason why the Wparameter is sometimes calculated only using information in the high energy wing
of the Doppler curve.
Intensity (arb. units)
Eg [keV]
S-parameter
W-parameter
Aw1
Aw2
As
AT
Figure 2.8: Schematic view on how to calculate the S- and Wparameters. The areas indicate the summation regions for the cal
AS and W = AW1 +AW2 .
culation of the line-shape parameters S = A
A
T
T
The S-parameter is a sensitive parameter to detect changes in low momentum contributions of the annihilation peak, which produce the smaller Doppler
broadening of the annihilation line. This means that the annihilation with valence electrons gives the largest contribution to the S parameter. The formation
of p−Ps also contributes almost entirely to the S-parameter because of its very
small momentum. On the other hand, the W-parameter is sensitive to variations
in high momentum contributions; thus, mainly annihilations with core electrons.
As an example of the importance in using the S and W-parameters in a Doppler
spectrum we may consider a defect-free and some defect-rich samples of the same
pure material. In a defect-free sample, the positron can be represented by a
Bloch state with its wave function overlapping the valence and core electrons. In
the material with defects, the positron can be trapped in a defect and its wave
30
2.3 Doppler shift or broadening of the annihilation radiation (DBAR)
function is localized, and although its wave function still overlaps with the valence
and core electrons, the fraction changes: the larger the open volume, the less the
wave function of the trapped positron will overlap with the core electrons, – which
have the highest momentum –, and therefore, the more it overlaps with the valence
electrons. Thus, the defect-rich sample should have the highest S- and the lowest
W-parameter.
2.3.2
Coincidence DBAR (CDBAR)
A substantial problem that limits the capability of the conventional DBAR technique is its relatively high background contribution. This is especially important
for the measurements of the high-momentum contribution of the 2D annihilation
peak –mainly at the low-energy side of the photo-peak–.
In 1976 Lynn et al. added a NaI(Tl) detector opposite to the Ge detector
to reduce the background by detecting both photons in coincidence [81]. This
method is still used today (see e.g. [82–87]) due to the relative small additional
cost compared to a single detector setup. It is used to reduce the background at
the 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 from
both detectors. This is conventionally called coincidence-DBAR (CDBAR) or
double-DBAR (DDBAR).
The coincident γ−quantum is almost anti-collinear to the primarily registered
annihilation photon and it serves to suppress the Compton background near the
511 keV energy which arises from the 1274 keV γ−ray of 22 Na. For comparison
two photo-peaks, one recorded by means of the typical DBAR and the other one
by means of the CDBAR techniques are shown in Figure 2.9(a). Thus, from this
figure it is clear that by using the coincidence Doppler broadening technique, the
background reduction is highly improved. An important consequence from the
drastic background reduction is that this method allows the observation of the
high momentum annihilations with core electrons and therefore the comparison
with theoretical calculations is possible.
In a CDBAR experiment a two-dimensional spectrum is recorded where the
axes represent the energy scales of the respective detectors.
For illustration, an example of a two-dimensional coincidence Doppler broadening spectrum is presented in Figure 2.9(b). Considering the momentum and
energy conservation of the annihilation process, an increase in the energy of an
annihilation γ in one detector according to the Doppler shift, is consequent with
a 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
2. EXPERIMENTAL TECHNIQUES IN POSITRON
ANNIHILATION
elliptical shape is explained.
530
E2 (keV)
520
510
500
500
(a) Comparison between a typical
photo-peak obtained by a single detector (open triangles) and a coincidence
setup (circles).
510
E1 (keV)
520
530
(b) Example of a two-dimensional
coincidence Doppler broadening spectrum.
Figure 2.9: Coincidence Doppler broadening spectrum
To obtain the corresponding one-dimensional spectrum, a projection onto the
E1 + E2 = 1022 axis is made. The projection includes events in a window along
the diagonal with 2m0 c2 − δ ≤ E1 + E2 ≤ 2m0 c2 + 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 produced
by coincidences of a 511 keV photon with a background photon.
For many years this method was not really used by the scientific community. It was only in the early-nineties that it was ‘rediscovered’ [90–93]. Then
it was demonstrated that in some cases CDBAR can be used for the chemical
identification of the atoms at the site of annihilation. Positron annihilation with
low-momentum valence or conduction electrons results in a small Doppler shift.
Annihilations with core electrons result in a large Doppler shift, contributing to
the wings of the 511 keV annihilation line. The momentum distribution of the
core electrons is typical for each chemical element. Therefore, the shape of the
core contributions form some kind of fingerprint of the atoms from which they
originate.
A more detailed information on the coincidence Doppler broadening technique
can 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 and
was further improved by DeBenedetti et al. [96] in 1949. This method involves
measuring θ, the small deviation from π radians in the angle between the two
annihilation γ−rays. As previously mentioned in section 2.3 on page 27, this
deviation is a consequence of the center-of-mass momentum of the annihilating
electron-positron pair.
The relationship between θ and the component of this momentum perpendicular to the direction of one of the γ−rays, which can be taken to be the longitudinalcomponent, PL , as in Figure 2.6, is given by (– also for the derivation, see Appendix
A; equation (A.8) –):
θ=
PT
m0 c
(2.8)
The sample-to-detector distance amounts typically to several meters (– 5 m typically –) so that γ−quanta from only a small solid angle are detected. Hence much
stronger sources compared with conventional PALS and DBAR techniques are required. On the other hand, angular resolution can be adjusted in the range 0.2 to
5 milliradians [97], which corresponds to the energy resolution of DBAR measurements in the range 0.05 to 1.3 keV. Thus this technique provides essentially the
same kind of information as DBAR, however, the momentum resolution is much
better.
An example of an angular correlation of annihilation radiation apparatus to
detect such small angular deviations is shown in Figure 2.10. Apart from the digitization and storage electronics, basically it consists of a pair of two-dimensional
position-sensitive gamma-ray detectors and a radioactive source, which is immediately adjacent to the sample being studied. The field of view of each detector is
in addition limited by lead collimators.
In media in which there is not a preferred axis of symmetry (e.g. in gases and
liquids in the absence of external fields) it is not necessary to use a two-dimensional
system (although in that case, the count rate will be lower). Instead, the onedimensional technique can be used, in which the position-sensitive detectors are
replaced by two single detectors, each with a long collimator placed in front of
it, giving integration over one of the components of the momentum. One of the
detectors is fixed whereas the other one is scanned through the angle θ.
33
2. EXPERIMENTAL TECHNIQUES IN POSITRON
ANNIHILATION
Detector 1
Detector 2
source
x
g
+
e
g
x
qx,y
sample
coincidence t
Py
digitization
and storage
Pz
Px
Figure 2.10: Scheme of the apparatus for two-dimensional angular correlation of annihilation radiation. The deviation angular, θ, of the
two γ−quanta is recorded by position sensitive detectors in a coincidence setup and stored in a two-dimensional memory array.
2.4.1
Relation between the para-positronium momentum
and the free-volume-hole size
The hole size can also be estimated from the electron-positron momentum distribution 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 superposition of Gaussian distributions whose components arise from the different annihilation channels. The narrowest component is associated to the self-annihilation
of para-positronium and the broader distributions to the free e+ and o-Ps pick-off
annihilations [98–101]. The narrow component reflects the momentum of the p-Ps
annihilating from a free-volume-hole. According to the Heisenberg uncertainty
principle (∆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 the
hole 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 21 is in mrad (1 keV = 3.913 mrad) and δR = 1.656 Å is an empirical
parameter. A graphical representation is given in Figure 2.11. In this figure the
effects of the extended Tao-Eldrup (ETE) model discussed in 2.2.3.2 on page 26
are not included. To my present knowledge no studies about these effects have
been reported in literature.
34
2.4 Angular correlation of annihilation radiation (ACAR)
5
4.5
θ1/2 (mrad)
4
3.5
3
2.5
2
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Free−volume hole radius R (nm)
Figure 2.11: Graphical representation of the relation between the p-Ps
momentum and free-volume-hole size.
35
2. EXPERIMENTAL TECHNIQUES IN POSITRON
ANNIHILATION
36
3
Interaction 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 not
suitable for thin-layer materials or near-surface experiments because in such conventional cases the positrons are immediately implanted into the sample material.
Mono-energetic positrons are therefore necessary in order to obtain a defined penetration depth. These positrons can easily be accelerated to well-defined implantation energies1 . This is the base for the low-energy, or slow, positron beams, also
referred to as Variable Energy Positrons (VEP).
A detailed knowledge of the positron interaction with solids and surfaces is of
fundamental importance for the interpretation of the results. For example, Schultz
and Lynn [103] have reviewed a variety of theoretical as well as experimental work
performed with positron beams. Thus, in this section we give a basic description
of the slow positron beam production through a discussion of the different interaction processes as moderation, through implantation into solids; the subsequent
thermalization and diffusion of the moderated positrons and also the emission into
vacuum of those positrons which reach the surface before being annihilated; and
their subsequent manipulation to form a beam.
1 According to the energy of the positrons, the positron beams can be divided into two
categories: The high-energy positron beams with typical energies of 1-4 MeV and the low-energy
positron beams ranging from 0-30 keV.
37
3. INTERACTION OF POSITRONS WITH SOLIDS AND
SURFACES
3.1.1
Introduction to positron moderation
The very first step in the production of a slow positron beam, namely the energy
moderation of positrons coming from a high-energy source, is based on typical
processes of positron interaction with solids in the bulk and at the surface.
High energy positrons are produced either as a result of nuclear decay, usually
of an artificially produced radioactive isotope (– as stated previously in section 2.1
on 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, most
experiments 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 solid
thermalize through collisions on a time scale of less than 10 ps. There the positrons
will diffuse around in this environment with an average lifetime of ∼ 100 ps. It
was realized by Madansky and Rasetti already in 1950 [104] that this time would
be sufficient for some of the thermal positrons to diffuse back to the surface where
they could be re-emitted. Unfortunately, due to contamination of the surface
their search for slow positrons diffusing out of thin 64 Cu activated foils failed; no
slow positrons were observed, even though one slow positron per 103 fast β + was
expected.
This method of making low-energy positrons has later become known as ‘moderation’. Thus, moderation is the name given to the technique of producing
a monochromatic positron beam from such a broad energy spectrum. In other
words, moderation is done with the purpose of optimizing the number of positrons
that can be emitted from the moderator foil into the vacuum. Those positrons
also have a low average energy and a narrow energy spread (–typically positrons
with kinetic energy in the eV range–).
Figure 3.1 shows a comparison of the energy spectrum of positrons emitted from
a 22 Na radioactive source to that of moderated positrons. From this figure, the
advantage of moderators becomes obvious: A positron beam of any desired energy
can be obtained just by accelerating the near-zero energy moderated positrons.
1 The phenomenon of pair production occurs when a high-energy photon interacts with an
atomic nucleus, allowing it to produce an electron and a positron without violating conservation
of momentum. This phenomenon, although interesting, is not used in this work, and therefore
will not be discussed here.
38
3.1 Slow Positron beams
-2
10
-3
Positron yield
10
Moderated positrons
10-4
-5
10
-6
10
-7
10
Typical b+
energy spectrum
10-8
-9
10
10-1
100
10
1
102
103
104
105
106
Positron energy [eV]
Figure 3.1: Positron energy spectra before and after the moderation
process, demonstrating the increased number of useful positrons for
low energy beam experiments. The curve is the normalized energy
distribution for β + particles emitted from a 22 Na source. The full bar
shows the efficiency of moderation
3.1.1.1
Positron re-emission
The first report (not published in a journal article) of slow positron emission was
made by Cherry in 1958 [105]. He reported that positrons with energies below 10
eV were emitted from a 64 Cu source, transmitted through mica, which was coated
with a thin layer of chromium. The conversion efficiency was approximately one
slow positron per 107 fast positrons. In 1968 Groce et al. [106] observed moderated
positrons 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 mean
energies a few eV higher than expected for a thermal distribution. This meant
that the positrons gained energy leaving the solid, giving rise to the name ‘negative
work function1 materials’. The theoretical explanation for this phenomenon (in
metals) was given by Tong in 1972 [107].
1 Work 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 surface
but still close to the solid on the macroscopic scale.
39
3. INTERACTION OF POSITRONS WITH SOLIDS AND
SURFACES
The work function, φ, for electrons and positrons has two contributions:
• One is the chemical potential of the particle (µ). For electrons this is the
energy required to lift an electron from the Fermi energy level into vacuum.
For the positrons a similar ‘binding energy’ arises mainly from the sum of
the repulsive interaction with the ion cores in the solid and the correlation
potential from the attractive interaction with the electrons of the solid.
• The second contribution to the work functions arise from the effect of the
solid surface dipole layer (D), which is repulsive for electrons and attractive
for positrons. At the surface the electron distribution tends to spill out into
the vacuum. Together with the ion cores inside the surface, these electrons
create a dipole layer at the surface. The effect of this layer on an electron
is to decelerate it as it leaves the solid, making it harder to remove. As
the positron has the opposite charge as the electron, the surface dipole has
the reverse effect on positrons as it does on electrons. That is, positrons
get accelerated by the surface layer which counters the binding effect of the
chemical 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 one
from the chemical potential, making the positron work function negative. This,
finally allows mono-energetic positrons to be re-emitted into the vacuum from the
surfaces of these materials, or to be emitted into the interior of a large open-volume
defect such as a void. This is important because positron re-emission forms the
basis of moderation.
Figure 3.2 illustrates a one-dimensional representation of the single-particle
potential energy of a positron in the near-surface region for the case where φ+ is
negative, so that escape of the thermalized positron from the solid into the vacuum
is energetically allowed.
Among the negative work function materials, tungsten (W(110)) has the highest 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 is
that the defects tend to attract and trap positrons, preventing them from diffusing
to the surface. To avoid this problem the moderator can be prepared by heating it
to high temperatures (annealing) which removes vacancies and dislocations. The
width of the energy distribution of positrons emitted from a well-prepared single
crystal is of the order of 0.5 eV (FWHM) or less. Popular alternatives to single
40
3.1 Slow Positron beams
Figure 3.2: Representation of the one dimensional potential for
a thermalized positron near the surface of a metal. The work
function φ+ is a combination of the bulk chemical potential µ+
and the surface dipole layer D. The positron chemical potential
contains a term Vcorr due to correlation with the conduction electrons and a term V0 due to the repulsive interaction with the ion
cores, shown as black dots. (Figure extracted from Schultz and
Lynn [103]).
crystals are poly-crystalline tungsten foils or commercially available meshes. For
the poly-crystalline tungsten foils and the meshes the moderation is in transmission, whereas for the single crystals the moderation is in reflection mode. However
being poly-crystalline, the efficiency of these moderators is lower (∼ 10−4 ) and the
energy width larger (1-2 eV FWHM).
To summarize, the principle of energy moderation by interaction with a solid is
simple:
1.
2.
3.
4.
An initial flux of fast positrons is implanted at some depth.
The positrons rapidly reach thermal equilibrium.
The thermalized positrons diffuse in the solid.
A certain fraction of thermalized positrons reach the free surface where
they can be re-emitted in the vacuum.
41
3. INTERACTION OF POSITRONS WITH SOLIDS AND
SURFACES
3.1.2
Beam transport
Applying a magnetic field along the direction of transportation is the simplest
way to guide a low-energy positron beam. If a positron in the beam has a velocity
component perpendicular to the beam direction it will be kept confined to the
beam by the cyclotron motion induced by the magnetic field. As a result the
positron will follow a spiraling trajectory along the magnetic field lines.
Since the positrons are so light and low in energy this guidance can be achieved
by using even a modest magnetic field, which can be produced with solenoids or
Helmholtz coils.
For some experiments however one may want to avoid having magnetic fields
present. An alternative is to use purely electrostatic guidance. The confinement
of the beam in this case is not as good as in a magnetic beam which may cause
loss of positrons during transport. However, the electrostatic beam optics are well
understood and can be simulated in advance to minimize any loss.
3.2
3.2.1
Positron beam interactions with solids and
surfaces
overview
When a slow positron beam hits the surface of a material, before being annihilated or ejected from the target surface, the positrons may undergo different
interactions (for more complete details, see for example [111]). These interactions
are schematically represented in Figure 3.3.
1. The first possibility is that some of the incident positrons will scatter elastically from the target and will form diffracted beams if the surface is a single
crystal. The rest of the incident beam may enter the solid.
2. Several energy loss mechanisms will cause the positrons that penetrate the
material to lose their kinetic energy; thus stopping and thermalizing them.
• Some of those thermal positrons may diffuse back to the surface where
the positron can be trapped in a two-dimensional state or in a nearsurface defect.
In the case of a material having a negative work function, the positron
may be emitted from the surface as a positron, or after forming positronium 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 and
therefore, it will be analyzed in more detail later on–.
• The thermalized positron that does not reach the surface may annihilate
with an electron of the material (free state) or may get localized during
diffusion by a positron trap such as defects, where it will eventually
annihilate. It may also de-trap from this trapped state, which occurs
in case of a shallow trap.
3. Scattering processes may cause that near-surface implanted positrons return
back to the surface before being thermalized. At the surface, these positrons
may annihilate inside the material (without even being fully thermalized) or
may be emitted directly from the surface as epithermal positrons or either
form positronium at the surface and escape as epithermal ‘hot’ positronium.
+
+
Non-thermal e
Epithermal e
Ps
g
Incident
positron
beam
Annhihilation
g
Capture
at defect
Diffracted
positrons
+
Slow e
Vacuum
Thermal e
+
Ps
Solid
Figure 3.3: Schematic representation of the possible positron interactions. Part of the incident positrons may be backscattered
and 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 again
where it can be emitted as a positron or positronium. Also not
thermalized positrons can return to the surface.
3.2.2
Positron backscattering
When a positron beam collides on a solid surface, some of the incident particles
will undergo a small number of large-angle collisions and leave the target with a
fraction of their original energy. These are termed ‘backscattered ’ positrons [112].
43
3. INTERACTION OF POSITRONS WITH SOLIDS AND
SURFACES
The backscattering probability depends on the incident energy of the positrons
and on the sample material. Knowledge of positron backscattering probabilities is
of practical use if positron annihilation is used to probe the surface and subsurface
regions of solids. The positrons which backscatter from a sample can be annihilated
at 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 entire
beta spectrum of energies from a radioactive source (for example, 0 to 540 keV
for 22 Na) [113, 114]. One attempt was made to select energy windows from the
beta spectrum using a magnetic spectrometer [115]. The first report of backscattering probabilities for truly mono-energetic positrons was for sub-3 keV positrons
interacting with Al foils [111], which formed part of a study of the penetration of
positrons through thin metallic foils.
More recent reports of Monte-Carlo simulations compared with experimental
results [116, 117]; showed that materials of higher atomic number cause a larger
fraction of backscattered positrons. Since these fractions are not negligible, one
has to take into account that these positrons annihilate outside the sample, leading
to a contribution in the spectra recorded by the detector.
Besides the Monte-Carlo simulations, a full description of the experimental
features and limitations, and a more detailed information on the positron and
electron backscattering from thick solids as a function of the incident and outgoing
angles 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 important 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 before
reaching the surface.
The penetration depth. Considering a number of positrons implanted into an
absorber, the penetration depth is the distance of the trajectory ‘endpoint’ of
the positrons from the entrance surface (it is, therefore, a random variable). The
penetration 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 with
the initial energy E thermalizing at depth z in the material.
1 also
referred to as stopping profile.
44
3.2 Positron beam interactions with solids and surfaces
Due to the difficulties in performing direct experimental measurements, researchers have often made use of computer simulations to gain information about
particle implantation. Monte Carlo simulations are in general, always reliable in
predictions of inelastic mean free path and stopping profiles of electrons. Monte
Carlo codes can also give some ideas on the positron path in a material which can
later on be supported by experimental work.
Obviously, implantation profiles depend on the energy spectrum of the incident positron flux. In the monoenergetic case, extensive Monte Carlo simulations [118–122], were found to be well approximated by the well-known Makhovian
implantation profile1 [123]:
m mz m−1
z
P (E, z) =
exp −
,
m
z0
z0
(3.1)
here the empirical shape parameter they find is m ≈ 1.9 and the parameter z0 is
a function of incident positron energy, given by:
m+1
z̄ = z0 Γ
m
(3.2)
where z̄ 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 page
738 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( 32 ) = 2π .
For illustration, Figure 3.4 shows some Makhov profiles of mono-energetic
positrons implanted in Al and Si and for three different values of implantation
energies. A more detailed listing for some materials can be found in reference [124].
Traditionally it has been the mean implantation depth z̄ that has been used
more widely to characterize P (E, z). This is somewhat unfortunate because the
parameter that is directly extracted from experiment is the median penetration
depth z1/2 2 .
From the Makhovian distribution z1/2 and z0 are related by
1
z1/2 = z0 (ln 2) m .
(3.3)
1 Named
after the Makhov’s original electron implantation experiments [123].
by combining Eq. (3.2) together with the following Eq. (3.3) and for m = 2, the
values of the the mean implantation depth (z̄) and median penetration depth (z1/2 ) are very
similar: z̄ = 1.064 z1/2 .
2 However,
45
3. INTERACTION OF POSITRONS WITH SOLIDS AND
SURFACES
0.018
0.016
E = 2 keV
Positron implantation profile
0.014
Al
Si
0.012
0.01
0.008
0.006
E = 5 keV
0.004
E = 8 keV
0.002
0
0
100
200
300
400
500
600
700
800
900
1000
Depth (nm)
Figure 3.4: Calculated Makhov profiles of mono-energetic
positrons implanted in Al (dashed lines, ρ = 2.7 g cm−3 ) and
Si (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 and
α = 4.0 µg cm−2 keV−n .
The dependence of the mean implantation depth on energy is assumed to be a
power law:
z̄(E) =
α n
E ,
ρ
(3.4)
which was originally developed for electron stopping. In Equation (3.4) the parameter ρ is the material density and α (with values often reported in units of µg
cm−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 energy
they 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
3.2 Positron beam interactions with solids and surfaces
Thus, although the material and energy dependencies of the empirical parameters α and n have been under some debate, the most frequently used values for
all 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 references [103, 111, 118, 119]).
According to Vehannen et al. [125], these parameters were considered to be
material independent.
3.2.4
Positron diffusion
If a positron is injected into a solid, the slowing down process will drop it close
to the bottom of the lowest energy band with a residual kinetic energy of the
order of kB T (where kB is the Boltzmann’s constant (1.38 × 10−23 J/K)). At these
energies, the de Broglie wavelength λth (∼ 60 Å at room temperature) is larger
than the interatomic distances in condensed matter (a few Å). This means that a
thermalized positron in a solid actually is to be seen as a propagating wave (see
the references [103, 124, 126] and [127]). However, unless the temperature is very
low and the host medium a very perfect crystal, the number of collisions with
phonons and imperfections during the positron lifetime τ can be so high, and at
such large angles [40], that any directional correlation of the trajectory with the
initial motion is, in practice, lost.
In these conditions, positron migration over distances greater than de Broglie
wavelength can be adequately described as a classical “random walk”. The de
Broglie wavelength (λth ) at temperature T is given by [128]:
λth
r
300
2π~
≈ (62 Å)
,
=p
T
3mp kB T
(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] allows us, therefore, to calculate the displacement of thermalized positrons in the
framework of the ‘diffusion approximation’, and to define the “diffusion coefficient”. Scattering mechanisms determine the positron mean free path hli:
3D+
hli = q
,
(3.6)
3kB T
m∗
where the mean free path is a function of temperature T , D+ is the positron
diffusion coefficient and m∗ is the effective positron mass. The positron diffusion
47
3. INTERACTION OF POSITRONS WITH SOLIDS AND
SURFACES
coefficient is related to the mobility µ+ by the Einstein relation:
µ+ =
eD+
,
kB T
(3.7)
with e the elementary charge (1.6 × 10−19 C).
The main process that determines the positron diffusion is the phonon scattering: the positron momentum distribution broadens due to phonon absorption by
the low-energy positron. In materials such as semiconductors and a lot of metals,
the scattering occurs by longitudinal acoustic phonons. This results in a temper1
ature dependent diffusion coefficient D+ ∝ T − 2 . The temperature dependence of
positron diffusion in several metals and semiconductors have been widely reviewed
by Schultz and Lynn [103].
In nonmetals, propagation of free positrons is not the only possibility; quasipositronium 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 of
information on solid excitations as probed by positrons [130].
In the framework of the diffusion approximation, it is a simple matter to solve
the 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 taken
normal to the boundary, the probability of reaching the surface is proportional
to [131, 132]:
Z ∞
z
F+ (E) =
P (E, z) exp −
dz,
(3.8)
L+
0
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 captured
in 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 of
F+ (E), with a monoenergetic positron beam of variable energy, gives access to
L+ .
The relation between the diffusion length and the diffusion coefficient (D+ ) is
given by
p
L+ = D+ τ ∗ ,
(3.9)
where τ ∗ = (τ −1 +κ)−1 is the mean time remaining in the propagating state which
is depopulated by annihilation at a rate τ −1 and by capture in a localized state at a
48
3.2 Positron beam interactions with solids and surfaces
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 positron
diffusion are the effect of the electron density enhancement in the vicinity of the
positive particle and the effect of the periodic lattice. Diffusion lengths are of the
order 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 timeindependent information. Thus, all processes to which thermal positrons are subjected, such as diffusion, drift, trapping at defects or free annihilation can be combined in a single one-dimensional equation. This steady-state positron diffusion
equation can be written as:
D+
d
d
c(z)
−
µ
E(z)c(z)
− λeff c(z) + I(E, z) = 0,
+
dz 2
dz
(3.10)
with c(z) the steady-state positron density as a function of depth z, I(E, z) the
number of positrons with an energy E implanted at a depth z, and E(z) a local
electric field.
In the diffusion equation, the positron trapping in defects is taken into account
and this leads to effective annihilation rate λeff :
λeff =
1
+ κ(z)
τb
(3.11)
Next to the bulk annihilations, represented by τb , positrons are trapped in defects
with a rate κ(z), which is a function of the specific positron trapping rate for
defects νt and the defect concentration nt (z):
κ(z) = νt nt (z)
(3.12)
In some materials, different specific positron trapping rates can be associated with
typical defect types like monovacancies and dislocations [134].
3.2.5
Epithermal positrons
The diffusion equation described in the previous subsection (Equation (3.10)) is
based completely on positrons that are thermalized. This is considered to be valid
for higher positron implantation energies if inelastic scattering processes dominate
elastic scattering and bulk trapping [135, 136]. In the case when the implantation
depth is smaller than the thermalization length Ltherm (of the order of 10 to 20
nm), one has to consider the epithermal positron effects. Therefore positrons
49
3. INTERACTION OF POSITRONS WITH SOLIDS AND
SURFACES
that are implanted with an energy lower than 1 keV have a high probability to
end up in an epithermal state, so as a consequence, there is an enhancement of
the Ps formation. This is the case specially for insulators and semiconductors
where thermalization is slower due to inefficient stopping processes. An attempt
for calculating the number of epithermal positrons that return to the surface was
first given by Britton et al. [137] in 1988 and was further improved by Kong and
Lynn [135, 136] who gave a more complete interpretation.
3.3
Experimental determination of the positronium fractions
As explained in subsection 1.2.2 on page 12, Ps can be formed under certain
conditions. While p-Ps annihilates into two anti-collinear 511 keV photons, o-Ps
decays in vacuum via the emission of three photons. The Ps atom has a zero
intrinsic momentum. Thus, p-Ps is observed as a narrow contribution to the
Doppler-broadened annihilation peak obtained by a Ge-detector –which causes
higher S-parameter values–.
Due to the conservation laws the total energy of the three annihilation photons
of o-Ps is equal to 1022 keV. This results in a continuous energy distribution up to
511 keV, which increases more or less linearly with increasing energy [12]. Thus, Ps
may 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 enhancement
of the photon energy spectrum in the 0 to 511 keV region. In that way, the
comparison of the number of counts in the peak region P with a chosen fixed area
of the Compton region C (i.e. the region at energies lower than 511 keV) gives
information about the amount of positronium.
The Ps fraction fPs can be deduced from the ratio R of counts accumulated in
different regions of a spectrum [25, 138, 139]:
R=
T −P
,
P
(3.13)
where T is the number of counts accumulated in the photon energy region below
511 keV and P the number of counts in the peak area around the 511 keV annihilation line (–for a better clarity; see the derivation of this and a modified version
of the next expression (Equation (3.14)) in the Appendix B–).
To determine the fraction of 3γ o−Ps annihilation a calibration should be done
where the Compton contribution is measured when no 3γ annihilation is present
and 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 were
normalized to equal height of the 511-keV peak. It represents the complete spectra
for the two conditions (0% and 100% Ps formation).
Figure 3.5: Annihilation spectra measured at an Al(110) surface with a
HPGe detector for situations representing 100% and 0% Ps [140]. The
first condition is realized by a low incident energy of 40 eV at a sample
temperature of 400 C, where all positrons are pushed back to the surface
for Ps formation, the second one (no-Ps) by a high incident positron
energy of 15 keV.
Thus, for the Ps fraction (fPs ) and with the use of Equation (3.13), the following expression is used:
"
fPs
P1 (R1 − Rf )
= 1+
P0 (Rf − R0 )
#−1
(3.14)
where Tf and Pf (intrinsic in Rf ) are the “total” and “photopeak” HPGe counting
rates for a given number of positrons annihilating per second and the subscripts 1
and 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
3. INTERACTION OF POSITRONS WITH SOLIDS AND
SURFACES
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 of
the incident energy, which can give information about the specific defect depth
profile.
With Equation (3.14), measurements of the Ps fraction as a function of the
positron implantation energy can therefore be fitted directly to obtain the positron
diffusion length (L+ ) and consequently the specific trapping rate.
3.4
Charging effects
In slow positron beam experiments (for positron implantation energies typically
below 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 total
gamma count rate, and (c) the positronium formation fraction.
Their measured S(z1/2 ) and the corresponding total photo-peak data are shown
in Figure 3.6. To translate E (in keV) into z1/2 , they used the relation z1/2 =
33E 1.6 nm.
(a)
(b)
Figure 3.6: (a) S vs z1/2 for an isolated polymer film. (b) Photo-peak count
rate for the same run as in 3.6(a). One run performed per day: H day 1; O
day 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 photopeak signal count rate is essentially unchanged for incident energies above about
5 keV. The constancy of the count rate is consistent with the reproducibility in
the same energy range (high E) in the S-parameter values in Figure 3.6(a). As E
decreases (and hence z1/2 ) the results for S for the day 1 sample indicate a steady
decrease. The authors claim that this may indicate that positrons are forming less
Ps in the first 10 – 50 nm below the surface. As the time elapses, the decrease
starts at progressively higher energies, and then S increases again at the lowest E
values. To give an interpretation the authors proposed the following model:
a subsurface electric field builds up with time under positron bombardment, as
electrons are annihilated. The increasing field strength progressively sweeps an
increasing number of positrons from depths of 100–500 nm towards the lower S
region just below the surface. At the lowest incident energies the positrons are
swept 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-peak
count rate (Fig. 3.6(b)) with time at lower incident energies, which suggests that
positrons are indeed being swept from the sample. After 100 h of exposure to
the positron beam over 70 % of positrons implanted at ∼1 keV are entering the
vacuum. Note, however that for z1/2 = 300 nm (E ∼ 4 keV) the signal rate shows
no significant decrease, whereas the effect on S is dramatic. This is consistent with
the model in which positrons are being swept to the near-surface region but not
out of the sample.
In summary, the results may point to field drifting of positrons towards the sample surface and they certainly have implications for the measurement of positronrelated parameters (such as S) for insulating samples.
The authors conclude that these problems can be overcome by taking lowenergy 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 later
time and, if necessary, remeasuring after a short aeration of the system.
53
3. INTERACTION OF POSITRONS WITH SOLIDS AND
SURFACES
54
PART II
EXPERIMENTAL DETAILS
55
56
4
Experimental set-up and Samples
description
A general description of the apparatus as well as a very short description of the
used 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 magnetically
guided positron beam in Ghent whose energy is variable between 0.1 and 30
keV [58]. All the specifications about the design of the Ghent positron facility
are given in reference [142].
In one of the experiments, some of the samples were also measured at the slow
positron beam of the Washington State University (WSU) [143]. In this magnetically guided slow positron beam, monoenergetic positrons can be accelerated from
a couple of eV up to an energy of 60 keV (–although in this thesis, the data are
shown only up to 26 keV–) with a total peak rate of ∼ 2 × 103 e+ /s. The HPGe
detector has a FWHM resolution of 1.6 keV at the 514 keV line of 85 Sr.
4.2
Doppler broadening
The annihilation radiation is detected by a High Purity germanium (HPGe) detector. It is coupled to a digital signal processor (DSP) unit. The digital signals
of the DSP unit are further processed by an interface card and then transferred
to 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 85 Sr. The DSP unit (model 2060 from Canberra) replaces the amplifier-ADC combination of a traditional high resolution
pulse processing chain [62]. In a conventional analog setup the energy signal is
processed, shaped and filtered by a shaping amplifier before being digitized by an
ADC. In the setup with the DSP unit, the energy signal is digitized immediately
after the preamplifier and stored in the MCA. Such a system has the advantage
that the trade-off between throughput and resolution is more favorable compared
to the ordinary amplifier-ADC combination. Furthermore there is less chance for
drifts and instabilities leading to an improved system stability. The settings of a
DSP can be saved. This leads to a higher flexibility (e.g. one can use different
settings for different experiments) and a better reproducibility which is extremely
important 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 the
detector has a calibration factor of 0.0332 keV/channel. The input count rate is
about 10 kHz. To obtain the best possible resolution a rise time of 5.6 µs and a
flat top of 0.8 µs for the trapezoidal filter function was chosen. This is equivalent
to a shaping time of 4 µs for a conventional shaping amplifier.
4.3
4.3.1
Polymer samples
Kaptonr
One of the selected materials for the experiment is Kaptonr Type HN2 , a polyimide film made by DuPont de Nemours [145]. Kapton is a material whose physical, electrical, and mechanical properties remain constant over a wide temperature
range (from 269◦ C to 400◦ C). The used Kapton sample has a thickness of 127 µm
and a density (ρ) of 1.42 g cm−3 .
Due to their properties, polyimides often replace glass and metals, such as
steel, in many demanding industrial applications. They are used for the chassis in
some cars as well as some parts under-the-hood because they can withstand the
intense heat and corrosive lubricants, fuels, and coolants cars require. They are
also used in circuit boards, insulation, fibers for protective clothing, composites,
1 Relative
efficiency at the 1332 keV line of 60 Co relative to a standard 3”-diameter, 3”-long
Thallium-doped Sodium Iodide (NaI(Tl)) scintillator.
2 A more complete description on the physical and chemical properties as well as the different
uses 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 as
well 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 a
molecule (drawn in blue) that has a general structure which looks like the one
shown in Figure 4.1.
Figure 4.1: Example of an imide group. There
are no special elements for R, R’ and R”. They
stand for any atom or group of atoms.
If the molecule shown above were to be polymerized1 the product would be a
polyimide. There are two general types of polyimides. In one type (so-called linear
polyimides) the atoms of the imide group are part of a linear chain. The second
of these structures is a heterocyclic structure where the imide group is part of a
cyclic unit in the polymer chain (where R’ and R” in Figure 4.1 are two carbon
atoms of an aromatic ring). Both types of polyimides are represented in Figure
4.2.
(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 the
aromatic heterocyclic polyimides, like the one on the right (Figure 4.2(b)).
In particular, in positron annihilation spectroscopy, a 7.5 µm-thick Kapton
is commonly used as a positron-source-supporting foil because of its satisfactory
1 Polymerization 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
4. EXPERIMENTAL SET-UP AND SAMPLES DESCRIPTION
resistance to radiation and also because it has the property of the complete inhibition of the positronium in the bulk and therefore avoids difficulties at the moment
of analysing the experimental data. Therefore, it is an important and interesting
material for research purposes.
4.3.2
Thin polymer films
Thin films are thin material layers ranging from fractions of a nanometer to several
micrometers in thickness.
From a physical point of view of the properties of the system, one can define
if a film is thin or not. Thin films have a behavior that differs from “bulk” behavior, with different structural attributes and/or different dynamics that result
in important differences in properties. This difference arises from the fundamental difference between the environment of molecules (or chain segments) at the
interface 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 between
interfaces into the material, but also on external parameters such as temperature.
When this influence expands over a significant proportion of a film, it can be
defined as thin.
In the case of polymers, the thicknesses of the “thin” films can be up to micrometers, because of the large size of the polymer chains. There are many known
techniques to prepare thin polymer films. In this thesis, we will only describe
spin-coating from a solution.
4.3.2.1
Spin Coating
In the process of spin coating, a polymer solution is deposited on a substrate, and
the substrate is then quickly accelerated to the desired spinning velocity (normally
1000 - 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 in
viscosity caused by solvent evaporation or until pressure effects cause the film to
reach an equilibrium thickness. The final thinning of the film is only due to solvent
evaporation [146]. The final thickness of the deposited polymer film depends on
several parameters [147, 148] among which are the initial viscosity of the polymer
solution, 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 is
the first time that the positron annihilation spectroscopy technique is performed
in self-supporting nanometric polymer films with the purposes discussed in the
next 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 = 100o C (DSC 10◦ C/min)) which was dissolved
in toluene to concentrations of 30, 50 and 70 mg/mL.
The polymer films were prepared by spin-coating on SiO2 wafers of 2 inches
diameter. As the concentration of the PMMA solution was always the same, the
coatings were performed by varying the spinning velocity from 500 to 4000 rpm
and with the spinning time of 30 seconds. Immediately after spin-coating, the
PMMA samples were dried at 110◦ C during ∼10 seconds. In the case of PS the
samples 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 measured with a surface profilometer (Talystep).
Table 4.1 collects all the information about the spin-coating of the samples and
also lists the measured thicknesses.
The preparation of each of the self-supporting films is described as follows: After
the 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 carefully
from the borders of the polymer1 .
The polymers used are very hydrophobic, while SiO2 is a very hydrophilic
surface. Thus, the polymer films are easily detached from silicon wafers when
immersed in a water bath because the water wets SiO2 much better than the
polymer 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.
1 According to literature [149], the samples should detach by themselves when immersed in
water. However, only the thickest PMMA film (listed in Table 4.1) was detached by itself after
24h.
61
4. EXPERIMENTAL SET-UP AND SAMPLES DESCRIPTION
Table 4.1: Spin coating: preparation of the thin polymer films.a,b .
Polymer Material
Concentration
(mg/mL)
PMMA
PMMA
PMMA
PMMA
PMMA
PS
PS
PS
PS
PS
PS
SBC 5%
SBC 5%
SBC 5%
SBC 5%
SBC 5%
30
50
50
70
50
50
c
Spin
velocityz
(rpm)
Measuredd
thickness
(nm)
4000
2000
3000
1000
500
3000
3000
1500
3000
800
1500
220 ± 10
310 ± 10
400 ± 20
480 ± 10
1700 ± 50
210 ± 10
340 ± 10
460 ± 10
650 ± 20
670 ± 20
960 ± 20
a
Immediately after the spin coating procedure, the PMMA samples were dried at
110◦ C during ∼10 seconds. The PS samples were not dried.
b
The spinning time for all the samples was 30 seconds.
c
The PMMA resist comes in a Spin Bowl Compatible solvent system (SBC) 5%.
The PS was dissolved in Toluene.
d
The thickness of each polymer film was measured with a surface profilometer
(Talystep).
z
The number ‘rpm’ is nominal. One sets certain spin velocity, but if the motor is
changed (as it was the case), the real spin velocity is no longer the same as it was
previously.
Figure 4.3: Floating PMMA film.
62
4.3 Polymer samples
Eventually, the floating film was picked up by an aluminum holder with a 3
cm diameter hole in its center and it was finally dried in a furnace at about 90◦ C
during 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
4. EXPERIMENTAL SET-UP AND SAMPLES DESCRIPTION
64
PART III
RESULTS AND DISCUSSION
65
66
5
Parameterization 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 of
fundamental importance for the interpretation of the results.
It has already been noted (–in the subsection 3.2.3 on page 44–) that the median
penetration depth as a function of the implantation energy, z1/2 (E), related to the
well-known Makhov distribution, P (E, z) (Equation (3.1)), can be parameterized
by means of the power-law z1/2 (E) = αρ E n , 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 most
frequently used empirical parameters.
In the case of polymers however, by analyzing the ortho-positronium yield from
positron 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 several
spin-coated polymers on Si substrates. However, these polymers were not detached
from the substrates.
It is expected that in a non-detached polymer (from the substrate) the interaction at the interface with the substrate would have a higher contribution of
annihilation of positrons in the polymer than in the case of self-supporting films.
67
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
Therefore, for the first time, positron transmission experiments have been performed on the free-standing poly(methyl-methacrylate) and polystyrene films of
nanometric thicknesses described in the subsection 4.3.3 on page 61 for incident
energies 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 these
positron transmission experiments, and so, we are able to compare our results with
the ones of Algers et al. and with the most commonly used parameters.
The calculation of positron implantation profiles is necessary for the analysis of
Doppler broadening of annihilation radiation (DBAR) experiments. The literature
is plenty of reports on experiments where the Doppler broadened line S-parameter
is investigated in function of the energy of the implanted positrons by means of a
Variable Energy Positron beam (VEP).
A few years ago, Coleman et al. [154] have suggested the idea of parameterizing z1/2 (E) from the S-parameter derived from DBAR measurements. They used
a technique based on the difference between positron annihilation lineshape parameters 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 this
chapter, the bi-layer method is avoided. From the measurements of the S-parameter
as a function of the positron implantation energy it is possible to obtain the values
for the parameters n and α that characterize the median penetration depth. The
analysis of the measured data shown in this chapter is a novel way, to parameterize
z1/2 (E). It is also worthwhile to emphasize here that this is the first time that
this type of measurements with such a purpose is performed in polymers which
additionally are free-standing nanometric films.
In addition, at the positron facility of the Washington State University and
for some of the samples, the same approach of Mills and Wilson of measuring the
transmission coefficient of positrons through thin films of known thickness [111] is
followed. The inspection of the peak count rate in these transmission experiments
are compared with the values obtained for the S-parameter experiments performed
in 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 thicknesses of the samples are calculated and compared with the experimental ones.
1 For
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 experimental facts described in this section.
- The the densities (ρ) of the samples were not measured. However, it is
expected that there is not a significant change with respect to the samples
used by Algers et al. [153]. Thus, for allowing the comparison with their
data, their density values were used. They were considered to be 1.197 g
cm−3 and 1.040 g cm−3 for PMMA and PS respectively.
- The experiments where the S-parameter is obtained as a function of the
positron implantation energy were performed at the variable energy positron
beam in Ghent [58].
- All the DBAR spectra were collected for several implantation energies from
0.1 to 21 keV. In these DBAR experiments two of the PMMA samples (310
nm and 480 nm in Table 4.1 on page 62) were measured at each positron
implantation energy for every 30 minutes, the other films were measured at
each positron implantation energy for every 10 minutes.
- The measurements were performed with the sample mounted in the vacuum
chamber in front of and perpendicular to the beam line and the Ge-detector
beside, at the sample position, but perpendicular with respect to the positron
beam axis. A scheme is shown in Figure 5.1.
Positron beam
Chamber walls
Sample
Beam axis
Detector axis
HPGe
detector
Figure 5.1: Scheme of the experimental DBAR setup.
- The absorption experiments where the count rate is taken into account were
done in only some of the samples (see Table 5.1 on page 82). They were
69
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
performed at the slow positron beam of the Washington State University
(WSU). In this experiment, the monoenergetic positrons were accelerated
from ∼ 0.06 keV up to an energy of 60 keV (–although in this thesis, the
data 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 which
can 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 PS
film (see Table 4.1 on page 62). All the data points were collected every 10 min
at 0.1 keV. It becomes obvious from the figure that as the time elapses the peak
counts decreases until certain saturation level. An explanation can be given with
the model proposed by Coleman et al. [141] (see the overview in the section 3.4
on page 52). Thus, this may be interpreted as a subsurface electric field builds up
with time under positron bombardment, as electrons are annihilated. Positrons
with low energy are swept through the surface, where they can form Ps which
decays in vacuum.
The fitting curve in Figure 5.2, which represents a charging time constant
τcharge is an exponential decay y(t) = y0 + A exp(−t/τcharge ). Here y(t) represents
the peak counts as a function of time. Fitting this equation to the data of Figure
5.2 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 = 0.1
keV for the 460 nm-thick PS film (from the same experiment as for the peak
counts). The significant increase in positronium formation is consistent with the
interpretation given above and therefore, serves as a support for it.
Thus, to reduce the influence of the charging effects on the results, the data in
the following experiments for this thesis are taken only from the first run (i.e. fresh
sample, see Coleman et al. [141] and Ito et al. [155]) so the charging effects are are
the lowest possible. Also the measuring time for each data point was relatively
short (10 minutes) in comparison to the charging time constant. A complete run
for 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 is
already 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
90000
PS
88000
Peak counts
86000
84000
82000
80000
78000
76000
0
100
200
300
400
500
600
700
800
Time (min) (10 min each point)
Figure 5.2: Charging test for the 460 nm-thick polystyrene
film at 0.1 keV. The fitting curve is an exponential decay
y(t) = y0 + A exp(−t/τcharge ). This fitting curve represents
the charging time constant which resulted to be τcharge =
184.8 min/60 min = 3.08 hours for polystyrene.
3.04
3.02
3.00
2.98
2.96
C/P
2.94
2.92
2.90
2.88
2.86
2.84
2.82
PS
2.80
2.78
0
100
200
300
400
500
600
700
800
Time (min) (10 min each point)
Figure 5.3: Compton-to-peak charging test for the 460 nm-thick
polystyrene film at 0.1 keV.
71
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
5.3
Analysis and results
All the implanted and transmitted positrons contribute to the S-parameter directly
obtained from the experimental data. The S-parameter was taken to be equal to
the integral of P (E, z) times S(z) from z = 0 to z = d (d = polymer thickness):
S=
R z=d
z=0
P (E, z)S(z)dz,
(5.1)
where S(z) represents the contribution to the S-parameter from a positron at a
certain depth z.
At the specific energy E1/2 in keV, 50% of the implanted positrons annihilate
in 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 polymer
film thickness d in nm. By plotting the film thickness d versus E1/2 , the shape of
z1/2 (E) was obtained and parameterized by means of the well-known power-law
equation z1/2 (E) = αρ E n .
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 the
sample position). In that case, these transmitted positrons do not contribute to
the S-parameter. In our case, as displayed in Figure 5.4, the extrapolation to high
energy values (50 keV) gives as a result that about 4.5% of the maximum of the
measured 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 the
same conditions, at different positron energies, with a dummy sample holder but
without a sample. The measured background resulted not to be the responsible
for the detected photo-peaks at high energies (it is displayed also in Figure 5.4).
1 For
a better clarity, see the definition of z1/2 on page 45.
72
5.3 Analysis and results
300.0
y=275.8
270.0
PMMA
Background
240.0
Peak counts (CPS)
210.0
180.0
150.0
120.0
90.0
y=12.5
60.0
30.0
0.0
0
5 10 15 20 25 30 35 40 45 50 55
Positron Energy (keV)
Figure 5.4: Experiment performed in Ghent. Peak counts: extrapolation to high energy values. The extrapolation to high energy
values gives as a result that about 4.5% of the maximum of the
measured curve (y = 275.8) is still detected. From the figure, it
becomes obvious that the measured background is not responsible
for this effect.
73
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
Thus, with the purpose to find out an explanation for this phenomenon, a different transmission experiment with one of the polymer films was performed. The
experiment consisted in measuring the S-parameter with teflon sheets everywhere
at the vacuum chamber walls were the sample is located. Thus, the ‘new’ walls of
the beam around the sample were internally “cladded” with Teflon.
The measurements of the S-parameter can therefore be compared with those
without clad. It was done because the material type of the chamber walls is
stainless steel and the characteristic S-parameter in Teflon is higher than in the
stainless steel.
The experiment is shown in Figure 5.5. When the walls were cladded with
Teflon, the constant level observed for the S-parameter at high energies (E >∼ 8
keV) 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 comparable (in good approximation) to the S-parameter value obtained for a standard
non-magnetic sample of stainless steel. This result suggests that the main contribution of the transmitted positrons comes from their annihilation in the chamber
walls.
74
5.3 Analysis and results
0.555
PMMA 220nm normal setup
PMMA 220nm + teflon clad inside
non magnetic Stainless Steel sample
Teflon sample
S-parameter
0.540
0.525
0.510
0.495
0.480
0
2
4
6
8
10 12 14 16 18 20 22
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 used
for the clad has been measured (4) and a standard non-magnetic stainless
steel sample was also measured (). This stainless steel sample was not of
the same material than the chamber walls but gives an indication that the
annihilation of positrons at high energies comes from the stainless steel.
All the points were measured during 20 minutes except the ones of the
stainless steel sample (10 min). The vertical dashed line highlights the
energy E1/2 at which S = S1/2 .
75
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
A simulation was also performed with the SIMION 7.0 program [156] in order
to 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 to
simulate electron and positron trajectories under the presence of electric and/or
magnetic fields. The simulation consisted in:
• An homogeneous magnetic field of about 65 gauss, which is about the same
as 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 because
we were interested in the behavior of the high energy positrons.
• These positrons ejecting from the sample at the same point but at angles
varying from 13 to 25 degrees. This is because according to the simulation,
the positrons that are ejected at angles lower than 13 degrees are efficiently
guided by the homogeneous magnetic field away from the sample so that
these annihilations can not be detected by the HPGe detector. On the contrary, positrons ejected with angles higher than 25 degrees always annihilate
at the chamber walls close to the sample. Therefore, these annihilations are
always detected by the HPGe detector.
From the simulation, the positrons with energies higher than 8 keV do not follow
the homogeneous magnetic field and annihilate in the chamber walls from approximately 5 to 10 cm from the sample position (depending on their ejection angle and
towards the direction of the beam axis). These annihilations might be detected
by the detector which has 4.89 cm of diameter and is centered with respect to the
sample 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 sample
and 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 the
positron beam facility in Ghent.
Another possibility to efficiently guide the transmitted positrons away from
the sample would have been by increasing the homogeneous magnetic field. From
the simulation ∼ 120 Gauss would have been enough. However, to generate these
field the current in the helmholtz coils has to be increased. The current would
have been that high for the helmholtz coils that they would have burned out.
The most important conclusion from the simulation is, therefore, that the
homogeneous magnetic field can not efficiently guide the positrons (with energies
higher than 8 keV) away from the neighborhood of the detector, and consequently
76
5.3 Analysis and results
the positrons annihilate in the chamber walls. This result hence, could support
the previous hypothesis. A screenshot of the simulation is shown in Figure 5.6.
Magnetic field
y
Ejected positrons
x
Sample position
z
Figure 5.6: SIMION simulation of the trajectory of the ejected
(transmitted) positrons in a homogeneous magnetic field of about
65 Gauss. In the simulation 60 positrons with energies varying
between 8 keV and 21 keV are ejected from the sample at the
same 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 different 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 clustering
point1 B representing the bulk value of the polymer film. Then, as the positron
incident 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 the
bulk 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 steel
and polymer). The evolution from B to C in Figure 5.7 indicates that in all the
measurements, for the PMMA as well as for the PS, positrons only annihilate in
the film and in the stainless steel.
1 A cluster point is a point in the S-W representation where it is clear that for different
implantation energies the positrons annihilate in an identical state.
77
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
0.56
B
PMMA
0.55
220 nm
310 nm
400 nm
480 nm
1700 nm
S-parameter
0.54
A
0.53
0.52
0.51
0.50
0.49
C
0.48
0.47
0.05
0.06
0.07
0.08
0.09
0.10
W-parameter
(a) S-W representation for PMMA
0.60
PS
B
210 nm
340 nm
460 nm
650 nm
670 nm
960 nm
0.58
S-parameter
0.56
0.54
A
0.52
0.50
C
0.48
0.46
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
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 samples. When increasing the positron implantation energy an evolution on
a fairly straight line is seen from a surface value A to a clustering point B
representing the bulk value of the polymer film. Then, for higher implantation energies the S-W plot evolves from the bulk polymer value B to a
cluster point C which represents the bulk value of the stainless steel (or
a combination of stainless steel and PMMA). The evolution from B to C
indicates that in all the measurements, for the PMMA as well as for the
PS, positrons only annihilate in the film and in the stainless steel.
78
5.3 Analysis and results
Figures 5.8(a) and 5.8(b) display the obtained S-parameters as a function of
the implantation energy for each film. In the figures, the upper and lower horizontal lines correspond respectively to the bulk polymer value and the bulk value of
the stainless steel (or a combination of stainless steel and polymer). The smooth
lines in the region between the upper and lower horizontal lines correspond to a
polynomial fitting grade 3. The fitting was performed with the purpose of extracting 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 (with
S1/2 ) of the polynomial fitting and not from the intercept of the lines that connect
the data points. The extracted energy values are listed in Table 5.1.
With the measured thicknesses and the obtained energies, the parameters α
and n that parameterize z1/2 (E) can, therefore, be found (z1/2 (E) = αρ E n ). In
Figure 5.5 on page 75, the vertical dashed line highlights the energy E1/2 at which
S = S1/2 . This is plotted to show that for both cases, i.e. with and without the
internal 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 performed at WSU. For comparison, the S-parameter of the same samples measured
in Ghent is also displayed. By inspecting the Figures, one can see that the two
experiments more or less overlap. Thus, by using the same procedure that was
performed 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
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
0.56
PMMA
220 nm
310 nm
400 nm
480 nm
1700 nm
0.55
S−parameter
0.54
0.53
0.52
S = S1/2
0.51
0.5
0.49
0.48
0.47
0
2
4
6
8
10
12
14
16
18
20
Positron energy (keV)
(a)
0.6
PS
210 nm
340 nm
460 nm
650 nm
960 nm
0.58
S parameter
0.56
S = S1/2
0.54
0.52
0.5
0.48
0.46
0
2
4
6
8
10
12
14
16
18
20
Positron energy (keV)
(b)
Figure 5.8: Experiment performed in Ghent. The figure shows the obtained S-parameters as a function of the implantation energy for the polymer films of (a) poly(methyl-methacrylate) (PMMA) and (b) polystyrene
(PS). The upper and lower horizontal lines correspond respectively to the
bulk polymer value and the bulk value of the stainless steel (or a combination of stainless steel and polymer). The line S = S1/2 yields the energy
at which 50% of the positrons have stopped in the films. The smooth lines
in the region between the upper and lower horizontal lines correspond to
a polynomial fitting grade 3. The energy values in Table 5.1 are thus
extracted from the intercept of the polynomial fitting with S = S1/2 .
80
5.3 Analysis and results
0.56
PMMA 400 nm WSU
PMMA 480 nm WSU
0.55
3
Normalised Peak rate (10 /s)
1.0
0.8
S-parameter PMMA 400 nm
S-parameter PMMA 480 nm
0.54
S-parameter
0.53
0.6
0.52
0.4
0.51
0.50
0.2
0.49
0.48
0.0
0 2 4 6 8 10 12 14 16 18 20 22 24 26
Positron implantation energy (keV)
(a)
2
4
6
8
10 12 14 16 18 20 22 24 26
0.60
1.0
S-parameter PS 670 nm Ghent
PS 670 nm WSU
0.8
0.58
0.56
0.6
0.54
0.4
0.52
0.50
0.2
S-parameter
3
Normalised Peak rate (10 /s)
0
0.48
0.0
0.46
0
2
4
6
8
10 12 14 16 18 20 22 24 26
Positron implantation energy (keV)
(b)
Figure 5.9: Transmission experiments performed at Washington
State University (WSU) to some of the free-standing polymer samples (filled squares). For comparison the resulting S-parameters
obtained in Ghent for the same samples is also displayed (empty
triangles). One can see that the two experiments more or less
overlap. This agreement gives support and a high degree of confidence that the findings are truthful.
81
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
Table 5.1: Extracted energy values E1/2 defined as the energy at which the median
penetration depth z1/2 (E) = polymer film thickness. At that point 50% of the
positrons have stopped in the films. The samples marked with star (?) were also
measured at Washington State University (WSU). The extracted energies from
these transmission experiments are also listed.
Polymer Material
PMMA
PMMA
PMMA (?)
PMMA (?)
PMMA
PS
PS
PS
PS
PS (?)
PS
Experimental
Thickness
(nm)
E1/2 from S at
Ghent
(keV)
220 ± 10
310 ± 10
400 ± 20
480 ± 10
1700 ± 50
210 ± 10
340 ± 10
460 ± 10
650 ± 20
670 ± 20
960 ± 20
4.75 ± 0.12
6.08 ± 0.09
6.52 ± 0.17
7.17 ± 0.06
13.73 ± 0.10
3.96 ± 0.03
5.39 ± 0.13
6.93 ± 0.10
7.79 ± 0.17
8.33 ± 0.17
9.45 ± 0.13
82
E1/2 from
transmission at WSU
(keV)
6.49 ± 0.08
6.96 ± 0.11
7.93 ± 0.07
5.3 Analysis and results
The values for the parameters α and n can be obtained as a result from a linear
regression of our data:
a
Y
z1/2
α
= En
ρ
⇒
bX
z }| { z }| { z }| {
log(z1/2 ρ) = log(α) + n log(E),
so it becomes linearized (y = a + bx). In that way the parameter n = b and the
parameter α is:
log(α) = a
⇒
α = 10a .
Thus, in order to obtain the values for the parameters α and n, the data points that
are listed in Table 5.1 on page 82 were plotted in the Figure 5.10. The error bars
of the data (PMMA (), PS (N) and transmission WSU (•)) were also taken into
account. 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 is
log(2) = 0.3. The lower limit log(0) = −∞. Thus, If we consider a certain data
value V with an upper value V + ∆V and a lower value V − ∆V ; the upper error
bars = 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 linear
regression log(z1/2 ρ) = 1.899 log(E) + 0.1340. For these fitting line, the errors of
the data on both axis were taken into account. With ρ the density, the resulting
fitting 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 Algers
et al. [153]. The corresponding fitting line (dashed line) was created by using
the 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 fitting
lines in Figure 5.10 are clearly different. As suggested in Subsection 3.2.3 on page
44 the differences does not arise from the shape of the profile. A probable reason
is that in the experiment of Algers et al., the spin-coated films were not detached
from the silicon substrate and subsequently the interaction at the interface with
the substrate would contribute to more annihilation of positrons in the polymer
than the expected in the self-supporting films.
In addition, the dotted line in Figure 5.10 was created with the most commonly 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 to
the dashed and solid lines are obvious. This suggest that the ‘standard’ values for
the parameters α and n are far from being the most appropriate for the analysis
of polymers.
83
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
2.8
2.6
2.4
log(z1/2 ρ)
2.2
2.0
1.8
PMMA thin films
PS thin films
Fit
PMMA Algers
PS Algers
Fit Algers
Common values
Transmission WSU
1.6
1.4
1.2
0.6
0.7
0.8
0.9
1.0
1.1
1.2
log(Energy)
Figure 5.10: Graphical representation of the power-law z1/2 (E) =
α n
ρ E according to the data of the transmission experiments. The results are compared with the results of Algers et al. [153] and with the
most commonly used values. The line that fits our data (filled squares
(), triangles (N) and transmission WSU (•)) is the result from a linear regression: log(z1/2 ρ) = 1.899 log(E) + 0.134; χ2 = 3.26. From
here that n = 1.90 ± 0.04 and α = 1.36 ± 0.04 µg cm−2 keV−n . The
data represented by open squares () and triangles (4) were taken,
respectively, from the PMMA and PS films measured by Algers et
al. [153]. The corresponding fitting line (dashed line) was created by
using the resulting values for the parameters α and n reported by Algers et al. The dotted line is created with the most commonly used
values for the Makhovian equation.
To clarify the legend in the figure, the circles (•) are the data corresponding to the transmission experiments performed in Washington
State University (WSU).
84
5.3 Analysis and results
It is also worthwhile to emphasize that for comparison a non-detached (from the
Si substrate) PMMA sample has also been measured. The obtained experimental
thickness was (220 ± 10) nm. The resulting S, S-W and W-parameters are shown
in Figure 5.11. The S-parameter has been analyzed with the well-known VEPFIT
program [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 the
film.
3. For comparison, the fitting was performed by employing the different values for the parameters α and n: the values proposed in Ghent-WSU, the
values proposed by Algers et al., and the most commonly used values.
Thus, the calculated thickness with the different values for the parameters α
and 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 suggest
that 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 = αρ E n ):
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 this
table.
For a visual interpretation, these different thicknesses of the samples are also
shown in Figure 5.12 for PMMA and in Figure 5.13 for PS.
85
86
-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
Non-detached PMMA thin film
Positron Implantation Energy (keV)
-2
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
0.545
0.545
0.555
0.550
S
0.550
0.555
0.560
Positron implantation energy (keV)
0.040
0.040
0.045
0.045
W
W
0.050
0.050
Figure 5.11: S, S-W and W-parameters of a Non-detached (from the SiO2 substrate) 220nm
PMMA thin film.
S
0.560
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
PMMA
PMMA
PMMA
PMMA
PMMA
PS
PS
PS
PS
PS
PS
PMMA (?)
PMMA (?)
PS (?)
Polymer Material
Experimental
Thickness
(nm)
220 ± 10
310 ± 10
400 ± 20
480 ± 10
1700 ± 50
210 ± 10
340 ± 10
460 ± 10
650 ± 20
670 ± 20
960 ± 20
400 ± 20
480 ± 10
670 ± 20
Obtained E1/2
(keV)
4.75 ± 0.12
6.08 ± 0.09
6.52 ± 0.17
7.17 ± 0.06
13.73 ± 0.10
3.96 ± 0.03
5.39 ± 0.13
6.93 ± 0.10
7.79 ± 0.17
8.33 ± 0.17
9.45 ± 0.13
6.49 ± 0.08
6.96 ± 0.11
7.93 ± 0.07
219 ± 31
351 ± 45
400 ± 62
480 ± 60
1648 ± 244
179 ± 18
321 ± 46
517 ± 69
646 ± 99
734 ± 112
933 ± 136
397 ± 51
453 ± 62
669 ± 86
Thickness
Ghent-WSU
(nm)
337 ± 65
514 ± 96
579 ± 121
682 ± 125
2070 ± 444
284 ± 43
482 ± 95
740 ± 143
904 ± 191
1014 ± 215
1258 ± 260
575 ± 107
648 ± 126
932 ± 177
Thickness
Algers et al.
(nm)
404 ± 78
600 ± 113
671 ± 141
781 ± 146
2209 ± 481
348 ± 54
570 ± 113
852 ± 166
1027 ± 218
1143 ± 244
1399 ± 293
666 ± 125
745 ± 147
1056 ± 204
Thickness
Common
values (nm)
Table 5.2: Comparison of the thicknesses (z1/2 ) of the thin polymer films obtained from the different values for the
parameters α and n that characterize the well-known power-law (z1/2 = αρ E n ) at the extracted energies E1/2 . The
used values of the parameters are: Ghent-WSU, n = 1.90(±0.04) and α = 1.36(±0.04) 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 Common values n = 1.60(±0.05) and α = 4.0(±0.3)
µg cm−2 keV−n . The extracted energy values from the Washington State University (WSU) experiments and their
corresponding thickness calculation are listed at the end of the table and marked with with a star (?).
5.3 Analysis and results
87
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
PMMA
2800
700
2600
600
2400
500
400
300
Thickness (nm)
Thickness (nm)
800
2200
2000
1800
1600
200
1400
100
900
800
700
700
600
500
400
Thickness (nm)
Thickness (nm)
800
600
500
400
Energy data From WSU
300
1000
1050
900
800
Thickness (nm)
Thickness (nm)
900
700
600
500
400
750
600
450
Energy data From WSU
300
Figure 5.12: Thicknesses of the PMMA samples obtained with the different values
for the parameters α and n compared with the experimental thickness values at
the extracted energy values E1/2 (from z1/2 (E) = αρ E n ).
The order of the data points, respectively, from left to right are: (1) experimental
thickness; (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
5.3 Analysis and results
PS
1400
600
400
300
1200
Thickness (nm)
500
1000
800
200
600
1100
1800
1000
1650
900
1500
800
700
600
Thickness (nm)
Thickness (nm)
Thickness (nm)
700
1350
1200
1050
500
900
400
750
1350
1200
Thickness (nm)
Thickness (nm)
1200
1050
900
750
1050
900
750
Energy data From WSU
600
600
Figure 5.13: Same caption that in the previous Figure 5.12 but in this case for the
PS films.
89
5. PARAMETERIZATION OF THE MEDIAN PENETRATION
DEPTH OF IMPLANTED POSITRONS IN FREE-STANDING
NANOMETRIC POLYMER FILMS
5.4
Summary
Positron transmission experiments have been performed on free-standing polystyrene
and poly(methyl-methacrylate) films of nanometric thicknesses.
The constants that parameterize the median penetration depth z1/2 (E) = αρ E n
have been found to be n = 1.90 ± 0.04 and α = 1.36 ± 0.04 µg cm−2 keV−n . These
values were successfully determined with the previous knowledge of the thickness
of the samples and with the data obtained from the transmission measurements
performed in two different positron facilities: (1) with the S-parameter as a function of the positron implantation energy and (2) with the peak rate obtained in
some of the samples.
As the results obtained from the experiments performed in Washington State
University resulted in a high agreement with the results obtained in Ghent, this
gives support and a high degree of confidence that the procedures and respective
findings are truthful.
The results suggest that special care has to be taken into account in selecting
correct values of the parameters when analyzing the experimental data as standard
values might lead to wrong results.
Our results seem to indicate that the parameters proposed by Algers et al. are
not valid in the case of self supporting films. However, when analyzing a nondetached sample from the Si substrate, their values are the ones that best fit our
data.
90
6
Determination of the positron
diffusion length in polymers by
analysing the positronium
emission
In this chapter are shown the results concerning some positron beam experiments
performed on the self-supporting poly(methylmethacrylate) (PMMA) film of 310
nm-thick (see Table 4.1 on page 62) and on the Kaptonr samples, both described
in Chapter 4.
The positronium (Ps) emission from the PMMA and Kapton surfaces is studied as a function of the positron implantation energy by using Doppler profile
spectroscopy and Compton-to-peak ratio analysis.
This experiment is also interesting because Ps is formed only in PMMA and
not in Kapton.
6.1
Introduction
Many solids are known to emit positronium when bombarded by low-energy positrons. The study of the positron motion is important for understanding the
interactions of positrons with matter. The Ps emission is a phenomenon particularly interesting in some metal oxides and in polymeric materials. An overview
about the several mechanisms at the basis of Ps emission in insulators can be found
in Reference [158]. The mechanisms of Ps formation at the materials surface have
been described by the following processes:
91
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
a. Implanted positrons can get trapped into a surface state that can be subsequently thermally activated into Ps emission [159].
b. Implanted positrons can reach the surface, capture an electron at the surface
and 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 the
surface where it is emitted [160].
The Doppler broadening of annihilation radiation (DBAR) technique (described
in section 2.3 on page 27) is one of the methods that may be used to study the
emission of para-positronium (p-Ps) (see for example [46]).
DBAR measurements performed in a longitudinal setup at the Variable Energy
Positron beam (VEP) in Ghent have been described in reference [161]. In these
setup 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 sample
has a linear momentum mainly away from the detector. This causes a red shift of
the p-Ps contribution in the annihilation spectrum. A ‘red-shifted’ contribution
means 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 side
of the 511-keV-line [161].
In this chapter it is proven that the detected photo-peak from DBAR experiments (after the background subtraction) can also be affected by the contribution
of the p-Ps emission at the high-energy side (blue-shift) or at the central part of
the photo-peak. A blue-shifted peak has an advantage over the red-shifted because
the Compton background contribution that appears at the low-energy tail of the
detected photo-peak can be avoided.
Two different specimen-detector geometries are thus proposed: The polymer
sample is located (1) at 45◦ and (2) perpendicular with respect to the positron
beam axis. The detector is located beside the sample position, but perpendicular
to the positron beam line.
For the PMMA sample, the following parameters are calculated and discussed in
this 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 diffusion
length and the efficiency for the emission of Ps by picking up an electron from the
surface were obtained.
6.2
Experimental
This section describes the experimental parameters that were not already given in
Chapter 4.
• All the experiments were performed at the variable energy positron beam in
Ghent [58].
• Two different specimen-detector geometries were performed for these DBAR
measurements. They are schematically represented in Figure 6.1:
1. In the first case the polymer sample was located at 45◦ with respect
to the positron beam axis. The HPGe-detector was located beside the
sample position, but perpendicular to the positron beam line (Figure
6.1(a)).
2. In the second case, the polymer sample was located perpendicular with
respect to the positron beam axis. The HPGe-detector was also located beside the sample position and perpendicular with respect to the
positron beam axis. The scheme is shown in Figure 6.1(b).
93
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
Sample
Orientation
Angle
Positron beam
Positron beam
Sample
Sample
45°
Beam axis
Beam axis
Detector axis
Detector axis
HPGe
detector
HPGe
detector
(a)
(b)
Figure 6.1: Experimental setup with the sample at (a) 45◦ and (b) perpendicular with respect to the beam axis.
⇒ For the 310 nm-thick PMMA sample:
- The DBAR spectra were collected for several implantation energies
from 0.1 to to 1.2 keV. The measurements were recorded at each
positron implantation energy for every 30 minutes.
⇒ For the Kapton sample:
- The DBAR measurements were recorded every 30 minutes minimum three times.
- Different implantation energies were used depending on the specimendetector 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 the
effect of charging of the sample. Three successive measurements
were done for testing the reproducibility and hence the effect of
charging:
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 1.1
keV to 0.1 keV at the same intervals. Finally for the third run,
the positron implantation energy was increased again in the
same way as for the first run.
The peak statistics as a function of the positron implantation energy 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.
38
36
34
1st
2nd
3rd
30
4
Peak Statistics (x 10 )
32
28
26
24
22
20
18
16
14
12
10
0
200
400
600
800
1000
Positron implantation energy (eV)
Figure 6.2: Comparison of the peak statistics as a function of
the positron implantation energy in Kapton for three successive
measurements. It was performed to test the reproducibility and
thus the effect of charging. The positron implantation energy was
increased during the first and third runs and was decreased in in
the second run. The setup used for this experiment was with the
Kapton perpendicular with respect to the beam axis (Fig. 6.1(b)).
Figure 6.3 shows the peak statistics as a function of time for a
Kapton sample. The experiment was performed with the sample
at 45◦ with respect to the beam axis. All the data points were
collected every 30 min at 0.1 keV. The fitting curve in the figure,
95
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
which represents a charging time constant τcharge is an exponential
decay y(t) = y0 + A exp(−t/τcharge ). Here y(t) represents the peak
counts as a function of time. Fitting this equation to the data of
Figure 6.3 resulted in a charging time constant of τcharge = 7.26
hours for Kapton.
In the following experiments the data are taken only from the first
run so the charging effects are are the lowest possible (see Coleman
et al. [141] and Ito et al. [155]). In addition, the influence of the
charging-up of the sample is only given below ∼300 eV and the
measuring time for each data point is relatively short (30 min) in
comparison to the charging time constant at these low energies. A
complete run for different implantation energies from 0.1 to 0.75
keV consists of 7 data files i.e. ∼ 3.5 hours, which is not even the
charging time constant. In addition, after 2 hours the sample is
already above 0.3 keV where the charging effect does not longer
affect the measurements (see Figure 6.2).
310000
Kapton
300000
Peak Statistics
290000
280000
270000
260000
250000
240000
230000
0
500
1000
1500
2000
2500
Time (min) (30 min each point)
Figure 6.3: Charging test for Kapton. The setup used for this
experiment was with the Kapton at 45◦ with respect to the beam
axis (Fig. 6.1(a)). The data points correspond to the peak statistics as as a function of time for implanted positrons at 0.1 keV. The
fitting curve is an exponential decay y(t) = y0 + A exp(−t/τcharge )
that represents the charging time constant. It resulted to be
τcharge = 7.26 hours for Kapton.
96
6.3 Analysis and results
• Time-of-flight (TOF) spectroscopy which uses a specialized setup [160] is
the basic method to investigate Ps emission in a direct way. In those experiments a detailed description of the energy distribution of the emitted p-Ps
can be obtained. Since we are only interested in the fraction and energy shift
of the emitted p-Ps from the sample surface, the detected photo-peak (by
only one Ge-detector) can (in first approximation) be fitted with a superposition of Gaussian distributions whose components arise from the different
annihilation channels [44, 101].
As Ps is only present in PMMA and not in Kapton, all the DBAR spectra
were independently analyzed with a sum of four Gaussians for the PMMA
film (see Fig. 6.4) and three Gaussians for the Kapton (Fig. 6.5). For the
analysis the DBAN program was used [162].
As the detected photo-peak has a big contribution from the Compton background at the low-energy side, it is necessary to subtract it and thus, a more
or 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 annihilation
peak and (3) a line with negative slope from the right of the peak. The
resolution function is convoluted and then a minimization by least square
method is performed1 .
Thus, once the stepwise background is subtracted, in the DBAN program
the 511-keV-line can be fitted with a sum of Gaussians (up to four). This
is convoluted with the resolution function of the detector, which is also
taken as a Gaussian, so the convolution results in a Gaussian. The observed
(FWHMf it ) values for the Gaussians are finally obtained.
6.3
Analysis and results
In Figures 6.4 and 6.5 all the contributions of annihilation of positrons with low
and high-momentum electrons are represented by Gaussians:
- In both cases (PMMA and Kapton) for the fitting of the detected photo-peak
the fixed parameters were:
a. The peak-shift of the component 1. It was centered with respect to the
511-keV-line (i.e., 0 keV in the figures).
1 For clarity about the code and functionality of the DBAN program, please contact his author
Nikolay Djourelov at [email protected]
97
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
b. A full width at half maximum (FWHM) of 1.1 keV for the component
3.
- The two main contributions (labeled with 1 and 2) have about the same
centroid and they describe fairly well the low and the high-momentum contribution 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 a
strong 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 is
oriented mainly towards the detector. This allows the detection of the
p-Ps emission as a narrow fly-away peak at the high-energy side of the
511-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 has
a big contribution of the Compton background which arises from the 1274
keV γ−ray of 22 Na. Therefore, a blue-shifted peak has an advantage over the
red-shifted because these Compton background contribution can be avoided.
- In the case of PMMA, the fourth contribution is centered with respect to
the 511-keV-line and has a narrow FWHM. It is identified as annihilation of
p-Ps in the bulk. This only applies to the PMMA because Ps is not formed
in bulk of Kapton, but it does in PMMA.
1 For the importance of the low and high momentum contribution of the annihilation of
positrons see e.g. the final paragraph of subsection 2.3.1 on page 29.
98
6.3 Analysis and results
5000
5000
4500
4500
4000
3500
Peak Counts
Peak Counts
3500
3000
2500
2000
3000
2500
2000
1500
1500
1000
1000
500
4
2
0
−4
1
4000
1
−3
−2
−1
0
4
500
3
3
2
1
2
3
4
0
−4
−3
−2
−1
0
1
2
3
Energy Shift(keV)
Energy Shift(keV)
(a)
(b)
Figure 6.4: Annihilation peak obtained for the PMMA sample for an
implanted positron energy of 467 eV.
(a) The sample is at 45◦ with respect to the beam axis. The fitting is
done with (1) a low momentum, (2) high momentum (dashed line), (3) a
contribution from the emitted p-Ps which is blue-shifted with respect to
the 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 fitting
is done with (1) a low momentum, (2) high momentum (dashed line), (3)
a centered contribution from the emitted p-Ps with respect to the 511keV-line and (4) annihilation of p-Ps in the bulk (dotted line).
99
4
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
4
14
x 10
12000
1
12
10000
1
8000
Peak counts
Peak counts
10
8
6
6000
4000
4
2000
2
2
0
−5
−4
−3
−2
−1
3
3
2
0
1
2
3
4
5
Peak shift (keV)
0
−5
−4
−3
−2
−1
0
1
2
3
4
Peak shift (keV)
(a)
(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 implanted positron energy of 103 eV. The fitting is done with (1) a low
momentum, (2) high momentum (dashed line) and (3) with respect to
the 511-keV-line, (a) a blue-shifted or (b) centered contribution from the
emitted p-Ps.
100
5
6.3 Analysis and results
Both, positron and Ps can diffuse back to the surface. First we will consider
the positrons and then Ps.
If Ps is formed in the bulk of the material, we may assume that the initial
distribution of Ps is equal to the positron implantation profile (see subsection 3.2.3
on page 44). We call fPs the fraction of positrons that form Ps. The remaining
fraction (1- fPs ) of positrons can diffuse back to the surface and may emerge as
o-Ps or p-Ps by picking up an electron from the surface. The fraction that captures
such an electron is fpu . The situation is schematically represented in Figure 6.6.
Vacuum
Sample
f pu
e+
Ps
F+(E)
Fp-Ps(E)
1-f Ps
e+
fs
(1/4)f Ps
f Ps
(3/4)f Ps
Fo-Ps(E)
Ps
fb
Fb(E)=[1- f p-Ps(E)]
Figure 6.6: Schematic representation of the positronium emission
from a sample surface.
The bulk Ps diffuses and is either trapped in free-volume sites or reaches the
surface whereupon it is ejected. As stated in Ref. [161], due to the self-annihilation
in these free-volume sites, only a fraction fb of p-Ps is observed as a narrow central
contribution in the Doppler line-shape.
h m i
m−1
Given the implantation profile P (E, z) = mzzm exp − zz0
, (with z0 =
0
m+1 −1
z̄ Γ m
), the probability of reaching the surface is proportional to [131,132]:
Z
Fj (E) =
0
∞
z
P (E, z) exp −
Lj
dz,
101
(6.1)
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
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 contribution in the detected photo-peak (in the PMMA case).
can be described by the following set of equations:
e
Ip-Ps
e
Io-Ps
b
Ip-Ps
1
=
[f F+ (E) + fPs fe Fp-Ps (E)] ,
4
3
=
[f F+ (E) + fPs fe Fo-Ps (E)] ,
4
1
=
fb fPs [1 − fe Fp-Ps (E)] ,
4
(6.2a)
(6.2b)
(6.2c)
where f = fpu (1 − fPs ) and fe represents the emission efficiency. Figure 6.6 is
also 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 peak
will be observed. In this case, the equations are reduced to:
e
Ip-Ps
= (1/4)F+ (E)fpu ,
e
Io-Ps
= (3/4)F+ (E)fpu
and
(6.3)
The experimental fraction of emitted o-Ps at implantation energy E is obtained
from a Compton-to-peak ratio analysis of the annihilation spectrum2 [138]:
exp
Io-Ps
−1
P (0) R(0) − R(E)
=α 1+
P (∞) R(E) − R(∞)
(6.4)
where P is the number of counts accumulated in the region centered around the
511-keV annihilation line and R = C/P is the ratio of the number of counts in a
1 For a brief description of the diffusion length refer to the subsection 3.2.4 on page 47 (the
diffusion length is defined after Equation (3.8)).
2 see also Section 3.3 on page 50 and Appendix B on page 117.
102
6.3 Analysis and results
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(∞) are
the 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 where
no o-Ps is detected by three-quantum annihilation.
6.3.1
Position of the p-Ps contribution in annihilation spectra
When p-Ps is emitted out of the sample in the vacuum, its distance to
the detector increases or decreases, depending on the detector-sample
position, before the annihilation takes place. This distance influences
the detection efficiency and thus the intensity of the p-Ps contribution
to the peak.
The distance between the sample and the annihilation position can be
estimated using the law for a statistical decay process:
t
p-Ps
−τ
N (t) = N0 e
(6.5)
with N (t) the number of particles that are not annihilated at time t and
N (0) the number of particles at time zero. τp-Ps is the mean lifetime of
p-Ps in vacuum, which is 124 ps. The time dependence can be replaced
by a place dependence x = vp-Ps t, with x the distance from the sample
to the annihilation position and vp-Ps the p-Ps velocity. This leads to1 :
−v
N (x) = N0 e
x
p-Ps τp-Ps
−
= N0 e
r Ep-Psx
me
τp-Ps
(6.6)
If one takes the representative numerical example of p-Ps emitted with
an 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, a
percentage of 99.9% of the p-Ps annihilates within the distance of 0.36
mm. For positrons with emission energies up to 5 eV this distance is
0.7 mm. This means a maximum decrease of the detection efficiency of
1% for the longest living Ps. Thus one can consider that the influence
on the detection efficiency for this distance is negligible.
For o-Ps the same equation can be used, but with a mean lifetime of
142 ns. Since its lifetime is approximately 1000 times larger than the
1 By simple calculations, one can determine the energy and velocity of the emitted p-Ps. See
for example the reference [163]
103
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
1
0.9
1 eV
5 eV
0.8
0.7
N/N0
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Distance from the sample (mm)
Figure 6.7: Free p-Ps population as a function of the distance
between the sample and the annihilation position. The p-Ps is
emitted with an energy of 1 eV and 5 eV.
p-Ps lifetime, most of o-Ps will annihilate within a distance of a few
centimeters. For this range the detection efficiency strongly depends
on 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 the
corresponding three-quantum annihilation is increased by a factor ≈ 2.26 with
respect to all two-quantum annihilation1 . Such an effect is taken into account by
adjusting the value of the proportionality constant α in Equation 6.4 in the way
that Io-Ps (0)/Ip-Ps (0) = 3.
With the sample at 45◦ (Figure 6.1(a)), the experimental data for the intensities
of the fly-away p-Ps, the fly-away o-Ps and the bulk p-Ps for the PMMA film are
shown in Fig. 6.8. The solid lines represent the simultaneous fitting of Eqs. (6.2) to
determine 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 for
the 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 that
the intensity of the bulk p-Ps increases as function of the positron implantation
energy mainly at the expenses of the intensity of the emitted p-Ps (blue-shifted
contribution).
2
solid angle is given by Ω = π d2r+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,
2
Ω1 = 0.27 sr and with d = 5 cm, Ω2 = 0.61 sr. The ratio is therefore, Ω
= 2.26.
Ω
1 The
1
104
6.3 Analysis and results
18
Positronium intensity (%)
16
14
12
10
8
6
4
2
0
−2
0
0.2
0.4
0.6
0.8
1
1.2
Positron implantation energy (keV)
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 (•) and
the intensity of the emitted o-Ps (4). The solid lines represent
the fit of Eqs. (6.2) to determine the positron diffusion lengths
(thermal (and epithermal)) L+ , Lp-Ps and Lo-Ps . The fitting was
performed using the values for the parameters found in the previous Chapter for the Makhovian equation (n = 1.90(±0.04) and
α = 1.36(±0.04) µg cm−2 keV−n ).
105
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
For comparison, a fitting was done by using the most frequently used values
of 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 proposed
by Algers et al. (n = 1.71(±0.05) and α = 2.81(±0.20) µg cm−2 keV−n ). All the
resulting values are summarized in Table 6.1.
Table 6.1: Comparison of the values obtained from the fitting of the experimental
intensities of the fly-away p-Ps, the fly-away o-Ps and the bulk p-Ps for the PMMA
film by using the the different values for the parameters α and n that characterize
the well-known power-law equation (z1/2 (E) = αρ E n ). 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) and α = 1.36(±0.04) µg cm−2 keV−n .
Common values
Algers et al.
Ghent-WSU
L+ (nm)
Lp-Ps (nm)
Lo-Ps (nm)
fPs
fpu
5.18 ± 0.20
3.46 ± 0.01
1.78 ± 0.22
12.68 ± 0.09
8.38 ± 0.09
3.63 ± 0.10
8.67 ± 0.08
5.82 ± 0.03
2.66 ± 0.09
40.07 ± 0.03
39.87 ± 0.03
39.24 ± 0.03
0.47 ± 0.08
0.47 ± 0.08
0.47 ± 0.08
From Table 6.1 it is clear that the values for the diffusion lengths are dependent
on the values of α and n used to describe the implantation profile. The values for
the total bulk Ps formation (fPs ) and the fraction of surface positrons that pick up
an electron (fpu ) do not depend on the different values for α and n used to describe
the implantation profile. From this it can be concluded that for determining the
diffusion lengths, special care has to be taken into account for describing the
implantation profile.
The experimental values for Lp-Ps and Lo-Ps are within the experimental error
not the same but they are comparable. This can be explained within a model as
proposed by Van Petegem et al. [161]:
The diffusion
length of positrons is related with the diffusion coefficient (D+ )
p
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 particle
into some sink. As in the bulk all positronium is efficiently trapped into
pthe free
volume sites, κ λ for both, o-Ps and p-Ps. Therefore, Lo-Ps ≈ Lp-Ps ≈ DP s /κ.
However, one has to be aware that this experiments are performed at very
low implantation energies (from 0.1 to to 1.2 keV) and epithermal positrons may
reach the surface, in which case it is the diffusion of a positron that is not in
thermal equilibrium. This may enhance the emission for very low energies, and
106
6.3 Analysis and results
thus shorten the apparent diffusion length L+ .
It is worthwhile to emphasize here that when the PMMA sample is perpendicular to the beam axis (Figure 6.1(b)), when increasing the positron implantation
energy, 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 4
in the same figure) because they are at the same position. Thus, this fitting was
not performed.
Figures 6.9 and 6.10 show the intensities of the fly away p-Ps and the fly away
o-Ps as function of the positron implantation energy for the Kapton sample as well
as the results from the simultaneous fitting of the Equations (6.3).
Positronium intensity (%)
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Positron implantation energy (keV)
Figure 6.9: Ps emission from the Kapton surface when the sample
is at 45◦ with respect to the beam axis as a function of the implantation 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 epithermal)) L+ . The fitting in the figure was performed using the
values for the parameters α and n in the Makhovian equation
found 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 with
increasing the implantation energy and at about 850 eV the intensity of the p-Ps
peak (i.e. the component 3 in Fig. 6.5) is too low to be clearly distinguished from
107
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
the broad central Gaussian (component 2 in Fig. 6.5).
20
Positronium intensity (%)
18
16
14
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Positron implantation energy (keV)
Figure 6.10: Ps emission from the Kapton surface when the sample is perpendicular with respect to the beam axis as a function
of the implantation energy. In the figure are shown the intensity
of the p-Ps (×), o-Ps (•) and the fit (solid lines) of Eqs. (6.3)
to determine the positron diffusion length (thermal (and epithermal)) L+ . The fitting in the figure was performed using the values for the parameters α and n in the Makhovian equation found
by Ghent-WSU in the previous Chapter (n = 1.90(±0.04) and
α = 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 the
surface and are emitted as Ps (fpu ) are listed in Table 6.2.
108
6.3 Analysis and results
Table 6.2: Comparison of the values obtained from the fitting of the experimental
intensities (at 45◦ and perpendicular with respect to the beam axis) of the flyaway p-Ps and the fly-away o-Ps for the Kapton sample by using the different
values for the parameters α and n that characterize the well-known power-law
equation (z1/2 (E) = αρ E n ) 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 surface
and 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) and
α = 1.36(±0.04) µg cm−2 keV−n . For the calculation of the diffusion coefficient
D+ , τ ∗ was considered to be 382 ps [133].
L+ (nm)
D+ (×10−4 cm2 /s)
fpu
◦
Common values
Algers et al.
Ghent-WSU
Common values
Algers et al.
Ghent-WSU
Sample at 45
8.22 ± 0.41
17.69 ± 1.76
5.51 ± 0.28
7.95 ± 0.81
2.41 ± 0.12
1.52 ± 0.15
Sample perpendicular
8.17 ± 1.06
17.47 ± 4.53
5.43 ± 0.71
7.72 ± 2.02
2.33 ± 0.31
1.42 ± 0.38
0.158 ± 0.003
0.155 ± 0.003
0.151 ± 0.003
0.250 ± 0.012
0.247 ± 0.012
0.241 ± 0.011
The differences of the values used for the parameters α and n for the data
analysis are clear when inspecting the Table 6.2. In the table, one can see that the
values for the parameters α and n have a big influence on L+ (or D+ ) depending
on the chosen model.
The only experiment found in the literature with the purpose of investigating
the positron mobility in Kapton was done by Brusa et al. [164]. They have analyzed
the drift of positrons under the action of an electric field. These experiments were
performed in the energy range between 0.85 keV and 2.5 keV. Our experiments
were performed with positrons with an energy below 0.85 keV where positronium
emission is possible.
This positron mobility experiment in Kapton resulted in a very small diffusion
coefficient: 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 values
listed in Table 6.2.
Both experiments (Ghent and Brusa’s) have different strengths and weaknesses
that can be summarized as follows:
109
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
1. Experiments performed in Ghent: The positron mobility is obtained by
means of a non-conventional method which is the positronium emission.
However, in Table 6.2, notice that once the values for the parameters α
and n that describe the implantation profile have been chosen, the analysis of two independent experiments (sample at 45◦ and perpendicular with
respect to the beam axis) produce the same results for L+ (within the errors) 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 at
the very low energy values E <∼300 eV).
b. If the model proposed by Coleman et al. [141] is valid (see a review
in Section 3.4), there might be an electric field drifting the positrons
towards the sample surface. This means that the positron mobility µ+
is affected by this electric field. As a consequence, the measured value
of the positron diffusion coefficient can be influenced by the charging
effects.
2. Experiments performed by Brusa et al.: The positron mobility is obtained in
a standard manner. By employing high implantation energies they measure
thermalized positrons and they can also avoid the influence of the charging
effects. However:
a. The authors measured the same Kapton sample type, from the same
manufacturer than for the experiments in Ghent. However, they used a
sample density of (ρ = 1.7 g cm−3 ). The experimental value measured
in Ghent for the sample density was 1.4 g cm−3 which is the same as
the 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 the
parameters α and n differ for metals and polymers the positron implantation profiles should be different for a bi-layered material that contains
metal-polymer.
c. They used the values for the parameters α and n that best fitted their
data. 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 values
found for polymers by Algers et al. and in Ghent-WSU (see the previous
chapter). The value found for α by Brusa et al. is even too high in
comparison 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 technique 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 charging and epithermal positrons. However, it appears as a promising technique that
in principle might be accessible for many positron laboratories.
6.4
Summary and Conclusions
The emission of Positronium as a function of the positron implantation energy
from a 310 nm-thick PMMA film and from a Kapton surface has been studied by
means of Doppler broadening, blue-shift spectroscopy and Compton-to-peak ratio
analysis.
In the analysis the detected photo-peak has been approximated by a sum of
Gaussians.
The narrow component attributed to the p-Ps emission is centered with respect
to the 511-keV-line when the sample and the detector are perpendicular to the
positron beam axis. However, it is shifted towards high energy when the sample
is at 45◦ with respect to the beam axis. This fact has been explained as the
linear momentum of the Ps emitted at the surface of the sample is oriented mainly
towards the detector causing a blue shift of the p-Ps.
From a detailed analysis of the experimental results, the thermal (and epithermal) positron diffusion lengths, the fraction of positrons that pick up an electron
from the surface and form positronium, and in the case of the PMMA also the
fraction fPs of positronium formed in the bulk can be obtained.
In the case of Kapton, the analysis of two independent experiments suggest
that the positron diffusion coefficient might not be as low as the one proposed by
Brusa et al. [164].
Special care has to be taken into account in selecting the correct values for
the parameters α and n that describe the implantation profile when analyzing the
experimental data as standard values might lead to wrong results.
The experiments performed on the Kapton sample also suggest that once the
values for the parameters α and n that describe the implantation profile have
been chosen, the analysis for two independent experiments (sample at 45◦ and
perpendicular) produce the same results for L+ (within the errors) which gives us
an 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 need
of a complicated setup. This, therefore, opens a new field of possible experiments
that might be accessible for many positron laboratories.
111
6. DETERMINATION OF THE POSITRON DIFFUSION LENGTH
IN POLYMERS BY ANALYSING THE POSITRONIUM EMISSION
112
A
Basic derivation for the
momentum measurements
In this appendix is presented the derivation necessary to obtain the energy of the
annihilation γ−rays and also the angular correlation of the two γ−ray decay, which
are 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 Doppler
νL
shift of the two γ−ray energies (see Fig. 2.6). the doppler shift is ∆ν
ν = c (with
PL
the longitudinal center-of-mass velocity νL = 2m
). For energy E = me c2 the
e
development is as follows:
As the energy of a photon is proportional to its frequency
∆E
E
⇒ ∆E
∆E
∆E
∆ν ∆ν
⇒ ∆E = ±
E
ν
ν
ν PL
L
= ±
E, but νL =
, then:
c
2me
!
PL
E, also E = me c2 , so that
= ±
2me c
!
PL
2
H
= ±
mH
ec 2
m
ec
∝
113
A. BASIC DERIVATION FOR THE MOMENTUM
MEASUREMENTS
∴ ⇒
A.2
∆E = ±
PL
c
2
(A.1)
angular correlation of the two γ−rays decay
In the center-of-mass system the energy of each annihilation γ−ray is exactly
me c2 = 511 keV, and the emission angle between the directions of the two photons
is 180◦ . In the laboratory system, this is true only if the positron-electron pair
has no kinetic energy. When slowing down in matter, most positrons thermalize
before annihilation and the momentum of the center of mass motion of an positronelectron 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 relevant
formulae will be quoted without derivation1 . In particular, the total energy, E, of
a particle having a rest mass, m0 and moving with a velocity v is given by
E=
mc2
1−
v2
c2
(A.2)
12
It is also known in special relativity that the energy and momentum are related
by:
E 2 = p2 c2 + m20 c4
(A.3)
Since the velocity of a photon is c and its energy E = hν = hc
λ is finite, we see
from (A.2) that we must take the mass of a photon to be zero, in which case we
observe from (A.3) that the magnitude of its momentum is
p=
E
h
=
c
λ
(A.4)
Now, we start with the derivation. For clarity, see Fig. 2.6. From the energy
conservation:
E1 + E2 = 2m0 c2 , but
Pc = E1 + E2 , so we have:
(A.5)
≈1
z }| {
PL c = E1 − E2 cos(θ)
⇒
PL c = E1 − E2 ,
and therefore:
1 For 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 + PL c,
E2 = E1 − PL c.
and also:
(A.6)
Now replacing Eq. (A.6) in (A.5):
2E2 + PL c = 2m0 c2 ⇒ E2
E1
PL c
, and also:
2
PL c
= m0 c2 +
.
2
= m0 c2 −
(A.7)
Now we make E = pc (from Eq. (A.4)):
⇒ P2 = m0 c −
PL
,
2
and also:
P1 = m0 c +
PL
.
2
On the other hand, by looking at the transverse components:
≈θ
z }| {
PT c = E2 sin(θ) +H
E1
0
H
⇒
PL c , but
PT c = θ m0 c2 −
2
⇒
∴ ⇒
PT c = θE2 ,
and replacing (A.7):
PL c
m0 c2 ,
2
PL c
≈ 0, so we have: PT c = θm0 c2
2
θ=
PT
m0 c
(A.8)
115
A. BASIC DERIVATION FOR THE MOMENTUM
MEASUREMENTS
116
B
Positronium fraction from the
Compton-to-peak ratio analysis
of the annihilation spectrum
Ps may be detected by studying the energy spectrum of its annihilation photons
because the 3 S1 state decays into 3γ’s with a continuous energy distribution in the
range 0 to mc2 = 511 keV, whereas positrons which annihilate with an electron of
a solid decay principally by 1 S0 overlap via 2γ’s with energy mc2 .
Assume that a fraction f of the positrons which annihilate in the target region
form Ps. Let Tf and Pf be the “total” and “photopeak” detector counting rates
for a given number of positrons annihilating per second. It is evident that
Tf = f Tk + (fk − f )T0 (B.1) , and
Pf = f Pk + (fk − f )P0 (B.2)
where the subscripts 1 and 0 refer to f = 1 and f = 0 respectively. To eliminate
any 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
Rf Pf
Tf − Pf
, now replacing (B.1) and (B.2):
Pf
= f Tk + (fk − f )T0 − f Pk − (fk − f )P0
=
substituting T = RP + P (from the ratio R):
Rf Pf
Rf Pf
Rf Pf
=
=
=
f (Pk + Rk Pk ) + (fk − f )(P0 + R0 P0 ) − f Pk − (fk − f )P0
H
H
HH
HH
f P
fk
P
f P
fk
P
k + f Rk Pk + 0 − fP
0 + fk R0 P0 − f R0 P0 − k −
0 + fP
0
H
H
H
H
f Rk Pk + fk R0 P0 − f R0 P0
117
B. POSITRONIUM FRACTION FROM THE COMPTON-TO-PEAK
RATIO ANALYSIS OF THE ANNIHILATION SPECTRUM
Replacing (B.2):
Rf (f Pk + (fk − f )P0 ) = f Rk Pk + fk R0 P0 − f R0 P0
f Pk Rf + fk P0 Rf −f P0 Rf = f Rk Pk + fk R0 P0 −f R0 P0
| {z }
| {z }
⇒ fk P0 Rf − fk R0 P0 = f P0 Rf − f Pk Rf + f Rk Pk − f R0 P0
fk (P0 Rf − R0 P0 ) = f (P0 Rf − Pk Rf + Rk Pk − R0 P0 )
fk (P0 Rf − R0 P0 )
f =
P0 Rf −Pk Rf + Rk Pk − R0 P0
| {z }
| {z }
f
f
f
"
∴ ⇒
f = fk
fk (P0 Rf − R0 P0 )
P0 (Rf − R0 ) + Pk (Rk − Rf )
#−1
"
P0 (Rf − R0 ) + Pk (Rk − Rf )
= fk
(P0 Rf − R0 P0 )
"
#−1
P0 (Rf − R0 ) + Pk (Rk − Rf )
= fk
P0 (Rf − R0 )
=
Pk (Rk − Rf )
1+
P0 (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 determined using the energy spectrum and calibration values for the counts in the
511-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 known
nonzero 3γ annihilation yield (fk ) (e.g. in Kapton). In Kapton Ps is not formed
in the bulk.
o−Ps which escapes from the sample will annihilate in the vacuum via 3γ annihilation, while the e+ annihilation is mostly via 2γ annihilation (the 3γ positron
annihilation cross section is only is 1/372 of the one of 2γ positron annihilation
and can be neglected). If there is Ps formation there will be 3γ o−Ps annihilation
with a fraction dependent on the o−Ps lifetime. The relative ratio of the 3γ/2γ
annihilation is known as Compton-to-peak ratio (sometimes this parameter is also
described 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. Science, 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 of
penetrating 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 an
electron-positron pair. PhD thesis, Yale University, 1950.
[11] J.A. Rich. Experimental evidence for the three-photon annihilation of an
electron-positron pair. Phys. Rev., 81:140, 1951.
[12] A. Øre and J.L. Powell. Three-photon annihilation of an electron-positron
pair. Phys. Rev., 75(11):1696, 1949.
119
BIBLIOGRAPHY
[13] M. Charlton. Positron physics. In Cambridge monographs on atomic, molecular and chemical physics, volume 173, Cambridge, 2001. Cambridge University Press.
[14] W.R. Johnson, D.J. Buss, and CO. Carroll. Single-quantum annihilation of
positrons. Phys. Rev., 135, 1964.
[15] J.C. Palathingal, P. Asoka-Kumar, G. Lynn, and X.Y. Wu. Nuclearcharge and positron-energy dependence of the single-quantum annihilation
of 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 spectrum
in 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 annihilation
of a positron-negatron pair. Phys. Rev., 96:691, 1954.
[21] S. Mohorivičić. Möglichkeit neuer elemente und ihre bedeutung für die astrophysic. 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 fine
structure of positronium. Phys. Rev., 84:601, 1951.
[26] J. Pirenne. Le champ propre et l’interaction des particules de dirac suivant
l’´lectrodynamique quantique. Arch. Sci. Phys. Nat., 28:233, 1946.
[27] C.N. Yang. Selection rules for the dematerialization of a particle into two
photons. Phys. Rev., 77:242, 1950.
120
BIBLIOGRAPHY
[28] S. Adachi, M. Chiba, T. Hirose, S. Nagyama, Y. Nakamitsu, T. Sato, and
T. 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. Measurement 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 quantities
and 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 decay
rate of ortho-positronium. Phys. Rev. A, 27:530, 1983.
[34] J.S. Nico, D.W. Gidley, A. Rich, and P.W. Zitzewitz. Precision-measurement
of 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 orthopositronium 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. Precision
redetermination of the fine-structure interval of the ground state of positronium 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 the
decay-rate of singlet positronium. Phys. Rev. Lett., 72:1632, 1994.
[39] I.J. Rosenberg, A.H. Weiss, and K.F. Canter. Positronium emission from
metal-surfaces. J. Vac. Sci. Technol., 17:253, 1980.
[40] R.M. Nieminen and J. Oliva. Theory of positronium formation and positron
emission 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-density
gases. In P.G. Coleman, S.C. Sharma, and L.M. Diana, editors, Positron
Annihilation, 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 formation
and 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 formation and diffusion in crystalline and amorphous ice using a variable-energy
positron beam. Phys. Rev. B, 32(11):7048, 1983.
[47] S.V. Stepanov and V.M. Byakov. Electric field effect on positronium formation in liquids. J. Chem. Phys., 116 (14):6178, 2002.
[48] S.V. Stepanov and V.M. Byakov. Physical and radiation chemistry of the
positron 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 in
molecular 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 molecules
and 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-positronium
in 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. Hasiguti
and K. Fujiwara, editors, Positron Annihilation, page 601, Tokyo, 1979.
Japan Institute Met.
[58] J. De Baerdemaeker and C. Dauwe. Development and application of the
Ghent pulsed positron beam. Appl. Surf. Sci., 194, 2002.
[59] S. DeBenedetti, F.K. McGowan, and J.E. Francis Jr. Self-delayed concidences 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 in
liquids 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, and
A. Siegle. Age-momentum-correlation (amoc) experiments by means of an
mev 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 dependence
of 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 in
fluids. World Scientific, Singapore, 1988.
[67] B. Jasińska, A.E. Koziol, and T. Goworek. Ortho-positronium lifetimes
in nonspherical voids. Journal of Radioanalytical and Nuclear Chemistry,
210(2):617, 1996.
[68] T. Goworek, K. Ciesielski, Jasińska B., and J. Wawryszczuk. Positronium
in large voids. silicagel. Chemical Physics Letters, 272:91, 1997.
123
BIBLIOGRAPHY
[69] T. Goworek, K. Ciesielski, Jasińska B., and J. Wawryszczuk. Lifetimes of
o-Ps in the pores of silica gel. Materials Science Forum (ICPA 11), 255257:296, 1997.
[70] T. Goworek, K. Ciesielski, Jasińska B., and J. Wawryszczuk. Positronium
states in the pores of silica gel. Chemical Physics, 230:305, 1998.
[71] T. Goworek, K. Ciesielski, Jasińska B., and J. Wawryszczuk. Mesopore
characterization by PALS. Radiation Physics and Chemistry, 68:331, 2003.
[72] M. Śniegocka, B. Jasińska, J. Wawryszczuk, Zaleski R., A. DeryloMarczewska, and I. Skrzypek. Testing the extended Tao-Eldrup model. silica
gels 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 method
to 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. Determination of pore size in mesoporous thin films from the annihilation lifetime of
positronium. J. Phys. Chem. B, 105(20):4657, 2001.
[76] K. Ito, H. Nakanishi, and Y. Ujihira. Extension of the equation for the
annihilation lifetime of ortho-positronium at a cavity larger than 1 nm in
radius. J. Phys. Chem. B, 103(21):4555, 1999.
[77] J.M.W. Dumond, D. A. Lind, and B.B. Watson. Precision measurement
of the wavelength and spectral profile of the annihilation radiation from
64
Cu 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 angular
distribution 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 between
positrons 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 of
positron-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 Dopplerbroadening 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 the
Doppler broadening of the annihilation line with efficient reduction of background. Nucl. Instr. & Meth. A, 378:337, 1996.
[85] U. Myler and P.J. Simpson. Survey of elemental specificity in positron annihilation peak shapes. Phys. Rev. B, 56:14303, 1997.
[86] H. Kauppinen, L. Baroux, K. Saarinen, C. Corbel, and Hautojärvi. Identification 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 of
positrons in a nanocomposite medium by 2-detector Doppler broadening
measurements. 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. Positronannihilation momentum profiles in aluminum: core contribution and the
independent-particle model. Phys. Rev. Lett., 38:241, 1977.
[89] J.R. MacDonald, K.G. Lynn, R.A. Boie, and M.F. Robbins. A twodimensional Doppler broadened technique in positron annihilation. Nucl.
Instr. & Meth., 153:189, 1978.
[90] D.T. Britton, W. Junker, and P. Sperr. A high resolution Dopplerbroadening spectrometer. Mat. Sci. For., 105-110:1845, 1992.
[91] M. Alatalo, H. Kauppinen, K. Saarinen, M.J. Puska, J. Makinen, P. Hautojärvi, and R.M. Nieminen. Identification of vacancy defects in compound
semiconductors 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, and
K.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 annihilations. Mat. Sci. For., 255-257:166, 1997.
[94] Steven Van Petegem. Positron annihilation study of nanocrystalline materials: a synergy of theory, experiment and computer simulations. Doctoral
Thesis, Ghent University, 2003.
[95] R. Beringer and C.G. Montgomery. Angular distribution of positron annihilation radiation. Phys. Rev., 61:222, 1942.
[96] S. DeBenedetti, C.E. Cowan, and W.R. Konneker. Angular distribution of
annihilation radiation. Phys. Rev., 76:440, 1949.
[97] R Krause-Rehberg and H.S. Leipner. Positron annihilation in semiconductors. 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. ElsevierAmsterdam, 1988.
[99] O.E. Mogensen. Positron Annihilation in Chemistry. Springer-Verlag, 1995.
[100] G. Dlubek, H. M. Fretwell, and M. A. Alam. Positron/positronium annihilation 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 Dopplerbroadened annihilation radiation of polypropylene-polystyrene alloys. Polymer, 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-offlight observation of low-energy positrons. Bulletin of the american physical
society, 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 metal
surfaces. Surface and Interface Analysis, 2(6):199, 1980.
[110] A. Vehanen, K.G. Lynn, P.J. Schultz, and M. Eldrup. Improved slowpositron 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, page
121. 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. Positron
transmission and scattering measurements using superposition of annihilation line shapes: Backscatter coefficients. Phys. Rev. A, 7:135, 1973.
[114] V.A. Kuzminikh and S.A. Vorobiev. Backscattering of beta-particles from
thick targets. Nucl. Instr. & Meth., 167(3):483, 1979.
[115] P.U. Arifov, A.R. Grupper, and H. Alimkulov. Coefficients of positron mass
absorption and backscattering. In P.G. Coleman, S.C. Sharma, and L.M.
Diana, editors, Positron Annihilation, page 699. North-Holland, 1982.
[116] J.S. Mäkinen, S. Palko, J. Martikainen, and P. Hautojärvi. Positron
backscattering 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 electron
and 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 electron
and positron slowing down in solids. ii. Appl. Phys. A, 35:51, 1984.
[120] E. Soininen, J. Mäkinen, D. Beyer, and P. Hautojärvi. High-temperature
positron 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 of
fast 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. Lowenergy contributions to positron implantation. J. Appl. Phys., 74:3479, 1993.
[123] A.F. Makhov. Penetration of electrons into solids, l– the intensity of an
electron 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 solid
surfaces. Rev. Mod. Phys., 66:841, 1994.
[125] A. Vehanen, K Saarinen, P. Hautojärvi, and H. Huomo. Profiling multilayer
structures with monoenergetic positrons. Phys. Rev. B, 35(10):4606, 1987.
[126] R.M. Nieminen, J. Laakonen, P. Hautojärvi, and A. Vehanen. Temperature
dependence 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 in
constrained media. Phys. Rev. A, 19:2317, 1979.
[128] P.J. Schultz and C.L. Snead. Positron spectroscopy for materials characterization. Metallurgical Transactions A, 21(5):1121, 1990.
[129] A. Dupasquier. Positron Solid-State Physics, page 510. Proceedings of the international 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 experiments
with slow-positron beams. Riv. Nuovo Cimento, 8(12):1, 1985.
[132] W Światkowski. 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 international school of physics Enrico Fermi Course LXXXIII. North-Holland,
Amsterdam, 1983.
[134] R.M. Nieminen and M. Manninen. Positrons in solids. In P. Hautojärvi,
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 epithermal
positrons in solids. i. Phys. Rev. B, 41:6179, 1990.
[136] Y. Kong and K. G. Lynn. Transport model of thermal and epithermal
positrons in solids. ii. Phys. Rev. B, 41:6185, 1990.
[137] D.T. Britton, P. C. Rice-Evans, and J.H. Evans. Epithermal effects in
positron 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. positronium formation at Ag(100), Ag(111), and Cu(111) surfaces. Phys. Rev. B,
22:99, 1980.
[140] J. Lahtinen, A. Vehanen, H. Huomo, J. Mäkinen, P. Huttunen, K. Rytsölä,
M. Bentzon, and P. Hautojärvi. High-intensity variable-energy positron
beam for surface and near-surface studies. Nucl. Instr. & Meth. B, 17:73,
1986.
[141] P. G. Coleman, S. Kuna, and R. Grynszpan. Slow positron implantation
spectroscopy of insulators: Charging effects. Materials Science Forum (ICPA
11), 255-257:668, 1997.
[142] Jérémie De Baerdemaeker. Defect characterization of sputtered, nitrided and
ion implanted materials using the new Ghent slow positron facility. Doctoral
Thesis, Ghent University, 2004.
[143] K.G. Lynn, B. Nielsen, and J. H. Quateman. Development and use of a thinfilm 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 spin
coating. 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 a
spin 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 or
salt solution on thin hydrophobic films. Langmuir, 20:7610, 2004.
[150] D. Freitas Siqueira, D. W. Schubert, V. Erb, M. Stamm, and J. P. Ama
to. Interface thickness of the incompatible polymer system PS/PnBMA as
measured 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 method
for the assembly of amino-terminated monolayers on SiO2 and the vapor
deposition of gold layers. Langmuir, 15:4520, 1999.
[152] A. T. Almeida, M. C. Salvadori, and D. F. S. Petri. Enolase adsorption onto
hydrophobic and hydrophilic solid substrates. Langmuir, 18:6914, 2002.
[153] J. Algers, P. Sperr, W. Egger, G. Kögel, and F. H. J. Maurer. Median
implantation depth and implantation profile of 3-18 keV positrons in amorphous polymers. Phys. Rev. B, 67:125404, 2003.
[154] P. G. Coleman, J. A. Baker, and N. B. Chilton. Experimental studies of
positron 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 variable
energy slow positron beam. Journal de Physique IV, 3(Colloque 4) supl. JP
II n9:131, 1993.
[156] D.A. David, EG & G Idaho. Simion 3d version 7.0 user’s manual. Idaho
National Engineering and Environmental Laboratory, Idaho Falls, Rev.5,
2000.
[157] A. vanVeen, H. Schut, J. deVries, R.A. Hakvoort, and M.R. Ijpma. Analysis
of positron profiling data by means of “VEPFIT”. In P.J. Schultz, G.R.
Massoumi, and P.J. Simpson, editors, Positron Beams for Solids and Surfaces, 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 powders of insulating oxides. In K. Tomala and E. Görlich, editors, Condensed
Matter Studies by Nuclear Methods, Proceedings of XXX Zakopane School of
Physics, page 275, Kraków, 1995. Institute of Physics, Jagielloian University
and H. Niewodniczanski Institute of Nuclear Physics.
[159] P. Sferlazzo, S. Berko, and K.F. Canter. Experimental support for physisorbed 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 of
positronium 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, and
D. Segers. Diffusion length of positrons and positronium investigated using 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 available
on request ([email protected]).
[163] Toon Van Hoecke. Construction and application of a source-based slow
positron beam. Doctoral Thesis, Ghent University, 2003.
[164] R.S. Brusa, A. Dupasquier, E. Galvanetto, and A. Zecca. Experimental
study 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 penetration
depth of implanted positrons in free-standing polymer films”. Ready to be
submitted.
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 at
the XVth International Conference on Positron Annihilation. Saha Institute
of Nuclear Physics. Kolkata, India. January 18 - 23, 2009.
3. Dirk I. Uhlenhaut, Florian H. Dalla Torre, Alberto Castellero, Carlos A.
Palacio Gomez, Nikolay Djourelov, Günter Krauss, Bernd Schmitt, Bruce
Patterson and Jörg F. Löffler.“Structural analysis of rapidly solidified Mg−
Cu−Y glasses during room-temperature embrittlement”. Submitted to PHILOSOPHICAL 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 on
free-standing nanometric polymer films”. Accepted for publication in MATERIALS SCIENCE FORUM in the conference proceedings of the 9th International 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. “Emission of positronium in a nanometric PMMA film”. APPLIED SURFACE
SCIENCE, Vol255, p. 197 (2008). (A1 journal).
doi:10.1016/j.apsusc.2008.05.235.
133
BIBLIOGRAPHY
6. C. A. Palacio, J. De Baerdemaeker, and C. Dauwe. “Determination of the
positron 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 study
of 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 probe
for the anisotropy of free-volume-holes in polymers”. PHYSICA STATUS
SOLIDI (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. “Positron
states in polypropylene and polystyrene at low temperature”. PHYSICA STATUS 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 the
consistency between positron annihilation lifetime and Doppler broadening
results in polypropylene”. PHYSICA STATUS SOLIDI (C) 4, No 10, 37103713 (2007). (C1 journal). doi:10.1002/pssc.200675732.
11. C. Dauwe, C. Bas, C. A. Palacio. “Formation of positronium: Multi-exponentials
versus blob model ”. RADIATION PHYSICS AND CHEMISTRY 76, p. 280284 (2007). (A1 journal). doi:10.1016/j.radphyschem.2006.03.051.
Awards
Honorable mention in the student poster competition. Award received during the
14th International Conference on Positron Annihilation (ICPA-14). It was held at
McMaster University, Hamilton, Ontario, Canada, July 23 − 28 / 2006.
134