2D-ACAR facility for analysis of nanocavities and
quantum dots
2D-ACAR facility for analysis of nanocavities and
quantum dots
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 7 oktober 2002 om 10.30 uur
door
Claudiu Valentin FALUB
Engineer in Physics, “Babeş-Bolyai” University, Cluj-Napoca, Romania geboren te Beclean, Roemenië
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. A. van Veen, Rijksuniversiteit Groningen,
Technische Universiteit Delft, promotor
Prof. dr. ir. H. van Dam, Technische Universiteit Delft
Prof. dr. S. Picken, Technische Universiteit Delft
Prof. dr. L. Niesen, Rijksuniversiteit Groningen
Prof. dr. P. Kruit, Technische Universiteit Delft
Prof. dr. L.D.A. Siebbeles, Technische Universiteit Delft
Dr. ir. P.E. Mijnarends, Technische Universiteit Delft,
Northeastern University, Boston
Dr. ir. P.E. Mijnarends heeft als begeleider in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.
Published and distributed by: DUP Science
DUP Science is an imprint of Delft University Press P.O. Box 98 2600 MG Delft The Netherlands Telephone: +31 15 2785678 Fax: +31 15 2785706 E-mail: Info@library.TUDelft.nl ISBN 90-407-2337-0
Keywords: positron annihilation / intense slow positron beam / 2D-ACAR / nanocavities / nanoclusters / quantum dots
Copyright © 2002 by C.V. Falub
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilised in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission from the publisher: Delft University Press.
1. Introduction 1
1.1. General 1
1.2. Positron physics 3
1.3. Positron annihilation spectroscopies 6
1.4. Intense positron sources 11
2. Theoretical aspects 17
2.1. Introduction 17
2.2. Momentum distribution 17
2.3. Positronium confined in a well potential 19
2.4. Positron transport in electric and magnetic fields 24
3. New experimental setup 29
3.1. Introduction 29
3.2. Conventional 2D-ACAR setup for bulk measurements 29
3.3. POSH, the Delft slow positron beam 38
3.4. The new POSH-ACAR setup for depth profiling studies 46
4. Metallic nanoclusters 65 4.1. Introduction 65 4.2. Li nanoclusters in MgO 65 4.3. Au nanoclusters in MgO 83 4.4. Conclusions 87 5. Nanocavities 91 5.1. Introduction 91
5.2. Spherical nanocavities in Si formed by ion implantation 92
5.3. Rectangular nanocavities in MgO 99
5.4. Nanocavities in Twaron 101
5.5. Conclusions 104
6. SiO2 based dielectric thin films 107
6.1. Introduction 107
6.2. Low-k dielectrics 107
6.3. MOS system 121
7. Concluding remarks and outlook 131 Summary 133 Samenvatting 135 Nomenclature 139 List of publications 143 Acknowledgments 145 Curriculum vitae 147
Introduction
1.1. General
Materials science involves the relationship between the internal structure and properties of materials (e.g. metals, electronic materials, polymers, ceramics, composites, biomaterials), and the processes of altering their structure and properties with the objective of designing and producing new materials that perform in predictable ways. The understanding of the microstructure of solids at the subatomic (electronic) and atomic level and the nature of defects at these levels is thus often crucial in the process of material fabrication. The study of these defects addresses issues of fundamental scientific and technological importance.
One of the branches of materials science research is the study of defects in materials. On an atomic scale the defects can be considered as tiny deviations from the normal positions of atoms and they often control the macroscopic properties of materials. For example, certain defects (e.g. nanopores in low-k dielectrics, defects in the Si/SiO2 system)
can improve or degrade the electronic properties of semiconductors used in computers and other digital equipment [1,2]. Therefore, a better understanding of the physics of defects is necessary for industry to improve the manufacturing of materials.
The positron is a positively charged particle and is repelled by nuclei. As a result it has a high preference for open volumes. In view of this preference, positron annihilation spectroscopy (PAS) has been developed over the last few decades into a successful non-destructive method for probing low atomic density regions (e.g. vacancies, clusters of vacancies, microcavities, open volumes) in materials over a wide range of depths, from the surface to depths of hundreds of nm [3]. PAS can thus provide information which is complementary to that provided by many other techniques, such as optical microscopy, neutron scattering, transmission electron microscopy, scanning tunneling microscopy and atomic force microscopy. Furthermore, PAS has been extensively applied to the study of thin films, layers of embedded nanoclusters and interfaces [3,4].
Positrons are produced either by nuclear reactions in radioactive isotopes or by pair production using high-energy gamma (γ) radiation. When injected into matter, the positron (e+) interacts with its surroundings prior to annihilating with an electron into γ rays. The characteristics of the emitted γ rays are different for positrons annihilating near a defect compared to positrons which annihilate in a defect-free material. By measuring the
positron lifetime, the energy distribution and the angular correlation of the annihilation γ rays (in one or two dimensions), one obtains information about the physical properties of the material under investigation. Because of their broad energy spectrum of a few 100 keV, the fast positrons emitted by a radioactive β+ source (e.g. 22Na) can only be used for bulk studies with the aid of conventional positron spectroscopies [5].
In recent years, the development of radio-isotope based mono-energetic slow positron (e+) beams has led to the study of interesting surface and near surface phenomena in thin films and ion implanted materials [6]. By varying the initial energy of the implanted positrons one can study defects at different depths. The importance of depth-profiling studies becomes clear if one considers layered systems obtained by various deposition or implantation techniques. In these systems defects may appear at the interface of the layers because of low mobility of the deposited atoms. In some cases the layer becomes amorphous and will show many defects. The relatively low intensity of the radioisotope based slow beams was offset by the use of efficient solid state detectors. The combination of depth profiling with the high-resolution two-dimensional angular correlation of annihila-tion radiaannihila-tion (2D-ACAR) method for the study of surfaces and layered systems, however, had to wait for the development of high-intensity slow positron beams. Nowadays, such beams are operational or under construction in several laboratories [7]. Depth profiling positron 2D-ACAR thin film studies have been performed for the first time at Brookhaven National Laboratory on the Si/SiO2 system employing a strong 64Cu source produced in a
high-flux beam reactor (HFBR) [8]. In the Netherlands, there was a joint effort of the Interfaculty Reactor Institute (IRI) and the Netherlands Energy Research Foundation (ECN) to build a depth-selective 2D-ACAR facility. The 2D-ACAR apparatus of ECN, equipped with multiwire proportional chamber (MWPC) detectors, had been used extensively to study the electronic momentum density of various systems, e.g. disordered
alloys, ferromagnets, high-Tc superconductors and heavy-fermion systems [9-15]. The
recent construction of a stable high-intensity reactor-based slow positron beam at the Delft 2 MW research reactor has proven to be successful [16,17]. The beam called POSH (“POSitrons at the Hoger onderwijs reactor”) has an intensity of ~ 108 e+/s. The positrons are obtained by pair formation induced by hard γ rays produced in the core of the reactor and extracted from the source region by means of electrostatic lenses. Transport of the positrons over a distance of ~ 25 m is achieved by an axial magnetic field of 0.01 T produced by a combination of solenoids and Helmholtz coils. Since March 2000 the POSH beam is coupled to a high-resolution 2D-ACAR setup which consists of parts of the ECN equipment and employs two Anger cameras up to 23 m apart [18-20]. In this configuration it is the first 2D-ACAR setup in the world using a continuous variable-energy slow positron beam of stable intensity. The facility was officially inaugurated in June 2000 and is a part of the Positron Centre Delft. After its completion many depth profiling 2D-ACAR studies of various materials of great interest in materials science have been performed, e.g. nanocavities in Si and MgO produced by ion implantation, metallic nanoprecipitates in MgO, SIMOX, low-k dielectric and silicon carbide thin films [21-27]. The aim of this thesis is to present the design and testing of the combined POSH/2D-ACAR facility and its applications in the field of materials science.
The structure of this thesis is as follows. Chapter 1 presents a general introduction into positron physics, positron annihilation techniques and intense positron sources. The
underlying theory of the thesis, dealing with momentum distributions, a simple quantum model for a positronium particle confined in a nanocavity and the optical design of the beam, is described in Chapter 2. Chapter 3 discusses the development of the novel depth selective 2D-ACAR facility. The newly built facility has been successfully applied to various types of materials and the results are collected in Chapters 4, 5 and 6. Chapter 4 describes Li and Au metallic nanoclusters embedded in a MgO matrix in terms of the electronic structure and the orientation relationship of the nanoclusters with respect to the host matrix. The experimental results in this chapter are supported by ab-initio quantum mechanical calculations. Chapter 5 presents results for materials containing structural (e.g. Twaron) or artificially created (e.g. Si, MgO) nanocavities. Chapter 6 describes the results for thin films of interest for microelectronics (e.g. low-k dielectrics and MOS). Finally, Chapter 7 presents the conclusions of this research and an outlook on future developments.
1.2. Positron physics
1.2.1. Positrons
The positron is the antiparticle of the electron. Hence, it has the same mass and the same spin (1/2), but opposite charge and magnetic moment. The positron has been postulated by Dirac in 1930 as the “negative” energy extension of his theory of electron energy levels [28,29]. Two years later Anderson [30] discovered the positron in his cloud chamber experiments. Positrons are unstable in matter where they annihilate with electrons predom-inantly via 2γ decay, both photons carrying an energy of 511 keV. In rare circumstances (e.g. when a positron is in the vicinity of a heavy nucleus) the positron and the electron can
undergo a 1γ decay process. The emission of more photons is also possible, but the
probability of such an event is small (the ratio between the probabilities of 3γ and 2γ decay in metals is 1/371 [31]).
When a positron is injected into a material it interacts with its surroundings and rapidly loses its energy by ionising collisions with atoms, collisions with conducting electrons and by phonon scattering, until it reaches thermal equilibrium. This process occurs within 10-11 s which is short compared to the mean lifetime of the positron of the order of 10-9-10-10 s [32]. This lifetime is so short that for realistic source strengths no more than one positron is present in the sample at any one time. At room temperature the wavelength associated with a thermalised positron is usually one order of magnitude larger than the typical lattice parameter. Hence, in a solid the positron behaves like a delocalised Bloch wave which in well-annealed crystals of metals and semiconductors exhibits maxima in the interstitial region due to core repulsion. In defected materials, however, the positron can be localised (trapped) in, or around, low-density regions such as vacancies, dislocations and microvoids. As a consequence, the positron lifetime becomes longer and the positron density is strongly enhanced at the position of a defect.
Positrons can be produced by the nuclear decay of isotopes or by pair production by high-energy gamma rays (Eγ > 1.022 MeV). Positrons emitted by a radioactive isotope (e.g. 22Na, 58Co, 64Cu) have continuous energy distributions ranging from 0 to a few
hundred keV. As a consequence, when positrons are implanted in a material, the implantation profile is broad, of the order of a few hundred microns. Since positrons are positively charged particles they are repelled by the atomic cores, thus leading to the possibility of their ejection from a surface due to their negative work function (φ+ < 0). The
positron work function for a solid is defined as the minimum energy required to remove a positron from a point inside the solid to one in the vacuum [3]. Owing to its interaction with the surrounding electrons, a positron can adopt an energy level that can be higher or lower than the energy level of the vacuum. As a consequence, the positron work function can be negative or positive in comparison to the electron work function which is nearly always negative. The effect of positron ejection from a surface is used in the production of slow positron beams. Examples of surfaces which emit positrons are clean metal surfaces, e.g. W(111), Ni(100), Cu(111), diamond, Si, SiC, GaN or the surfaces of rare-gas solids.
1.2.2. Positronium
In some liquids and insulating solids, the positron can capture an electron prior to annihilation, forming positronium (Ps), a hydrogen-like positron-electron bound state with a binding energy in vacuum of ~ 6.8 eV. Predicted classically by Mohorovicic [33] in 1934 and quantum mechanically by Ruark [34] in 1945, the Ps atom was experimentally discovered in 1951 by Deutsch who found a long-lived component in a lifetime spectrum that was nearly independent of gas pressure [35].
The spectroscopic differences between Ps and hydrogen originate in the particle-antiparticle nature of Ps (equality of the positron and electron masses and of the magnitudes of their magnetic moments), which lead to the possibility of self-annihilation. The ground states of Ps consist of a singlet 1S0 state (S = 0; ms=0) with the spins oriented
antiparallel, called para-positronium (p-Ps), and a triplet 3S1 state (S = 1; ms=0, ±1) with
the spins oriented parallel, called ortho-positronium (o-Ps). Para-Ps has a lifetime in vacuum of ~ 125 psec and decays via two γ rays, while o-Ps has a lifetime in vacuum of ~
142 nsec decaying into three or more γ rays. In the absence of external electric and
magnetic fields the relative amount of p-Ps:o-Ps formed is 1:3. In a low density gas the long-lifetime component approaches that characteristic of o-Ps, while at higher densities or in condensed matter the interaction of Ps with atomic electrons leads to a decrease in the 3γ decay mode associated with a shortening of the long lifetime. This decrease is due to the so-called “pick-off” process in which the positron of o-Ps annihilates during a collision with an atomic electron that has an antiparallel spin, and thus annihilates via a 2γ process. Magnetic field induced 2γγγγ annihilation
In the presence of an external magnetic field of flux density B the p-Ps state (1Φ0) and the
ms=0 substate of o-Ps (3Φ0) are mixed in so-called ortho-like-Ps and para-like-Ps states
[36-38]: 0 1 2 0 3 2 1 1 1 Φ + + Φ + = Φ+ y y y , (1.1)
0 1 2 0 3 2 1 1 1 Φ + + Φ + − = Φ− y y y , (1.2) with 1 1+ 2 + = x x y and E B x B ∆ = µ4 . (1.3)
Here, µB is the Bohr magnetron and ∆E = 8.4×10-4 eV is the hyperfine splitting between
the p-Ps state and the o-Ps states. The annihilating rates of the new states are:
1 2 2 3 2 1 1 1 Γ + + Γ + = Γ+ y y y (1.4)
for ortho-like-Ps and
1 2 3 2 2 1 1 1+ Γ + + Γ = Γ− y y y (1.5) for para-like-Ps, where Γ3 = (142 ns)-1 and Γ1 = (125 ps)-1 are the annihilation rates of o-Ps
and p-Ps, respectively. Both these new states can undergo 2γ and 3γ annihilations. Since Γ1
is three orders of magnitude larger than Γ3, even a weak magnetic field for which y << 1
causes a large fraction of ortho-like-Ps atoms to self-annihilate into 2γ. The annihilation characteristic of para-like-Ps for small y, however, is not so much affected by the magnetic
field. This effect is called magnetic field induced o-Ps→p-Ps conversion or magnetic
quenching of o-Ps.
Positronium annihilates subsequently with a characteristic signature, and can be detected by studying the annihilation γ’s in different manners: i) by measuring the 3γ to 2γ yield ratio with a multiple-coincidence γ detector system, ii) by measuring the positron lifetime, iii) by measuring the energy spectrum of one of the annihilation photons, iv) by measuring the momentum distribution of the electron-positron pairs, v) by studying the decrease in the 3γ yield due to the Zeeman mixing of Ps states when a magnetic field is applied [39-41]. In this thesis the attention will be focused on iv).
Positronium is usually formed in regions with a low electron density such as gases or liquids. In gases Ps formation takes place by radiationless electron capture as long as the kinetic energy (T) of the positron is in the energy region Ei −6.8≤T ≤ Ei [eV], called the Ore gap, where Ei is the ionisation energy of the gas atom or molecule [31]. Ps can also be
formed in some insulating materials and organic materials with large open volumes where an electron is captured in the ionisation trail (“spur”) created by the slowing down of the positron [42]. In these materials p-Ps still has a lifetime of 125 psec, but the lifetime of o-Ps is reduced to a few nanoseconds owing to pick-off processes. Since the Pauli exclusion principle is not operative for a Ps atom (Ps is a boson), the Ps atom becomes quickly ther-malised and thus resides near the bottom of its own band as a virtually momentum-free system. As a consequence, self-annihilation of Ps contributes a sharp peak in a 2D-ACAR spectrum. In metals and semiconductors, thermalised positrons annihilate from delocalised Bloch states without forming Ps.
With respect to the spatial extension of the Ps wave function (ΨPs), two different types
of Ps can be distinguished:
of the medium. Figure 1.1-(a) shows the experimental 2D-ACAR momentum distribution of localised p-Ps atoms formed in radiation produced microvoids in vanadium [5];
ii) delocalised Ps, when ΨPs(r) extends over a region containing a large number of
elementary cells of a crystalline solid. Figure 1.1-(b) shows the experimental 2D-ACAR momentum distribution of delocalised p-Ps formed in an oriented crystal of SiO2 [5].
Figure 1.1: a) Positronium localised in radiation produced microvoids in vanadium. b) Delocalised positronium in an oriented crystal of SiO2 (from Berko [5]).
In some circumstances (e.g. when Ps is trapped in a large void) the positronium atom formed can not loose its energy by collisions with atoms and molecules, thus remaining non-thermalised or “hot” Ps [21]. Also, when positrons are implanted at shallow depths in metals or semiconductors, emission of energetic Ps atom from the surface may occur [43,44].
1.3. Positron annihilation spectroscopies
There are three major positron techniques that use the annihilation photons in order to gain information about the electrons: a) two-dimensional angular correlation of annihilation radiation (2D-ACAR); b) Doppler broadening of annihilation radiation (DBAR) experiments; and c) positron lifetime experiments. The first two techniques give informa-tion on the momentum distribuinforma-tion of electrons while the third one gives informainforma-tion on the electron density. Other techniques use different properties of the positron to probe material properties, e.g. low-energy positron diffraction (LEPD), positron channeling, positron annihilation induced Auger electron spectroscopy (PAES), but they are not the subject of this thesis [3,45].
Figure 1.2: Two-photon annihilation process in: a) center-of-mass frame; b) laboratory frame .
1.3.1. Two dimensional ACAR
The measurement of the angular correlation of the annihilation radiation (ACAR) is one of the means for studying positron annihilation in solids. This technique measures the angular deviation from collinearity of the two coincident annihilation photons. In the center-of-mass frame the total momentum of the e+-e- pair is zero and since the energy and mo-mentum before and after annihilation are conserved the two photons are emitted in oppos-ite directions, each one having an energy equal to m0c2 (see Fig. 1.2-a). In the laboratory
frame, however, the e+-e- pair carries a total momentum p. Since the positron is thermal and annihilates mainly with weakly bound electrons, the momentum p is relatively small (~ 10-2 m0c) and, as a consequence, it will slightly change the picture found in the
center-of-mass frame. Hence, the components of the momentum perpendicular to the photon emission axis, px and py, will generate small angular deviations from collinearity (see Fig.
1.2-b) that can be written as follows:
c m px 0 sin ~ θ = θ , (1.6) c m py 0 sin ~ φ = φ . (1.7)
Figure 1.3 shows a schematic diagram of a typical 2D-ACAR setup. It consists of two large-area position sensitive gamma detectors, a central source-sample vacuum chamber and data acquisition electronics. In the central chamber a positron beam is guided by an axial magnetic field to the sample of interest. The annihilation quanta are detected in the two detectors and the information concerning their individual points of arrival is converted by the acquisition electronics into (θ, φ) addresses which are stored in an incrementally updated 2D discrete array. In this way one obtains the two-dimensional projection of the three-dimensional momentum density ρ2γ(p) [46]:
(
( ))
( , ) ) , ( ) , ( N px py dpz R px py N θ φ = =∫
ρ2γ p ⊗ , (1.8)where px, py, pz are the components of the total momentum p ( pz is not measured) and ⊗R
Figure 1.3: Schematic diagram of a typical 2D-ACAR setup showing the principal components.
Figure 1.4: 2D-ACAR spectrum for Si (100).
Typical resolutions obtainable are of the order of 1 mrad. The samples are usually oriented with one of their main crystallographic directions parallel to the integration axis of the setup. Figure 1.4 shows an example of a 2D-ACAR spectrum for an oriented crystal of Si(100). The structure in the spectrum reflects the crystalline features of the momentum density of the electrons in Si. In the older one-dimensional angular correlation of annihilation radiation method (1D-ACAR) only one component of the total momentum was measured:
(
( ))
( ) ~)
(pz 2 dpxdpy R pz
N
∫∫
ρ γ p ⊗ , (1.9)where ⊗R corresponds to the convolution with the one-dimensional angular resolution
R( pz) of the setup. Since the 2D-ACAR method maps the momentum distribution of the
electrons with a high-resolution, the method has important applications, e.g. Fermi surface topology, physics of defects in materials, etc.
1.3.2. Doppler broadening of annihilation radiation
In addition to the angular deviation of the annihilation photons, discussed in the previous subsection, there is a shift of the energy of the photons from its average value of 511 keV, caused by the component of the momentum parallel to the direction of γ-ray emission. The energy shift, δE, from 511 keV can be expressed by:
2 / z cp E = δ , (1.10)
where pz is the longitudinal component of the momentum of the e+-e- pair. Since one half
of the electrons moves towards the detector while the other half moves away from it, the energy shift results in a Doppler broadening of the 511 keV line by several keV’s which can be measured by Ge-detectors. The best detectors have a resolution of 1.1 keV at 511
keV which corresponds to a momentum of 4.3×10-3m0c. Owing to the small distance
between the detector and the sample, the Doppler broadening experiment integrates over the transverse component of the momentum and the measured spectrum is a one-dimen-sional projection of the three-dimenone-dimen-sional momentum density:
(
dp dp)
Rp N E
N( )= ( z)~
∫∫
ρ2γ(p) x y ⊗ , (1.11)where ⊗R denotes convolution with the energy resolution of the Ge-detector.
Figure 1.5 shows a typical Doppler broadening spectrum consisting of a broadened 511 keV peak. Due to the double integration and convolution with the resolution of the detectors the spectrum is smooth and featureless. However, one can derive valuable infor-mation from this spectrum by defining two regions of interest: the low-momentum central part corresponding to annihilations of positrons with valence electrons, and the high-momentum tails corresponding to annihilations with more tightly bound electrons, e.g. core electrons. Thus, one can characterise the shape of the 511 keV peak with the aid of two parameters, S and W, defined as follows [47,48]:
C B A B S + + = , (1.12) F E D F D W + + + = , (1.13)
where A, B, C, D, E and F are the areas of the energy windows defined in Fig. 1.5. Therefore, the S parameter is more useful for monitoring the presence of defects in a material, while the W parameter gives the signature of that material. In order to compare the absolute values of the S and W parameters (influenced by the settings of the energy windows and the resolution of the detector) obtained with different setups, these parame-ters are divided by the S and W values of a reference material, e.g. defect-free Si.
Figure 1.5: Definition of the S and W parameters in a Doppler broadening spectrum. In view of the positron affinity for open volumes, the Doppler broadening of annihilation radiation technique in combination with slow positron beams has been extensively applied to the depth profiling of defects in materials [3,4]. In these experiments the S and W parameters are measured as a function of the incident positron energy, E. The implantation of a slow positron beam can be approximated by a Makhovian profile [49]:
− = − m m m z z z mz E z P 0 0 1 exp ) , ( , (1.14)
where m is a shape parameter usually set to 2 and z0 is related to the mean implantation
depth given by:
n E z nm z ρ α = π = 0 2 ] [ . (1.15)
Here ρ represents the density of the material in g/cm3 and α and n are two material independent constants with values of 40 g cm-2 keV-1.62 and 1.62, respectively.
The S vs. E and W vs. E curves for Doppler broadening data are analysed with the aid of the VEPFIT program [50] which assumes the system studied to consist of a stack of different positron trapping layers, each layer being characterised by a thickness, an S parameter and a positron diffusion length. The S parameter for a given energy E is then given by:
bulk n i bulk surface surface i i E S f E S f E S f E S
∑
= + + = 1 ) ( ) ( ) ( ) ( , (1.16)where fi(E) is the fraction of positrons implanted at energy E which after thermalisation
and diffusion annihilate in the layer i, Si denotes the corresponding value of the lineshape
parameter in layer i, and bulk and surface refer to bulk and surface contributions. Owing to the property of linearity of the S and W parameters, one can obtain a direct interpretation of the experimental results by plotting S and W data as a trajectory in an S-W diagram. In this case the running parameter is the positron implantation energy. The advantage of the S-W diagram is that it allows one to distinguish annihilations from different types of defects which on the basis of the S and W parameter alone could not be detected. In a Doppler broadening measurement the sensitivity to annihilations with core-electrons is reduced due to the relatively high background in the spectrum. However, this background can be reduced by placing a second Ge detector opposite to the first one at the same distance from the sample [51]. In this way the peak to background ratio is improved by a factor of almost 1000 compared to a single detector system at the expense of much lower count rates.
1.3.3. Positron annihilation lifetime
In comparison with the previously discussed positron annihilation spectroscopies, which measure projections of the momentum density of positron-electron pairs, the positron lifetime technique measures the time difference between the emission of a positron and its annihilation with an electron of the material under study. In this experiment a positron source is used which by emission of a simultaneous γ ray provides a zero time signal at the moment at which the positron enters the sample. Typical values of positron lifetimes are 100-200 ps for bulk materials, corresponding to free-positron annihilation. When the size of the open volume is large enough to accommodate a Ps atom, the lifetime is considerably higher (~ ns). The lifetime spectrum can be described as a sum of decaying exponentials:
(
)
∑
= − = N i i i i t I t F 1 / exp ) ( τ τ , (1.17)where N represents the number of different annihilation states, τi the lifetime of positrons
corresponding to the state i and Ii the intensity of that state. Contemporary positron lifetime
setups commonly use non-moderated 22Na sources. Therefore the information extracted
from a lifetime spectrum is only global due to the uncertainty in depth at which the positrons annihilate. However, positron lifetime equipment combined with a slow positron beam is now operational or under construction in a few laboratories [52,53].
1.4. Intense positron sources
Positron spectroscopies for depth resolved material analysis, at the surface or near surface, require monochromatic positron beams. Low intensity (< 106 e+/s) positron beams based on commonly used radioactive isotope sources used in positron physics (22Na, 58Co) have
been extensively applied to the investigation of surface or near surface material properties, e.g. surface structure or depth distribution of defects at surfaces and buried interfaces. Low-intensity beams are sufficient to perform many of the positron annihilation spectros-copies (e.g. Doppler broadening of annihilation radiation and positron lifetime), but the application of the 2D-ACAR method and positron microscopy to the study of surfaces and layered systems had to wait for the development of high-intensity slow positron beams (~ 108 e+/s). Nowadays, such beams in which positrons are created by pair production induced by high-energy γ-rays are operational or under construction in several laboratories [7], and as these facilities are commissioned the number of applications of positron annihilation spectroscopies in the field of materials science will increase.
A slow positron beam system usually consists of the following components: i) a primary positron source with a continuous energy distribution; ii) a positron moderator located near the primary source, and iii) a beam transport system. In order to increase the intensity of a slow positron beam one or more of these components will have to be improved. The beam transport systems are highly efficient so there is little room for improvement. The role of the positron moderator is to slow down the fast positrons, e.g. when fast positrons are implanted into a well-annealed sample, those within the diffusion length (~ 100 nm) from the surface can diffuse back and are ejected nearly mono-energet-ically as a result of their negative work function. Typical moderating efficiencies are modest, of the order of ~ 10-4-10-3, and these limitations are based on the physics of positron interaction in the moderating system and the geometry of the moderators. The major factor to achieve intense slow positron beams is therefore the primary source strength, which has practically no physical limit. In the following, different approaches adopted at different laboratories to achieve strong sources of moderated positrons are presented.
1.4.1. Sources based on relatively short-lived radioactive isotopes
a) Reactor based sources. A high-current positron beam can be achieved by activation of
63
Cu in a nuclear reactor to 64Cu which decays under the emission of a positron. Lynn et al. [54] built an intense beam relying on the activation of a strong (800 Ci) source of copper in the High Flux Beam Reactor at Brookhaven. The positron emitting 64Cu, obtained via the nuclear reaction 63Cu(n,γ)64Cu, was evaporated onto a single-crystal W(110) substrate in an external vacuum system to provide a self-moderating copper film. The system has operated successfully for several years and has produced many useful data for atomic and solid state physics including 2D-ACAR on thin films. Disadvantages were the operational efforts. Every three days the source had to be refreshed due to the 12.8 h decay time of the 64Cu radioisotope.
b) Accelerator based sources. Irradiation of materials with accelerator ion beams can produce intense positron emitting radionuclides [55]. An example is the irradiation of 12C with a high current of low energy deuterium ions which produces 13N. By interchanging the carbon target between the deuteron beam and a positron moderation stage an intense quasi-continuous slow positron beam can be obtained. Since compact linear accelerators
are available, this technique is promising for producing inexpensive high-current positron beams. A first attempt to produce such a beam was undertaken at Brandeis University [7].
1.4.2. Sources based on pair production
Pair production is the most commonly used method for producing high-current positron beams. When photons with an energy higher than 1.022 MeV interact with the fields surrounding heavy nuclei (e.g. tungsten) that energy can be converted into e+-e- pairs. The probability of pair production increases when the photon energy increases. The fast positrons are then separated from the electrons and moderated in order to obtain an intense slow positron beam. Two primary designs are available: a) linac based positron beams, and b) nuclear reactor based positron beams.
a) Linac based positron beams. Successful intense positron beams are linear accelerator (linac) based positron beams with intensities over 109 e+/s. Positrons are generated in a high-Z target (e.g. tungsten) hit by high-energy (~ 100 MeV) electrons. Bremsstrahlung photon generation and subsequent pair formation yield fast positrons that can be moderated by tungsten foils. An example of such an intense positron beam was developed at the 100 MeV electron linac at Lawrence Livermore National Laboratory [56]. Other designs are available or under development at different laboratories [6]. Linac based positron beams are inherently pulsed. A quasi-continuous beam can be obtained by trapping the positrons from the linac in a magnetic bottle and then releasing them over the period between linac beam pulses.
b) Reactor based positron beams. The advantage of these beams is that they are practically continuous over time. Pair production can be obtained in nuclear reactors in two different ways: i) Capture of high-energy gamma rays in thin tungsten moderator foils. A successful continuous intense positron beam based on this principle is POSH (“POSitrons at the Hoger onderwijs reactor”). This beam has been recently developed at the Interfaculty Reactor Institute in Delft and provides a constant intensity of 0.8×108 e+/s to a 2D-ACAR target chamber [9,10]. The design of the beam will be discussed in detail in Chapter 3. ii) Neutron capture gamma rays from cadmium foils. An alternative source of high-energy photons for pair production of positrons consists of the gamma rays generated when thermal neutrons are captured in cadmium foils via the nuclear reaction 113Cd (n,γ) 114Cd. Consequently, the gamma rays produce electron-positron pairs in tungsten foils. An intense beam facility based on neutron capture gamma rays is under development at the Research Reactor Munich [57].
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Theoretical aspects
2.1. Introduction
The first part of this chapter shortly describes the underlying methodology used to determine the 2D-ACAR momentum distributions from Korringa-Kohn-Rostoker (KKR) ab-initio calculations. The second part of the chapter describes a simplified method for calculating the size of cavities with particular shapes (e.g. rectangles and spheres). The wave function of p-Ps is determined analytically by solving the Schrödinger equation for the bound electron-positron particle confined in infinite rectangular and spherical well potentials. The method is an approximation since the potential barrier of a cavity is in reality finite and has a more complicated shape which allows the overlap of the Ps wave function with the external electrons. Furthermore, the Ps atom is treated here as a point particle and possible effects caused by its two-particle internal structure are thus neglected [1]. The last part of the chapter presents the underlying theory of the computer codes CERN-POISSON and POEM, used to design the focusing section of the slow positron beam target in the 2D-ACAR setup presented in Chapter 3.
2.2. Momentum distribution
The measured two-dimensional (2D) ACAR spectrum can be expressed as the following integral of the momentum distribution of the annihilation photon pair ρ2γ(p) [2]:
(
( ))
( , )) ,
(px py dpz R px py
N =
∫
ρ2γ p ⊗ , (2.1)where px, py, pz are the components of the total momentum p (pz is not measured) and ⊗R
denotes the convolution with the 2D angular resolution R(px, py) of the setup. In the
non-relativistic limit, |p| << m0c, with m0 the positron rest mass and c the velocity of light, the
momentum distribution of the photon pair, ρ2γ(p), is given by [3]:
∑ ∫
Ψ π = ρ γ − ⋅ j ep j i e d c r02 2 2 (p) r pr (r) , (2.2) where r0 = e²/m0c² is the classical electron radius, the summation is over all occupiedelectron states j and ep(r)
j
Ψ is the two particle wave function when the electron and
) ( ) ( ) ( ) (r r j r j r ep j =Ψ Ψ γ Ψ + . (2.3)
Here Ψ+ (r) and Ψj (r) are the positron and electron wave functions and γj (r) is the
enhancement function, which takes into account the electron-positron correlation effects. The exact solutions of Ψ+ (r), Ψj (r), and γj (r) are intractable quantum mechanical
prob-lems. In practice, many approximations have to be made in order to describe the mo-mentum distribution of the annihilating e--e+ pair. Due to the screening of the positrons by an electron cloud in a real solid, e--e+ correlation effects occur which are essential in ex-plaining positron lifetime results [5]. The momentum dependence of these effects, how-ever, is relatively weak [6,7]. One of the simplest approximations is the independent-par-ticle model (IPM) which neglects the e--e+ correlation, i.e. γj (r) = 1. In this approximation
the shape of the momentum distribution of the annihilating e--e+ pair is reasonably well described if only the sp valence electron contribution is considered. Since e--e+ correlation effects are not exactly known, approximate methods incorporating many-body effects into the calculation of the momentum distribution have been developed, e.g. the Local Density Approximation (LDA) [8,9] and the Generalised Gradient Approximation (GGA) [10,11].
In the present thesis, the theoretical momentum distributions for some elementary crystals (e.g. Li, Au, MgO) were evaluated in the IPM approximation with the aid of the Korringa-Kohn-Rostoker (KKR) method [3,12-15]:
( )
∑
∑
× Ω π = ρ + + + γ j k E f E f ( ) ( ) 4 ) ( 2 4 2 p(
)
[
]
× − − − − − ⋅ − − − ×∑∑
∑
µ + + µ µ + n L L L E E S C i ( ) ( ,2) exp n n μ n K k p K k p k b K k p(
)
[
]
× + − + ⋅ + − ×∑
∑
ν + + + + + ν + + + 2 ' 2 ' ν ' ν ) , ( ) ( exp L L L E E S C i n n n K k K k k b K k j E E LL L LL L M C C M C C = − µν ΛΛ στ + + τ Λ στ + ΛΛ + ∗ + σ Λ ν µν ∗ µ ×∑∑
∑∑
1 ' ' ' ' ' ' ) ( ) ( ) ( ) (k ! k k ! k (2.4)In this equation all positron-related quantities have been distinguished by the superscript +,
Ω is the volume of the unit cell, f+ represents the Maxwell-Boltzmann distribution for the
low-density positron state, and f represents the Fermi function of the electrons. Further-more, k+(≈ 0) is the wave vector of the positron, k the wave vector corresponding to the
momentum p of the electrons, Kn a reciprocal-lattice vector, and M! =dM /dE and C are
the energy derivative of the KKR matrix M and its eigenvectors, respectively. The summa-tions are over all occupied bands j up to the Fermi level and over all reciprocal-lattice vec-tors Kn. In practice, the plane-wave expansion of the positron wave function converges
relatively rapidly which limits the number of reciprocal-lattice vectors in the summation. The various 2D-ACAR distributions were obtained by projection onto the desired plane.
2.3. Positronium confined in a well potential
In this section, the attention is focused only on the part of the 2D-ACAR distribution attributed to self-annihilation of p-Ps atoms trapped in open volumes (e.g. nanocavities or free space in polymers). This part consists of a narrow peak due to the low momentum of the thermalised e--e+ pair (see Fig. 1.1 in Chapter 1). The underlying physical mechanism
of p-Ps confinement is the repulsive interaction between Ps and the surrounding atoms arising from the electron exchange contribution, which leads to a localised state of p-Ps in a confining potential [16-18].
In materials where the p-Ps atom is delocalised (e.g. quartz, alkali-halides), the contribution from the self-annihilation of p-Ps can be easily identified from the overall spectrum since it is very sharp (see Fig. 1.1-(b) in Chapter 1 and Fig. 3.10 in Chapter 3). As a consequence of the Heisenberg principle, when a p-Ps atom is localised in a single void (e.g. polymers, porous semiconductors), the momentum of the e--e+ pair becomes higher than in the case of delocalised Ps. In this particular case the identification of the p-Ps contribution is not so straightforward. However, one can decompose the 2D-ACAR spectrum into a crystalline part Ncrystal corresponding to annihilations of positrons with
valence electrons carrying crystalline information, and a part N* corresponding to
annihilations of positrons with core electrons, self-annihilation of p-Ps and pick-off annihilation of o-Ps: ) , ( ) , ( ) , (px py Ncrystal px py N* px py N = + . (2.5)
For oriented crystalline materials the term Ncrystal can be separated by analysing the
aniso-tropy of the total spectrum. There are also materials exhibiting a non-symmetric p-Ps con-tribution due to self-annihilation of p-Ps in elongated cavities, but the corresponding anisotropy is found close to the centre of the distribution and thus can be easily separated from the total anisotropy. The problem remains now to separate the p-Ps contribution from the part N*( px,py). In practice (e.g. a polymeric material), it is shown that the 2D-ACAR
spectrum, remaining after subtracting the crystalline contributions, can be fitted by a sum of several gaussians:
∑
= + − = n i y x p p a i i y x p I e p N , 1 ) ( *( , ) 2 2 , (2.6) where Ii and ai are parameters related to the heights and shapes of the gaussiancompo-nents. The full width at half maximum (FWHM) of each component is 2(ln 2/ai)½. The
narrowest component obtained from the fit is attributed to the self-annihilation of p-Ps. In order to determine a relationship between the size of the cavity and the FWHM of the narrow component in the 2D-ACAR spectrum one has to solve the quantum problem of a p-Ps particle confined in a cavity. The infinite well potential model is often used in quantum mechanics to represent a situation in which a particle moves in a restricted region of space under the influence of forces which hold it in that region. Although by using this simplified model some detail of the momentum distribution (e.g. penetration into the surrounding medium) is lost, it retains the essential feature of binding the particle by forces of a certain strength to a region of a certain size. Depending on the shape of the cavity one can have different types of potential wells: rectangular, spherical or ellipsoidal. These cover most of the situations encountered in practice (e.g. rectangular cavities in ion
im-planted MgO, spherical cavities in porous Si, ellipsoidal cavities in Twaron or in elongated polymers). In the present work only the rectangular and the spherical potential wells are discussed, since the elongated cavities in Twaron can be considered to a first approxima-tion as rectangular. The ellipsoidal well potential model has been used by some authors to probe the anisotropic hole properties in polymers from positron lifetime data [19,20].
2.3.1. Infinite rectangular well potential
The Schrödinger equation for a p-Ps particle confined in a potential V(r) is: ) ( ) ( ) ( ) ( 2 2 2 r r r r + Ψ = Ψ Ψ ∇ − V E mPs " , (2.7) where " is Planck’s constant h/2π, mPs = 2m is the mass of p-Ps atom (m is the rest mass
of the electron), ∇2 is the Laplacian operator, Ψ(r) is the wave function of p-Ps,
z y
x 1 1
1
r =x⋅ +y⋅ +z⋅ is the position vector, and E is the total energy of the system.
Figure 2.1-(a) presents the infinite well potential corresponding to the rectangular box shown in Fig. 2.1-(b). The potential is represented only along the x direction, but has the same shape along the other two major directions, y and z:
∞ ≤ ≤ − ≤ ≤ − ≤ ≤ − = elsewhere , 2 / 2 / , 2 / 2 / , 2 / 2 / , 0 ) , , (x y z a x a b y b c z c V (2.8)
Solving Eq. 2.7 for the potential presented in Eq. 2.8 one obtains by separation of variables the wave function of p-Ps atom in real space:
π− π ⋅ π− π ⋅ π− π = Ψ z c n n y b n n x a n n V z y x n 3 3 2 2 1 1 2 sin 2 sin 2 sin 8 ) , , ( , (2.9)
where V=abc is the volume of the rectangular box and n1, n2, n3 = 1, 2, 3, … are quantum
numbers of the p-Ps states. Figure 2.2 presents four wave functions for a p-Ps atom confined in a rectangular box of 1.7×1.0×1.0 nm3 in the special case n1= n2= n3= n. The
ground state of p-Ps is obtained for n = 1:
c z b y a x V z y x = π π π
Ψ( , , ) 8 cos cos cos (2.10)
In order to calculate the momentum density one computes the p-Ps wave function in momentum space by applying the Fourier transformation to Eq. 2.10:
( ) ∫
⋅ − Ψ π = Ψ p r r p r d e i " " ( ) 2 1 ) ( 3/2 (2.11) 2 2 2 2 2 2 2 3 2 cos 2 cos 2 cos 4 ) , , ( − ⋅ − ⋅ − ⋅ π = Ψ " " " " " " " z z y y x x z y x p c cp p b bp p a ap V p p p π π π (2.12)Figure 2.1: a) Infinite square well potential. b) Positronium atom in a rectangular box.
Figure 2.2: Four bound-state wave functions of a p-Ps atom confined in a rectangular box of 1.7×1.0×1.0 nm3.
The 2D-ACAR momentum distribution can be expressed as: 2 2 2 2 2 2 2 2 2 cos 2 cos ) , , ( ) , ( − π ⋅ − π ∝ ρ = ∞
∫
∞ − " " " " y y x x z z y x y x p b bp p a ap dp p p p p p N (2.14)Figure 2.3 presents the calculated 2D-ACAR momentum distribution calculated for ground state p-Ps in a rectangular box of dimensions 1.7×1.0×1.0 nm3, based on Eq. 2.14. It is observed that the distribution becomes narrower in the direction px where the cavity is
elongated and broader in the direction py where the cavity dimension is smaller.
Solving the equation N( px, py) = ½N(0,0), in
or-der to find the px and py coordinates where N( px, py) is at half height, one obtains a
one-to-one relationship between the FWHM of the p-Ps momentum distribution along the px and py
directions and the dimensions of the rectangular cavity [21]: ) mrad ( 886 . 2 nm) ( x FWHM a = (2.15) ) mrad ( 886 . 2 nm) ( y FWHM b = (2.16)
The total energy of the p-Ps atom is quantised:
+ + = 232 2 2 2 2 2 1 188 . 0 ] [ c n b n a n eV En (2.17)
Figure 2.3: Calculated 2D-ACAR mo-mentum distribution for the ground state of a p-Ps atom in a rectangular box of 1.7×1.0× 1.0 nm3.
2.3.2. Infinite spherical well potential
A positronium atom confined in an infinite spherical well potential is a good approxima-tion for many applicaapproxima-tions found in practice (e.g. Ps in free volume holes in polymers, Ps in cavities in porous silicon, etc.) [18]. Let us consider a p-Ps particle confined in a spherical well with radius R having an infinite potential barrier (see Fig. 2.4). The radial part of the Schrödinger equation can be written as [22]:
0 ) ( ) ( ) 1 ( d d 2 2 2 2 2 = Ψ − + − + − V r E r r l l r mPs n " , (2.18)
where mPs = 2m is the mass of p-Ps atom (m is the rest mass of the electron), n and l are the
principal and orbital quantum numbers, respectively, En are the energy levels of p-Ps and V(r) is the spherical well potential:
∞ ≤ ≤ = elsewhere , 0 , 0 ) (r r R V (2.19)
Solving Eq. 2.18 one obtains for the ground state (n = 0, l = 0) wave function of p-Ps the following expression: π ≤ ≤ π = Ψ elsewhere , 0 0 , sin 1 2 1 ) ( R r R r r R r (2.20)
Figure 2.4: Infinite spherical well potential.
Applying a Fourier transformation to the wave function in Eq. 2.20 one obtains: ) ( sin ) ( 2 2 2 / 3 x x x R x − π = Ψ " , (2.21)
where x is the dimensionless quantity pR/". The momentum distribution ρ(x) and the 1D-ACAR curve N( p) are:
2 2 2 3 2 ) ( sin ) ( ) ( − π = Ψ = ρ x x x R x x " (2.22)
∫
∫
∞ ⋅ ∞ − π π = → ρ π = θ θ " " mcR p dx x x x R N p p p p N 2 2 2 2 2 1 sin 4 ) ( ' d ) ' ( ' 2 ) ( , (2.23)where c is the velocity of light and θ is the deviation from collinearity. The FWHM of the narrow component can be obtained by numerically solving the equation:
) 0 ( 2 1 ) (FWHM N N = (2.24)
The only numerical solution found is
(
mcR/")
⋅FWHM =4.296, which leads to the fol-lowing one-to-one relationship between the FWHM of the narrow component in the 1D-ACAR spectrum and the diameter of the cavity [18]:] mrad [ 3.32 nm) ( 2 FWHM R = (2.25)
The last equation can be used to determine the cavity diameter in porous materials by measuring the FWHM of the narrow component in the 1D-ACAR distribution. Eqs. 2.15, 2.16 and 2.25 show that the proportionality factor between the size of the cavity and the FWHM of the p-Ps momentum distribution is somewhat larger for the spherical well potential than for the rectangular well potential. This is a consistent result since the larger volume of a rectangular cavity generates a narrower momentum p-Ps distribution.
2.4. Positron transport in electric and magnetic fields
A slow positron beam is usually transported from the moderator to the target with the aid of magnetic and/or electrostatic fields (see Chapter 3). The beam passes through acceler-ators and lenses of various designs. In order to achieve the optimum combination of transmission factor and beam properties at the target, the positron transport system should be designed based on accurate calculations of positron kinetics in electric and magnetic fields.
2.4.1. Calculation of non-uniform electric and magnetic fields
The electric and magnetic fields in the various components along the beam transport system presented in Chapter 3 are calculated with the aid of the CERN-POISSON program [23]. This program package, developed at CERN, consists of a set of computer programs that solve the equations of Poisson and Laplace for two-dimensional magnetostatic and electrostatic potential problems. Material properties may be linear or non-linear and polar-ised materials (e.g. permanent magnets or dielectric media) are allowed.
The magnetostatic scalar and vector potential problems
The magnetostatic scalar (Φm) and vector ( A ) potentials obey the following equations (all
quantities are given in SI) [24]:
(
∇Φ)
=0 ⋅ ∇ µ m (2.26) J A c π = ∇× × ∇ 1 4 µ , (2.27)where µ represents the magnetic permeability, c the velocity of light, and J the electric current density. With these potentials and given the distributions of µ(r) and J(r), CERN-POISSON finds the spatial distributions of the magnetic field and the magnetic flux density: m Φ −∇ = H (2.28)
A
B=∇× (2.29)
The electrostatic scalar potential problem
The electrostatic scalar potential Φe obeys the equation of Poisson [24]:
(
ε∇Φ)
=− πρ⋅
∇ e 4 , (2.30)
where ε is the dielectric constant and ρ is the electric space charge density. In regions of space where there is no charge density and epsilon is constant, the scalar potential satisfies the equation of Laplace:
0
2Φ =
∇ e . (2.31)
With the potential Φe, CERN-POISSON finds the components of the electric field strength
and their spatial variation:
e Φ −∇ =
E . (2.32)
In order to determine the electric and magnetic fields the system with specific components (e.g. coils, solenoids, electrostatic lenses, permanent magnets, etc.) has to be defined in regions and subregions where the material properties are known. The first stage is to construct a triangular mesh representing the discretisation of the system. Next, one solves iteratively for the vector potential at these mesh points using discretised forms of the equations of Poisson and Laplace and the necessary boundary conditions.
The main modules of the CERN-POISSON program are LATTCR, POISCR and TRIPCR. The LATTCR produces the triangular mesh based on user input data describing the problem (e.g. geometry, boundaries, sources) which is directed into POISCR, the program which solves the potential problems described earlier. TRIPCR plots the triangular mesh as well as region boundaries generated by LATTCR and equipotential lines derived from the POISCR module.
2.4.2. Positron transport
A Monte Carlo program POEM (POlarised beam simulator in Electric and Magnetic fields) developed by Kumita et al. [25] has been used to simulate the trajectories of the slow positrons in beam transport systems. This program is based on GEANT, software developed at CERN which simulates the passage of elementary particles through matter. The relativistic equation of motion and spin precession for a particle with rest mass m and charge e moving in an electric field E and a magnetic field B are [24]:
(
v E)
E B v E v = ⋅ + × = e t c e t m d d ; d d (2.33)(
)
(
)
× + γ γ − − ⋅ + γ γ − − γ + − × = s B v B v v E s 1 2 1 1 1 2 1 1 1 2 d d 2 g c g c g mc e t , (2.34)where v denotes the velocity of the particle, c the velocity of light, s the spin of the particle, g the giromagnetic factor of e+ and γ = 1/[1-(|v|/c)²]½. Basically, for a given
electromag-netic configuration, POEM solves the differential equations 2.33 and 2.34 using the Runge-Kutta method. However, in this work only the part simulating the positron traject-ories has been used; one sums over the spins. The program uses a specific input file con-taining the configuration of the beam transport system, e.g. the length and radius of the beam transport line, the number of particles, the longitudinal and transversal energy of the particles at the entrance, and the initial polarisation of the particles. Additionally, POEM uses two input files given by the CERN-POISSON program which contain the electric and magnetic fields in the beam line. POEM generates a number of histograms containing the beam profiles at entrance and target, positron trajectories and spin precessions, and the spin distribution at the target. Furthermore, the program gives an output file containing the transversal positions and the energies of the particles at the entrance and target. From this file one can derive the energy distribution at the target. The energy of the particle is given by two values corresponding to the transversal and horizontal components of the velocity vector of the particle. Therefore, one can follow the evolution of the angular distribution of the beam along the beam line (see also Chapter 3).
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