Dottorato di Ricerca in Fisica - XXV CICLO - infn.it un forte impulso, dovuto al crescente interesse per i processi di reazione di- ... intrinseche dei nuclei interagenti e/o processi

UNIVERSITA DEGLI STUDI DI MESSINA

Dottorato di Ricerca in Fisica - XXV CICLO

THE DEVELOPMENT OF EXPERIMENTALTECHNIQUES TO INVESTIGATE SUB-BARRIERFUSION REACTIONS INDUCED BY WEAKLY

BOUND NUCLEAR BEAMS

MARIA FISICHELLA

Tutor: Prof. L. Torrisi

Co-tutor: Prof. M. Lattuada

PhD Coordinator: Prof. L. Torrisi

Referees: Prof. A.C. Shotter

Dr A. Di Pietro

Settore Scientifico Disciplinare FIS/01

2010/2012

Abstract

Lo studio delle collisioni nucleari a basse energie ha subito negli ultimi anni

un forte impulso, dovuto al crescente interesse per i processi di reazione di-

retti e di fusione ad energie intorno e al di sotto della barriera coulombiana.

L’origine di tale interesse e legata ai risultati di alcuni esperimenti, effettuati

all’ inizio degli anni 80, riguardanti reazioni di fusione indotte da nuclei sta-

bili, nei quali e stato riscontrato un forte aumento della sezione d’urto di

fusione ad energie al di sotto della barriera coulombiana, rispetto alle pre-

visioni teoriche fornite da calcoli di penetrazione di barriera singola. Dal

punto di vista semiclassico infatti, ad energie attorno alla barriera Colom-

biana e possibile trattare il processo di fusione utilizzando semplici modelli

come quello di penetrazione di barriera unidimensionale (1D-BPM). Ad en-

ergie piu basse rispetto la barriera e stato osservato che tale trattazione non

e piu valida. L’innalzamento della sezione d’urto osservato in molti sistemi

rispetto il modello 1D-BPM e stato attribuito all’ “accoppiamento”del canale

di fusione con i possibili gradi di liberta del sistema (ad esempio eccitazioni

intrinseche dei nuclei interagenti e/o processi diretti, come trasferimento di

neutroni o breakup).

Ciascun accoppiamento produce una rimodulazione della barriera coulom-

biana, cosicche il processo di fusione e regolato da una distribuzione di bar-

riere, che e la somma di tutte le barriere rimodulate, le quali agiscono modifi-

cando in modo sostanziale i coefficienti di trasmissione (ovvero la probabilita

di tunnelling). La possibilita di estrarre distribuzioni di barriere di fusione

da dati sperimentali ha permesso di identificare i canali che agiscono da

stati-porta per la fusione. Diversi esperimenti hanno meso in luce il legame

esistente tra il processo di fusione e la struttura dei nuclei interagenti sul

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Abstract

processo di fusione. Si e inoltre stabilita l’ importanza del processo di trasfer-

imento di neutroni sulla sezione d’urto di fusione sotto barriera, sebbene la

comprensione quantitativa dell’ accoppiamento con questo particolare canale

non sia ancora del tutto chiara a causa dei calcoli complessi che sono necessari

per tenere conto di tale processo. L’idea che il processo di trasferimento di

neutroni porti ad un innalzamento della sezione d’urto di fusione e stata spie-

gata da un modello semiclassico, secondo cui un trasferimento intermedio di

neutroni con Q-valore positivo puo portare ad un guadagno in energia cinet-

ica del sistema e quindi ad un innalzamento della probabilita di penetrazione

della barriera e della stessa sezione d’urto di fusione. La maggior parte degli

esperimenti, realizzati per investigare i possibili effetti del trasferimento di

neutroni sulla sezione d’urto di fusione, studiano sistemi simili caratterizzati

da Q-valori per trasferimento di neutroni diversi. Confrontando sistemi sim-

ili si ci aspetta che sia piu facile isolare gli effetti del trasferimento. Nelle

reazioni di fusione dei sistemi Ca+Zr, e stato osservato un innalzamento ad

energie sotto la barriera per il sistema 40Ca+96Zr rispetto agli altri sistemi.

Questo comportamento e stato attribuito dagli autori proprio alla presenza di

canali di trasferimento di neutroni con Q-valore positivo piu grande rispetto

a quello osservato negli altri sistemi studiati. Tuttavia, in un lavoro recente,

riguardante la fusione di sistemi come Ni+Sn e Te+Ni ad energie simili alla

barriera, le sezioni d’urto di fusione misurate risultano molto vicine per tutti

i sistemi, anche se questi presentano Q-valori per il trasferimento di neutroni

molto diversi. Questa nuova osservazione implica che nuovi esperimenti sono

necessarie per investigare l’eventuale relazione esistente tra il trasferimento

di neutroni ed il processo di fusione ad energie al di sotto della barriera

coulombiana.

Ad accrescere ulteriormente l’interesse per queste reazioni di fusione ad en-

ergie vicine alla barriera ha contribuito in modo rilevante l’avvento delle

facilities per la produzione di fasci di ioni radioattivi (RIBs). Gli studi si

sono concentrati in particolare sulle reazioni indotte dai nuclei con alone,

caratterizzati da un core fortemente legato e da uno o piu nucleoni che occu-

pano stati di particella singola con energia di legame molto bassa e piccolo

momento angolare. La piccola energia di separazione che caratterizza i nu-

4

Abstract

cleoni piu esterni di questi nuclei e causa di una distribuzione di densita

di materia spazialmente estesa, per cui la funzione d’onda corrispondente a

questi stati di particella singola mostra una lunga coda che si estende ben

oltre il core compatto del nucleo. Esempi di nuclei dotati di alone di neutroni

sono il 11Be, l’ 6He e il 11Li. Vi sono inoltre nuclei, come il 8B, dotati di un

alone di protone, il quale presenta caratteristiche analoghe a quello di neu-

trone. Alla luce di quanto osservato per i sistemi stabili ci si aspetta che, la

presenza di un “alone di neutroni ”in nuclei esotici leggeri, possa portare, in

reazioni indotte da tali nuclei, ad un aumento della sezione d’urto di fusione

specialmente ad energie sotto barriera. Nonostante negli ultimi anni siano

stati effettuati diversi esperimenti per studiare la dinamica delle collisioni

indotte da nuclei ad alone, la realizzazione di tali esperimenti presenta delle

difficolta legate alle basse intensita con cui questi fasci sono disponibili. Per

questo motivo lo studio di tali fenomeni ha riguardato anche esperimenti

con nuclei stabili debolmente legati, come il 9Be, il 6Li ed il 7Li. Infatti,

sebbene questi nuclei non possiedano una struttura ad alone, ci si aspetta

comunque che a causa della loro bassa energia di legame, anche in questo

caso, l’accoppiamento al continuo abbia influenza sulla dinamica delle col-

lisioni. Per quanto riguarda gli effetti di una simile struttura sulle reazioni

di fusione bisogna considerare due categorie di effetti: statici e dinamici. I

primi sono dovuti alla distribuzione diffusa di materia che influenza il poten-

ziale proiettile-bersaglio riducendo la barriera e quindi innalzando la sezione

d’urto di fusione. Gli effetti dinamici sono generati dall’ accoppiamento del

moto relativo bersaglio-proiettile non solo a stati legati ma anche a stati del

continuo (stati del breakup) o ad altri canali di reazione (come ad esempio

il transfer). I modelli teorici concordano con l’idea che la grande estensione

spaziale della densita di materia possa causare un forte aumento della sezione

d’urto di fusione.

Il ruolo del breakup, ad energie sotto la barriera, e al centro di un intenso

dibattito scientifico. La bassa soglia di breakup implica che l’interazione

coulombiana o l’interazione nucleare possano facilitare la dissociazione del

proiettile; da una parte alcuni autori sostengono che cio potrebbe causare un

aumento della probabilita di breakup e una conseguente diminuzione della

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Abstract

probabilita di fusione, dovuta al minor numero di nuclei proiettile disponibili

per la fusione. Secondo altri autori invece una diminuzione dello scattering

elastico dovuta all’apertura del canale di breakup non implica una dimin-

uzione del flusso disponibile per la fusione, poiche tale processo e anche legato

all’apertura di canali inelastici che causera una variazione nell’altezza della

barriera di fusione. Questi autori, per investigare il ruolo del breakup sul

processo di fusione, hanno eseguito dei calcoli di canali accoppiati (CC). In

particolare, e stato sviluppato un metodo che tiene conto dell’ accoppiamento

agli stati del continuo, il metodo Continuum Discretized Coupled Channels

(CDCC). Esso si basa sul fatto che, sotto determinate condizioni, e possibile

approssimare il continuo con un numero finito di stati discreti contigui con-

sentendo di calcolare effetti di accoppiamento non solo a stati legati ma anche

a stati nel continuo. Questi calcoli prevedono che gli effetti dinamici abbiano

come risultato quello di inalzare la sezione d’urto di fusione ad energie sotto

barriera rispetto al caso senza accoppiamenti.

Per quanto riguarda il ruolo del breakup ad energie sopra barriera molti i

modelli concordano sul fatto che, in reazioni su bersagli pesanti, i processi

diretti e in particolare il breakup, dominano la sezione d’urto di reazione

producendo una riduzione della sezione d’urto di fusione. Con lo scopo di

studiare l’effetto del breakup sul canale di fusione sono stati realizzati diversi

esperimenti utilizzando nuclei stabili debolmente legati su bersagli di massa

leggera, media e pesante. Le reazioni su bersagli pesanti hanno mostrato

una soppressione della fusione completa ad energie sopra la barriera coulom-

biana rispetto al modello di penetrazione di barriera singola (1D BPM) o

rispetto a dati sperimentali riguardanti collisioni indotte da nuclei ben legati

sullo stesso bersaglio. Questo comportamento e stato spiegato come dovuto

al fatto che, in presenza di un forte campo Coulombiano, il proiettile de-

bolmente legato si divide nelle sottostrutture costituenti, riducendo cosı la

probabilita di completa fusione. In accordo con questa spiegazione ci si as-

petterebbe che la soppressione dipenda dalla massa del bersaglio. In realta,

per nuclei sufficientemente pesanti (Z > 40), la soppressione e stata osservata

essere indipendente dalla massa del bersaglio e dipendente dalla probabilita

di breakup del proiettile (tanto piu bassa e la soglia di breakup tanto mag-

6

Abstract

giore e la soppressione sopra barriera). In esperimenti con bersagli leggeri

non e stata osservata nessuna soppressione della sezione d’urto di fusione

completa, attribuibile alla bassa soglia di break-up del proiettile.

Con lo scopo di indagare ulteriormente su queste problematiche e stato pro-

posto lo studio, oggetto di questa tesi di dottorato, delle reazione di fu-

sione indotte da isotopi di Li su isotopi di Sn: 6Li+120Sn, 7Li+119Sn and8Li+118Sn. Lo studio di questi sistemi permette di investigare ulteriormente

sul ruolo delle reazioni di trasferimento di neutroni sul processo fusione.

Queste reazioni sono infatti caratterizzate da Q-valori diversi per il trasfer-

imento di uno o due neutroni. Il sistema 6Li+120Sn presenta i Q-valori di

trasferimento di uno o due neutroni piu piccoli. La caratteristica piu interes-

sante di tali sistemi e che i loro canali di ingresso sono molto simili. Infatti,

scegliendo diverse combinazioni di isotopi di Li e Sn, abbiamo assicurato

che la massa ridotta dei sistemi e quindi la loro velocita di avvicinamento

nel sistema del centro di massa siano molto vicine e il raggio di massimo

avvicinamento e uguale per tutti i sitemi. In questo modo, le differenze tra i

quattro canali di ingresso sono principalmente legate ai diversi Q-valori per

il trasferimento di neutroni.

Lo studio di questi sistemi ha inoltre permesso di investigare il comporta-

mento della sezione d’urto di fusione completa ad energie sopra barriera in

una regione di massa del bersaglio mai investigata prima.

Il presente lavoro riguarda in particolare lo studio delle reazioni di fusione6Li+120Sn e 7Li+119Sn realizzate presso i Laboratori Nazionali del Sud (Cata-

nia) ad energia vicine alla barriera coulombiana (da 16 MeV a 24 MeV). Per

misurare la sezione d’urto di fusione a queste energie abbiamo utilizzato i fasci

Tandem ed una tecnica di attivazione che si basa sulla rivelazione dei raggi X

atomici emessi in seguito al decadimento radioattivo per cattura elettronica

dei residui di evaporazione prodotti nel processo di fusione. Tale tecnica e

particolarmente adeguata per la misura della sezione d’urto di fusione ad en-

ergie sotto barriera. Infatti a queste energie i residui di evaporazione prodotti

non hanno sufficiente energia per lasciare il bersaglio, cosicche una loro rive-

7

Abstract

lazione diretta non e possibile. Scegliendo in maniera adeguata il bersaglio,

in modo che la maggior parte dei residui di evaporazione siano radioattivi

e decadano per cattura elettronica, e possibile misurare la sezione d’urto di

produzione di tali residui misurando i raggi X caratteristici emessi in seguito

al processo di decadimento. Tale tecnica si basa su due step: l’irraggiamento

del bersaglio e la misura dei raggi X-caratteristici. Alla fine dell’ irraggia-

mento, il bersaglio viene posizionato davanti ad un rivelatore Si (Li) per

misurare i raggi X emessi dai residui di evaporazione. Poiche i raggi X atom-

ici caratterizzano gli elementi ma non gli isotopi, e possibile discriminare i

vari isotopi prodotti monitorando l’attivita di ciascun elemento in funzione

del tempo. Dai fit delle curve di attivita sperimentali, noto l’andamento dell’

intensita di fascio incidente in funzione del tempo, e possibile ottenere la

sezione d’ urto di produzione di ciascun isotopo. Dalla somma delle sezioni

d’urto di produzione dei vari residui di evaporazione si ottiene la sezione

d’urto di fusione totale.

I risultati ottenuti mostrano che ad energie al di sopra della barriera coulom-

biana per entrambi i sistemi, come atteso, e stata osservata una soppressione

della sezione d’urto di fusione rispetto al modello di penetrazione di barriera

singola. In entrambi i casi la soppressione e confrontabile con quella osser-

vata per i sistemi piu pesanti precedentemente studiati. Per quanto riguarda

il confronto dei due sistemi ad energie sotto la barriera, per comprendere

l’effetto dei diversi Q-valori di trasferimento di neutroni, dall’intervallo di

energie investigato non si puo dedurre nessun innalzamento della funzione di

eccitazione del 7Li rispetto a quella del 6Li imputabile al Q-valore di trasferi-

mento piu grande. Per evidenziare questo effetto e necessario infatti scendere

ad energie piu basse, dove ci si aspetta che l’effetto sia piu forte.

Lo studio effettuato nell’ ambito del presente lavoro ha richiesto lo sviluppo

della tecnica sperimentale descritta, che in prospettiva, permettera di mis-

urare le sezioni d’urto di fusione indotte dai fasci instabili 8Li e 9Li. In genere

per questo tipo di misure invece di irradiare singoli bersagli si presferisce ir-

radiare stack di bersagli, alternati ad altrettanti assorbitori. La scelta di

irraggiare un insieme di bersagli permette infatti di misurare piu punti della

funzione di eccitazione con una sola energia del fascio: questo metodo e

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Abstract

particolarmente adatto nel caso in cui si lavori a basse energie o con fasci

instabili perche permette di ridurre notevolmente il tempo necessario ad es-

eguire la misura. Nonostante questo notevole vantaggio, l’utilizzo di uno

stack di bersagli presenta anche alcune difficolta legate alla determinazione

dell’ energia “effettiva” per ciascuna sezione d’urto misurata. Nota infatti

l’energia del fascio incidente su un bersaglio, la sezione d’urto di fusione che

si misura e relativa alla distribuzione di energie all’ interno dello stesso. In

prima approssimazione si puo pensare di attribuire alla sezione d’urto mis-

urata l’energia del fascio al centro del bersaglio. In realta questa procedura

puo essere considerata corretta solo ad energie superiori alla barriera coulom-

biana dove la sezione d’urto di fusione ha una debole dipendenza dall’energia.

Per energie inferiori alla barriera avendo la sezione d’urto un andamento espo-

nenziale ciascuna energia all’interno del bersaglio avra un peso differente e

quindi la semplice media delle energie da luogo ad una non corretta asseg-

nazione dell’energia media alla quale associare la sezione d’urto misurata.

Ovviamente la presenza dello stack “amplifica” ulteriormente questo prob-

lema, a causa dello straggling energetico che subisce il fascio man mano che

attraversa i vari bersagli e assorbitori.

Un ulteriore fattore che contribuisce a degradare la risoluzione energetica del

fascio e la presenza di disuniformita nel bersaglio. Per la prima volta, in

questo lavoro, e stata effettuata un’ analisi dettagliata sui bersagli utilizzati

e sono stati presi in considerazione i possibili effetti che la presenza di dis-

uniformita puo comportare nella determinazione della funzione di eccitazione

di fusione. Si e quindi sviluppato un metodo sperimentale che permetta di

determinare la distribuzione di spessori all’ interno del bersaglio e una pro-

cedura analitica attraverso la quale, tenendo conto di questi fattori, si riesca

a determinare l’energia effettiva alla quale associare correttamente ciascun

punto della sezione d’urto misurata. Questo problema puo essere superato

nel futuro mettendo a punto tecniche diverse di fabbricazione dei bersagli

che permettano di realizzare bersagli uniformi.

9

Abstract

10

Contents

Introduction 13

1 The fusion process 21

1.1 Basic concepts of the fusion process . . . . . . . . . . . . . . . 21

1.2 Compound Nucleus Formation . . . . . . . . . . . . . . . . . . 25

1.3 Compound Nucleus Decay . . . . . . . . . . . . . . . . . . . . 29

1.4 Beyond the single-barrier model . . . . . . . . . . . . . . . . . 31

1.5 Barrier distribution . . . . . . . . . . . . . . . . . . . . . . . . 34

1.5.1 Extraction of Experimental Barrier Distributions . . . 37

1.6 Coupled Channel Equations . . . . . . . . . . . . . . . . . . . 38

1.7 Sub-barrier enhancement and neutron transfer process . . . . 41

1.7.1 The 40Ca+90,94,96Zr reactions . . . . . . . . . . . . . . . 42

1.7.2 Fusion of Sn+Ni and Te + Ni . . . . . . . . . . . . . . 45

1.8 Fusion and break-up in collisions induced by weakly bound

nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.8.1 Theoretical models . . . . . . . . . . . . . . . . . . . . 49

1.9 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 51

1.9.1 Fusion reactions involving heavy targets . . . . . . . . 53

1.9.2 Fusion reactions involving medium and light targets . . 54

1.10 Motivation of the PhD project . . . . . . . . . . . . . . . . . . 57

2 Experimental technique and setup 65

2.1 Activation technique for fusion measurements . . . . . . . . . 65

2.2 Target activation . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . 66

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Contents

2.2.2 Readout electronics . . . . . . . . . . . . . . . . . . . . 70

2.3 Characteristic X-rays measurement . . . . . . . . . . . . . . . 72

2.3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . 72

2.3.2 Readout Electronics . . . . . . . . . . . . . . . . . . . 74

2.4 Si(Li) energy calibration . . . . . . . . . . . . . . . . . . . . . 76

2.5 Si(Li) detector efficiency . . . . . . . . . . . . . . . . . . . . . 77

2.6 Characterization of the Si(Li) detector . . . . . . . . . . . . . 78

2.6.1 Determination of the detector crystal position . . . . . 78

2.6.2 Determination of detector effective area . . . . . . . . . 79

2.7 Si(Li) efficiency for Tellurium kα X-rays . . . . . . . . . . . . 79

3 Data analysis 85

3.1 The radioactive decay law and the production cross-section . . 86

3.2 X-ray spectra analysis . . . . . . . . . . . . . . . . . . . . . . 86

3.3 Activity curve analysis . . . . . . . . . . . . . . . . . . . . . . 91

3.4 Beam intensity profile . . . . . . . . . . . . . . . . . . . . . . 93

3.5 ER production cross-section extraction . . . . . . . . . . . . . 95

4 Detailed study of the stack activation method 101

4.1 Drawbacks of the activation technique . . . . . . . . . . . . . 101

4.2 Target uniformity . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.3 Determination of the target thickness distribution . . . . . . . 112

4.3.1 Determination of the Nb and Ho thickness distribution 112

4.3.2 Determination of the Sn thickness distribution . . . . . 112

4.4 Analytical procedure to determine the effective energy . . . . 115

5 Fusion excitation functions for the 6Li+120Sn and 7Li+119Sn

reactions 123

5.1 Complete fusion suppression factor at energies above the barrier123

5.2 Comparison of 6Li+120Sn and for 7Li+119Sn fusion excitation

functions and possible role played by the n-transfer Q-value . 130

Conclusions and future perspectives 133

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Contents

Acknowledgments 136

Bibliography 138

13

Introduction

The study of the fusion process between two nuclei at energies close to the

Coulomb barrier has been the object of extensive experimental and theoreti-

cal efforts in the past 30 years. In former times, it was attempted to describe

the fusion of two heavy ions by the 1D Barrier Penetration Model (1D-BPM),

which involves as variable the radial distance between the centers of mass of

the two ions [1, 2]. The effective potential between the two colliding ions

is the sum of the nuclear, Coulomb and centrifugal potentials. For a fixed

incident angular momentum L, the effective potential as a function of the

distance between the two nuclei shows a maximum corresponding to a spe-

cific distance between the two colliding nuclei.

Some experiments [3, 4, 5, 6, 7] involving medium-mass stable systems, per-

formed during the eighties, showed that this simple model is not appropriate

to describe the fusion process at energies below the Coulomb barrier. In

particular, they found that at these energies the fusion cross-section was

enhanced with respect to 1D-BPM prediction. This enhancement was ex-

plained in terms of couplings to the possible degrees of freedom (as intrinsic

excitation of the nuclei, transfer processes, ...) of target and projectile. It was

suggested that each degree of freedom had as result a remodulation of the

potentialin barrier. Therefore by considering all of these degrees of freedom,

the fusion process will not be governed by a single barrier but by a distribu-

tion of barrier, which is the sum of all the remodulated barriers, some lower

and some higher than the single one. The distribution of barriers leads to an

enhancement of the cross-section since the tunneling or overcoming the lower

barriers is much more probable than penetration through the single barrier.

The possibility to extract fusion barrier distributions from accurate data [8]

15

Introduction

brought a substantial advance in the identification of the relevant channels

acting as doorways to fusion [9]. In particular, various experiments have

shown the importance of static deformations [10, 11, 12, 13] and of complex

surface vibrations [14, 15]. Moreover, the importance of neutron transfer in

increasing the sub-barrier fusion probabilities was also established, but quan-

titative understanding of the coupling strengths remained elusive due to the

need to rely on complex theoretical calculations.

The idea that neutron transfer process would lead to an enhancement of the

fusion cross-section was explained by a semiclassical model [16], according to

which an intermediate neutron transfer with positive Q-value may lead to a

gain in kinetic energy of the colliding nuclei and, thus, to an enhancement of

the barrier penetrability and therefore of the fusion cross-section.

According with this semiclassical model, most of the experiments, performed

to study the effects of neutron transfer on the fusion cross-section, compared

systems with positive Q-value for neutron transfer with their isotopic coun-

terparts with no positive Q-value. By comparing similar systems one could

expect that it is easier to isolate the effects of transfer couplings with respect

to inelastic channels (which are always present). As an example, in the fu-

sion reactions of Ca+Zr systems, a sub-barrier enhancement was observed

in 40Ca+96Zr with respect to the other systems. This behaviour has been

attributed, by the authors, to the presence of neutron transfer channels with

positive Q-value larger than in the other systems studied [17]. However in a

recent work [18] concerning fusion in the Sn+Ni and Te+Ni systems similar

sub-barrier fusion cross-sections have been measured although these systems

present very different Q-values for multi-neutron transfer.

Other fusion measurement are needed to search for the causes which lead to

an enhancement of the fusion cross-section, and in particolar to investigate

the possible role of neutron transfer process.

In the last years, the development of facilities for the production of Radioac-

tive Ion Beams (RIBs) has opened up opportunities to investigate fusion

reactions involving unstable nuclei. In particular, the studies were focused

on the so called halo nuclei. (such as 6He, 11Li, 11Be, 19C or 8B) which are

much more loosely bound than stable ones. Their outer nucleons are char-

16

Introduction

acterized by binding energy which range from 0.1 to 1 MeV. The structure

of halo nuclei is expected to favor the direct processes, due to the low bind-

ing energies of valence nucleons. Moreover, the halo might be expected to

strongly affects the probability of fusion at low energies, due to their larger

extension. Differently from the case of reactions of well bound nuclei, in

the reaction involving these weakly bound nuclei also the coupling with the

break-up channel has to be taken into account.

With the aim of investigating the role played on fusion cross-section by the

break-up process, many experiment with the stable weakly bound nuclei (as9Be, 6Li, 7Li) have been performed. Despite these nuclei do not present an

halo structure one could expect that the high-probability of break-up might

influence the reaction dynamics as in the case of reactions involving halo nu-

clei. These beams are very suitable for studying the role of break-up since,

thanks to their high intensities, they allow to perform measurements with

good statistics and precision.

From the theoretical point of view, different models agree that the spatial

extension of the matter density of these weakly bound nuclei (both stable

and unstable) might cause an enhancement of the fusion cross-section. In-

stead, an intense debate arose about the possible effect of break-up, mainly

because the way the mechanism was described. A low break-up threshold

implies that the Coulomb or nuclear interactions can easily split the nucleus

into core and valence nucleons. Some authors argued [19] that this would

enhance the break-up probability and, in turn, hinder the fusion of the whole

exotic projectile with the target nucleus, due to a decrease of the number of

nuclei available for fusion. However, other authors [20], argued that a de-

crease of the elastic scattering due to break-up, does not imply a increase of

the flux available for fusion. In order to investigate the role of the break-up

process of halo and weakly bound so called Continuum-Discretised Coupled-

Channel (CDCC) calculations were developed. In this model the coupling

not only to bound but also to continuum states (break-up states) have been

considered. These calculations predict that dynamical effects enhance the

sub-barrier total fusion cross-section with respect to the no-coupling case.

At energies above the barrier, all the models agree that the direct processes

17

Introduction

and in particular the break-up, dominate the cross-section of reaction, pro-

ducing a reduction of the complete fusion cross-section. A lot of fusion mea-

surements involving the stable weakly bound nuclei with light, medium and

heavy mass targets have been performed [21, 22, 23, 24, 25, 26] at ener-

gies above the barrier. The fusion measurements in reaction involving heavy

targets (Z ≈ 62-83) have shown a suppression of the complete fusion cross-

section at energies above the barrier with respect to the 1D-BPM predictions

one or with respect to experimental data concerning collisions induced by well

bound nuclei on the same targets (see e.g. [24]). This suppression has been

attributed to the break-up probability of the projectile: it increases by in-

creasing the break-up probability of the projectile. It is less clear how this

suppression varies with the charge of the target (see e.g. [21]).

In order to investigate further the possible effect of direct reactions, as neu-

tron transfer and/or break-up, on fusion mechanism, the study of 6Li+120Sn,7Li+119Sn, 8Li+118Sn and 9Li+117Sn fusion reactions, at energies around the

Coulomb barrier, was proposed

These reactions are characterised by very similar structures of the interacting

nuclei and entrance channels and they present different Q-values for one- and

two- neutron transfer. Thus this study could help in increasing our under-

standing about the role played by the transfer on fusion reactions at energies

below the Coulomb barrier.

Moreover, by studying these reactions at energies above the barrier it is possi-

ble to investigate the complete fusion suppression above the barrier in a target

mass range never studied before. For the reaction 6Li+120Sn, 7Li+119Sn and8Li+118Sn, the suppression factor will be extracted from the comparison of

the measured complete fusion excitation function with respect to the predic-

tion of the Single Barrier Penetration Model.

The present work concerns, in particular, the study of fusion reaction for the6Li+120Sn and 7Li+119Sn systems in the energy range around the Coulomb

barrier (from 16 MeV to 24 MeV), performed at Laboratori Nazionali del

Sud (Catania).

18

Introduction

Since the direct detection of ER, produced in the collision of a low energy

light projectile onto a medium mass target, due to the low kinetic energy of

the ER produced, is not possible, we measured the fusion excitation func-

tions by using an activation technique, based on the off-line measurement of

the atomic X-ray emission following the electron capture decay of the evap-

oration residues produced in the reactions.

The fusion excitation functions were measured by activating both the single

foils and stacks of targets. The technique of the stack is particular useful

in the case of the radioactive beams, because it is possible to extract the

cross-section at different energies without changing the beam energy, thus

reducing the beam time needed to perform an excitation function measure-

ment with the very low intensity radioactive beams. Despite of this great

advantage, this technique presents several drawbacks which if not considered

may lead to a wrong determination of the fusion excitation function. Since

this technique will be used for the measurement with the 8Li and 9Li radioac-

tive beams, we have used the data of the present experiments also to analyse

in detail and to solve the problems related with the use of this technique. In

particular, for the first time, in this work, the problems due to the presence

of non-uniformities in the target will be investigated. It has been developed

an experimental method to determine the target thickness distribution and

an analytical procedure which allows to take into account the effects due to

the stack and the foil non-uniformity on the fusion excitation function.

The structure of the thesis is as follow. In Chapter 1 an introduction

on the fusion process will be done. Then a survey of main concepts and

theoretical tools needed for the study of the fusion process will be given.

Some experimental data, reported in the literature, will also be discussed in

detail to better focus on the studied topics of this work. At the end of the

chapter the motivation of the present work will be discussed in detail. In

the Chapter 2 the experimental apparatus and the technique used will be

described. In Chapter 3 the data analysis will be discussed. In Chapter

4 the drawbacks of the stack technique will be presented. A detailed de-

scription about the developed procedure for determining the target thickness

19

Introduction

distribution and the fusion excitation function will be done. The fusion ex-

citation functions for the two studied system, 6Li+120Sn and 7Li+119Sn, will

be shown in Chapter 5. Each of them will be compared with the 1D-BPM

prediction to extract the suppression factor. From the direct comparison of

the two fusion excitation function it will be estimated the possible effect of

the neutron transfer Q-value. Finally, in Chapter 6 the conclusions and the

future perspectives will be presented.

20

Chapter 1

The fusion process

1.1 Basic concepts of the fusion process

Fusion reactions in nuclear physics are not only of central importance for

stellar energy production and nucleosyntesis, but also because they provide

new insights into reaction dynamics and nuclear structure.

By using a simple model, nuclear reactions can be described by considering

the two nuclei as rigid spherical objects that interact via a potential which

depends only on the relative distance (R) between the centre-of-mass of two

nuclei.

The total potential between the target and projectile nuclei for the L-th par-

tial wave is given by the sum of Coulomb, centrifugal and nuclear terms:

V (r, L) = VCoul(r) + Vnucl(r) + Vcentr(r, L) = V0(r) + Vcentr(r, L)(1.1.1)

where

VCoul(r) =

ZpZte2

rr ≥ R

ZpZte2

2R3 (3R2 − r2) r < R(1.1.2)

with Zp and Zt respectively the atomic number of the projectile and of the

target and R = r0(A1/3t + A

1/3p ) the sum of their radii, e2 = 1.44 MeV · fm.

r0 is a constant usually having typical values ranging from 1.2 to 1.3 fm and

At and Ap are the mass numbers of the interacting nuclei.

21

1. The fusion process

Vcentr(r, L) =~2L(L + 1)

2µr2(1.1.3)

Vnucl(r) = −V01

1 + er−Ra0

(1.1.4)

where V0 is the depth of the well and a0 the diffusness parameter and µ is

the reduced mass of the system

In figure 1.1.1 an example of effective potential for two colliding nuclei is

shown, for different angular momenta, L. This potential is taken as the sum

of the nuclear Woods-Saxon, the Coulomb and the centrifugal potentials. For

Figure 1.1.1: For the system 18O+120Sn, the sum of the nuclear, Coulomb, andcentrifugal potentials are shown for the indicated values of the angular momentumL. The horizontal line marks E = 87 MeV [27]

.

a fixed incident angular momentum L, the effective potential is characterized

by a local maximum and a local minimum (or pocket). The maximum, V LB ,

occurs at a distance, RLB, where the repulsive and attractive forces balance

each other. V LB and RL

B are known as height and position of the potential

22

1.1 Basic concepts of the fusion process

barrier. The barrier is also characterized by the curvature ~ωL, which is

defined by the relation (1.1.5):

ω2L = − 1

µ

d2V (r, L)

dr2|r=RL

B(1.1.5)

For L = 0 the maximum of the potential is called Coulomb barrier. In the

following VB, RB and ω will represent respectively the height, the position

and the curvature of the barrier for L = 0. From the figure 1.1.1 we can

see that the higher the angular momentum the higher the barrier and conse-

quently the shallower the potential well, until for a certain Lc, called critical

angular momentum, the pocket disappears.

Fusion is referred as the ability of the system to overcome or tunnel through

the barrier: if the system is trapped inside the pocket for a sufficiently long

time the two nuclei can fuse completely and form a compound nucleus (CN),

which has a number of nucleons equal to the sum of the nucleons of the

participants. According to this Single Barrier Penetration Model (SBPM),

known also as one-dimensional Barrier Penetration Model (1D-BPM), since

the separation is the only degree of freedom which is taken into account in the

fusion process, it is not possible to have fusion for angular momenta larger

than Lc, due to the absence of the pocket. Actually, the processes which ac-

complish the fusion (like mutual excitation, particle and cluster transfer, and

interpenetration) can occur not only in the pocket but more generally as the

nuclei approach each other, with the result that some of the kinetic energy

of the system is converted into excitation energy. Thus, by the time the bar-

rier region is reached, the kinetic energy may already be reduced, facilitating

the formation of the compound nucleus even when the initial orbital angular

momentum exceeds Lc.

Nuclei formed in heavy-ion fusion reactions (that is in reactions involving

nuclei with a mass number A > 4) are highly excited, spinning objects which

decay before being detected. The excitation energy of the compound nucleus

is determined by the reaction Q-value and the kinetic energy; the spin (angu-

lar momentum) distribution is generated by the range of impact parameters

contributing to fusion. According to the famous assumption of Bohr [29]

23


Figure 1.1.2: Schematic representation of fusion. After its formation the com-pound nucleus releases its excitation energy and its spin emitting particles andgamma rays and leaving an evaporation residue or fission fragments [28]

a fusion reaction proceeds in two stages: first, the formation of the com-

pound nucleus through the absorption of the incident particle by the target

nucleus; second,the disintegration of the compound nucleus by nuclear evap-

oration [30] (by emitting neutrons, protons, α particles, other light charged

particles) or by undergoing fission [31] (see figure figure 1.1.2). When the

excitation energy of the residual nucleus is below the particle separation en-

ergy, the compound nucleus de-excites by emission of a cascade of γ-rays

until it reaches its ground states. The compound state has a mean life (10−20

- 10−18 seconds) long, compared with the time a nucleon take to cross the

nucleus itself (10−22 - 10−21 seconds). During this time the excitation energy

of the compound nucleus is uniformly distributed on all its nucleons. This

intermediate rearrangement is so complex that one can suppose that the CN

”forgets” the particular formation channel except, of course, for the restric-

tions imposed by energy, angular momentum and parity conservation. The

decay modes of the CN are governed by statistical factors associated with

the CN, which are assumed to be independent of the mode of CN formation

but nevertheless must be in accordance with the conservation law of energy,

24

1.2 Compound Nucleus Formation

angular momentum and parity.

The separation of a nuclear reaction into two “indipendent“ stages permits

us to express the cross-section of a reaction of the type A+ a → C∗ → B + b

as:

σ(a, b) = σa(ε)Pb(E∗) (1.1.6)

where σa(ε) is the cross-section for the absorption of the particle a with

kinetic energy ε by the target nucleus A to form the compound nucleus C∗.

Pb(E∗) is the probability of disintegration of C∗ into the final state B + b

and E∗ is the excitation energy of the CN.

The experimental verification of the Bohr hypothesis was accomplished by

S. N. Ghoshal in 1950 [32]. He studied two reactions populating the same

compound nucleus, 64Zn∗, and measured the cross-sections of three different

decay channels, as shown below:

p +63 Cu 63Zn + n

64Zn∗ −→62 Cu + n + p

α +60 Ni 62Zn + 2n (1.1.7)

He demonstrated that by choosing the energy of the proton and of the α-

particle such to produce the compound nucleus with the same excitation

energy, the relative yield of the different decay channels are the same (figure

1.1.3) even if the formation channels are different. They only depend from

the excitation energy of the compound nucleus.

In the next sections the processes of compound nucleus formation and

decay will be treated in detail.


The aim of this section is to derive a simple formula to calculate the fusion

cross-section. A simple estimate of the fusion cross-section is obtained by

the one-dimensional potential model, described in the first section, where

one considers the degree of freedom only of the relative motion between the

colliding nuclei, and it deals with the Schrodinger equation with a static, i.e.,

25


Figure 1.1.3: Cross sections for the reactions shown in 1.1.7. The scales of theupper axis (energy of the protons) and lower axis (energy of the α particle) wereadjusted to correspond to the same excitation energy of the compound nucleus[32]

energy and angular momentum independent, potential for the radial part of

the relative motion for each partial wave and assumes that the fusion oc-

curs when the two nuclei come across the potential barrier into the inner

region, where a potential pocket exists. Once the potential is given, the

direct numerical solution of the Schrodinger equation gives the barrier trans-

mission probability, TL(E), i.e., the fusion probability, which can be well

approximated by using the Wentzel-Kramers-BrillouinWKB formula, valid

for energies both above and below the barrier [33, 34]:

TL(E) = [1 + e2KL(E)]−1 (1.2.8)

26


where the WKB penetration integral is

KL(E) =

√2µ

~2

∫ rL2

rL1

dr[VL(r) − E]12 (1.2.9)

V (r, L) is defined by the relation 1.1.1. In 1.2.9 rL1 and rL2 are the inner and

the outer classical turning points at the potential barrier for the L-th partial

wave potential barrier (see figure 1.2.4). The transmission coefficients are

Figure 1.2.4: Interaction potential between two nuclei as a function or theirseparation distance

related to the partial fusion cross-sections σL by

σL(E) = πλ2(2L + 1)TL(E) (1.2.10)

where λ is the de Broglie wavelength of the relative particle, i.e. it is the

reciprocal of the wavenumber. The cross-sections for complete fusion are ob-

tained by summing over all contributing partial waves (impact parameters):

σfus(E) =∑L

σL(E). (1.2.11)

There are several analytic approximations to the exact quantum tunneling

method discussed above. The approximations are based on the observation

that the barriers are nearly parabolic in shape. The Coulomb plus nuclear

27


parts of the potential V (r, L) (see equation (1.1.1) can be replaced by an

inverted harmonic oscillator potential to give:

V0(r) = V LB +

1

2

d2VL(r)

dr2(r −RL

B)2 (1.2.12)

and by taking into account the relation 1.1.5

V0(r) = V LB − 1

2µω2

L(r −RLB)2. (1.2.13)

In this approximation the transmission coefficient can be written [35]:

TL(E) = [1 + exp[2π

~ωL

(V LB − E)]]−1 (1.2.14)

The (1.2.14) is called Hill-Wheeler formula. In (1.2.14) VL, denotes the bar-

rier height for the Lth partial wave, ~ωL the corresponding barrier curvature

and E the bombarding energy. Ignoring the L dependence of ~ω and of

the barrier position RB and assuming that the L-dependence of V LB is given

only by the difference of the centrifugal potential energy, one can obtain the

following formula for the fusion cross-section [36]:

σW (E) =πR2

B

E

~ω2π

ln[1 + exp[2π

~ω(E − VB)]]. (1.2.15)

It is known as Wong formula. VB is the height of the fusion barrier for the

s-wave scattering. In the classical limit, where ~ω → 0, or if E >> VB the

Wong fusion cross-section reduces to the following expression:

σfus(E) = πR20(1 − VB/E) (1.2.16)

while if E << VB, then

σW (E) =πR2

B

E

~ω2π

exp[2π

~ω(E − VB)]. (1.2.17)

The observed fusion cross-section shows indeed 1 − const/E dependence as

suggested by equation (1.2.16) for some energy region above the Coulomb

barrier (region I) [37]. Figure 1.2.5 illustrates the global energy dependence

of the fusion cross-section. The potential model explains also the behaviour

at high energies (region III). Since the number of partial waves which have

a potential pocket is limited, the fusion cross-section should tend to zero as

28

1.3 Compound Nucleus Decay

Figure 1.2.5: Energy dependence of the fusion cross-section.

the bombarding energy approaches continue to increase. The behaviour at

intermediate energies (region II), where the fusion cross-section descends, has

been discussed, e.g.,in [38] by introducing the concept of a statistical Yrast

line and a critical distance. An important result in equation (1.2.15 is that

only three parameters, RB, VB and ~ω, completely determine the excitation

function of the fusion cross-section over the whole energy region.

1.3 Compound Nucleus Decay

In this section, equations which are the basis for the Hauser-Feshbach sta-

tistical model of nuclear decay will be derived. A complete treatment of the

statistical model in heavy-ion reaction in given in [39]. The Hauser-Feshbach

theory describes the statistical de-excitation of a compound nucleus treat-

ing the decay as completely independent from the formation process. This

approximation could be considered as correct since the compound nucleus

survives for a time interval sufficient to make equiprobable all the quantum

states that the CN may occupy. The process is regulated by the conservation

of energy, angular momentum and parity. In this section the probability P

that a nucleus, with an excitation energy Ex, with a total angular momentum

J and parity π, decays by emitting a particle or a radiation will be deduced.

The procedure will be first to consider for each decay mode the partial rate of

29


decay (i.e. decay per unit time) for the different channels and modes. These

rates can then be converted to normalized probabilities once the total decay

rate and total number of open channels has been obtained. As we will see

the important quantities for determining the cross-section for the population

of a given channel are the transmission coefficients and the level densities of

the mother nucleus and of the daughter one. In the Hauser-Feshbach theory,

for the level density calculation, the classical Bohr and Mottelson equation

[40] is used:

ρ(E, J) =2J + 1

12

√a(

~2

2I)3/2

1

(E − Erot)2exp2[a(E − Erot)

1/2] (1.3.18)

where Erot = (~2/2I)J(J +1) and I is the rigid body moment of inertia. The

modes of decay are labeled by the type of radiation or the type of products

emitted.

First γ-ray decay is considered. The average rate at which an ensemble of

nuclei with initial excitation energies ranging from Ei to Ei + dE, angular

momentum Ji and level density ρ(Ei, Ji) emits γ radiation of energy εγ and

multipolarity λ to produce nuclei with final state energies from Ef to Ef +dE

and angular momentum j may be written [41, 42]

Rγ(Ei, Ji;Ef , j)dE = Cλ(εγ)ε2λ+1γ

ρ(Ef , j)

ρ(Ei, Ji)dE (1.3.19)

where Cλ is a quantity which depends on the multipolarity of the emitted

radiation, εγ is the photon energy and the last term represents the final to

initial levels density.

Let us consider now an ensemble of nuclei in equilibrium with energies from Ei

to Ei + dEi and angular momenta Ji that evaporates particles µ with kinetic

energy ε, spin s, orbital angular momentum l, and leaving the residual nuclei

with excitation energies from Ef to Ef + dEf and spin j. The average rate

of emission, summed over the orbital angular momentum, is [43]:

Revap(Ei, Ji;Ef , j, s)dE =1

h

j+s∑S=|j−s|

ji+S∑l=|Ji−S|

Tl(ε)ρ(Ef , j)

ρ(Ei, Ji)dE (1.3.20)

where S = j + s is the channel spin. The energies Ei and Ef are linked by

Ei = Ef +Sµ + ε, where Sµ is the separation energy for a particle type µ. Tl

30

1.4 Beyond the single-barrier model

is the optical model transmission coefficient for a formation of a compound

nucleus in a time-reversed reaction.

Differently from the previous cases, the decay rate for fission does not depend

on the level densities or other statistical properties of the residual nuclei.

Rather, it depends on properties of compound nucleus at the saddle point

configuration (the point where the nucleus becomes committed to fission).

The fissioning nuclear system thus passes through a transition state where

most of the energy has gone into deformation and the energy available for

intrinsic excitation may be quite small. Thus fission is treated in a manner

almost analogous to particle decay if in (1.3.20) we understand the final

state whose energy is denoted by Ef to be of the transition states. The

transmission coefficients are taken to be unity if the total available energy is

in excess of the fission barrier and zero otherwise. Thus:

Rfiss(Ei, Ji;Ef , j)dE =2Ji + 1

h

ρ(Ef , j)

ρ(Ei, Ji)(1.3.21)

and

Ef = Ei − EB(Ji) − ε (1.3.22)

EB(Ji) is the fission barrier or saddle-point energy and ε is the kinetic energy

at the saddle point.

The average total rate R(Ei, Ji)dE at which a nucleus in a state Ei, Ji decays

is the sum of the rates for all possible transitions that depopulate that state,

that is:

R(Ei, Ji) = Rγ + Revap + Rfiss. (1.3.23)

The probability that any given channel, x, will be populated , P (Ei, Ji;x),

is just

P (Ei, Ji; x) =R(Ei, Ji;x)

R(Ei, Ji). (1.3.24)


The single-barrier penetration model (SBPM) successfully describes fusion

cross-sections for light nuclei [37]. For heavier systems the prediction of

31


the SBPM works rather well at and above barrier energies, but at energies

below the barrier the cross-sections are under-predicted by orders of mag-

nitude, in spite of the fact that the nucleus-nucleus potentials are known

with reasonable precision.[44, 28]. This behavior was observed for the first

time in fusion experiments involving stable nuclei: 16O+148,150,152,154Sm [3],40Ar+144,148,154Sm [4] and 58Ni+58Ni, 58Ni+64Ni, 64Ni+64Ni reactions [5, 6, 7].

In figure 1.4.6 fusion excitation functions for 16O+Sm [45] and Ni+Ni reac-

tions are shown. Balantekin et al. [46] showed the limitations of the simple

Figure 1.4.6: (a) Fusion excitation functions for 16O on Sm isotopes [45] showa marked increase of cross-section with increasing mass number and deformation.(b) The difference in the energy dependence of the cross-sections for the 58Ni+58Ni,58Ni+64Ni, and 64Ni+64Ni reactions [5, 6, 7] might depend on transfer of nucle-ons which affects the fusion process. The dashed lines show the calculated crosssections assuming a single barrier at energy B0.

1D-BPM model, extracting an effective potential by inverting the experimen-

tal data. For the heavier systems this procedure yielded unphysical internu-

clear potentials, while those for lighter ions matched well with phenomeno-

logical models. This demonstrated conclusively that the discrepancies were

not due to the use of the wrong form of potential, rather the problem was

32


that the two interacting ions were considered as structureless objects and not

as two colliding many body systems. In addition, the experimental data of

fusion on Sm (figure 1.4.6(a)) show a rather strong isotopic dependence, that

goes much beyond the simple radius scaling. The increased cross sections

for deformed nuclei such as 154Sm are easy to understand geometrically, by

using a semi-classical interpretation. Compared with the spherical problem,

the Coulomb barrier is lower when the projectile approaches the ”pole” of the

deformed Sm target and is higher when it approaches the ”equator”. Taking

account of all possible target orientations this yields a distribution of barrier

heights, some lower and some higher than the single barrier. The distribu-

tion of barriers leads to an enhancement of the cross-section, since tunneling

or overcoming the lower barriers is much more probable than penetration

through the single barrier. From this simple semiclassical picture it is clear

that the interaction between 16O and 154Sm must include at least one other

degree of freedom, namely the orientation of the deformed target.

Also for the Ni+Ni reactions it has been shown that the fusion excitation

function for 58Ni+64Ni decreases more gradually than for the other two sys-

tems (figure 1.4.6(b)) at energies below the Coulomb barrier. Such a behavior

could not be explained by simple scaling of the isotope sizes. In this case,

the fact that among the three systems, 58Ni +64 Ni is the only one with a

positive Q-value transfer reaction, led to the proposal [47] that specific re-

action channels can affect the fusion process. This recognition inspired the

idea that coupling between the relative motion and any nuclear degree of

freedom (not only static deformation or low-lying collective excitations but

also vibrational modes [48, 49], transfer processes [5, 6, 7, 47] and neck for-

mation [50]) might influence in a significant way the fusion probability. As in

the case of the rotational nucleus, even if the intuition is less clear, one can

imagine that any coupling leads also to a barrier distribution. A complete

discussion of the barrier distribution is made in [9]. In the next section, the

concept of the distribution of barriers will be discussed briefly.

33


1.5 Barrier distribution

Despite of the simplicity of the concept, the subject of fusion barrier dis-

tributions was rarely discussed until it was shown [8] that an experimental

fusion barrier distribution could be determined from measured fusion cross-

sections.

Rowley et al. [8] demonstrated that the distribution of barrier heights in

a reaction could be extracted directly from a fusion excitation function by

using the second derivative of Eσfus as a function of the energy E.

A clear interpretation about the relation existing between the barrier dis-

tribution and the fusion cross-section is given in [51]. These authors, to

elaborate on the physical significance of the distribution of the barriers, used

as a starting point the relation between the fusion cross-section σfus(E) and

transmission probability for the s-wave potential barrier, T0 [52, 53]:

Eσ(E) ∝∫ E

−∞dE ′T0(E

′). (1.5.25)

Differentiating equation 1.5.25 twice, one finds that the energy derivative of

the s-wave transmission probability is approximatively proportional to the

second energy derivative of the quantity Eσ:

dT0(E)

dE∼ d2

dE2[Eσ(E)] (1.5.26)

For a completely classical system, T0 is unity above the barrier and zero be-

low; hence the quantity dT0(E)/dE will be a delta function peaked when E

is equal to the barrier height, as shown in figure 1.5.7(left side). Quan-

tum mechanically (figure 1.5.7-right side), since the transmission probability

smoothly changes from zero at energies far below the barrier to unity at

energies far above the barrier, the sharp peak is broadened. It has been

demonstrated [8] that this broadened peak presents a near gaussian distri-

bution with a width of 0.56~ω. In the case of the two channel coupling,

the quantity dT0(E)/dE is further broadening, owing to the overlapping of

the two peaks, and it can be taken to represent the distribution of barriers,

due to the coupling to extra degrees of freedom, as depicted in figure 1.5.8.

34

1.5 Barrier distribution

Figure 1.5.7: Classical (on the left) and quantum-mechanical (on the right)transmission probabilities for a one-dimensional potential barrier [51].

Thus, according to the relation 1.5.26 the second derivative of d2(Eσ)/dE2

reproduces the barrier distribution, D(E):

1

πR2

d2(Eσfus)

dE2= D(E) (1.5.27)

The degree to which this relation holds has been demonstrated by [45]. In

figure 1.5.9 a set of discrete barriers is shown as vertical lines whose posi-

tions are the barrier energies, relative to the average barrier B0, and whose

lengths represent the probability of encountering that barrier. The dashed

line represents the distribution obtained by smoothing the discrete distribu-

tion with a near Gaussian function with a width of 0.56 ~ω, as predicted

in [8]. In this calculation the radius and curvature ~ω of each barrier are

assumed to be the same. The solid line shows a more realistic calculation of

d2(Eσfus)/dE2 where the barrier radius and curvature depend on the angular

momentum l~ and the barrier energy is included. The differences between the

two curves in figure 1.5.9 demonstrate the degree to which the second deriva-

tive of d2(Eσ)/dE2 differs from the smoothed barrier distribution. Since the

35


Figure 1.5.8: Classical (on the left) and quantum-mechanical (on the right)transmission probabilities for a two-channel coupling. V0 is the height of the one-dimensional potential barrier coupled to these channels[51].

differences are small, it is possible to refer to d2(Eσ)/dE2 as the barrier dis-

tribution.

Prior to the introduction of the concept of the experimental barrier distri-

bution and the first measured distributions, fusion excitation functions were

thought to be smooth and featureless curves, providing a poor test of the-

oretical models [54, 55]. Since then, it has been demonstrated that fusion

excitation functions exhibit subtle structures that carry unique information

on the interactions of the two nuclei at the fusion barrier. For example, the

first measurements made with the aim of extracting fusion barrier distribu-

tions were for the fusion of 16O with the statically deformed nuclei 154Sm

[10] and 186W [13]. It was found that the measurements were sensitive to

the detailed shape of the nuclei since the experimental barrier distributions

were only reproduced when both the β2 and β4 were optimised. The values

giving the best fits to the data [45] were close to those determined by other

methods, showing that the barrier distribution was potentially a valuable

tool to predict the nuclear structure and the dynamic of the reactions.

36

1.5.1 Extraction of Experimental Barrier Distributions

Figure 1.5.9: Barrier distributions associated with a set of discrete barriers withweights represented by the height.The dashed line shows the smoothing effect ofbarrier penetration whilst the solid curve is the second differential of d2(Eσ)/dE2,from a realistic calculation.

1.5.1 Extraction of Experimental Barrier Distributions

The distribution D(E) can be extracted from experimental excitation func-

tions using the point-difference approximation:

D(E) ≈ 2Eσ(E) − Eσ(E + ∆E) − Eσ(E − ∆E)

∆E2(1.5.28)

Obviously, a small energy step ∆E approximates the analytical derivative

better than a large one. However, for a fixed relative experimental uncer-

tainty of the cross-section, δ, the absolute experimental uncertainty of D(E)

is given by

∆D w δEσ

∆E2(1.5.29)

It follows that a large ∆E reduces the uncertainty ∆D. Thus an optimum of

information can be gained from the experimental data when D(E) is extracted

for different values of ∆E ranging from the smallest possible to large energy

steps of the order of the width of the barrier distribution.

37


1.6 Coupled Channel Equations

As it was discussed, to describe the fusion process correctly it is necessary

to take into account the coupling among the different degrees of freedom

(intrinsic excitation of the nuclei, transfer processes, ...) which might influ-

ence the fusion probability. The theoretical method to take into account the

contribution of a number of reaction channels to the fusion cross-section is

the coupled channel formalism (CC). This method consists in describing the

wave function of the system as a sum of a number of components equal to

the number of intrinsic quantum states involved [56, 57]:

|Ψ(+)(α0k0)〉 =∑α

|χ(+)(α0k0)〉|α〉 (1.6.30)

The notation |Ψ(+)(α0k0)〉 indicates that the collision is initiated in channel

α0 with wave vector k0.

The Schrodinger equation for the system is

(E −H)|Ψ(+)(α0k0)〉 = 0 (1.6.31)

H is the total Hamiltonian of the system:

H = h + K + U (1.6.32)

where h is the intrinsic Hamiltonian, K ≡ −~2 52 /2µ is the kinetic energy

of the relative motion and U is the potential. The eigenstates of the intrinsic

Hamiltonian, |α〉 satisfy the Schrodinger equation,

(εα − h)|α〉 = 0 (1.6.33)

and the orthonormality relation

〈α′|α〉 = 0. (1.6.34)

Let’s insert equations (1.6.30) and (1.6.32) into (1.6.31), and take the scalar

product with each intrinsic state 〈α|. We then get the coupled-channel equa-

tions:

(Eα −K − Uα|χ(+)(α0k0)〉 =∑α′

Uαα′|χ(+)(α0k0)〉. (1.6.35)

38

1.6 Coupled Channel Equations

where Eα = E − εα is the energy of the relative motion in channel |α〉 and

the potential U has been divided in Uα and Uαα′ which are respectively the

intrinsic potential of the channel and the interaction potential.

In the coordinate representation, this equation is written as

[Eα +~2

2µO2 − Uα(r)]χα(r) =

∑α′

Uαα′(r)χα′(r) (1.6.36)

Equation 1.6.36 actually represents an infinite set of similar coupled equa-

tions. Usually, one truncates the infinite set of equations to a relatively

few channels which are expected theoretically or known experimentally to

be strongly coupled and either neglects the rest or represents their effect by

including an imaginary part in the channel potential Uα. This truncation is

called closed coupling approximation.

To qualitatively demonstrate the CC effect on the fusion cross-section, it is

reported a two channel example suggested by Dasso et al. [58, 59]. They

considered coupled-channel equations as given by equation 1.6.36 in the spe-

cial case where the entrance channel, α = 0, is coupled to a single additional

channel, α′ = 1. They assume, furthermore, that the potentials Uα and Uαα′

are the same for the two channel considered and neglect any other channel

coupling. The coupled-channel equations thus reduce to the following two:

[E0 +~2

2µ52 −Ueff (r)]χ0 = Ucoupl(r)χ1 (1.6.37)

[E0 +~2

2µ52 −Ueff (r)]χ1 = Ucoupl(r)χ0 (1.6.38)

If the Q-value for populating the excited states is negligible, i.e. is zero,

E0 = E1 = E and the equations above are separated by introducing the

waves function [58]

χ±(r) =1

2[χ0(r) ± χ1(r)] (1.6.39)

into

[E +~2

2µ52 −Ueff (r) ± Ucoupl(r)]χ1 = 0. (1.6.40)

39


Since the entrance channel contains 50%of each eigenchannel considered, the

total flux transmitted for a partial wave L is

T =1

2[T+(L) + T−(L)] (1.6.41)

where TL is the flux transmitted by the potential

Ueff (r) =~2L(L + 1)

2µr2± Ucoupl(r) (1.6.42)

Figure 1.6.10: Upper plot: splitting of the potential barrier by the couplinginteraction. Lower plot: influence of splitting upon the transmission probabilities.The broken curve is the result of penetration through the unmodified barrier ofheight Vb. The full curve gives the penetrability in the presence of the couplinginteraction.

In other words, the total transmission is a weighted sum of the transmis-

sion for the potentials Ueff + Ucoupl and Ueff − Ucoupl, that is the potential

barrier encountered by the incident flux in the no-coupling situation is split-

ted into two barriers by the coupling F (figure 1.6.10) The total transmission

for the coupling and no-coupling cases is also shown in the figure 1.6.10 as a

function of the energy E. For energies below the no-coupling barrier the trans-

mission is increased while for energies above this barrier the transmission is

40

1.7 Sub-barrier enhancement and neutron transfer process

decreased. Thus, coupling enhances fusion below the barrier and diminishes

fusion above the barrier. The splitting into only two effective barriers with

equal weight in this case follows from the simple model discussed. More

generally one could think that the effect of the couplings is to open up the

original barrier into a manifold of effective barriers. In particular the num-

ber of the barriers is expected to be equal to the number of strongly coupled

channels. The weight of each channel depends on the type of coupling in-

volved and on the reaction dynamics. The fusion cross-section would then

result from a weighted average.

1.7 Sub-barrier enhancement and neutron trans-

fer process

The systematic enhancement of the sub-barrier fusion cross-section, observed

for heavy stable nuclei, with respect to one-dimensional calculations has been

explained in terms of couplings to internal degrees of freedom of target and

projectile, hence the sensitivity of the fusion process to nuclear structure has

been recognized [28]. The possibility to extract fusion barrier distributions

from accurate data [8] has brought a substantial advance in the identifica-

tion of the relevant channels acting as doorways to fusion [9]. In particu-

lar, various experiments have shown the importance of static deformations

[10, 11, 12, 13] and of complex surface vibrations [14, 15].

Moreover, the importance of neutron transfer in increasing the sub-barrier

fusion probabilities was also established, but quantitative understanding of

the coupling strengths remained elusive due to the need to rely on complex

theoretical calculations.

The idea that neutron transfer process would lead to an enhancement of the

fusion cross-section is supported by a semiclassical model proposed by Zagre-

baev [16], according to which an intermediate neutron transfer with positive

Q-value may lead to a gain in relative kinetic energy of the colliding nuclei

and, thus, to an enhancement of the barrier penetrability and therefore of

the fusion cross-section.

41


Most of the experiments performed to study the effects of neutron trans-

fer on the fusion cross-section compared systems with positive Q-value for

neutron transfer with their isotopic counterparts with no positive Q-value.

By comparing similar systems one could expect that it is easier to isolate

the effects of transfer couplings with respect to inelastic channels (which are

always present). Many reactions have been studied including 40Ca+46,48,50Ti

[60], 32,36S+110Pd [61], 40Ca+116,124Sn [62], and 40Ca+90,96Zr [63, 64], and

barrier distributions have been extracted to investigate the effects of two-

or more neutron transfer channels with positive Q-values on fusion. In the

following, the 40Ca+90,96Zr [63, 64] measurements will be discussed as typical

example of measurement in which it has been possible to observe the effects

of multineutron transfer on fusion cross-sections.

In addition a recent work [18] concerning the fusion of Sn+Ni and Te+Ni

systems will be briefly discussed. The latter measurements are of particular

interest because the conclusions about the influence of neutron transfer on

fusion differ from previous works.

1.7.1 The 40Ca+90,94,96Zr reactions

To study the effects of neutron transfer on fusion Timmers et al. [63, 64]

chose the 40Ca+90,96Zr reactions since the structure of the two targets are

very similar (they are both spherical nuclei, the lowest 2+ and 3− states are

at comparable excitation energies and have similar transition probabilities).

Moreover, the projectile is a magic nucleus and so its influence on fusion was

expected to be small. According to these properties, the authors came to

the conclusion that the main difference between the two reactions was the

neutron transfer Q-values. Up to eight neutrons can be transferred from 96Zr

to 40Ca with positive Q-value. The corresponding Q-values in the closed shell

system 40Ca+90Zr are all negative and thus transfer might be expected to be

suppressed. In table 1.1 the Q-values for neutron transfer channels, for these

two reactions and for the other reactions which we will discuss in this section,

are reported. Figure 1.7.11(a) shows the measured excitation functions for40Ca+90,96Zr on a reduced-energy scale. A very large relative enhancement is

42

1.7.1 The 40Ca+90,94,96Zr reactions

System +1n +2n +3n +4n +5n +6n40Ca+90Zr -3.61 -1.44 -5.86 -4.17 -9.65 -9.0540Ca+94Zr +0.14 +4.89 +4.19 +8.12 +3.57 +4.6540Ca+96Zr +0.51 +5.53 +5.24 +9.64 +8.42 +11.6248Ca+90Zr -6.82 -9.79 -17.73 -22.67 -31.93 -37.6048Ca+90Zr -2.71 -2.82 -6.63 -8.69 -13.87 -17.00

Table 1.1: Q-values (MeV) for g.s → g.s. neutron pick-up transfer channels forvarious Ca+Zr systems.

Figure 1.7.11: The fusion excitation functions and barrier distributions for40Ca +90,96 Zr reactions. The coupled-channel calculations for 90Zr (solid ) and96Zr (dashed ) including up to four-phonon excitations of the Zr nucleus are shown.Data from [63]; figure from [9]

observed for 40Ca+96Zr at low energies. The barrier distribution is plotted in

figure 1.7.11(c). The barrier structure and the fusion excitation function for

43


the reaction 40Ca+90Zr could be explained in terms of the coupling to single-

and double-phonon excitations of 90Zr (solid lines in figure 1.7.11(a),(b)). In

contrast, these couplings could not explain the behavior of 40Ca+96Zr fusion

cross-section (dashed curves in figure 1.7.11(a),(c)). The authors attributed

the differences to the presence of strong coupling to few- and possibly multi-

neutron transfer channels in 40Ca+96Zr with very positive Q-values.

Later, other authors [65] suggested that the observed difference in the cross-

section could be a consequence of the strong octupole vibration in 96Zr, which

was not taken into account by Timmers et al. This latter argument is not

supported by more recent data [17] obtained using a 48Ca projectile, which is

magic and more rigid than 40Ca (where a strong octupole vibration exists) on

the same two zirconium targets. In this case, the large difference between the

sub-barrier cross-sections disappears. What has changed is that for a 48Ca

beam there are no positive Q-value neutron-transfer channels with either Zr

isotope, as it shown in table 1.

In order to consolidate any possible conclusions arising from these observa-

tions, Stefanini et al. [66] decided to measure near- and sub-barrier fusion

cross-sections in the intermediate case 40Ca+94Zr. The Q-values for few-

neutron pick-up channels in 40Ca+94Zr are very similar (positive) to those

in 40Ca+96Zr and the octupole state in 94Zr is significantly weaker than in96Zr, so the difference in the interplay of collective excitations and transfers

should be reflected in the results.

The 40Ca+94Zr data are shown in the Eσfus vs energy plot on a reduced

scale in figure 1.7.12, together with the other four systems 40,48Ca+90,96Zr.

The 40Ca+94,96Zr systems are greatly enhanced at low energies, with respect

to the other three, and remarkably similar to each other. This similarity

of the two systems can be seen also by observing the experimental fusion

barrier distribution in figure 1.7.13. The three distributions of 40Ca+94Zr

and of 40,48Ca+90Zr are remarkably similar near the barrier, with two main

separated peaks, the lower-energy one being more intense. The authors un-

derlined that this evidence indicates the dominant influence of couplings to

low-lying inelastic excitations in the Zr isotopes in that energy range. The40Ca+94,96Zr systems present a low-energy tail which is not observed in the

44

1.7.2 Fusion of Sn+Ni and Te + Ni

Figure 1.7.12: Fusion excitation functions for Ca+Zr systems [66]

other three systems which is responsible for the very similar behavior of the

two corresponding excitation functions at sub-barrier energies. The authors

concluded that additional coupling modes [9, 45] are needed in order to ex-

plain the behavior of 40Ca+94,96Zr systems. They proposed to identify this

mode with neutron transfer with positive Q-value, which is available only in

these two systems.


The systematic approach of Stefanini et al. provided relatively clear evi-

dence of the presence of transfer coupling effects in a model-independent

way. Recently, the influence of the transfer was investigated with a model

independent approach for Sn+Ni and Te+Ni systems by Kohley et al. [18].

The Q-values for different numbers of neutrons transferred in the reactions

45


Figure 1.7.13: Fusion barrier distributions for Ca+Zr systems [66]

are shown in figure 1.7.14(b). In figure 1.7.14(a) the reduced fusion excita-

tion functions for Sn+Ni and Te + Ni systems are compared.

This comparison shows, within the statistical uncertainty, that no signifi-

cant differences between the reduced excitation functions is observed. The

correlation between sub-barrier fusion enhancement and large Q-values for

multineutron transfer, observed in previous experiments with lighter systems

does not seem to be present in the Sn+Ni and Te+Ni reactions. Koley et al.

[18] suggested the possibility that the increasing importance of deep-inelastic

and quasifission reactions may be playing a role in the Ni+Sn and Te+Sn

cases.

46


Figure 1.7.14: Reduced fusion excitation functions for 132Sn+58Ni [18],132Sn+64Ni [67, 68], 58Ni+124Sn [69, 70], 64Ni+124Sn [71], and 64Ni+118Sn [71]reaction systems. (b) g.s. → g.s. Q-values as a function of the number of neutronstransferred from the Sn to Ni nucleus for each of the reaction systems presentedin (a). The legend shown in (a) also applies to (b). [18]

It is important to underline that the data of the Sn+Ni and Te+Ni reactions,

shown in figure 1.7.14, have been reduced by normalizing the cross-section

with respect to the quantity πR2 and by dividing the energy for the height of

the Coulomb barrier, VBass obtained by the Bass model. The cross-section for

the Ca+Zr systems are compared without any reduction of the cross-section

and the procedure used for the normalization of the energy is different from

the previous case.

Then a possible reason for the different conclusions reached in the two cases

could be attributed to the different kinds of comparison. In any case, as it

was already discussed in the literature by [72, 73], this type of comparison

should always be done by following a general procedure, in order to avoid

conclusions which could result dependent to the particular analysis proce-

dures used by different authors.

The last discussed measurement has demonstrated that the relation between

transfer and fusion process is still an open issue. In conclusion, other fusion

measurements are needed to search for the causes which lead to an enhance-

ment of the fusion cross-section, and in particular to investigate the possible

role of the neutron transfer process.

47


1.8 Fusion and break-up in collisions induced

by weakly bound nuclei

As it was discussed in the previous sections, in reactions induced by heavy

well bound nuclei, near-barrier fusion is governed by the structure of the

interacting nuclei and the coupling to the direct nuclear processes, such as

inelastic excitations and nucleon transfer [28, 9]. For these systems, gener-

ally, the coupling of the relative motion to these internal degrees of freedom,

excluding processes involving transfer, successfully explains the enhancement

of fusion cross sections with respect to the one-dimensional barrier penetra-

tion model (1D-BPM) calculations at sub-barrier energies [9]. The situation

gets more complicated in reactions involving weakly bound nuclei, since other

degrees of freedom may influence the fusion cross-section; in particular the

break-up of the projectile, have to be taken into account. Weakly bound

nuclei are characterized by low break-up thresholds, cluster structure and

diffuse matter distribution. There are weakly bound nuclei either stable, like6Li, 7Li, 9Be, or unstable, like the halo nuclei 6He, 11Li and 11Be. For reac-

tions involving these types of projectiles direct reactions (like break-up and

transfer) are expected to be favored. Concerning the fusion reactions, there

are two different effects which could influence this process: static (linked to

the diffuse structure of such nuclei) and dynamic effects. Dynamic effects are

due to the coupling of the relative motion of projectile and target to their

intrinsic excitations or to other reaction channels (for example break-up and

transfer processes).

Owing to the low intensities of the radioactive ion beams currently available,

experimental investigation of reaction mechanisms with unstable beams is

still difficult. In contrast, precise fusion cross-section measurements can be

carried out with the readily available high-intensity beams of stable weakly

bound stable nuclei. Before giving a summary of the most important results

obtained so far by using stable weakly bound projectiles, it is important to

give some definitions. In the case of fusion reactions induced by these nu-

clei two independent fusion processes can be distinguished: Complete Fusion

(CF) and the InComplete Fusion (ICF). These two types of fusion processes

48

1.8.1 Theoretical models

are connected to the dynamics of the projectile fragments. From a theo-

retical point of view the CF is defined as the process in which all nucleons

from the projectile and the target fuse forming the compound nucleus. In

the case of ICF only a part of the mass of the projectile is captured by the

target, while one or more fragments fly away from the interaction region. Ex-

perimentalists tend to define complete and incomplete fusion as absorption

of all the charge of the projectile or of a part of it, respectively. The sum

of the CF and ICF cross-sections is usually defined total fusion (TF) cross-

section. Incomplete fusion has been described in a two-step process picture

where the projectile is first broken into two or more fragments by Coulomb

and/or nuclear forces and then some of the fragments penetrate the barrier

and fuse with the target. Strictly speaking, events where all fragments fuse

with the target nucleus after break-up are also possible (and experimentally

indistinguishable from complete fusion): in this case this is referred to as

sequential complete fusion (SCF). A schematic representation of the fusion

and break-up processes that can take place in a collision induced by weakly

bound projectile is shown in figure 1.8.15.

In the following the CDCC model used for taking into account the role of

the break-up process on fusion will be described. Then some of the most sig-

nificative experimental results obtained by using stable weakly bound nuclei

on heavy, medium and light mass targets will be described.

1.8.1 Theoretical models

A theory for reactions induced by weakly bound nuclei should take into ac-

count both static and dynamic effects. All the authors agree that the presence

of the diffuse structure which characterises these nuclei affects the shape of

projectile-target potential, reducing the Coulomb barrier and therefore en-

hancing the fusion cross-section. A lot of theoretical studies have been done

in order to understand the role of these dynamic effects on the fusion cross-

section with halo nuclei. In particular, an intense debate arose about the

possible effect of break-up, mainly because of the way the mechanism was

described. Theoretically, the inclusion of the break-up channel represents a

49


Figure 1.8.15: Schematic representation, from [74], of the fusion and break-upprocesses that can take place in the collision of a weakly bound projectile. Forsimplicity is assumed that the break-up produces two fragments.

particular challenge. Break-up is essentially an excitation of the nucleus to

an energy higher than the corresponding threshold; if the calculation treats

it in the same way as an inelastic excitation to a bound state, it will indeed

lead to an enhancement of the tunnelling probability. However, the excita-

tion is into the continuum, in many cases not even to a resonance and thus

with no well-defined values for the energy and spin of the final state.

The role of the break-up process of a weakly bound projectile on fusion

has been investigated via Coupled Channel (CC) calculations. Recently in

these calculation in order to properly describe the reaction dynamics for

collisions involving weakly bound nuclei the coupling involves not only ex-

citation to bound states but also to continuum states (break-up states).

Continuum-Discretized Coupled-Channel (CDCC) calculations discretise the

continuum into energy bins ([75, 76]). As an example, in the framework of

CDCC calculation Hagino et al. [77] investigated the effect of the break-up

in 11Be+208Pb system. These calculations included the initial 2s1/2 bound

50

1.9 Experimental results

state of the last neutron in 11Be, and the break-up channel, but neglected

continuum–continuum couplings, in order to simplify the calculations. Their

results are reported in figure 1.8.16. They concluded that coupling to con-

Figure 1.8.16: Fusion cross-section for the reaction 11Be+208Pb as a functionof the bombarding energy in the center of mass frame. The thin slid curve showsthe results of the one-dimensional barrier penetration. The solid and the dashedlines are solutions of the coupled-channel equations for the CF and CF+ICF,respectively [77].

tinuum causes (1) an enhancement of the complete fusion cross-section at

energies below the average barrier (as in the familiar case of heavy tight nu-

clei, where inelastic excitation is involved), and (2) a suppression of the CF

at energies above the barrier. Diaz-Torres arrived at the same conclusion in

[78], by performing calculations where continuum-continuum couplings were

included.

1.9 Experimental results

As it was already mentioned, beams of stable weakly bound nuclei which

have low break-up threshold are very suitable for investigating the role of

51


break-up because, thanks to their high intensities, they allow measurements

to be performed with good statistic and precision. A lot of fusion measure-

ments involving these nuclei with light, medium and heavy mass targets have

been performed. Table 1.2 itemises the most energetically favorable break-

up channels for some stable weakly bound nuclei, with the relative break-up

Q-value. In the last column, for each nucleus, are listed the targets which

were used so far to perform the fusion measurements.

ProjectileBreak-up Break-up Heavy Medium/light

configuration Q-value targets targets6Li α + d -1.47 MeV 90Zr [79] 12,13C [80, 81, 23]

144Sm [22] 16O [82, 83]159Tb [21] 24Mg [84]208Pb [85] 27Al [86]209Bi [87] 28Si [88]

59Co [89]64Zn [90, 86]

7Li α + t -2.45 MeV 159Tb [91] 12,13C [80, 81, 23]165Ho [25] 16O [82, 83]209Bi [87] 24Mg [84]

27Al [86, 92]28Si [93]59Co [89]64Zn [90, 86]

9Be 8Be+n -1.67 MeV 89Y [94] 27Al [95, 86, 96]→ α + α + n 124Sn [97] 64Zn [98, 90]

144Sm [99]208Pb [100]209Bi [101]

10B 6Li+4He -4.46 MeV 159Tb [91]209Bi [102]

11B 7Li+4He -8.66 MeV 209Bi [102]

Table 1.2: Some fusion reactions performed by using as projectile a weakly boundnucleus.

52

1.9.1 Fusion reactions involving heavy targets

1.9.1 Fusion reactions involving heavy targets

In reactions involving weakly bound projectiles and heavy targets a reduction

of the CF cross-section is observed at energies above the barrier,by comparing

the CF cross-section both with calculation (CC or single barrier penetration

calculation) or with reactions induced by well bound nuclei.

As an example, the fusion excitation function for the 9Be+208Pb system [87]

is reported in figure 1.9.17, with the results of CC calculations. The com-

Figure 1.9.17: The excitation function for complete fusion (filled circles) for thereaction 9Be+208Pb from [100]. The dashed line is the result of a CC calculationthat ignores break-up effects. The full line is the same calculation scaled by a 0.68factor. The empty circles give the sum of measured CF and ICF cross-sections.

plete fusion excitation functions show a suppression at energies above the

barrier both with respect to the 1D-BPM and to CC calculations of about

30%. Such a reduction was attributed to the projectile break-up in the strong

Coulomb field of the target nucleus, followed by ICF. This idea is supported

by the fact that the measured incomplete fusion cross-sections is very similar

to the cross-section ”missing” from complete fusion. The empty circles in

53


figure 1.9.17 represent the sum of CF and ICF. It is possible to see that the

predictions of the coupled channel fusion calculations match the measured

values, suggesting a direct relationship between the flux lost from fusion and

the incomplete fusion yields.

A complete systematic study of the suppression factor for CF (FCF ) as a

function of the product of the charge of projectile and target (ZPZT ) for

different reactions involving weakly bound nuclei has been reported recently

by Gasques et al. [102]. After this work other measurements have been

performed, to investigate this systematic further. The results for these new

systems, along with other reactions reported earlier [102], are shown in fig-

ure 1.9.18. It is possible to observe that, except for the case of 9Be, the

suppression factor is more or less independent of the product ZPZT . In fig-

ure 1.9.19 the FCF , for reactions induced on 209Bi targets, has been reported

as a function of the break-up threshold of the projectile. A remarkably con-

sistent correlation is seen, suggesting that the break-up process might have

an influence on fusion cross-section at energy above the barrier: the lower

the break-up threshold is the larger is the FCF .

1.9.2 Fusion reactions involving medium and light tar-gets

Reactions on medium and light mass targets have produced different results.

For 6,7Li and 9Be on 64Zn [103] the CF seems not to be affected by the break-

up process, no reduction of the fusion cross-section was observed within the

experimental uncertainties. For 9Be+64Zn collision the total reaction cross-

section was found almost equal to the total fusion cross-section. On the

contrary, the reaction induced by 6,7Li on the same 64Zn target have shown

a total reaction cross-section much larger than the total fusion cross-section

[103].

In the study of reactions between light-weakly-bound nuclei, apparently con-

tradictory results have been published. As an example according to [104],

fusion excitation functions in collisions between weakly bound nuclei, show

suppressed values at energies above the barrier. For the system 6Li+9Be the

54

1.9.2 Fusion reactions involving medium and light targets

Figure 1.9.18: The complete fusion suppression factor FCF at above-barrierenergies as a function of the charge product of projectile and target (adopted from[102]). The reactions considered are 6Li incident on 90Zr [79], 144Sm [22], 159Tb[21], 208Pb [85], and 209Bi [87] targets; 7Li incident on 159Tb [91], 165Ho [25], and209Bi [87] targets; 9Be incident on 89Y [94], 124Sn [97], 144Sm [99], 208Pb [100],209Bi [101]; 10B incident on 159Tb [91] and 209Bi [102] targets; and 11B incidenton a 209Bi [102] target.

authors find values of the fusion cross-section which are about 50% lower

than the calculated ones. However other authors [83, 23], studying similar

reactions as [104] do not find hints indicating the presence of strong suppres-

sion effects on the fusion cross-section.

It is worth noting that, the fundamental difference between the reaction per-

formed on heavy and light targets is that in the former case, where the emis-

sion of charged particles is hindered by the presence of a strong Coulomb bar-

rier, the CF and ICF contributions can be usually clearly separated since they

produce evaporation residues of different charge (see e.g. [24, 99, 85, 21]). In

contrast, for fusion of weakly bound nuclei on medium or light mass targets,

the same residue isotope can be populated by the CF and ICF contributions,

55


Figure 1.9.19: FCF for reaction induced on 209Bi as a function of the projectilebreak-up threshold

making their separation much more difficult. Therefore, in all of the inves-

tigations with low-Z targets, the sum of the complete and incomplete fusion

(i.e. the total fusion) is obtained and compared with theoretical expecta-

tions.

However, a clear evidence that fusion cross-section for light systems is not

influenced by the break-up process was obtained by comparing 10B+12C and13N+9Be fusion cross-sections above the barrier [105]. The two colliding sys-

tems are very similar, but in the latter both the projectile and the target are

weakly bound nuclei. If, due to the low biding energies, the system breaks-up

before fusion occurs, one may expect to observe a suppression of the fusion

cross-section for 13N+9Be when compared to 10B+12C. The experimental ER

relative yields for the two reactions were found very similar and in good agree-

ment with the predictions of the statistical model (see figure 1.9.20). This

excludes the possibility of having a strong incomplete fusion contribution in

the 13N-induced reaction even if weakly bound nuclei are involved.

56

1.10 Motivation of the PhD project

Figure 1.9.20: Calculated (histogram) and measured (symbols) ER charge dis-tribution in % for the reactions a) 10B+12C and b) 13N+9Be [105]


This PhD work is part of a wider project which aims at studying the 6Li+120Sn,7Li+119Sn, 8Li+118Sn and 9Li+117Sn fusion reactions at energies around the

barrier for investigating the possible effects of the neutron transfer and break-

up on the fusion process.

In the present work the study and the results concerning the first two systems

will be presented. Both the experiments were performed at the Laboratori

Nazionali del Sud (Catania), by using an activation technique based on the

off-line measurement of the atomic X-ray emission following the electron cap-

ture decay of the evaporation residues (ER) produced in the reactions. The6Li and 7Li beams were delivered by the SMP Tandem Van de Graaff accel-

erator, with an average beam intensity of about 1010 pps.

The choice of these Li+Sn combinations seems to be particularly suitable for

investigating the role of the neutron transfer at energies below the Coulomb

barrier. As it was explained in the previous sections, the effect of this par-

ticular channel on the fusion process is not yet clear because it is not trivial

to develop a model which takes into account the coupling to transfer. From

an experimental point of view, the best way to investigate the role played

by neutron transfer is to study fusion reactions for systems characterized by

similar structures, for which the influence of inelastic channels on the fusion

57


Reactions Q, 1n pickup from Li Q, 2n pickup from Li6Li+120Sn 0.51 MeV -12.3 MeV7Li+119Sn 1.86 MeV 2.36 MeV8Li+118Sn 4.45 MeV 6.30 MeV9Li+117Sn 5.26 MeV 9.71 MeV

Table 1.3: Q-values (MeV) for 1n and 2n transfer in 6,7,8,9Li+120,119,118,117Snreactions.

process is similar. In this way differences arising from a comparison of the

fusion excitation functions are imputable to coupling to the transfer channel.

However, so far, contradictory results have been reported in the literature.

Some authors (see e.g.[17, 66]) concluded that the observed enhancement

in some systems can be related to the positive sign of the Q-value for neu-

tron transfer. On the contrary, in recent fusion measurements of Sn+Ni and

Te+Ni systems [18], no particular differences have been observed among the

sub-barrier fusion excitation function for these systems, even if they are char-

acterised by very different Q-values for multi-neutron transfer.

Our systems present different Q-values for neutron transfer (see table 1.3)

and are charaterised by similar structure. The target nuclei are spherical and

proton shell closed and thus the influence of the target inelastic excitation

on fusion should be comparable for all the Sn isotopes under investigation.

This means that for these systems the influence of target excitation upon

fusion probability should be comparable. In our case, the nuclei involved

in the different fusion reaction not only present similar structures but, by

choosing different combinations of Li and Sn isotopes, we assured that also

the kinematic conditions in the entrance channel are as close as possible.

The reduced mass of the systems and therefore their approaching velocity

in the center-of-mass system is very similar and the distance of closest ap-

proach (Rmin) is the same for all of them (see figure 1.10.21). Therefore by

comparing their fusion excitation functions below the barrier, we would like

to isolate possible effects that could be attributed to the different neutron

transfer Q-values. The compound nucleus formed in the Li+Sn fusion reac-

58


Figure 1.10.21: Schematic view of the collisions, object of the present work, andthe related n-transfer Q-values.

59


configuration Q-value (MeV)6Li α + d -1.4757Li α + t -2.4688Li 7Li + n -2.0329Li 8Li + n -4.063

7Li + 2n -6.096

Table 1.4: For each Li isotope, the channel with lowest break-up threshold andthe respective Q-values are listed.

tions is the same for all the systems, i.e. the 126I. At the energies at which

we are interested in, the 126I decays by evaporating only neutrons. This is an

important feature since it guarantees the discrimination between the com-

plete and incomplete fusion processes and thus the correct determination of

the complete fusion excitation function.

A feature of our systems, with respect to those previously measured, is the

low break-up threshold for the projectiles. In table 1.4 the channel with

lowest threshold of break-up for each projectile are reported along with the

respective Q-value. Thus, in these reactions also the coupling with the break-

up channel has to be taken into account. CDCC calculations will help in

isolating the contribution due to the break-up channel.

If on one hand the projectile break-up may cause some complications for

interpretation involving the role played by the neutron transfer, on the other

hand these measurements can help to investigate further the role of break-up

at energies above the barrier. As discussed in the previous sections, measure-

ments of fusion in which low break-up threshold projectiles are involved have

shown a suppression of the complete fusion cross-section at energies above

the barrier with respect to the 1D-BPM or experimental data concerning

collisions induced by well bound nuclei on the same targets. It has been

understood that there is a dependence of the complete fusion suppression

factor FCF on the break-up probability of the projectile (figure 1.9.19): the

lower is the break-up threshold the higher is the suppression factor. For 6Li

and 7Li it has been observed that the CF suppression factor is independent

60


Figure 1.10.22: Schematical representation of a stack set-up, used for fusionmeasurement at low energy or with radioactive nuclei.

on the charge of the target (see fig 1.9.18), in the mass range investigated so

far (Z > 60).

With the study of the 6Li+120Sn and 7Li+119Sn at energies above the barrier

it is possible to investigate further the behavior the 6Li and 7Li complete

fusion suppression factor on a different target charge.

The study of these two fusion reactions has allowed not only to provide part

of the data for the survey but also to investigate in detail the experimental

problems linked to the activation technique. This kind of technique is gener-

ally used for measuring the fusion cross-section at energies around and below

the Coulomb barrier. At these energies, in fact, the direct detection of ER is

very difficult since, owing to the their low kinetic energy, a large fraction of

them will not come out from the target or, if they do, their energy is often

too small to detect them. The only way to detect these ER is by using an

indirect technique, which consists in the detection of particles and radiation

originated in the decay of the residual nuclei which are stopped inside the

irradiated target.

In many cases, rather than a single target, a stack of targets, that is an

ensemble of targets each one alternated with a catcher (figure 1.10.22), is ir-

radiated. This setup is particularly suitable in experiments with low-intensity

radioactive beams because it allows a significant reduction in the beam time

needed to perform an excitation function measurement; this is an important

61


factor since radioactive beams are generally of low intensity. By using a

stack rather than a single target, it is possible to extract the cross-section

at different energies without changing the beam energy and also to extend

the fusion cross section measurements to energies lower than the incident

beam energies available. The catcher placed after each target is used to stop

the small fraction of residues which emerge from the previous target and to

increase the average difference in beam energy for the different targets.

Despite this great advantage, this technique presents several drawbacks which,

if not considered, may lead to the wrong determination of the fusion exci-

tation function. The main drawback of the activation technique is related

to the determination of the effective energy, Eeff , that is the energy which

should be associated with the measured cross-section, for the reconstruction

of the fusion excitation function. The fusion cross-section measured in a

thick target is in fact the result of the integration over a finite energy range.

Usually authors associate with the measured cross-section the beam energy

calculated at the centre of the target. This approximation could be consid-

ered good at energies above the barrier where the fusion cross-section does

not have a strong energy dependence. At energies below the barrier, where

the cross-section varies exponentially with the energy, the effective energy

can be determined only by using an averaging procedure, by weighting the

energy not only with the trend of the fusion excitation function but also with

the energy distribution inside the target.

The width of the energy distribution inside the target depends on the energy

loss in it and on the finite energy dispersion of the incoming beam. The de-

termination of the energy distribution is not trivial when using a stack since

it depends not only on the features of the single target, but also from the

ones of the targets placed upstream the considered one. In fact it is affected

by the energy distribution of the beam emerging from the previous targets.

Another factor which could influence the determination of the effective en-

ergy in a target is the target non-uniformity, which contributes further to

give a different weight to the energies inside the target.

Since we will use the stack setup for the reactions induced by the radioactive8Li and 9Li beam, we are interested in the investigation of this problems. A

62


part of this thesis has been devoted to investigate the effects that the pres-

ence of the stack and of possible target non-uniformities could generate on

the fusion excitation function.

The writer took part to the preparation and to the experiments and has been

responsible for the data analysis.

63

Chapter 2

Experimental technique andsetup

2.1 Activation technique for fusion measure-

ments

As it was already mentioned, the best way to measure the fusion cross-section

at energies around and below the Coulomb barrier is to use an activation

technique, based on the detection of particles and radiation originated in the

decay of the residual nuclei which are stopped inside the irradiated target. In

this work fusion cross-section was measured by using an activation technique

based on the off-line detection of the atomic X-ray emitted after the electron

capture (E.C.) decay of the ER produced in the reaction. This technique

was used for measuring the fusion cross-section of the systems 6He+64Zn

[106, 107]. By using the statistical code PACE 4 we estimated the expected

yields for different evaporation channels. These calculations show that the

majority of the yield corresponding to complete fusion for the two reactions

under study is shared between 123I, 124I which corresponds to the 3n and

2n evaporation channels. These two residues are unstable against electron

capture decay.

The experimental technique consists of two steps: the target activation and

the measurement of the characteristic X-rays.

During the activation step, a target and a catcher, placed behind it, are irra-

65

2. Experimental technique and setup

diated with the beam. The thickness of the catcher is chosen in order to stop

completely the fraction of residues which emerges from the target. Possible

reactions induced by the beam on the catcher do not represent a problem

since the associated X-ray energies are different from the ones corresponding

to reactions on the target. At the end of the irradiation time, the target and

catcher are placed in front of an X-ray detector for measuring the activity of

the ER

2.2 Target activation

2.2.1 Experimental setup

The activation step of the measurement was performed in the CT2000 scat-

tering chamber of LNS with the 6Li and 7Li beams delivered by the SMP

Tandem van de Graaf accelerator, with an intensity of about 1010pps. In

figure 2.2.1 a schematic representation of the experimental setup used during

the activation procedure is shown.

Figure 2.2.1: Sketch of the experimental setup for the activation procedure. Thebeam passes through a thin Au foil in order to perform current normalization withthe Rutherford scattering. A catcher follows the Sn target in order to stop therecoiling ER.

For the 6Li+120Sn reaction we have measured the fusion cross-sections

66


Set-up Beam Incident Sn Catcher Catcher Irradiation

Energy thickness element thickness time(MeV) (µg/cm2) (mg/cm2)

STACK 1 6Li 25 470 93Nb 2.4 80 min

6Li 23.81 480 93Nb 2.46Li 22.59 490 93Nb 2.46Li 21.31 495 93Nb 2.4

STACK 2 6Li 21 550 93Nb 1.95 2 days6Li 19.85 485 93Nb 1.946Li 18.86 480 93Nb 1.956Li 17.74 430 93Nb 1.92

SINGLE 6Li 17.5 440 93Nb 2.47 1 day

SINGLE 6Li 18.7 460 93Nb 2.495 12 h

SINGLE 7Li 25 420 165Ho 2.33 80 min7Li 23.8 500 165Ho 2.30 80 min7Li 22.6 450 165Ho 2.30 80 min7Li 21.3 482 165Ho 2.42 80 min7Li 19.8 440 165Ho 2.35 10 h7Li 18.7 430 165Ho 2.24 20 h7Li 17.4 425 165Ho 2.28 48 h

Table 2.1: Summary of the targets stacks used in the activation step. Thethickness have been measured by alpha particle energy loss

by using both the stack and the single foil configuration. By comparing the

measured cross-section with the two different setups we aim to investigate

the possible experimental problems due to using a stack configuration rather

than a single target. This information is of great importance in view of the

measurement with the unstable 8Li and 9Li beams, where the use of a stack

becomes necessary.

The fusion cross-section measurement for the 7Li+119Sn reaction was per-

formed by using a single foil. The used targets and catchers with their thick-

nesses are listed in table 2.1. The thicknesses of the targets and catchers

have been obtained by measuring the energy loss of 5.48 MeV α particles,

67


emitted by an 241Am source, in traversing the foil. The energy loss has been

measured as the difference between the mean energy of the α peak measured

with and without the target. The error in the calculations was estimated by

using two energy loss programs (SRIM, LISE) which agreed to each other

within 1% for the 197Au, 119,120Sn and 165Ho foils and within 2% for 93Nb

foils.

In the seventh column of table 2.1 the irradiation time for each stack/target is

shown. The irradiation time needed to maximize the production of an evap-

oration residue is equal to about 3 - 4 times the life time, τ , of the nuclide of

interest. In fact, as shown in figure 2.2.2, at this time the formation rate of

Figure 2.2.2: Number of produced radiactive nuclei as a fuction of the irradiationand decay time

an ER becomes comparable with the rate with which the nuclei, previously

created, decay. Activating for a longer time does not yield a greater gain in

the production of these radioactive nuclei. However, to have good statistic, it

is not always necessary to reach the secular equilibrium. For example, if the

cross-section of the ER is large, enough activity can be produced in a shorter

time. Moreover, for ER having lifetimes of few days or longer reaching the

secular equilibrium would need an irradiation time that is too long. There-

fore the choice of the irradiation time is a compromise between the different

life-times and cross-sections of the ER produced in the reaction. In any case,

68


at low energies, where the fusion cross-section is expected to be small, it is

important that the beam current is optimized to the available experimental

time.

As it will be shown in the next chapter, to extract the production cross-

section it is necessary to measure the number of incident particles on the

target. This value can be obtained simply by integrating the charge accu-

mulated in a Faraday cup, placed downstream the target. This method is

suitable for measuring fusion cross-sections in a collision in which the pro-

duced evaporation residues have a much longer life time with respect to the

irradiation time. In this case it is not necessary to know the beam profile

of irradiation. On the contrary, the extraction of the beam intensity profile

with time is important in the case of ER with a half-life shorter than, or

comparable with, the duration of the activation period. In fact, at the end

of the activation, the total amount of short-lived elements present in the tar-

get is the result of the competition between the formation of new ER by the

fusion process and their decay during the activation period. The beam inten-

sity measurement as a function of the time has been performed by detecting

the elastically scattered particles from a thin gold foil (120 µg/cm2) placed

before the stack on the beam line, using two 1000µm Surface Barrier silicon

detectors (see figure (2.2.1)). These two monitor detectors were symmetri-

cally placed at ±20 with respect to the beam line and at a distance of about

80 cm from the Au foil. Since the elastic scattering at this angle follows

the Rutherford law, the beam intensity can be extracted by the well-known

cross-section formula. By using two symmetrical monitors it is possible to

reduce systematic errors due to mechanical misalignments and small beam

position shifts. In the stack-holder, in addition to the stack, a beam stopper

is placed in order to prevent the scattered particles emerging from the stack

interacting with the monitor detectors.

In order to measure the elapsed time, a pulser with 5Hz frequency was also

triggering the data acquisition system.

69


2.2.2 Readout electronics

The acquisition electronic is sketched in figure 2.2.3. The signals from the

monitors are processed by charge preamplifiers, then shaped and amplified

by Ortec 572 amplifiers and sent to the ADC. The Ortec 474 fast amplifiers

provide a timing signal for each channel. A 5 Hz pulser was used to generate

the reference for the elapsed time. The pulse generator signal was split into

two signals: one signal was sent to an amplifier and then to the ADC, the

other signal was sent to a discriminator.

The two timing signals and the pulser were sent to a discriminator and the

output sent to a total OR to produce the valid event signal. Two counters

were used to monitor on-line the beam intensity using the scattered particles

detected by the monitors.

A scaler was used to check the acquisition dead time by looking at the ratio

between the total number of valid events and the acquired ones. The total

number of events from the pulser and the monitors were also counted by the

scalers. A manual latch was used to start and stop the acquisition and scalers

at the same time.

70

2.2.2 Readout electronics

Figure

2.2.3:Electronic

chain

usedto

process

thesignals

from

thedetectors

71


2.3 Characteristic X-rays measurement


In order to measure low-energy gamma rays or X-rays, detectors with a deep

active layer are needed to maximize the probability of photoelectric absorp-

tion. Si-Li drifted detectors are suitable for this purpose thanks to their

thick depletion layers that varies from 5mm to 10mm. Moreover, their en-

ergy response to low energy X-rays is linear and their energy resolution is

sufficiently good ( 250 eV) to allow a very good separation of the X-ray en-

ergy components. In Si(Li) detectors the compensated region is sufficiently

thick that, at room temperature, the fluctuations in the leakage current can

be a significant source of noise. For this reason, and also to avoid the pos-

sibility of gradual redistribution of the drifted lithium at room temperature,

these systems have to be continuously maintained at low temperature, typ-

ically that of liquid nitrogen temperature (77 K). They are insulated from

the surrounding ambient by vacuum.

For the present experiment the activity was measured with two Si-Li drifted

detectors (which are labeled for convenience Detector 1 and Detector 2) and

their technical details, provided by the manufactor, are summarized in table

2.2.

DIMENSIONS

Dectector Active Diameter 16 mm

Detector sensitive depth 5.50 mm

Detector-to-Be window 7 mm

Beryllium-window diameter 18 mm

ABSORBING LAYERS THICKNESS

Beryllium 0.05 mm

Gold 200 A

Silicon dead-layer 0.1µm

Table 2.2: Technical data of the two Si(Li) detectors used.

The detectors were surrounded by a lead shield in order to reduce the

72


background, as shown in figure 2.3.4.

Figure 2.3.4: (left) The experimental apparatus for the X-ray detection. A leadshield surrounds the detector in order to reduce the background. (right) Theactivated stack is positioned above the beryllium window at the detector top usingthe stack holder.

Immediately after the end of each activation, the stack is taken from

the reaction chamber and each foil is placed in front of one of the Si(Li)

detectors. A sketch of the experimental setup is shown to the right of figure

2.3.4. Each activated foil is placed in a target holder in order to fix its

position with respect to the detector and reduce all possible uncertainties due

to geometrical factors. The target holder allows for five distances between

the activated foil and the beryllium window (from about 4 mm to about 112

mm). The indetermination on the distance value is ±1 mm.

In fact for all the activity measurements only the lower position of the stack

holder was used. The exact knowledge of the geometry of the entire apparatus

is needed in order to correctly determine the detector geometric efficiency.

73


2.3.2 Readout Electronics

The energy signals coming from each Si(Li) detector are readout by a charge

sensitive preamplifier. The energy signals from the detector preamplifiers

were shaped and amplified by Ortec 672 amplifiers and acquired by an 8-

channels CAMAC ADC. The second output from the detector preamplifiers

was processed by a second amplifier in order to generate a timing signal. All

timing signals were sent to a discriminator. A total OR of the discriminator

outputs was used to give the valid event trigger for the acquisition. A scaler

was used to check the acquisition dead time by the difference between the

total number of valid events and the acquired ones. A latch was used to start

and stop the acquisition and the scalers at the same time. A 5 Hz pulser

was used to generate the reference for the elapsed time. The pulse generator

signal was split into two signals: one signal was sent to an shaping amplifier

and then to the ADC, the other signal was sent to a fast time amplifier and

then to a discriminator to be used to generate the trigger signal. Using the

pulser, one can easily correct the measured activities for the dead-time, as

it will be explained in the next chapter. In figure 2.3.5 a schematics of the

electronics for the two Si(Li) detectors is shown.

74

2.3.2 Readout Electronics

Figure

2.3.5:Electronic

chainusedto

process

andacquirethesignals

from

thedetectors

duringtheX-ray

smeasurement.

75


2.4 Si(Li) energy calibration

In order to calibrate the energy response of the Si(Li) detectors over a wide

range of energies, two X-ray sources were used: a Fe and a Cd source which

emit the kα and kβ lines of Mn and Ag, respectively. The energies of the

X-rays emitted from the two sources are listed in table 2.3:

Fe(keV) Cd (keV)

kα 5.894 22.101

kβ1 6.489 24.92

kβ2 25.455

Table 2.3: Energies of the X-rays used for the energy calibration.

The energy calibration was obtained from a linear best-fit, i.e. E =

a ∗ ch + b performed on the correlation between the X-ray peak position in

channel, evaluated by a Gaussian fit and the corresponding X-ray energy. A

calibrated X-ray energy spectrum is shown in figure 2.4.6.

Figure 2.4.6: A Si(Li) calibrated energy spectrum of the Fe and Cd X-ray sources.

76

2.5 Si(Li) detector efficiency

2.5 Si(Li) detector efficiency

As it will be discussed in the next chapter, to extract the activity for each

ER produced in the fusion-evaporation process, the total X-ray detection

efficiency must be known. This is defined as that fraction of events emitted

by the source which are actually registered by the detector i.e.

εtot =events registered

events emitted by source(2.5.1)

The absolute efficiency can be factorised into two parts, the intrinsic effi-

ciency, εintr and the geometrical efficiency,εgeo:

εtot = εintrεgeo (2.5.2)

The intrinsic efficiency usually depends on the detector material, the radi-

ation energy, and the physical thickness of the detector in the direction of

the incident radiation. The geometric efficiency depends entirely on the ge-

ometrical configuration of detector and source. In figure 2.5.7 the intrinsic

efficiency given by ORTEC for the used Si(Li) detectors is shown.

Figure 2.5.7: Si(Li) detector intrinsic efficiency versus incident X-ray energy

77


2.6 Characterization of the Si(Li) detector

The geometric characteristics of the whole experimental apparatus is funda-

mental in order to correctly calculate the total detection efficiency.

By using a 55Fe source a test was undertaken to cross-check the values of the

distance between the Be-windows and the detector surface and of the size of

the detector as given in the manufacturer data sheet.

2.6.1 Determination of the detector crystal position

By placing the 55Fe source in the different positions of the target holder, the

efficiency was determined as a function of the distance between the source and

the detector. The measured efficiency was corrected for the X-ray absorption

by the Be-window, Au-layer and Si dead-layer. By using a Montecarlo simu-

lation code the expected value of the geometrical efficiency was calculated for

each position of the source. If the geometry determined by the simulation is

correct, the ratio of the total efficiency, measured by relation (2.5.1), to the

geometrical one should be the same at all heights. In fact this ratio represent

the intrinsic efficiency of the detector which is independent from any geomet-

rical factor. We have found that this occurs only for a value of the distance

between the Be-window and the crystal surface which is Dexp = 7.6 ± 0.1

mm, i.e. larger than the one tabulated in the data sheets (see table 2.2).

As it is possible to see from figure 2.5.7 the intrinsic efficiency of these detec-

tor for the X-ray energy emitted by the 55Fe source (5.86 keV) is expected

to be ≈ 100%. We have determined a value of (95 ± 2)% for the detector

2 and of (65 ± 2)% for detector 1. The efficiency value has been obtained

by averaging the values obtained at different positions. The respective error

has been calculated as the standard deviation of these values. It has been

found that this difference between the two detectors is due to the fact that

the active surface of the detector 1 is smaller that the value reported in the

manufacturer data sheet, as it will be explained in the following.

78

2.6.2 Determination of detector effective area

2.6.2 Determination of detector effective area

The effective detector surface has been evaluated by studying how the effi-

ciency for the X-ray of 55Fe varies by varying the opening (from 1 to 13 mm)

of a diaphragm which is placed in front of the Be window [108]. A graphical

representation of the obtained intensities for the 5.86 keV line vs the irradi-

ated detector area is given in figure 2.6.8 for detector 1 and for detector 2

respectively. As expected, for small irradiated areas the intensity increases

linearly. At a certain opening, the intensity does not increase anymore. This

identifies the sensitive detector area that corresponds to the intensity satu-

ration point. It is interesting to note that for the detector 2 the active area

corresponds to that given by the manufacturer (200 mm2). For the other

detector the area is much smaller (125 mm2). This result explains the lower

efficiency of detector 1 with respect to the other one.

Other authors have already observed this kind of inefficiency. They have

explained such behavior as being due to a very thick silicon dead layer at the

edge of the crystal [108].

2.7 Si(Li) efficiency for Tellurium kα X-rays

The X-rays which are best matched to this experiment are the kα characteris-

tic X-rays of Tellurium (27.30 keV), produced in the complete fusion reaction

in which we are interested in. As it is possible to see from figure 2.5.7 the

intrinsic efficiency in the 27 keV region depends strongly on the energy. The

best way to determine the εtot is to measure it with a calibrated source of

the same or near the energy of the tellurium X-rays.

The only calibrated sources available to us in this energy region were a 241Am

source, which emits X-rays of 26.34 keV and a 137Cs source which emits X-

rays of 32 keV (kα X-ray) and of 36.32 keV (kβ X-ray). Thus, in order to

determine the total efficiency for the energy region of interest the following

procedure was adopted.

For the three different energies of the available calibrated sources, the total

efficiency was measured by using the relation 2.5.1, and calculated by using

79


Figure 2.6.8: Rate of the 55Fe X-rays (5.86 keV) vs irradiated area of detector 1(up) and detector 2 (down) area.

80


Figure 2.7.9: Measured efficiency of the 241Am and 137Cs calibrated sources, forthe two Si(Li) detectors used in the experiment.

a Montecarlo simulation program. It is important to underline that, as it

was seen in the case of the 55Fe source, it was found that the total efficiency

of the detector 1 is always lower than for detector 2 (see figure 2.7.9).

By taking as reference the detector 2, we compared the calculated ef-

ficiency with the measured value and found that the calculated value well

reproduced the experimental one, for all the three considered cases. This

agreement between the simulation and the measured value for the calibra-

tion source gave us confidence in using the Montecarlo simulation code to

calculate the efficiency of the kα X-rays of the Tellurium whose energy is

close to the one of the calibration source.

In the following, some details concerning the Montecarlo simulation will be

described.

As it has been mentioned at the beginning of the section, the total efficiency

of a detector can be factorized into two terms (see relation (2.5.2)), the ge-

ometrical and the intrinsic efficiency. To take into account the geometrical

effects we have to know: (1) the geometry of the activated region in the tar-

get, (2) the position of this area with respect to the centre of the detector,

(3) the distance between the target and the detector.

81


Concerning the geometry and the position of the activated area in the tar-

get, for targets activated with long activation time, this region was visible

to the eye. A circular diffused spot with a diameter of 2 ± 0.5 mm, quite

well centered could be seen. Thus, for determining the εgeo the diameter of

the activated area was set equal to 2 mm and placed perfectly centered with

respect to the centre of the detector.

As distance between the Beryllium window and the detector surface, we set

the one that we have measured, Dexp (see previous section) and not the one

reported in the manufacturer data sheets.

Concerning the intrinsic part of the efficiency, it represents the fraction of the

X-rays impinging on the detector which interacts by photoelectric absorption

in the crystal. The expression for the intrinsic efficiency of the crystal for a

mono-directional and mono-energetic beam is:

fphoto = (µphoto/µtot)[1 − exp(−µtotT )] (2.7.3)

where µphoto and µtot are respectively the photoelectric and the total linear

attenuation coefficients for the silicon and T the detector thickness. It should

be noted that the expression 2.7.3 is valid only in case of highly directional

beam, in which the cross-sectional area of the beam at the detector’s back

surface is smaller than (or equal to) the cross-sectional area of the detector.

When the beam has a significant divergence, there will be a distribution of

detector thicknesses ”seen” by the photons impinging on the crystal and the

use of a single value, T , in equation 2.7.3 is no longer accurate. Moreover,

due to the divergence of the beam it could be also that a fraction of the beam

emerge from the side of the crystal, so the traversing path will be different

from the detector thickness, T .

In order to take into account the different absorption probabilities for each

path, an absorption probability function was inserted in the simulation pro-

gram, given by the relation 2.7.3 where instead of T there is, as free param-

eter, the path length.

The Montecarlo simulation was also used to evaluate the uncertainty on the

total efficiency, by determining how the efficiency is influenced by the geo-

metrical uncertainties of the experimental system. This has been undertaken

82


by estimating the geometrical efficiency variation due to the variation of the

distance r between the irradiated spot and the centre of the target and by

varying the distance, D, between the target and the detector surface.

Our experimental apparatus, for the measurement of the characteristic X-

rays, has been constructed in such a way that the position of the activated

region with respect to the centre of the detector and the distance between

the detector surface and target could be measured accurately. But in each

measurement there are some systematic errors which are difficult estimate.

For example, even if there is a good beam collimation system, small displace-

ments with respect to the target centre are possible. Moreover, it was found

that due to the way the targets were manufactured, when inserted into the

target-holder the distance between them and the detector surface can vary

by ±1 mm.

The only way to take into account these types of systematic errors is by as-

sociating them with the geometric efficiency error.

Let us assume D0 as the nominal distance between the detector surface and

the target: D0 = Dexp+DBe−T , where Dexp is the measured distance between

the beryllium windows and the detector surface, and DBe−T is the distance

between the beryllium windows and the target, that has been measured by

a caliber.

In figure 2.7.10 the geometrical efficiency is plotted as function of D − D0,

that is a function of the actual target displacement from the nominal value

D0. Each colored curve in the graphic has been calculated for each different

position, r, of the irradiated spot with respect to the centre of the target

(r0 = 0).

Observing figure 2.7.10, one can conclude that the geometric efficiency

depends strongly on the distance D between the detector and the target:

the small indetermination (±1 mm) in the relative distance between the

detector and target corresponds to a considerable indetermination in the

geometric efficiency (∆εD ∼ 10%). The effect due to the indetermination on

the displacement of 1 mm of the activate region with respect to the centre is

smaller (∆εr ∼ 2%).

The possible effect on the geometrical efficiency due to the variation of 1 mm

83


Figure 2.7.10: Geometric efficiency variation obtained by changing the relativeposition (D−D0) between the target and the detector surface. The coloured linesrepresent the geometric efficiency obtained by varying the relative position, r− r0,between the centre of the activated region and the centre of the target.

of the diameter of the irradiated area has also been estimated: this resulted

in a variation in efficiency (∆εs) of about 5%.

The total error on the geometric efficiency is the combination of the errors

due to the spot radius ∆εs, the spot displacement with respect to the centre

of the target ∆εr and the target-detector distance ∆εD:

∆ε =√

∆ε2s + ∆ε2D + ∆ε2r (2.7.4)

The total detection efficiency for the detector 2 for the Tellurium kα X-ray

energy has been estimated to be equal to 0.0326 ± 0.0058.

84

Chapter 3

Data analysis

In this chapter the analysis of the experimental data will be discussed. In

particular the procedure used for the extraction of the ER production cross-

sections, σER, will be described. The fusion cross-section will be obtained by

summing the contributions of all ER produced inside the target:

σfus(E) =∑i

σERi(E) (3.0.1)

The compound nucleus formed in both fusion reactions studied in the present

thesis is 126I. In table 3.1 the evaporation residues, predicted by the statisti-

cal code CASCADE, with their half-lives and evaporation channels are listed.

Residues Decay chain t1/2124I 2n 4.17 d123I 3n 13.22 h

Table 3.1: Evaporation residues, predicted by the statistical model code CAS-CADE, produced in both fusion-evaporation reactions 6Li+120Sn and 7Li+119Sn.

It is worth emphasizing again that the expected evaporation residues in

this study are produced only by neutron evaporation: this ensures that the

discrimination between complete and incomplete fusion is possible.

In the following it will be described the analysis procedure used for: 1) iden-

tifying the evaporation residues produced in the complete fusion reaction; 2)

determining the activity at the end of the target irradiation time for each

ER; 3) determining the beam intensity as a function of time.

85

3. Data analysis

Finally, the production cross-section for each ER will be extracted and the

relative yield for the different residues will be compared with the one pre-

dicted by the statistical model code CASCADE [109].

3.1 The radioactive decay law and the pro-

duction cross-section

Let’s consider a target where N0 radioactive nuclei are present at time t=0.

If no nuclei are introduced into the sample then the number of undecayed

nuclei present at a certain time t is given by

N(t) = N0e−λt (3.1.2)

where λ is called decay constant and the relation (3.1.2) is known as expo-

nential law of radioactive decay. The law in this form is of limited usefulness

from an experimental point of view, because N is a very difficult quantity to

measure. Instead of counting the number of undecayed nuclei in the sample,

it is easier to count the number of decays (by observing the emitted radia-

tions) that occur in a time t. Let’s define the activity of a radioactive nuclide,

at a certain time t, as the rate at which decays occur in the sample:

A(t) = λN(t) = A0e−λt (3.1.3)

where A0 = λN0 is the initial activity at time t=0. It is easily demonstrated

that if the time interval, t, during which the activity is measured, is much

smaller than the radioactive nuclide half-life, the activity is given by the ratio

between the number of decays measured in that time interval and the time

interval:

A(t) =Decayed nuclei in a time interval ∆t

∆t(3.1.4)

3.2 X-ray spectra analysis

The first steps in the data analysis for the fusion cross-section measurement

are the identification of the evaporation residues produced in the reaction

86


and the determination of their activity. As it was explained in the previous

chapter, since the direct detection of ER is not possible, this information can

be obtained by detecting the characteristic X-rays emitted in the electron

capture decay of these evaporation residues, by using Si(Li) detectors. Two

Si(Li) detectors were used and each target was measured in turn in both the

detectors in order to minimize the probability of systematic errors due to the

experimental setup. The irradiated targets were measured off-line for about

six months.

In figure 3.2.1 typical X-ray spectra for the 6Li+120Sn and for the 7Li+119Sn

reactions are shown. In the spectrum for 6Li+120Sn, it is possible to observe

Figure 3.2.1: Typical X-ray spectra for the 6Li+120Sn (on the left) and 7Li+119Sn(on the right) reactions. In the spectrum corresponding to the 6Li induced reaction,besides the X-ray produced in the E.C. decay of the ER produced in the interactionwith the target, it is possible to distinguish the ones coming from the E.C. decayof the ER produced in fusion reactions beam with the Nb catcher.

peaks in two different energy ranges. Those at higher energies are the X-ray

peaks which correspond to the E.C. decay of evaporation residues produced

in the 6Li+120Sn reaction; the ones at lower energies correspond to the ER

produced in the reaction of the same beam with the Nb catcher. The latter

peaks are at energies low enough to be separated from the peaks of interest.

In the case of 7Li+119Sn reaction, the catcher material is holmium. In the

energy spectrum we can only see the contribution of ER produced in the7Li+119Sn reaction, because the energies of the X-ray coming from the E.C.

87

3. Data analysis

decay of ER produced in the fusion reaction with 165Ho catcher are much

higher than the ones of interest.

In figures 3.2.2 and 3.2.3 are shown some X-ray spectra for 6Li+120Sn and

for 7Li+119Sn reactions, respectively, in the energy region of interest, and for

different times after irradiation.

Figure 3.2.2: X-ray spectra measured off-line, 20 minutes, 2 hours, 13 hours and6 days after the end of the irradiation, for the reaction 6Li+120Sn at ELAB= 25MeV. It is possible to distinguish the kα and kβ X-rays emitted in the E.C. decayof Iodine and Antimonim produced respectively in the complete fusion process andin the incomplete fusion process of the d with the target.

It is possible to distinguish the kα and kβ X-rays produced in the decay

of Iodine (the only element which can be produced after complete fusion) to

Tellurium. In the present experiment, the analysis was performed only on

the kα lines, taking into account that they represent about the 80% of the

88


total X-ray emission.

Figure 3.2.3: X-ray spectra measured off-line, 20 minutes, 2 hours, 24 hours and6 days after the end of the irradiation, for the reaction 7Li+119Sn at ELab= 25MeV. It is possible to distinguish the kα and kβ X-rays emitted in the E.C. decayof Iodine and Antimonim produced respectively in the complete fusion process andin the incomplete fusion process of the t with the target.

In the spectra it is also possible to distinguish the kα and kβ peaks of Sb,

produced by incomplete fusion of the deuteron (in the case of 6Li) or tritium

(in the case of 7Li) fused with the target, which then decays by E.C. into Sn.

The experimental activity in counts/hour has been measured by using the

relation (3.1.4), that is

Aexp(t) =N(t)

∆T(3.2.5)

In order to extract the activity it is necessary that this can be considered

constant in the time interval of the measure. This condition was fulfilled in

89

3. Data analysis

the data analysis.

The number of decays, N , was obtained by subtracting from the total number

of counts of the peak of interest, C, the number of background events, B. As

can be seen, the background level in figures 3.2.2 and 3.2.3, is very small. The

background under the Kα peak has been evaluated by integrating a region

near the peak of interest but where there is no evidence of other peaks. In

such a way, the average background contribution per channel was extracted

and then multiplied by the number of channels where the peak of interest

contributes. The uncertainty on C and B corresponds to the statistical error:

δC =√C ; δB =

√B (3.2.6)

The uncertainty on N is given by applying the error propagation on the

relation N = C −B

δN =√

[δC2 + δB2] =√C + B (3.2.7)

The duration of the acquisition time, ∆T , is easily determined by the follow-

ing relation:

∆T (s) =P

5Hz(3.2.8)

where P is the number of pulser counts acquired.

The uncertainty of the activity extracted by relation (3.1.4) is obtained by

the statistical error of the peak integral N by assuming no uncertainty on

the time:

δAexp =δN

∆T(3.2.9)

Using the acquired pulser counts to determine the run time, also allows

the correction of the measured activities for the dead-time. Let D be the

dead time correction factor, that is the same for N and P .

The activity is given by:

A =N ·D∆T

=N · 5 ·DP ·D

=N · 5

P, (3.2.10)

that does not depend on the dead time.

90

3.3 Activity curve analysis

3.3 Activity curve analysis

Since, the resolution of the X-ray spectra is such that it is not possible to

resolve the isotopic shift, from the X-ray lines only different elements can be

identified. Each isotopes contributes to the overall activity with its charac-

teristic half-life, hence the overall activity as a function of the time is the

sum of several exponential contributions. This means that contributions of

different isotopes can be unfolded by following the activity of the X-ray lines

as a function of the time.

In figures 3.3.4 typical activity curves for the 6Li+120Sn and 7Li+119Sn reac-

tions are plotted: the measured activity at a time tmiis plotted against the

time tmi, defined as

tmi= t1i +

t2i2

(3.3.11)

where t1i is the time elapsed between the end of irradiation and the be-

ginning of the activity measurement, and t2i is the time interval in which the

activity was measured.

The open dots and closed squares represent respectively the data of de-

tector 1 and 2. In order to plot together the data from the two detectors

the measured activity has been corrected for the detector efficiency. As it

was explained in section 2.8 the efficiency εT for detector 2 was determined

by using a combination of calibration sources and a Montecarlo simulation.

Since the geometry of detector 1 is not well characterized, for this detector

the efficiency was determined by normalizing the data to those of detector 2.

In figure 3.3.4 the ratio

Aexp

εT(3.3.12)

is plotted as a function of time, and where εT is the total efficiency of the

detector. The error of each point is determined by applying the error prop-

agation given by the relation 3.3.12:√(δAexp

εT)2 + (

AexpδεTε2T

)2 (3.3.13)

91

3. Data analysis

Figure 3.3.4: Typical activity curves for the 6Li+120Sn (on the top) and for the7Li+119Sn (on the bottom) reaction. The data corresponding to the detector 1(open dots) have been normalized to the ones of the detector 2 (closed square).See text for details.

92

3.4 Beam intensity profile

In figure 3.3.4, it is possible to observe three different slopes which char-

acterize these curves. To identify the isotopes, the activity curve was fitted

with the following function:

A(t) = A01expe−λ1t + A02expe

−λ2t + A03expe−λ3t (3.3.14)

The fits have been performed by leaving as free parameters A01,A02,A03 and

assuming λ1 = λ123I and λ2 = λ124I since 123I and 124I were the residues

expected to be produced in the fusion-evaporation process. Concerning the

third component, initially we assumed λ3 = λ125I , since 125I could be pro-

duced by the evaporation of 1 neutron. By using this component we were

not able to fit the experimental data.

The X-ray energy emitted by the E.C. decay of a particular Iodine isotope

corresponds to the one of the daughter isotopes Tellurium. However, Tel-

lurium isotopes can be produced also in the incomplete fusion of α with the

target nucleus; in particular 123Te is produced in the case of 6Li+120Sn reac-

tion and the 121Te in the case of 7Li+119Sn. These two nuclei are produced

from the 1 neutron and 2 neutron evaporation channels; they are metastable

(with half-life respectively of 119 days and 164 days) and decay by internal

conversion thus emitting X-rays. In figure 3.3.5 are shown the fits of the

activation curves, previously shown in figure 3.3.4, by using 123Te and 121Te

isotopes as third component. By fitting the activation curves for each ER

one obtains A0exp, which is its activity at the end of the irradiation time.

Of course, the experimental activity is only a fraction of the real activity of

the residue. The values that we have obtained have already been corrected

for the detector efficiency, but they have also to be corrected for the fluores-

cence probability, kα, since the electron capture decay is in competition with

other decay modes.

3.4 Beam intensity profile

As it was explained in section 3.2, in order to extract the cross-sections for

the production of various short-lived nuclides, it was necessary to monitor the

beam current as a function of time, during the activation run. To determine

93

3. Data analysis

Figure 3.3.5: Fit of the activity curve: it is possible to disentangle the contribu-tion due to the 123I and 124I produced the and 122Te (top) and 121Te (bottom).

94

3.5 ER production cross-section extraction

the incident beam current, during the activation, the elastic scattering from a

thin gold foil at small angles, where the scattering is known to be Rutherford,

was measured by using two monitor detectors (see figure 3.4.6). At the same

time a time signal produced by a pulse generator with a fixed and stable

frequency has also been acquired. By using this pulser signal as a time

reference, we have stored elastic scattering events as function of time.

The beam current Nb is linked to the scattered particles by the relation:

Nb =M

Nt(dσdΩ

)Ruth∆Ω∆T(3.4.15)

where M is the number of counts under the elastic peak in a time interval

∆T , Nt is the number of atoms per cm2 in the gold foil, (dσdΩ)Ruth is the

Rutherford cross-section in the laboratory system and ∆Ω is the solid angle

covered by the monitor detector

∆Ω ∼=π(φ

2)2

d2(3.4.16)

where φ is the monitor collimator diameter and d its distance from the

gold target.

As an example, in figure 3.4.6 the beam current profile of one7Li+119Sn run

is shown. This figure shows the two profiles extracted from the monitor 1

and monitor 2 (see figure 2.2.1 for the set-up).


The half-lives of the ER which we are interested in are, in general, compara-

ble to the duration of the irradiation time.

By knowing how the current varies with time, it is possible to evaluate the

number of radioactive nuclei which decay during the irradiation time. The

number of radioactive residual nuclei produced in an irradiation time inter-

val, ∆tj, is proportional to Nb(∆tj)Nt∆tj, where Nb(∆tj) is the number of

incident particles at time ∆tj, and Nt is the number of target atoms per cm2.

The number of nuclei that decay in the same time interval ∆tj, is propor-

tional to Nj−1e−λ∆tj , where Nj−1 is the number of nuclei in the target at time

95

3. Data analysis

Figure 3.4.6: Beam profile extracted from the monitor detectors for a 7Li+119Snrun.

∆tj−1. The number of the total produced nuclei for each unit of cross-section

is

N0 = Ni(∆t1)Nt∆t1 +∑j=2

[Ni(∆tj)Nttj + Nj−1e−λ∆tj ] (3.5.17)

and so the corresponding activity for unit of cross-section is

A0 = λN0. (3.5.18)

Then, the production cross-section for a radioactive nucleus whose life-time

is shorter than or comparable to the irradiation time can be obtained as the

ratio between the measured activity A0exp and the activity per unit of cross-

section constructed from the knowledge of the beam intensity as a function

of time, A0

σER =A0exp

kαA0

(3.5.19)

96


where kα is the fluorescence probability taken from [110]. By using the

previous relation we have determined the production cross-section for 123I

and 124I. The results are reported in tables 3.2 and 3.3.

6Li+120Sn123I 124I

ECM(MeV ) σ ± δσ(mb) σ ± δσ(mb)

23.68 343.05 ± 51.46 38.39 ± 5.7621.61 240.75 ± 36.11 40.88 ± 6.13

21.51 145.5 ± 21.88 38.13 ± 5.72

20.38 72.425 ± 10.86 29.13 ± 4.37

19.84 55.16 ± 8.27 24.77 ± 3.72

18.85 18.225 ± 2.73 12.99 ± 1.95

17.83 3.57 ± 0.54 4.54 ± 0.68

16.77 0.43 ± 0.06 1.025 ± 0.15

Table 3.2: Production cross-section of ERs produced in the 6Li+120Sn reaction.

7Li+119Sn123I 124I

ECM(MeV σ ± δσ(mb) σ ± δσ(mb)

23.53 436.60 ± 65.49 33.38 ± 5.0122.38 314.75 ± 47.21 31.09 ± 4.66

21.25 211.60 ± 31.74 28.51 ± 4.28

20.02 87.34 ± 13.10 17.16 ± 2.57

18.60 21.65 ± 3.25 7.10 ± 1.06

17.57 4.58 ± 0.69 2.49 ± 0.37

16.33 0.44 ± 0.07 0.40 ± 0.06

Table 3.3: Production cross-section of ERs produced in the 7Li+119Sn reaction.

The uncertainties in the cross-section reflect the statistical and the sys-

tematic errors arising from the determination of the beam current, the de-

tection efficiency, the decay branching ratios and the target thickness. The

97

3. Data analysis

uncertainty on the experimental activity is given by the relation (3.3.13).

The uncertainty on the decay constant and on the fluorescence probability

are the ones reported in the NNDC database. The uncertainties on the num-

ber of incident particles corresponds to the statistical and systematic error

on the normalization of (dσelast/ dσRuth) in each time bin, ∆tj (∆tj = 1 min),

and it is about 5%. The error in the target atoms per cm2 arises from the

uncertainty in the target thickness measurement and is about 2%.

The experimental relative yields for the produced residues have been com-

pared with the prediction of the statistical model code CASCADE. In figure

3.5.7 the comparison is shown for some of the measured energies for the6Li+120Sn and 7Li+119Sn reactions. An excellent agreement between the

experimental data and the calculation predictions can be observed.

98


Figure 3.5.7: ER production relative yields for the 6Li+120Sn (left) and7Li+119Sn (right) reactions compared with the prediction of the statistical modelCASCADE.

99

Chapter 4

Detailed study of the stackactivation method

4.1 Drawbacks of the activation technique

As it was discussed in Chapter 2, the fusion cross-sections for the 6Li+120Sn

and 7Li+119Sn sytems were measured by using an activation technique, based

on the off-line measurement of the atomic X-rays emission following the elec-

tron capture decay of the residues produced in the fusion-evaporation reac-

tions.

It was also explained that in the case of measurements performed at low

energies or by using low-intensity radioactive beams instead of irradiating

one target per time it is necessary to irradiate a stack of targets (see figure

1.10.22). In this way it is possible to extract the cross-section at different

energies without changing the beam energy, thus reducing the beam time

needed to perform an excitation function measurement.

Being the next step of the project to perform the experiments with radioac-

tive 8Li and 9Li beams, we planned to measure the fusion cross-section for

the 6Li+120Sn reaction activating both single foils and stacks of targets. In

this way, the acquired data were useful both for investigating the behavior of

the complete fusion excitation function at energies around the barrier and for

studying in detail the stack activation technique, focusing on its drawbacks

and trying to find the necessary solutions.

The main drawback of the activation technique is related to the determina-

101

4. Detailed study of the stack activation method

tion of the effective energy, Eeff , that is the energy at which the measured

a cross-section should be associated. The fusion cross-section measured in a

thick target is in fact the result of a integration over a finite energy range.

Usually authors associate with the measured cross-section the beam energy

calculated at the middle of the target. This approximation is not completely

correct, in particular at energies below the Coulomb barrier, since it does not

take into account that each energy, inside the target, has a different weight

due to the energy dependence of the fusion cross-section, σ(E), and of the

energy distribution inside the target, D(E), which is not constant.

Thus, to determine the effective value, Eeff , for each measured cross-section,

one needs an averaging procedure which takes into account the σ(E) and the

D(E) energy dependence:

Eeff =

∫ Ef

EiE · σ(E) ·D(E)dE∫ Ef

Eiσ(E) ·D(E)dE

(4.1.1)

To our knowledge this kind of approach has been adopted for the first time by

Wolski et al. [111], who used the stack activation technique to measure the

fusion excitation function for the 6He+206Pb system. The authors assumed

that the energy distribution inside the target was gaussian, centered at an

energy in the middle of the target. The width of the distribution, inside

each target, was determined by postulating a linear relation of the energy

dispersion as a function of energy. By knowing the width of the energy distri-

bution before and after the stack, they determined this relation empirically.

The integration range [Ei, Ef ] was calculated by a simple energy loss calcu-

lation assuming a monoenergetic beam impinging in the target. As it will be

explained in the following, this calculation leads to a wrong determination

of the range, especially if a stack of targets is used. The obtained effective

energy Eeff was shifted towards higher energy with respect to the energy

value calculated at the centre of the target. The authors also observed that

the lower the energy the larger was the shift. This behavior is mainly due

to the energy dependence of the excitation function since the cross-section

decreases exponentially while traversing the target.

According to our studies, the approach of Wolski et al.[111] presents some

102


limitations. First of all, they determined the energy distribution inside the

target in a too approximate way. Intuitively the choice of a gaussian energy

distribution could seem correct since it reflects the statistical nature of the

energy loss process, but the assumption that the sigma varies linearly with

the energy and the way in which they determined the relation are arguable.

Moreover we found that in some cases the energy distribution inside the tar-

get is not gaussian. Figure 4.1.1 shows the simulated energy distributions,

expected for a single Sn foil of 0.5 mg/cm2, irradiated with 6Li beam of 18.45

MeV with a resolution ∆E/E equal to 5·10−3, and for a Sn foil placed in a

stack. The simulations were performed by using the SRIM Monte-Carlo pro-

gram [112].

In particular we used the TRIM (Transport of Ions in Matter) program. It

is a Monte-Carlo calculation which follows the ion into the target making

detailed calculations of the energy transferred to every target atom collision.

In the case of the stack the incoming energy distribution has been chosen

such that the average energy before the considered stack was the same as for

the single foil irradiation. As it is possible to see, in the case of single foil

the energy distribution inside the target has not a gaussian shape. We found

that this kind of distribution is typical of target irradiated by a beam whose

input energy distribution Full Width at Half Maximum (FWHM) is smaller

than the energy loss , ∆E, in traversing the target. On the contrary when

∆E is larger than the FWHM, the energy distribution inside the target has

a gaussian shape, as in the case of the stack irradiation.

It is worth to be underlined that not only the shape but also the range of the

two distributions are different. This means that, even if the average incom-

ing energy and the target thickness are the same, the measured cross section

is integrated in different energy regions in the two cases. This difference is

due to the different width of the two incoming distributions. In view of this

observation, the integration range in equation 4.1.1 cannot be determined by

simply performing an energy loss calculation, as it was done in [111], since

it would lead to an underestimation of the real range. As example, the re-

gion inside the red lines in figure 4.1.1 represents the energy range estimated

by performing an energy loss calculation, without taking into account the

103


Figure

4.1.1:Sim

ulated

energy

distrib

ution

insid

eatarget

assumingincomingbeam

with

anenergy

resolution

of5·10

−3

(top)an

dabeam

incid

enton

atarget

after

passin

gthrou

ghasta

ck.See

textfordetails.

104


finite width of the energy distribution. This is a good approximation in the

case of single foil irradiation but it is half of the range observed in the stack

case. Thus, if the finite energy width of the incoming beam is not taken into

account one gets a wrong determination of the effective energy and conse-

quently of the fusion excitation function. Of course this effect becomes more

and more important by increasing the number of targets in the stack used.

It is important to emphasize that the energy spread in the distribution shown

in figure 4.1.1 is only due to the statistical nature of the energy loss process.

A further contribution to the broadening of the energy distribution can arise

from the non-uniformity of the foil thickness which will cause a variation in

the average energy loss and will result in an additional energy spread in the

distribution inside the target and also in the outgoing beam.

Wolski et al. [111] reported that their targets are non-uniform, but the effect

of the foil thickness non-uniformity on the beam energy distribution in the

stack targets was not taken into account explicitly. A systematic account

of such non-uniformities and the effect on the energy distribution in target

stack systems was never reported previously in literature.

In the present PhD work an accurate analysis has been undertaken for inves-

tigating the problem of target non-uniformity and the effect that this might

have on the energy distribution inside the target and, consequently, on the

measure of the fusion excitation function. As it will be shown in next sections,

due to the target non-uniformity the determination of the energy distribu-

tion inside the target is not trivial. The presence of target non-uniformities

not only can lead to a broadening of the gaussian distribution but also it can

dramatically change the expected shape for the energy distribution inside the

target. Of course, these effects are amplified when the stack is used, since the

energy distribution inside the target does not depend only on the features of

the single target, but also on the ones of the targets placed upstream of the

considered one.

105


4.2 Target uniformity

Interesting information on the uniformity of the foils used for our measure-

ments have been obtained by the analysis of the α residual energy spectra of

transmitted α particles, which we used for measuring the thicknesses of the

foils.

The targets used to perform the fusion measurements of 6Li+120Sn and7Li+119Sn have been made by evaporating Sn enriched isotope on rolled foils

of 93Nb (in the case of 120Sn) or 165Ho (for 119Sn). In figures 4.2.2 the energy

spectra of α-particles, emitted by an 241Am source, passing through respec-

tively a 93Nb, a 165Ho, a 120Sn+93Nb and a 119Sn+165Ho foils are shown (black

line); in red are shown the spectra obtained by SRIM simulations considering

a uniform target with the measured average thickness.

As it is possible to see the simulation never reproduces the experimental

data, but while in the case of the 93Nb and 165Ho foils the difference be-

tween the two measured and simulated curves is the width of the gaussian

distributions, in the case of 120Sn+93Nb (119Sn+165Ho) foils, the shape of

the experimental distributions are not Gaussian anymore. It seems that the

non-uniformity is not something random in the surface of the target, as one

would expect, but it seems the α-particles experience large variation in the

thickness of the target.

Some rough calculations have been done for estimating the range of thick-

nesses that the α should pass through for producing this type of spectra. In

figures 4.2.3(a) the spectra for α-particles passing through the 93Nb (red line)

and the 120Sn+93Nb (black line) are shown together. The spectra have been

normalized with respect to the area for the sake of comparison. In the blue

axis, the 120Sn thickness (in mg/cm2 and in µm) which α-particles should

pass through in order to produce the measured residual energy, is shown.

According to this calculation the thickness of the Sn varies from about 100

µg/cm2 (0.140 µm ) to 1.5 mg/cm2 (2 µm).

In figure 4.2.3(b) the same plot has been made for the 119Sn+165Ho target:

the Sn thickness range is almost the same also for this target.

Since α-particle spectra present the same behavior both in the case of 93Nb

106


Figure 4.2.2: Experimental (black line) and simulated (red line) energy spectra ofα-particles emitted by an 241Am source passing through a (a) 93Nb, (b) 165Ho, (c)120Sn+93Nb and (d) 119Sn+165Ho foil. Black and red lines represent respectivelythe experimental spectrum and the simulated one obtained by using the SRIMprogram. See text for details.

107


Figure 4.2.3: Energy spectra of α-particles passing through a (a) 93Nb (∼ 1.90mg/cm2 ) and 120Sn+93Nb foil; (b) 165Ho (∼ 2.4 mg/cm2 ) and 119Sn+165Hofoil. The two spectra have been normalized with respect to the area for sake ofcomparison. In the blue axis, the 120Sn thickness, which the α-particles shouldpass through for producing the observed residual energy, is reported.

108


and 165Ho substrate, this suggest that this large energy straggling does not

depend so much on the substrate but is maybe characteristic of the evap-

orated Sn. To be sure, we performed the same kind of measurement on a64Zn (284 µg/cm2) target evaporated on a rolled 165Ho (2.345 µg/cm2) foil.

In figure 4.2.4 the spectra for α-particles passing through the 165Ho and the64Zn+165Ho targets are shown. In this case both spectra present a Gaussian

shape, confirming the idea that the straggling observed in the case of Sn is

characteristic of the evaporated Sn. As we have already observed, the exper-

imental width is larger with respect to the simulated one, due to the target

non-uniformities. In order to study the shape of these spectra we decided

Figure 4.2.4: Experimental (black line) and simulated (red line) energy spec-tra of α-particles emitted by an 241Am source passing through a (a) 165Ho and(b) 64Zn+165Ho foil. Black and red lines represent respectively the experimentalspectrum and the simulated one obtained by using the SRIM program.

to analyse the surface of our targets by performing a Scanning Electron Mi-

croscopy (SEM) analysis by using a Zeiss Supra 25 microscope. The SEM

analysis has been performed in a laboratory of the Institute of Microelectron-

ics & Microsystems (CNR-IMM) at the Dipartimento di Fisica e Astronomia

of the University of Catania. In figures 4.2.5-4.2.6 the SEM images of the

targets are reported.

By looking at these pictures one can notice that:

109


Figure 4.2.5: SEM plan view of the 93Nb and 165Ho side of target.

1. The surface of both substrates (Ho and Nb) does not show any particular

features on the scale of µm. The observed surface morphology is typical of

rolled foils and small defects seems randomly distributed on the foil surface.

2. In the Sn surface, it is possible to distinguish structures of different sizes,

which varies from some hundreds of nm to about 2 µm. The size of this

structures seems to be compatible with the shape of the α-spectra, shown in

figures 4.2.3. To be sure that this structures are Sn structures we performed

also a Rutherford BackScattering (RBS) analysis on our samples. From this

analysis we did not obtain any information on the thickness of the Sn and of

the substrate, since it was not possible to resolve Sn from the substrate, but

from the RBS spectrum it was clear that on the surface is present only Sn.

3. The Zn surface is characterized by the presence of hexagonal structures

whose size is about 200 nm, uniformly distributed on the surface.

In the light of the SEM results the shapes of the measured alpha-spectra

become clear. In the case of Nb, Ho and Zn the presence of non-uniform

structures randomly distributed on the surface causes a broadening of the

gaussian distribution expected in the case of uniform foil. Instead, in the

case of the Sn, crystals of different sizes are present in the target and the

shape of the distribution is dominated by the presence of these crystals.

As it will be shown in the following, this kind of non-uniformities have a non

negligible effect on the determination of the fusion excitation function. For

110


Figure 4.2.6: SEM view of the 120Sn side of a target evaporated on 93Nb (top)and 165Ho (middle), used for the 6Li+120Sn and the 7Li+119Sn reactions. Thebottom image is a SEM view of the 64Zn side of a target evaporated on 165Ho.

111


studying this effect it is necessary to determine the thickness distribution of

the target. In the next section the method that we developed to obtain this

information will be described.

4.3 Determination of the target thickness dis-

tribution

Two different procedures were used for determining the thickness distribution

of the catchers and the Sn foils; both procedures were based on the analysis

of the energy spectra of the transmitted 241Am α particles.

4.3.1 Determination of the Nb and Ho thickness dis-tribution

In the case of Nb or Ho catcher foils the contribution due to the foil non-

uniformity can be determined by using the following relation:

FWHM2exp = FWHM2

coll + FWHM2res + FWHM2

non−unif (4.3.2)

where FWHMexp is the FWHM of the experimental transmitted alpha en-

ergy spectra, FWHMcoll , FWHMres and FWHMnon−unif are respectively

the FWHM due to the statistical nature of the energy loss process, of

the intrinsic detection resolution (which is about 40keV), and of the foil

thickness non uniformity. From the (4.3.2) it is possible to determine the

FWHMnon−unif and thus to estimate the thickness variation, ∆t, needed to

explain such a straggling from the relation:

FWHMnon−unif

∆E=

∆t

t(4.3.3)

where ∆E is the energy loss in a uniform target of thickness t. With this

procedure we have estimated that the non-uniformities for 93Nb and 165Ho

foils are about the ±7%.

4.3.2 Determination of the Sn thickness distribution

As a first step SRIM simulations were performed to determine the expected

residual energy spectra for alpha particles emitted by an Am source, which

112

4.3.2 Determination of the Sn thickness distribution

traverse a Sn+Nb foil, varying the Sn thickness from 0.01 mg/cm2 to 1.5

mg/cm2 at steps of 0.015 mg/cm2. The 93Nb thickness, was given as input in

the simulation program, including the measured non-uniformity. The energy

spectra obtained for each step were summed giving to each a weight such as

to reproduce the experimental energy distribution. In figure 4.3.7 the experi-

mental energy distribution measured after the α traversing a 120Sn+93Nb foil

is shown together with the simulated one obtained by applying the described

procedure. The ratio between the areas of each weighted distribution and

Figure 4.3.7: (black line) Typical experimental energy spectra of α-particleswhich traverse a Sn+Nb(Ho) foil. (red line) Simulated energy spectra, obtainedby taking into account the Sn foil non- uniformity.

the total distribution represents the probability, P, to have a certain target

thickness. In figure 4.3.8 a typical plot of the probability P versus the Sn

target thickness is reported. As a first approximation, it seems reasonable to

ascribe a linear fit to the data.

In order to check the procedure developed for determining the Sn thick-

113


Figure 4.3.8: Sn thickness distribution.

ness distribution, we measured the residual energy spectrum of a 21 MeV6Li beam after traversing one and three target foil combination, by placing a

detector behind the target. The beam was first scattered by a thin 197Au foil.

The foils and the detector were placed at 20 in order to reduce the number

of incident particles. A 3 mm collimator was placed, in front of the foils, in

order to reduce the kinematic broadening of the scattered beam. The exper-

imental residual energy spectra corresponding to one and three target foils

measurements are shown in figures 4.3.9. As it is possible to see, the beam

energy distribution after traversing the first foil of the stack (120Sn+93Nb)

is not gaussian. The shape of this distribution is dominated by the target

non-uniformity, as it was previously observed for the α spectra distributions.

Concerning the beam distribution after the third foil of the stack, its shape

is more similar to a gaussian distribution, but its width is 3 times wider than

the one expected if the foil were uniforms.

By using the Nb and Sn thickness distribution, determined from the α-

analysis of the spectra described above, we performed SRIM simulations, for

114

4.4 Analytical procedure to determine the effective energy

determining the expected energy spectra for 21 MeV 6Li beam after travers-

ing one and three foils (see figure 4.3.9). The agreement that we obtained

between the experimental data and SRIM simulations is very good. We

believe that it is particularly interesting that the laboratory α-test has ef-

fectively given reasonable account of what expected with an actual beam.

In fact, it will be shown in the following that for the correct determination

of the fusion excitation function, one needs to know the actual target thick-

ness distribution and not just the average thickness, both in the case of the

single foil and stack irradiation. The work described in this thesis demon-

strates that this thickness measurement can be done independently from the

fusion measurement, without requesting any additional beam time, which is

of fundamental importance for experiments with radioactive beams.

4.4 Analytical procedure to determine the ef-

fective energy

From the analysis presented in the above sections, it is clear that the presence

of target non-uniformities must have an important effect in the determination

of the effective energy and thus of the fusion excitation function.

In this section, the details of the procedure developed for determining the

effective energy will be described. The procedure will be applied in the case

of the fusion excitation function of the 6Li+120Sn system. It is important to

remember that for this reaction the fusion measurement has been performed

by using the stack configuration (two stacks were used composed of four

target foils) and also some points were measured by irradiating single foils.

In this way the comparison of the two sets of data should show the possible

effects linked to using a stack of targets.

To determine the respective effective energy, for each measured cross-section,

it has been necessary to modify the relation (4.1.1), for taking into account

also the target non-uniformity. In fact, since the targets are characterized by

a certain thickness distribution, the integration range becomes a function of

115


Figure 4.3.9: Experimental (black line) and simulated (red line) residual energyspectra of a 21 MeV 6Li beam, after traversing one (a) or three (b) 120Sn+93Nbfoils. See text for details.

116


the target thickness t and the relation (4.1.1) can be written as:

Eeff =

∫ tfti

E(t) · σ[E(t)] ·D[E(t)] · ρ(t)dt∫ tfti

σ[E(t)] ·D[E(t)] · ρ(t)dt(4.4.4)

where ti and tf are the minimum and maximum target thickness, ρ(t) is

the target thickness distribution determined by the procedure described in

the previous section and D is the energy distribution inside the target. For

each target, the energy distribution inside was obtained by performing SRIM

simulations. The SRIM code, which usually takes into account only the

energy dispersion due to the statistical nature of the energy loss process, was

implemented by inserting in it the thickness distributions, ρ(t), determined

by the α particle residual energy measures described in section 4.2.

It has been observed that, for a correct reconstruction of the effective energy,

it is also important to take into account the finite width of the incoming beam

distribution, since it influences the width of the integration range and the

energy distribution inside the target. Of course, this correction is important

also in the case of a single foil but it becomes extremely important in the case

of the stack irradiation since the beam energy distribution becomes larger and

larger, owing to the energy spread, while traversing the foils.

The determination of the effective energy can be done by applying an iterative

procedure. Since σ(E) is not known a priori, a first approximation for the

effective energy is obtained from the fit of the experimental fusion excitation

function obtained by plotting the measured fusion cross-sections versus the

expected energy calculated in the centre of the target. The deduced values

of σ(E) are then used in equation 4.4.4 to obtain Eeff and so a new σ(E) is

determined. The procedure is repeated until the convergence of the energy

values is found.

To validate the procedure, the Wong formula (1.2.15) determined for the6Li+120Sn system, by using as RB, ~ω, and VB value respectively of 10.35

fm, 4.28 MeV and 19.25 MeV, was used as a starting point. The Wong fusion

cross-section value inside each target was extracted by using the following

117


formula:

σ′

W =

∫ tfti

σW [E(t)] ·D[E(t)] · ρ(t)dt∫D[E(t)] · ρ(t)dt

(4.4.5)

where D(E(t)) is the expected energy distribution inside the actual target

and ρ(t) is the target thickness distribution as measured by α-particles (see

section 4.3.2). In figure 4.4.10 the cross-sections are reported as a function of

the energy calculated at the centre of the target, ET . The filled symbols cor-

respond to the stack data (the triangles and inverted triangles represent the

two different stacks used), the empty diamonds are the data corresponding

to the single foils measurements. It is immediately clear that associating the

fusion cross-section with the energy at the centre of the target (as usually

authors have done until now) leads to a wrong determination of the fusion

excitation function. For each stack, the discrepancy between the fusion ex-

citation function and the Wong curve becomes larger and larger when the

number of foils, traversed by the beam, increases.

In figure 4.4.10(bottom part) the Wong cross-sections are reported as a func-

tion of the effective energy, Eeff , determined by the relation (4.4.4), which

takes into account the different weight of the energies inside the target due to

the shape of the σ(E), D(E) and the thicknesses distribution of the target,

ρ(t). By applying this procedure the data of the single foils and of stack

measurements converge and they reproduce very well the Wong curve. This

result validates the procedure developed in this work for determining the

fusion excitation function.

We have applied this procedure for determining the fusion excitation function

for the 6Li+120Sn system. In the table 4.1 the complete fusion cross-section,

obtained by summing the ER production cross-sections of all Iodine isotopes,

are listed with the corresponding effective energy.

In figure 4.4.11 the fusion excitation function of the 6Li+120Sn system is

plotted as a function of the effective energies. The filled symbols are the

data corresponding to the stack measurements (the triangles and inverted

triangles represent the two different stacks used), the empty diamonds are

the data corresponding to the single foils measurements.

118


Figure 4.4.10: Fusion excitation function obtained from 4.4.5 plotted vs theenergy in the middle of the target (top) and vs the effective energy determinedfrom 4.1.1.

119


6Li+120SnEeffCM(MeV ) σ ± δσ(mb)

stack 1 23.53 381.45 ± 56.76

22.39 281.63 ± 38.17

21.25 183.99 ± 26.46

19.99 101.56 ± 14.85

Stack 2 19.96 87.84 ± 11.96

18.60 33.71 ± 4.73

17.47 8.5 ± 1.25

16.31 1.64 ± 0.36

Single 17.62 7.21 ± 1.22

Single 16.46 1.32 ± 0.32

Table 4.1: Fusion cross-section for the 6Li+120Sn reaction for each activationenergy.

Figure 4.4.11: Fusion excitation function of 6Li+120Sn reaction. The full tri-angles correspond to the data obtained by using the stack set-up. The emptydiamond are the data corresponding to the single foils irradiation.

120


The data corresponding to the single foils and stack measurements agree

within the error bars.

In summary these results show that great care must be exercised in deter-

mining fusion excitation functions using a stack target arrangement. This is

especially true at low energies where much of new physics interest lies.

121

Chapter 5

Fusion excitation functions forthe 6Li+120Sn and 7Li+119Snreactions

As it was mentioned previously, the cross-section for complete fusion, defined

experimentally as the capture of all the charges of the Li projectile, was ob-

tained by summing the ER production cross-sections of all Iodine isotopes,

(listed in table 3.2 and 3.3). The values of the measured cross-sections for

the 6Li+120Sn and for 7Li+119Sn have been reported in table 4.1 and 5.1 re-

spectively. The 6Li+120Sn cross-sections at the two lowest energy considered

in this section are the ones corresponding to single foil measurements.

The study of these two fusion reactions should help to investigate further

the effects that direct reactions, like break-up and transfer, could generate

on the fusion process.

5.1 Complete fusion suppression factor at en-

ergies above the barrier

So far, in order to investigate the specific role played by the break-up, re-

actions induced by the least bound stable nuclei (6,7Li, 9Be) have been per-

formed on light, medium and heavy targets (see table 1.2). In reactions

on heavy targets a reduction of the CF cross-section is observed at energies

above the barrier, with respect to calculations (CC or single barrier pene-

123

5. Fusion excitation functions for the 6Li+120Sn and 7Li+119Snreactions

7Li+119Sn

EeffCM(MeV ) σ ± δσ(mb)

23.51 469.97 ± 63.23

22.35 345.84 ± 46.38

21.23 240.11±32.02

20.00 104.49± 13.99

18.57 28.74 ± 3.87

17.54 7.07 ± 1.10

16.30 0.84 ± 0.11

Table 5.1: Fusion cross-section for the 7Li+119Sn reaction for each activationenergy.

tration calculation) or to reactions induced by well bound nuclei. Such a

reduction was attributed to the projectile break-up in the strong Coulomb

field of the target nucleus. Intuitively, one could expect that the break-up

probability, and thus the suppression of the complete fusion, might depend

on the atomic number of the target due to the Coulomb field. Actually, it has

been observed that for heavy targets the CF cross-section reduction depends

on the break-up threshold of the projectile (see figure 1.9.19), but it does not

seem to depend on the Z of the target, in the mass range investigated (Z >

62).

In reactions on medium and light mass targets the CF cross-section seems

not to be affected by the break-up process and no reduction was observed

within the experimental uncertainties (see for details section 1.9).

The present study extends the investigation of the possible relation between

the suppression factor and the atomic number of the target in a mass range

never investigated before.

In figures 5.1.1 and 5.1.2 the complete fusion excitation functions for the6Li+120Sn and 7Li+119Sn reactions are plotted and compared with the pre-

diction of a 1D Barrier Penetration Model (black continuous line). The 1D-

BPM fusion excitation functions have been calculated by using an optical

model approach.

124

5.1 Complete fusion suppression factor at energies above the barrier

The potential consisted of a double-folding real part and a Woods-Saxon

imaginary part with parameters W =50 MeV, RW = 1.0 × (A1/3P + A

1/3T )fm,

aW =0.3 fm [113], where AP and AT are the masses of the projectile and

target, respectively. The very short-range imaginary potential accounts for

fusion absorption of the flux that penetrates the Coulomb barrier [114]. The

Coulomb potential of a uniformly charged sphere of radius RC = 1.25 ×(A

1/3P + A

1/3T )fm was used in the calculations. The double-folding potential

was calculated using the M3Y effective interaction in the form given in [115].

The matter densities for 6,7Li and 120,119Sn were obtained from the charge

distributions for these nuclei taken from [116] and [117, 118], respectively.

The matter density for 7Li was taken from [119]. The double-folding poten-

tial was calculated with the code DFPOT [120] and the fusion cross-sections

were calculated using the code FRESCO. [121].

It can be seen from figure 5.1.1 and 5.1.2 that for both the reactions, at

above-barrier energies the measured complete fusion cross-section lies below

the 1D-BPM predictions. The underestimation of the 1D-BPM prediction

at energies below the barrier may result from neglecting coupling to the

excited states of the target and projectile and/or to the other different chan-

nels. To obtain some information about the possible couplings to inelastic

System VB (MeV) RB (fm) a (fm) V0 (MeV) r0 (fm)6Li+120Sn 19.25 10.35 0.63 29 1.207Li+119Sn 19.38 10.45 0.63 28.25 1.20

Table 5.2: The potential parameters for the two systems used in the CCFULLcalculation

excitations at energies below the Coulomb barrier, coupled-channel (CC) cal-

culations were performed by using the CCFULL code [122]. This code solves

the Schrodinger equation and the coupled equations exactly. The isocen-

trifugal approximation is used in the program, according to which one can

replace the angular momentum of the relative motion in each channel by

the total angular momentum. The fusion cross-sections are calculated us-

125


Figure 5.1.1: Complete fusion excitation function for 6Li+120Sn (black circles)compared with the prediction of a 1D Barrier Penetration Model (continuous blackline). The blue line represents the result of the CC calculation considering onephonon coupling to the first two vibrational states of the 120Sn. Also the couplingto the first excited state of projectile has been considered (red line).

ing an incoming wave boundary condition inside the Coulomb barrier. The

nuclear potential is taken to be of a Wood-Saxon form. The input poten-

tial parameters (V0, r0,and a) for the CCFULL code are listed in table 5.2

for both the reactions. The diffuseness parameter a was obtained from the

Wood-Saxon parametrization [123] of the Akyuz-Winther potential [124]. To

obtain the appropriate potential depth V0 we used first the CCFULL code in

the no-coupling limit. V0 was varied accordingly so that the corresponding

cross-section agrees with the 1D-BPM obtained using the FRESCO code.

The first set of coupled channel calculations was performed including coupling

to the quadrupole (multipolarity λ = 2) and octupole (λ = 3) one-phonon

vibrational states of the target, listed in table 5.3. In the even-even 120Sn

126


Figure 5.1.2: Complete fusion excitation function for 7Li+119Sn (black circles)compared with the prediction of a 1D Barrier Penetration Model (continuous blackline). The blue line represents the result of the CC calculation considering onephonon coupling to the first two vibrational states of the 119Sn (see details in thetext). Also the coupling to the first excited state of projectile has been considered(red line).

these states correspond to the first 2+ and 3− states at excitation energies

of 1.17 and 2.40 MeV, respectively [11]. In the even-odd nucleus 119Sn these

excitation modes can be viewed as the 118Sn core vibrations and have to be

coupled to the spin of the valence neutron in s1/2. For the mode of multipo-

larity λ this coupling leads to two nearly degenerate states with spins λ−1/2

and λ+1/2 at an excitation energy close to the excitation energy of the core.

There will be two states at energies 0.95 and 1.35 MeV for the quadrupole

mode, and two states at energies 2.25 and 2.50 MeV for the octupole mode

[125]. The deformation parameters β for these states are given in table 5.3.

Because of the limited model space of CCFULL, in the case of 119Sn, each

127


Nuclei Ex (MeV) λ β120Sn 1.17 2 0.107

2.40 3 0.150

119Sn 0.95 2 0.1051.35

2.25 3 0.116

2.50

Table 5.3: The deformation parameters, excitation energies and multipolaritiesof the 119Sn and 120Sn. The excitation energies and β values for 119Sn and 120Snare taken from [125] and [11] respectively.

of the two modes is represented by one state with an excitation energy equal

to the average energy of the two nearly degenerate states.

The results of these calculations are shown in figures 5.1.1 and 5.1.2 (blue

line). The effect of the coupling to the excited states of the target on the

fusion cross-section improves the agreement with the experimental data, be-

low the barrier, but it is not sufficient to reproduce them. It has been found

that further inclusion of two-phonon states did not significantly change the

calculated fusion excitation function.

In addition to coupling to target excited states, possible coupling with the

first excited state of the projectile has also been considered in both the re-

actions. As suggested in [24] rotational couplings with β2 of 0.87 and 0.80

for 6Li and 7Li, respectively, were considered. As it is possible to see from

figures 5.1.1 and 5.1.2 (red line), this coupling leads to an overestimation of

the data over the whole range. It seems that coupling to the weakly bound

projectile has a strong influence in fusion.

Using the data above the Coulomb barrier, a complete fusion suppression

factor (FCF ) of about 35% in the case of 6Li and about 20% in the case of7Li has been estimated, with respect to the 1D-BPM predictions. The error

has been calculated by applying the error propagation procedure.

The FCF for 6Li and 7Li induced reactions on different targets are compared

in figures 5.1.3 and 5.1.4, using the present data and those reported in the

128


literature [79, 22, 21, 85, 87, 91, 25, 87] The values of FCF obtained in this

work are in good agreement with the one previously measured for heavier

targets. Our measurements confirm that for heavy targets the FCF appears

to be independent of the atomic number (ZT ) of the target nucleus. In the

case of the 6Li projectile, a measurement, recently undertaken, involving a

Zr target has been repoted [79]. Some authors [24] have argued that this

Figure 5.1.3: Complete fusion suppression factor for measurements induced bythe 6Li weakly bound nucleus.

suppression is strictly linked with the break-up threshold of the projectile, by

measuring the incomplete fusion cross-section. It was found that the sum of

the complete plus incomplete fusion cross-section would saturate the above

barrier fusion cross-section calculated with the CC or 1D-BPM model (see

figure 1.9.17).

In our case since a fraction of the residues produced as a result of the incom-

plete fusion is stable, it is not possible to perform this type of analysis.

New complete fusion measurements with lower atomic number targets are

needed to investigate further the fusion suppression factor especially to low

129


Figure 5.1.4: Complete fusion suppression factor for measurements induced bythe 7Li weakly bound nucleus.

ZT values.

5.2 Comparison of 6Li+120Sn and for 7Li+119Sn

fusion excitation functions and possible

role played by the n-transfer Q-value

Another aim of this work is to provide a further contribution to the inves-

tigation of the possible influence of entrance channel n-transfers Q-value on

sub-barrier fusion reactions.

In the few last years experimental evidences have shown a dependence of the

sub-barrier fusion cross-section on the sign of the neutron transfer Q-value:

in particular an enhancement of the sub-barrier cross-section has been ob-

served for reactions with a positive n-transfer Q-value [66]. This idea has

been supported by a semi-classical model of sequential fusion [16] in which

an intermediate rearrangement of valence neutrons having positive Q-value

130

5.2 Comparison of 6Li+120Sn and for 7Li+119Sn fusion excitation functionsand possible role played by the n-transfer Q-value

leads to a gain in kinetic energy of the colliding nuclei in the entrance channel

and, thus, to an enhancement of the barrier penetrability and, consequently,

of the fusion cross-section.

The study of the Li+Sn systems is particularly suitable for studying the role

of transfer channels on fusion. These reactions are characterised by very

similar structure of the interacting nuclei and entrance channels, but they

present different Q-values for one- and two- neutron transfer (see table (1.3)).

If the assumed relationship between the Q-value and the fusion process were

correct then for the reactions studied in this thesis we should observe, at

energy below the barrier, an enhancement of the 7Li+119Sn fusion excitation

function with respect to the 6Li+120Sn one.

In figure 5.2.5 the comparison between the measured 6Li+120Sn and 7Li+119Sn

excitation functions is shown.

Figure 5.2.5: 6Li+120Sn (blue circles) and 7Li+119Sn (red circles) fusion excita-tion functions reduced as σ/πR2

B vs Ecm − Vb.

131


The data have been reduced by dividing the fusion cross-section by the

square of the barrier radius (R2B) and subtracting from the energy the height

of the Coulomb barrier Vb. Above the barrier the 7Li excitation function is

higher because of the different suppression factors, discussed in the previous

section. It seems that by reducing the beam energy such an enhancement be-

comes smaller. Therefore sub-barrier fusion enhancement in the 7Li induced

collisions due to larger n-transfer Q value cannot be deduced from these data.

A step further in the understanding of the dynamics might be achieved by

extending the fusion excitation function measurements and by measuring

the 8Li+118Sn and 9Li+117Sn, where the Q-values for neutron transfer are

positive and significantly larger than for the 7Li+119Sn.

132

Conclusions and futureperspectives

The present thesis concerns the study of 6Li+120Sn and 7Li+119Sn fusion reac-

tions performed for investigating the effects that direct reactions, like break-

up and transfer, could generate on the fusion process, at energies around the

Coulomb barrier.

Both the experiments were performed at the Laboratori Nazionali del Sud in

Catania and the beams were delivered by the SMP Tandem van de Graaf ac-

celerator. The fusion excitation functions were measured in an energy range

from about 16 MeV to about 24 MeV, by using an activation technique,

based on the off-line detection of the atomic X-rays emitted after the elec-

tron capture decay of the E.R. produced in the reaction. This technique is

particularly useful when measuring at energies close and below the Coulomb

barrier since at these energies the evaporation residues produced after the

fusion process do not have enough energy to leave the target and so to be

detected directly. In the case of the 6Li+120Sn not only single targets but

also stacks of targets were irradiated. The stack setup is particularly suitable

in experiments where the fusion cross-section is measured at energies below

the barrier or with low-intensity radioactive beams because it allows a signif-

icant reduction in the beam time needed to perform the experiment. Since

this PhD work is part of a wider project which includes the study of the8Li+118Sn and 9Li+117Sn systems, we were interested in a detailed analysis

of this experimental technique that in these experiments is mandatory.

Even if a lot of measurements have been done by using this technique, its

drawbacks were not treated in detail and taken into account in the deter-

133

Conclusions and future perspectives

mination of the fusion excitation function. For a correct determination of

the fusion excitation function with the stack technique, it should be consid-

ered the energy dependence of the fusion cross-section, especially at energies

below the barrier, and also the energy distribution inside the target. In prin-

ciple, the shape and the width of the energy distribution inside the target

depend on the statistical nature of the energy loss process. It depends on

the incoming energy beam distribution and thus in the case of the stack it

depends also on the features of other targets placed upstream. It has been

shown that another important factor which could influence the shape of the

energy distribution inside the target is the presence of non-uniformities.

For the first time, in this PhD work, an accurate analysis has been done

concerning the target non-uniformity and the effect that this could have on

the determination the fusion excitation function. It has been developed a

technique to determine the target thickness distribution and an analytical

procedure which allows to determine the fusion excitation function, by tak-

ing into account the energy dependence of the fusion cross-section and of

the beam energy distribution inside the target and the possible presence of

target non-uniformities. It has been clearly shown how important is to con-

sider these corrections for the proper determination of the fusion excitation

function.

Although the correctness of the developed procedure has been demonstrated,

it is clear that at lower and lower energy the stack effect on the fusion cross-

section becomes stronger and stronger. It would therefore be appropriate

not only to correct it but also to minimize these effects. Nothing can be

done to reduce the energy spread due to the statistical nature of the energy

loss process, but an accurate analysis of the target non-uniformity could be

helpful in order to avoid further broadenings due to target non-uniformities.

By performing the SEM analysis of the Sn targets, it has been found that

the non-uniformities present in our target are typical of the evaporated Sn.

A possible solution for making more uniform targets could be to produce

them with a different technique. In the future, we plan to try the sputtering

technique, which for other materials has allowed to produce targets much

more uniform than those obtained by evaporation.

134

Conclusions and future perspectives

The fusion excitation functions for the 6Li+120Sn and 7Li+119Sn have been

both determined by applying the correction procedure.

At energies above the barrier, the comparison of the measured fusion excita-

tion functions with the respect to the ones predicted by the 1D-BPM model

allows to investigate the possible complete fusion cross-section suppression,

which is generally attributed to the combined effect of the low break-up

threshold of the projectile and the Coulomb field generated by the target.

The strong Coulomb field causes the projectile to break-up before fusion

occurs, thus reducing the fusion probability. The CF suppression factors

estimated for the 6Li+120Sn and 7Li+119Sn reactions are in agreement with

the ones previously measured in reactions involving targets heavier than the

Sn, confirming that, for heavy targets, the FCF appears independent on the

atomic number (ZT ) of the target nucleus.

At energies below the barrier, both the fusion excitation functions overesti-

mate the 1D-BPM prediction By considering the coupling with the first two

vibrational states of Sn, the agreement between the two curves is improved

but the data are still underestimated by the CC calculations. An overestima-

tion is instead obtained if also the coupling with the first rotational state of

the projectiles is included. Since 6Li and 7Li are weakly bound projectile, the

enhancement at sub-barrier energy might be also due to the coupling with

the break-up channel, as it has been already observed in other fusion reac-

tions involving this kind of projectile. To investigate this hypothesis CDCC

calculation will be performed in the future.

By comparing the two fusion excitation functions, at energy below the bar-

rier, the possible influence of the positive neutron transfer Q-value on the

fusion excitation function has been investigated, but no definite conclusion

can be drawn from our data.

We plan to extend the measurements at lower energies. Moreover in order

to look for a possible evidence of transfer coupling effects, a systematic ap-

proach is needed. For this reason it would be interesting to measure the

fusion cross-sections for the systems 8Li+118Sn and 9Li+117Sn characterized

by positive Q-values for neutron transfer, larger than for the 7Li+119Sn.

135

Acknowledgements

At the end of my thesis I would like to thank all the people who made this

work possible and an unforgettable experience for me.

To begin with, I would like to thank Professor Torrisi for having accepted to

be my tutor and for his availability.

I would like to express my deepest gratitude to Professor Lattuada, Dr Alessia

Di Pietro and Dr Pierpaolo Figuera, for all their support, valuable advices

and encouragement throughout my Ph.D. studies. I thank them for the

guidance and great effort they put into training me in the scientific field.

It is my pleasure to thank Professor Shotter for offering me such an interesting

topic of investigation. The enthusiasm he has for his research was contagious

and motivational for me. I am grateful to him for the discussions that helped

me sort out the technical details of my work, helping me to understand and

enrich my ideas.

I also would like to sincerely thank Dr Mile Zadro who helped me with his

advices, discussions and suggestions.

It is also a pleasure to mention my good friends and colleagues, Domenico,

Francesco, Giacomo and Manuela. I would like to thank them for all the

discussions job and not-job related, for their friendship and support.

A special thank to Franck for his understanding and patience during my

research and writing this dissertation. His support, in particular, during the

final stages of this Ph.D. was so appreciated.

Lastly, I would like to thank my parents and my sister Tiziana who supported

me in all my choices with their love and encouragement.

137

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Dottorato di Ricerca in Fisica - XXV CICLO - infn.it un forte impulso, dovuto al crescente interesse per i processi di reazione di- ... intrinseche dei nuclei interagenti e/o processi