Study of a silicon detector for the determination of …...Universit`a degli Studi di Trieste...

Universita degli Studi di Trieste

Facolta di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Specialistica in Fisica

Tesi di Laurea Specialistica

Study of a silicon detectorfor the determination

of the annihilation pointof an antiproton

Studio di un rivelatore al silicio perla determinazione del punto

di annichilazione di un antiprotone

Relatore: Laureando:Dott. Germano Bonomi Florian Zenoni

Correlatore:Prof.ssa Anna Martin

Anno Accademico 2009-2010

Kirk: Like Lazarus. Identical, yet bothLazarus. Except one is matter and the otherantimatter. If they meet...Spock: Annihilation Jim. Total, complete,absolute annihilation.

Star Trek: The Original Series

Contents

Introduzione i

Introduction v

1 Toward the antihydrogen 11.1 Theoretical revolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outstanding experimental proofs . . . . . . . . . . . . . . . . . . . . . . 41.3 Motivations to build an antiatom . . . . . . . . . . . . . . . . . . . . . . 71.4 Is there room for antigravity? . . . . . . . . . . . . . . . . . . . . . . . . 15

2 The AEgIS experiment 192.1 Antihydrogen, here we are . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 AEgIS generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 The design of AEgIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Positron accumulator . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Transfer of positrons into the main magnet . . . . . . . . . . . . 292.3.3 The magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.4 The diluition refrigerator . . . . . . . . . . . . . . . . . . . . . . 302.3.5 The trap system . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Positronium formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Handling antihydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5.1 Antihydrogen production by charge exchange . . . . . . . . . . . 372.5.2 Acceleration and deceleration of Rydberg atoms . . . . . . . . . 382.5.3 The gravity measurement . . . . . . . . . . . . . . . . . . . . . . 39

3 Antiproton-matter interaction 473.1 Antinucleon-nucleon cross-sections . . . . . . . . . . . . . . . . . . . . . 47

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1.2 p-nucleus interaction . . . . . . . . . . . . . . . . . . . . . . . . . 483.1.3 p-nucleus cross-sections (A ! 3) . . . . . . . . . . . . . . . . . . . 543.1.4 Pion multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 p energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.1 The Barkas e!ect . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 The AEgIS beam line 634.1 Description of the beam line . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 The p-beam counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3 Other detectors in AEgIS . . . . . . . . . . . . . . . . . . . . . . . . . . 69

i

ii CONTENTS

4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.4.1 Particle generation . . . . . . . . . . . . . . . . . . . . . . . . . . 704.4.2 Beam counter in the ‘sun’ . . . . . . . . . . . . . . . . . . . . . . 714.4.3 Beam counter near the Malmberg-Penning trap . . . . . . . . . . 754.4.4 Final considerations on the choice of the detector . . . . . . . . . 784.4.5 Study of the aluminium degrader . . . . . . . . . . . . . . . . . . 78

5 Simulation of the annihilation physics 835.1 Chiral Invariant Phase Space (CHIPS) Physics list . . . . . . . . . . . . 83

5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.1.2 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . 855.1.3 Antiproton-nuclear simulation annihilation at rest . . . . . . . . 86

5.2 Secondary particles multiplicity . . . . . . . . . . . . . . . . . . . . . . . 865.3 p stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.4 Deposition length and energy released . . . . . . . . . . . . . . . . . . . 89

5.4.1 Deposition length . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4.2 Energy released . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 Conclusions 101

A The Virtual Monte Carlo 103A.1 Geant3 and Geant4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A.2 Introduction to VMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104A.3 The VMC concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104A.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.5 Use of VMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107A.6 Geant3 VMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109A.7 Geant4 VMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109A.8 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.9 Description of the simulation program parameters . . . . . . . . . . . . 113

B Semiconductor detectors 117B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117B.2 Semiconductor properties . . . . . . . . . . . . . . . . . . . . . . . . . . 118B.3 Charge carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119B.4 Migration of charge carriers in an electric field . . . . . . . . . . . . . . 120B.5 E!ect of impurities or dopants . . . . . . . . . . . . . . . . . . . . . . . 122B.6 The np semiconductor junction . . . . . . . . . . . . . . . . . . . . . . . 123B.7 Reverse biasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124B.8 Fully depleted detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Bibliography 127

Ringraziamenti 133

Introduzione

Nessuno ha mai lasciato cadere una sola particella di antimateria, eppure la maggiorparte dei fisici e convinta che essa cadrebbe al suolo esattamente come la materia co-mune. Le loro convinzioni sono basate su due idee saldamente radicate: il principio diequivalenza debole della gravita e la simmetria tra materia e antimateria espressa nelteorema CPT. Tuttavia, secondo alcuni fisici teorici e possibile che il primo di questiprincipi possa rivelarsi falso. Sulla base di determinati modelli, esperimenti come quellidi Lorand Eotvos non escludono infatti che gli e!etti di questa violazione possano es-sere visibili. Tuttavia, se si impiega la materia comune per e!ettuare misure in campogravitazionale, le deviazioni risultano di entita trascurabile.Per questi motivi un esperimento in grado di misurare l’attrazione gravitazionale dell’an-timateria potrebbe essere di grande importanza per la comprensione della gravita quan-tistica. Da un diverso punto di vista, esso rappresenterebbe un coronamento per lalunga tradizione degli esperimenti sull’antimateria. Questo percorso, costellato da pre-mi Nobel, fu inaugurato nel 1931 da Carl D. Anderson con la scoperta del positronenei raggi cosmici. Continuo poi con la scoperta dell’antiprotone al Bevatron nel 1955,grazie agli sforzi di Owen Chamberlain e Emilio Segre; inoltre pochi anni dopo, graziea Hans Dehmelt, un altro successo venne segnato con l’intrappolamento di un singolopositrone. Al CERN, la costruzione dell’acceleratore LEAR negli anni ’80 consentı laprima accumulazione di antiprotoni, ponendo le basi per una nuova branca della fisica:lo studio degli antiatomi, ovvero atomi interamente composti da antimateria. All’iniziodel nuovo secolo, l’anti-idrogeno divenne una realta grazie ad un miglioramento dellestrutture a disposizione (specialmente con l’Antiproton Decelerator). Diversi successisperimentali, al CERN e al Fermilab, andrebbero quindi menzionati, partendo dallaprimissima collaborazione, PS-210, fino ad ATHENA, la cosiddetta “fabbrica di antia-tomi”.E’ proprio su questo straordinario background che un nuovo esperimento, chiamatoAEgIS (Antimatter Experiment: Gravity, Interferometry and Spectroscopy), sta per ve-dere la luce al CERN. In preparazione dal 2007, esso iniziera le prime acquisizioni datialla fine del 2011. Lo scopo primario dell’esperimento AEgIS e la misura diretta dellacostante di accelerazione di gravita g dell’anti-idrogeno. Accanto alle grandi aspettativeche la comunita di fisici teorici ha nei confronti di queste misure, esse rappresentano unpunto di svolta anche dal punto di vista della capacita di manipolare l’anti-idrogeno. Aldi la della sua formazione e intrappolamento, nessuno ha mai e!ettuato misure direttesu di esso, mentre nel futuro prossimo i test di CPT attraverso la spettroscopia nonsaranno ancora probabilmente possibili.Nella prima fase dell’esperimento sara condotta una misura dell’accelerazione gravita-zionale con una precisione relativa dell’1%. Cio avverra mediante l’osservazione dellospostamento verticale di un fascio di anti-idrogeno che ha attraversato un deflettometro

i

ii Introduzione

di Moire, la controparte classica di un interferometro. Il deflettometro di Moire hagia dato prova di se, essendo stato utilizzato in passato con atomi e molecole. I passiessenziali che portano alla produzione di anti-idrogeno e alla misura di g con AEgISsono i seguenti: produzione di positroni (e+) da una sorgente e immagazzinamento inun accumulatore Surko-type; cattura e immagazzinamento di antiprotoni da AD in unatrappola di Malmberg-Penning cilindrica; produzione di positronio (Ps) tramite bom-bardamento di una materiale nanoporoso mediante un flusso intenso di e+; eccitazionedel Ps in uno stato di Rydberg con numero quantico principale n = 30, . . . , 40; ricombi-nazione di anti-idrogeno (H) con reazione risonante di scambio carica tra Ps di Rydberge antiprotoni freddi; formazione di un fascio di H tramite accelerazione Stark con campielettrici disomogenei; determinazione di g con un deflettometro di Moire accoppiato conun rivelatore sensibile alla posizione. La vera scommessa di AEgIS e chiaramente quelladi riunire in un solo apparato tecniche sperimentali molto diverse, le quali hanno perogia dimostrato di funzionare in settori di ricerca indipendenti.Lo scopo di questa tesi e quello di elaborare una simulazione Monte Carlo, concentratasulla linea del fascio di AEgIS. Gli antiprotoni vengono forniti da AD, in bunch da" 107 particelle ciascuno, aventi energia cinetica di 5.3 MeV. In particolare, il compitoprincipale e quello di simulare un rivelatore al silicio, chiamato beam counter, il cuiruolo e quello di monitorare il fascio e il suo centramento, in vista dei test prelimina-ri. In aggiunta al beam counter, le simulazioni si concentrano anche sullo studio di undegrader di alluminio, avente la seguente funzione. Per essere catturati dalla trappoladi Malmberg-Penning, gli antiprotoni devono essere il piu lenti possibile. L’energia ci-netica del fascio di AD, di 5.3 MeV, e quasi 3 ordini di grandezza troppo grande perle necessita della trappola. Il miglior modo per rallentare le particelle da accumularee far loro attraversare un foglio di alluminio. La massima energia desiderata per gliantiprotoni e di 10 keV; e quindi importante capire che un errore di pochi micron suldegrader e su"ciente per compromettere la conseguente formazione di anti-idrogeno:se lo strato e troppo sottile, il fascio mantiene troppa energia, e non ci sono abbastanzaantiprotoni con EK < 10 keV; al contrario, se lo strato e troppo spesso, gli antiprotoniperdono tutta la loro energia, e terminano la loro corsa nel materiale. Ad ogni modo,il setup dell’esperimento e stato studiato per poter modificare lo spessore di alluminiomicron per micron, fino ad un massimo di 30 m. E’ tuttavia necessario fornire tramitele simulazioni un valore minimo per l’alluminio, in modo che un degrader principalepossa essere fissato permanentemente nel setup, immediatamente a monte della trap-pola. Tali test preliminari occuperanno probabilmente la Collaborazione al CERN pertutta la prima parte dell’acquisizione dati.Sono state considerate due possibili forme per il rivelatore al silicio, e due di!erenti sitisono previsti per esso. La simulazione, scritta in C++ nel framework del Virtual MonteCarlo (VMC), e pensata per essere estesa alle altre porzioni del setup sperimentale. Inparticolare, sara in futuro necessario implementare il rivelatore sensibile alla posizione,posto in fondo all’apparato.La simulazione di antiprotoni su nuclei di silicio non e probabilmente mai stata eseguitain un range di energia cosı basso. Per quel che ne sappiamo, il generatore di eventiCHIPS (Chiral Invariant Phase Space), incluso come Physics list in Geant4, e l’unicomodo per conseguire una tale simulazione Monte Carlo. Conoscere il comportamentodei prodotti di annichilazione giochera un ruolo di primo piano nella progettazione delrivelatore di posizione, in quanto la risoluzione desiderata e di 10 m. Per questa ra-gione, e necessario che i risultati forniti da CHIPS vengano confermati. La molteplicitadelle particelle secondarie e stata cosı confrontata con i dati sperimentali presenti in

iii

letteratura, con particolare attenzione a quella dei pioni carichi. In questo modo sonostati discussi il range di molti di questi prodotti di annichilazione, insieme all’energiada loro rilasciata.In conclusione, i risultati di questa tesi possono essere sfruttati per migliorare il gene-ratore di eventi. Ma soprattutto, il presente lavoro di tesi e dedicato nell’analizzare ilfascio di antiprotoni lungo il suo asse, dal loro punto di generazione fino all’entrata nellatrappola di Malmberg-Penning. Tutte queste considerazioni si concretizzano attraversola scelta dei parametri fondamentali del beam counter, e nella sua precisa disposizionenell’apparato sperimentale, cosı come nel fornire indicazioni sullo spessore ideale deldegrader di alluminio.

iv Introduzione

Introduction

No-one has never let fall a single particle of antimatter. Yet, most of the physicists areconvinced that it would fall down exactly like common matter. Their arguments arebased upon two deep-rooted ideas: the weak equivalence principle of gravitation, andthe quantum field symmetry between matter and antimatter, expressed by the CPTtheorem. According to some theorists, it may be possible that the former of these ideascould be false. In their opinion, it cannot be excluded that the e!ects of such a violationare visible in Lorand Eotvos’ experiments. However, when adopting common matter toperform measurements in the gravitational field, deviations are negligible.Thus, an experiment capable to measure the gravitational attraction of antimatter couldbe of great importance to the understanding of quantum gravity. From an other pointof view, it would represent the crowning achievement of a tradition on antimatter ex-periments. This path, truly sparkling with Nobel prizes, was started in 1931 by CarlD. Anderson with the discovery of positron in cosmic rays. It then continued with theuncovering of antiproton at Bevatron in 1955, among others, by Owen Chamberlain andEmilio Segre; thanks to Hans Dehmelt, another success was marked years later with theentrapment of a single positron. At CERN, the construction of the LEAR apparatus inthe ’80s led to the first storage of antiprotons, preparing the ground to a new branchof physics: the study of antiatoms, that is, atoms entirely made by antimatter. At thebeginning of the new century, antihydrogen became a reality after an upgrade of thefacilities (especially with the Antiproton Decelerator). Several successful experimentsat CERN and Fermilab must be mentioned, from the very first one, PS-210, up toATHENA, the so-called ‘antimatter factory’.It is upon this extraordinary background that a new experiment at CERN, called AEgIS(Antimatter Experiment: Gravity, Interferometry and Spectroscopy), is about to seethe light of day. In preparation since 2007, it will start the first data acquisition atthe end of 2011. The primary scientific goal of AEgIS is the direct measurement ofthe Earth’s gravitational acceleration g on antihydrogen. Beside the high expecta-tions coming from the theoretical physics community, this measure would represent abreakthrough in the capacity of handling antihydrogen. Apart from its formation andstoring, no direct measurement has never been performed on antihydrogen, and CPTtests through spectroscopy will probably not be possible in the near future.In a first phase of the experiment, a gravity measurement with 1% relative precision willbe carried out by observing the vertical displacement of the shadow image produced byan antihydrogen beam as it traverses a Moire deflectometer, the classical counterpartof a matter wave interferometer. The Moire deflectometer has a good track record,since it has already been used with atoms and molecules. The essential steps leading tothe production of anti-hydrogen and the measurement of g with AEgIS are the follow-ing: production of positrons (e+) from a Surko-type source and accumulator; capture

v

vi Introduction

and accumulation of antiprotons from the AD in a cylindrical Malmberg-Penning trap;production of positronium (Ps) by bombardment of a nanoporous material with an in-tense e+ pulse; excitation of the Ps to a Rydberg state with principal quantum numbern = 30, . . . , 40; recombination of antihydrogen (H) by resonant charge exchange betweenRydberg Ps and slow antiprotons; formation of an H beam by Stark acceleration withinhomogeneous electric fields; determination of g in a two-grating Moire deflectometercoupled with a position-sensitive detector. It is clear that the bet of AEgIS is to puttogether very di!erent experimental techniques, some of which are already establishedin independent areas of reasearch.The aim of this thesis work is to fully implement a Monte Carlo simulation, focused onthe antiprotons beam line. Antiprotons will be delivered from AD, in bunches of " 107

p each, with a kinetic energy of 5.3 MeV. In particular the main task is to simulatea silicon detector, named beam counter, whose role is to monitor the beam and itscentering, especially in preliminary phase tests.In addition to the beam counter, simulations are also focused on the study of an alu-minium degrader. To be captured by the Malmberg-Penning trap, antiprotons beammust be as slow as possible. The AD beam kinetic energy of 5.3 MeV is almost 3 ordersof magnitude too large for the needs of the trap. The best way to slow down particles tobe stored, is to degrade them through an aluminium foil. The maximum energy wishedfor antiprotons is 10 keV. It is now important to realize that an error of a few micronson the degrader is su"cient to compromise the subsequent formation of antihydrogen:if the layer is too thin, the beam preserves too much energy, and there are too few an-tiprotons with EK < 10 keV; in reverse, if the layer is too thick, the antiprotons lose alltheir energy and are stopped in the material. Anyhow, the experiment setup is equippedto modify micron per micron the degrader thickness. It is however necessary to providethrough simulations a minimum value for this layer, since an aluminium degrader mustbe permanently placed in the setup, upstream of the trap. These tests will probablyoccupy the AEgIS Collaboration at CERN for the first part of data acquisition.Di!erent detector designs have been considered, and two possible locations for it areavailable. Furthermore, this simulation program, written in C++ in the framework ofthe Virtual Monte Carlo (VMC), is intended to be extended to other portions of theexperimental setup. In particular, it will be necessary to implement the final positionsensitive detector.Simulating antiprotons annihilation on silicon nuclei have probably never been per-formed in this range of energy. As far as we know, Chiral Invariant Phase Space(CHIPS) event generator, included in Geant4 as Physics list, is the only way to at-tempt such a Monte Carlo simulation. Knowing the behaviour of annihilation productswill play a primary role for the design of the already quoted sensitive position detector,as the resolution is wanted to be kept at 10 m. For this reason, it is hence importantto validate or not the results given by CHIPS Physics list. The multiplicities of sec-ondary particles have been compared with data present in literature, with a particularattention dedicated to charged pions. Then, the range of several of these annihilationproducts, as well as the energy they release in silicon, have been discussed.Summarizing, this thesis then represents a contribution that could help to make im-provements in the event generator. Bust most of all, the current work analizes theantiproton beam along the beam axis, from their point of income in the apparatus, tillthe entrance of the Malmberg-Penning trap. All these considerations are put into ac-tion by the choice of the fundamental parameters of the beam counter, and the preciselocation in the experimental setup, as well as indications about the ideal thickness for

vii

the aluminium degrader.

viii Introduction

Chapter 1

Toward the antihydrogen

To describe the history of quantum mechanics and particle physics is necessarily todescribe the history of antimatter, at least until the ’70s, when the formulation of theStandard Model, and the consequent attention to the massive vector bosons W± andZ0 (let alone the rush on the Higgs Boson) lead to a gap between the physics of low andhigh energy. This is why, since antimatter holds a role of primary importance in ourknowledge of the Universe, it is critical to keep in mind the scientific and human pointof view on the fundamental steps that led to the physics we deal with here. Since themain character of the experiment this work is linked with is by far the antihydrogenatom, in this chapter we present a historical review, both theoretical and experimental,of the components of the antiatom: the antiproton and the positron.Then the motivations to all the e!orts in place to build an antiatom will be given, and inthe end we will briefly investigate the room left by modern physics cornerstones to thepossibility of antigravity. Since the gravitational measurement of the AEgIS experimentwould be the first one to directly check the weak equivalence principle violation with abody not made of matter, a theoretical incursion in this field cannot be avoided.

1.1 Theoretical revolutions

The Dirac equation

The birth of antimatter occurred with the Dirac equation. This relativistic quantummechanical wave equation was formulated by British physicist Paul Dirac in 1928 [1]. Itprovides a description of elementary spin- particles, such as electrons, consistent withboth the principles of quantum mechanics and the theory of special relativity.The wave equation is necessarily of the form:

i!!

!t= HD!, (1.1.1)

where HD is a Hermitean operator of state-vector space, and ! is the four-componentwavefunction [2]. Taking into accout certain hypoteses (e.g. the equation must be offirst order with respect of the spatial variables, the Hamiltonian must be invariant undertranslation, etc.), the energy operator can be written in the form:

HD = ! · p + "m, (1.1.2)

1

2 Toward the antihydrogen

where the operator p has the significance indicated by the correspondence rule, i.e.p = #i$, m is the rest mass of the electron, and ! % (#x, #y, #z) and " denote 4Hermitean operators acting on the spin variables alone. Therefore, in absence of theelectromagnetic field, the equation states:

!

i!

!t# ! · p# "m

"

! = 0, (1.1.3)

that is the Dirac equation for the free electron originally proposed by Dirac himself.However, one can use a more convenient form, more symmetrical with respect to thespace and time coordinates, called the explicitly covariant form of the Dirac equation:

#i!$µ!µ! + m! = 0, (1.1.4)

where $µ are the Dirac matrices. One defines:

$µ % ($0, $1, $2, $3) % ($0, "), (1.1.5)

$0 % ", " % "!. (1.1.6)

To solve the Dirac equation in the absence of a field is equivalent to finding the eingen-solutions of the Hamiltonian HD. Such solutions are plane waves, that is, functions ofthe form

u(p)eip·r,

where u(p) is a four-component spinor independent of r. It is determined by theeigenvalue equation:

Hu(p) = Eu(p). (1.1.7)

Calculations give:H2 = p2 + m2. (1.1.8)

The only possible eigenvalues of H are therefore the two values ±#

p2 + m2, i.e.:

E = %Ep (% = ±1) (1.1.9)

Ep =#

p2 + m2 (1.1.10)

As they stand, the negative energy solutions have no physical significance. If it werepossible to completely decouple the positive and negative energy states the latter couldsimply be ignored. Such, however, is not the case. Consider, for example, the com-plete spectrum of the hydrogen atom. Owing to the coupling of the electron with theelectromagnetic field, there is always a possibility of a radiative transition from a givenstate of the atom to a state of lower energy. Consequently an electron in one of thebound states of the hydrogen atom can, even if isolated, make quantum jumps to statesof negative energy with emission of one or several photons; further, since the spectrumhas no lower bound, the hydrogen atom has no stable state.In order to avoid these di"culties Dirac has made the well-known suggestion of thehole theory, by postulating the existence of a negative energy ‘sea’ of electrons. In thispicture, one of these electrons can make a transition from the sea to a state of positiveenergy. The ‘hole’ of negative energy then appears as a particle of positive mass +mand charge #e.

1.1 Theoretical revolutions 3

The hole theory is only a first step in the direction of the correct theory of the quan-tized electron field. It has the merit of providing simple pictures and therefore serve asa guide in the elaboration of the correct theory. But pitfalls and contradictions appearwhen it is pushed too far.

The need for antimatter

Dirac’s equation had pointed the way towards antimatter, but it was Robert Oppen-heimer who really saw the full vision [3]. Dirac actually seriously considered the protonas a candidate for the positively charged particle that had emerged from his equations.1

Dirac’s aim was simply to quieten the persistent questioners, allowing him to get onwith explaining his profound ideas and to leave the question of the mass as a ‘detail’ tobe solved later.However, Oppenheimer pointed out that the positive particle could not be the proton,for if it were, then hydrogen atoms would self-destruct. The arguments that implied anelectron and its positive counterpart can emerge from the vacuum, could be applied inreverse: play the film backwards and it would show the pair mutually annihilating, dis-appearing into gamma rays. So if the positive particle were identified with the proton,hydrogen atoms would survive only so long as the proton did not meet the electron:not just hydrogen, but all matter would vanish in a flash of light.Dirac immediately realized the power of Oppenheimer’s criticism and accepted that hispositive electron was indeed something entirely new. In September of 1931 he publishedthe following conclusion [4]:

A hole, if there were one, would be a new kind of particle, unknown to exper-imental physics, having the same mass and opposite charge to an electron.We may call such a particle an anti-electron. We should not expect to findany of them in nature, on account of their rapid rate of recombination withelectrons, but if they could be produced experimentally in high vacuum theywould be quite stable and amenable to observation.

In fact, it was the birth of the modern idea of antimatter.In his 1931 paper he also made clear that in his theory “there is a complete and perfectsymmetry between positive and negative electric charge [. . .]. If this symmetry is reallyfundamental in nature, it must be possible to reverse the charge of any kind of particle”.Thus he predicted that an antiproton, a negatively charged massive mirror of the proton,should also exist.Dirac was awarded with the Nobel Prize in 1933, with Werner Heisenberg and ErwinSchrodinger “for the discovery of new productive forms of atomic theory”.

Reinterpretating the negative energies

It is trivial that classically, negative energies for free particles appear to be completelymeaningless [5]. In quantum mechanics, however, we represent the ampliude of aninfinite stream of particles, say electrons, travelling along the positive x-axis with 3-momentum p by the plane wavefunction:

& = Ae!i(Et!px)/!, (1.1.11)

1In 1928 the particle picture was simple: matter is made from negatively charged electrons andpositive protons. In this relatively cosy world-view, there was no need for further particles or anydesire for them; the anti-electron had no place.


where A is a normalization constant. We define the angular frequency as ' = E/!,and the wavenumber as k = p/! [5]. As t increases, the phase advances in the directionof increasing x. Formally, however, equation (1.1.11) can also represent particles ofenergy #E and momentum #p travelling in the negative x-direction and backwards intime (i.e. replacing Et by (#E)(#t) and px by (#p)(#x)). Such a stream of negativeelectrons flowing backwards in time is equivalent to positive charges flowing forward,and thus having E > 0. Hence, the negative energy particle states are connected withthe existence of positive energy antiparticles of exactly equal but opposite electricalcharge and magnetic moment, and otherwise identical.

1.2 Outstanding experimental proofs

A positive result from cosmic rays

On the other side of the ocean, Robert Millikan at California Intitute of Technology(Caltech), the man who coined the name ‘cosmic rays’, had his own theories about theorigins of this extraterrestrial radiation. In 1930 he suggested to his research student,Carl Anderson, that he build a magnet powerful enough to deflect them [3]. Andersondiscovered to his surprise that the cosmic rays contained both negative and positivecharged particles in about equal numbers.His interpretation was that the negatives were electrons and the positives, protons.However the images in Anderson’s photographs did not really fit with this. Lightweightparticles such as electrons leave thin wispy trails, quite di!erent from the dense trailsof bulky protons. Most of the trails in Anderson’s pictures looked like electrons, and sohe suggested that those that curved ‘the wrong way’ were not due to positively chargedparticles rushing downwards but were instead electrons moving upwards. Millikan didnot like this, and with his judgement skewed by his prejudices on the nature of cosmicrays, insisted that even though the trails were thin and not thick, they must nonethelessbe caused by downward moving protons.Anderson settled the debate by putting a lead plate across the middle of the chamber.If a particle passed through the plate, it would lose energy and so would have a tightercurve afterwards than it had had before entering. This way there would be no argumentabout whether they were travelling downwards or upwards; it would also determine onceand for all the sign of their charges: positive down-mover or negative up-mover.He soon found several examples of such ‘positive electrons’, particles that were clearlymuch lighter than protons, coming down from above, and had enough confidence togo public. The editor of Science News Letter published a photograph of one of thetracks in the December 1931 edition, and coined the name ‘positron’ (see Fig. 1.1).It has been known as such ever since. In 1936, Anderson won the Nobel Prize “forhis discovery of the positron”. Whereas he was the first to identify a positron, it wasPatrick Blackett and Giuseppe Occhialini in the Cavendish Laboratory at Cambridgewho confirmed its existence without a doubt a few months later, and explained whereit had come from. Their big idea was to put one Geiger counter above a cloud chamber,and another one below. By connecting the Geiger counters to a relay mechanism, theelectrical impulse from their simultaneous discharges triggered the cloud chamber anda flash of light captured the tracks of the cosmic rays on film.They observed the first examples of the production of e+e! pairs in cosmic ray showers(see Figure 1.2). Thus, they realized that the positrons were being formed as a resultof collisions between the cosmic rays and atoms in the chamber. Albert Einstein’s

1.2 Outstanding experimental proofs 5

Figure 1.1: The discovery of antimatter [6]. The picture shows the track of a positron observed byAnderson in a cloud chamber placed in a magnetic field and exposed to cosmic rays. Note that themagnetic curvature of the track in the upper half of the chamber is greater than that in the lower half,because of the loss of momentum in traversing the metal plate; hence the particle was proved to bepositively charged and travelling upwards (a fact that initially confused Anderson).

equation, E = mc2, implies that energy (E) can be converted into mass (m) andBlackett and Occhialini had for the first time demonstrated the creation of matter, andantimatter, from radiation. Blackett and Occhialini’s paper was sent in February 1933[7]. Blackett was awarded with the Nobel prize in 1948 “for his development of theWilson cloud chamber method, and his discoveries therewith in the fields of nuclearphysics and cosmic radiation”.

A new target: the antiproton

Dirac originally wrote his equation in order to describe the electron. However, it appliesequally well to a proton or a neutron, as the nucleons are fermions too.2 Therefore, theequation implied that the proton and neutron also have antimatter counterparts: theantiproton and antineutron. The problem was that as the proton is nearly two thousandtimes heavier than an electron, so would an antiproton be that much heavier than apositron, which means that much more energy is needed to make it. While antiprotonsdo occur in cosmic rays, they are much rarer and harder to identify.In 1950, an ambitious plan took hold at Berkeley in California to build an acceleratorthat would speed protons such that when they smashed into a target, there would beenough energy to produce an antiproton. The machine was known as the BeVatron.The nominal maximum energy of the protons in the Bevatron was 6.2 GeV. This is justslightly above the threshold energy of 5.6 GeV needed to produce antiprotons from freeprotons. If the target nucleon is in a nucleus and has some momentum the threshold islowered.A small team consisting of Owen Chamberlain, Emilio Segre, Clyde Wiegand, and TomYpsilantis had first turn on the new BeVatron. They decided to adopt a method of

2The neutron was discovered by James Chadwick in 1932 (another Nobel Prize winner, in 1935).


Figure 1.2: Creation of an electron and positron. A high energy cosmic ray has knocked an electronout of an atom – this gives the gently curving trail from the top to bottom left of the image. There isenough energy also to create an electron and positron, which give the tightly curving counter-rotatingspirals at the top of the image. Lower down the picture a further electron and positron are createdwhich depart leaving the inverted vee shape.

discovery that consisted in the determination of the mass and charge of the particlefrom measurements of momentum and velocity. Two independent methods were used– time-of-flight and Cherenkov counters. This was tricky because antiprotons wouldbe very rare and overwhelmed by the production of lighter particles such as showers ofelectrons and positrons, and of pions.A schematic diagram of the apparatus is shown in Figure 1.3. A double focusing spec-trometer, based on a suggestion by Oreste Piccioni, was used to transport the beam ofparticles along its predetermined path. Two scintillation counters, S1 and S2, separatedby a distance of about 12 m were used to measure the time of flight, and two Cherenkovcounters, C1 and C2, were used to discriminate between the antiproton and the pionsin the beam. Having a momentum of 1.19 GeV/c (" = 0.78 for a proton-mass particle)the slower moving antiprotons required about 51 ns to travel the 12 m distance betweenS1 and S2, 11 ns more than the pions. Therefore, the ability to detect the antiprotonsrelied on the ability of the electronic circuits to measure time di!erences of about 11

1.3 Motivations to build an antiatom 7

Figure 1.3: Bevatron: diagram of experimental arrangment.

ns. Eventually, to reduce the background, the Cherenkov counters were used to triggeronly the antiprotons events in the time-of-flight measure.Their idea worked, and sixty antiprotons were detected. The experiment showed thatthe new particle had a mass within 5 percent of that of the proton mass. It was asuccess and in 1955 they announced the discovery [8]. Chamberlain and Segre wonthe Nobel in 1959, “for the discovery of antiproton”. One of the other teams led byOreste Piccione that had entered the competition also gained success themselves withthe discovery of the antineutron in 1957.

1.3 Motivations to build an antiatom

A very important concept in physics is the symmetry or invariance of the equationsdescribing a physical system under an operation – which might be, for example, atranslation or rotation in space. Intimately connected with such invariance propertiesare conservation laws – in the above cases, conservation of linear and angular momen-tum. Such conservation laws and the invariance principles and symmetries underlyingthem are the very backbone of physics. The transformations to be considered can beeither countinous or discrete. Our attention will be focused on the latter.Another principle of prime importance that is susceptible to be involved in antimatterexperiments is the weak equivalence principle, also known as the universality of freefall, or the Galilean equivalence principle. It will be discussed afterwards.

Parity

The operation of the spacial inversion of coordinates (x, y, z & #x,#y,#z) is an exam-ple of a discrete transformation. This transformation is produced by the parity operatorP , where:

P&(r) = &(#r) (1.3.1)


Repetition of this operation clearly implies P2 = 1, so that P is a unitary operator.The eigenvalue of the operator (if there is one) will be ±1. While the intrinsic parityof a fermion is a matter of convention, the relative parity of a fermion and antifermionis not. Fermion-antifermion pairs (e.g. pp) can be created in a reaction so that theintrinsic parity of the pair must be a measurable quantity. The Dirac theory of fermionsrequires particle and antiparticle to have opposite intrinsic parity. This prediction waschecked experimentally by Chien-Shiung Wu and Irving Shaknov in their observationson positronium [9], an ‘atomic’ bound state of e+ and e!.The strong and electromagnetic interactions are parity-conserving. In 1956, following acritical review of the experimental data then available, Lee and Yang (both were Nobelawarded in 1957) came to the conclusion that weak interactions were not invariantunder space inversion, i.e. they did not conserve parity. This was largely on the basisof the fact that the K+ meson could decay in two modes: K+ & 2(, K+ & 3(, inwhich the final states have opposite parities (even and odd respectively). To test parityconservation, an experiment was carried out by C.S. Wu and collaborators in 1957,employing a sample of 60Co at 0.01 K inside a solenoid [10]. At this low temperature,a high proportion of the 60Co nuclei (spin J = 5) are aligned with the field. The cobaltundergoes "-decay to 60Ni" (J = 4), a pure Gamow-Teller transition. The relativeelectron intensities along and against the field direction were measured. The degree of60Co alignment could be determined from the angular distribution of the $-rays from60Ni". The results found for the electron intensities were consistent with a distributionof the form:

I()) = 1 + ## · pE

= 1 + #v

ccos), (1.3.2)

where # = #1, # is a unit spin vector in the direction of J ; p and E are the momentumand energy of the electron and ) is the angle of emission of the electron with respectto J . The asymmetry from ‘top to bottom’ of the intensity in (1.3.2) implies that theinteraction as a whole violates parity conservation. The first term (unity) does notchange sign under reflection – it is scalar (even parity). #, being an axial vector, doesnot change sign either, while the polar vector p does. So the product # ·p changes signunder reflection. It is a pseudoscalar, with odd parity, and the presence of both termsin the intensity implies a parity mixture. Conservation of the z-component of angularmomentum in the above transition implies that the electron spin must also point in thedirection of J , so that if now # denotes the spin vector, the intensity is the same ofEquation 1.3.2.Representing the intensities for # parallel and antiparallel to p by I+ and I!, the netlongitudinal polarisation is:

P =I+ # I!

I+ + I!= #

v

c. (1.3.3)

Experimentally,

# =

$

+1 for e+, thus P = +v/c#1 for e!, thus P = #v/c

(1.3.4)

The quantity:

H =# · p|p| = #1 (1.3.5)

is called the helicity (or handedness). It measures the sign of the component of spin ofthe particle, jz = ±!, in the direction of motion (z-direction). The z-component of spinand the momentum vector p together define a screw sense, as in Figure 1.4. H = +1


Figure 1.4: A neutrino has LH polarisation, while a antineutrino is RH.

corresponds to a right-handed (RH) screw, while particles with H = #1 are left-handed(LH).The result (1.3.4), if applied to a neutrino (m = 0), implies that such a particle mustbe fully polarised, P = +1 or #1. Here E = |p| and the neutrino is in a pure helicitystate, P % H = ±1.Helicity is a well-defined, Lorentz-invariant quantity for a massless particle, for the sim-ple reason that such a particle travels at velocity c. In making a Lorentz-transformationto another reference frame of relative velocity v < c, it is therefore impossible to reversethe helicity. It turns out that solutions of the Dirac equation with finite mass term,such as electrons, are not pure helicity eigenstates but some admixture of LH and RHfunctions. However, provided they are extremely relativistic, massive (anti)fermions aremostly (LH)RH. Ultimately, that is the reason of the parity violation in "-decays.

Charge conjugation and CP symmetry

As the name implies, the operation of charge conjugation reverses the sign of the chargeand magnetic moment of a particle, leaving all other coordinates unchanged. In rel-ativistic quantum mechanics the term ‘charge conjugation’ implies the interchange ofparticle and antiparticle, e.g. e! & e+.Strong and electromagnetic interactions are found experimentally to be invariant un-der the charge conjugation operation. This trasformation is produced by the chargeconjugation operator, C. Like invariance under the parity operation, it is broken inweak interactions. Under the C operation, a LH neutrino " will transform into a LHantineutrino ". Such a state is not found in nature. However, under the combinedoperation CP, a LH neutrino "L transforms into a RH antineutrino "R. So while theweak interactions respect neither P nor C invariance separately, they are eigenstates ofthe product CP:

CP |"L' & |"R' . (1.3.6)

Until 1964, it was believed that CP symmetry was respected. Actually this statementis only very nearly true: in that year Christenson and collaborators discovered thatthe long-lived neutral kaon, normally decaying into three pions with CP eigenvalue #1,could also, with probability 2 · 10!3, decay into two pions with CP = +1 [11]. Let’s seethis violation with more details.The K mesons are in the form of two isospins (I3 = ± 1

2 ) doublets, of strangenessS = ±1. The S = ±1 doublet consists of K+ (= us) and K0 (= ds), while theirantiparticles, with S = #1, are K! (= us) and K0 (= ds) respectively. All decay tonon-strange particles and/or leptons, through the #S = ±1 weak interaction.


As a consequence, starting o! with a pure K0 beam one can end up, after a few meters,with a beam of mixed strangeness, i.e. K0 and K0. Indeed, since both are neutral,both can decay to pions by the weak interaction, with |#S| = 1. Thus, mixing canoccur via (virtual) intermediate pion states. The state of superposition of both K0 andK0 at a time t can be written:

|K(t)' = #(t)|K0' + "(t)|K0'. (1.3.7)

We know that objects that decay by weak interactions are eigenstate of CP, not ofstrangeness S. The operation of CP on the states K0 and K0 is as follows:

CP|K0' & *|K0', CP|K0' & *#|K0', (1.3.8)

where *, *# are arbitrary phase factors, which we can define as * = *# = 1. Clearly|K0' and |K0' are not CP eigenstates, but we can form the linear combinations:

|KS' =

%

1

2

&

|K0' + |K0''

, (1.3.9)

|KL' =

%

1

2

&

|K0' # |K0''

. (1.3.10)

The nomenclature KS and KL stands for short and long lifetimes to decay, as discussedbelow. We have therefore that:

CP|KS' & |KS', CP|KL' & #|KL'. (1.3.11)

Unlike K0 and K0, distinguished by their mode of production3, KS and KL are distin-guished by their mode of decay:

|KS' & 2( (CP = +1), +S = 0.893 · 10!10 s (1.3.12)

|KL' & 3( (CP = #1), +L = 0.517 · 10!7 s. (1.3.13)

As anticipated, in 1964 an experiment by Christenson and collaborators first demon-strated that the long-lived state we have called KL could also decay to (+(! with abranching ratio of order 10!3.Denoting the CP = +1 amplitude by K1 and the CP = #1 amplitude by K2, the KL

and KS amplitudes are then the admixture:

KL =1

#

1 + |%|2(K2 + %K1) (1.3.14)

KS =1

#

1 + |%|2(K1 # %K2) (1.3.15)

where % is a small parameter quantifying the CP violation. The degree of CP violationis usually quoted as the amplitude ratio:

|*+!| =ampl(KL & (+(!)

ampl(KS & (+(!)= (2.29 ± 0.02) · 10!3. (1.3.16)

3In pp annihilation the nature of the neutral kaon produced is identified via the charged K and (secondaries produced in association.


Similarly, the measured amplitudes for 2(0 decay stand in the ratio:

|*00| =ampl(KL & (0(0)

ampl(KS & (0(0)= (2.28 ± 0.02) · 10!3. (1.3.17)

This is called indirect CP violation. But a direct one can also occur, where a violation%# takes place in the decay process itself. This can arise as follows. Weak hadronicdecays generally obey a #I = 1/2 rule, for the change in isospin of the hadrons in thedecay process. But #I = 3/2 transitions, although strongly suppressed, seem also tooccur. We can write:

*+! = |*+!| ei!+! ( % + %# (1.3.18)

*00 = |*00| ei!00 ( % # 2%#. (1.3.19)

A total of four experiments to date (1999) find when averaged an e!ect quoted at asignificance level of over five standard deviations:

%#

%= (2.2 ± 0.4) · 10!3. (1.3.20)

On this basis direct CP violation appears to have been established [5].

CPT invariance

The phenomenon of CP violation is linked to violation of invariance under time reversal,T , through the very important CPT theorem. The theorem states that all interactionsare invariant under the successive operation of C, P and T taken in any order.The proof of the CPT theorem is based on very general assumptions and impacts thewhole of particle physics. In 2002, Oscar Greenberg proved that CPT violation impliesthe breaking of Lorentz symmetry [12]. Consequences of the CPT theorem that may beverified experimentally relate to the properties of particles and antiparticles: they musthave the same mass and lifetime and equal and opposite electric charges and magneticmoments. These results would follow from C invariance alone, if it helds universally,but since weak interactions violate C and CP symmetry the prediction rests on themore general theorem.Various experiments have been performed comparing di!erent properties, such as themagnetic moment, the charge to mass ratio and the mass, of a particle and its an-tiparticle [13]. The best limits are summarized in Figure 1.5. In particular from thecomparison of the gyromagnetic moment of e! and e+ a limit of 10!12 could be de-duced in terms of relative precision, while the K0 # K0 mass di!erence sets a limit of10!18. However according to some theoretician both measurements are model depen-dent. Indeed in specific models CPT may be broken without a!ecting the gyromagneticmoment [14, 15] while the limit extracted from the K0 # K0 mass di!erence assumesthe validity of the standard model which does not contain itself a mechanism to violateCPT. The role of the antihydrogen arises from the fact that, di!erently form a singleantiparticle, it is a complete system perfectly suitable for a direct test of CPT. Usingthe same experimental methods of hydrogen spectroscopy high precision comparisonof the two system may be achieved. Indeed antihydrogen is a mostly electromagneticsystem (weak interaction, or parity violating, e!ets are small and the same for hydrogenand antihydrogen) making the comparison matter-antimatter less model dependent.


Figure 1.5: Relative precision of various CPT tests performed comparing di!erent properties of aparticle and its antiparticle counterpart [13].

The weak equivalence principle (WEP)

Aside from discrete transformations, as foretold, there is another concept in physics thatis suitable to be tested in presence of antimatter. It is known as the weak equivalenceprinciple (WEP).WEP is the oldest and the most trusted principle of contemporary physics [16]. It statesthat the trajectory of a point mass in a gravitational field depends only on its initialposition and velocity, and is independent of its composition, or equivalently, that alltest particles at the alike spacetime point in a given gravitational field will undergo thesame acceleration, independent of their properties, including their rest mass. Its rootsgo back to the time of Galileo Galilei and his discovery of ‘universality of the free fall’on Earth. His tools were an inclinate plane to slow the fall, a water clock to measureits duration, and also a pendulum, to avoid rolling friction. These observations werelater improved by Christiaan Huygens. A deeper understanding was achieved by IsaacNewton who explained ‘universality of the free fall’ as a consequence of the equivalenceof the inertial mass mI and the gravitational mass mG. Indeed, if the inertial massentering in his second law might not be precisely the same as the gravitational massappearing in the law of gravitation, we would have to write Newton’s second law as:

F = mIa, (1.3.21)

and write the law of gravitation as

F = mGg, (1.3.22)


where g is a field depending on position and other masses. The acceleration at a givenpoint would be:

a =

!

mG

mI

"

g (1.3.23)

and would be di!erent for bodies with di!erent values for the ratio mG/mI ; in particularpendulums of equal length would have periods proportional to (mI/mG)1/2. Newtontested this possibility by experiments with pendulums of equal length but di!erentcomposition, and found no di!erence in their periods. This result was later verifiedmore accurately by Friedrich Wilhelm Bessell in 1830. Then, in 1899, Lorand vonEotvos succeeded by the mean of a torque balance in showing that the ratio mG/mI

does not di!er from one substance to another by more than one part in 109.Einstein was very impressed with the observed quality of gravitational and inertialmass, and it served him as a signport toward the Principle of Equivalence.4 It sets verystringent limits on any possible nongravitational forces that might exist. For instance,any new kind of electrostatic force in which the number of nucleons plays the role ofcharge would have to be much much weaker than the gravitation [17]. In 1964 a groupunder R. H. Dicke at Princeton improved Eotvos’ method, concluding that “aluminiumand gold fall toward the sun with the same acceleration, the accelerations di!ering fromeach other by at most one part in 1011” [18]. It has also been shown (with very muchless precision) that neutrons fall with the same acceleration as ordinary matter [19],and the gravitational force on electrons in copper is the same as on free electrons [20].Unlike many other principles, which lose importance with time, the WEP has evenincreased its importance: it is presently the cornerstone of Einstein’s General Relativityand of modern Cosmology. It should rather be called ‘the principle of giants’.Is it ever possible to violate the WEP? Could gravitational interaction be di!erent fromwhat is known so far, involving for example repulsion instead of attraction? Even ifarguments against antigravity have been given by numerous authors and the majorityof the scientific community gives poor credit to a di!erent scenario, no experimentaltest of the WEP has been performed with particles and antiparticles. The hypotesis ofa repulsion between matter and antimatter would also imply a strong revision to ourcosmological knowledge.The first, and up to now, last attempt to measure the gravitational acceleration ofantiproton was PS-200 experiment at CERN. In the absence of better ideas, PS-200was based on the use of TOF (time-of-flight) technique [21].The experiment [22] was designed to obtain a su"ciently large number of antiprotonsat 4 K, and to determine g through time-of-flight measurement. A schematic diagramof the experimental setup is shown in Figure 1.6. The 5 MeV energy of the extractedLEAR (Low Energy Antiproton Ring, a facility to be discussed in Chapter 2) beamwas lowered to a few tens of keV, an energy suitable for trapping particles, by passingthe beam through a thin foil. The catching trap, the first stage of the trapping system,was designed to accept a complete beam burst from the degrading foil at several tensof keV, and to store it for the period of time necessary for electron cooling to lowenergies. At this time the particle bunch were ejected from this trap and transferredinto a smaller size cooling and launching trap. Here the particles were cooled to 4 K (0.3meV), using resistive cooling, and were then be launched into the drift tube in burstscontaining 100 particles each. This method involves measuring the distribution of flight

4The Einstein equivalence principle states that the weak equivalence principle holds, and that “theoutcome of any local non-gravitational experiment in a freely falling laboratory is independent of thevelocity of the laboratory and its location in spacetime”.


Figure 1.6: PS-200 apparatus scheme. Beyond the cut-o! time no particle can reach the detector,because the slowest particles don’t have enough energy to came up to the acceleration grid, beforegravitation stops their upward motion. The experiment would have measure separately and afterwardscompared cut-o! times related to antiprotons and hydrogen ions (black curve). The latter are chargedand have nearly the same mass of protons. If antimatter would undergo a gravity force greater thanthat exerted by common matter, antiprotons would have a cut-o! time lower than that of hydrogenions (colored curve).

time through the drift tube, thereby determining a cuto! time tc = (2L/g)1/2 (about0.4 s) from which g can be extracted. Measurements using antiprotons were comparedwith measurements using H! ions in order to eliminate many systematic errors. Themain purpose of the drift tube was to shield the slowly moving particles from externalelectric fields. However, due to di"culties in this particular task, for the electromagnetice!ects are much higher than the gravitational ones, no results have been achieved.If we consider the di"culties that arise because of the electromagnetic e!ects, it isstraightforward that a neutral system like antihydrogen is free from problems associatedwith such interactions and consequently is perfect for a direct test of the gravitationalinteraction of antimatter with the Earth field.Summarizing, antihydrogen is the ideal candidate to study the symmetries of nature,better than antiprotons or positrons alone.

1.4 Is there room for antigravity? 15

1.4 Is there room for antigravity?

Speculation concerning possible violation of WEP may be divided in two groups [23].The first (and older) group consists in a number of di!erent theoretical scenarios forminimal violation of the WEP. The universality of gravitational attraction is not beingquestioned (so, there is no room for antigravity), but in some cases gravitational andintertial mass may slightly di!er [24].The second group of speculations, (which has appeared in the last decade of the 20thcentury) predicts gravitational repulsion between matter and antimatter, i.e. antigrav-ity as the most dramatic violation of WEP.There are three new lines of argument in favour of antigravity.

The intriguing possibility that CP violation observed in the neutral kaon decaymay be explained by the hypothesis of a repulsive e!ect of the earth’s gravitationalfield on antiparticles [25].

The Isodual theory of antimatter [26].

The growing evidence (since 1998) that the expansion of the universe is acceler-ating rather than decelerating. Antigravity may be a possible explanation of thisaccelerated expansion of the universe [27].

The concept of antigravity

From the particle-physics point of view, general relativity is a theory of gravity wherethe force is mediated by a tensor (spin-two) particle with the charge being the mass.Therefore, the force is always attractive. On the other hand, classical and quantumelectromagnetism both have two charges, positive and negative. The forces are mediatedby a vector (spin-one) field which produces an attractive force between opposite chargesand a repulsive force between like charges.Many physicists who worked in the late ’30s and early ’40s gradually understood thatcharge-forces mediated by even-integer spin bosons are always attractive (scalar, tensor,etc.) whereas forces mediated by odd-integer spin bosons can be both attractive orrepulsive, depending upon whether the charges are opposite or alike.Gravity and antimatter were a combination which stimulated everyone. Could therebe ‘antigravity’ (or tensor-antigravity)? By this is meant that matter and antimatterrepel each other due to a tensor gravitational interaction, with the sign of Newton’sconstant reversed. It is worthwhile to note the question of the dominance of matterover antimatter in the universe. One can conclude this from the lack of observedelectron-positron annihilation photons in cosmic rays. Since one expects the equationsof physics to be CPT-invariant, one has to wonder why this asymmetry exists. We viewthis predominance of matter as a problem of the early universe. We now know that CPviolation exists and it is stronly conjectured that baryon and lepton numbers are notconserved individually, although they are perhaps conserved in the combination B-L.Therefore, early universe scenarios where this CP violation allows the universe to evolveto a dominance of matter over antimatter are considered.

Some (very brief) theoretical ideas on gravity and antimatter

The two great triumphs of modern theoretical physics are quantum mechanics (as fi-nalized in quantum field theory) and classical general relativity. However, at some level


these two theories are incompatible: in quantum mechanics one is taking a many-pathpoint of view, whereas in general relativity one takes a space-time geodesic point ofview, that requires precise information on position and momentum coordinates in afashion inconsisten with the former. Particle physicists think that general relativitymust be modified in order to attempt to unify gravity with the other forces of nature.A generic feature of these models is that the normal spin-two graviton can have twotypes of new partners: a spin-one (gravivector) and a spin-zero (graviscalar) [24]. Thesenew partners are in general massive (finite range) and have coupling strengths on theorder of the normal gravity which can be composition-dependent.Scherk [28] realized the deep theoretical and experimental implications of the two part-ners of the graviton. He discussed these ideas in terms of experimental limits on viola-tions of the inverse-square law and the principle of equivalence (Eotvos’ experiments).The simplest linearized classical potential between two point masses m1 and m2 is ofthe form:

V = #!

Gm1m2

r$1$2

"

(

[2(u1 · u2)2 # 1] )

)

qv1qv2(u1 · u2)e!r/v +

)

qs1qs2e!r/s

*

,

(1.4.1)where ui is the four-velocity

ui = $i(1, "i). (1.4.2)

In Equation (1.4.1), the first term arises from normal graviton exchange. The rangesof the gravivector and graviscalar are v and s, respectively. The summation signssymbolically indicate that in principle there could be many partners of each spin, eachhaving its own charge and range.The vector charge per unit mass is given by qv. It is expected to be composition-dependent, such as would be the case if it were baryon number or lepton number perunit atomic mass. The sign in front of the vector exchange term is critical to antimatter.It reflects the fact that the force is repulsive between like charges (matter and matter),but attractive between opposite charges (matter and antimatter). In contrast, the forceassociated with scalar exchange is always attractive, as it is always the case for even-spin exchange as in normal (tensor) gravity.Taken to the static limit and assuming, for simplicity, that there is only one vector andone scalar partner of the graviton, the static potential is:

V = #Gm1m2

r(1 ) ae!r/v + be!r/s), (1.4.3)

where a and b represent the products of the vector and scalar charges of the two parti-cles. The overall signs have been arranged so that both a, b ! 0.Experiments on interactions between matter and matter will be sensitive to the dif-ference of the two terms in a and b, i.e. to |a # b|. Limits on this di!erence willnot necessarily be applicable to antimatter-matter experiments, for which the sign ofa changes, and which are thus sensitive to |a + b|. Specific (proof-of-principle) modelshave been constructed in which a precise cancellation takes place for matter-matterinteractions, while leaving matter-antimatter interactions unconstrained.It should be noted that in quantum gravity scenarios under the static potential regimeof Equation (1.4.3), antimatter always fall as the same rate as, or faster than, matterdoes towards the earth. It never falls up, as in the case with “antigravity”. It nevereven falls slower than matter (see Figure 1.7).

1.4 Is there room for antigravity? 17

Figure 1.7: New gravity bosons would induce a di!erence in the forces that are exerted betweenmatter and matter and between matter and antimatter. The graviton and the graviscalar wouldproduce attraction in both cases, but the gravivector would produce repulsion for matter and attractionfor antimatter. If the gravivector and graviscalar e!ects are almost the same, they should cancel outfor common matter interactions, while they should add up for matter and antimatter interacionts. Soantimatter should fall down with a velocity 14% higher (and maybe more) compared to matter.

Chapter 2

The AEgIS experiment

The concept of antimatter has become so familiar in physics that the production ofantiprotons, positrons and other antiparticles is considered as a routine. However thestudy of the bound states of antimatter is at its dawn, since these states are so di"cultto create and manipulate. Recently, great improvements have been achieved in handlingantimatter thanks to the experiments at LEAR accelerator first (from 1982 to 1996),then to the so-called ‘second generation’ antimatter experiments since 2000 at the AD,such as ATHENA, ALPHA, ATRAP and ASACUSA (see Section 2.1).As anticipated in Chapter 1, one of the main reasons of interest in the antihydrogenatom is the possibility to study a composite (and stable) system, entirely composedof antimatter. From a simple comparison of some properties with the correspondingquantities of the hydrogen atom, it is possible to obtain some ‘direct’ tests of theCPT symmetry. The set of the energetic levels of the hydrogen atom presents anextremely complex structure, when analized with su"cient resolution. Some quantitieshave been recently investigated with a great experimental precision: the transitionfrequency between the states 1S and 2S, for instance, has seen an increment of therelative precision in its measurement of three orders of magnitude in the last decade,up to the present limit of " 3 · 10!13 [29].If we could apply the same spectroscopical techniques in the case of the antihydrogen aswell, and obtain results of similar precision, limits on the validity of the CPT theoremcould be obtained. The main goal would be to achieve limits comparable (if not better)to the best ‘direct’ CPT tests.There is another experiment with similar grounds but a di!erent goal, binded this timenot to CPT but to the weak equivalence principle. This experiment, named AEgIS(see Section 2.2), manipulates antiprotons from the AD to build antihydrogen andeventually to perform gravity measurements. In this Chapter we will briefly review themain features of the antihydrogen experiments, with a specific attention to AEgIS. Asanticipated, this experiment – the most recent of all those cited, as the data acquisitionwill start in Fall 2011 for a few-weeks period – has no very-short-term spectroscopy aims,as it concentrates on the measurement of the acceleration constant g of antimatter. Theimportance of this quantification is not lesser than the former ones, since, as we havediscussed, the question of antigravity is strongly linked to fundamental symmetries.

19

20 The AEgIS experiment

Figure 2.1: The first antiprotons trap consisted in simple copper electrodes separated by glass spacers.

2.1 Antihydrogen, here we are

Before 1995 it is possible that not even a single atom of antimatter had ever existed inthe history of the universe.1 When positrons and antiprotons in cosmic rays encounterone another, they are moving so fast that they continue on their separate ways ratherthan lingering and combining into atoms.To produce antihydrogen from its constituents, antiprotons and positrons are needed. Ifit is relatively easy to obtain positrons from "+ radioactive decays, much more di"cultis the creation of a usable sample of antiprotons. To capture and store antiparticles insmall-scale containers you need them to be as still as possible. So teams of scientists andengineers at CERN used their experience and built a storage ring where the antiprotonswere slowed down. Indeed the history of antihydrogen started with the construction ofLEAR (Low Energy Antiprotons Ring) machine at CERN, ultimated in 1982.In the middle of the ’80s, Gerald Gabrielse and collaborators took up the challenge ofextracting antiprotons from LEAR and capturing them in the Penning trap shown inFigure 2.1. Penning traps use a strong homogeneous axial magnetic field to confineparticles radially and a quadrupole electric field to confine the particles axially. Thestatic electric potential can be generated using a set of three electrodes: a ring and twoendcaps. This potential produces a saddle point in the centre of the trap, which trapsions along the axial direction. The electric field causes ions to oscillate (harmonically inthe case of an ideal Penning trap) along the trap axis. The magnetic field in combinationwith the electric field causes charged particles to move in the radial plane with a knownmotion. The trap was about 15 cm long. When the antiprotons entered, the voltagewas increased, which created an electrical barrier like shutting a trapdoor. He capturedantiprotons for the first time in 1986 [30], and within three years he was able to storesixty thousand of them for four days. But it was not enough, for the aim was forprecision, the long term storage of a few antiprotons. In 1991 a hundred antiprotonswere stored for several months and by 1995 the goal of trapping a single antiproton wasachieved.In the ’90s LEAR could deliver ‘low energy’ (600 MeV/c of momentum) antiprotonsto various experiments. One of them, PS210, exploited the interaction between theantiprotons and jets of xenon atoms. With a relative probability of 10!19 a particularreaction could give birth to an antihydrogen atom. The experiment measured 9 events[31], the first antihydrogen atoms ever produced (see Figure 2.2). In the following years

1AMS experiment is currently trying to establish a new upper limit of 10!6 for the antihe-lium/helium flux ratio in the Universe, an improvement of three orders of magnitude over AMS-01,su"cient to reach the edge of the expanding Universe and resolve the issue definitively.

2.1 Antihydrogen, here we are 21

Figure 2.2: The PS210 experiment exploited the interaction between antiprotons and jets of xenonatoms. The latter crossed the LEAR antiproton beam. In fact, any antiproton passing close enough toa heavy atomic nucleus can create an electron-positron pair; in a tiny fraction of cases (! 10!19), theantiproton would bind with the positron to make an atom of antihydrogen. The antihydrogen, beingneutral continued its path on a straight line. A ionization field then divided it in its components. Theantihydrogen signal consists in a simultaneously detection of an antiproton and a positron.

a similar result was obtained at Fermilab thanks to the E862 experiment with a finalsample of about 100 events [32]. This way of producing antihydrogen was clearly lowe"cient (still 1 H per 10!19 p) and, since the antiprotons were let to fly through a xenonjet, the antihydrogen was too fast to make them usable for any further study.When LEAR ended its operation in 1996, the ‘low energy’ antiproton physics communitypushed for a new machine able to deliver even lower energy p. From the ashes ofLEAR, at CERN, the AD machine (Antiproton Decelerator) rised (Figure 2.3) andstarted operation in 2000. The 26 GeV proton beam (about 1.5 · 1013 protons/bunch)from the Proton Synchrotron impacts an iridium production target from which theantiprotons are extracted and injected at 3.5 GeV/c inside the AD decelerating ring.In this way around 2 · 107 antiprotons every " 100 s, with a momentum of 100 MeV/c(5.3 MeV in kinetic energy), were at hand for antimatter physics. Thanks to theavailability of such a p beam, a second generation of antihydrogen experiments wasplanned and built: ATHENA (see Figure 2.4) and ATRAP. The specific goal was thecreation of antihydrogen in electromagnetic traps for future spectroscopy studies. In theAD hall, in the same period (around the year 2000), another experiment, ASACUSA,was also starting operation for the study of the properties of the antiprotonic-helium[33]. Being di!erent on the experimental apparatus, the basic concept of both ATHENAand ATRAP was the following:

1. collect antiprotons from the AD and positrons from a 22Na radioactive source;

2. store them separately using trap electrodes voltages to confine the charged parti-cles axially and a magnetic field to confine them radially;

3. mix them inside a so called ‘nested Penning trap’. ATHENA and ATRAP haveused such a two-traps system. In this device, collisional cooling continues until theinitially hot antiprotons reach a very low velocity relative to the cold positrons.After they are cooled, the interaction of the antiprotons and positrons can becontrolled through potentials tuning. In the form used here, the nested Penningtrap is an outer potential well for antiprotons, within which is nested an inverted


Figure 2.3: Schematic overview of the AD (Antiproton Decelerator) hall at CERN. The numbers from1 to 4 indicate the main stages of the low antiprotons production. Antiprotons are generated as a 26GeV beam of ! 1013 protons collides over a target. The beam comes from the PS accelerator (point 1).When it enters the AD, the beam is formed by 5 ·107 antiprotons with a moment of 3.57 GeV/c (point2). From here, with several stages and two di!erent cooling techniques (stochastic cooling and electroncooling), the beam is slowed down (point 3) first to 2 GeV/c, then to 300 MeV/c, and eventually to100 MeV/c: at that point it is extracted from the ring (point 4). The whole cycle requires 140 s, andit is able to provide bunches of ! 107 particles, with impulses having 200 ns of nominal length.

well for positrons. The wells are generated by applying potentials to a stack ofcylindrical ring electrodes made of gold-plated copper, as shown in Figure 2.5;

4. monitor the blending to discover evidence of antihydrogen atoms. In particularwhen antihydrogen is formed, being neutral, it escapes the mixing region and itannihilates on the trap walls. A space-time coincidence of antiproton and positronannihilations is thus a clear sign of production.

In 2002 ATHENA first, and ATRAP few months later, indeed reported the first creationof ‘cold’ antihydrogen [34, 35]. ATRAP also demonstrated the possibility to create Hatoms via a charge exchange technique [36]. In this technique an antiproton interactswith a positronium (Ps), a bound state of a positron and an electron, and strips thepositron to form an antiatom (see Section 2.2). When formed with antiprotons at rest,the antihydrogen is generated with a velocity distribution dominated by the antiprotontemperature, thus, in principle, very cold antihydrogen can be formed. So, antiatomswere finally available in traps. Unfortunately such antiatoms were escaping the trap assoon as they were produced, so they were unusable for any further study. Furthermore,they needed to be confined before trying to investigate them with laser spectroscopyfor a CPT test. In 2005 and 2006 AD was shut down in a saving plan for the LHC.It was the right time for an upgrade. The place of ATHENA was taken by a newexperiment, ALPHA, while the ASACUSA collaboration also added to its goals anti-hydrogen physics. The ‘next step’ for these experiments was to try to confine both

2.2 AEgIS generalities 23

Figure 2.4: A scheme of the ATHENA apparatus. The antiprotons trap (left) accumulates antiprotonsbunches coming from the decelerator. When a su"cient number of them has been accumulated, theyare transfered into the recombination trap. The trap keeps antiprotons and positrons, available fromthe positrons accumulator (right, in the same magnet), allowing the formation of antihydrogen atoms(central region). Positrons emitted from a 22Na source (in the far right) are slowed down by a passagethrough a thin layer of solid neon and by collision with gaseous nitrogen. Slow positrons are captured bya Penning trap. When enough positrons have been accumulated, nitrogen is sent o! and the positronsare transfered in the recombination trap.

charged particles and neutral atoms in the same region. For plasma confinement thecylindrical symmetry has to be preserved while for atom confinement the magnetic fieldmust be minimal in the center. The solution proposed by ALPHA and ATRAP wasto superpose a radial magnetic multiple trap upon a solenoidal field. ASACUSA onthe other hand decided to avoid di"cult trapping and intend to make an antihydrogenbeam, since it can count on a Radio Frequency Quadruple Decelerator (RFQD) able tofurther decelerate the AD antiproton beam down to 50 keV. In November 2010, Naturepublished a letter from the ALPHA collaboration: from the interaction of about 107

antiprotons and 7 · 108 positrons, 38 annihilation events were observed consistent withthe controlled release of trapped antihydrogen from the magnetic trap [37]. However, asfar as experiments stand, the CPT test (spectroscopy) with antihydrogen seems not yetfeasible, at least in the very near future. In the next years (the AD machine operationhas been approved and financed up to 2016) the three experiments on the floor, willtry hard to achieve trapping, or beam production, to perform the first spectroscopicmeasurements. In particular ATRAP and ALPHA aims at measuring the 1S-2S tran-sition (see [38] for a possible scheme) while ASACUSA intend to carry out microwavespectroscopy of the ground-state hyperfine structure [33].

2.2 AEgIS generalities

Being di!erent from the other experiments, the primary scientific goal of AEgIS (An-timatter Experiment: Gravity, Interferometry, Spectroscopy; Figure 2.6 depicts theapparatus with a portion of the Antiproton Decelerator (AD)) is to perform, for thefirst time, a direct measurement of the gravitational acceleration g on a beam of anti-hydrogen (H) [39]. This gravity measurement, with 1% relative precision planned, willbe carried out by observing the vertical displacement of the shadow image produced byan H beam as it traverses a Moire deflectometer. To obtain such a result three mainsteps are needed:

1. to produce antihydrogen, with a technique based on the charge exchange reaction


Figure 2.5: A nested Penning trap. Antiprotons bounce back and forth between the boundariesof a great deep potential well, with a little hill rising in the center of it. Positrons, having oppositecharge, ‘see’ an inverted potential. Thus the hill becomes a depression on the top of a large hill.Depression traps the positrons in the same region of space where the antiprotons are, so thatantihydrogen may form.

between Rydberg positronium and cold antiprotons;

2. to form it into a beam, taking advantage of the sensitivity of the Rydberg atomsto electric field gradients to form a beam of antihydrogen: an appropriate electricfield is applied to sectors of the trap electrodes (Stark accelerator) immediatelyafter the formation process to give a velocity boost of the order of a few hundredm/s in the horizontal direction to the antihydrogen atoms;

3. to measure its fall: the Moire deflectometer is mounted downstream of the bore ofthe magnet and outside of the magnetic field. The beam is directed towards thegratings of the Moire deflectometer. The vertical position and the arrival time ofeach antihydrogen atom are measured.

The AEgIS design is based on the experience of the ATHENA and ATRAP experimentsat the AD, on a series of tests and developments performed by AEgIS members on mat-ter systems and on simulations of several critical processes (charge exchange productionof antihydrogen, antihydrogen acceleration process and propagation through the Moiredeflectometer, resolution of the position sensitive detector located at the end of theMoire deflectometer). The proposed gravity measurement with the Moire deflectome-ter is made feasible by merging in a single experimental apparatus technologies alreadydemonstrated separately and including some reasonable additional development. Mag-netic trapping of the antiatoms is not necessary to perform the gravity measurement inthe first phase of the experiment.This gravity measurement with 1% precision, in spite of the modest precision, is scien-tifically relevant, as it would represent the first direct measurement of a gravitational

2.2 AEgIS generalities 25

Figure 2.6: The AEgIS apparatus with a portion of the AD facility.

e!ect on an antimatter system. Additional physics results concerning Rydberg spec-troscopy can be obtained.In a second phase of the experiment higher precision gravity measurements will becomefeasible. CPT spectroscopy is also of great interest for AEgIS and it is included in thelong term scientific goal, even if the design of the apparatus and the technical choicesare focused on the gravity measurement.In order to carry out the proposed measurement, the following steps are required:

Antiprotons delivered by the AD are trapped in a Malmberg-Penning trap mountedin a cryostat inside the bore of a 5 T magnetic field and cooled by electron coolingdown to sub eV energies. The antiprotons are then transferred into a second trapinside a lower magnetic field where they are cooled to 100 mK.

A bunch (with a duration of tens of ns and few mm size) of more than about108 positrons, accumulated in a Surko-type device in about 200 s, is transferredfrom the accumulator into a trap mounted inside the same magnetic field as theantiproton trap. Here the bunch, compressed in space and time with standardnon-neutral plasma techniques, is sent on an appropriate porous target materialwhere positronium in the fundamental state is produced with high e"ciency.

The positronium cloud emerging from the target is excited by two laser pulsesinto a specific Rydberg state (with quantum number n between 20 and 30).

The trap containing the cold antiproton cloud is mounted very close to the positro-nium production target. Cold (100 mK) antihydrogen atoms with a predictable


Figure 2.7: The AEgIS apparatus as of May 2011. Only a portion of the antiproton beam line is inplace.

population of excited states are produced during the time in which the positro-nium traverses the antiproton cloud. While the preparation of the antiprotonsand positronium cloud requires a few hundreds of seconds, the production of theantihydrogen atoms in AEgIS will be pulsed and it will happen within a short,experimentally known, time interval less than 1 s.

Antiprotons that have not recombined can be quickly transfered back toward thecatching region before applying the Stark accelerating electric field.

The Rydberg atoms should decay toward the fundamental state (or the 2S state)during their flight from the end of the Stark accelerator to the first Moire deflec-tometer grating. If needed, the decay rate can be accelerated by appropriate laserfields.

A beam of antihydrogen eventually takes place, and the measurement of theshadow deflection caused by the Moire gratings is performed.

In Figure 2.7 is reported a photograph of the AEgIS apparatus, as of May 2011. Pre-liminary tests should start in Fall 2011, for a data acquisition of a few weeks.

2.3 The design of AEgIS 27

Figure 2.8: Section of AEgIS. The numbers from 1 to 7 indicate the main sections of the apparatus,to be deepened in the next sections. Point 1 indicates the path followed by antiprotons coming fromAD with 5.3 MeV of kinetic energy, while 2 indicates the portion dedicated to positrons from thesource to the main 5 T magnet (point 3). Point 4 represents the Malmberg-Penning trap that capturesantiprotons with a kinetic energy of 10 keV (this trap is of course preceded by a degrader). It storespositrons as well, to cool them at temperature of 4 K. The separated beams of positrons and antiprotonsare then on their way to the 1 T magnet (point 6), where is set the porous target intended to formpositronium, and the Penning trap for antiprotons (point 5). In this trap, kept at 100 mK by a diluitionrefrigerator, is formed antihydrogen by charge-exchange reaction. The beam of antiatoms (point 7) isthen accelerated by inhomogeneous electric fields through Stark e!ect.


2.3 The design of AEgIS

The main components of the AEgIS apparatus are a Surko–type positron accumulator(see Section 2.3.1), a 5 T superconducting magnet (we will refer to it as main magnet,see Section 2.3.3), a 1 T magnet, a dilution refrigerator (see Section 2.3.4), several elec-tromagnetic traps for charged particles (see Section 2.3.5), the grating system for thegravity measurement (see Section 2.5.3), various types of detectors (see Section 4.2) andlaser systems. Figure 2.8 depicts the section of the main setup, except for the gratingsystem which is not represented.The low temperature inside the main magnet provides the cryogenic environment neces-sary for handling and cooling charged particles. The traps for the charged particles, thetarget for the positronium formation and the electrodes for the Rydberg antihydrogenacceleration, as well as the gratings of the Moire deflectometer and the antihydrogenposition detector at its end are mounted inside the cryostat. A lower temperature of100 mK is needed only in the small region where the antihydrogen will be formed; tem-perature of ( 4 K is su"cient in other regions of the apparatus. A dilution refrigeratorwill cool the interested zone to the required 100 mK.Positrons must be injected inside the main magnet from the same side as the antipro-ton beam because the other side of the apparatus has to be free to allow the gravitymeasurement.Electromagnetic traps for charged particles are widely used in various sections of theapparatus: to accumulate positrons in the Surko–type accumulator, to catch and coolantiprotons and to prepare the positrons for positronium production. They are basicallymade by a series of cylinders of appropriate length and radius to which static voltagesare applied to ensure the axial trapping while a uniform magnetic field along the trapaxis provides the radial confinement.

2.3.1 Positron accumulator

As anticipated, positrons will be accumulated using a Surko–type positron accumulator.The techniques were pioneered at the University of California San Diego positron [40, 41]and electron [42, 43, 44] groups; a device of this type has been used with success in theATHENA experiment and the technology is now so well established that a commercialversion of the system is available [45].

The operation of the positron accumulator is based on the bu!er gas capture andcooling of positrons in a Malmberg-Penning trap. Positrons emitted from a radioactive22Na source are moderated using solid neon (see Figure 2.9). Moderators are usuallygrown at a temperature of 7 K with ultra-pure neon admitted at a pressure of 10!4 mbarfor few minutes. After the moderator a slow positron beam with typically 2 · 105 e+/smCi is obtained. Using a 40 mCi source, 8 · 106 e+/s are expected. The accumulatortraps and cools this continuous beam of slow positrons guided into the trapping regionusing axial magnetic field transport.

In order to remain trapped, positrons have to lose some axial energy during theirtravel before reaching the end of the trap. Electronic excitation of the nitrogen gasis the cooling mechanism (see Figure 2.10). Once trapped the positrons continue tolose energy in collisions with the gas, finally residing in the potential well formed bythe voltages applied to the large diameter trap electrodes. A di!erential pressure ismaintained along the trap axis. The trap potential well minimum is located in the


Figure 2.9: A 22Na source (22Na is a positron emitter) is located behind a parabolic cone. This coneis cooled to 7 K using a two-stage refrigerator. By admitting neon gas into the source chamber, a layerof solid neon is deposited on the cone. This acts as a moderator for the positrons, which are emittedfrom the neon surface with an energy spread of approximately 1 eV (FWHM) at a mean energy of !2 eV. This compares to a mean energy of ! 0.2 MeV from the radiocative source, with the energy ofthe positrons ranging from 0 to 0.5 MeV. This process has an e"ciency of 0.1%.

region with the lowest pressure. A magnetic field of 0.15 T is normally used. This isrealized with non-superconducting coils. Standard non–neutral plasma manipulationtechniques (rotating wall [42, 44]) are used to radially compress the positron cloud thusincreasing the storage time and the number of accumulated positrons. The compressionwill be applied during the accumulation phase following the experience of ATHENA.More than 108 positrons are expected to be accumulated in 200-300 s.

2.3.2 Transfer of positrons into the main magnet

After accumulation, positrons will be transferred into the trap located inside the mainAEgIS magnet.Before beginning the transfer procedure the gas in the accumulator must be pumped outto limit the gas flux in the ultrahigh vacuum (UHV) and cryogenic region in the mainmagnet. The vacuum chamber of the main magnet and of the accumulator are separatedby a valve which is opened only at the time of positron transfer. Pulsed magnets andan appropriate electrostatic guiding system will be used to drive the positrons into thetrap located in the 4 K region in the main magnet. Depending on details of the transferprocedure, positrons will reach the trapping system with a kinetic energy of severaltens or hundreds eV. The high magnetic field provides a cooling mechanism for lightparticles through the emission of cyclotron radiation (the cooling time constant is about1 s). The positron cloud will reach a thermal equilibrium state at a temperature of theorder of 4 K.

2.3.3 The magnets

A high value of the magnetic field is needed to trap and cool antiprotons in an e"-cient way while low magnetic field values are desirable in the region where the Starkacceleration of the Rydberg antihydrogen atoms takes place. The field homogeneity isan important parameter to allow reaching long plasma storage times and low plasmatemperature. The first field at 5 T with a relative inhomogeneity of less than 10!4

(within a cylinder of length 30 cm and radius 3 cm) houses the antiproton catching andcooling trap and the UHV positron storage trap (catching and cooling region). The


Figure 2.10: Positrons lose energy by electronic excitation of the nitrogen molecules and they becometrapped in the third stage in less than 1 ms. They continue to lose energy through vibrational androtational excitation of the molecules, cooling to the electrode temperature in approximately 1 s.The e"ciency of the trap is approximately 25%, measured by comparing the number of positronsaccumulated in the trap to the moderated positron flux from the radioactive source.

length of the main magnet bore is 1.3 m and its inner diameter is 15.4 cm. The secondregion at 1 T is separated from the former by 16 cm. This region, with a magnetichomogeneity better than one part in 105 (within a cylinder of radius 1 cm and length4 cm), is in the antihydrogen formation region. The bore end should be at a distanceof the order of 30 cm from the center of the homogeneous region at 1 T.The inhomogeneity I(,, z) is defined as follows:

I(,, z) =

+

+

+

+

Bz(,, z) # B0

B0

+

+

+

+

, (2.3.1)

where B0 is the field value at the center of the given volume and Bz(,, z) is the z-component of the magnetic field in the ,, z position.The bore of the 1 T magnet will be at 4 K. A dilution refrigerator, able to reach 100mK in the region where antihydrogen is formed and completely independent of the onehosting the coils of the magnet, will be inserted inside the bore. The second antiprotontrap is of course mounted inside this cryostat. Its purpose is to prepare the antiprotonto the charge-exchange reaction with the Rydberg positronium.The free space between the inner bore surface and the external part of the trap cryostatallows laser access to the interaction region. The external flanges of the magnet areused to fix the supports of the mirrors designed to deflect the laser beams from theaxial to the perpendicular direction. The length of the second magnet bore is 1.3 mand its inner diameter is 22.4 cm.

2.3.4 The diluition refrigerator

The principle of operation of a dilution refrigerator is based on the quantum propertiesof liquid mixtures of the two He isotopes (3He and 4He). When such a mixture is cooled


Figure 2.11: The Malmberg-Penning trap of AEgIS where AD antiprotons and positrons from theSurko-type accumulator are stored and manipulated.

to 0.86 K it separates into two distinct phases: a concentrated 3He phase and a diluted3He phase. The key point is that even at zero temperature the 3He concentration inthe diluted phase of the mixture is finite (6.4%). Below 0.5 K, the superfluid 4He ofthe diluted phase is in a fundamental state with a negligible entropy. For that reason,4He behaves as a ‘vacuum’. In the mixing chamber, the lighter concentrated phaseforms above the diluted phase. A cooling e!ect is created by 3He atoms passing fromthe concentrated phase to the diluted phase. This cooling e!ect can be viewed as an‘evaporation’ of 3He. This process produces cooling of the two phases in the mixingchamber. The solubility limit (6.4%) allows passage of the 3He atoms into the dilutephase even at the lowest temperatures, thus maintaining a large cooling power. The 3Heatoms are extracted by pumping on the still at a temperature a little below 1 K. Thedilution process runs in a closed cycle. Before being re-introduced into the cryostat,the evaporated 3He is cooled down to 4.2 K in a liquid helium Dewar.2 Then the 3He isliquefied: finally the liquid 3He is cooled at the still and in the heat exchangers beforeagain reaching the mixing chamber.The mixing chamber is then the point at the lowest temperature in the system. In theAEgIS setup the mixing chamber will be installed very close to the trap region whereantihydrogen will be formed. The rest of the system will be placed outside the mainvacuum chamber to optimize the e"ciency of the cryostat without a!ecting the UHVvacuum of the trap region.The heat load in the cryostat is dominated by the radiation coming from regions athigh temperature. In fact the cryostat cannot be completely closed because antiprotonsand positrons have to be injected from a 4 K region in their traps installed inside thecryostat. The two holes necessary for the incoming particles (from the AD and positroninjection side) bring radiation into the cold region. In addition windows for the laseraccess are needed in the region at 100 mK where positronium is formed.The second contribution is the conductive flow of heat through cables and apparatussupports, estimated at ( 250 W. The CERN cryogenic group is defining the detailsfor the final design of the system to garantee in any case the temperature of 100 mK.

2.3.5 The trap system

Malmberg-Penning traps of cylindrical shape are used to handle the charged particlesand plasmas. They are cylindrically-symmetric devices used to confine non-neutral

2A Dewar flask is a vessel designed to provide very good thermal insulation. The Dewar flask wasnamed after its inventor, the Scottish physicist Sir James Dewar (1842-1923).


plasmas with both a uniform, axial magnetic field to confine the plasma radially, andelectrostatic fields at the ends of the trap to confine the plasma axially. These trapsusually have cylindrical electrodes (similar to a hollow cathode configuration) to providethe electrostatic fields, allowing diagnostic access (see Figure 2.11 for the AEgIS case).Because the plasma is non-neutral, there is a self-electric field which causes the plasmato E * B rotate. Antiprotons, electrons and/or positrons are accumulated, cooled,manipulated and detected in these traps. Each trap electrode receives static or time-variable voltages. Two parallel series of electrodes are mounted inside the cryostat closeto each other: one of them is used to handle the antiprotons while the second one isdevoted to positrons. The trap radius will typically be 1 cm.The standard operating procedure for antiprotons in AEgIS will consist of capture inthe catching trap, cooling by collisions with a preloaded cloud of electrons, stackingof many AD shots, and possibly radial compression of the antiproton cloud followedby transfer into the antihydrogen formation region (see Figure 2.12). Here antiprotonswill be cooled to sub-Kelvin temperatures. Any antiprotons that have not recombinedduring the production of antihydrogen will be reused through suitable electrode voltagemanipulations to move them from the recombination region to the catching region. Theantiprotons delivered by AD in bunches of about 2.5 · 107 particles within 100 ns andwith a kinetic energy of 5.3 MeV, traverse a few foils acting as energy degrader. Bysuitably pulsing the voltages of the catching trap electrodes a fraction of the antiprotonswith energy in the keV region can be caught. The maximum antiproton energy isrelated to the voltage potential VHV applied to the electrode that ends the trappingsection. The potential of the last trap electrode of the catching trap is initially set toVHV. Antiprotons traversing the last foil of the degrading system with axial energylower than eVHV are reflected from this potential and captured by ‘closing the trap’,that is applying a voltage VHV to the entrance electrode before they bounce back andreturn to it. Typically the trap has to be closed after a time interval of 500-700 nsfrom the antiproton’s arrival at the entrance. The antiproton arrival time is measuredusing the antiproton beam monitor detector described in Chapter 4.2. The numberof captured antiprotons increases with applied trap potential. Typically in ATHENAaround 104 antiprotons were captured at 5 kV for an incident AD flux of 2.5 ·107/pulse.To maximize the number of available antiprotons, the AEgIS catching trap is designedto sustain voltages up to 10 kV.Cooling of the high energy antiprotons through Coulomb collisions between them andan electron cloud preloaded in the catching trap has been largely demonstrated [46].Although the electrons are heated by this process, they e"ciently cool themselves byemission of cyclotron radiation in the 5 T magnetic field. Ideally, the two species ofparticles will reach a final equilibrium temperature equal to that of the environment.The cooling process is usually described by the di!erential equations:

dTp

dt= #Tp # Te

+c, (2.3.2)

dTe

dt=

np

ne

Tp # Te

+c# Te # Tt

+e, (2.3.3)

where Te and Tp are the electron and p temperatures, Tt is the unperturbed electrontemperature, ne and np are the electron and p densities, +e is the synchrotron and +c


(a) (b)

(c) (d)

Figure 2.12: A cold electronic cloud is preloaded in the trap. Incoming antiprotons enter in the trapthrough a thin aluminium layer. (a) Aluminium slows down part of the antiprotons. (b) An electricpotential wall reflects backwards in the trap slowed down antiprotons. High energy antiprotons escapeto the right. (c) An other potential wall is quickly raised at the left end, so that antiprotons bounce backand forth between the two extremities. (d) Antiprotons give energy to electrons e eventually occupythe center of the trap. The left wall is lowered to let free entrance to the next bunch of antiprotons.

the electron cooling time. The latter is given by:

+c =3mempc3

8(2()1/2nee4 ln(-)

!

kTp

mpc2+

kTe

mec2

"3/2

. (2.3.4)

Here mp and me are the p and electron masses, e is their electrical charge and - isgiven by:

- =4(#0ne

!

kT

e2

"3/2

. (2.3.5)

The solution of these equations shows that 104 antiprotons having energies in the keVrange can be cooled down to less than a few eV within a few tenths of a second if theyoverlap completely with an electron cloud of density around 107 # 108 cm!3.The cooling process is not exponential and its rate increases very rapidly while the an-tiproton energy decreases. At the end of the cooling process, antiprotons and electronsshare the same volume. The electrons can then be ejected from the trap by applyingappropriate electric pulses of about 100 ns duration which do not a!ect the heavierantiprotons.Summarizing, in the following are some informations about specific traps of the appa-ratus:

Traps in the high field region. In the homogeneous magnetic field region at


5 T is mounted the trap in which catching, cooling and stacking of the antipro-tons coming from the AD is realized and the trap where the positrons transferredfrom the Surko accumulator are stored and manipulated. Once every " 200 s apositron bunch accumulated in the Surko–type device is injected into this trapand is cooled there through cyclotron radiation emission down to the cryogenictemperature of 4 K.

Traps in the low magnetic field. Antiprotons and positrons will be transferredinto traps located in the ultra-cold region (100 mK) at 1 T by properly shaping theelectric field along their path. The manipulations of the positron cloud before andafter the transfer from high to low magnetic field have the purpose of producinga positron bunch with a radius of re+ ( 1 mm. This bunch has to be acceleratedtowards the porous target with a tunable kinetic energy of some keV. The finaltrap region where the antiprotons should reach the 100 mK temperature is aPenning trap.

2.4 Positronium formation

Positronium will be obtained in AEgIS by sending a bunch of positrons on a suitabletarget acting as positronium converter with high e"ciency. Rydberg positronium willthen be formed by two step laser excitation of the resulting cloud of ground state ortho-positronium. The velocity of the Rydberg positronium must be of the order of some104 m/s corresponding to a kinetic energy of about 10 meV. We will refer to this ascold positronium.The AEgIS design is focused on positronium emitted in reflection geometry, that ison positronium emitted from the same side of the target where positrons are injected.It is important to underline that in the AEgIS apparatus the converter target will bemounted in a cryogenic environment with temperature of " 100 mK.

Mechanisms of positronium formation

Positronium in vacuum is normally produced by implanting positrons with a kineticenergy of the order of several hundred eV or few keV into a solid target (converter).Slowing down at thermal energies occurs rapidly in comparison with annihilation. Ther-mal or epithermal positrons can be re-emitted into the vacuum as positronium atomsafter capture of an electron. The Ps/e+ yield and the energy distribution of the emittedPs depend on the nature of the converter material, and, for a specific material, on theimplantation depth and on the temperature of the target.Ground state positronium is formed with equal likelihood in the singlet state (para-Ps,p-Ps, with spin 0) or in one of the three triplet states (ortho-Ps, o-Ps, with spin 1). Theself-annihilation lifetime of p-Ps is short, 125 ps, and it mainly occurs with the emissionof two $ with 511 keV. O-Ps in vacuum is required to annihilate at least into 3 $ witha total energy of 2 · 511 keV and this process has a longer characteristic lifetime of 142ns. We are interested only in the fraction of positronium emitted as o-Ps, since thelifetime of p-Ps is too short to allow its laser excitation before its decay. As a reference,positronium emitted from the converter material with a velocity of 5 · 104 m/s needs afew tens of ns to reach a distance from the surface of the order of a few mm where the

2.4 Positronium formation 35

laser excitation can easily take place.In a magnetic field the triplet state with spin component Sz = 0 is mixed with thesinglet state, resulting in a reduction of the self annihilation lifetime. The lifetime in a1 T magnetic field is reduced to about 15 ns while about 45 ns are expected in a 0.5 Tfield. On the contrary the lifetime of the triplet states with Sz = ±1 are not a!ectedby the magnetic field, therefore the maximum expected yield reduction in a magneticfield of 1 T is 1/3 of the o-Ps fraction.Positronium production occurs in metals and semiconductors as well as in insulatormaterials but the production mechanisms are somewhat di!erent.In metal and semiconductors, positronium formation is only a surface process originat-ing from positron back-di!usion to the surface followed by electron capture. Thermal-ized positrons can produce positronium by an adiabatic charge transfer reaction at anytemperature, provided that the positronium formation potential W is negative:

W = .! + .+ # 6.8 eV < 0, (2.4.1)

where 6.8 eV is the Ps ground state binding energy and .! and .+ are, respectively,the work functions of the electron and of the positron for the converter material.In addition to adiabatic emission, thermally activated formation has been observed.This additional process is dominant when the target temperature is of the order of sev-eral hundred kelvin and it is interpreted in terms of surface traps in which the positronsreside but from which they may be desorbed as positronium.In insulators surface formation of positronium by thermal positrons is unlikely since thebinding energy of the positronium atom is normally insu"cient to compensate for theextraction of the positron and of the electron (W > 0). However the thermalisation ofpositrons in an insulator is less e"cient than in a metal, thus a larger flux of positronsreturning to the surface of the insulator with su"cient kinetic energy to form positro-nium can be expected.In addition positronium can be formed in the bulk, it can reach the surface and thenbe emitted into the vacuum. Ps is formed during the slowing down of e+, mostly whenthe e+ energy is in the interval between Egap #Esolid and Egap (the so-called Ore gap).Egap is the energy necessary to excite an electron of the insulator from the valence tothe conduction band and Esolid is the binding energy of the e+e! system in the solid.In general Esolid < 6.8 eV.Bulk positronium formation is also possible when a positron encounters a spur electron,i.e. an electron raised in the conduction band by the positron itself during its slowingdown.The two alternatives described above (surface or bulk formation) depend on the tem-perature of the sample only indirectly, through temperature e!ects on migration andtrapping. Thus the e+ Ps conversion yield may be expected to stay high even at cryo-genic temperatures.

Positronium formation in porous materials

Positronium formation in porous materials is specially interesting. A material can havepores not connected to the surface or a network of pores (ordered or not ordered) con-nected to the surface. Positronium formed in the bulk can di!use into a pore or it canbe formed at the surface of the pore. If the pores are connected to the surface of thematerial then the positronium can escape toward the vacuum following the pore chan-nels and colliding with the pore walls (see Figure 2.13). The energy spectrum of the


Figure 2.13: Ps formation in porous films.

emitted positronium depends on the energy of the positronium entering the pore, on thenumber of collisions with a pore surface and on the mean energy loss for each collision.The depth in the bulk where positronium is formed depends on the e+ energy and withan appropriate design of the pore geometry and by controlling the implantation depththrough the positron implantation energy, it is possible to tailor the energy spectrumof the emitted positronium to match the required values.Figure 2.13 shows schematically the mechanism of positronium production in porousmaterials.The use of a metallic converter would certainly be more e"cient from the point of viewof cooling, since a single collision of a positronium atom with a free electron at thesurface of the metal can produce a fractional energy loss of 50% thereby reducing thenumber of necessary collisions to less than about 100. However pick-o! annihilationlosses in a metal are expected to strongly reduce the flux of Ps emerging from the chan-nels.Experimental evidence of high yield emission of thermalized positronium from a poroussilica (SO2) target kept at room temperature has been obtained in an experiment at-tempting precision measurements of the vacuum decay rate of the triplet Ps [47].Among several available materials for the cold positronium formation target, the focusof this proposal is on the use of porous materials like silica or porous alumina films,as they potentially could provide long enough contact for Ps atoms to cool down to atemperature of a few ten K.

2.5 Handling antihydrogen

The essential steps leading to the production of antihydrogen and the measurement ofits gravitational interaction in AEgIS are explained in the following sections. Figure2.14 represents the portion of the setup on which we will now concentrate.

2.5 Handling antihydrogen 37

Figure 2.14: Schematic view of the AEgIS Moire deflectometer (not to scale). The cylinders representthe electrodes of the trap for charged particles in which the antihydrogen will be produced. The firstcloud represents the antihydrogen before the acceleration and the second one the cloud at the time t0when the accelerating electric field is switched o!. The two gratings and the detector are shown.

2.5.1 Antihydrogen production by charge exchange

The production of antihydrogen in AEgIS is based on the charge exchange reactionbetween antiprotons and Rydberg positronium:

Ps" + p & H"

+ e!, (2.5.1)

as seen in Figure 2.15. The use of this reaction was proposed some time ago [48]and recently demonstrated by the ATRAP collaboration [49]. The method by whichthe antihydrogen production will be implemented in AEgIS however significantly di!ersfrom that of ATRAP (see Figure 2.15). Furthermore, charge exchange reactions betweenRydberg atoms and ions are largely studied in atomic physics.The main reasons that make this reaction interesting for the AEgIS design are:

the large cross section which is of the order of a0n4 where a0 = 0.05 nm is theBohr radius and n is the principal quantum number of the Ps;

the expected distribution of the quantum states of the produced antihydrogen.The antiatoms are produced in Rydberg states with a predictable state populationstrictly related to that of the incoming positronium. The range of final quantumstates is reasonably narrow. Thanks to the sensitivity to electric field gradientsof these Rydberg atoms a beam can be formed by accelerating the atoms with atime dependent inhomogeneous electric field;

the possibility to experimentally implement the reaction in such a way that verycold antihydrogen can be produced. To maximize the e"ciency in the use of


(a) AEgIS (b) ATRAP

Figure 2.15: The ATRAP Collaboration has introduced a method laser-controled to make antihy-drogen atoms without needing recombination traps (like ATHENA). Antiprotons and positrons arecaptured in adjacent traps, and positroniums transfer positrons to antiprotons. The sequence of thereactions should guarantee that the resulting antihydrogen has a limited energy. The AEgIS apparatususe a porous target as an electron source. It houses as well the formation of positrons.

the antihydrogen and the quality of the beam it is in fact important that thetransverse velocity be as low as possible.

The main contribution to the antihydrogen velocity comes of course from the initial an-tiproton velocity. AEgIS is designed to prepare antiprotons with a thermal distributionof 100 mK (that is with a mean velocity of 41 m/s): the resulting antihydrogen atomsshould be produced with a velocity of a few tens of m/s.

2.5.2 Acceleration and deceleration of Rydberg atoms

In recent years various techniques to control the translational motion of samples ofdipolar atoms and molecules in the gas phase have been developed. Of particularrelevance to the formation of a H beam in the AEgIS experiment is the Rydberg Starkacceleration/deceleration technique which has been demonstrated experimentally at theUniversity of Oxford for H2 molecules [50] and at ETH Zurich for argon [51] and atomichydrogen [52], and has led to the realization of components with optical analoguessuch as a Rydberg atom lens [53], a mirror [54] and both two-dimensional and three-dimensional traps.These experiments rely on the force exerted on an electric dipole by an electric fieldgradient. The principle of the deceleration method can be understood as a conversionbetween kinetic and potential energy that takes place when a Rydberg particle moves inan inhomogeneous electric field. The experiment is thus the electric field analogue of the


famous Stern-Gerlach experiment. If an H atom is excited to a state the energy of whichis lowered by an electric field (‘red-shifted’ or ‘high-field seeking’ state) it deceleratesas it flies out of this field, whereas an atom excited to a blue-shifted, low-field seekingstate accelerates. The maximum induced electric dipole moment for a Rydberg stateof a given principal quantum number scales with n2 and at n = 30 in atomic hydrogenhas a value of " 1300 D.3 To first approximation the energy levels of the H atom in anexternal, homogeneous electric field of magnitude F are given, in atomic units, by

E = # 1

2n2+

3

2nkF (2.5.2)

where k = n1 #n2, where n1 and n2 are the parabolic quantum numbers related to thethe spherical quantum number n, l by n = n1 +n2+ |m|+1. k runs from #(n#1# |ml|)to (n#1# |ml|) in steps of two and ml is the azimuthal quantum number. The manifoldof these Rydberg Stark states with principal and azimuthal quantum numbers n = 30and ml = 0 in atomic hydrogen is displayed schematically as a function of electric fieldstrength. In Figure 2.16 the vertical axis indicates the detuning from the zero-fieldposition of the n = 30 Rydberg state. The states represented that exhibit the largestdipole moment are the outermost Stark states for which k = ±29. If one excites an Hor H atom to the k = +29 state, for example at the point labeled A in Figure 2.16,which is shifted higher in energy by the electric field, and lets the excited atom moveout of the electric field, its internal energy will decrease and the atom will accelerate.Similarly if one excites the k = #29 state under the same conditions the atom willdecelerate as it moves out of the field. The corresponding gain/loss in kinetic energy isequal to /E = 3

2nk/F in atomic units.

2.5.3 The gravity measurement

Extremely accurate gravity measurements on cold atoms have been carried out by usingatom interferometry: a relative sensitivity to the Earth’s gravitational acceleration g of10!10 after two days of integration time has been reported with an atomic interferometerbased on cold Cs atoms [55]. A similar experiment could be designed with antihydro-gen atoms but presently such a high sensitivity is not reachable, mostly because theexperimental procedures able to achieve very low temperatures ( K or below) routinelyobtained with matter atoms have not yet been developed for antihydrogen atoms. Whilethe high sensitivity o!ered by atom interferometry remains a long term scientific goalfor AEgIS and work is already in progress in the collaboration to address some of theexperimental issues, the experiment itself proposes a measurement of g based on theuse of a classical Moire deflectometer which is feasible with available state-of-the-artantihydrogen manipulation techniques. Such an apparatus was built and operated byM. K. Oberthaler and collaborators in 1996 [56], and successfully tested with an argongas source to measure the local gravitational acceleration; AEgIS proposes an upgradeddesign of it. The temperature of the detector will be around 140 K. The main reasonfor having the grating cold is to avoid that hot radiation from their surface reachesthe cold antiproton trap. As a reference, the grating will have a radius of 20 cm andtheir distance will be 40 cm. These values can still be tuned. A proper magnetic fieldshielding will externally surround the grating system.

3The debye (symbol D) is a CGS unit of electric dipole moment. It is defined as 1·1018 statcoulomb-centimeter (1 C/

"4(!0 = 2 997 924 580 statC).


Figure 2.16: Stark structure of the n = 30, m = 0 state of atomic hydrogen as a function of electricfield strength. The vertical axis indicates the detuning from the energy position of the field-free Rydbergstate in units of inverse centimeters (1 cm!1 # 0.12 · 10!3 eV).

A classical deflectometer is usually made of three gratings; if the slit widths are suf-ficiently large di!raction can be neglected since the motion of the atoms is purelyclassical. Through the gratings, the Moire deflectometer allows to enhance the sensitiv-ity in the measurement of the gravity deflection of the antihydrogen beam traveling inthe horizontal direction. Indeed assuming a path length of about 1 m and a horizontalvelocity of 500 m/s, the deflection is only 19.6 m. This deflection would be impossibleto determine with a beam with a final radial diameter of the order of 10 cm, due to theradial expansion (EKbot " a few m/s) during the flight path from the production pointto the end of the deflectometer. The gravity induced phase shift however can be seenas the atom beam’s deflection during the flight between the two gratings, measured inunits of the grating vector.The upgraded design proposed by the AEgIS collaboration is based on the use of a

position sensitive detector in place of the third grating, as depicted in Figure 2.17. Thisinnovative design of the Moire deflectometer allows the use of antiatoms in an e"cientway and thus reduces the measurement time. This solution cannot be adopted in thecase of the atom interferometer because the required position resolution is too high,and because of course it does not annihilate.The ultimate sensitivity of the classical deflectometer is inferior to that of an atominterferometer because the grating wave-vector is smaller, but its significant advantageis that a collimated beam is not required.The presence of the first two gratings at distance L produces a periodical structure inthe number of the atoms N(x) arriving at distance L from the second grating. Here x


Figure 2.17: Configuration of the Moire deflectometer.

is the coordinate in the direction of the gravity force. In the experiment described in[56] the atomic density modulation was detected, as in the case of the interferometerexperiments, by using the third grating as filter: the third grating is moved in thevertical direction by a fraction /x of the grating period and the total number of atomspassing through is detected vs /x. The number of atoms N(/x) is a periodical functionof /x with period ke! = 2(/a, where a is the grating period.The comparison of N(/x) with and without including the gravity force shows that thefall of the atoms due to gravity induces a shift in N(/x) exactly given by the followingequation as in the atom interferometer:

/0g = ke! · gT 2, (2.5.3)

where /0g is the phase shift due to the acceleration of gravity g, ke! is the grating wavevector, and T is the time spent by the atom between consecutive di!raction gratings.The e!ect here is purely classical.In the proposal of the AEgIS experiment are shown the results of simulations of thebehaviour of the Moire deflectometer coupled to a position sensitive detector. They arepresented again hereafter.Let us consider Figure 2.18. The fraction fN of atoms arriving on a surface of radius

w at a distance l from the source is given by the ratio of the surface taken by the atoms,A = ((vtt)2, where t is the time equal to l/vh, over the area of the detector, A# = (w2:

fN =A

A#=

w2

l2v2

h

v2t

, (2.5.4)

where vh is the horizontal velocity and vt is the transverse velocity. Assuming l ( 1m, vh ( 500 m/s vt = 50 m/s we easily see that a grating radius and a detector sizew ( 10 cm are needed to detect all the atoms.The simulation results shown below refer to L = 40 cm (grating-grating distance anddetector-second grating distance), a grating period a of 80 m and an opening fraction


Figure 2.18: Computing the minimum size of the sensitive-position silicon detector.

f = 0.3 (defined as the ratio between the surface in which there is material, and theone where there is not). The distance between the first grating and the center of theantihydrogen cloud before the acceleration is assumed to be Ls = 30 cm. Typically aradial velocity corresponding to 100 mK and a radial beam extension of 1 cm radiusare considered. See again Figure 2.14 for a schematic view of the setup.Figures 2.19 and 2.20 show that the Moire deflectometer does not require a collimated

beam. Figure 2.19 shows the number of atoms arriving at the position of the detector(here we assume infinite position resolution) in three extreme cases without includingthe force due to gravity. The second plot is obtained assuming that the antiatoms havea transverse temperature of 100 mK and that they originate from a point-like source.The last plot is the result obtained launching antiatoms with 100 mK from an extended(3 cm radius) source. In Figure 2.20 the gravity force is included. The gravity inducedshift of the N(x) function is evident in all three cases.It has been verified with the simulations that the minimum in the number of detectedcounts is insensitive to the radial antihydrogen velocity and to the radial section of theantihydrogen beam within the range of parameters of our interest. Of course the radialvelocity influences the number of atoms arriving on the detector and so it must be keptas low as possible.To recover g from the measured 0g value it is necessary to know the time of flight T ofthe antiatoms between the two gratings. If all have the same velocity then T = L/vh,where vh is the horizontal velocity, in the range of few hundreds m/s. If antiatomshaving di!erent axial velocities are grouped together then each one of them contributesto the signal with its phase shift. The result is similar but an e!ective value T 2

e!has to be used instead of T 2 = L2/v2

h. The correct g value can be obtained usingT 2

e! = +T 2'. In the simulations presented in the proposal, the fit result obtained with10 m detector resolution is 9.8± 0.13. The corresponding value obtained with infinitedetector resolution is 9.8 ± 0.1, as reported in Figure 2.21. Assuming a productionrate of useful antihydrogen of 1 Hz and access to antiprotons for 25% of a day, themeasurement would require of the order of two weeks of beam time.


Figure 2.19: Number of atoms detected at distance L from the second grating versus x/a. The forcedue to gravity is not included. The 3 plots refer to di!erent initial conditions. From top to bottom theplots are obtained with a radial extended source with vt = 0; with a point-like source where vt takenfrom a 100 mK Maxwell distribution; with an extended source and 100 mK radial temperature.


Figure 2.20: As in Figure 2.19 but including the force due to gravity.


Figure 2.21: Phase shift as a function of the time-of-flight $T 2% (s) between the two gratings of theMoire interferometer. Top plot: assuming a 10 m detector resolution; Bottom plot: assuming perfectdetector resolution.

Chapter 3

Antiproton-matter interaction

Being tied to the current requirements of the AEgIS experiment, this thesis work isfocused on the interaction of antiprotons at very low energies on a silicon detector,used as a beam monitor, as well as the antiparticles’ electromagnetic interaction inaluminium, since a study of a beam degrader is needed. To preface such topics, it isvery useful to deepen the interactions that rule the annihilation of p on nucleons andnuclei. It will be clear very soon that it is very di"cult to establish an annihilationtheory from first principles, as all existing fit of the experimental data are made onan empirical basis. Therefore, we will limit ourselves to a qualitative description ofthe undergoing processes, as for instance the list of the di!erent cross-sections in play,or the implications of the final state interactions (FSI). We will underline the mainexperimental evidences available, such as the secondary multiplicities, with some carededicated to those of the charged pions. Then, in the next chapters, we will be able tocompare them with the simulation results.A cornerstone of the radiation matter interaction, the Bethe Bloch formula, will thenbe briefly presented. It represents an excellent starting point to evaluate the energyloss by a charged particle in a layer of material. We cannot prescind from it, if we wantto outline the behaviour of an antiproton beam through a thin aluminium layer. Thistopic holds a most important position in AEgIS, and its underestimation could lead toserious damages in the outcome of the experiment, in particular in the formation ofantihydrogen.

3.1 Antinucleon-nucleon cross-sections

3.1.1 Introduction

Proton-antiproton annihilation is used to denote the process that occurs when a sub-atomic particle, a proton, collides with its respective antiparticle, the antiproton. Anni-hilation is defined as ‘total destruction’ or ‘complete obliteration’ of an object, having itsroot in the Latin nihil (nothing). Since energy and momentum must be conserved, theparticles are not actually made into nothing, but rather into new particles, sometimesincluding photons. Antiparticles have exactly opposite additive quantum numbers fromparticles, so the sums of all quantum numbers of the original pair are zero. Hence, anyset of particles may be produced, the total quantum numbers of which are also zero,as long as conservation of energy and conservation of momentum are obeyed. The in-

47

48 Antiproton-matter interaction

teraction of slow antiprotons with nuclei is presently the only way to investigate theinteraction of antimatter with matter at low energies.The annihilation sets free an enormous amount of energy (1880 MeV) in the volume ofonly one or two nucleons. During a low-energy collision, proton-antiproton annihilationusually produces a collection of pions or related light mesons. High-energy particlecolliders, like the Tevatron, sometimes collide protons with antiprotons at very high en-ergy, and these collisions can produce a long list of heavy particles. The vast majorityof proton-antiproton events are mediated by the strong interaction, i.e. from a quarktheory point of view, they involve only rearrangements of quarks, and like-flavor quarkpair creation or destruction. This results primarily in various mesons, primarily pions,and kaon pairs are also observed. Rarer events include a photon. (Of course, many ofthe primary decay products, notably neutral pions, later decay into gamma-ray pho-tons.) Unlike in electron-positron annihilation, proton-antiproton annihilation to twophotons is expected to be extremely rare; one sensitive search for such events [57] sawan excess of signal-like events but did not claim this as a direct detection.Antinucleon (N = n, p) colliding in flight with free nucleons (N = n, p) undergo elasticand inelastic processes [58]; the latter ones include: charge-exchange (pp , nn), annihi-lation (NN & mesons and hyperons), and inelastic reactions (NN & NN plus mesonsand hyperons). The annihilation with production of mesons has no energy thresholdand is the dominant process at low momenta (say below 1 GeV/c). As the interactionenergy increases, many reaction channels open. With a probability much smaller thanfor the annihilation into hadrons, pp pairs annihilate also into $X pairs (X = $ or me-son) and electron-positron pairs.In the following, we shall use these notations for the cross-sections:

elastic scattering: 1el

charge-exchange: 1cex

annihilation: 1a

annihilation into charged particles: 1ach

annihilation into neutral particles: 1a0

inelastic scattering (p in the final state): 1in

inelastic reactions: 1r = 1cex + 1a + 1in

reactions without charged particles: 10

total cross-section: 1t = 1el + 1r

Most of the existing data concern the pp interaction and only few ones the np interaction.The most reliable trend of the pp cross-sections of the di!erent processes below 15 GeV/cis displayed in Figure 3.1, 3.2 and in Table 3.1.1. Figure 3.3 reports the annihilationcross-section in the energy range of our interest, while Figure 3.4 shows cross-sectionsat very low energies.

3.1.2 p-nucleus interaction

In the pp case, the annihilation cross section can be obtained by direct measurements orby measurements of 1ach, 1a0, 1cex and 1in according to the reaction 1a = 1ach +1a0 =1r # 1cex # 1in. Note that, below 775 MeV/c, 1in = 0.There are several sets on data on 1ach. They can be fitted by a function of the type:

1ach = A + B/p. (3.1.1)

However, taking nuclei instead of a simple proton allows to study the interaction of p(n) both with p and with n, thus evidentiating its dependence on the isospin.

3.1 Antinucleon-nucleon cross-sections 49

Figure 3.1: pp cross-sections up to 15 GeV/c. Most reliable behaviours (upper figure) and contribu-tions to the total cross-section from the di!erent processes (lower figure). The experimental points aretaken from [58] and references therein.

p may interact with nuclei in flight or at rest. In a conventional frame, the p-nucleusinteraction in flight includes:

1. elastic scattering;

2. inelastic reaction with p and nuclear fragments in the final state (nucleus excita-tion, pick-up and break-up reactions, etc.);

3. charge exchange (pp , nn);

4. inelastic scattering with p, mesons and hyperons in the final state like on freenucleons;

5. annihilation like on free nucleons;

6. final-state interaction (FSI) between the annihilation products and the residualnucleons; mesons may be absorbed and the residual nucleus may break also in agreat number of fragments.


Figure 3.2: pp cross-sections in the interval 200-900 MeV/c. Most reliable behaviours (upper figure)and contributions to the di!erent cross-sections to the total one (in percentage, lower figure).

The threshold of the reaction 3 increases with the mass number A and those for reactions4 decrease with A. Some of the previous reactions can occur in cascade, so that ingeneral residual nuclei, nuclear fragments, nucleons, mesons and hyperons are presentin the final products. In the following we shall indicate with 1bu the cross-section of anyprocess the only result of which is breaking up the nucleus and with 1exc that of anyprocess which only excites it. All these processes contribute to the inelastic reactions,so that we redefine:

1r = 1cex + 1a + 1bu + 1exc. (3.1.2)

In the annihilation 5 pions are produced in the average, both in correlated and inuncorrelated ways; in the latter way they are produced through heavier mesons (,, ',. . .) which then decay into pions (see Figure 3.5). The mean pion energy, about 230MeV, is in the region of the /-resonance, so that pions passing through the nucleusare very likely to be scattered or absorbed, transferring high excitation energy to the


Figure 3.3: Annihilation cross-sections in the very low momentum region. np interaction (!) anddashed line. pp interaction (&), full line; dot-dashed line 1r'1cex. In the inset, the measured quantitiesare shown in terms fo 1". For the sake of completeness also the p annihilation cross-sections on 3Heand on 4He are shown. No other result exists in this momentum region.

nucleus. The decay mechanism of the residual nucleus is a consequence of the depositedexcitation energy E". If it is small (E" - 1 # 2 MeV per nucleon), the basic decaymechanism is the evaporation of particles from the coumpound nucleus. If E" is close tothe total binding energy of the nucleus (E" " 5#7 MeV per nucleon), the multifragmentdecay mechanism dominates.An important consequence of FSI is that the pp annihilation events can assume featuresof the pn annihilations and viceversa. This is due to the (N charge-exchange reactions


Figure 3.4: Values of the total pp annihilation cross section at low energy, multiplied by the incomingbeam velocity, as a function of the p incident momentum [59]. The full line is the total annihilationcross section, the dashed line represents the S-wave contribution. In the inset the low energy region ismagnified.

and to the ( absorption reactions:

(+n , (0p,

(!p , (0n,

(!pp , np,

(!pn , nn,

(+pn , pp,

(+nn , np,

(3.1.3)

which transforms p into n and viceversa and change the relative numbers of (+, (! and(0. We recall that in pp annihilations equal numbers of (+ and (! are produced andin pn annihilations the number of (! exceeds that of (+ by 1. Another consequenceof FSI is that hyperons can be produced at energies lower than the thresholds on freenucleons; for instance, through the rescattering of kaons which are captured by nucleons(KN & -().Beside the above ones, also nonconventional processes are expected in nuclei, that is


Equation GeV/c(1) 1t = (41.1 ± 1.17) + (77.2 ± 2.91)p!(0.68±0.14)+

+(0.293 ± 0.0245) log2 p # (1.82 ± 0.337) log p > 1(2) 1t = 65.55 + 53.84/p < 1(3) 1cex = 20.6 # 14.5p 0.2 - 1(4) 1cex = #0.39 + 7.5p 1 - 3(5) 1cex = (18.8 ± 0.902)p!(2.01±0.04) 3 - 35(6) 1a = 1r # 1cex 0.2 - 0.775(7) 1a = (0.532± 0.210) + (63.4 ± 0.513)p!(0.71±0.03) 0.775 - 22.4(8) 1r = (38 ± 0.17) + (38 ± 0.257)p!(0.96±0.06)#

#(0.169 ± 0.0145) log2 p 0.306 # 175(9) 1el = (10.6 ± 0.354) + (53.1 ± 1.25)p!(1.19±0.10)+

+(0.136 ± 0.0119) log2 p # (1.41 ± 0.135) log p > 2(10) 1el = 1t # 1r

Table 3.1: Most reliable expressions for pp cross-sections vs. p momentum. Equation (1) fits dataobtained before 1984 in the interval 5.0 - 4.32 · 105 GeV/c. Equation (8) fits data in the interval0.306-175.0 GeV/c; it works satisfactorly also at lower momenta. Equation (9) fits data in the interval2.0 - 1.59 · 105 GeV/c. Equation (7) from data on deuterium. Simple parametrizations below 7 GeV/cdeviate somewhat from those given here below 0.5 GeV/c.

processes which cannot occur on free nucleons, as the annihilation involving more thanone nucleon (B > 0 annihilation). Examples of them are the so-called Pontecorvoreactions. Considering the interaction of p with deuterons the Pontecorvo reactions canbe summarized by the two-final-body reaction:

p 2H & M + X, (3.1.4)

where M stands for a meson (strange or not strange) and X for a baryon (nucleon,baryonic resonance or hyperon). These processes have been considered from di!erentpoint of views, both in terms of final-state absorption or virtual mesons and of quarkdegrees of freedom. The quark picture deals with delocalization of the nucleon quarks.The momentum of the two final particles and the mass of M reflect their space correla-tion between the bound nucleons: the lower the momentum (the higher the mass), the

Figure 3.5: Schematic picture of p annihilation on the nuclear surface. The Figure shows the di!er-ence between uncorrelated pion production (1) and pion production by heavy mesons (2), where theprobability that all pions leave the nucleus is higher.


larger the characteristic distance between the nucleons. The first measured Pontecorvoreaction was:

p 2H & p + (!. (3.1.5)

In the case of annihilation on nuclei, Pontecorvo reactions can be summarized by thereaction:

p + nucleus & (M + X) + spectator nuclear fragments. (3.1.6)

The simplest example is the annihilation without production of pions:

p 3He & p + n. (3.1.7)

B > 0 annihilations are expected to contribute to the high-energy tail of the momentumspectra of the protons emitted from the nuclei and a higher amount of strange particlesthan in the pp annihilation. As far as proton spectra are concerned, theoretical analyseslead to opposite conclusions about the existence of B > 0 annihilations. The uncertaintypersists also after the measurements and their analysis, in spite of the claimed evidenceof such annihilations. A di"culty in detecting B > 0 annihilations could be related totheir low probability.Because of the high annihilation cross-section, p annihilate with high probability onthe surface of the nuclei, at a radius at which the nuclear density has about 10% ofthe central density, and only a small fraction penetrates deeply into the nuclei. Thisfraction increases with p energy.Annihilation at rest are similar to those in flight, but with some specific features, whichmerit to be described shortly. Antiprotons passing through a material are slowed downfurther and further and are finally captured into atomic orbits with high principalequantum number n and high orbital quantum number. From there they cascade downby emission of Auger electrons and X-rays. Once the p have reached levels betweenn = 9 and 4 (depending on the charge of the nucleus), they annihilate on one nucleonat the nuclear surface. In some cases all the annihilation pions either leave the nucleuswithout scattering or knock out nucleons via quasi-free scattering or pion-nucleon chargeexchange. In such cases a weakly excited compound nucleus is produced, which emitsone or two nucleons or only $-rays. The occurrence of these facts depends on the nucleardensity at the nuclear periphery and on the type of pion emission. An uncorrelatedemission, which is isotropic at rest, may lead to more pions interacting with the nucleus,resulting in the subsequent emission of many nucleons, while a correlated emission ofpions may favour residual nuclei which have lost only few nucleons.

3.1.3 p-nucleus cross-sections (A ! 3)

In the interaction with the nuclei, p undergo Coulomb and nuclear scattering. Let uswrite the scattering amplitude in the form:

A()) = fC()) + fN()), (3.1.8)

where fC includes single and multiple scattering by the (screened) Coulomb field of thenucleus. We define:

1C =

, 4"

0|fC())|2 d2, (3.1.9)

1N =

, 4"

0|fN())|2 d2, (3.1.10)


1int =

, 4"

02.f"

C())fN())d2. (3.1.11)

The total nuclear cross section is:

1t = (4(/k)/fN(0$) = 1N + 1r, (3.1.12)

where 1r is the reaction cross-section, which includes all the processes other than theelastic one (annihilation, break-up, nuclear excitation, charge exchange, etc.).

3.1.4 Pion multiplicity

Below 775 MeV/c the pions are due only to the annihilation process. As a guideto facilitate the presentation of the experimental data, we point out very schematicallysome facts determining the features of the annihilation pions emitted from nuclei withina conventional frame. They are:

the number of protons (Z) and neutrons (N);

the di!erent probabilities of the annihilation on a proton and on a neutron;

FSI, in particular the pion-nucleon charge exchange and the pion absorption afterthe annihilation.

As a general trend, the production of negative pions is favoured as A increases for twomain reasons. First, the percentage of neutrons increases with A and this favours theoccurrence of pn annihilation which produces on average 1(+ and 2(! against 1.5(+

and 1.5(! as in the pp annihilations. Second, if the number of the neutrons is higherthan that of the protons, the charge exchange reactions:

(+n & (0p, (0n & (!p (3.1.13)

are more favoured than the reverse ones, as well as the absorption reactions:

(+nn & pn, (+pn & pp, (3.1.14)

with respect to:(!pp & pn, (!pn & nn. (3.1.15)

Therefore, owing to FSI, the (+ should disappear with higher probability than the (!.On the basis on the above considerations, some useful relations can be written. To thisaim, we define the following quantities:

n"!(pn), n"!(pp), n"+(pn), n"+(pp): mean numbers of negative and positive pi-ons produced in annihilations on single neutrons and singole protons, respectively;

P (pn), P (pp): probabilites that p annihilate on single neutrons and single protons,respectively, with P (pn) + P (pp) = 1;

P (+), P (#): probabilites that positive and negative pions survive after FSI;

P ((+), P ((!): probabilites that the events have at least one positive (negative)pion among the final products.


The mean number per event of positive and negative pions delivered from nuclei withmass number A(N, Z) can be given approximately by the following relations:

n"! = P (#)P (pn)Nn"!(pn) + P (pp)Zn"!(pp)

P (pn)N + P (pp)Z, (3.1.16)

n"+ = P (+)P (pn)Nn"+(pn) + P (pp)Zn"+(pp)

P (pn)N + P (pp)Z. (3.1.17)

The fractions are the mean numbers of positive and negative pions produced in theannihilations in nuclei. Assuming for the sake of simplicity:

n"!(pn) = 2, n"+(pn) = 1, n"+(pp) = n"!(pp) = 1.5, (3.1.18)

and setting:

R =P (pn)

P (pp), (3.1.19)

N"! =2R + 1.5Z/N

R + Z/N,

N"+ =R + 1.5Z/N

R + Z/N,

(3.1.20)

Equations (3.1.16) and (3.1.17) become:

n"! = P (#)N"! ,

n"+ = P (+)N"+ .(3.1.21)

The quantities N"± are the mean numbers of charged pions which should be emittedfrom the nuclei in the absence of FSI, that is if P (#) = P (+) = 1. It is interesting toevaluate them as well as their ratio and di!erence by comparison with the correspondentexperimental quantities, which are a!ected by FSI. Considering the nuclei in which dataexist, Z/N varies from 1 down to 0.63 for 238U, excepted 3He for which Z/N = 2. Rvaries in the interval 0.4 - 1. Assuming R = 0.8, a good value for liquid deuterium andother targets, N"! increases by 3.5% from deuterium to uranium and N"+ decreases by4.7%. The behaviours of N"! and N"+ are shown in Figure 3.6. Correspondingly, theratio (N"+/N"!) decreases by 8% and the di!erence (N"! # N"+) increases by about20%. R being fixed, the above variations depend only on the neutron excess.To point out the main common features of the behaviour of the multiplicities as afunction of the nucleus mass number, we consider the multiplicity of the charged pions(n"± = n"++n"!) at rest, whose data are richer in statistics and cover with a significantdensity the whole nuclear mass range.Starting from A = 2, the ( multiplicity decreases quickly as A increases for A < 80 andis almost constant above (Figure 3.7).

3.2 p energy loss

In general, two principle features characterize the passage of charged particles throughmatter:

3.2 p energy loss 57

Figure 3.6: (! (•), (+ (&), and (0 (!) mean multiplicites distributions vs. A. The full lines areguide for the eyes. The dashed lines give the behaviour of N((!) and N((+) according to Equations(3.1.20) and (3.1.21); that is, they are multiplicities estimated neglecting FSI.

a loss of energy by the particle and

a deflection of the particle from its incident direction.

These e!ects are primarily the result of two processes:

inelastic collisions with the atomic electrons of the material;

elastic scattering from nuclei.

These reaction occur many times per unit path length in matter and it is their cumu-lative result which accouunts for two principal e!ects observed. These, however, are byno means the only reactions which can occur, since other processes include emission ofCherenkov radiation, nuclear reactions and bremsstrahlung. In this thesis work, sincewe deal with antiprotons, we will take into account the nuclear reactions, that include ofcourse annihilation. However the other two are very rare, in comparison to the atomiccollision processes.The inelastic collisions are almost solely responsible for the energy loss of heavy parti-cles in matter. We can consider a heavy particle with a charge ze, mass M and velocityv passing through some material medium and we can suppose there is an atomic elec-tron at some distance b from the particle trajectory. We assume that the electron isfree and initially at rest, and furthermore, that it only moves very slightly during theinteraction with the heavy particle so that the electric field acting on the electron maybe taken at its initial position. Moreover, after the collision, we assume the incidentparticle to be essentially undeviated from its original path because of its much largermass (M 0 me). Trying to calculate classically the energy gained by the electron, oneobtains:

#dE

dx=

4(z2e4

mev2Ne ln

$2mv3

ze23, (3.2.1)


Figure 3.7: (± multiplicity distributions vs. A. The full line is the result of a best-fit calculation.The dashed line is the behaviour expected neglecting FSI according to Equations (3.1.20) and (3.1.21).

where $ = (1#"2)!1/2 and 3 is the mean frequency, averaged over all bound states, ofthe electron. This is essentially Bohr’s classical formula. It gives a reasonable descrip-tion of the energy loss for very heavy particles such as the $-particle or heavier nuclei.However, for lighter particles, e.g. the proton, the formula breaks down because ofquantum e!ects. It nevertheless contains all the essential features of electronic collisionloss of charged particles.The correct quantum mechanical calculation was first performed by Bethe, Bloch andother authors. In the calculation the energy transfer is parametrized in terms of mo-mentum transfer rather than the impact parameter. This, of course, is more realisticsince the momentum transfer is a measurable quantity whereas the impact parameteris not. However, two corrections are normaly added: the density e!ect correction 4,and the shell correction C. The formula obtained is then:

#dE

dx= 2(Nar

2emec

2,Z

A

z2

"2

-

ln

!

2me$2v2Wmax

I2

"

# 2"2 # 4 # 2C

Z

.

, (3.2.2)


with:

re: classical electron radius ,: density of absorbing materialme: electron mass z: charge of incident particle in units of eNa: Avogadro’s number " = v/c of the incident particleI: mean excitation potential $ = 1/

#

1 # "2

Z: atomic number of absorbing material 4: density correctionA: atomic weight of absorbing material C: shell correctionWmax: maximum energy transfer

in a single collision

Given the Bethe Bloch formula, we might be tempted to calculate the mean range of aparticle of a given energy, T0, by integrating the dE/dx formula,

S(T0) =

, T0

0

!

dE

dx

"!1

dE. (3.2.3)

This yields to the approximate pathlength travelled. Equation (3.2.3) ignores the e!ectof multiple Coulomb scattering, however, which causes the particle to follow a zigzagpath through the absorber. Thus, the range, defined as a straight-line thickness, willgenerally be smaller than the total zigzag pathlength.As it turns out, however, the e!ect of multiple scattering is generally small for heavycharged particles, so that the total path length is, in fact, a relatively good approxi-mation to the straight-line range. In practice, a semi-empirical formula must be used:

R(T0) = R0(Tmin) +

, T0

Tmin

!

dE

dx

"!1

dE, (3.2.4)

where Tmin is the minimum energy at which the dE/dx formula is valid, and R0(Tmin)is an empirically determined constant which accounts for the remaining low energybehavior of the energy loss. Figure 3.8 shows some typical range-energy curves fordi!erent particles calculated by a numerical integration of the Bethe-Bloch formula[60]. From its almost linear form on the log-log scale, one might expect a relation ofthe type:

R 1 Eb. (3.2.5)

This can also be seen from the stopping power formula, which at not too high energies,is dominated by the "!2 term,

#dE

dx1 "!2 1 T!1, (3.2.6)

where T is the kinetic energy. Integrating, we thus find:

R 1 T 2, (3.2.7)

which is consistent witch our rough guess. A more accurate fit in this energy range, infact, gives:

R 1 T 1.75, (3.2.8)

which is not too far from our simple calculation. This is only one of many theoreticaland semi-empirical formulas which cover many energy ranges and materials.


Figure 3.8: Calculated range curves of di!erent particles in aluminium.

3.2.1 The Barkas e!ect

Although slowing down of fast charged particles in matter has been studied for morethan half a century, there is still a need for a better understanding of the energy-lossprocess. This not only bears on practical questions of predicting the slowing-down, butalso fundamental questions related to the basic atomic-collision processes involved arestill unresolved [61].A di!erence in the stopping power for positive and negative particle, known as theBarkas e!ect, has been known for a long time [62], but for many years lack of suitableantiparticle beams has prohibited quantitative studies of the e!ect. The Barkas e!ectcan also be deduced by comparisons of the stopping power for protons, alpha particlesand lithium nuclei [63], but the interpretation of these data is not as straightforward forlight targets as particle/antiparticle data. Furthermore the antiparticle measurementscan be performed at much lower energies than with alpha particles and lithium nucleiowing to electron capture e!ects. Actually, the absence of electron capture for negativeparticles makes their stopping powers better suited for comparisons with calculations.The basic stopping power formula in the high energy limit is the Bethe formula:

#dE

dx=

4(e2NZ2

mv2Z2

1L0, (3.2.9)


which is derived in the first Born approximation. Hence the stopping power is propor-tional to the square of the projectile charge Z1. In Equation 3.2.9, v is the projectilevelocity, N the target density and Z2 the target atomic number. The Bethe stoppingfunction, which is independent of Z1, may be written as:

L0 = ln

-

2mv2

I

!

1 # v2

c2

".

# v2

c2# C

Z2, (3.2.10)

where I is the mean ionization potential and C/Z2 the so-called shell corrections. For-mally, one may generalize the Bethe equation to include the higher-order Z1 termis inthe stopping function:

L = L0 + Z1L1 + Z21L2 + · · · , (3.2.11)

where L1 and L2 are the Z1-independent coe"cients of the Z31 and Z4

1 terms in thestopping power. Higher order terms can be neglegted at MeV energies or above.A Z3

1 (Barkas) correction to the Bethe formula for the harmonic oscillator was firstcalculated for distant collisions only. The e!ect is related to the polarization of themedium induced by the projectile. The impact parameter discriminatoring betweenclose and distant collisions is of the order of the radius of the harmonic oscillator. Thedistant-collision Barkas e!ect is related to the polarization of the medium induced bythe projectile. This explains qualitatively why negatively charged particles have a lowerstopping power than positively charged ones. It was also shown that the Z4

1 correction(the so-called Bloch term) is significant at velocities where the Z3

1 correction is.

Chapter 4

The AEgIS beam line

The AEgIS beam line is the heart of the Monte Carlo simulations. After a descriptionof the sensitive volumes implemented all along the beam line, and some considerationsover the position-dependent magnetic field, the role of the main character of the thesis,the beam counter, will be discussed. Some mentions to the other AEgIS detector will begiven as well, even if they are mostly on preliminary studies stage. The beam counter isa particular silicon detector that will be used to monitor the centering of the incomingantiproton beam from AD, and possibly also its intensity. It is very important tostudy this device, because the first part of data acquisition in AEgIS will be a beamdebugging period. Indeed, it is on the ground of the beam line that we have worked totest di!erent options. Two locations are possible for the detector: one is in an easilyreachable vacuum chamber, external from the magnets and refrigerator complex; theother is next to the aluminium degrader and the trap for the antiprotons. Most ofthe combinations between detector designs and locations will be explored. At the end,merits and defects of each of them will be discussed, and fundamental parameters ofthe beam counter will be given. Several tests have been performed, setting di!erento!sets in the generation of the beam: average value of the generation point, angularo!set of the beam and its divergence will be modified. The aim of these analysis is notto study a device that should reconstruct correctly a whole series of beam parameters.The beam parameters coming from the AD team are to be trusted. Instead, the role ofthe beam counter is to distinguish between acceptable and non-acceptable cases. Beamanomalies can arise from many sources: if part of the hardware of the beam line has abreak-down, one has that antiprotons completely miss their target. The beam countermust then detect if, say, the missed antihydrogen formation is due to the lack of theincoming antiprotons. It must not carry out fine-tuning or refinements of the beamposition.Eventually, we present here the study of the aluminium degrader. This layer will beplaced at the entrace of the trap. Its aim is to prepare antiprotons storage, and to slowthem down to a kinetic energy EK < 10 keV. The importance of knowing precisely thethickness of this layer of material will be pointed out. It should nevertheless be clearfrom now that without a correct study of the aluminium degrader, the production ofantihydrogen is compromised.

63

64 The AEgIS beam line

4.1 Description of the beam line

In Figure 4.1 are reported some details of the plan already showed in Figure 2.8. Sucha zoom is necessary to fully understand the role of all devices that have been alreadydiscussed, and those to be presented in this chapter. In fact, it can be stated that thethesis is entirely focused on this portion of setup, more precisely from the generationpoint of the antiparticles, at -240 cm (taking the origin in the space between the twomagnets), down to the entrance of the trap, at -106 cm.

The p will arrive from AD, will enter the AEgIS beam line where they will haveto go through two layers of material set: the beam counter (monitoring and possiblycounting) and an aluminium foil to reduce the kinetic energy in order to maximize thenumber of p with EK < 10 keV. Both the beam counter and the degrader will be in placefor p trapping and moved during e+ manipulations, thus will be mounted on movablesystems. Another very important element to be considered, when the behaviour ofantiprotons in this setup is planned, is the AEgIS magnetic field, shown in Figure 4.2.The magnetic field grows until it reaches 5 T. The trap is positionned in a place wherethe field is already constant. The duty of this field is to focus the antiprotons in frontof the degrader and the trap, to strongly reduce the trasversal size of the beam. Theshape of the magnetic field toward the 1 T magnet is currently under study.In the Monte Carlo, several sensitive volumes have been introducted. In the followingtheir coordinates are summarized. When a detector is labelled as ‘fake’, it means thatit exists only in the simulation, because it has been defined taking vacuum (that isA " Z " 10!15) as the material that fills its volume. Indeed, it is a simple way totake a snapshot of the beam profile on that coordinate, without disturbing it throughscattering or other interactions. The following table is taken from an ouput file of asimulation run, involving a beam counter set near the trap. The position of the fakedetectors is maintained throughout all the simulations. Every fake detector is 1 mthick.

Device z-coordinate (cm)Fake detector 0 -239.90Fake detector 1 -172.90Fake detector 2 -172.00Fake detector 3 -168.00Fake detector 4 -150.00Fake detector 5 -140.00Fake detector 6 -130.00Fake detector 7 -120.00Beam counter -111.00Degrader -110.00Fake detector 8 -109.88Fake detector 9 -106.00Fake detector 10 -100.00

Some observations are necessary. The first fake detector is placed 1 mm downstream ofthe generation point of the antiprotons. The second one is set in the ‘sun’, to monitorthe beam just after one of the possible positions of the beam counter (currently it is notthe case). Detector 8 is set 1 mm downstream of the aluminium degrader to underlinethe e!ects of the multiple Coulomb scattering.

4.1 Description of the beam line 65

(a) Detail from Figure 2.8. Portion of the beam line (first part).

(b) Detail from Figure 2.8. Portion of the beam line (second part).

Figure 4.1: Zoom of the AEgIS plan, showing the portion of the beam line of our interest throughoutthis chapter. Origin of the beam axis is between the two magnets (z = 0 cm). Point 1 indicates thevacuum chamber known as the ‘sun’. Aluminium foils a few microns thick are inserted inside thischamber to study a correct degradation of the antiprotons. These foils are automatically retractedwhen positrons cross the ‘sun’, because otherwise they would interact with them. The ‘sun’ is alsoone of the two possible rooms in which the silicon beam counter could be placed, at z = '173 cm.Speaking of which, point 2 represents the other possibility to place the silicon detector, at -111 cm.One centimeter downstream, there is the main aluminium degrader. Number 4 points the Malmberg-Penning trap and its electrodes, from -87 cm to -74 cm. For the sake of completeness, point 5 showsthe 5 T main magnet, point 6 the space between the magnet, and point 7 the 1 T magnet.


Figure 4.2: The AEgIS magnetic field profile, showed in kG vs. cm, drawed from its actual map,and fully implemented in the simulation program. Point (A) is placed at -173 cm and indicates theposition of beam counter within the ‘sun’. Position (B), placed at -111 cm refers to the position of thebeam counter close to the degrader (-110 cm), next to the beginning of the trap. The di!erence of themagnetic field intensity between the two cases is evident.

4.2 The p-beam counter

The main task of this section is the description of a particular detector: the beamcounter. Although it does not really count the beam particles, in the inner jargonof the experiment it has been called in this way.1 It is devoted to the control of thecorrect centering of the beam, to contribute to diagnose the steps of the production ofthe antihydrogen beam.The other important duty here is to describe the setup of the AD beam implemented inthe simulation, that is mostly the thick degrading aluminium foil, but also the severalsensitive detectors made of vacuum in the simulations, so that they not disturb thebeam. The former has the role to maximize the number of p with a kinetic energylesser than 10 keV, so that they can be captured by the Malmberg-Penning trap. Thelatter are used in the simulation to monitor the beam section at various coordinates.We recall that the AD will deliver to the AEgIS experiment a p beam (about 3 · 107ps)with a momentum of about 100 MeV/c (5.3 MeV in energy) in bunches of about 200ns every 200 s. To monitor the beam intensity and the beam alignment a silicon beamcounter can be used, and two possible designs are considerated here.

The first design can be similar to the one used in the ATHENA experiment [64],previously adopted by the Crystal Barrel experiment, and that proved to be veryuseful for the beam diagnostic. It should consist of a 55 m thick p-i-n silicondiode (see Section B.8 for further informations on this particular junction), pro-duced by MICRON (Great Britain), 15 mm in diameter and segmented with onecircular center pad (38.5 cm2) and four equally large surrounding pads (34.6 cm2),each connected to an individual signal line (see Figure 4.3). The voltage requiredto fully deplete the diode is 4.5 V. The beam counter would be located in frontof the antiproton catching trap and is able to operate between 10 and 300 K, ina vacuum of 10!8 mbar and in a 5 T magnetic field. The average energy loss of5.3 MeV antiprotons in silicon has been estimated to be about 11.4 keV per mof silicon [61], thus creating 3200 electron-hole (e-h) pairs, compared to 80 e-h

1Actually it is more a beam monitor, since at this stage of the experiment design as well as in thesimulations, no ‘one-by-one’ counting of the antiprotons is planned.

4.2 The p-beam counter 67

Figure 4.3: Photograph of the ATHENA 67 m thick silicon counter mounted in a PCB frame,very similar to the 55 m one to be used in AEgIS. The contacts to the individual pads aremade using ultrasonic wire-bonds.

Figure 4.4: Measurement of the integrated signal versus bias voltage with and without thickdegrading foil in the beam. The image plots show beam profiles measured at 100 V: the grayrectangles correspond to the five pads of a beam counter identical of the one considered, but 67m thick, white corresponds to zero integrated signal and black to 1.5 · 10!6 V s.

pairs for minimum ionizing particles. Thus around 650 keV are released in suchbeam counter. In order to detect this high instantaneous current generated bysuch energy loss in the silicon pads, a readout system was developed where thesignal current is read directly across a 100 $ protection resistor and fed into anADC. The signal from the silicon beam counter can also be used to trigger the pcatching trap. The 5 pad configuration ensures the possibility of measuring thealignment of the p beam, being sensitive to vertical and horizontal displacementsof the beam.Figure 4.4 shows the integrated signal measured as a function of the bias voltage

with and without degrader foil 1 m before the beam counter. The measurementwas taken during the run of ATHENA with 3 T and at 2 10 K. The signal in-creases as the bias voltage is raised above the depletion voltage but starts to showa plateau at around 30-40 V. This indicates that at lower bias voltages a largefraction of the charge pairs recombine, but are separated at higher voltages.


Figure 4.5: Schematic overview of a portion of AEgIS, the same already presented in Figure 4.1.Two positions are possible for the beam counter: one is in the vacuum chamber the so-called ‘sun’; theother is just before the aluminium degrader and the Malmberg-Penning trap.

It should be pointed out that if this detector will be used, its 55 m would con-tribute to the degrading of the beam. In fact, this detector would be permanentin the setup, because its thickness is too small to allow the antiproton annihila-tion. The total degrading would then consist in the beam counter added to thealuminium foil.

The second possible design is very di!erent. It consists of a square silicon paddetector, 300 m thick, divided in several sensitive pads. The surface of thisdetector, as well as the number of pads are fundamental parameters to be obtainedfrom the simulations. The advantage of this option is that silicon detectors 300 mthick are well known and commercially available. In this case, however, since theantiprotons would all stop in the device, it would be removed once the preliminarytests to center the beam are completed. Thus exclusively the aluminium foil wouldbe in charge of the slowdown of the particles.

Two di!erent positions for the beam counter, whatever it will be, are planned in thesetup (see Figure 4.5).

The first possibility is to put the beam counter in the main magnet, at -111 cm,one centimeter before the degrader and consequently very near to the Malmberg-Penning trap. The detector would be placed on a mechanism that allows to folddown the device parallel to the surface of the magnet bore. It should be taken intoaccount in fact, that since the AEgIS setup has a design totally turned forward,it cannot reproduce the scheme of ATHENA. In other words, the ingredients ofantihydrogen, positrons and antiprotons, must travel on parallel roads, and cannotface each other. This is why there is a mechanism that make the silicon beamcounter detect the antiprotons, and let pass the positrons undisturbed.

Another choice is to install the beam counter at -173 cm in the so-called ‘sun’,a vacuum chamber " 60 cm upstream of the degrader. This vacuum chamberhas service and partly diagnostic purposes. It comprehends a series of tools thatwould be too di"cult to handle inside the cryostat. For instance, here is set acamera to monitor the phosphor screen of the micro-channel plate, which detectsthe positron beam. The ‘sun’ is " 15 cm thick, and his diameter is " 30 cm long.It is a structure set outside the system of the magnets, and it is the best choiceto place a device, if we want it to be easily reachable.

4.3 Other detectors in AEgIS 69

As the beam counter disturbs the beam, since the antiprotons lose energy in the firstcase, and completely annihilate in the second one, as foretold it is intended be removedduring data acquisition only in the second case.

4.3 Other detectors in AEgIS

Several types of detectors integrated in the AEgIS setup allow monitoring and step-by-step optimization of the sequence of the various particle manipulations leading tothe antihydrogen beam. Some of these detectors are of the type widely used in nuclearand particle physics while others are very common in the atomic and molecular physicscommunity or in the non-neutral cold plasma physics field. A very important detectorto be used in the AEgIS experiment is the antihydrogen detector.To achieve AEgIS main objective, H must be produced in a stable way and an anti-hydrogen detector thus plays an essential role. It would allow 3-D imaging of the p/Hannihilation [65] and thus the monitoring of p/H losses and H production. The presenceof the antihydrogen detector, together with the pulsed antihydrogen production schemewould also allow a direct measurement of the antihydrogen velocity after its production.The time of production of antihydrogen is known to the uncertainty in the arrival timein the antihydrogen cloud of the Rydberg positronium (" 1 s). With a velocity of 100mK of " 40 m/s, a distance to the nearest surface of 1 cm, and an expected produc-tion rate of " 100 atoms, the H annihilation time distribution covers a range of " fewhundred s, during which " 100 annihilations take place. A single hit time resolutionof " 1 s will then allow measuring the velocity distribution of the ensemble of theproduced H atoms. Its design is underway. Although this can be achieved by using thedesign of the ATHENA antihydrogen detector (see Figure 4.6), the conception of theone that will be used in AEgIS is still uncertain. It is nevertheless known that it willbe set inside the bore magnet, in the area that surrounds the 100 mK Penning trap.

Another important device, whose design is under study, is the position sensitive g-measurement detector, able to measure the antihydrogen vertical coordinate and itsarrival time. In other words, it will be dedicated to the measurement of the fall of theantihydrogen beam itself. Its requirements are a position resolution of 10 m to ensurea 1% uncertainty on the gravity determination. Given the geometrical dimensions ofthe Moire deflectometer region, a detector with a 20* 20 cm2 sensitive region is desir-able. The detector should also be fully operational at a temperature of at most 140 K.Given such requirements, a silicon -strip detector has been considered to be the bestoption. The simulation program developed in this thesis work is intended to get readyto the future implementation of this crucial detector. The validation of the Physics listin the MeV range of energy, to be presented in Sections 5.3 and 5.2, is also intended tosmooth out its utilization to come.

4.4 Simulation results

All simulations, except when specified, have been performed with Geant4 VMC with thePhysics list CHIPS, to be discussed in section 5.1. Several results have been achieved,like the determination of the beam section with the presence of the AEgIS magneticfield, at several positions along the z-axis. In particular, the simulation of the beamdetection by the silicon detector has been performed.Furthermore, the presence of a thin aluminium foil has been implemented in the setup.


Figure 4.6: A scheme of the ATHENA antihydrogen detector. When they form in the recombinationtrap, antihydrogen atoms, being neutral, tend to exit from the electromagnetic trap and collide withthe container walls. Antiprotons and positrons annihilate here, producing three high energy pions anda couple of gamma rays. The layers of the particle detector surrounding the recombination area detectthese particles.

The aim is to establish what would be the best thickness that let the maximum numberof 5.3 MeV antiprotons composing the beam, to slow down at a kinetic energy lowerthan 10 keV. In fact, this is the upper energy required to the antiprotons to be capturedin the Malmberg-Penning trap, set in the main magnet.

4.4.1 Particle generation

The primary particles are generated at z = #240 cm. The origin of this coordinatesystem is set between the two magnets, and the beam is oriented toward the z-axis.Even if the AD provides " 107 antiprotons per bunch, in the simulation the primariesare generated by choice one-by-one so, from now on, one antiproton corresponds to oneevent.Every antiproton has a kinetic energy of 5.3 MeV, since it is the actual mean energyof the AD beam. The other parameters come as well from the AD team, and theyare adopted in the simulation. The x and y coordinates have independent Gaussiandistributions, with mean 0 cm, and standard deviation 1xy % 1x = 1y of 0.19 cm; 1xy

could thus be considered as an estimate of the transverse beam size. Its value dependsfrom the distance s from the focal point, and comes from the following equation:

1xy = 1(s) =#

"(s)5, (4.4.1)

where "(s) is defined by an empirical equation coming from the AD team:

"(z) = "(0) +s2

"(0), (4.4.2)

4.4 Simulation results 71

where "(0) = 0.5 m is beta function at focal point (that is, in this reference system,when s = 0). In the definition of 1xy 5 is the emittance (almost constant). Thedivergence is defined as:

vx

vz=

5

1(s). (4.4.3)

From the divergence, that defines the apex angle of the beam cone, one can obtain thestandard deviation of the particles momentum:

1p = sin

!

vx

vz

"

pz. (4.4.4)

Summarizing, the AD beam parameters are the following:

5 1 (in units of % mm rad)1xy 0.19 cmvx/vz 1.64 mrad1p 1.64 · 10!4 GeV

A kinetic energy of 5.3 MeV corresponds to a momentum of " 100 MeV/c in the case ofan antiproton (m = 938.3 MeV). In fact, since the relativistic equation of energy holds,it is straightforward that:

pc =#

E2 # m2c4, (4.4.5)

where p is the total momentum of the particle, and E the total energy (that is the sumof the kinetic contribution and the mass at rest). After having generated px and py

through independent Gaussian distributions, with mean 0, and 1px = 1py % 1p, thevalue of pz is deduced from the equation:

pz =/

p2 # p2x # p2

y, (4.4.6)

where, once again, p is the total momentum.It must be pointed out that these values are believed to be reliable. The maximumdeviation from these parameters is included in a factor of 2. If the beam goes beyondthese variances, it is because of major misalignments, due to malfunctioning or damageddevices (such as magnets) in the beam line coming from AD. So, in this context nomiddle ways are foreseen. In the following results coming from our simulations, severalbeam configurations have been investigated. Some of them represent cases of extremediscrepancy, to be considered unrealistic if the beam line is correctly operating.

4.4.2 Beam counter in the ‘sun’

As foretold, once the particle are generated, they must interact with some materialto be slowed down. The main character of this portion of simulation program is thesemiconductor beam counter. In our program, this detecor is made of pure silicon, anddoes not take into account any dopant, nor wafer structure.The main concern of a detector monitoring a particle beam is to check the beam align-ment with AEgIS z-axis. The baricenter coordinates can be computed from the energyreleased in the silicon sensitive pads. These coordinates are defined as:

Bx =

0ni=1 xp,iEp,i0n

i=1 Ep,i, (4.4.7)


where Bx is the x coordinate of the baricenter, xp,i is that of the i-th pad center (inboth cases the baricenter is easily found), Ep,i is the total energy released in the samei-th pad and n is the number of pads. Note that in Equation 4.4.7, the energy acts asa weight, so the baricenter is the result of nothing but a weighted sum. Of course, By

may be computed in an analogous way.In this Section, we answer to the following question: is it possible to place a beamcounter inside the ‘sun’? If it is, which design should we choose?

Square beam counter

We recall that two possible detector designs were considered. Here we will focus on thesquare beam counter, 300 m thick. The size is to be optimized in the simulation, as wellas the number of pads. No dead zone is considered. For the reasons already discussed,this detector would be set in the AEgIS vacuum chamber (the so-called ‘sun’). Its centeris nominally centered in (x, y, z) = (0, 0,#173) cm. Our detector would consequentlybe situated about 35 cm upstream of the main magnet’s bore.Several simulations have been run to highlight the beam shifts on the detector when

position o!sets are given to antiproton generation. Three of them are reported in Figure4.7. They prove that a 4 * 4 pads configuration, each of them having a side 0.5 cmlong, would be an excellent option to fully reconstruct the position of the beam. Inthat case, the whole side on the detector is 2.1 cm long. On the graphs, (xg, yg) arethe mean value of the generation coordinates (we recall that they follow independentsGaussian distributions), while (xr , yr) are the reconstructed coordinates. The errorassociated with the single coordinate of the pad is the side of the pad itself, 0.5 cmdivided by

312, so 1.4 mm. Error propagation gives the same error for Bx % xr. So

1x = 1y = 1.4 mm. Being more specific: (a) the generation of the beam is centered;its position is well reconstructed: (xr , yr) = (0, 0) mm; (b) an o!set of 1 mm in bothdimensions has been given: (xr, yr) = (1.1, 1.1) mm are compatible with the valuesgenerated; (c) once again, (xr, yr) = (#2.1,#2.0) mm are an excellent reconstructionfor (xg, yg) = (#2.0,#2.0) mm. The same occurs for (d) and (e), where the shifts setare respectively of (#3.0,#3.0) and (5.0, 1.0). Once again, it is important to point outthat there are no parameters other than the reconstructed baricenter that can expressif the beam center is in the desired position. The aim of the beam counter is not toextrapolate beam parameters to be used elsewhere, but simply to make sure that theantiprotons are present (so that the beam has no major problems) and that they willenter in the trap.In Figure 4.8 cases (a) and (b) show two excentric cases if compared with the formerones. In the first graph the number of pads per size has been doubled, so that wehave now 64 pads. This has been done to see if some advantage can be reached in thebaricenter reconstruction. On the right, the detector has been cut in such a way topreserve the original 4* 4 design but with the total surface divided by 4, which resultsin a 1.05* 1.05 cm2 beam counter. In particular: (a) the o!set (xg , yg) = (#2.0,#2.0)mm has been kept, the reconstruction is of course very good, but it is clear that thereis no advantage in this new configuration (and the cost would be much higher, due tothe need of increase fourfold the readout electronics, etc.); (b) this case showed that asmaller detector (1 * 1 cm2) cannot be taken into account since most of the beam islost. Furthermore, the baricenter reconstruction gets worse, due to the heavy loss ofinformation, and it is less satisfying. This configuration show no advantages, comparedto the original one.


(a) (xg , yg) = (0, 0) mm. (b) (xg, yg) = (1, 1) mm (c) (xg , yg) = ('2,'2) mm

(d) (xg, yg) = ('3,'3) mm (e) (xg, yg) = (5, 1) mm

Figure 4.7: Three di!erent o!sets for the antiproton beam, as seen by the 16 pads detector. Theupper graphs represents the beam ‘as it is’ in the simulation. The lower shows the reconstruction ofthe beam based on the energy released in each pad. Color scale is not absolute. Number of eventsgenerated in each case: 105.


(a) (xg, yg) = ('2, .2) mm, 8(8pads

(b) (xg, yg) = ('2,'2) mm, 4(4 pads but the size of the pad ishalved

Figure 4.8: (a) and (b) are two excentric cases that strenghten the choice of a 4 ( 4 pads, 2.1 cm (2.1 cm configuration.

Circular beam counter

The other possibility is to take a detector very similar to the one used in the ATHENAexperiment, excepted for the thickness: 55 m instead of 67. This beam monitor has 5pads, with one circular set in the center, and the other four arranged like daisy petals.The main di!erence between this detector and the squared one, is that antiprotonsseldom annihilate in 55 m of silicon. Figure 4.9 shows the energy deposited in thisthickness by one antiproton. We recall that the whole device has a diameter of 15 mm,giving a surface of 1.77 cm2. Its dimensions make it unsuitable to be positionned in thevacuum chamber, as we can see from Figure 4.10 (a). In fact, in the ‘sun’ the beamcovers almost the whole detector surface, enlarging thus the risk to lose parts of thebeam. The same o!sets presented for the squared beam counter have been used. In the(b) case, (xr, yr) was found to be (0.7, 0.7) mm and in the (c) case (xr, yr) = (#1.7,#1.8)mm. A small position o!set such as the (b) and (c) cases provoke the loss of animportant part of the incoming particles. The baricenter calculations therefore su!ersof a lack of precision, not so much for (d) (corresponding to the well reconstructed (d)case of Figure 4.7), since it is (xr , yr) = (2.8, 2.8) mm, but for (d): (xr, yr) = (3.5, 1.3)mm (same case as Figure 4.7). It is also straightforward that another cause of thisworsening is due to the fact that 5 pads provide severely less informations than 16, inthe bargain with a smaller surface (less than 2 cm2 against more than 4 cm2). It ismore desirable to be sure that the main part of the beam hit the central pad: it iswithin a radius of " 3 mm that the electron cloud used for cooling is set out. Anyway,


Figure 4.9: Energy released in 55 m of silicon by one antiproton with 5.3 MeV of kinetic energy.Along this thickness, annihilation is a very rare event (! 10!5 of probability)

besides the reducted dimension of the beam counter, there is another important issueto be underlined. In fact, there’s more to say about the choice to put the circular beamcounter in the vacuum chamber. In Figure 4.11 the sections of the beam for all thevacuum detectors set in the simulations has been reported, to see how the beam radialsize varies with the z-coordinate. It is evident that the beam counter set in the ‘sun’causes a multiple Coulomb scattering. The risk is obviously to lose part of the beamduring the segment between the ‘sun’ and the entrance of the trap: indeed, mechanicala restriction of the beam line is foreseen around -130 cm. The red circle shown inthe pads contained in Figure 4.11 represents the limits of the trap entrnace. However,at -110 cm the magnetic field is so strong that its focusing properties bring back theparticles on the axis, but not as much as in the other configuration. We shall see it inthe next section.

4.4.3 Beam counter near the Malmberg-Penning trap

The another position considered is close to the trap, at -110 cm, just before the alu-minium degrader that slows the 5.3 MeV antiprotons down to 10 keV of kinetic energy.The fate of those particles is to fall into the Malmberg-Penning trap, set a few centime-ters downstream, in the main magnet bore. There are two reasons for it is not possibleto set the squared beam counter close to the trap. The first is that there, the space isprobably not enough for a 300 m device. The second, and the most important, is thatsince such a silicon thickness stops all the antiprotons, it would be necessary to removethe beam counter once the preliminary tests are over. Unfortunately, the area next tothe trap is not easily accessible at all, unlike the ‘sun’.On the contrary, the circular beam counter is much more suitable in this site. Here thebeam profile is completely di!erent. In fact, it is the region where the AEgIS magneticfield starts to reach its maximum value, that is 5 T. This intensity is maintained in


(a) (xg , yg) = (0.0, 0.0) mm;(xr , yr) = (0.0, 0.0) mm

(b) (xg , yg) = (1.0, 1.0) mm;(xr , yr) = (0.7, 0.7) mm

(c) (xg , yg) = ('2.0,'2.0) mm;(xr , yr) = ('1.7,'1.6) mm

(d) (xg , yg) = ('3,'3) mm;(xr , yr) = ('2.8,'2.8) mm

(e) (xg , yg) = (5, 1) mm;(xr , yr) = (3.5, 1.3) mm

Figure 4.10: Detection of the AD beam in the ‘sun’ by ex ATHENA beam counter 55 m thick. Thesame o!sets as the squared beam counter case have been applied, for better confrontation purposes.As usual the upper graphs represents the beam profile, and the lower the beam reconstruction basedon the pad signals. The remodeling of the beam profile is made filling the pads’ shape pixel by pixel,and it is intended to imitate a possible live monitor. Once again, the color scale is not absolute.


Figure 4.11: Section of the beam at di!erent coordinates along the beam axis. The beam counter55 m thick is here set into the ‘sun’, at -173 cm. The aluminium degrader is 161 m thick in thissimulation run, and it is placed at -110 cm. One can observe from one side the consequences of themultiple Coulomb scattering generated by the silicon detector; on the other, the focusing properties ofthe magnetic field are once more evident. At z = '109.9 cm the red circle represents the size of thebeam counter. At z = '106 cm, it is the trap entrance, having a radius of 15 mm.

an almost constant way, downstream of this position, along all the main magnet. Thesituation can be visualized thanks to the graphs in Figure 4.12. The first novelty thatmust be pointed out is that the beam profile is reduced to a small spot, whose diameteris less than 1 mm. This can be very surprising, but it should not be neglected thathere the 5 T magnetic field came into action. Indeed, the first series of three graphshow growing position o!sets, from 3 mm to so much as 2 cm, that is an enormous shiftif compared to the detector size and, above all, to &xy, the standard deviation of thegeneration point. Even in this case – here named (c) – the spot doesn’t quit the centerpad. Of course, such a deviation is totally unlikely for the AD beam parameters, butit robustly prove the e!ectiveness of the magnetic field focusing properties. To deviatethe beam, one is forced to act on other parameters, such as the polar angle of the beamand its divergence. These cases – (d), (e) and (f) – are instructive since one realizesthat the magnetic field gives, at first sight, unexpected positions to the particles goinginto the trap. It can be seen that a positive angular o!set brings the particles in thenegative quandrant of the scheme. Obviously, in the magnetic field charged particlesundergo the Lorentz force, which is perpendicular to their velocity. Their motion hasthen the shape of a spiral. This behaviour proves that even if the beam su!ers of greatangular o!sets, the beam still hits the central pad. This pad, as foretold, representsmore or less the surface in which the interaction with the electron cloud in the trap islikely.


Having such focusing properties, one may wonder why it is then necessary to keep abeam counter in the apparatus. This answer, once more, is to diagnose severe malfunc-tions in the beam, in which case one would observe no signal at all.

4.4.4 Final considerations on the choice of the detector

Taking into account these results, as well as the presence of a magnetic field with strongfocusing properties near the degrader, it is very likely that the practical configurationswill be one of the following:

if the 300 m thick detector is chosen, it would be placed in the ‘sun’. Since it stopsall the incoming antiprotons, from a logistical point of view it would be a morefavourable situation as it would be easily removed. Of course, no measurementsmay take place if no antiproton survives the beam counter. Furthermore theplace near the trap entrance is limited. The squared pad configuration is alsomore appropriate to monitor the beam centering when the particles are not yetfocused;

otherwise, the ex ATHENA beam counter would be placed near the degrader andthe trap. As the 55 m thick detector doesn’t stop the antiprotons, it wouldbe a permanent device in the apparatus. Of course, an automatical mechanismis in place to let the positrons to enter in the trap undisturbed. To have thedetector upstream in the ‘sun’ would mean to deal downstream with multipleCoulomb scattering, and there would be a risk to lose part of the beam before theMalmberg-Penning trap. These issues are evident looking at Figure 4.11. Eachof the pad contains the scatter plot of the coordinates x and y at the entrance ofthe 11 fake detectors. We recall one more time that if the beam counter is in the‘sun’, its coordinate on the beam axis is -173 cm. The coordinate of the detectorsare written in red on each pad. From -150 cm to -120 cm, the multiple Coulombscattering provoked by the beam counter in the vacuum chamber is glaring. Evenif at -109.9 cm, that is, just after the aluminium degrader, the magnetic field hasalready focused the beam, the risk to lose a fraction of the particles in meanwhileis too dangerous.As we have seen from the simulations, in 55 m of silicon 5.3 MeV antiprotonsrelease about 700 keV. So the ex ATHENA detector would take part in the degra-dation of the beam, that would be finalized by an aluminium foil.

The circular beam counter is already available from members of the former ATHENACollaboration. It is then very likely that it will be used in AEgIS, positionned next tothe degrader and the trap.

4.4.5 Study of the aluminium degrader

When the antiprotons arrive in front of the main magnet, they must be slowed downto less than 10 keV to be captured by the Malmberg-Penning trap. This trap has aradius of 15 mm, and if the circular beam counter is placed before the degrader, 100%of the antiprotons have coordinates suitable to enter in the trap. To obtain low energyantiprotons, the simplest solution is to place a foil of some material, such as aluminium,and simply exploit the matter-radiation interaction ruled by the Bethe-Bloch regime.In fact, the processes into play are merely electromagnetic. If the circular detector ischosen, as discussed, it will not be removed even during the data acquisition. In fact,


(a) (xg , yg) = ('3,'3) mm;(xr , yr) = (0, 0) mm

(b) (xg , yg) = (1, 1) cm;(xr , yr) = (0, 0) mm

(c) (xg, yg) = (2, 2) cm;(xr , yr) = (0, 0) mm

(d) ("x, "y) = (25, 10) mrad,divergence is 1.64 mrad (stan-dard)

(e) ("x, "y) = (25, 10) mrad, di-vergence is 2(

(f) ("x, "y) = (25, 10) mrad, di-vergence is 3(

Figure 4.12: Detection of the AD beam near the trap by ex ATHENA beam counter 55 m thick.The pads enlightened in cases (a) and (c) are due to some annihilation products that have releasedenergy outside the center pad. Cases from (d) to (f) shows a the profile of a beam generated with someangular o!set, and a growing divergence.


Figure 4.13: Detail of the AEgIS design showing the positions of the degrading aluminium foils.

it would participate in the degradation of the beam, since antiprotons lose " 700 keVinside it. To find the best aluminium degrader thickness, it is then su"cient to placeone main layer of aluminium (say, of 150 m) and then to have the possibility to addseveral thin foils, if needed. In such a way, it would be possible to move freely in a rangeof 20-30 m with a great sensitivity. The set of foils would be of this kind: one shouldconsist in layers 1, 2, 3, 4, 5 m thick, the other should include thicknesses of 10, 15,20, 25 and 30 m. Of course, layers can be combined. These ‘thin’ degraders will bemounted on two movable systems (sliders) in the ‘sun’ and a trapping e"ciency curvewill be performed by the experiment in the first phase of the data taking. A thickerdegrader (whose value has to be found thanks to this study) will stay in place closeto the trap. See Figure ?? for a precise sight of these locations. Some informationson how analysis has been done is available in Figure 4.14. The study of the optimumthickness of this aluminium layer is very important to the AEgIS apparatus becauseonly a little fraction of antiprotons, say a few percent, is destined to have the rightenergy. A small error of three or four m would be fatal, as it would compromisethe capture of particles, and consequently the formation of antihydrogen in a su"cientquantity. The first weeks of the data acquisition will in fact be dedicated, among otherduties, on the experimental measurement of the fraction of the antiprotons that has theright energy to be trapped.In the following we present the study of the thickness of the aluminium degrader,performed with Geant4, involving the usual Physics lists CHIPS. We do know thatCHIPS includes the standard Physics List of Geant4, EmStandard, so the simulationof electromagnetic processes is guaranteed. The graphs have been built focusing onthe ‘fake detector’ made of vacuum standing 1 mm downstream of the aluminium foil.In particular, the kinetic energy of the incoming particles, that is, those just exitedfrom alumium, has been observed. In the graphs the number of antiprotons having anenergy lesser than 10 keV is shown, normalized to the total number of particles in asingle run, that is 105. On the y-axis is then showed a percentage, while on the x-axisis represented the thickness of the material included in that specific simulation run.The fit is performed using a Gauss distribution. It should be pointed out that thereis no reason in considering the distribution authentically Gaussian, and in fact the '2

value indicates indeed that it is not the case. However, we are not interested in aperfect fit of the simulation values, but only in a rough estimate of the mean value of


Figure 4.14: An overview of the simulation results related to the beam monitoring, and the degradingof the antiprotons. In this run the degrader is 161 m thick. The upper graph is built as a scatterplot that put the x-coordinate of the antiproton at the entrance of each detector, solid or fake, vs.its z-coordinate. Then it allows to visualize the positions of the 11 fake detector, of the silicon beamcounter and of the aluminium degrader. The red rectangle depicts the dimensions and position of theMalmberg-Penning trap. Lower, at left is represented the x'y coordinates of the beam. The red circlerepresents the trap entrance, having a radius 15 mm long. The middle histogram shows the kineticenergy of the antiprotons at trap entrance. The integral of the number of antiprotons having theirkinetic energy lesser than 10 keV is computed here. It is the value that will fill the graphs included inthe next figures. The last graph, at right, shows the radial kinetic energy at trap entrance. It is anestimate of the spread of the beam. In other words, it contains an information on the likelihood to loseantiprotons outside the trap.

this distribution. This mean would represent the optimum value for which a maximumnumber of antiprotons can be captured in the trap. The possibility in the apparatus toadd aluminium foils is limited to steps of 1 m each. In fact, two sets of layers should beavailable in the ‘sun’. The first includes thicknesses of a few microns: 1 m, 2 m, 3 m,4 m and 5 m. The second one would start from 10 m, in steps of 5 m. To quantifythe full width half maximum (FWHM) of the Gaussian is nevertheless important, asit gives an important information on which minimum thickness we should start doingmeasurements. In such a way, up to 30 m from the minimum value can be studiedwith 1 m resolution. Let us consider the simulation presented in Figure 4.15. Themean of the Gaussian distribution is 175.4 m. The FWHM is related to the variance12 of the Gaussian distribution in the following way:

FWHM = 23

2 ln 21 2 2.351. (4.4.8)


Figure 4.15: Number of antiprotons with maximum kinetic energy of 10 keV normalized to the totalnumber of events in the simulation run, versus the thickness of the aluminium foil. CHIPS Physics listis employed.

Thus, having 1 2 2 m, FWHM 2 5 m. Then if these results were trusted, it wouldbe reasonnable to place an aluminium layer having a thickness 2 * FWHM inferior tothe mean, that is 165 m, close to the trap entrance. The study of the p-trap e"ciencycurve adding, as anticipated, the ‘thin’ degraders up to a total of 30 m. Once the bestvalue will be experimentally measured, all the degrading material would be placed closeto the trap entrance, except maybe for 1 or 2 m additional degraders, still in the ‘sun’.Further investigations should be focused on the determination of the statistical uncer-tainties; that is, one should wonder if the simulations response is stable. It should befigured out if activating or deactivating specific physical processes (in particular thosethat come into play at very low energy) a!ect the simulation results. However, sincethe aim of this study is to obtain a safety factor for the minimum layer thickness, errorson this estimated value lose somehow importance.

Chapter 5

Simulation of the annihilationphysics

As far as we know, the CHIPS Physics list is the only generator of events able to re-produce the p-nucleon and p-nucleus annihilation. CHIPS developers claim that it isavailable for all hadrons, in all energy ranges. Is it really true? Cross section measure-ment in the MeV range, that is very low energy range, are quite rare and it is not easyto perform direct comparisons. No experiment has exploited CHIPS in this way before.It is thus necessary to try to validate some basics results given by the Monte Carlo.Doing simulations for the AEgIS experiment turned out to be a strong testing groundfor CHIPS in the low energy range limit. The results will be discussed with the CHIPSdeveloper’s team.Before presenting our considerations, an introduction to CHIPS is given, as well asan overall description on what are its operating principles. Comparisons were madebetween the antiproton range in silicon obtained through simulations, and the oneextracted from a semi-empirical model. Since they do not fit together, we kept on in-vestigating other quantities. We mainly focused on the secondary multiplicities, payinga particular attention to those of the charged pions. Eventually, graphs on the range ofparticles in silicon were obtained, as well as the energy released by them.

5.1 Chiral Invariant Phase Space (CHIPS) Physicslist

5.1.1 Introduction

The CHIPS computer code is a quark-level event generator for the fragmentation ofhadronic systems into hadrons. In contrast to other parton models, CHIPS is non-perturbative and three-dimensional. It is based on the Chiral Invariant Phase Space(ChIPS) model which employs a 3D quark-level SU(3) approach. Thus Chiral InvariantPhase Space refers to the phase space of massless partons, hence only light (u, d, s)quarks can be considered and all phase space integrals are simple.1 The c, b, and t

1A phase space is a space in which all possible states of a system are represented, with each possiblestate of the system corresponding to one unique point in the phase space. For mechanical systems, thephase space usually consists of all possible values of position and momentum variables. In quantum

83

84 Simulation of the annihilation physics

quarks are not implemented in the model directly, while they can be created as a resultof the gluon-gluon or photo-gluon fusion. The basic idea of the model, a hadroniza-tion of quark-partons in nuclear matter (one quark-parton into one outgoing hadron)appeared in 1984, when secondary spectra of hadrons in high energy nuclear reactions(pions, protons, neutrons, deuterons, etc.) were found to be a reflection of the quark-gluon spectra. The model si based on the hypotesis of asymptotic freedom of QCDwhich assumes that chiral symmetry is locally restored inside hadrons. In quantumfield theory, chiral symmetry is a possible symmetry of the Lagrangian under which theleft-handed and right-handed parts of Dirac fields transform independently. The chiralsymmetry transformation can be divided into a component that treats the left-handedand the right-handed parts equally, known as vector symmetry, and a component thatactually treats them di!erently, known as axial symmetry. The local chiral symmetryrestoration lets us to consider quark-partons as massless particles and integrate suchcomplicated processes as the invariant phase space distribution and the quark exchange(quark fusion) mechanisms of hadronization.The main parameter of the CHIPS model is the critical temperature Tc 2 200 MeV.The probability of finding a quark with energy E drops with the energy approximatelyas e!E/T , which is why the heavy flavors of quarks are suppressed in the Chiral Invari-ant Phase Space. The s quarks, which have masses less then the critical temperature(ms ( 100 MeV), have an e!ective suppression factor in the model.The critical temperature Tc defines the number of 3D partons in the hadronic systemwith total energy W . If masses of all partons are zero then the number of partons ncan be found from the equation W 2 = 4T 2

c (n # 1)n.In CHIPS the interactions between hadrons are defined by the Isgur quark-exchangediagrams, and the decay of excited hadronic systems in vacuum is treated as the fusionof quark-antiquark or quark-diquark partons. An important feature of the model is thehomogeneous distribution of asymptotically free quark-partons over the invariant phasespace, as applied to the fragmentation of various types of excited hadronic systems.The CHIPS event generator may be applied to nucleon excitations, hadronic systemsproduced in e+e! and pp annihilation, and high energy nuclear excitations, amongothers. Despite its quark nature, the nonperturbative CHIPS model can also be usedsuccessfully at very low energies. It is valid for photon and hadron projectiles and forhadron and nuclear targets. Exclusive event generation models multiple hadron pro-duction, conserving energy, momentum, and other quantum numbers. This generallyresults in a good description of particle multiplicities, inclusive spectra, and kinematiccorrelations in multihadron fragmentation processes. Thus, it is possible to use theCHIPS event generator in exclusive modeling of hadron cascades in materials.In the CHIPS model, the result of a hadronic or nuclear interaction is the creation of aquasmon which is essentially an intermediate state of excited hadronic matter. Whenthe interaction occurs in vacuum the quasmon can dissipate energy by radiating parti-cles according to the quark fusion mechanism. When the interaction occurs in nuclearmatter, the energy dissipation of a quasmon can be the result of quark exchange withsurrounding nucleons or clusters of nucleons, in addition to the vacuum quark fusionmechanism.In this sense the CHIPS model can be a successful competitor of the cascade models,because it does not break the projectile, instead it captures it, creating a quasmon, and

field theory, a phase space integral has usually the following form:R d3p

(2")32Ep=

R d4p

(2")4(2()4(p2 '

m2) )(p0).

5.1 Chiral Invariant Phase Space (CHIPS) Physics list 85

then decays the quasmon in nuclear matter.

5.1.2 Fundamental concepts

The CHIPS model is an attempt to use a set of simple rules which govern microscopicquark-level behavior to model macroscopic hadronic systems with a large number ofdegrees of freedom. The invariant phase space distribution as a paradigm of thermal-ized chaos is applied to quarks, and simple kinematic mechanisms are used to modelthe hadronization of quarks into hadrons. Along with relativistic kinematics and theconservation of quantum numbers, the following concepts are used:

Quasmon: in the CHIPS model, a quasmon is any excited hadronic system; it canbe viewed as a continuous spectrum of a generalized hadron. At the constituentlevel, a quasmon may be thought of as a bubble of quark-parton plasma in whichthe quarks are massless and the quark-partons in the quasmon are homogeneouslydistributed over the invariant phase space. The traditional hadron is a particledefined by quantum numbers and a fixed mass or a mass with a width. Thequark content of the hadron is a secondary concept constrained by the quantumnumbers. The quasmon, however, is defined by its quark content and its mass, andthe concept of a well defined particle with quantum numbers (a discrete spectrum)is of secondary importance. A given quasmon hadronic state with fixed mass andquark content can be considered as a superposition of traditional hadrons, with thequark content of the superimposed hadrons being the same as the quark contentof the quasmon.

Quark fusion: the quark fusion hypothesis determines the rules of final statehadron production, with energy spectra reflecting the momentum distribution ofthe quarks in the system. Fusion occurs when a quark-parton in a quasmon joinswith another quark-parton from the same quasmon and forms a new white hadron,which can be radiated. If a neighboring nucleon (or the nuclear cluster) is present,quark-partons may also be exchanged between the quasmon and the neighboringnucleon (cluster). The kinematic condition applied to these mechanisms is thatthe resulting hadrons are produced on their mass shells. The model assumesthat the u, d and s quarks are e!ectively massless, which allows the integrals ofthe hadronization process to be done easily and the modeling decay algorithm tobe accelerated. The quark mass is taken into account indirectly in the masses ofoutgoing hadrons. The type of the outgoing hadron is selected using combinatoricand kinematic factors consistent with conservation laws. In the present version ofCHIPS all mesons with three-digit PDG Monte Carlo codes up to spin 4, and allbaryons with four-digit PDG codes up to spin 7/2 are implemented. The generalform of PDG codes is a 7-digit number: ±nnrnLnq1

nq2nq3

nJ . In composite quarksystems (mainly mesons and baryons) nq1!3

are quark numbers used to specifythe quark content, while the rightmost digit nJ = 2J + 1 gives the system’s spin.

Critical temperature: the only non-kinematic concept of the model is thehypothesis of the critical temperature of the quasmon. This has a 40-year historyand is based on the experimental observation of regularities in the inclusive spectraof hadrons produced in di!erent reactions at high energies. Qualitatively, thehypothesis of a critical temperature assumes that the quark-gluon hadronic system(quasmon) cannot be heated above a certain temperature. Adding more energy


to the hadronic system increases only the number of constituent quark-partonswhile the temperature remains constant. The critical temperature is the principalparameter of the model and is used to calculate the number of quark-partons ina quasmon. In an infinite thermalized system, for example, the mean energy ofpartons is 2Tc per particle, the same as for the dark body radiation.

5.1.3 Antiproton-nuclear simulation annihilation at rest

In the CHIPS algorithm antiprotons annihilate with the peripheral nucleons which arerelatively far from the dense nuclear surface [66]. The annihilation step is simulated bythe CHIPS nucleon-antinucleon annihilation algorithm. The secondary mesons interactwith the residual nucleus and create quasmons. The solid angle in which the secondarymesons are absorbed is a parameter. The created quasmons must be hadronized insidethe same nucleus. The parallel hadronization of a few quasmons in the same nuclearenvironment is a challenging task because each quasmon can be below the mass shell.From the reference [66], it has been found out that the measured low energy protonsin the antiproton-carbon annihilation look to be suppressed by factor two in respectto simulation. It cannot be explained by the Coulomb barrier as its value is smallfor carbon. The experimental spectra of pions reach 1 GeV. The spectra of negativepions are always lower than the spectra of positive pions, because the negative pionsare attracted by the nucleus and easily absorbed, while the positive pions are pushedaway by positive nuclei. This e!ect is reproducted by simulations of the same reference.Branching ratios of various measured [57] exclusive channels as shown in Figure 5.1.

In the next sections, confrontations beween some significant results obtained withCHIPS and experimental data will be performed. In particular, the multiplicity ofsecondary particles, that is the products of annihilation, will be checked, as well as therange of particles in silicon and the energy they release in it.The CHIPS event generator simulates yields of charged nuclear fragments, but thecorresponding data have not been found by the developers. In any case, applyingCHIPS physics list to the need of the AEgIS experiment is an excellent occasion to testit under well-defined conditions. In particular CHIPS here is forced to operate at verylow energies, and this may lead to great help for further developments.

5.2 Secondary particles multiplicity

Table 5.1 summarizes the total measured yields of several nuclear fragments for 1 pannihilation at rest on nuclei with di!erent A [58, 67]. When the data were taken, inorder to be identified, the particle had to cross three detectors, and had to stop in oneof the last two of them. I.e. it had to have an energy Emax

1 (m, z) < E - Emax3 (m, z),

where Emaxi (m, z) is the maximum energy a particle of mass m and charge z could have

and still stop in detector i. This is why an energy range of observation is reported inTable 5.1. Since the data in our possession don’t cover all the nuclei of the element tablebut only a few of them, we may only guess the likely values for the silicon case. Thisparticular one is important for us, since in this portion of the simulation silicon playsan important role. Thus we assume 28Si to be in the middle of the 12C and 40Ca data.For the sake of clarity, other simulations have been performed with a beam countermade up of the following elements: hydrogen (to simulate pp annihilation), 28Si, 12Cand 40Ca. The latter two are included in the literature table. The thickness of thedetector has been modified from time to time to guarantee an interaction probability

5.2 Secondary particles multiplicity 87

Figure 5.1: Branching probabilities for di!erent channels with three-particle final states in proton-antiproton annihilation at rest. The points are experimental data [57] and the histogram is from theCHIPS Monte Carlo, from Geant4.9.4 Physics Reference Guide.

of 1; in the hydrogen case, its density has even been strongly increased for this verypurpose. Here are the results, generated with CHIPS physics list and normalized onthe number of events simulated in a run (105). To be closer to the literature results,their same kinetic energy range has been applied to the following particle species. Theyare reported in Table 5.1.

Products Energy range (MeV) 12C 28Si 40Cap 6-18 0.6260 1.6056 1.63962H 8-24 0.1054 0.1549 0.07053H 11-29 0.0155 0.0295 0.03223He 36-70 0.0005 0.0014 0.00584He 36-70 0.0023 0.0124 0.0082

The results for p-C annihilation have been compared in Figure 5.2. The main di!erencesconsist in:

the number of protons emitted;

the particles having A ! 3: for lighter nuclei, we would expect more tritons thanalphas.

These results have been discussed with CHIPS developers, and they may result from aparticular non-optimized CHIPS parameter: this parameter sets the ratio between the


Particle Approximate (MeV) 12C 40Cap 6-18 0.233(2)(18) 0.742(3)(38)2H 8-24 0.093(1)(7) 0.181(2)(9)3H 11-29 0.045(1)(3) 0.057(1)(3)3He 36-70 0.0172(4)(13) 0.0222(2)(12)4He 36-70 0.0114(3)(9) 0.0218(5)(11)6He 39-89 0.00025(5)(2) 0.00045(7)(3)8He 44-90 0.000041(18)(3) 0.00014(4)(1)Li 61-96 0.00017(4)(2) 0.00075(9)(4)

Table 5.1: Total measurement yields of p, 2H, 3H, 3He, 6He, 8He and Li per p emitted after an-nihilation at rest on nuclei with A = 12 ' 63 [58, 67]. The former number within parenthesis is thestatistical error and the latter the systematic one. Isotopically enriched targets (%): 12C (98.9), 40Ca(96.9), 63Cu (! 98).

hpartEntries 5Mean 1.196RMS 0.4898

a.u.0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

hpartEntries 5Mean 1.196RMS 0.4898

p d t 3He α

Secondaries pbar-C (CHIPS) hpart-litEntries 5Mean 1.701RMS 1.018

a.u.0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

hpart-litEntries 5Mean 1.701RMS 1.018

p d t 3He α

Secondaries pbar-C (literature)

Figure 5.2: Left graph: secondary particles per event generated by Geant4 using CHIPS, comingfrom the p-C annihilation. Right: same as left, but the values are taken from [67]. In both cases,particles are selected according to the ranges reported in Table 5.1. The main di!erence between themconsist in the number of protons emitted.

annihilations that occur on the surface on the nucleus, and those in which the antipro-tons enter in the nucleus core.We also studied the (± multiplicity. We have observed that the sum of charged pions(data on neutral pions are very poor) are quite close to the values present in literature.The quantity from simulations is in fact ! 2.8 on average. It should be pointed outthat one expects more (! than (+, as already discussed in Section 3.1.4, and CHIPSgives the opposite. Once more, these values have been given to CHIPS developers. Thefollowing table summarizes the literature pion mean multiplicities emitted from annihi-lations on di!erent nuclei ([58] and references therein), and reports the sum of chargedpions multiplicity for the annichilation on former targets, and in addition data from Hand 4He targets. The density of the latter gases has been strongly incremented, in away that annihilations occur at rest even in these cases.

5.3 p stopping 89

1H 4He 12C 28Si 40Can"! 1.53(2) 1.77(6) 1.57(2) - -n"+ 1.53(2) 1.33(3) 1.22(1) - -n"± 3.06(3) 2.98(9) 2.79(2) - -n"! (CHIPS) 1.74 1.10 0.58 0.67 0.74n"+ (CHIPS) 1.74 1.58 1.71 1.69 1.68n"± (CHIPS) 3.48 2.68 2.29 2.36 1.42

Some squares are empty because we could not find any experimental data. Once more,the results presented here were sent to the CHIPS team, at CERN.

5.3 p stopping

Reminding section 3.2, calculated range curves for protons in aluminium are known.We consider protons instead of antiprotons, since the mass is the same. In additionwe take into account a mean kinetic energy of (5.3 - 0.7) MeV = 4.6 MeV, because weare in a configuration in which the antiprotons have already interacted with the beamcounter 55 m thick.To find the correspondent range starting from such an energie value is not simple, merelylooking at Figure 3.8. Instead, since we know that a linear form on the log-log scaleexists and it is R = kT 1.75, we may deduce the proportionaly constant by looking for amore trasparent energy value: for instance 5 MeV, that is also very close to 4.6 MeV. At5 MeV, the proton curve intersects a range in aluminium of 5 · 10!2 g/cm2. Extractingk, and knowing the density of alluminium (( = 2.70 g/cm3) it can be computed thatthe range of a proton or antiproton, neglecting the polarization of the medium, shouldbe 160 m.In Geant4, these expectations are not met. Figure 5.3 represents the range curve ofantiprotons in aluminium, according to CHIPS. As can be seen, for small thicknesses,all (or practically all) the particles manage to pass through. As the range is approachedthis ration drops. The curve slopes down over a certain spread of thicknesses. Thisresult is due to the fact that the energy loss is not in fact continuous, but statistical innature. The mean value of the distribution is known as the mean range, and correspondsto the midpoint of the descending slope. This is the thickness at which roughly half theparticles are absorbed. More commonly, however, what is desidred is the thickness atwhich all the particles are absorbed, in which case the point at which the curve dropsto the background level should be taken. This point is usually found by taking thetangent to the curve at the midpoint and extrapolating to the zero-level. This value isknown as the extrapolated or practical range. Four points of the slopping down curvehave been considered and fit through a linear relation. The equation obtained is:

y = #0.169x + 30.12. (5.3.1)

The intercept between this straight line and the x axis is x ( 178 m. So accordingto the quoted semi-empirical model, CHIPS lacks of reliability. These results will bediscussed and deepend with the help of the CHIPS team.

5.4 Deposition length and energy released

In the following, especially in light of future studies for the g-sensitive detector, we willpresent the results of the antiproton beam interaction with the square beam counter.


Figure 5.3: Number of antiprotons that overlook the trap entrance, normalized on the total numberof AD beam antiproton bunch, vs. the degrader thickness. A linear fit has been done in the pointsfrom 174 m to 177 , in order to find the intercept with the x-axis, representing the extrapolatedrange in the material. The simulations have been done using CHIPS.

This is the only case in which this kind of analysis makes sense, since conversely tothe ex ATHENA beam counter, it has a su"cient thickness to annihilate almost allthe incoming antiparticles. The reaction gives therefore su"cient informations to beanalized. In particular, the energy released by the primary and secondary particles willbe shown, as well as the length (called deposition length) along which that energy wasreleased inside the material.

The upper-left graph in Figure 5.4 shows the total energy released by all particlesin one event. A clear threshold can be seen at the same value of antiproton’s kineticenergy, that is 5.3 MeV; the first peak is not the trivial case in which the antiprotoncompletely stops in the material, since he we deal with antimatter. Indeed, this peakrepresents all the events in which the antiproton annihilate and produces solely neutralparticles. Of course, particles having zero charge do not interact in silicon, so theyrelease no energy. This peak is so high (about 1300 events) because events havingdi!erent origin accumulate there. A few values are still present under this threshold,though they are hardly visible due to the linear scale of the y axis. They representsevents in which antiprotons undergo simple Coulomb scattering. Beyond the 5.3 MeVthreshold takes shape a structure, whose elements will be delineated in the followinggraphs.Focusing on the range of energies involved, we can compare our results with a previ-

ous measurement performed by McGaughey and collaborators [68]. The situation wassomehow di!erent since the measurements were performed with a beam of antiprotonwith a momentum of 608 MeV/c (about 170 MeV of kinetic energy) on a silicon detector1 mm thick. Results can be seen in Figure 5.5. Anyhow the order of magnitude of theenergies released is the same.The upper-right graph shows what is the percentage of the total energy released inan event by the di!erent species of particles involved. Correspondingly to the alreadydiscussed rich secondary particles composition, it reveals once more that alpha particles

5.4 Deposition length and energy released 91

Figure 5.4: General informations on p-Si annihilation. The detector is 300 m thick. See text forfurther informations.

play a role too much important in these simulations, since we would not expect such acontribution (up to 40%) from heavy particles. Every column is devoted to one or moreparticle species, in order: antiproton (the mother), charged pions, protons, deuterons,tritons, alphas, heavy nuclear fragments (with A ¿ 4), and other particles that don’t fitin the former categories, such as photons, electrons, positrons, kaons and some 3He.The last two graphs hold a great importance for our aims. What message do theybring? The first one shows the number of pad that are hit in one event: in most of thecases, they are only one or two. These informations go in the right direction. In fact,since we would like to reconstruct the beam profile, the main hope is that the totalenergy released during an event is as much as possible localized in the pad hit by theantiproton. A signal that would spread on the adjacent pads would cause much moretroubles in the baricentering of the beam. So, we wanted to study how the energy re-leased is distributed among the hit pads and the adjacent ones. This is why the secondgraph adds an important information: it states that even when some energy is releasedin pads that are not the original one, it can be quantified as only 14% on average. Wecan then conclude that most of the energy is released in the pad hit by the antiproton.Of course, each of the annihilation products presented above releases energy in thesilicon detector. There are not only secondary particles, because if one thinks at thepion generation mechanisms (correlated and uncorrelated), and at the electrons andgammas showers, many levels of generation may be included. From now on, the follow-ing histograms are filled paying attention not anymore to the single pads; it is not the


Figure 5.5: Relative number of annihilations vs energy deposition in the silicon detector for theproduction of protons from the annihilation of 608 MeV/c antiprotons, from reference [68]. The solidhistogram represents the experimental data.

energy collected by pads that is depicted. Instead, informations are now based on thetracks, in other words on what happens to a single particle once it is generated until itis no longer tracked by the simulation tool. Therefore the detector design is no moreinfluential. This study is necessary to understand what kind of signal one should expectfrom an annihilation reaction in a silicon detector for future developments.

5.4.1 Deposition length

Using the same approach, one may analize the length in the material along which theenergy has been released by each track. We call this quantity deposition length. Let usnow start with some analysis of the deposition length and consider Figure 5.6.

1. The first, upper-left graph, concerns the antiprotons. It can be easily deducedthat CHIPS stops the antiprotons at (256±8) m in the material. Such a peakedresult is due to the monochromatic generation of the p by the Monte Carlo.

2. The upper-right histogram is devoted to charged pions. Neutral pions (andneutrons as well) are of course excluded from this analysis since particles withzero electric charge have no interest in the context of a semiconductor detector.A structure made of two distributions are evident.

3. Here too the two-peak behaviour is visible. Protons and deuterons lengthdistributions are shown at the bottom of the Figure.

If we look now Figure 5.7 one sees that the second peak has disappeared in the threehistograms. The reason is simple. As the antiproton annihilate at 250 m in thematerial, the distance forward to the end of the detector is 300 m - 250 m = 50 m.The two distributions clearly visible in the first case are two populations of particles.The first has a threshold at 50 m, and contains all the pions/protons/etc. goingforward: in fact, 50 m is the minimum distance they have to cross in order to exit


Figure 5.6: Deposition length for primary particles (antiprotons), charged pions, protons anddeuterons.

from the material. Since the distribution is isotropic, the tail is very long. The seconddistribution, having its threshold exactly in the annihilation point of the antiproton,must cross at least 250 m before exiting (see Figure 5.8).

The next graphs (Figure 5.9) represents the same quantity in the cases of tritons,alphas, heavy nuclear fragments (with A > 4), and eventually other particles. Wediscuss these results in the following order.

1. Triton’s track length. Approximally the same structure than former histogramsis visible: there is a peak having its threshold at " 50 m, and another one at" 250 m. The reason is always the same, related to the annihilation point of theantiproton.

2. Alpha’s track length. A shape including a small peak at " 50 m and a largepopulation before it is showed. There is no peak at " 250 m because the alphas,having such a mass, lose all their energy and stop in the material long before.

3. Heavy nuclear fragments’ track length. The distribution of these heavyparticles is peculiar. Nevertheless, it can be seen that most of them stop in arange included between 1 and 2 m. Thus, the risk that some of this energywould be released in adjacent pads is negligible.

4. Other particles’ track length. Once again, there are two peaks, one at " 250m, the other at " 50 m.


Figure 5.7: Same as Figure 5.6, for forward tracks (pz > 0).

In the following, we show for forward tracks (pz > 0) (Figure 5.10). As expected, thesecond peak of tritons and other particles has disappeared. Heavy’s histogram has nodi!erence, since all these fragments stop soon in the material. In the alphas case, thepopulation of the histogram has now lowered, and the 50 m peak is more evident.If the particles that exit the material are now excluded, this peak totally disappears(Figure 5.11). The same is certainly true for tritons.

5.4.2 Energy released

Here are presented the histograms related to the energy released in silicon by primaryand secondary particles. Let us start from Figure 5.12.

1. Antiprotons. The energy released by antiprotons in annihilation is 5.3 MeV.The global mass/energy available in the process is not represented here, becauseit is part of the histograms devoted to the secondary particle species involved.

2. Pions. Two Landau-like distributions are clearly recognizable: the first is relatedto the pions going forward, the second to those going backward.

3. The same can be stated for protons and deuterons, even if the second distribu-tion is hidden under the tail of the first one.

It should be pointed out that the distributions are not exactly Landau distributions (orconvolutions of Landau and Gauss distributions as well). In fact, there are two aspects


Figure 5.8: Generic annihilation event. Particle going forward (green) must cross not less than 50m, while those going backward (blue) cross more than 250 m.

that complicate the histograms: the secondary particles don’t constitute a monochro-matic beam, and furthermore, the layer crossed is never the same, as the emission ofparticles is isotropic.Excluding as usual the particles going backward, one obtains the histograms presentedin Figure 5.13, where each of them reports one single Landau-like distribution.

In Figure 5.14 are reported the energy released by tritons, alphas, heavy nuclearfragments and other particles.

1. Triton histogram is made by two distributions,

2. Alpha histogram has a complicate structure.

3. Heavy histogram is made by several distributions, since it groups many particlespecies.

4. Other histogram reports one main distribution populated by electrons and positrons.The sharp peak at " 3 keV is the energy released in silicon by photons, that arethe decay products of (0.

We can once more exclude the particles going backward (Figure 5.15), and those thatdon’t stop in the material (Figure 5.16). At the end, the alpha’s graph is much moresmooth. Summarizing, the entirety of informations extracted from energy releasesand deposition lengths will be useful for the design of the final detector. For instance,the strip dimensions depend from the ability to reconstruct the baricenter. The latterdepends on how much energy is released and in how many strips does it distributes. Toknow which species of particle release energy ((, p, rather than the recoil nucleus) isalso a valuable information.


Figure 5.9: Deposition length for tritons, alphas, heavy nuclear fragments (A > 4) and other particles.


Figure 5.10: Same as Figure 5.9, for forward particles (pz > 0)

Figure 5.11: Deposition length for tritons, alphas, heavy nuclear fragments (A > 4) and otherparticles. Particles going backwards, that is with pz < 0, and those who stop in the material, havebeen excluded.


Figure 5.12: Energy released by antiprotons, pions, protons and deuterons.

Figure 5.13: Same as Figure 5.12, for forward (pz > 0) particles.


Figure 5.14: Energy released by tritons, alphas, heavy nuclear fragments and other particles.

Figure 5.15: Same as Figure 5.14, for forward particles (pz > 0).


Figure 5.16: Same as Figure 5.15, but particles that don’t stop in the material are excluded.

Chapter 6

Conclusions

The AEgIS experiment will begin data acquisition in the autumn of 2011, for a fewweeks. The topics of this thesis have been designed in order to help achieving thepreparation of the experiment setup for this upcoming first stage. Decisions and en-hancements on the Monte Carlo simulations have been done following the feedbacks ofthe Collaboration. Thus, our work has been mainly focused on the aspects that willallow to perform the first part of the experiment, cited above. That is:

the design and location of the silicon beam counter;

the study of the aluminium degrader, to be placed next to the p trap;

the validation of CHIPS Physics list, compared to experimental results.

At this stage, what do we have made certain?

Several configurations for the beam counter have been analized. Two locationsare possible: the first is in the vacuum chamber, a place easily accessible, wheredevices can be reached and removed if necessary; the second is close to the entranceof the p trap. Two designs for the detector have also been considered: a squaredbeam counter 300 m thick, in which case its main parameters, such as its side andthe number of pads on it, have been determined; the other design is circular, 55 mthick and has one central pad, surrounded by other four pads, arranged as flower’spetals. Eventually, two configurations have been found to be possible. The first isto place the squared detector, 2.1* 2.1 cm2 with a 4* 4 pads configuration, intothe vacuum chamber, where the magnetic field is very low. In fact, it would bevery well adapted to reconstruct the baricenter of the beam when its transverseside has not been reduced yet by the magnetic field. Furthermore, since it stopsevery antiproton, it should be easily removable in view of the data acquisitionstage. The second configuration implies to put the circular beam counter, havinga diameter of 15 mm, next to the degrader and to the entrance of the p trap.Being smaller than the squared one, this detector can fit in an easier way in thearea close to the trap’s entrance. Furthermore, here the beam has been focusedby the magnetic field, and its spot is only " 1 mm large, so that it doesn’t need tobe detected by big devices. On the other side, to put it in the ‘sun’ would provokemultiple Coulomb scattering and would mean to deal with the risk to lose part ofthe beam, in the segment between the beam counter and the degrader.

101

102 Conclusions

Since the circular beam counter is already available, it is very likely that the latteroption will be adopted.

For the determination of the aluminium layer thickness, the simulations havebeen done considering the most probable configuration for the beam counter.The circular detector has been placed at -111 cm, 1 cm upstream of the degrader.A transmission curve has been obtained counting the number of antiprotons thatcross the layer and have a final kinetic energy lower than 10 keV. The thicknessthat optimizes this number has been obtained through a Gaussian fit. Di!erentresults have been obtained from di!erent versions of Geant4. These surfaceduncertainties reflect the uncertainties of the experimental datas at " 5 MeV ofkinetic energy, which are very few. The most conservative choice is to consider thefit that gives the minimum value. We suggest therefore to place an alluminiumdegrader of 160 m next to the trap, and to provide the possibility to add twodi!erent sets of thin layers, depending on the requirements. One set should consistin layers of a few microns: 1 m, 2 m, 3 m, 4 m and 5 m. The thicknessof the layers from the other set should start from 10 m in steps of 5 m (10,15, 20, 25). In such a way, up to 30 m can be added to the main degrader.This additional thin degraders must be placed in the ‘sun’, during preliminarytests. Once the optimal value maximizing the number of cold antiprotons will beexperimentally found, the ultimate set of layers will be moved close to the trap,in order to avoiding multiple Coulomb scattering.

Having seen that Geant4 results using CHIPS or the standard Physics list Em-Standard may give results departing from semi-empirical models, we have decidedto keep on investigating other quantities. It is likely that simulation of antipro-tons may need further developments at very low energies (that is, at the order of afew MeV). To highlight where should the possible improvements be directed, theestimate of parameters related to the annihilation products has been performed.Some issues emerged in the confrontation with secondary multiplicities, in thereaction p-nucleus. Despite the pions multiplicity is approximately similar to theexperimental ones, way too much nuclear fragments are produced in the reaction.It looks like that the antiproton disintegrates the nucleus, instead of reacting on itssurface with a single nucleon. Additionnaly, the range of primary and secondaryparticles in the material have been found, as well as the energy released by eachparticle species. Contacts with the CHIPS developers have been established, andfurther investigation are necessary with their help.

Summarizing, a whole simulation focused on the AEgIS beam line has been imple-mented. This program is intended to be extended to the portion of the apparatusplaced in the 1 T magnet, and furthermore to the g-sensitive final silicon detector. Thephysics validation part of the thesis is meant to be a preliminary study to optimize itsmain parameters, such as the strips width, the pitch, etc. Indeed, these parametersdepend on information like the length covered by secondary particles, and the energythey release in silicon.

Appendix A

The Virtual Monte Carlo

A.1 Geant3 and Geant4

GEANT is the name of a series of simulation software designed to describe the passageof elementary particles through matter, using Monte Carlo methods. The name is anacronym formed from ‘GEometry ANd Tracking’. Originally developed at CERN forhigh energy physics experiments, today GEANT has uses in many other fields.The first version of GEANT dates back to 1974. Versions of GEANT through 3.21 werewritten in FORTRAN and eventually maintained as part of CERN Program Library.Since about 2000, the last FORTRAN release has been essentially in stasis and receivesonly occasional bug fixes. GEANT3 is, however, still in use by some experiments. Mostof GEANT3 is available under the GNU General Public License, with the exception ofsome hadronic interaction code contributed by the FLUKA collaboration.Geant4 was developed by the RD44 collaboration in 1994-1998: it is the successor ofthe GEANT series of software toolkits developed by CERN, and the first to use Objectoriented programming (in C++). Its development, maintenance and user support aretaken care by the international Geant4 Collaboration.Geant4 includes facilities for handling geometry, tracking, detector response, run man-agement, visualization and user interface. For many physics simulations, this meansless time need be spent on the low level details, and researchers can start immediatelyon the more important aspects of the simulation. Following is a summary of each ofthe facilities listed above:

Geometry is a description of the physical layout of the experiment, includingdetectors, absorbers, etc., and considering how this layout will a!ect the path ofparticles in the experiment.

Tracking is simulating the passage of a particle through matter. This involvesconsidering possible interactions and decay processes.

Detector response is recording when a particle passes through the volume of adetector, and approximating how a real detector would respond.

Run management is recording the details of each run (a set of events), as well assetting up the experiment in di!erent configurations between runs.

Geant4 o!ers a number of options for visualization, including OpenGL, and afamiliar user interface, based on Tcsh.

103

104 The Virtual Monte Carlo

A.2 Introduction to VMC

The Virtual Monte Carlo (VMC) [69] provides a simulation framework which is inde-pendent of any concrete Monte Carlo (MC) and is based on the ROOT system [70].Geant3, Geant4 and Fluka have already been integrated.1

The main advantage with VMC is that the user can run the same simulation pro-gram with three di!erent transport MCs. The obvious consequence is that the di!erentmodels can be compared and better understood. VMC also facilitates the use of lessuser-friendly tools such as Geant3 and Fluka and – being inspired from Geant3 – it isalso suitable for users starting with existing Geant3 applications.The integration of the ROOT geometrical modeller gives the user a useful means forgeometry definition, browsing, visualization and also verification directly within thescope of the VMC.The concept of VMC has been gradually developed by the ALICE Software project[71]. From the beginning, the ALICE collaboration has adopted a strategy for thedevelopment of the simulation framework that would allow a smooth transition fromthe currently used transport code, Geant3, to new ones Geant4 and Fluka. Instead ofmaintaining the Geant3 based code written in FORTRAN and developing in parallela new framework, based on a new simulation program, the user code was graduallymigrated from FORTRAN to C++ and a general C++ interface to a transport MCwas developed.The VMC development went through the following phases:

1. The C++ class, TGeant3, providing access to Geant3 data structures (commonblocks) and functions was introduced. This provided a starting point for a fullmigration of the user code from FORTRAN to C++.

2. The abstract C++ class, AliMC, was defined as a generalization of TGeant3.This gave the initial step for the development of the Geant4 interface and theexplicit Geant3 dependencies in the user code were also taken away. However,the implementations of the AliMC interface for both Geant3 and Geant4 weredependent on the ALICE software.

3. The interfaces to the user Monte Carlo application were introduced. The depen-dence of the implementations of the AliMC interface on the ALICE software couldthen be removed and the VMC was also made available to non-ALICE users.

A.3 The VMC concept

With the VMC concept the user Monte Carlo application can be defined independentlyof a specific transport code (see Figure A.1). It can then be run with all supportedMonte Carlos, without changing the user cose, i.e., the geometry definition, the detec-tor response simulation and input or output formats. The selection of a concrete MonteCarlo (Geant3 or Geant4) is made dynamically at run time.

1Actually Fluka is no longer available in the VMC framework since March 2010. A controversial mailsent recently to users by FLUKA Scientific Committee reports: ‘[. . .] We regretfully have to announcethat the use of the FLUKA-VMC interface is no longer permitted or supported, with the only exceptionof the Alice collaboration. Whichever further use of that interface will be considered an infringementto the FLUKA license.’

A.4 Design 105

Figure A.1: The VMC concept.

The VMC is base on the ROOT system, which is used mainly for scripting and dynam-ical loading of libraries. Once the VMC application has been defined, simulations canbe run interactively from the ROOT UI or using ROOT macros.

A.4 Design

In order to completely decouple the user code from the concrete Monte Carlo, theinterfaces to both the Monte Carlo itself and to the user application code have beenintroduced as shown in Figure A.2. In the following, all interfaces will be discussed indetail.

Virtual MC

The Virtual MC interface (class TVirtualMC) was the first interface written and it isthe most robust one. It has been defined as the generalization of Geant3 functions forthe definitions of simulation tasks and it provides:

methods for building and accessing geometry,

methods for building and accessing materials,

methods for setting physics,

methods for accessing transported particle properties during stepping,

methods for run control.

The implementations of the Virtual MC for concrete transport programs are part ofthe VMC distribution and are provided for the user. At the present time, the Geant3VMC and the Geant4 VMC are in distribution.


Figure A.2: The VMC design.

Virtual MC Application

The Virtual MC Application interface (class TVirtualMCApplication) is the interfaceto a user application code. It defines user actions at each stage of a simulation run:Construct geometry, Init Geometry, Generate Primaries, Begin Event, Begin Primary,Pre Track, Stepping, Post Track, Finish primary and Finish event. The implementationof the Virtual MC Application completely defines the user application and has to beprovided by the user.

Virtual MC Stack

The Virtual MC Stack interface (class TVirtualMCStack) defines the interface to a userdefined particle stack. Users can choose one of the concrete stack classes provided inthe VMC examples or can implement their own stack class.

Virtual MC Decayer

The last interface in the VMC, the Virtual MC Decayer (class TVirtualMCDecayer),defines the interface to the external decayer. The implementation of this interface by auser is optional.

A.5 Use of VMC 107

A.5 Use of VMC

The user VMC application code is written by implementing the MC Application class.In the vase of very simple applications the user can write everything in the one class.In more complex cases it can be convenient to define the user application class as acomposition of more action classes.In this section, three examples of our user code based on the VMC will be given.

Geometry construction

The geometry definition is implemented in the detector construction class. It hasevolved in two phases [72]:

In the first implementation, the VMC provided Geant3-like functions for buildinggeometry. This facilitated the move to VMC for Geant3 users, however thisinterface is limited to the features available in Geant3.

A new approach came with the introduction in ROOT of its own geometricalmodeller, TGeo, independent from existing simulation tools, which provides IO,visualization and verification tools. The old way of defining geometry via VMCfunctions is deprecated and new VMC users are encouraged to start from TGeo.

In the following, an example of a geometry definition, taken from our simulation pro-gram, is given. The geometry is based using directly the ROOT geometrical modellerby calls to the TGeoManager class.

void MyMCApplication::ConstructGeometry(){// Create tracker box volumefsilBCX = 2.1;fsilBCY = 2.1;fsilBCZ = 0.03;Double_t silSize[3], silPos[3];silSize[0] = fsilBCX/2.;silSize[1] = fsilBCY/2.;silSize[2] = fsilBCZ/2.;TGeoVolume *silBC = gGeoManager->Volume("silBC", "BOX", fmedSi,

silSize, 3);

// Place tracker box volumefBCposZ = -173.;silPos[0] = 0.;silPos[1] = 0.;silPos[2] = fBCposZ;

TGeoRotation *silRot = new TGeoRotation("silRot", 0., 0., 0.);TGeoCombiTrans *silCombi = new TGeoCombiTrans(silPos[0], silPos[1],

silPos[2], silRot);top->AddNode(silBC, 1, silCombi);

}


In the first block, a volume named ‘silBC’ of the shape box with side of 2.1 cm and300 m thick, is created and associated with a material defined by the tracking mediumidentifier ‘fmedSi’. In the second block, this volume is placed at the position (-173 cm, 0cm, 0 cm) in the model volume named ‘top’, with the desired rototranslation containedin the matrix called ‘silCombi’.It is possible as well to build geometry with Geant3 style. The VMC uses the Geant3system of default physical units.

Primary particles

In the next example, always taken from our source, we show how to define primaryparticles. This is done by calls to the Virtual MC Stack interface. The particle type(antiproton, electron, . . . ) is defined using the PDG encoding and particle static prop-erties (mass, charge, · · · ) are taken from the particle database in ROOT (representedby the TDatabasePDG class).

void MyMCApplication::GeneratePrimaries(){// Define particle properties:// PDG encoding: pdg// position: vx, vy, vz, t// momentum: px, py, pz, e// ...

// Add particle to MC StackfStack->PushTrack(toBeDone, -1, pdg, px, py, pz, e, vx, vy, vz,

tof, polx, poly, polz, kPPrimary, ntr, 1., 0);}

Detector response

Eventually, our example of a user stepping function is shown. This function is called byMC at each step. In this example the properties of the particle transported are obtainedvia calls to the Virtual MC interface and then saved in the user own hits objects. Forlarge detectors it is reccomended to delegate this function to stepping functions definedin subdetector classes.

void MyMCApplication::Stepping(){// Get current volume IDInt_t copyNo;Int_t id = gMC->CurrentVolID(copyNo);

// Check if step is performed in the sensitive volumeif(id != fSensitiveVolumeID) return false;

// Get track positionDouble_t x,y,z;gMC->TrackPosition(x,y,z);

// Get energy deposit

A.6 Geant3 VMC 109

Double_t edep = gMC->Edep();

// Create user hitmySD->AddHit(x,y,z,edep);

}

A.6 Geant3 VMC

The Geant3 program was written to describe the passage of elementary particles throughmatter. Originally designed for high energy physics experiments, it has also found ap-plications outside this domain in the area of medical and biological sciences, radiopro-tection and astronautics. The first version was released in 1974 and the system wasdeveloped with some continuity over 20 years till the las release 3.21 in 1994. It hasbecome a popular and widely used tool in the HEP community.The Geant3 VMC, which implements the Virtual MC interface to the Geant3 program,is provided within a single package ‘geant3’ together with Geant3.21 itself. The ‘geant3’package is available from the ROOT web site.As the Virtual MC was largely inspired by Geant3, its implementation for Geant3 wasstraightforward and has no limitations.Besided the implementation of the Virtual MC, the Geant3 VMC also includes theGeant3 Geometry Browser, a GUI which provides a variety of functions, namely:

visualization of the geometry volumes tree,

drawing of volumes and interactive setting of drawing options,

browsing material and tracking medias parameters,

browing applied cuts and activated physics processes,

plotting of dE/dx and cross-sections for a selected physical process.

The implementation is based on the ROOT GUI classes.

A.7 Geant4 VMC

The Geant4 project was started in 1994, the first production version was released in 1998and the system is continously under development by the Geant4 Collaboration. Its areasof application include particle and nuclear physics experiments, medical, accelerator andspace physics studies.The Geant4 VMC, which implements the Virtual MC interface for the Geant4 program,is provided within the ‘geant4 vmc’ package and requires a prior Geant4 installation.The ‘geant4 vmc’ package is available from the ROOT site.The implementation of the interface to MC for Geant4 was presented at CHEP 2001conference. The design, the implementation and also the problems arising from theG3toG4 approach and their foreseen solutions were discussed. Despite improvementsand design changes, the structure and the components of the package presented therecan be found in the current Geant4 VMC package. The major change applied sincethen was that the dependencies on the ALICE classes in AliGeant4 have been replacedby the dependencies on the interfaces to a user application. This meant that all classes


from AliGeant4 could be moved to the experiment independent part, TGeant4, thathas been then renamed Geant4 VMC.The VMC interface provides a common denominator for all implemented MCs andcannot cover all commands available in a Geant4 user session through the Geant4 UI.Switching between the ROOT UI and the Geant4 UI gives the VMC user the possibilityof working with the native Geant4 UI when needed or desired.It is also possible to process a foreign command or a foreign macro in both UIs, forexample the ROOT commands and macros can be processed in the Geant4 UI and viceversa.The Geant4 VMC is under test by several collaborations who adopted the VMC code:ALICE, CBM, PANDA, Opera, MINOS, and of course AEgIS.

Geometry definition

In a similar way as the Geant3 VMC, the Geant4 VMC also includes the Geant4 Ge-ometry Browser. It was implemented in an analogous way. It provides the same func-tionality for browsing geometry. It does not include the panels that allow to browsethe activated physical processes and their characteristics. The support for Geant3-likeVMC functions is in Geant4 VMC implemented using the G3toG4 package in Geant4.The support for TGeo in Geant4 VMC is realized in two ways:

By geometry conversion from TGeo to Geant4. It was first, in 2003, implementedwith a specific roottog4 converter, provided with Geant4 VMC, which was later,in 2005, replaced with use of the Virtual Geometry Model, provided as a toolexternal to Geant4 VMC.

By direct interaction between the TGeo navigator and the Geant4 simulationtoolkit. By the end of 2006, Geant4 VMC was integrated with the G4Root pack-age, which provides the interface between the TGeo navigation and Geant4 sim-ulation toolkit.

Users have the possibility to choose between Geant4 native navigation and G4Rootnavigation, which are available in both cases to define geometry: via the VMC interfaceor via TGeo. The possible selections are: VMCtoGeant4, VMCtoRoot, RootToGeant4,Root, Geant4, where the first word refers to the geometry input and the second oneto the navigator to be used. The last option is reserved for geometry defined via theGeant4 geometry model, which is supported in Geant4 VMC, too. Once the optionsselected are specified in the user code, the necessary packages are automatically con-nected and the selected navigator is activated.

Tracking media

The tracking medium in VMC has the same meaning as in Geant3. It represents a set oftracking parameters associated to a material: sensitivity flag, parameters for magneticfield, maximum step, etc.As Geant4 did not adopt the Geant3 concept of tracking media, in Geant4 VMC aspecialization of the G4UserLimits class is used to hold the relevant information fromuser defined tracking media: the step limit, the vector of cut values and the vector ofprocess controls. The user limits are then used by the special processes, to apply userdefined values in tracking.

A.7 Geant4 VMC 111

Physics selection

Geant4 does not have any default particle or physics process. Users have to definethem explicitly in their application. The VMC defines its own identifiers associatedwith given physics processes. First, the processes are associated with Geant3-like flagnames, which can be used by users for activation or inactivation of a selected process.Then, the TMCProcess codes are defined for getting the information during trackingvia TVirtualMC.The special processes implement the following VMC features:

Special controls processThis process implements the activation and inactivation of selected processes viaTVirtualMC using the VMC control flags. The flag names and the flag values havethe same meaning as in Geant3. Below we give the declaration of the TVirtualMCfunction and an example of its use for the global activation of the Comptonprocess:

TVirtualMC::SetProcess(const char* flagName, Int_t flagValue);gMC->SetProcess("COMPT", 1);

Special cuts processIn Geant4, cuts are applied as cuts in range per particle and region. In VMCinstead, there are introduced Geant3-like cuts in energy which are applied globallyor per tracking medium. In Geant4 VMC, Geant3-like cuts are implemented bythe special cuts process and user limits. These cuts are applied as tracking cuts,not as a threshold. Below we give the declaration of the TVirtualMC function andan example of its use for setting the global energy 1 MeV cut to gamma particle:

TVirtualMC::SetCut(const char* cutName, Double_t cutValue);gMC->SetCut("CUTGAMA", 1e-03);

The special processes are not activated by default, they have to be activated by theuser explicitly.

Primary generator and VMC stack

The VMC provides the interface for the stack of particles, which has to be implementedby the user. Users can also re-use the stack implementations from the VMC exam-ples. The particles in the VMC stack are of the ROOT TParticle type. The primaryparticles are first filled by the user application in the user VMC stack as TParticle ob-jects, then they are transformed by Geant4 VMC in Geant4 objects and passed to theGeant4 kernel. This sequence is shown in Figure A.3. The secondary particles, createdby Geant4 physics processes, are stored by Geant4 VMC in the user VMC stack. Bydefault, storing happens at the beginning of tracking a secondary particle. Optionally,users can choose to store a particle already in the step of the parent track, just whenthe secondary particle is produced. The storing of secondary particles can be also in-activated.In VMC, users also have the possibility of adding particles to the VMC stack duringtracking. In this case, the particle added by the user is considered as a secondary par-ticle of a current particle tracked. This feature is used by the ALICE collaboration


Figure A.3: Stacking of primary particles.

to simulate the generation of feedback photons in an avalanche close to a multiplica-tive wire. In Geant4 VMC, this functionality has been implemented in the specialTG4StackPopper process, which monitors the VMC stack, and if a new particle in thestack is detected, it is popped from the VMC stack and passed to the Geant4 tracking.

A.8 Distribution

The VMC is distributed with the ROOT system. It consists of the following packages:

mc: the core package (interfaces)

geant3: Geant3.21 + Geant3 VMC

geant4 vmc: Geant4 VMC

examples

The ‘mc’ package is directly included in ROOT, ‘geant3’ and ‘geant4 vmc’ are availablefrom the ROOT CVS server as independent modules and the package with the examplesis provided within ‘geant4 vmc’. The tarballs with sources are also available from theVMC Web page.All the simulations have been performed with the last recommended versions up-to-date.That is:

ROOT, version 5.28/00c (April 15, 2011)

Geant4.9.4.p01 (February 25, 2011)

A.9 Description of the simulation program parameters 113

Geant4 VMC 2.11 (December 22, 2010)

To work with an updated framework is very important in this kind of simulations. Wemay even say that they are ‘pioneristic’ (not always in the positive meaning of the term),because it is possible that CHIPS is not still adapted to a very low energy ranges.For this reason, every new release of the packages bring new features that may be cru-cial. CHIPS model has been exploited for several aspects in some Physics list sinceseveral years, but a Physics list named CHIPS entirely based on the model has beenimplemented in Geant4, only since the 9.3 version. That’s why the simulations pre-sented in the proposal are performed with Geant3, forcing the annihilation reactionsinside the source.In the release notes of Geant4.9.3 CHIPS, among other updates it is descibed as ‘aphysics-lists which uses the CHIPS model for nuclear interactions of all energies andparticles’, and it is said that ‘the CHIPS model has been extended with hadronic inter-actions covering all energies for all hadronic particles; in addition hadron- and lepton-nuclear reactions are extended to high energies (" TeV). [. . .] CHIPS ion-ion elasticscattering model is also now available’. In Geant4.9.4, anti-baryon-nucleon elastic scat-tering was added to the CHIPS model, that is a whole new process related (also) toantiprotons.

A.9 Description of the simulation program parame-ters

The simulation program has been written in C++ language, in the framework of VirtualMonte Carlo and Geant4. It was build on the basis of the examples elaborated by IvanaHrivnacova and included in the VMC package. Our program has been initially built tosimulate only the AEgIS beam counter, but it is intended to become one of the mainsimulation programs of the experiment. It may also include with no trouble the Moiredeflectometer, and the position sensitive silicon detector. In fact, there is no real needto simulate positrons and the formation of antihydrogen. Antiprotons can be generatedwith the same radial and axial energy distribution than antihydrogen and gravitate inthe same way (the mass of the positron is negligible). Luckily, in simulations there areno inconveniences related to electromagnetic e!ects: antiprotons may fall undisturbed.Of course, since Geant does not simulate gravitation, an ad hoc electric field must beimplemented in order to reproduce the same e!ects.The top macro, named macroRun.C, is run by the user and give the simulation a seriesof important parameters. Here they are:

RunNumber is an integer that labels the run in progess. This number is asso-ciated to all the output files generated.

nEvents is the number of events (say, antiprotons) that are generated, one at atime, throughout the run. Usually the results presented in the following sectionsare obatined with 105 events.

g3org4 is an integer with two possible values: 3 and 4. It is used as the conditionif Geant3 or Geant4 must be called. As explained, VMC is a sort of layer betweenROOT and the simulation tools. To switch from one to another is quite simple,as the main concern is merely to load the correct libraries when needed.


TracksLim is the maximum number the annihilation products, whose informa-tion we want to save at the end of the run. The total number of tracks is neverthe-less generated by the software. This number is generally kept at 200. Informationslost in this way are most probably those of the gammas, positrons and electrons.

visual is just a boolean variable that can be switched on/o!, as it ables/disablesthe graphics tools of VMC.

the integer BeamCounter has two possible values: 1 and 2. It has been im-plemented as it allows with one single variable to choose between two detectordesigns: squared or circular, as described in Section 4.2.

After the base libraries of G4VMC have been loaded, g3org4 is immediately put to useto load the Geant3 libraries in one case, and those of Geant4 in the other. Here theuser loads a source and for this he must make a choice coherent with the definition ofgeometry used in the simulation macros. The possibilities are the following:

if he uses Geant3:

– g3Config.C: configuration macro for G3 with native geometry;

– g3tgeoConfig.C: configuration macro for G3 with TGeo geometry (default).

if he uses Geant4:

– g4Config.C: configuration macro for G4 with native geometry navigation(default);

– g4tgeoConfig.C: configuration macro for G4 with TGeo geometry navigation.

The configuration macro used here is g4tgeoConfig.C, since this (newer) way is recom-mended for new users, and the way via VMC is kept for a backward compatibility. Hereis defined TG4RunConfiguration, that admits two options. The first is the navigationoption, and in the following are commented the di!erent options:

geomVMCtoGeant4: geometry defined via VMC, G4 native navigation. The in-terfaces to functions for geometry definitions provided by VMC were stronglyinspired by Geant3; the implementation of these functions in Geant4 VMC istherefore made with use of the G3toG4 tool provided by Geant4.

geomVMCtoRoot: geometry defined via VMC, ROOT navigation. The imple-mentation of this option uses TGeoMCGeometry class provided in the root/vmcpackage. The same class is used also by TGeant3TGeo for supporting user geom-etry defined via VMC.

geomRoot: geometry defined via ROOT, ROOT navigation. If geometry is definedvia ROOT and G4Root navigation is selected, Geant4 VMC only converts theparameters defined in TGeoMedium objects to Geant4 objects.

geomRootToGeant4: geometry defined via ROOT, G4 native navigation. Thegeometry defined via TGeo is converted in Geant4 geometry using the externalVirtual Geometry Model (VGM), which has replaced the old one-way convertersfrom Geant4 VMC (G4toXML, RootToG4), removed from Geant4 VMC with theversion 1.7. In the VGM, these convertors has been generalized and improved.

A.9 Description of the simulation program parameters 115

geomGeant4: geometry defined via Geant4, G4 native navigation. User Geant4detector construction class can be passed to Geant4 VMC via user defined runconfiguration class (see below ). If Geant4 VMC is built with VGM, geometrycan be exported in ROOT using the built-in command: /vgm/generateRoot andreused in Geant3 or Fluka (when available) VMC simulation.

The second option, a very important one, required by the configuration macro is knownas the Physics list. A Physics list contains all the physical processes and models tobe used in the simulation. There are several dozens of them, for many case records,sometimes with very general purposes, and other times very specific. Here are the mainareas of application: high energy physics calorimetry, high energy physics tracking, LHCneutron fluxes, linear collider neutron fluxes, shielding applications, low energy dosi-metric applications, medical and military neutron applications, undeground physics, airshowers, etc.For instance, LHEP Physics list are based, with many variants, on a parametrised mod-eling for all hadronic interactions for all particles. The parametrised model is an im-proved version of the Gheisha model. These lists combine the high energy parameterised(HEP) and low energy parameterised (LEP) models describing inelastic interactions forall hadrons. The modeling of elastic scattering of a nucleus and of capture of negativestopped particles and neutrons proceeds via parameterised models.The Physics list used in our simulations is CHIPS (Chiral Invariant Phase Space), andis deepened in Section 5.1.Coming back to macroRun.C, the local program library is loaded (so first of all youhave to compile your program!). Then, at last, the local MCApplication is created.The rest of the source code is pretty standard: in any case the basis of the simulationprogram has already been discussed.

Appendix B

Semiconductor detectors

This thesis work is mainly focused on the simulation of a silicon detector, used tomonitor a beam of incoming antiprotons. Thanks to the Geant3 and Geant4 tools,there is no specific need to simulate the microscopical processes that underlie in asemiconductor device, such as the correct equation of electron-hole pairs, Gaussiandi!usion in the material, etc. Furthermore, our purposes are strongly linked withthe needs of the AEgIS Collaboration and they allow us to ignore all the simulationsdedicated to the electronic noise, the read-out configuration, the signal digitalization,and so on, since they would have no impact in the discrimination between di!erentdetector designs. It is nevertheless crucial to know which device we are dealing with.That is why an introduction on the main features of the semiconductor detectors ismandatory in this context.

B.1 Introduction

Semiconductor detectors, as their name implies, are based on crystalline semiconductormaterials, most notably silicon and germanium. These detectors are also referred to assolid-state detectors, which is somewhat older term recalling the era when solid-statedevices first began appearing in electronic circuits. While work on crystal detectorswas performed as early as the 1930s, real development of these instruments first be-gan in the late 1950s. The first prototypes quickly progressed to working status andcommercial availability in the 1960s. These devices provided the first-high resolutiondetectors for energy measurement and were quickly adopted in nuclear physics researchfor charged particle detection and gamma spectroscopy. In more recent years, however,semiconductors devices have also gained a good deal of attention in high energy physicsdomain as possible high-resolution particle track detector.The basic operating principle of semiconductor detectors is analogous to gas ionizationdevices. Instead of gas, however, the medium is now a solid semiconductor material.The passage of ionizing radiation creates electron-hole pairs (instead of electron-ionpairs) which are then collected by an electric field. The advantage of a semiconductor,however, is that the average energy required to create an electron-hole pair is some 10times smaller than thar required for gas ionization. Thus the amount of ionization pro-duced for a given energy is an order of magnitude greater resulting in increased energyresolution. Moreover, because of their density is 1000 times greater than that for a gas,they have a greater stopping power than gas detectors.

117

118 Semiconductor detectors

Scintillation detectors o!er as well one possibility of providing a solid detection medium.One of the major limitations of scintillation counters is their relatively poor energy res-olution. The chain of events that must take place in converting the incident radiationenergy to light and the subsequent generation of an electrical signal involves many in-e"cient steps. Therefore, the energy required to produce one information carrier (aphotoelectron) is of the order of 100 eV of more, and the number of carriers created ina typical radiation interaction is usually no more than a few thousand.The only way to reduce the statistical limit on energy resolution is to increase the num-ber of informations carriers per pulse. That is why semiconductor detectors are devicesso advantageous. In addition to superior energy resolution, solid-state detectors canalso have a number of other desirable features. Among these are compact size, rela-tively fast timing characteristics, and an e!ective thickness that can be varied to matchthe requirements of the application. Drawbacks may include the limitation to small sizeand the relatively high susceptibility of these devices to performance degradation fromradiation-induced damage.

B.2 Semiconductor properties

The period lattice of crystalline materials causes an overlapping of the electron wave-functions, and establishes allowed energy bands for electrons that exist within thatsolid. They are actually regions of many discrete levels which are so closely spacedthat they may be considered as a continuum. The energy of any electron within thepure material must be confined to one of these energy bands, which can be separatedby gaps or ranges of forbidden energies, in which there are no available energy at all.A simplified representation of the bands of interest in insulators or semiconductors isshown in Figure. A lower band, called valence band, corresponds to those outer-shellelectrons that are bound to specific lattice sites within the crystal. In the case of siliconor germanium, they are part of the covalent bonding that constitute the interatomicforces within the crystal. The next higher-lying band is called the conduction band andrepresents electrons that are free to migrate through the crystal. Electrons in this bandcontribute to the electrical conductivity of the material. The two bands are separatedby the bandgap, the size of which determines whether the crystal is classified as a semi-conductor or an insulator. The number of electrons within the crystal is just adequateto fill completely all available sites within the valence band. In the absence of thermalexcitation, both insulators and semiconductors would therefore have a configuration inwhich the valence band is completely full and the conduction band completely empty.Under these circumstances, neither would theoretically show any electrical conductivity.For insulators, the band gap is usually 5 eV or more, whereas for semiconductors, thebandgap is considerably less, 2 1 eV (see Figure B.1.The width of the gap is determined by the lattice spacing between the atoms. These

parameters are thus dependent on the temperature and the pressure. At normal tem-peratures, the electrons in an insulator are normally all in the valence band, thermalenergy being insu"cient to excite electrons across this gap. When an external electricfield is applied, therefore, there is no movement of electrons through the crystal andthus no current. In a semiconductor, the energy gap is intermediate in size such thatonly a few electrons are excited into the conduction band by thermal energy.

B.3 Charge carriers 119

Figure B.1: Energy band structure of conductors, insulators and semiconductors.

B.3 Charge carriers

At 0 K, in the lowest energy state of the semiconductor, the electrons in the valencebond all participate in covalent bonding between the lattice atoms. Both silicium andgermanium have four valence electrons so that four covalent bonds are formed. Atnormal temperature, however, the action of thermal energy can excite a valence electroninto the conduction band leaving a vacancy (called a hole) in its original position. Thecombination of the two is called an electron-hole pair. The electron in the conductionband can be made to move under the influence of an electric field. The hole, representinga net positive charge, will also tend to move in an electric field, but in a directionopposite that of the electron. The motion of both this charges contributes to theobserved conductivity of the material.In a semiconductor, electron-hole pairs are constantly being generated by thermaleenergy. At the same time, there are also a certain number of electrons and holes whichrecombine. Under stable conditions, an equilibrium concentration of electron-hole pairsis established. If ni is the concentration of electrons (or equally holes) and T thetemperature, then:

ni =#

NcNv exp

!

#Eg

2kT

"

= AT 3/2 exp

!

#Eg

2kT

"

, (B.3.1)

where Nc is the number of states in the conductor band, Nv the number of states inthe valence band, Eg the number of gap at 0 K and k the Boltzmann constant. Nc

and Nv can be calculated from Fermi-Dirac statistics and each can be shown to vary asT 3/2. Making this dependence explicit then gives the right-hand side of (B.3.1) wherethe constant A is indepedent of temperature but characteristic of the material.Typical values for ni are on the order of 2.5 · 1013 cm!3 for Ge and 1.5 · 1010 cm!3

for Si at T = 300 K.1 As reflected by the exponential term, the probability of thermalexcitation is critically dependent on the ratio of the bandgap energy to the absolutetemperature. Materials with a large bandgap will have a low probability of thermal ex-citation and consequently will show the very low electrical conductivity characteristics

1This should be put into perspective, however by noting that there are on the order of 1022atoms/cm3 in these materials. This means that only 1 in 109 germanium atoms is ionized and 1in 1012 in silicon! Despite the large exponents, therefore, the concentrations are very low.


of insulators. If the bandgap is as low as several eV, su"cient thermal excitation willcause a conductivity high enough for the material to be classified as a semiconductor.In the absence of an applied electric field, the thermally created electron-hole pairsobserved at any given time is proportional to the rate of formation. From Equation(B.3.1), this equilibrium concentration is a strong function of temperature and will de-crease drastically is the material is cooled.2

After their formation, both the electron and the hole take part in a random thermalmotion that results in their di!usion away from their point of origin. If all electrons (orholes) were initially created at a single point, this di!usion leads to a broadening dis-tribution of the charges as a function of time. A cross section through this distributionwould be approximated by a Gaussian function with standard deviation 1 given by:

1 =3

2Dt, (B.3.2)

where D is the di!usion coe"cient and t is the elapsed time. Values for D can predictedfrom the relationship:

D = µkT

e, (B.3.3)

where µ is the mobility of the charge carrier, k is the Boltzmann constant, and T is theabsolute temperature. At 20 $C (293 K), the numerical value of kT/e is 0.0253 V.

B.4 Migration of charge carriers in an electric field

Under the action of an externally applied electric field, the drift velocity of the electronsand holes through a semiconductor can be written as:

ve = µeE

vh = µhE,(B.4.1)

where E is the magnitude of the electric field and µe and µh are the mobilities of theelectrons and holes respectively. For a given material, the mobilities are functions of Eand the temperature T . For silicon at normal temperatures, µe and µh are constant forE < 103 V/cm, so that the relation between velocity and E is linear. For E between103 # 104 V/cm, µ varies approximately as E!1/2, while above 104 V/cm, µ varies as1/E. At this point the velocity saturates approaching a constant value of about 107

cm/s. Physically, saturation occurs because a proportional fraction of the kinetic en-ergy acquired by the electrons and holes is drained by collisions with the lattice atoms.Many semiconductor detectors are operated with electric field values su"ciently highto result in saturated drift velocity for charge carriers. Because these saturated veloc-ities are of the order of 107 cm/s, the time required to collect the carriers over typicaldimensions of 0.1 cm or less will be under 10 ns. Semiconductor detectors can thereforebe among the fastest-responding of all radiation detector types.At temperatures between 100 and 400 K, µ also varies approximately as T!m, wherem depends on the type of material and on the charge carrier. Values of m in siliconare m = 2.5 for electrons and m = 2.7 for holes, while in germanium, m = 1.66 forelectrons and m = 2.33 for holes.

2Because the ionization potential for gases is typically 15 eV or more, the probability of a thermallygenerated ion pair is negligibly small in gas ionization chambers, event at room temperature.

B.4 Migration of charge carriers in an electric field 121

While there is considerable scatter in the experimental data, for noncompensated ma-terial (no counter doping) for heavily doped substrates (i.e. 1018 cm!3 and up), themobility in silicon is often characterized by the empirical relationship:

µ = µ0 +µ1

1 #(

NNref

*# , (B.4.2)

where N is the doping concentration (either ND or NA), and Nref and # are fittingparameters. At room temperature, the above equation becomes, for majority carriers[73]:

µn(ND) = 65 +1265

1 +&

ND

8.5·1016

'0.72 ,

µp(NA) = 48 +447

1 +&

NA

6.3·1016

'0.76 .(B.4.3)

For minority carriers [74]:

µn(NA) = 232 +1180

1 +&

NA

8·1016

'0.9 ,

µp(NA) = 130 +370

1 +&

NA

8·1017

'1.25 .(B.4.4)

These equations apply only to silicon, and only under low field.The mobilities, of course, determine the current in a semiconductor. Since the cur-rent density J = ,v, where , is the charge density and v the velocity, J in a puresemiconductor is given by:

J = eni(µe + µh)E, (B.4.5)

where we have substituted (B.4.1) for v and used the fact that current is carried byboth electrons and holes. Moreover, J = 6E, where 6 is the conductivity; comparisonwith (B.4.5), therefore, gives us the relation:

6 = eni(µe + µh). (B.4.6)

In addition to their drift, the charge carriers will also undergo the influence of di!usionmentioned in the previous section. Without di!usion, all charge carriers would travel tothe collecting electrodes following exactly the electric field lines that connect their pointof origin to their collection point. The e!ect of di!usion is to introduce some spread inthe arrival position that can be characterized as a Gaussian distribution whose standarddeviation can be predicted by combining Eqs (B.3.2), (B.3.3), and (B.4.1):

1 =

%

2kTx

eE, (B.4.7)

where x represents the drift distance. In small-volume detectors, a typical value for 1would be less than 100 m. This di!usion broadening of the charge distribution limitsthe precision to which position measurements can be made using the location at whichcharges are collected at the electrodes in semiconductor detectors.


Figure B.2: (a) Addition of donor impurities to form n-type semiconductor materials. The impuritiesadd excess electrons to the crystal and create donor impurity levels in the energy gap. (b) Additionof acceptor impurities to create p-type material. Acceptor impurities create an excess of holes andimpurity levels to the valence band.

B.5 E!ect of impurities or dopants

In a pure semiconductor crystal, the number of holes equals the number of electrons inthe conduction band. Such material is called an intrinsic semiconductor and

ni = pi. (B.5.1)

The quantities ni and pi are known as the intrinsic carrier densities. Intrinsic hole orelectron densities at room temperatures are 1.5 · 1010 cm!3 in silicon, and 2.4 · 1013

cm!3 in germanium. The properties of an intrinsic semiconductor can be describedtheoretically, but in practice they are virtually impossible to achieve. The electricalproperties of real materials tend to be dominated by the very small levels of residualimpurities. Nevertheless, this (approximated) balance can be changed by introducing asmall amount of impurity atoms having one more or one less valence in their outer atomicshell. For silicon and germanium which are tetravalent, this means either pentavalentatmos or trivalent atoms. These impurities integrate themselves into the crystal latticeto create what are called doped or extrinsic semiconductors.If the dopant is pentavalent, in the ground state the electrons fill up the valence bandwhich contains just enough room for four valence electrons per atom. Since the impurityatom has five valence electrons, an extra electron is left which does not fit into this band.This electron resides in a discrete energy level created in the energy gap by the presenceof the impurity atoms. This level is extremely close to the conduction band beingseparated by only 0.01 eV in germanium and 0.05 eV in silicon. At normal temperatures,therefore, the extra electron is easily excited into the conduction band where it willenhance the conductivity of the semiconductor (see Figure B.2). In addition, the extraelectrons will also fill up holes which normally form, thereby decreasing the normal holeconcentration. In such materials, then, the current is mainly due to the movement ofelectrons. Holes, of course, still contribute to the current but only as minority carriers.Doped semiconductors in which electrons are the majority charge carriers are called

B.6 The np semiconductor junction 123

n-type semiconductors.If the impurity is now trivalent with one less valence electron, there will not be enoughelectrons to fill the valence band. There is thus an excess of holes in the crystal. Thetrivalent impurities also perturb the band structure by creating an additional state inthe energy gap, but this time, close to the valence band. Electrons in the valence bandare then easily excited into this extra level, leaving extra holes behind. This excessof holes also decreases the normal concentration of free electrons, so that the holesbecome the majority charge carriers and the electrons minority carriers. Such materialsare referred to as p-type semiconductors.Regardless of the type of dopant, the concentration of electrons and holes obey a simplelaw of mass action when in thermal equilibrium. If n is the concentration of electronsand p is the concentration of holes, then their product is:

np = n2i = AT 3 exp

!

#Eg

kT

"

, (B.5.2)

where ni is the intrinsic concentration given in (B.3.1). Since the semiconductor isneutral, the positive and negative charge densities must be equal, so that:

ND + p = NA + n, (B.5.3)

where ND and NA are the donor and acceptor concentrations. In a n-type material,where NA = 0 and n 0 p, the electron density is therefore:

n ( ND, (B.5.4)

i.e., the electron concentration is approximately the same as dopant concentration.Using (B.5.2), then, the minority carrier concentration is:

p ( n2i

ND. (B.5.5)

From (B.4.6), the conductivity or resistivity of a n-type material thus becomes:

1

,= 6 ( eNDµe. (B.5.6)

An analogous result is found for p-type materials. As an example, assume we have siliconwith a donor density of 1013/cm3, which will also be the concentration of conductionelectrons. Then the resistivity will be:

, =1

eNDµe

=1

(1.6 · 10!19 C)(1013/cm3)(1350 cm2/V · s)= 463 2 · cm.

(B.5.7)

B.6 The np semiconductor junction

The functioning of all present-day semiconductor detectors depends on the formation ofa semiconductor junction. Such junctions are better known in electronics as rectifying


diodes, although that is not how they are used as detectors. Semiconductor diodes canbe formed in a number of ways.For instance, the formation of a pn-junction creates a special zone about the interfacebetween the two materials. Because of the di!erence in the concentration of electronsand holes between the two materials, there is an initial di!usion of holes towards the n-region and a similar di!usion of electrons in the p-region. As a consequence, the di!usingelectrons fill up holes in the p-region while the di!using holes capture electrons on the n-side. Recalling that the n and the p structures are initially neutral, this recombination ofelectrons and holes also causes a charge build-up to occur on either side of the junction.Since the p-region is injected with extra electrons it thus becomes negative while then-region becomes positive. This create an electric field gradient accross the junctionwhich eventually halts the di!usion process leaving a region of immobile space charge.Because of the electric field, there is a potential di!erence across the junction. This isknown as the contact potential. The energy band structure is thus deformed, with thecontact potential generally being on the order of 1 V.The value of the potential 7 at any point within the region of the junction can be foundby solution of Poisson’s equation:

$27 = #,

5, (B.6.1)

where 5 is the dielectric constant of the medium, and , is the net charge density. Inone dimension, Equation (B.6.1) takes the form:

d27

dx2= #,(x)

5, (B.6.2)

so that the shape of the potential across the junction can be obtained by twice inte-grating the charge distribution profile ,(x).Where a di!erence in electrical potential exists, there must also be an electric field E.Its magnitude is found by taking the gradient of the potential:

E = #$7 (B.6.3)

which, in one dimension, is simply:

E(x) = #d7

dx. (B.6.4)

The region of changing potential is known as the depletion zone or space charge regionand has the special property of being devoid of all mobile charge carriers. And, in fact,any electron or hole created or entering into this zone will be swept out by the electricfield. This characteristic of the depletion zone di particularly attractive for radiationdetection. Ionizing radiation entering this zone will liberate electron-hole pairs whichare then swept out by the electric field. If electrical contacts are placed on either end ofthe junction device, a current signal proportional to the ionization will then be detected.The analogy to an ionization chamber thus become apparent.

B.7 Reverse biasing

While the pn-junction described above will work as a detector, it does not present thebest operating characteristics. In general, the intrinsic field will not be intense enough

B.8 Fully depleted detectors 125

Figure B.3: A solid state detector is formed by a semiconductor crystal on which is applied apotential di!erence. When a particle release energy in the central region, free from holes (black circles)and electron (red circles), it frees by ionization several electron-hole pairs. Under the potential applied,they move in opposite directions, giving rise to an electric impulse having an amplitude proportionalto the particle energy.

to provide e"cient charge collection and the thickness of the depletion zone will besu"cient for stopping only the lowest energy particles. Better results can be obtainedby applying a reverse-bias voltage to the junction, i.e., a negative voltage to the p-side.This voltage will have the e!ect of attracting the holes in the p-region away from thejunction and towards the p contact and similarly for the electrons in the n-region (seeFigure B.3). The net e!ect is to enlarge the depletion zone and thus the sensitive volumefor radiation detection – the higher the external voltage, the wider the depletion zone.Moreover, the higher external voltage will also provide a more e"cient charge collection.The maximum voltage which can be applied, however, is limited by the resistance ofthe semiconductor. At some point, the junction will breakdown and begin conducting.

B.8 Fully depleted detectors

A generalized solution for the thickness of the depletion region is:

d (!

25V

eN

"1/2

, (B.8.1)

where d is the entire distance over which the space charge extends, N represents thedopant concentration (either donors or acceptors) and V is just the value of the appliedreverse bias. Thus, the width of the depletion region associated with a p-n junctionincreases as the reverse biase voltage is increased. If the voltage can be increased farenough, the depletion region eventually extends across virtually the entire thickness ofthe silicon wafer, resulting in a fully depleted (or totally depleted) detector. Because ofthe several advantages this configuration presents over partially depleted detectors, the


fully depleted configuration is the preferred type in most applications.In the usual case, one side of the junction is made up of a heavily doped n+ or p+ layer3,or alternatively, a surface barrier. The opposite side of the junction generally consistsof high-purity semiconductor material that is only mildly n- o p-type. The reasonthat high-purity material is important is reflected in Equation (B.8.1). For a givenapplied voltage, the depletion depth is maximized by minimizing the concentration ofdoping impurities on the higher-purity side of the junction. Thick depletion regionscan therefore only be obtained by starting from semiconductor material with the lowestpossible impurity concentration. Also, with a large di!erence in the doping levels, thedepletion layer essentially extends only into the high-purity side of the junction. Theheavily doped layer can then be very thin, providing an entrance window for weaklypenetrating radiations.In Figure, we assume that we have such a junction formed between a heavily dopedp+ surface layer and a high-purity n-type silicon wafer. As the reverse bias voltageapplied to the detector is raised from zero, the depletion region extends further fromthe p+ surface into the bulk of the wafer. For low values of the voltage, the wafer isonly partially depleted and the electric field goes to zero at the far edge of the depletionregion. Between this point and the back surface of the wafer, a region of undepletedsilicon exists in which there is no electric field. This region represents a very thick deadlayer from which charge carriers are not collected. For all practical purposes, partiallydepleted detectors are therefore only sensitive to charged particles incident on the frontsurface.If the applied voltage is increased further, the depletion region may be made to extendall the way to the back surface of the wafer. The voltage required to achieve thiscondition is sometimes called the depletion voltage. Its value is found by setting thedepletion depth d in Equation

Vd =eNT 2

25. (B.8.2)

When this stage is reached, a finite electric field exists all the way through the wafer,and the back dead layer thickness is reduced to that of the surface electrical contactthat is employed. Once the wafer is fully depleted, raising the applied voltage furthersimply results in a constant increase in the electric field everywhere in the wafer. Atvoltages much larger than the depletion, the electric field profile therefore tends tobecome more nearly uniform across the entire wafer thickness. Under these conditions,the detector is sometimes said to be over-depleted. Because of the advantages of havinga high electric field everywhere within the detector active volume, virtually all totallydepleted detectors are operated at su"cient voltage to achieve this condition.Let’s considerate the case in which intrinsic or perfectly compensated material is usedfor the wafer. In this case the distinction between the two contacts disappears and theelectric field is uniform throughout the entire wafer. The detector is fully depleted evenfor very low values of applied voltage. This configuration is known as p-i-n.Fully depleted silicon detectors are very useful as transmission detectors for incidentparticles that have su"cient energy to pass completely through the wafer. The pulseamplitude then indicates the energy lost by the incident radiation during its transitthrough the decise. Totally depleted silicon detectors are commercially available inthicknesses from about 50 to 2000 m.

3Impurity concentrations in these materials can be as high as 1020 atoms/cm3, so that they arehighly conductive.

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Ringraziamenti

Donner a nos enfants le desir de savoir, eveillerleur curiosite. Les traiter aussi en personnes apart entiere, qui comprennent bien plus que nele croient les adultes. Vous les fortifierez ainsiet ils vous en sauront gre.

Date ai nostri bambini il desiderio di sapere,

suscitate la loro curiosita. Trattateli inoltre

come persone a tutti gli e!etti, che capiscono

molto di piu di quanto non credano gli adulti.

In questo modo li renderete piu forti, e ve ne

saranno riconoscenti.

Albert Barille (1921-2009)

Ci sono cose che resteranno: la solidarieta tra compagni; un docente che ti trasmet-te metodo e passione; l’emozione nell’aver capito un concetto sottile; le pause di sole,sempre troppo brevi, tra un’ora di lezione e l’altra; la determinazione che sta in unvoto rifiutato; la stanchezza olimpica dopo aver sostenuto un esame a Miramare (unoqualunque); la soddisfazione di avere scalato certe montagne, che finiscono per farticonoscere te stesso.Per questo ringrazio Germano: e stato, se non perfetto, il relatore migliore che potessiavere. La sua fiducia, anche “internazionale”, rappresenta per me il piu bel banco diprova degli ultimi sei anni; spero che le nostre strade si incroceranno di nuovo. Rin-grazio poi la Prof.ssa Martin per l’entusiasmo e il sostegno che ha dimostrato lungotutto l’arco della tesi. Ringrazio il Prof. Camerini per aver dato concretezza all’avviodi questo lavoro, e per molto altro ancora. Ringrazio anche la Dott.sa Gregorio, peravermi salvato da acque tempestose in almeno tre occasioni. Un grazie particolare vainfine all’Ing. Luca Dassa, che mi ha fornito con tutta la disponibilita possibile glischemi tecnici dell’esperimento, accompagnate da spiegazioni sempre chiarissime.Ai compagni di trincea, Andrea C. e Fabrizio L., per avermi dato un’amicizia che spessonon ho meritato. A Marina, perche a lei e alla sua generosita devo molto piu di quelche crede. A Nicolo, perche non si e mai tirato indietro. A Monja, che pur sapendotutto di me continua a volermi bene. A Davide B., che con la sua energia incredibileha il merito di stupirmi ogni volta. A Mirco, che ha semplicemente segnato il momentopiu positivo della mia vita universitaria. A Davide S., di una gentilezza unica, di quelliche gli nasce dentro. A Giulio, che restera fino alla fine un mistero bu!o. A Laura,Irene e Damiana per avere dato calore e colore alle giornate passate insieme. Ai mieicoinquilini Fabrizio O., Marco, Nicola e Simone per avermi salvato dal purgatorio.

133

134 Ringraziamenti

Grazie ancora ad Anna, Maria, Paolino, Ale e la Fra. Avervi al proprio fianco e unafortuna.Ai miei genitori, perche devo a loro quel che c’e di buono in me.

Alla mia Elisa, prigioniera di questa splendida chimera, cui dedico questa tesi, e selo vorra anche molti giorni a venire: non solo perche e la diretta responsabile di questotraguardo, ma perche mi ha restituito un volto nel quale riconoscermi.

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