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Thesis for the degree of Doctor of Philosophy

Exploring the Exotic

Experimental investigations far from stability

Karin Markenroth

Department of Experimental PhysicsChalmers University of Technology

and Goteborg UniversityGoteborg, Sweden, 2001

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Cover: An ∆E − E identification plot from 12C+p scattering.

Exploring the Exotic- Experimental investigations far from stability

Karin Markenroth

c© Karin Markenroth, 2001

ISBN 91-7291-070-4ISSN 0346 - 718x Doktorsavhandlingar vid Chalmers tekniska hogskola,

Ny serie nr 1753

Subatomic Physics, Department of Experimental PhysicsSE–412 96 Goteborg, Swedenfax: +46–(0)31–772 32 69

Reproservice, Chalmers BibliotekGoteborg 2001

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Exploring the Exotic

Experimental investigations far from stability

Karin Markenroth

Subatomic physics groupDepartment of Experimental PhysicsChalmers University of Technology

and Goteborg UniversityOctober 2001

Abstract

The exploration of exotic nuclei is one of the most intriguing and fastest expandingfields in modern nuclear physics. In the extreme conditions of unsymmetricneutron-proton distributions and small binding energies many nuclear phenomenanot encountered closer to stability have revealed themselves, often as a surprise tothe nuclear physics community. The prime example is the development of halostructures as the binding energy diminishes. To comprehend the new features ofthe nuclear world that are discovered as the drip-lines are approached, reliable andunambiguous experimental data are needed. The work in this thesis summarizes anumber of experiments performed along two main lines.

Papers 1 and 4 deal with studies of resonances in unbound nitrogen isotopes.These systems are interesting also for comparing with its mirror systems, studies ofproton decay and shell structures far from stability. These experiments, theirresults and the method used are described in sections 2, 3 and 4.

To study halo systems longitudinal momentum distributions and cross sections aremeasured in one-nucleon removal processes where the surviving fragment isdetected. Three experiments of this type are reported in papers 2, 3, 5 and 6.Some background to the papers, descriptions of the experiments and a summary ofthe results are given in sections 5, 6 and 7 while a personally coloured glimpse intothe future is given in section 8.

Keywords: driplines, resonances, elastic resonance scattering, breakup reactions,inbeam γ spectroscopy, longitudinal momentum distributions, cross sections

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PublicationsThis thesis is based on work reported in the following papers:

1. Study of the unbound nucleus 11N by elastic resonance scatteringL. Axelsson, M. J. G. Borge, S. Fayans, V. Z. Gol’dberg, S. Grevy,D. Guillemaud-Mueller, B. Jonson, K.-M. Kallman, T. Lonnroth,M. Lewitowicz, P. Manngard, K. Markenroth, I. Martel, A. C. Mueller,I. Mukha, T. Nilsson, G. Nyman, N. A. Orr, K. Riisager, G. V. Rogatchev,M.-G. Saint-Laurent, I. N. Serikov, O. Sorlin, O. Tengblad, F. Wenander,J. S. Winfield and R. WolskiPhys. Rev. C 54 (1996) R1511

2. Longitudinal momentum distributions of 16,18C fragments afterone-neutron removal from 17,19CT. Baumann, M. J. G. Borge, H. Geissel, H. Lenske, K. Markenroth,W. Schwab, M. H. Smedberg, T. Aumann, L. Axelsson, U. Bergmann,D. Cortina-Gil, L. Fraile, M. Hellstrom, M. Ivanov, N. Iwasa, R. Janik,B. Jonson, G. Munzenberg, F. Nickel, T. Nilsson, A. Ozawa, A. Richter,K. Riisager, C. Scheidenberger, G. Schrieder, H. Simon, B. Sitar, P. Strmen,K. Summerer, T. Suzuki, M. Winkler, H. Wollnik and M. ZhukovPhys. Lett. B 439 (1998) 256

3. New results on the halo structure of 8BM. H. Smedberg, T. Baumann, T. Aumann, L. Axelsson, U. Bergmann,M. J. G. Borge, D. Cortina-Gil, L. Fraile, H. Geissel, L. Grigorenko,M. Hellstrom, M. Ivanov, N. Iwasa, R. Janik, B. Jonson, H. Lenske,K. Markenroth, G. Munzenberg, T. Nilsson, A. Richter, K. Riisager,C. Scheidenberger, G. Schrieder, W. Schwab, H. Simon, B. Sitar, P. Strmen,K. Summerer, T. Suzuki, M. Winkler and M. ZhukovPhys. Lett. B 452 (1999) 1

4. Crossing the dripline to 11N using elastic resonance scatteringK. Markenroth, L. Axelsson, S. Baxter, M. J. G. Borge, C. Donzaud,S. Fayans, H. O. U. Fynbo, V. Z. Goldberg, S. Grevy, D. Guillemaud-Mueller,B. Jonson, K.-M. Kallman, S. Leenhardt, M. Lewitowicz, T. Lonnroth,P. Manngard, I. Martel, A. C. Mueller, I. Mukha, T. Nilsson, G. Nyman,N. A. Orr, K. Riisager, G. V. Rogatchev, M.-G. Saint-Laurent, I. N. Serikov,N. B. Shul’gina, O. Sorlin, M. Steiner, O. Tengblad, M. Thoennessen,E. Tryggestad, W. H. Trzaska, F. Wenander, J. S. Winfield and R. WolskiPhys. Rev. C 62 (2000) 034308

5. One-nuclon removal cross sections for 17,19C and 8,10B

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D. Cortina-Gil, T. Baumann, H. Geissel, H. Lenske, K. Summerer,L. Axelsson, U. Bergmann, M. J. G. Borge, L. M. Fraile, M. Hellstrom,M. Ivanov, N. Iwasa, R. Janik, B. Jonson, K. Markenroth, G. Munzenberg,F. Nickel, T. Nilsson, A. Ozawa, K. Riisager, G. Schreider, W. Schwab,H. Simon, C. Scheidenberger, B. Sitar, M. H. Smedberg, T. Suzuki andM. WinklerEur. Phys. J. A 10 (2001) 49

6. New proof of the 8B ground state configurationD. Cortina-Gil, K. Markenroth, F. Attallah, T. .Baumann, J. Benlliure,M. J. G. Borge, K. Boretsky, L. Chulkov, U. Datta Pramanik,J. Fernandez-Vazquez, C. Forssen, L. M. Fraile, H. Geissel, J. Gerl,F. Hammache, I. Itahashi, R. Janik, S. Karlsson, H. Lenske, S. Mandal,M. Meister, M. Mocko, G. Munzenberg, Y. Ohtsubo, A. Ozawa,Y. Parfenova, V. Pribora, K. Riisager, H. Scheit, R. Schneider, K. Schmidt,G. Schreider, H. Simon, B. Sitar, A. Stoltz, P. Strmen, K. Summerer,I. Szarka, S. Wan, H. Weick and M. ZhukovSubmitted to PRL (September 2001)

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Contents

1 Introduction 1

1.1 Basic concepts in nuclear physics . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 The driplines . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Exotic nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Halo nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Other exotic phenomena . . . . . . . . . . . . . . . . . . . . . 13

1.3 Production of radioactive beams . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 Inflight separation (IFS) . . . . . . . . . . . . . . . . . . . . . 17

1.3.2 Isotope separation online (ISOL) . . . . . . . . . . . . . . . . 18

1.3.3 Summarizing the production techniques . . . . . . . . . . . . . 19

1.4 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Reaction studies . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.2 Decay studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.3 Techniques for unbound nuclei . . . . . . . . . . . . . . . . . . 21

1.5 Applications of RIB’s . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5.1 Nuclear astrophysics . . . . . . . . . . . . . . . . . . . . . . . 21

1.5.2 Probing the physics beyond the standard model . . . . . . . . 22

1.5.3 Solid state physics . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5.4 Nuclear medicine . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Elastic resonance scattering 27

2.1 Theoretical description of scattering . . . . . . . . . . . . . . . . . . . 28

2.2 Thick versus thin targets . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Conventional versus inverse geometry . . . . . . . . . . . . . . . . . . 31

2.4 Elastic resonance scattering using inverse geometry and thick gas targets 32

2.4.1 Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.2 Competing reactions . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.3 Applicability to radioactive ion beams . . . . . . . . . . . . . 35

2.4.4 ERSIT summary . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Investigations of proton-rich nitrogen isotopes 39

3.1 Physics motivation for studying 11N . . . . . . . . . . . . . . . . . . . 39

3.2 Physics motivation for studying 10N . . . . . . . . . . . . . . . . . . . 42

3.3 Introduction to the experiments . . . . . . . . . . . . . . . . . . . . . 43

3.4 The first GANIL experiment . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 44

viii CONTENTS

3.4.2 Background sources . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 The NSCL experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5.2 Background sources . . . . . . . . . . . . . . . . . . . . . . . . 48

3.6 The second GANIL experiment . . . . . . . . . . . . . . . . . . . . . 49

3.6.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6.2 Background sources . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Results of 10,11,13N resonance experiments 51

4.1 Data treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Interpretation using a potential model . . . . . . . . . . . . . . . . . 52

4.3 The Thomas-Ehrman shift . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Results for 13N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5 Results for 11N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5.1 Interpretation of the data . . . . . . . . . . . . . . . . . . . . 58

4.5.2 Comparing with other experiments . . . . . . . . . . . . . . . 63

4.5.3 Comparing with theory . . . . . . . . . . . . . . . . . . . . . . 65

4.5.4 Summarizing 11N . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6 Preliminary results for 10N . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6.1 Points of the analysis . . . . . . . . . . . . . . . . . . . . . . . 68

4.6.2 Reviewing earlier results . . . . . . . . . . . . . . . . . . . . . 69

5 Break-up reactions: what, why and how 71

5.1 What are break-up reactions? . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Why do break-up reactions work as probes? . . . . . . . . . . . . . . 72

5.3 How are break-up reaction measured? . . . . . . . . . . . . . . . . . . 76

5.3.1 Cross section measurements . . . . . . . . . . . . . . . . . . . 76

5.3.2 Momentum distributions . . . . . . . . . . . . . . . . . . . . . 77

5.3.3 In-beam γ spectroscopy . . . . . . . . . . . . . . . . . . . . . 79

5.3.4 Other techniques . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Break-up experiments of halo nuclei at GSI 83

6.1 The FRagment Separator FRS . . . . . . . . . . . . . . . . . . . . . . 83

6.2 Detectors and calibrations . . . . . . . . . . . . . . . . . . . . . . . . 84

6.2.1 Time of Flight (ToF) . . . . . . . . . . . . . . . . . . . . . . . 85

6.2.2 Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.3 Positions and angles . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.4 γ–detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2.5 Beam and system diagnostics . . . . . . . . . . . . . . . . . . 89

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6.3 Longitudinal momentum distributions p// . . . . . . . . . . . . . . . 89

6.4 Cross section measurements . . . . . . . . . . . . . . . . . . . . . . . 92

6.5 γ Coincidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7 Results of the FRS break-up experiments 95

7.1 Physics motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.1.1 Reasons to study 17,19C . . . . . . . . . . . . . . . . . . . . . . 95

7.1.2 Reasons to study 8B . . . . . . . . . . . . . . . . . . . . . . . 97

7.2 Results of p// measurements . . . . . . . . . . . . . . . . . . . . . . 98

7.2.1 p// distributions of 40Ar fragments . . . . . . . . . . . . . . . 98

7.2.2 p// investigations of 17,19C . . . . . . . . . . . . . . . . . . . . 99

7.2.3 p// and γ investigations of 8B . . . . . . . . . . . . . . . . . . 104

7.2.4 p// and γ studies of 18-24O . . . . . . . . . . . . . . . . . . . . 107

7.3 Cross section measurements . . . . . . . . . . . . . . . . . . . . . . . 108

8 Outlook 111

9 Acknowledgements 113

A Appendices 115

A.1 Abbreviations of facilities and accelerators . . . . . . . . . . . . . . . 115

A.2 Some basic concepts and definitions . . . . . . . . . . . . . . . . . . . 118

A.3 Kinematic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.3.1 Transformation c.m. ⇔ lab. . . . . . . . . . . . . . . . . . . . 119

A.3.2 The volume element in lab. and c.m. . . . . . . . . . . . . . . 122

A.3.3 Additional useful relations . . . . . . . . . . . . . . . . . . . . 122

A.4 Some ion optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.5 FRS messhutte FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

A.5.1 Momentum ⇔ kinetic energy . . . . . . . . . . . . . . . . . . 126

A.5.2 Magnetic rigidity ⇔ kinetic energy . . . . . . . . . . . . . . . 127

A.5.3 Magnetic rigidity ⇔ momentum . . . . . . . . . . . . . . . . 127

A.5.4 Velocity ⇔ kinetic energy . . . . . . . . . . . . . . . . . . . . 128

A.5.5 Velocity ⇔ magnetic rigidity . . . . . . . . . . . . . . . . . . . 128

A.6 Layouts of accelerator facilities . . . . . . . . . . . . . . . . . . . . . 130

x LIST OF FIGURES

List of Figures

1.1 An image of the “Russian doll structure” of matter. . . . . . . . . . 1

1.2 The chart of nuclides. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The isobaric A = 125 chain. . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Differences in matter density for halo nuclei and normal nuclei. . . . . 9

1.5 Deduced radii for low-Z isotopes, including several halo nuclei. . . . . 10

1.6 The known halo nuclei. . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 The areas within which 1n-halos and 2n-halos can be found. . . . . . 12

2.1 Schematic description of the ERSIT method. . . . . . . . . . . . . . . 34

2.2 Spectrum of α-cluster states in 20Ne. . . . . . . . . . . . . . . . . . . 36

3.1 The A = 11 mass chain and some properties of its isotopes. . . . . . 39

3.2 Systematics of the lowest states for the N = 7 isotones. . . . . . . . . 41

3.3 The scattering chamber used in the 10,11,13N experiments . . . . . . . 44

3.4 The layout of the LISE spectrometer at GANIL. . . . . . . . . . . . 45

4.1 The lowest energy levels of 13N and 13C. . . . . . . . . . . . . . . . . 55

4.2 The excitation function for 13N as measured by the ERSIT method. . 57

4.3 The excitation function of 11N measured by elastic resonance scattering. 60

4.4 The 11N excitation function after different ∆E − E cuts. . . . . . . . 61

4.5 The amplitudes of the 11N partial waves 1s1/2, 0p1/2 and 0d5/2. . . . . 62

4.6 The phase shifts of the 11N partial waves 1s1/2, 0p1/2 and 0d5/2. . . . 62

4.7 The 11N spectrum from the angular detector. . . . . . . . . . . . . . . 63

4.8 Comparing the energy levels of 11N and 11Be. . . . . . . . . . . . . . 64

4.9 The proton energy spectrum with and without condition on the ToF. 68

4.10 The ∆E − E identification for 9C+p scattering. . . . . . . . . . . . . 69

4.11 Laboratory energies of 9C+p scattering. . . . . . . . . . . . . . . . . 70

5.1 Cartoon descriptions of different break-up reactions. . . . . . . . . . . 74

6.1 The layout of the FRS and the detector set-up used. . . . . . . . . . 83

6.2 Identification Z vs. A/Z plot in F4. . . . . . . . . . . . . . . . . . . . 87

6.3 The NaI array in F2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.4 Schematic illustration of the p// measurements at the FRS. . . . . . . 90

7.1 p// distributions of 39Ar and 39Cl. . . . . . . . . . . . . . . . . . . . . 99

7.2 p// distributions of 16C and 18C. . . . . . . . . . . . . . . . . . . . . . 100

7.3 The Doppler corrected γ spectrum of 7Be fragments. . . . . . . . . . 105

7.4 The ptotal// and pexc.// distributions of 7Be fragments. . . . . . . . . . . 106

A.1 Defining the kinematic variables in two-body elastic scattering. . . . . 120

A.2 A detailed figure of the LISE3 spectrometer. The scattering chamberis placed in D6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A.3 The accelerator facility GANIL with experimental areas. . . . . . . . 131

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A.4 The accelerator facility NSCL after the upgrade. . . . . . . . . . . . 132

A.5 Layout of the A1200 spectrometer at NSCL. . . . . . . . . . . . . . . 132

A.6 The accelerator facility GSI . . . . . . . . . . . . . . . . . . . . . . . 133

List of Tables

1.1 Summarizing the four interactions. . . . . . . . . . . . . . . . . . . . 3

1.2 Different modes of radioactive decay. . . . . . . . . . . . . . . . . . . 6

1.3 A sample of RIB facilities worldwide. . . . . . . . . . . . . . . . . . . 16

3.1 Experimental results for low-energy states of 10Li. . . . . . . . . . . . 42

3.2 The performed measurements for 10,11,13N. . . . . . . . . . . . . . . . 45

4.1 Parameters used for the best fit to the 11N excitation function. . . . . 59

4.2 Experimental and theoretical results for 11N. . . . . . . . . . . . . . . 66

7.1 Properties of neutron rich carbon isotopes. . . . . . . . . . . . . . . . 96

7.2 Summary of p// widths (FWHM) obtained in the FRS runs. . . . . . 103

7.3 Some important data on 8B and 7Be. . . . . . . . . . . . . . . . . . . 104

7.4 One-nucleon removal cross sections measured at the FRS. . . . . . . . 108

A.1 Abbreviations for facilities mentioned in the thesis. . . . . . . . . . . 115

A.2 Some basic concepts in nuclear physics. . . . . . . . . . . . . . . . . 118

xii LIST OF TABLES

A few hints on how to read this thesis

Sections 1.1 and 1.5 are written for a general audience who are interested in gettinga taste of the field, and those sections assume a minimum of physics knowledge.Useful definitions and concepts are summarized in appendix A.2. Section 1.1 givesa broad introduction to nuclear physics in general and exotic nuclei in particular.A few applications of present-day experimental nuclear physics are described in 1.5.

The remaining sections of the introduction, 1.2–1.4, give further insight to thepossibilities and problems that characterize experimental nuclear physics of exoticnuclei, describing the halo phenomenon and ways to produce radioactive ionbeams, for example. These sections require some knowledge on physics, forexample basic notions of quantum mechanics.

Sections 2–8 treat the specific work of this thesis in detail, and are mainly aimedat scientists working with similar questions. Most of the work is also found in theappended papers.

Karin, Goteborg 2001

1

1 Introduction

Sitting outside my tent I look out over the high Andes. The snowcapped summitsseem to touch the clear blue sky. At lower altitude, the steep slopes are coveredwith harsh grass and small, sturdy flowers and lead down to the thick forest wherethe leaves are dripping with moisture. Thin clouds hang in midair, right in front ofme. The roaring from the river deep below is almost visible in the still border zonebetween heaven and earth. In this exotic landscape, nuclear physics should feelvery far away. But hiking through the jungle on narrow footpaths and strugglingto climb high mountain passes in many ways resemble the difficulties encounteredwhen trying to do physics in the exotic parts of the nuclear chart, where much isunknown and surprises lurk after every turn of the winding road.

1.1 Basic concepts in nuclear physics

This section intends to place nuclear physics on the size scale of Nature as well asintroduce a few basics ideas and concepts to novices of the field. Some definitionsused throughout the text are summarized in table A.2. The world around us

atomic physics

nuclear physics particle physicsstrings?membranes?

material physicschemistry

increasing size

increasing energy

Figure 1.1: An image of the “Russian doll structure” of matter.

consists of things and substances as rocks, water and grass, all with specific colourand texture. However, Nature is built up much like a Russian doll where each shellcontains still smaller constituents. Each level has its own length and energy scale,its own phenomena and models that explain them. Physics has in the last centuryventured deep into matter, reaching smaller and smaller sizes using higher andhigher energies and connecting models that describe the different stages, but we donot yet know what the innermost doll hides.

Let us take the Russian doll of Nature apart, step by step. Opening the outershell, we find that all matter, from the steam in the high summer clouds to thegreen leaves in the forest is built of atoms. The atoms in their turn have of a

2 Introduction

number of point-like electrons whose energy levels determine the chemicalproperties of the atom. The electron cloud determines the size of the atom.

Looking inside the nuclear core reveals its composition of Z protons withcharge +1, and N neutrons, which are electrically neutral. These nuclearconstituents are with a common name called nucleons. As each nucleon is about2000 times heavier than an electron, the atomic mass is concentrated in thenucleus. The total number of nucleons is called the massnumber, A. A neutralatom has as many electrons as protons, but the proton number Z defines theelement X. An isotope is identified by its massnumber and chemical element1, andis denoted AX. A typical feature of nuclei is that they have about the samedensity, ≈1014 g/cm3. This value is comparable to the density of a neutron star,an object that actually can be modelled as a giant nucleus consisting only ofneutrons. In this respect, a nucleus can be thought of as a bag of hard marbles –adding more nucleons will increase the volume and radius of the nucleus, but thedensity will remain constant. From this picture, the relation in eq. 1.1 between themassnumber and the radius of a nucleus can be understood. The radius parameterr0 is fitted from experiments [1].

r = r0 · A1/3, r0 ≈ 1.2 fm (1.1)

Nucleons in their turn have internal structure in form of quarks, point particleswith charge +2/3 or −1/3. Theories of strings, super symmetries and M-theoriestry to describe quarks, electrons and other subatomic particles in terms of evenmore fundamental building blocks. An interesting and popular account of thelatest developments in this area is found in [2].

This imaginary doll play, schematically illustrated in fig. 1.1, places nuclear physicsbetween atomic physics and particle physics both in energy and length scale.

A few words about the fundamental forces in nature are in order. Their basicproperties are summarized in table 1.1. A force is an interaction between objects,for example nucleons. It is often described in terms of potentials. A potential dipimplies that the interaction lowers the total energy of the interacting system. Oncethe system has fallen into this potential well, it takes energy to lift it above thebarrier and into a region where they are free from each others influence. Weusually divide the fundamental interactions into four forces which are mediated byforce particles. The weakest force by far is the gravitational force, carried by thegraviton which is not yet observed. The graviton is massless and has infinite range,and can thus influence macroscopic objects even on astronomical distances. It isresponsible for the structure of the galaxies as well as the return to Earth of a

1To stress the proton or neutron number, its possible to write AZX or even A

ZXN .

3

Table 1.1: Summarizing the four interactions. The strength are normalized to 1 for the elec-tromagnetic force. More information is found for example on the Particle AdventureWebsite [3].

Force Rel. strength Range (m) Mediated by: Acts on:

Gravitation 10−38 ∞ graviton allWeak force 10−7 ≈ 10−15 W±, Z0 quarks, leptonsElectromagnetism 1 ∞ photon charged particlesStrong force 20 ≈ 10−15 gluon quarks, gluons,

baryons

struck golf ball. Also the electromagnetic force, acting between charged particlesby exchanging photons, has infinite range. The weak force is responsible forβ-decay by changing neutrons into protons and vice versa, while the strong forceholds nuclei together and governs α-decay and nuclear reactions (see tables A.2,1.2). Theoreticians have managed to give a unified description of the weak andelectromagnetic forces, and it is generally believed that all four forces will beunified at extremely high energies. The holy grail of theoretical physics is thisunification of the forces in a Grand Unified Theory (GUT). Present candidates aredifferent string theories and M-theories.

To get an overview of a territory, it is useful to have a map to navigate by. Innuclear physics, the chart of nuclides, fig. 1.2, positions all nuclei on a N–Z grid,giving a layout over the isotopes we know. When examining this map of thenuclear landscape it is seen that the stable nuclei are rather few and lie on a linewhich is often called the stability line. All others are unstable and decay, seefurther section 1.1.1.

This narrow band can be viewed as our home on the isotopic continent. The worldwe live in is built from the stable elements and a few naturally occurringradioactive isotopes which have so long lifetimes that they have persisted since thecreation of the solar system. These longlived unstable nuclei decay to stableisotopes via long decay chains, producing a natural radioactivity. Adding to thenatural radioactivity is the radioisotopes produced by cosmic rays in theatmosphere. The science of nuclear physics begun more than one hundred yearsago with the discovery of this background radiation. Pioneering experimental andtheoretical work was done in 1900–1930. In the 1930’s and 1940’s, nuclear physicsevolved dramatically, and the field has continued to increase in size complexityever since. The development is driven by a close interplay between experiment,theory and technology. New technologies permit experimental discoveries that

4 Introduction

demand theoretical explanations, and theoretical predictions call for studies thatmight require improved technologies.

As all three musicians in a trio must perform in harmony, so are these three keyfields in physics all necessary for progress.

1.1.1 Radioactivity

Why are so few nuclei stable, and what is radioactivity? To answer this question,we must first understand why nuclei exist at all. The answer lies in Natures urge tominimize its total energy. If the A nucleons of a nucleus are free (non-interacting),they have a mass A · u. If they interact with each other, their energy changes fromits free value. If the interaction reduces the total energy of the system, theA-nucleon nucleus is formed. The binding energy B is the amount of energyreleased as the nucleus is molded, which is also how much energy it will cost tobreak it up again. The larger the binding energy, the tighter the nucleus is bound.

Neutron

dripline

Proton

dripline

Z

N

2

2 8

8

20

2850

82

162

20

28

50

82

114

184

126

Halo states

Proton emitters

Superdeformation

Superheavy

elements

p-n pairing

New shell structures

Extreme neutron

matter

Doubly magic

nuclei r-process

s-process

rp-process

Figure 1.2: The map over the continent of bound nuclear isotopes. Some areas and phenomenawhich are currently being investigated are marked.

5

As shown in the famous eq. 1.2, mass and energy are related through the speed oflight c, and thus the mass M of the nucleus is less than that of the free nucleons.

E = m · c2 (1.2)

The binding energy is simply the difference between the mass of the free nucleonsand the mass of the nucleus, multiplied by c2.

For a given massnumber A, N and Z can be varied to find the combination thatresults in the minimum energy. This combination corresponds to the stable isotopefor this massnumber. There can be several minima for a certain A, resulting inseveral stable isotopes. The numbers of neutrons and protons that minimize theenergy for a certain A is determined by the characteristics of the strong force. Atypical isobaric chain is shown in fig. 1.3. The energy is decreased if the isotopes tothe left and right of Z = 52 decay to the stable 125

52Te. These decays are mediated

β -decay

β+de

cay

A = 125

48 50 52 54 56 58 Z

Ene

rgy

(MeV

)

20

15

10

5

0

In

SnSb ITe

Xe

Ba

Cs

La

Ce

Cd

Figure 1.3: The isobaric A = 125 chain with the stable 125Te at the bottom of the parabola. Thevertical axis shows the total energy E = mc2 of the nucleus.

by the weak force, and the process is called β-decay for historical reasons. Forneutron rich nuclei, the decay changes a neutron into a proton while nuclei withtoo many protons transform a proton into a neutron, as shown generally intable 1.2. The decay thus conserves the massnumber, and the nuclei slide down theenergy-Z parabola in fig. 1.3 one step at a time until they come to rest at thebottom. For the process to conserve charge and lepton number (table A.2), anelectron and a neutrino must also be created. These decay products carry awaythe released energy as kinetic energy.

6 Introduction

Table 1.2: Different modes of radioactive decay.

α-decay Governed by the strong forceAZX ⇒ 4

2He + A-4Z-2Y Decay mode for heavy nuclei, A>140

β-decay Governed by the weak forceβ−: A

ZXN ⇒ AZ+1XN -1 + e− + ν Decay mode for neutron rich nuclei

β+: AZXN ⇒ A

Z-1XN+1 + e+ + ν Decay mode for proton rich nucleiec : A

ZXN + e− ⇒ AZ-1XN+1 + ν Electron capture, competes with β+

γ-decay Governed by the electromagnetic forceAX∗ ⇒ AXg.s. + γ Excited nuclei can emit photons

fission Governed by the strong forceAZX ⇒

≈A/2≈Z/2X +

≈A/2≈Z/2X +

xnPossible for heavy nuclei, A>220

In conclusion, radioactivity is the process of reducing the energy of nuclearsystems. The excess energy is released in form of kinetic energy of the decayproducts. These energetic emissions are called nuclear radiation. The energyreduction is achieved by converting less bound isotopes into more tightly boundnuclei by changing the Z/N ratio. This can be done by changing neutrons intoprotons or vice versa (β-decay), or, for heavy nuclei, by emitting a 4He cluster(α-decay). A decay mode which is possible for some very heavy nuclei, A>220, isfission where the nucleus splits in two parts and a number of neutrons are emitted.A nuclear power plant produces electricity from such energetic neutrons emitted infission reactions.

Another form of radioactivity, where N and Z are not changed, is γ-emission. Justas electrons have excited states in atoms, so have nucleons in the nucleus. Bygiving energy to a nucleus, it can reach an excited state, denoted by AX∗. Thenucleus can decay from this excited state to its ground state by emitting photons.These light packages have much too short wave length to be visible by the humaneye, while de-excitations of electrons in the atom produce the light we see.

1.1.2 The driplines

What limits the isotopic continent in fig. 1.2, i.e., what defines its shoreline?

Let us start at the bottom of the energy parabola for a given A, for example infig. 1.3, and climb up the slope to the left by changing protons into neutrons. Foreach step higher on the ladder, the binding energy, B, of the nucleus will besmaller than the one in the previous step while the total energy (mass) will be

7

larger. At some point, B will become negative. This means that instead of gainingenergy by forming the nucleus, it will cost energy and the nucleus would be heavierthan the sum of the free nucleon masses. Such an isotope can gain energy bysimply disintegrating, and will do so on the time scale of the strong force,≈10−22 s. The system is unbound, and we have reached the sea of unboundnuclear matter on the neutron rich side of the nuclear chart. This shoreline iscalled the neutron dripline. The opposite side of the island is limited by the protondripline, which is reached by removing neutrons from our stable starting point.The term dripline comes from a simple picture of what happens if one tries to addmore neutrons to a nucleus at the neutron dripline: they will drip off again.

However, this image is an oversimplification. The properties of the strong forcemake the edges of the chart of nuclides jagged, as is seen in fig. 1.6. A strikingexample is the Z = 3 isotopes. The isotope 9Li is radioactive, but bound. It cannot bind one additional neutron, and B for 10Li is negative. However, the pairingbetween two neutrons combined with the 9Li potential is enough to make 11Libound with a very small two-neutron separation energy, S2n = 295 keV. Thisenergy is small enough for the two last neutrons to create a large halo around the9Li core, see further section 1.2.1. The pairing force is an important part of theinteraction close to the edge of the nuclear chart. Both 9Li and 11Li are driplinenuclei, i.e., they terminate the isobaric chains for A = 9 and 11, respectively. TheA = 10 chain is on the neutron rich side ended with 10Be. Similar pairingbehaviour can be seen for protons, for example for the N = 7 isotones.

The extreme neutron rich side of the isotopic continent is Terra Incognita forZ & 30. Nuclei are synthesized along the nuclear dripline in astrophysicalenvironments, making it an interesting part of the map (see section 1.5.1).However, it is impossible to reach the neutron dripline for anything but thelightest nuclei in present-day laboratories due to its demand for extremely neutronrich beams as well as targets and/or complicated reactions. There exist manytheoretical models to calculate masses, but while these agree with each other andwith experimental results for known masses, their predictions for unknown nucleirapidly diverge when extrapolating from known regions. So far, there is no way tojudge if any model is better than others, and there are no reliable predictions ofthe position of the neutron dripline. The proton dripline on the other hand, isexperimentally mapped up to high proton numbers.

1.2 Exotic nuclei

With the development of modern accelerator techniques and computationaltechniques, it is now possible to take nuclear physics far away from the area close

8 Introduction

to stability both in theory and in practice. Nuclear systems far from stability arein a geographical analogy called exotic nuclei. The epithet “exotic” signalssomething which is different from known situations and difficult reach but which isexciting and full of surprises, notions that fit the concept of exotic nuclei very well.This section will briefly describe the ways that lead to those exotic regions andsome of the phenomena encountered once we get there.

1.2.1 Halo nuclei

Almost two decades ago, the nuclear physics community was startled asexperiments unexpectedly showed that the fundamental connection betweenmassnumber and volume (eq. 1.1) was not valid for some exotic nuclei, or in otherwords that the nuclear matter density was not constant after all. The nuclear halophenomenon was discovered.

The name is borrowed from the light phenomenon that can be seen around the sunor moon, as the image of a halo nucleus is a thin neutron veil around the core.Halo states result from a proximity to the top of the potential well that makes theprobability for tunnelling through the potential barrier large, and an extremely bigportion of the wavefunction can be located outside the classical turning point. Thiswill give the matter density a long, dilute tail which makes the system remarkablybig, as is schematically shown in fig. 1.4. The size is reflected in large nucleonremoval cross sections and narrow momentum distributions. Since tunnelling is apurely quantum mechanical phenomenon, the halo has no classical analogy. Thepotential barrier is lowest for neutrons with ` = 0 which do not feel neitherCoulomb nor angular momentum barriers, and so the most pronounced halos areexpected for neutrons in s-states. When the angular momentum barrier appearsfor ` > 0, the tunnelling probability, and thus the halo character of the system, isreduced. Proton halos are always hindered by the Coulomb barrier, and protonhalos are not anticipated to be very large even for ` = 0 orbits in low-Z nuclei.

There is no distinct definition of a halo system, which is hardly surprising sincethere is no sharp threshold above which the phenomenon occurs, but there is rathera gradual transition from normal matter densities to halo structures. However, it isclear that to be labelled a halo, a nucleus must have a pronounced cluster structurewhere one or several clusters have extremely small separation energies that allowsthem to tunnel into the classically forbidden region, giving the large, dilute matterdistribution which is the trademark of the system. The few-body character leadsto prominent single-particle structure of the states. One could define halos interms of size or in the portion of the wavefunction residing outside the classicalturning point (for example 50%), but since there are no physical reasons for any

9

particular value, such definitions seem rather arbitrary. An analytical definition isthat the radius of the system diverges as the binding energy goes to zero. This is astrict requirement, fulfilled only for neutrons in ` = 0 or 1 orbits, and not directlyapplicable for real nuclei since they have a finite binding energy. For neutrons with` > 1 and all proton orbits, the radii can get large but remain finite as the bindingenergy disappears. The analytical conditions can be used to measure the degree ofhalo character of a nucleus as compared to the limit of Shalo = 0. Analyticalconditions for halos are investigated for example in refs. [4–7].

The earliest direct experimental indications of the halo phenomenon came in 1985,when the interaction cross sections (σI) of several neutron rich isotopes of helium,lithium and beryllium were measured by Tanihata et al. at LBNL [8–10]. Theinteraction cross section is here defined as the summed reaction probability for allreaction channels where the proton and/or neutron number of the projectile ischanged in the interaction. From σI, the root-mean-square matter-radius of thenucleus, Rm

rms, can be deduced using a reaction model. The radii for 6,8He, 11Liand 11,14Be were discovered to be much larger than what expected. The effect isparticularly pronounced for 11Li, which shows a radial increase of 30% compared to

r cn

r ch

radius

normal nucleus

halo nucleus

matter density

potential depth

SnN

ShN

ρ0

Figure 1.4: The figure illustrates some qualitative differences between halo nuclei and normal nu-clei. Shown are the matter density distributions, the positions of the loosest boundnucleon in the potential well (SnN , ShN ) and the classical turning points (rnc , rhc ). Thedensity at r = 0 is denoted ρ0. The dash-dotted lines and superscripts h refer a halostate and the dashed lines and n’s to the properties of a normal level. A schematicpicture of a halo nucleus and a normal-density nucleus with the same massnumber isalso shown. In reality, the potential itself is also different for the two cases.

10 Introduction

HeLi

BeB r = r0 A

1/3

Mass number

Rrm

s(fm)

4 6 8 10 12 14 16

3.5

3

2.5

2

1.5

Figure 1.5: The radii for low-Z isotopes, including several halo nuclei such as 11Li, 11,14Be and6,8He. The radii are deduced from refs. [8–10] using Glauber theory.

its closest bound neighbour, 9Li. The data together with the normal behaviourr = r0A

1/3 are shown in fig. 1.5. However, it is important to remember that theseradii depend on the reaction model used when extracting them from σI . As anexample, a re-analysis of the cross section data from [8–10] in a Glauber model(see section 5.2) that treated the few-body character of the weakly bound systemsexplicitly gave 5–14% larger radii for 11Li, 11Be and 8B than a standard Glauberanalysis [11, 12]. Present day experiments probe not only global features of haloslike cross sections, but also the structural details, such as spectroscopic factors andweights of wavefunction components. The arsenal of experimental tools includemomentum distributions, γ-ray experiments, complete kinematics break-upmeasurements and β-decay studies, many of which will be discussed in subsequentsections.

The nuclei which exhibit halo properties appear in the light-mass part of the chartof nuclides, fig. 1.6. The first interpretation of the experimental findings as aneutron halo structure was given in 1987 [13], where a simple model consisting of adineutron and a 9Li core was used to deduce qualitative properties of such systems.An important recognition is that the halo properties are mainly determined by theasymptotic tail of the wavefunction, which can be described by a Yukawawavefunction. The expression for an s-wave is shown in eq. 1.3. Note the influenceof the separation energy on the system.

ψ(r) =1√2πκ

e−κr

r, κ =

√2µSx~

(1.3)

The parameter µ is the reduced mass of the system, and κ is the inversewavelength. The fact that the halo phenomenon was not predicted by any nuclear

11

N

Z

H1 2H

He He He He

Li Li Li

Be Be Be

He He3 4 6 8

11

14119

5 7

Li6 7

B B B B BBB11108

C C12 13 C CC

1816

2119 ?

1n halo

1p halo

stable

2n halo (or more)

unbound

10

17?

22

19?13Be

Figure 1.6: The low-Z part of the nuclear chart. Known and suggested halo nuclei are marked asindicated in the figure. White fields are β-decaying isotopes.

theory implies the challenge that exotic nuclei pose to standard theories. Modelsthat work well close to stability are at the driplines exposed to hard tests of weakbinding, extreme neutron–proton asymmetries and clustering effects. However,development of new theoretical models and adaption of existing theories to thespecific conditions in halo nuclei have given increasingly elaborate models whichare able to probe structural details as well as global properties. Halo nuclei arewell described by few-body models, which are naturally well suited to investigatecluster systems. Also models initially used in the standard fields of nuclearphysics, such as shell models, quantum Monte Carlo shell models, nuclearmany-body theory and Glauber theory, have been used to treat halo nuclei withdiffering amounts of success. To compare experimental and theoretical results, it isimportant to recognize that nuclear structure and reaction mechanism effects aremixed in the experimental data. Progress in reaction theory has been necessary toproperly reproduce the experimental results theoretically. Some reviews on halonuclei which cover theoretical as well as experimental topics are refs. [6, 14–17].

A deeper understanding of halo nuclei requires knowledge of the structure andinteractions in their subsystems. For one-nucleon halos, this might seem trivialsince the obvious clusters of those nuclei is the two-body system core–halo nucleon,and their interaction is given by the halo nucleus itself. However, this initial simpleview of halo nuclei as consisting of inert clusters has had to give way for a morecomplex picture. It has been shown that the internal structure of the coreinfluences the properties of the halo system. One example is the necessity toinclude core-excited states in the description of halo nuclei in order to getagreement between experiment and theory. For example, the description of 11Beclearly benefits from the inclusion of 10Be excited states,and the 19C results are

12 Introduction

non-reproducible if the excited state at 1.62 MeV in 18C are ignored.Anotherexample is the one-proton halo 8B, where a three-body model which explicitlytreats the 7Be core as a 3He–4He cluster gives a 8B wave-function that succeeds inreproducing the experimental longitudinal momentum distributions and therelative weight of the wavefunction corresponding to an excited 7Be core [18] (seesection 7.2.3).

The 2n-halos can be divided into two classes:

• Cases where the core–n subsystem is bound. Few, if any, clear-cut cases existin this category. This fact simply reflects the threshold character of the halophenomenon which makes the n–n pairing vital to the binding.

• If all 2-body subsystems are unbound while the three-body system is bound,the 2n-halo is called Borromean after a heraldic symbol of three Italianprinces. Examples in this class are 11Li, 6He and 14Be.

The conditions for the cases are illustrated in fig. 1.7. All 2n-halos have the

bound

2-body

An

bound 3-body Ann

bound2-body

nn

No bound states

An, nn or Ann

Vnn

VAn

BORROMEAN

REGION

A

n nVnn

VAn VAn

Figure 1.7: The areas within which 1n-halos and 2n-halos can be found. The figure is reproducedwith kind permission [16].

subsystem n–n, which is known to be unbound, but has a low-lying s-state close tothe threshold [1]. The knowledge about n–n and other n–core systems is limited bythe specific difficulties in performing experiments in neutron rich parts of thenuclear chart. Since neutrons are β-unstable it is practically impossible to makeneutron targets. Also the making of neutron beams is difficult as they are

13

electrically neutral and accelerator techniques can not be used. Fission reactorsproduce large amounts of low-energy neutrons, but to get higher energies one hasto use nuclear reactions. One classic method is the Ra–Be powder source where9Be absorb α’s from the decay of 222Ra, producing 13C∗ excited above thethreshold for neutron emission. However, for all such processes, the intensity islow. Using radioactive ion beams, transfer reactions can often be employed withbetter results. One example is the recent investigation of excited states in the n–nsystem using a deuteron beam on tritium target [19].

These experimental problems are the reason that the unbound subsystems core–nof 2n-halos, for example 10Li, are hard to investigate. A common method is to usetransfer reactions to produce the unbound system, and then detect the decayproducts. Another way is to use a secondary beam of the 2n-halo in question andstudy one-neutron removal reactions. If the removed neutron is a halo neutron, thesystem left is the unbound subsystem which will decay, and the decay propertiescan then be used to find information about the subsystem2. An indirect techniqueis to investigate resonances in the mirror system of the interesting case and usemirror symmetries to deduce its properties. The mirror system will lie on orbeyond the proton dripline. A method that can be used to study unbound protonrich systems is elastic scattering in inverse geometry, further described in section 2.

Halo nuclei are fascinating study objects whose properties depend both oncontinuum coupling effects, nuclear shell structure and clustering phenomena. Theunderstanding of the mixing of these effects have increased tremendously since thehalo was discovered, but the answers continue to raise additional questions. Thecoming of new theoretical predictions and experimental possibilities promise tokeep this an active field for many years to come.

1.2.2 Other exotic phenomena

There is a multitude of phenomena which exhibit themselves in nuclei far fromstability, and Mother Nature certainly still has some surprises in store for thenuclear physicists of the future. Below, glimpses are given of a selection of areasthat currently attract interest and are experimentally studied using radioactive ionbeams.

• Superheavy nuclei: Theoretical models predict a possible island of stability inthe region of the next double spherical shell closure, which should be situatedaround Z = 114–126 and N = 184 (see fig. 1.2). These so-called SuperHeavy

2These stripping reactions include n-transfer reactions, but also diffraction dissociation and Coulombdissociation where the neutron is not transferred to the target but simply removed from the halo.

14 Introduction

Elements (SHE’s) have become a goal towards which many laboratoriesstrive, the most successful ones so far being GSI and JINR. The experimentaltechnique that has given the largest production cross sections of these heavynuclei is cold fusion of heavy-ion beams and targets. After the target area,the interesting reaction products are guided through a separator system andimplanted in a position sensitive detector, where α’s from the decay of aheavy ion are detected. Subsequent α–decays give decay chains whichprovide links to known isotopes, which can give unambiguous identificationof the implanted element. If the produced nucleus is very neutron rich, thedecay chain will not end at a known mass but in the white area of the map.This opens new gates to the neutron dripline, but makes the identification ofthe possible SHE’s more uncertain.

Both the separation and the detection system has to be extremely efficientsince only a few SHE’s are produced in several months of beamtime. Toselect these from the enormous number of non-interacting beam nuclei anduninteresting reaction products, and to efficiently and unambiguously detectthe decay products of the superheavy isotopes is a serious challenge. Themost proton rich nuclei synthesized so far are are Z = 112, 114, and116.Comprehensive reviews over the field of superheavy nuclei and ways toproduce them are given in refs. [20, 21].

• The driplines: To experimentally locate the driplines, especially on theneutron rich side, is an important subject of study. The work on extendingthe chart of nuclides is heavily dependent on accelerator intensity upgradesand development of neutron rich beams such as 48Ca, as well as newproduction techniques that reach further out in the unknown landscape.

• Nucleosynthesis paths: Nuclear physics is naturally connected toastrophysics through the questions of universal isotopic abundances,nucleosynthesis and stellar burning. Nuclei are created close to the neutrondripline by competition between neutron capture and β-decay (see furthersection 1.5.1). Nuclei on the proton dripline are important for thecorresponding proton capture reactions (the rp-process). The necessaryconditions for these paths might be found in explosive stellar burning sites.Reviews on nuclear astrophysics are found in for example refs. [22, 23].

• Neutron skins: How will nuclear matter properties change when N Z?One possibility is that the neutron density extends further than the chargedensity, and form a neutron skin on the surface of the nucleus [24]. Suchbehaviour has been seen in Na and Mg isotopes [25]. Skins should not beconfused with the halo phenomenon, even though cases as the 4n-halo 8Hehave been described in both pictures. A halo is a few-body system with

15

strong single-particle structure, while a neutron skin is a more collectiveproperty.

• Shell structures: The nuclear potential will change shape at the driplines ascompared to close to stability. This influences the ordering of the orbits, andthe shell structure with its gaps as we know it from textbooks is notobviously valid for exotic nuclei. There is evidence for N = 16 being a newmagic number close to the neutron dripline, while the shell gaps at N = 8and 20 seem to weaken or disappear [26, 27]. The sd-shell intruder state inthe N = 7 nuclei 11Be, 10Li and 9He, as well as the much investigated islandof inversion in the region of neutron rich Ne, Na and Mg are examples offailures of the standard shell model when facing extreme situations. Theregion of superheavy nuclei is another area where shell model predictions aretested.

• Proton emitters: On and beyond the proton dripline, the studies of protonemitters is a hot topic. Protons are confined to a nucleus by the Coulombbarrier, and so the nucleus can have a rather long lifetime, ms–s, even if it isunbound with respect to proton emission. Using γ-detection techniques, it iseven possible to study excited states in ground-state proton emitters.Mechanisms of two-proton emission are also investigated (one example is31Ar [28]), and the search for multi-proton emitters is ongoing. A review onproton emission is found in ref. [29]

• N = Z line: After passing through the stable nuclei for light isotopes, theN = Z line follows the proton dripline rather closely for medium-heavysystems. This means that experiments can study isospin symmetryproperties in very exotic nuclei, for example by measuring the differences ofenergy levels in mirror pairs

[A

Z + 1XN −AZ XN + 1

]. Another phenomenon that

can be investigated is neutron-proton pairing which is exceptionally strong innuclei around N = Z, giving a contribution to the binding energy which isoften called the Wigner term.

1.3 Production of radioactive beams

The discoveries of the above mentioned and other exotic phenomena have beenpossible largely due to the development of Radioactive Ion Beams (RIB’s) ataccelerator facilities. To get study radioactive species with short halflives, theymust first be produced in the laboratory. There are two main ways to producebeams of exotic nuclei, both with some modifications. These techniques areInFlight Separation (IFS) and Isotope Separation OnLine (ISOL). The different

16 Introduction

beam properties make the two techniques complementary, and different types ofexperiments are performed at facilities utilizing the IFS and ISOL methods. Asample of the facilities worldwide that provide users with RIB’s and whatproduction mechanism they use is given in table 1.3, and a more complete list withlinks is found on the web page of the EURISOL project [30]. A general review overmethods to produce RIB’s and what new physics that can be investigated withthose beams is found in ref. [31], and a popular review is found in ref. [32].

Table 1.3: A by no means complete selection of the facilities worldwide that provide RIB’s, andsome laboratories that are planned or under construction. Explanations of the acronymsare given in table A.1.

Facility Lab. (Country) Type Comment

Operating facilities

A1900 NSCL (US) IFS Upgrade from A1200 in 2000, Bρ=6 TmCRC UCL (B) ISOL+PA ERIB up to ≈ 100 MeVFLNR JINR (RU) IFS Several set-upsFRS GSI (D) IFS High energies, ≤2.0 GeVHRIBF ORNL (US) ISOL+PA Fisson fragments and fusion evaporationIGISOL JYFL (FI) ISOL Low-E beamsISAC TRIUMF (CA) ISOL+PA Up to 1.5 MeV/u, first runs in 2000ISOLDE CERN (CH) ISOL Operational since 1965, low-E beamsLISE3 GANIL (F) IFS ERIB=5-95 MeV/uLISOL KUL (B) ISOL Operational since 1974, low-E beamsRIPS RIKEN (JP) IFS Energies up to 135 MeV/uSPIRAL GANIL (F) ISOL+PA ERIB= 1.7–25 MeV/u, start in 2001.

Facilities that are planned or under construction

EURISOL Europe ISOL Joint EU project, still site-independentMAFF Munich, (D) ISOL Acceleration of fisson fragmentsREX-ISOLDE

CERN (CH) ISOL Post-acceleration up to 2.2 MeV/u,scheduled for operation during 2001

RIA USA ISOL+IFS

Still a site-independent study, proposedfor 2007

RIbf RIKEN (JP) IFS Phase I ready 2002, Phase II ready 2008SPIRALII GANIL (F) ISOL Neutron rich beams produced by fast

neutrons generated in deuteron break-upSuperFRS GSI (D) IFS Upgrade, superconducting magnets

17

1.3.1 Inflight separation (IFS)

In the IFS scheme, a heavy ion beam at high energy (40 – 2000 MeV/u) is collidedwith a target which of the optimal material and thickness for production of thewanted isotopes. This usually corresponds to the beam suffering an energy loss ofonly some percents. The projectiles react in the target, and the fragments continuewith essentially the same energy/nucleon as that of the primary beam.

Several reaction channels are simultaneously open, but the one most oftenfavourable for production is projectile fragmentation. This is a peripheral processin which one or several nucleons are shaved off and the fragments continue withthe beam velocity. The production is well described both by macroscopicabrasion-ablation models and microscopic nucleon-nucleon scattering models. Inprinciple, all ions lighter than the initial beam are produced, but only the onesreasonably close to the beam have high enough intensities to create secondarybeams that are useful for more than particle identification. Also, it is moreprobable to create neutron deficient isotopes, a limitation which becomes anoticeable obstruction when reaching for the neutron dripline.

Some of these nuclei can instead be produced by projectile fission that givesneutron rich nuclei of medium masses. Specific nuclei can be efficiently producedin selective direct reactions, suitable if they have large cross sections and narrowangular distributions. Coulomb dissociation through excitation of the Giant DipoleResonance has high cross sections, but mainly produces nuclei with one neutronless than the primary beam, giving secondary beams on the proton rich side. Muchwork has gone into improvement of the predictions of production rates infragmentation reactions. One result of this is the EPAX code [33].

After the target area, a magnetic separator system is set up to select the wantedisotopes from the mix of nuclei produced in the target-projectile reactions. Theseparator consists of dipole magnets which select ions by their mass-to-chargeratio, see eq. 1.4.

Bρ =mv

q∝ v

A

Z(1.4)

At IFS energies, light ions are completely stripped of all electrons so that theircharge is equal to the proton number. Heavier ions will have a charge statedistribution after the target, giving a risk of losing nuclei with the wrongmass/charge ratio and instead getting an admixture of neighbouring isotopes. Tosolve this, a profiled degrader can be inserted between the dipole sections. Thisseparates nuclides with different charge due to their different energy losses. Thiscombination of sandwiched magnetic and energy-loss separation is commonly

18 Introduction

called Bρ−∆E −Bρ technique. A degrader is normally used also for totallystripped ions to increase the isotopic resolution and get the cleanest beam possible.Crossed E and B fields work as velocity filters, and purify the beams further.Quadrupole and higher order magnets are used to focus the beams at differention-optical planes in the separator. Due to the high energy of the primary beam,the secondary beam will be kinematically focussed in the forward direction,enhancing the transmission through the spectrometer. When the RIB is preparedat the final focal plane, the beam can be used for experiments or be sent to otherexperimental set-ups. A review on the separation techniques for secondary beamsis found in ref. [34].

A majority of the RIB facilities constructed in the 80’ies use inflight separation toproduce exotic beams. One reason for this is that it is relatively easy to constructan IFS-extension at an already existing heavy-ion accelerator facility. Anadvantage of the IFS method is that the limit on the lifetime of the secondarybeam ions is determined only by the flight-path through the spectrometer and thespeed of the beam. Typically, nuclei with lifetimes down to the microsecond regioncan be studied. This allows for identification of very exotic species, such as 48Niwhich was recently shown to be bound by fragmenting a 58Ni beam on aNi-target [35].

The high energies and large emittance of the secondary beams are not well suitedfor all experiments. Many IFS facilities have been extended with or are planningstorage rings with coolers, where the energy spread of ion beams as well as thebeam energies are reduced by for example electron cooling. This opens newexperimental possibilities at these laboratories.

A thorough review of the IFS scheme that covers its possibilities and limitations aswell as existing and planned facilities is found in ref. [36].

1.3.2 Isotope separation online (ISOL)

While one can say that most first and second generation RIB-facilities used theIFS method of beam production, ISOLDE being one exception, the thirdgeneration seems to belong to ISOL laboratories, in most schemes extended with apost-accelerator.

In the basic ISOL scheme, a primary beam is impinging on a thick target, forexample UC2. The primary beam can be thermal neutrons, heavy ions or protons,but so far the highest intensities have been reached using high energy protons (atISOLDE 1.4 GeV protons from the PS Booster are used). High-energy protonsproduce radioactive nuclides through spallation, fission and target fragmentation.

19

The beam is stopped in the thick target, and the reaction products are extractedusing radiochemical methods, and subsequently ionized in some sort of ion sourcewhich can be in a unit with the target or after a transfer line. Element specificproperties, such as ionization potential and diffusion constant, play crucial rolestogether with the target temperature. This makes it obvious that beams of someelements will be harder to produce than others, but developments, on e.g. the laserion source [37] and new target geometries [38], are constantly being made.

After extraction from the ion source, a magnetic separator system purifies thebeam, which can then be sent to experimental areas. Depending on the selectivityof the target and ion source, the beams are often quite pure to begin with, and theseparator systems are not as complicated as in an IFS facility, where the separatorreally creates the secondary beam out of a collection of fragments. Present dayISOL facilities give very low energy beams, typically around 60 keV. Theseenergies are favourable for e.g. β-decay studies, but also set limits on whatexperiments that can be performed. At many new facilities, such as SPIRAL andREX-ISOLDE, post-accelerators will be installed. The secondary beam energieswill then then have energies ranging from one up to some tens of MeV/u. This willof course give completely new experimental options at ISOL laboratories.

The beams from ISOL facilities are often of higher intensity the IFS counterpart,and the beam quality is usually superior (the emittance is smaller and purityhigher). The low-energy beams can easily be implanted in foils and their decayproperties studied. However, extraction of the ions from the target and thesubsequent ionization and acceleration takes time, and the lifetime of the studiednuclides have to be on the order of at least milliseconds, depending on the chemicalproperties of the element and which type of ion source that can be used. A reviewof the ISOL scheme and existing as well as planned facilities is given in ref. [39].

1.3.3 Summarizing the production techniques

In their basic appearance, IFS and ISOL facilities are different in several ways. IFSgives high energy secondary beams of mixed isotopes, which in principle haverather large emittance. The high energies focus the beams relativistically, butmuch care has to be taken to purify them isotopically. ISOL facilities on the otherhand give pure, low-energy beams with good spatial properties. The big effort hereis extracting the reaction products from the target-ion source. In new laboratories,methods are developed to cover the energy gap between the basic techniques.Post-accelerators at ISOL facilities increase beam energies, while beam cooling andstorage rings at IFS facilities decrease the energy and emittance of those beams. Incombination with ion source developments, experimentalists will in a near future

20 Introduction

have available beams of a few keV up to several GeV/u, having high intensities,good purity and good emittance. To increase the number of beams, many projectsare dedicated at investigating alternative production mechanisms. One example isSPIRALII, where the plan is to use neutron-induced fission and extract the fissionfragments. In all, these prospects promise many exciting discoveries in the comingdecades.

1.4 Experimental techniques

What experimental techniques are used to study exotic nuclei? This questioncould easily be the topic of a PhD thesis in itself, so below only the main classesare listed. More details on the techniques used in the investigations relevant forthis thesis are given in the relevant sections, as well as in the appended papers 1–7.

1.4.1 Reaction studies

The technique that has come to dominate the studies of nuclei far from stability isreaction studies. This can be called an active method where the nucleus isproduced and then forced to interact with a target so that the exotic species canbe investigated through their reaction properties. IFS-beams are apt for many ofthese experiments, even if some of them can also be performed at ISOL facilities.There are a multitude of reactions that can be studied, probing different featuresof the radioactive nuclei. Some reaction studies, like measurements of crosssections and masses, basically probe global properties while for examplemomentum distributions, γ-ray studies and transfer reactions mainly investigatestructural details. In this thesis, examples are given in sections 5–7.

1.4.2 Decay studies

Another way to study exotic nuclei is through their radioactive decay. This couldbe viewed as a passive method, since the ions are allowed to decay unperturbedafter the production. These experiments are well suited for ISOL beams. Manydripline nuclei have extremely large Qβ± values. This, in combination with thedecreasing particle separation energies in the daughter as one goes away fromstability, often makes it energetically allowed for the β-decay to populate particleunbound states in the daughter, resulting in a multitude of β-delayed particlesfollowing the decay. Investigations of for example branching ratios of differentdecay modes, angular correlations of the breakup products and strength functions

21

can give important insights into the structure of exotic nuclei. A review on thesubject is found in [40].

1.4.3 Techniques for unbound nuclei

When investigating unbound systems, a combination of reaction methods anddecay studies must naturally be used. The unbound nucleus has to be createdthrough some reaction, for example by nucleon transfer or scattering, and itsproperties can only be probed through its decay products. In this thesis, examplesof this class of experiments are given in chapters 2–4.

1.5 Applications of RIB’s

Contrary to the first impression, there are many fields other than nuclear physicsthat can benefit from the radioactive beam facilities and the nuclear physicsresults obtained there.

1.5.1 Nuclear astrophysics

All nuclei in the universe are created at some time and place, and nuclear physicsis connected with astrophysics mainly through the nucleosynthesis processes.Astrophysical models calculate the abundances of different isotopes and comparewith observed numbers. As input, the theories need reaction cross sections andhalflives, which are measured by nuclear physicists. The initial nucleons wereproduced in the first three minutes after the Big Bang. The conditions during thisshort period allowed for production of 1H (≈ 75%), 4He (≈ 24%) and traces of 2H,3He and 7Li [41]. The first generation stars were built of these materials, and otherelements were produced in stellar fusion, explosive burning and other astrophysicalenvironments. In stars, hydrogen nuclei fuse to form helium in an energy releasingprocess. Once a star begins to run out of hydrogen fuel, the created helium nucleican start to fuse. In this way, it is energetically favourable to produce elements upto iron. Once the star reaches the last stage, it has run out of nuclear fuel andcollapses. Much of its mass, and thus the formed elements, will be ejected intospace during this process [42].

Nuclei heavier than iron are formed in competitions between nucleon absorptionand β-decays. An isotope ejected into space by dying stars can in neutron richenvironments absorb a neutron, and thus form a β-decaying nucleus. If theneutron flux is large, the nucleus can absorb several neutrons before decaying, and

22 Introduction

in this way isotopes further towards the neutron dripline are formed along what iscalled the r-process path (r for rapid). If there are few available neutrons, it ismore likely that the isotope will β-decay to a nucleus with Z+1 before capturinganother neutron. This so called s-process (s for slow) proceeds closer to thestability line. In a proton flux, similar processes can occur along the protondripline (rp-process). The calculated isotopic abundances are heavily dependent onabsorption cross sections and β-decay halflives from the nuclear physics side, aswell as particle fluxes, temperatures and pressures in different situations from theastrophysics side [22].

A long standing nuclear astrophysics question is the solar neutrino problem. Thisconcerns the large discrepancy (30-50%) between the theoretical and experimentalvalues of the solar neutrino flux. In this context, much attention has been directedtowards one single nucleus, 8B, since the β-decay of 8B is responsible for allhigh-energy neutrinos from the sun. Explaining the discrepancy by neutrinooscillations (see next section), does not make detailed knowledge of the 8B reactionrates less important, since the neutrino data have low statistics and understandingof the underlying solar processes will allow reductions of the error bars on theoscillations. For further discussion on 8B in context of the solar neutrino flux, seesection 7.1.2.

1.5.2 Probing the physics beyond the standard model

Radioactive beam facilities can also give results of importance to fundamentalphysics, for example tests of the standard model. Examples are investigations ofthe unitarity of the Cabibbo-Kobayashi-Maskawa matrix which describes thecoupling of different quarks. The superallowed Fermi 0+⇒0+ transitions betweenanalog T = 1 states should all have the same ft–value if the weak vector couplingconstant GV is constant, as the CVC hypothesis implies. However, theexperimental data have to be corrected to eliminate radiative and isospindependent effects. The unitarity of the CKM-matrix can be tested by the elementsof first row, whose squares should add up to 1. The measurements done withisotopes with A <55 has given a result which deviates from 1 with more than 2.2σ,see eq. 1.5.

V 2ud + V 2

us + V 2ub = 0.9968 ± 0.0014 (1.5)

Is this deviation a sign of insufficient understanding of the correction factors or ofnew physics? Since the corrections grow larger with larger Z, a goal of RIBfacilities is to produce suitable A > 55 isotopes with high enough intensity. Oneexample is 74Rb, which is being studied at ISOLDE and ISAC.

23

Returning to the solar neutrino problem, this can be resolved if the neutrinososcillate, i.e., change flavour. For this to be possible, they must have mass, whichis also beyond the standard model. Experimentally, such oscillations have recentlybeen reported [43], but there is a need for more controlled measurements.Producing a neutrino beam and sending it to one (or several) measuring station(s)hundreds of kilometers away would give more definite answers. The directedbeams could be made with known intensities, as opposed to the present situationwhere the solar neutrino flux is a quantity to be measured and not be used asinput. Future RIB accelerator facilities could be used to create high intensityneutrino beams from decaying muons. There are also ideas to use the RIB’sthemselves, since their decays produce neutrinos. One example would be 6He,which can be produced with high intensity.

1.5.3 Solid state physics

Using the fact that different isotopes in most situations are chemically equivalent,radioactive ions can be sent into matter as spies. They function as stable atoms,but signal their location by emitting radiation which can be detected. This is usedby researchers wanting to understand for example doping or contamination effectsin semiconductors or superconductors. Doping with a radioisotope givesinformation on how atoms diffuse through the material, and even on the latticesites occupied by dopants or contaminants. Radioactive beam facilities are used toproduce the radionuclides, and it is often advantageous to implant the beams inthe test material at the facility and make experiments there.

Accelerated beams are also used to manipulate materials. Applications areinvestigations of track formation and radiation damages is materials, or productionof extremely narrow tunnels in substances and study the properties of suchnanotubes.

1.5.4 Nuclear medicine

In 1895, Rontgen discovered the properties of X–rays passing through the body.This started the interdisciplinary field of nuclear medicine, which provides bothdiagnostic and therapeutic tools. Radionuclides are treated by the body as thecorresponding stable isotope, and can thus be used to examine bodily functions. Acommon technique to check the functioning of the thyroid gland is to monitor itsuptake of radioactive 131I with a gamma camera.

Positron Emission Tomography (PET) can measure for example blodflow through

24 Introduction

different parts of the brain. Shortlived, positron emitting isotopes are injected intoa patient and the two anti-parallel photons emitted when positrons annihilate withelectrons are observed with detectors placed around the patient. The radioisotopesbest suited for medical use are not found in nature, but have to be created usingaccelerators. ISOL facilities in many places supply hospitals with radionuclides asa part of their program. It is desirable that radioisotopes injected into the bodyhave halflives on the order of minutes to hours, and so the laboratories have to besituated rather close to the hospitals.

The most known use of nuclear physics in medicine is cancer therapy. If enoughenergy is deposited in a molecule, ionization can occur, giving the moleculedifferent chemical characteristics. If this happens in a cell, it could mean that anenzyme stops working or that DNA is damaged. The aim of radiation treatment isto subject tumour cells to ionizing radiation that damages them beyond repair.The obvious problem is that the tumour is embedded in healthy tissue, which willalso be harmed by the radiation. To administer a lethal radiation dose to thetumour but minimize the exposure of the healthy cells is a balancing act. Theenergy of the incoming beam will be deposited as a function of penetration depth,and this curve differs for different types of radiation. The energy of the beam canbe adjusted so that the peakis located within the tumour, which will then get ahigher dose than the surrounding tissue. To further minimize the dose to thehealthy tissue, the beam direction is often rotated so that it goes through differentparts of the body on its way to the tumour.

The radiation types most often used to carry the energy into the tumour arephotons and electrons. However, heavier charged particles have many advantagesand proton therapy has been tested in several places with positive outcomes. Forprotons and heavy ions, the energy deposition function will have a sharper peakthan for photons. But, so far the apparatus needed is larger and more complicatedthan the standard hospital machines. In Japan, an accelerator complex dedicatedto cancer therapy was built some years ago. The hospital uses heavy ion beams,and the results have been encouraging. At GSI in Germany, a medical test facilityhas been constructed in connection to the existing accelerators, and clinical trialson non-operational brain and spinal cord tumours have been performed for someyears. High energy 12C beams are used, and to minimize the dose to the healthytissue, the fact that the beam is well defined in space is utilized. Steering magnetsare used to scans the beam over the cross section of the tumour in two dimensions,like the electron beam from the cathode tube scans a TV-screen. Then the energyis slightly changed to vary the depth of the Bragg peak location, and the scanningis repeated. This procedure makes it possible to distribute the dose very exactlyspatially. It is possible to use radioactive beams in the treatment. One advantagewould be that the decay energy would be mainly deposited in the tumour as well

25

as the kinetic energy. If positron emitting ions are used, a real time PET scangives direct feedback on the process. One problem is that RIB’s are usually not aswell defined in space or in energy as stable beams.

An old technique of cancer treatment is to make the tip of a needle radioactive andinsert this directly into the tumour. This method is obviously limited to tumoursclose to the skin, and modern research tries to replace the needle with a carriermolecule that would take the radioactive isotope to the tumour and attach itselfthere. To find molecules that bind selectively to both tumour cells and suitableradioactive isotopes is a challenge to interdisciplinary researchers.

26 This page is intentionally blank. Feel free to fill it.

27

2 Elastic resonance scattering

Resonances are in a classical sense eigenfrequencies of vibrational motion,characterized by large amplitudes. A well known example of resonant oscillation isthe dramatic collapse of the Tacoma bridge in 1940. Due to an unluckycombination of winds and the aerodynamic properties of the bridge, it was driveninto oscillation at a resonance frequency until it was torn apart.

In a quantum mechanical and nuclear physics context, resonances are unboundstates that have a finite life time even though they lie above the threshold fordisintegration. Quantum mechanically, the mere presence of a potential wellresults in a passing wave having a transmission coefficient smaller than unity. Ifpotential barriers such as angular momentum barriers and Coulomb barriers areadded to the spherical potential, the transmission is reduced further as the barrierheight grows larger than the energy of the wave. The transmission probability isdetermined by the thickness and height of the barrier relative to the energy of thestate. If the resonance is situated close to the top of the potential barrier thetunnelling probability will be large and the lifetime τ of the state will be short, asan incident nucleon can both enter and leave the state with high probability. Thelife time is related to the width Γ of the state as Γ = ~/τ . For states with `=0,neutrons feel no potential barrier and the states become very broad and difficult todetect. For protons, the Coulomb barrier contains the wave inside the well for asomewhat longer time. If ` > 0, the increased angular momentum barrier tends tokeep the wave inside the nucleus, giving more longlived resonances.

Nuclei at the driplines have few bound states, and in order to get a completepicture of nuclear structure far from stability unbound states have to be studied.Indeed, as previously mentioned, some of the most interesting isotopes have nobound states at all.

Studies of levels in the continuum is not new, but resonances have beeninvestigated since the 1930’s when narrow states in reactions using thermalneutrons were discovered. Discrete resonant states could be anticipated from thequantum mechanical theory being developed at this time, but the observedneutron resonances in heavy nuclei were much sharper than expected. This ledNiels Bohr to formulate the hypothesis of compound nuclei, which assumes thatthe decay of the compound system projectile+target is independent of theformation channel. A formalism for describing resonances was developed by Breitand Wigner while studying absorption of thermal neutrons [44].

A specific case of resonances are cluster states, which exhibit a high degree ofcluster structure. One type is α-cluster states that are well described as a core+α

28 Elastic resonance scattering

system. Such states can be found at high excitation energies in nuclei where thecore is tightly bound. As an example, there are α-cluster states in 16O that have toa high degree have 12C+α structure.

Investigations aiming at nuclei at or beyond the dripline have to modify thetraditional experimental methods or find new ones to reach their goals. When itbecame feasible to produce exotic beams, the first objective was to study theproperties of the exotic ions themselves. However, RIB’s can also be used toproduce even more exotic species. An example is investigations of nuclei beyondthe driplines. These are by definition unbound and have to be probed via theirresonance states.

Transfer reactions where one or several nucleons are interchanged between targetand beam is one method to reach unbound states. This process is generallywritten A+y(a+x, A+x)a+y, where A+y is the target nucleus, a+x is theprojectile and x and y are the transferred nucleons1. A specific example is a chargeexchange reaction with y= n and x= p (or vice versa), that results in a moreneutron rich nucleus. Very large beam intensities are available if stable beams areused, but this advantage quickly disappears when performing multi-nucleontransfer reactions aimed at producing dripline nuclei. The cross section fortransferring i nucleons decreases rapidly with the number i, so that for most casesthe production of an exotic nucleus AX will gain from doing a one-nucleon transferwith a radioactive beam A+ 1X instead of a multi-nucleon transfer in stable beam.For discussions on the use transfer reactions for investigations of dripline nuclei,see for example ref. [45] and references therein.

2.1 Theoretical description of scattering

Elastic scattering has been extensively used as a tool to study resonances for manyyears . In elastic scattering, the exit channel of the reaction is identical to theentrance channel. This implies that no energy has excited the target or projectile.One consequence is that the flux of the outgoing beam is the same as that of theincoming one, and that their wave functions have equal amplitudes. The incomingand outgoing waves can thus differ only by a phase shift δ`.

The cross section for the elastic resonance scattering is coherently added to thedirect reaction where the incident nucleon is scattered off the mean potentialwithout forming a resonant state2 and, in the case of charged incident particles, to

1If x = 0 the reaction is usually called pick-up, and if y = 0 we have a stripping reaction. Fordefinition of the shorthand notation, see eq. 2.10.

2This is called potential scattering, shape-elastic scattering or hard sphere scattering.

29

the Coulomb scattering amplitude.

It is convenient to express the elastic scattering cross section as a function of thephase shifts δ`. The derivation of these formulas can be found in forexample [1, 46, 47], and only a short summary is given below. The cross sectioncan be written as a function of the scattering amplitude f(θ), eq. 2.1.

dσ

dΩ= |f(θ)|2 (2.1)

Expressions for f(θ) can be found from the asymptotics of the scattered wavefunction. Eq. 2.2 gives the scattering amplitude for spinless projectile and targetnuclei [47].

f(θ) =1

2ik

∞∑`=0

(2` + 1)(e2iδ` − 1)P`(cos θ) (2.2)

Squaring eq. 2.2 and integrating over the solid angle, using the orthogonality of theLegendre polynomials, gives the elastic cross section in eq. 2.3 from which it is seenthat the cross section is maximal when δ` = π/2. This is commonly takes as adefinition of the position of a resonant state ER [47].

σel(E) =π

k2

∞∑`

(2` + 1)|e2iδ` − 1|2 (2.3)

The values of the phase shifts can be found by matching the wave functions for theinternal and external regions for each `. The shape of the resonance is given bydδ`(E)/dE. We can expand δ`(E) around δ`(ER) = π/2 and define the width interms of dδ`(E)/dE.

δ`(E) = δ`(ER) − (ER − E)dδ`dE

∣∣∣E=ER

(2.4)

Γ = 2(dδ`

dE

)−1

(2.5)

Combining eqs. 2.4 and 2.5 yields [δ`(E)− δ`(ER)]−1 ≈ tan δ`. Using this andtrigonometric relations together with eq. 2.3 and |e2iδ` − 1|2 = 4 sin2 δ`, gives thefollowing expression for the cross section around the resonance:

σel(E) =π

k2(2` + 1)

Γ2el

(ER −E)2 + Γ2/4(2.6)

This is the Breit-Wigner formula for an isolated resonance in the elastic channelwith partial width Γel. If the projectile and target have spins sp and st, the factor

30 Elastic resonance scattering

(2` + 1) should be replaced with the statistical factor g(J) which describes thecouplings of the spins to the channel spin J [47].

g(J) =2J + 1

(2st + 1)(2sp + 1), J = ` + st + sp (2.7)

The influence from potential scattering can be included by introducing an extraphase and amplitude. Extending the formalism to spins different from 0 introducesa dependence on the azimuthal angle ψ and contributions from substates withm ≥ 0, described by associated Legendre functions Pm

` (cos θ) in addition to theLegendre polynomials P`(cos θ). The expressions for the elastic scattering crosssection of protons on spinless nuclei in eqs. 2.8a and. 2.8b can be found in forexample [47, 48] and are cited below. The general formulas for target spin 6= 0 canbe found in [46].

dσ

dΩ= |A(θ)|2 + |B(θ)|2 , (2.8a)

where

A(θ) =zZ

2µv sin2(θ/2)e(i~/µv) ln(1/ sin2(θ/2)) +

1

2ik

∞∑`=0

[(` + 1)(e2iδ+

` − 1) + `(e2iδ−` − 1)]

×e2iσ`P`(cos θ)

(2.8b)

and

B(θ) =i

2k

∞∑`=0

(e2iδ+

` − e2iδ−`

)e2iσ`P 1

` (cos θ) (2.8c)

where e2iσ` is defined by

e2iσ` =Γ(`+ 1 + i

k

)Γ(` + 1− i

k

) (2.8d)

The first term in A(θ) represents Coulomb scattering, and the other terms in A(θ)and B(θ) are due to nuclear scattering. Partial waves with ` > 0 have to beseparated into two components since there are two possible values for the coupledspin, J = `± 1

2. The phase shifts associated with these contributions are denoted

δ+` and δ−` , respectively. The phase shifts δ±` are the sum of the hard sphere phase

shifts -φ` and the resonant nuclear phase shifts β`:

δ±` = β±` − φ` (2.9)

31

Other notations introduced are σ` for the Coulomb phase shift, z and Z for theproton and target charges, v for their relative velocity, µ for the reduced mass ofthe system and k for the magnitude of the wave vector. For the case of a spinlessprojectile, B(θ) ≡ 0 and A(θ) is simplified by δ+ ≡ δ− ≡ δ.

2.2 Thick versus thin targets

Elastic scattering has traditionally been performed by scattering high-resolutionbeams of light ions off thin, solid target foils. Some examples of such experimentsusing protons or α’s are found in refs. [49, 50]. In tandem accelerators it is notuncommon to reach energy spread of the beam below 1 keV. Such well definedbeams and the use of thin targets give excitations functions with excellentresolution. However, scanning large energy regions with energy steps of a few keVis very tedious.

To make the experiments more efficient in terms of beam time, methods to usethicker targets have been developed. A thicker target means that the beam willscan an energy region due to its energy loss in the target, and larger energy stepscan be taken. The price one has to pay is a deteriorated energy resolution. Anexample of elastic backscattering on thick target is found in [51].

2.3 Conventional versus inverse geometry

We can write a general reaction process where a projectile a hits a target A at restin the laboratory frame, and with reaction products B and b as:

a + A⇒ B + b or, in shorthand, A(a, b)B (2.10)

B usually denotes a heavy residue that stays in the target, while b is ejected andcan be detected. Traditionally, reaction experiments are carried out using a lightion beam such as H or He and a heavy ion target. Investigations of 13N throughelastic proton scattering would for example be performed using a proton beam anda thin 12C target, a reaction denoted 12C(p,p)12C [52]. Detecting energies andangles of the scattered protons as a function of the beam energy give informationabout the 13N resonances. This geometry became the standard mainly for practicalreasons, since early accelerators could only accelerate light ions.

However, changing places of beam and target can give several interestingexperimental advantages. We can consider the 13N experiment mentioned above ininverse geometry3, p(12C,12C)p. Then, a beam of 12C impinges on a proton target

3Inverse geometry reactions are usually symmetrically written a(A,B)b even if b is detected.

32 Elastic resonance scattering

and the scattering angle and energy of the proton is detected. If a thin target isused, the scattered 12C can in principle be detected as well, depending on itsenergy loss in the target. Comparing the energies of the scattered protons inconventional and inverse kinematics, we find that the proton energies are about afactor 4 larger in inverse geometry (see A.3). This is an experimental advantagewhen searching for low-lying resonances.

The protons will in inverse geometry be scattered in a more narrow cone (lab.)than in the case of a proton beam. A larger part of the solid angle can therefore becovered, and the detection efficiency is increased. This is seen from thetransformation of c.m. angles into the laboratory frame, see appendix A.3.1.

If the target is made so thick that it stops the beam, it is possible to makemeasurements at 0lab., which corresponds to 180c.m. . At this angle, both Coulomband potential scattering that interfere with the elastic scattering are minimal.

Proton targets will often be needed for experiments in inverse kinematics. Purehydrogen targets can be made in liquid phase and gas phase, and even solidhydrogen targets have been developed.However, the pure hydrogen targets rely onrather specialized technologies (liquid and solid) and safety issues is a concern(especially for gas targets). The only option is often to use either solid plasticcompounds or hydrogen-rich gas targets as substitutes for proton targets. Thisrequires background runs which should be subtracted from the data, a procedurewhich introduces uncertainties and increases the error bars.

Note that the use of secondary beams in reaction experiments implies inversegeometry, simply because there is no possibility to make a target out of the fewand shortlived exotic nuclei produced. It is somewhat confusing that mostexperiments involving RIB’s are performed “inversely”, but, as is the case withmany historical names, the term has stayed on.

2.4 Elastic resonance scattering using inverse geometry

and thick gas targets

The experiments described in sections 3 and 4 used a technique of ElasticResonance Scattering in Inverse geometry using Thick gas targets (ERSIT) [53, 54].This method combines the high cross sections of elastic scattering with theenergy-scanning efficiency of a thick target. The inverse geometry brings forwardlyfocused reaction products and the possibility to measure at 180 c.m. angle.

The method consists of a heavy-ion beam at rather low energy, around 10 MeV/u.The beam is guided into a scattering chamber, filled with a gas rich in the light ion

33

to be detected, for example H or He. The gas acts simultaneously as beam stopperand scattering target. In the back of the scattering chamber, detectors forregistration of the energy and angle of the scattered light ions are placed. Theincoming beam ions are continuously slowed down in the gas, and if their energy atsome point along the path corresponds to a resonance in the compound system,the cross section for elastic scattering increases dramatically and can exceed 1 b.The light scattered nuclei experience a much smaller stopping power than theheavy ions and pass through the gas to reach the detectors. The gas pressureshould be adjusted so that the beam is stopped in the gas, preferably as close tothe detectors as possible in order to avoid that low-energy scattered ions lose theirenergy in the gas before being detected.

Advantages of a gas target as compared to a solid target are mainly itshomogeneity and the possibility to easily modify the pressure and thus the targetthickness. This facilitates changes of beam and/or beam energy during the run.

Due to the two-body nature of the reaction, detection of energy and angle isenough to uniquely determine the c.m. energy of the scattering, and thus theheavy-ion energy in the entrance channel, see section A.3. A point worth stressingis that this also corresponds to a specific interaction point in the chamber, sincethe beam ions scan the energy region from Ebeam down to 0 due to the slowingdown in the gas. As a result of this, each proton energy also corresponds to aspecific solid angle. This is schematically drawn in fig. 2.1, where examples ofenergies and corresponding distances and excitation energies are given by therulers. After detecting the light-ion energy, it is possible to use energy loss tablesto find the interaction point. With this information, one can correct the detectedion energy for its energy loss in the gas, and also find the scattering angle, ifdifferent from 0lab..

2.4.1 Energy resolution

The above discussion is strictly valid only for a mono-energetic beam, which isnever possible to obtain. Even if the beam from the accelerator would be infinitelynarrow, its energy spread will increase as the beam is slowed down in the target.How much does this fact influence the energy resolution? We assume that thebeam spread at some point along the interaction path is ∆E. This energy widthwill result in protons with the same energy coming not from the same point butfrom within a distance interval ∆x. The size of this interval depends on ∆E as

34 Elastic resonance scattering

well as the energy loss dE/dxM of the beam ions:

∆x =∆E(dEdx

)M

(2.11)

Here and in the following, M and m denote the projectile and scattered particlemasses, respectively. The corresponding spread of detected proton energies ε willalso depend on the proton energy loss dE/dxm:

ε = ∆E

(dEdx

)m(

dEdx

)M

(2.12)

Using the Bethe-Bloch energy loss formula and the relation between the velocitiesfor the scattered proton and the beam ions (vmlab and v), see appendix A.3.1, wefind an approximate expression for ε:

ε ≈ ∆E

4

z2

Z2

M +m

M≈ ∆E

4

z2

Z2(2.13)

As an example, we consider 10C+p scattering with ∆E = 5 MeV, for which casewe get a contribution of around 35 keV to the proton energy resolution from thebeam spread. This is on the same order as the contribution from the detectorenergy resolution. Clearly, the spread of the beam does not significantlydeteriorate the resolution. Also, transforming the resolution from the lab. frame toc.m. will decrease the resolution by a factor 4 (see section A.3).

ν

10C

10B

Sidetectore+

CH4 gas

10C 10C

10C

p

p

10C

Eres

Eres

10C

Distance (cm)

Elab.10C (MeV)

Eex.11N (MeV)

0 10 20 30 40

90 77 62 45 20

8.1 6.4 5.6 3.7 1.8

Figure 2.1: The figure illustrates the ERSIT method schematically for the case of 10C+p. Therulers show the relation between distance, heavy ion energy and excitation energy.The numbers are taken from the first GANIL run. Note that the resonance positionsindicated are not the true ones. It is seen that the solid angle is dependent on theinteraction position, and therefore Ω = Ω(E).

35

2.4.2 Competing reactions

The backbone of the technique is that the elastic cross section at a resonancedominates over all background sources. The possible exception is inelasticresonance scattering, a channel witch has to be considered for each case as itdepends on the excited states of the specific projectile. However, elastic andinelastic resonance scattering can be experimentally distinguished from each other.Since energy goes into exciting the projectile to an energy E∗, the inelasticscattering reaction which gives protons of the same energy as the elastic channelhas to take place for a higher projectile energy. If E∗ is smaller than the kineticenergy of the projectile, E∗ E, the detected proton energy for the inelasticscattering can be written [55](notations following appendix A.3):

Emlab(E

∗) = 4m ·M

(m +M)2

(E − E∗

2

M +m

m

)(2.14)

For Emlab(E

∗) to be the same as Emlab(g.s.), the projectile energy has to be higher by

an amount E:

E≈E∗

2

(M +m)

m(2.15)

Since this corresponds to the reaction taking place farther from the detector, thetiming will be different for the two processes. If the excited state is not too low inenergy the elastic and inelastic channels can be separated in a two-dimensionalplot of ToF versus proton energy.

The ERSIT method was proposed at Kurchatov Institute and used there and atJYFL to study resonances with α-cluster structure in O, Ne, Mg and Si isotopes.An example of such a spectrum is shown in fig. 2.2, where narrow resonances withlarge cross sections are measured. For these experiments, the scattering chamber isfilled with helium gas. Stable beams, or beams close to stability, together with theinherent high efficiency of the technique ensure high statistics, and remarkably wellresolved, narrow peaks were seen in all excitation spectra [53, 56]. Theinterpretations of these experiments are simplified since both target and projectilehave spin 0, and only the angular momentum of the resonance enters into theexpression for the differential cross section.

2.4.3 Applicability to radioactive ion beams

Radioactive ion beams are well suited for use in elastic scattering reactions onthick targets. The relatively low intensities of RIB’s require reactions with large

36 Elastic resonance scattering

6 8 10 12 14 16

400

800

dσ/dΩ (mb/sr)

Eex. (MeV)

16O + α

Figure 2.2: Spectrum of α-cluster states in 20Ne measured from 20Ne+α. The spectrum is pub-lished in [56].

cross sections and setups with high efficiency. Low-energy secondary beamsproduced by IFS facilities usually have a large energy spread. This puts someconstraints on the experimental methods that can be used. Elastic resonancescattering in inverse geometry using thick targets is well suited for use with RIB’s.The elastic resonance reactions can have cross sections around 1 b, and theresonances are broad enough so that a limited resolution is not a problem. Thefact that the experiment is performed in inverse geometry makes thetransformation from the laboratory system to the c.m. system favourable, sincethe lab. resolution and energy spread will be reduced by approximately a factor 4in the c.m. system (see section A.3).

A problem when using the method is that the desired beam energy is in the regionof a few up some ten MeV/u, and while RIB energies produced by in-flightmethods are much larger than that, the ISOL energies are too small. The solutionto slow down the IFS beams using thick production targets and/or degradersdecreases the intensities by orders of magnitude. However, this obstacle is onlytemporary, since new facilities which combine ISOL-production withpost-acceleration are being started up as this is written. In Louvain-la-Neuve, aISOL with a cyclotron post-accelerator has been operational since 1989, and elasticscattering of RIB’s in inverse geometry have been applied to studies of resonancesof astrophysical interest. The solid polyethene targets used did not stop the beam,

37

so the 0lab. distribution was not measured [57, 58]. The SPIRAL project at GANILwill provide RIB’s in the energy range 1.7-25 MeV/u, which will be ideal forERSIT measurements. Actually, an ERSIT proposal was chosen to be the firstexperiment at SPIRAL4, and more experiments of this type are intended at thefacility. Another method of generating RIB’s by resonance reactions was used atKurchatov [54, 59].

2.4.4 ERSIT summary

Summarizing, the ERSIT method has several features advantageous forexperimental studies of low to medium energy resonances in exotic nuclei. The lowintensities of RIB’s are counteracted by the large cross sections characteristic ofelastic scattering, the forwardly focussed scattered protons and the scanning of theenergy region by every beam ion due to the thick target. The target thickness alsoallows for measuring at 180c.m. where other scattering amplitudes have minima.The rather large energy spread of RIB’s does not deteriorate the energy resolutionof the technique considerably. The inverse geometry makes the proton energieslarger than in conventional measurements, and the transformation from the lab. tothe c.m. improves the resolution. A further benefit is that the theory of elasticscattering has been in use for many years and is well understood.

There are also some potential problems with using the technique. One is thedifficulty to produce radioactive beams at low energies and high intensities, aproblem soon to be overcome in ISOL facilities with post-accelerators such asSPIRAL. Another obstacle is that the interpretation of the data becomes moredifficult when using projectiles and targets with spins different from 0.

4A run on p(18Ne,18Ne)p was performed the first week in October 2001.

38 This page is intentionally blank. Feel free to fill it.

39

3 Investigations of proton-rich nitrogen isotopes

In this section, physics motivations to the experiments performed on 10,11N aregiven before the three measurements carried out so far are described. The resultsare summarized in the next section.

3.1 Physics motivation for studying 11N

The A = 11 mass chain has several intriguing members as shown in fig. 3.1.

The bound member of this chain with the lowest proton number is 113Li, which lies

on the dripline and was shown to be particle stable already in 1966 [60]. However,the interest in 11Li did not explode until the mid-eighties. The measurement of itslarge interaction cross section σI [8] together with the interpretation of this resultas a 2n–halo structure [13], was the starting point of the field of nuclear halophysics. It is still considered a benchmark case for Borromean 2n-halos, and sinceits halo properties were discovered it has been the subject of hundreds oftheoretical and experimental publications. Its ground state structure is composedof roughly equal amounts of s- and p-wave relative motion between the neutronsand the core [61, 62]. However, as will be discussed later, the theoretical modelsare hampered by the lack of information about the interactions in the unboundsubsystem 9Li+n.

Li311

11

4Be

511

B

11C6

711

N

t = 20.4 min1/2

t = 13.8 s1/2

t = 8.5 ms1/2

1n halointruder state

proton unboundintruder state

= 1.4 MeVΓ

Stable

80 %

2n halo

abundancenatural

Figure 3.1: The A = 11 mass chain and some properties of its isotopes.

The shell structure of 114Be has intrigued physicists for four decades. The inversion

of its ground state and first excited state was discussed already in the sixties [63].

40 Investigations of proton-rich nitrogen isotopes

Considering the simple shell model, the last neutron in 11Be should be placed inthe 0p1/2 orbit. However, experiments have proved that the 1s1/2 level in thisnucleus comes down in energy from the sd-shell into the p-shell, constituting a socalled intruder state1. This gives the ground state spin Iπ = 1/2+ instead of theanticipated 1/2−. The 1/2− level is instead found at an excitation energy of320 keV as the only bound excited state, situated only 164 keV below thethreshold for neutron emission. The valence neutron in 11Be is mainly in the 1s1/2

orbit, and this combined with the low Sn = 504 keV allows a 1n-halo to developaround the 10Be core. An indication of an extremely large radius of 11Be was givenalready in 1983 when the B(E1) strength of the transition between 1/2− and 1/2+

was measured and found to be the largest seen between bound states [64]. 11Bewas for over a decade the only known 1n-halo nucleus, and has been extensivelystudied both theoretically and experimentally.

The initial picture of a halo nucleus was a few-body system whose properties werenot influenced by internal degrees of freedom of the clusters. The determination ofthe details of the 11Be ground state configuration has been an important tool ininvestigating to which degree this assumption holds. In the simplestapproximation, ψ(11Beg.s.) can be written as |10Be(0+)⊗ n(0 s1/2)〉. However, 10Behas an excited 2+ state at 3.37 MeV, which can couple to a 0d5/2 valence neutron.This coupling can give a contribution to the 11Be ground state 1/2+, which shouldthus be written:

|11Beg.s.(1/2+)〉 = αs|10Be(0+) ⊗ n(0 s1/2)〉+ αd|10Be(2+)⊗ n(0d5/2)〉 (3.1)

The αx coefficients are the spectroscopic amplitudes. The admixture of the secondcomponent has been calculated to between 7% [65] and 40% [66]. Experimentally,the situation can be probed using one-neutron removal reactions combined withinbeam γ spectroscopy, a technique described in sect. 5.3.3. Such experiments wererecently performed at NSCL, and the results support the picture of a dominating|10Be(0+)⊗ n(0 s1/2)〉 configuration with a small admixture of the core excited 2+

state coupled to a valence 0d5/2 neutron [67]. A DWBA analysis of thep(11Be,10Be)d reaction experiment gave a relative weight for the core-excitedcomponent of 16% [68].

The intruder state has been explained for example in ref. [69] as a combined effectof (i) reduced single-particle energies close to the driplines, (ii) admixture of 10Be∗

in the 11Beg.s. and (iii) Pauli blocking for the negative parity state. Fig. 3.2 showshow the splitting of the 1/2− and 1/2+ changes for the N = 7 isotones when goingfrom the proton to the neutron dripline. The level inversion occurs for 11Be, andrecent experimental evidence suggests that the trend continues for the unboundN = 7 nuclei 10Li [70] and 9He [71], as was suggested theoretically in [69]. The

1Also the terms “parity inversion” and “level inversion” are found in the literature.

41

15O 12B 9He10Li11Be13C14N

8765432

-12

-14

-10

-8

-6

-4

-2

0

2

Z

Eex.−Sn(M

eV)

1/2+

1/2−

Figure 3.2: Systematics of the lowest states for the N = 7 isotones. The onset of inversion comesat 11Be and the trend continues for 10Li and 9He.

most proton rich member of the A = 11 chain is 117N, mirror nucleus to 11Be. This

nuclear system is an interesting case, and it is well suited for investigation with theelastic resonance scattering-method described in the previous section. 11N isunbound with respect to proton emission, and not much was known about it beforeour experiments. The obvious question to answer was of course whether the levelinversion observed in 11Be would show up also in its mirror partner on the protondripline. Another issue was if the halo structure of 11Be would somehow manifestitself also in an unbound, proton rich system, and how the energy differences of thelevels would shift in going from the neutron to the proton drip line. However, itseems difficult to talk about true halo properties for unbound systems.

The lowering of IAS2 energies in the proton-unbound partner in a mirror pair, theso called Thomas-Ehrman shift, was observed for the 13N–13C pair already in thefifties [72, 73]. The exact mechanism of the shift is still in question, which makespredicting level energies in mirror partners difficult. Knowledge of the 11Nstructure allows for comparisons with the well known 11Be, thus adding a datapoint to adjust theoretical models with.

The positions and widths of the states in 11N are also relevant in the context oftwo-proton emission from 12O. If this decay is sequential it goes through aresonance in 11N. If this channel is not energetically open, the decay is forced to besimultaneous. Being one of the lightest 2p-emitters, 12O has recently attractedexperimental [74, 75] and theoretical interest, not least due to the disagreement ofthe ground state width (see [76] and references therein).

2Isobaric Analog States, i.e., states in isobars obtained by charge conjugation and thus having thesame isospin T.

42 Investigations of proton-rich nitrogen isotopes

The first experiment concerning 11N was performed in 1974 using the14N(3He,6He)11N reaction [77]. The results indicated a resonant state at 2.24 MeV,which was interpreted as the first excited state 1/2− rather than the ground state.

3.2 Physics motivation for studying 10N

Table 3.1: Experimental results for low-energy states of 10Li. The energies are given in MeV, andGS denotes “Ground State”. The fourth column gives the orbit of the last neutron orthe ground state spin for 10Li. Energies of are given relative to the 9Li+n threshold.

Ref. ER ΓR Iπ Comments

[78] 0.80±0.25 1.2±0.3 — 9Be(9Be,8B)10Li[79] 0.15±0.15 < 0.4 (s1/2)

11B(π−,p)10Li[80] ≥ 0.15 or ≈ 2.5 — (s1/2) GS, or from 9Li excited state[81] 0.42±0.05 0.15±0.07 1+ 9Be(13C,12N)10Li and

0.80±0.08 0.30±0.10 2+ 13C(14C,17F)10Li reactions[82] (> 0.100) (< 0.230) (s/p) Weak evidence for this state,

0.538±0.062 0.358±0.023 s1/211B(7Li,8B)10Li used

[83] < 0.05 — s1/2 By 11Li break-up, as < −20 fm[84] 0.21±0.05 0.12±0.10

0.05 s1/2 By 11Li break-up0.62±0.1 0.6±0.1 p1/2 at relativistic energies

[85] 0.42 — p1/2 From 11Li break-up[86] < 0.05 — s1/2

18O+9Be, as < −20 fm0.540 — p1/2

[87] 0.500±0.06 0.400±0.06 p1/29Be(9Be,8B)10Li

[88] 0.24±0.04 0.10±0.07 1+ 9Be(13C,12N)10Li0.53±0.06 0.35±0.08 2+

[70, 89] < 0.05 — s1/211Be+9Be, as < −20 fm

The unbound 10Li has been mentioned several times in the previous paragraphsand in earlier sections. The interest in 10Li originates in its importance as one ofthe two-body subsystems of 11Li. Another important property is its being a N = 7nucleus with a possible inversion of the lowest levels, which if proved will indicatethe trend of the intruder 1s1/2 state energy after 11Be. However, detection ofneutron s-wave levels close to the threshold are difficult. Since there are neitherCoulomb nor centrifugal barriers to contain the wave, the state will be very broadand technically it will not be a resonance as the phase shift does not pass π/2. Theexperimental challenges are reflected in the contradicting results for the position,width and quantum numbers for the ground state of 10Li as is seen in table 3.1.

43

These are reasons for investigating its mirror system, 10N. This could give adifferent perspective on the issue, and the results might help to resolve some of theambiguities for 10Li.

The anticipated states are 1−, 2− and 1+, 2+, resulting from couplings between the9C 3/2− ground state and a valence nucleon in either 1s1/2 or a 0p1/2 orbits.

3.3 Introduction to the experiments

In the following sections, three runs in which the proton-rich nitrogen isotopes10,11,13N were investigated will be described, while the results will be presented insection 4. The studies were used the ERSIT technique described in section 2. Asummary of the measurements done is given in table 3.2.

Two different spectrometers were used for these measurements, the LISE3 atGANIL and the A1200 at NSCL, MSU. The spectrometers are used as IFseparators which produce a beam of radioactive isotope through projectilefragmentation and separates the products using Bρ−∆E − Bρ separationtechniques.

In the back of the chamber, Si-detectors were placed to detect scattered protons.A kapton foil3 separated the gas volume from the beamline vacuum. The thicknessof this foil differed between the runs, but is on the order of tens of µm. Theentrance window has a diameter of 20 mm.

In all runs, the scattering chamber schematically drawn in fig. 3.3 was used. It canbe filled with 0.0241 m3 of gas at a pressure of maximum 2 atmospheres. Filledwith CH4 gas, it constitutes a thick proton rich target, sustituting for a purehydrogen target. To determine the influence of the C atoms in the target,background measurements were done with CO2 gas. In the analysis, we assumethat 12C and 16O give similar contributions to the background. They are bothN = Z nuclei of roughly the same size, having large proton separation energies,Sp(

12C)=15.96 MeV and Sp(16O)=12.13 MeV. The CO2 measurement will also

disclose any reactions due to beam contaminants, reactions in the kapton foil orother possible sources related to the beam and setup. Estimating the number ofcarbon-like nuclei in the CO2 volume gives a scaling factor which in addition to thenumber of projectiles should be used when normalizing the background run.

3The homogeneity of this plastic H10C22N2O5 foil is better than 10%. Kapton is the commercialname for the chemical name polypyromellitimide-polyamide.

44 Investigations of proton-rich nitrogen isotopes

3.4 The first GANIL experiment

The aim of this first run was to investigate the lowest levels of 11N, and inparticular to deduce the ground state spin. Another intention was to test themethod of elastic resonance scattering on thick gas targets in inverse geometryusing a radioactive beam. The experiment was carried out at the LISE3spectrometer at GANIL, shown in fig. 3.4. It consists of two 30 bending dipolemagnets with Bρmax = 4.3 and 3.2 Tm, respectively. Between the dipoles is theintermediate focal plane where an achromatic degrader can be inserted. Theflightpath from the production target to the final focus is 43 m. A velocity filtercompletes the isotopic selection. The LISE3 spectrometer is described in detail inref. [90].

3.4.1 Experimental setup

To produce 11N as a compound system using elastic proton scattering, a beam of10C was used. The scattering chamber was filled with CH4 that acted as protontarget. The primary beam of 12C6+ at 75 MeV/u was fragmented in a 7962µmthick, rotating Be target combined with a 400 µm thick Ta target situated at theentrance of LISE3. The intensity of the 12C beam was about 2·1012 ions/s. The10C fragments were selected by the LISE3 spectrometer using an achromatic Bedegrader at the intermediate focal plane (on average 220 µm thick) and theWien-filter with 50 kV fields after the last dipole. The kapton foil at the entranceof the scattering chamber had a thickness of 80 µm, corresponding to1.14 mg/cm2. Before entering the chamber, the beam passed a Parallel PlateAvalanche Counter (PPAC), which was used as beam monitor. The integratedbeam current was later used to deduce the cross section.

Kaptonfoil

secondary beam

movableα source

gas Si-detectors

12.5o

25o

ToF(stop)

∆E, E, ToF(start)

55 cm~

Figure 3.3: The scattering chamber used in the elastic resonance scattering experiments investi-gating proton-rich nitrogen isotopes. The detector angles are defined with respect tothe center of the chamber. The ToF detector was a PPAC or an MCP detector.

45

Table 3.2: The experiments on proton-rich nitrogen isotopes from which results are reported inthis thesis. In all runs, a primary beam of 12C at 75 MeV/u produced the secondarybeams through projectile fragmentation. Ebeam is the seconday beam energy where itenters the active target volume. This is behind the kapton foil for all cases except the9C runs, where it is behind the entrance detector. The error in pressure is ±5 mbar.

Expt. Facility Beam Ebeam (MeV/u) Target P (bar) Isotope

E252 LISE3 12C 6.25 CH4 0.450 13N(1996) GANIL 12C 6.25 CO2 0.240 backgr.

10C 9.0 CH4 0.815 11N10C 9.0 CO2 0.470 backgr.

97030 A1200 12C 5.1 CH4 0.470 13N(1998) NSCL 10C 7.4 CH4 0.490 11N

E252a LISE3 12C 7.1 CH4 0.640 13N(2000) GANIL 10C 8.1 CH4 0.560 11N

9C 11.0 CH4 0.780 10N9C 11.0 CO2 0.420 backgr.

Three Si-detectors, all 2.5 mm thick and with area 300 mm2, were placed in theback of the chamber. The most important one is the one placed at 0 in thelaboratory system (lab.), corresponding to protons backscattered at 180 in thecenter-of-mass system (c.m.). The other detectors were placed at angles 12.5 and25 degrees (lab.) where the angles are defined with respect to the center of thechamber. The thickness of the detectors limited the accessible energy region, since2.5 mm of Si stops protons with energies below 20 MeV and protons above this willonly deposit a part of their energy in the detector. 20 MeV protons correspond to

Target

D1 D2

D3 D4

degrader

EB

S1

S2S3

F1

F2

A /ZvAv/Z

23

selection in

to the

chamber in D6

primary beam from the cyclotron

secondary beam

Figure 3.4: The layout of the LISE spectrometer at GANIL.

46 Investigations of proton-rich nitrogen isotopes

5.6 MeV excitation energy for 11N (5.5 MeV for 13N). The protons that punchthrough the detector will give a background below the maximum energy.

Apart from energy-information from the Si-detectors, Time of Flight-data (ToF)was recorded for the three flight paths PPAC–detector, RF–detector, andRF–PPAC, all started by the last of the two signals (RF is the Radio Frequency ofthe cyclotron).

The energy calibration was done using a triple α-source (244Cm, 241Am and 239Pu)which was placed on a movable arm inside the chamber. It remained there for theentire run, allowing for “middle-of-the-run” calibration checks.

The energy of the 10C beam was measured using the central Si-detector in theevacuated chamber. It was found to be to be 9.0 MeV/u, with a FWHM 1.5 MeV.The optimum pressure was experimentally determined by slowly increasing the gaspressure from vacuum until the beam was not seen in the detector. To completelystop the secondary beam just in front of the detectors, 816±5 mbar was needed.The 10C intensity was around 7000 ions/s according to the PPAC, which at thisintensity and ion charge has an efficiency close to 100%.

An additional calibration, as well as an important check of possible systematicerrors, was provided by measuring the known excitation function of 13N.This wasdone by degrading the primary beam to 6.25 MeV/c on the production target, andsubsequently scatter it in the methane target at a pressure of 240±5 mbar, whichwas optimized using the same procedure as for the 10C beam. The resultingexcitation energy spectrum clearly shows a peak from two known closely lyingresonances in 13N, see fig.4.2. These two resonances interfere and can therefore notbe resolved. Their tabulated properties are a 3/2− resonance at 3.50 MeV,Γ=62 keV, and a mainly single particle 5/2+ state at 3.55 MeV, Γ=47 keV, [91].The first unbound state in 13N lies only 422 keV above the 12C+p threshold andwill be effectively drowned by the Coulomb scattering, preventing its detection inthis experiment. The resonances above 5 MeV (the lowest at 6.4 and 6.89 MeV)could no be seen due to the limited energy region. There was also a background ofα particles contaminating the 12C beam.

3.4.2 Background sources

The standard beam diagnostics4 observed small admixtures of d, α and 6Li withthe same velocity as the 12C degraded beam, while no contaminant particles wereseen in the 10C beam. Background sources in the specific setup include scattering

4The diagnostic detectors are Si-telescopes that can be inserted into the beamline. ToF–∆E iden-tification is also available for the 43 m flightpath through LISE3.

47

in the proton rich kapton foil at the entrance window, proton producing reactionswith the carbon nuclei in the CH4 molecules, and beam ions that penetrate the gastarget.

The contributions from these sources were measured by detecting protonsproduced when the 10,12C beams were sent into the scattering chamber filled withCO2 gas of adequate pressure. This run containes all setup related background,and was after normalization subtracted from the data taken on the CH4 target.

Since 10C is a β+ emitter with a half life of 19.26 s and Qβ+ = 3.648 MeV,positrons constitute part of the background. However, the positron emission has auniform time distribution and a maximum energy deposit of 1.25 MeV in 2.50 mmSi. It is therefore easy to select the protons in a two-dimensional spectrum ofToF(PPAC–detector) versus detected energy. Energies below 1.25 MeV is in theenergy range of Coulomb scattered protons, and this peak is anyway unreachablein this investigation.

Considering possible inelastic resonance processes, we note that the first excitedstate in 10C is a 2+ state at 3.35 MeV. Eq. 2.15 gives E ≈ 20 MeV, which wouldgive a ToF difference of a few ns between the elastically and inelastically scatteredprotons with the same recoil energies. No such events were seen in our data.

3.5 The NSCL experiment

The original purpose of the measurement at the A1200 spectrometer, was tomeasure the system 10N. However, the 9C intensity was found to be much toosmall at the low energy required, so the beam time was used to re-measure theexcitation function of 11N with an improved setup. The A1200 spectrometer isdescribed in ref. [92], and shown in fig. A.5. Its principles are similar to those ofLISE3, but each magnetic section consists of two oppositely bending 22.5 dipoles,and their Bρmax = 7.2 Tm. In connection to the NCSL Coupled CyclotronProject, the A1200 has been upgraded to the A1900 [93]. The first round ofexperiments for the upgraded facility was approved in the middle of May 2001.

3.5.1 Experimental setup

12C4+ at 75 MeV/u was used as primary beam. Targets of 1455 mg/cm2 Be and1962 mg/cm2 Al were used to produce the secondary beams of 12C6+ and 10C6+,respectively. No degrader was used since insertion of any matter in the secondarybeam reduced the intensity too much. Also, the velocity filter RPMS was not used

48 Investigations of proton-rich nitrogen isotopes

due to transmission problems of the low energy ions. The same scattering chamberwas used as in the GANIL experiment, now with a 25 µm kapton window.

Two main improvements were made in the detector setup as compared to the firstexperiment. One was the use of ∆E −E detector telescopes, permittingunambiguous identification of the detected particles. The other important changewas the increased thickness of the E-detectors to be able to cover the completeenergy range scanned by the beam. The E-detectors were 5 mm thick, more thanenough to detect the maximum proton energy of 25 MeV. The areas of alldetectors are 300 mm2. The ∆E detectors were used as ToF detectors and triggers.

Choosing the thickness of the ∆E detectors is somewhat of a balancing act.Ideally, both detectors in the telescope should registers all events, since otherwisethe possibility of making selections in the identification plot is lost. If the detectoris too thin, the high energy protons will deposit an amount of energy that lies inthe background noise and escapes detection. If on the other hand the ∆E is toothick, the low energy protons will be lost from the identification plot since they arenot registered in the E–detector. The low energy part of the spectrum could alsobe distorted since low energy protons going through the ∆E would loose muchenergy in the gas between the two detectors. The low lying resonances are themost interesting, and so it was decided to use ∆E detectors 100 µm thick. Both∆E and E detectors had an area of 300 mm2.

The energy of the 10C beam after the foil was 7.4 MeV/u, and its intensity was2000 pps, as measured by the PPAC’s in front of the scattering chamber. A CH4

pressure of 490±5 mbar was used for the 10C beam.

To test everything, a short, high-intensity run was made with a degraded 12Cbeam at energy 5.1 MeV/u after the kapton foil. This data were used for checkingthe setup and the energy calibration, which was primarily done using a tripleα-source. For this background run, the CH4 pressure was 470±5 mbar.

3.5.2 Background sources

The background sources are the same as for the first measurement, apart frompossible differences in the beam composition. The telescope set-up reduces thebackground as there is now a possibility to select proton events, and discriminateα’s, deuterons, projectiles and other contaminants.

49

3.6 The second GANIL experiment

Having performed the measurements described above, the measurement of 10N wasthe next step. To discover any systematical errors, some beamtime was also usedfor 11,13N. Again the experiment was carried out at the LISE3 spectrometer atGANIL.

3.6.1 Experimental setup

The primary 12C6+ beam at 75 MeV/u was fragmented in SISSI, instead of usingthe target location in LISE3, since this gives a higher secondary beam intensity.The primary beam current was about 2.9 µA at its maximum (≈ 3·1012pps). Thetarget was 1100 mg/cm2 C, and a 103.5 µm Be achromatic degrader was insertedbetween the LISE3 dipole magnets. The Wien filter with 50 kV fields was used topurify the beams. The PPAC detectors previously used were replaced by an MCP(Multi-Channel Plate) was placed in front of the chamber as beam counter. A25 µm thick Kapton foil separated the gas target from the beamline vacuum. Thisdetector unfortunately failed in the beginning of the run, so the cross section for11,13N have to deduced using the primary beam current which was calibrated withthe beamline detectors.

The detector setup tested at A1200 was used again, this time with 4.5 mm and5.0 mm thick detectors behind the 100 µm thick ∆E’s. The failure of the MCP leftus without ToF(entrance-detector) for the 9C beam, needed to remove β-delayedprotons (see below). Instead, a 100µm thick Si ∆E detector was mounted insidethe chamber, close to the entrance window. The ToF between this detector andthe firing one could then be used to select scattered protons. In fig. 4.9, the protonspectrum for the 9C beam is showed before and after applying the ToF-cut. This∆E detector was also used as beam monitor for the 9C beam.

The energies of the secondary beams when entering the gas target were 7.1, 8.1and 11.2 MeV/u for 12C, 10C and 9C, respectively. The intensities of 10C and 9Cbeams reached 60000 and 2000 pps outside the chamber for a maximum primarybeam current ∼ 2.8µA. The beam intensity was reduced around 30-50% in thecentral chamber detector.

3.6.2 Background sources

A specific background source for the 9C case is its decay products. The decay of 9Cis well known, see for example a recent experiment at ISOLDE [94] and references

50 Investigations of proton-rich nitrogen isotopes

therein. Its lifetime is 126.5 ms and the Qβ+ value is 16.5 MeV. This large Q-valuepermits the decay to feed several states in the daughter 9B, all of which areunbound. These states break up into p+α+α via intermediate states such asp+8Be, p+8Be∗ and α+5Li. This means that in addition to the e+, each decay willgive β-delayed protons and α’s. The β-delayed protons have a larger intensity thanthe elastically scattered protons, and can not be excluded simply by a cut in theenergy spectra due to their high energies. However, the β-delayed protons reachthe detectors on a much longer time scale than the scattered protons, a fact whichcan be used. The idea when setting up the experiment was to use theToF(MCP–detector) to select the scattered protons. As the MCP failed, we insteadinstalled a ∆E detector at the entrance, which was used for ToF. The ToF betweenthis detector and the firing one could then be used to select scattered protons. Infig. 4.9, the proton spectrum for the 9C beam is showed before and after applyingthe ToF-cut. This ∆E detector was also used as beam monitor for the 9C beam.

This experiment is still under analysis and all results shown in the subsequentsection are preliminary.

51

4 Results of 10,11,13N resonance experiments

The treatment of the data from the resonance scattering experiments is in anatural way divided into two parts. First, the raw data is analyzed to obtain theexcitation functions, i.e., the differential cross section dσ/dΩ as a function ofexcitation energy of the compound nucleus. This process is described insection 4.1. Once the excitation functions are derived, they have to be interpreted.This is done using a potential model which is described in section 4.2. Theexperimental results for the nitrogen isotopes, together with discussions andcomparisons with results obtained by others, are presented in sections 4.4 – 4.6 aswell as in papers 1 and 4.

4.1 Data treatment

In a first step, spectra of proton energies in the laboratory frame are obtained afterenergy calibrations and summing of the ∆E and E energies in the case of telescopesetups. Some conditions should be set on the events to minimize the backgroundin the final spectra. For the experiments described here, event selections were inall cases made in the two-dimensional spectra of ToF(RF–detector)1 versus energy.When telescopes were used, this ToF-cut was combined with proton events selectedin the ∆E versus E plot, and in the 9C measurement a condition was put on theToF(entrance–detector), see figures in coming sections for examples. Applyingspatial cuts corresponding to the size of the entrance window of the chamber didnot change the spectra in the two runs where position information was available.

The resulting spectra are then transformed into excitation functions by convertingthe energy axis from detected proton energy to excitation energy of the compoundnucleus, and the y-axis from counts to differential cross section. To achieve this,the proton energy has to be treated bin by bin. The scheme uses energy-loss tablesfor the projectiles and protons in CH4 and CO2 gas.

For the first part of the procedure, the experimental input needed is Sp of thecompound system, the beam energy after the kapton foil, the detector diameter,and the total target thickness which corresponds to the pressure and distance fromthe entrance window to the detector. Then an iteration over reaction position inthe chamber, from the window up to the maximum range of the beam, isperformed. For each position, the angle and the solid angle to the detector arecalculated and the energy of the projectile is found from the relevant energy losstable by interpolation between neighbouring values. When the beam energy is

1When telescope detectors were used, the ∆E gave the time signal.

52 Results of 10,11,13N resonance experiments

determined, the excitation energy for the compound system and the energy of thescattered proton are calculated using the formulas found in section A.3. The lowenergies, β . 0.15, justify using non-relativistic kinematics for thesetransformations. Knowing the remaining distance to the detector, the protonenergy that will be detected is given by the energy loss table. The positiontogether with the corresponding excitation energy, detected proton energy, angle tothe detector, solid angle, and target thickness are written into an array, theposition is stepped up, and the procedure is repeated. Once the position is greaterthan the total range of the projectiles, the iteration stops and the next part of theprocedure is initiated. This part analyzes the experimentally detected protonenergies using the table previously obtained. For each energy bin, the matchingvalue of the detected energy is found in the array (interpolating when needed), andthe corresponding values of all parameters are registered. Last, the number ofcounts for this energy bin is used with the integral number of beam particles, thesolid angle and the total number of protons in the target to calculate thedifferential cross section by applying the formula in eq. 4.1. Nd and Nt denote thenumber of detected particles and the total number of beam ions, while ∆Ω is thesolid angle of the detector, t is the target thickness and n is the number of targetnuclei per unit volume in the gas target.( dσ

dΩ

)c.m.

=1

4 cos θlab.

(dσ

dΩ

)lab.

=Ndetected

Ntotal · t · n ·∆Ω · 4 cos θlab.(4.1)

The results are written to a file, from which the excitation function is plotted onceall bins have been treated. This histogram looks quite different from the rawspectrum, the main distinction being the enhancement of the high-energy part ofthe spectrum. This is due to these protons seeing a much smaller solid angle thanthe low-energy protons coming from close to the detector. For an example, seefig. 3 in paper 4.

4.2 Interpretation using a potential model

The low-` resonance states in the unbound nitrogen isotopes are situated close tothe top of the potential well. This makes them broad, and several resonancesoverlap, producing interference effects.

To draw conclusions from the excitation function, comparisons with a theoreticalmodel has to be made. We choose to use a potential model, where no imaginaryterms were included since inelastic processes are assumed to be negligible. Theadvantage of this approach is the relative simplicity and good understanding of thetheory, which gives relatively transparent results. The theory assumes single

53

particle states, and once the basic structure of the isotopes is understood, theremay be a need for more detailed descriptions which allow for different componentsof the wave functions. However, we know that the three lowest states in 11Be havemainly single particle nature [95], and it is a reasonable assumption that the sameis true for its mirror 11N.

The potential used to calculate the single particle orbits for a proton in a corepotential is given in eqs. 4.2a–4.2c. It consists of central and spin-orbit potentialsin the standard Woods-Saxon (WS) parameterization and a Coulomb potentialtaken as a charged sphere with radius Rc.

V =V0(`j)

1 + e[r−R0(`j)]/a0(`j)+ ` · sV`s

r

(λπ/2π)2

a`s

e[r−R`s]/(a`s)[1 + e(r−R`s)/(a`s)

]2 +`(` + 1)~2

2µr2+ Vc

(4.2a)

where Vc is defined as:

Vc =

zZ

2

e2

4πε0Rc

(3− r2

R2c

): r < Rc

zZ

r

e2

4πε0: r > Rc

(4.2b)

and the radii are taken to be:

R0 = r0(`j)A1/3, Rc = rcA1/3, R`s = r`sA

1/3 (4.2c)

In these equations, µ = Mm/(M +m), and λπ = h/mπ0c, while M,Z and m, z arethe projectile and target masses and charges, respectively. The potentialparameters r0 ,`s,c , V0 ,`s , and a0 ,`s are varied within reasonable limits to get thebest fit to the data. The parameters V0 , r0 and a0 were allowed to be different fordifferent partial waves. The Schrodinger equation for the potential is solved usingthe standard Fox-Goodwin method, which is similar to the Runge-Kutta method.The Coulomb wave functions are calculated using the recipe of ref. [96], and thecross section is given by eqs. 2.8a–2.8d. All interference effects are taken intoaccount in the total excitation function. While the interference has to be takeninto account when fitting the experimental spectrum, the resonance parameters areextracted from the separate partial waves. We define the position ER(`j) andwidth ΓR(`j) of a resonance using the amplitude of the corresponding partial waveat r = 1 fm as a function of excitation energy. The peak of this function is definedas the resonance position and the FWHM of the peak is taken to be the resonancewidth.

54 Results of 10,11,13N resonance experiments

4.3 The Thomas-Ehrman shift

Since the relative energy shift in mirror nuclei was first discussed in connectionwith 13N [72, 73], and the concept is of interest for the mirror pairs 11N -11Be and10N -10Li, the phenomenon will be briefly addressed in this section.

The term “Thomas-Ehrman shift” (TES) has during the 50 years since itsintroduction been used in different senses in the literature. The original papersfrom which the name comes [72, 73] discuss the lowering of the proton 1/2+–levelrelative to the neutron 1/2+–level in the mirror2 13C. More generally, the issuediscussed in those papers was the relative displacement of the protonsingle-particle orbits as compared to the neutron analog states in a core+nucleonmodel for the case of an unbound proton level, and that is the meaning of the termTES adopted in this thesis. The citation below and the shadowed region infig. 4.1) illustrates this definition.

“It is investigated to what extent a change in boundary conditions at the nuclearsurface due to Coulomb wavefunction distortion in the external region can explainthe relative displacement of the first excited states in 13C and 13N.” [72]

For the case 13C -13N, the core is the same (12C). For our cases, 11N -11Be and10N -10Li, the cores are mirror pairs as are the nuclei, but the cores have retainedthe mirror symmetries to a higher degree, with the same energies of the firstexcited states.

Assuming charge independence of the nuclear force, the energy differences betweenlevels in mirror nuclei is due to the electromagnetic term of the Hamiltonian. Thisis manifested as the neutron–proton mass difference and a Coulomb-repulsionterm, which is well approximated with a spherical charge distribution.

Espherec =

3

5

e2

4πε0

Z(Z − 1)

Rc(4.3)

In a pair of tightly bound mirror nuclei, the Coulomb energy will move the protonpotential up by a constant amount as compared to the neutron potential, but thewave functions will be the same. This results in the energy levels of the moreproton rich nucleus being higher by ∆ as compared to the neutron rich member,but having the same ordering and spacing.

∆ = ∆Ec − (mp −mH)c2 (4.4)

Mirror nuclei should thus have identical level schemes after subtraction of theCoulomb energy. This behaviour is observed in nuclei close to the stability line.

2The energies of the excited states are taken relative the the ground state of the nucleus, implicitlyassuming equal radii of the bound ground states.

55

But mirror pairs only exist for light nuclei as the N = Z line crosses the protondripline at Z ≈ 40, and hence stable mirror pairs are few. Since the N = Z line

−3/2+

7/2+

3/2+ 5/2+

5/2

6

1/2−01/2−

3/2−5/2

6.864

1/2+

+

+

5/2+3.854

3/2−3.6851/2+3.089

5/2

751500

9

115

7.9

7.166.90

7.38

0.13

0.0160.0690.14

C13

T = 1/2T = 1/2

31

60

48

1.9435

2.369

3.503.54

0

N13

0.54

0.031

0.21

6.38 11 0.0031

Sp

Sn4.946

TES

Figure 4.1: The lowest states of 13N and 13C. The ground states are put at equal energies, and allvalues are taken from [91]. The numbers to the left of the levels are the energies of thestates in MeV. On each resonance, the width is given in keV to the left and the reducedwidth to the right. The latter is a measure of the degree of single particle nature thelevel exhibits. Apart from the levels, the one-proton (one-neutron) separation energyis shown, and the Thomas-Ehrman shift is indicated.

goes on the proton rich side and the states in the proton rich partner are higherthan in the neutron analog, the high-Z nucleus becomes unbound before its mirrorisobar. It is thus common that the proton rich member of a mirror pair only isunbound or only has one bound state. When the last proton in the proton richmember is unbound or very loosely bound, the valence proton has a largerprobability to tunnel through the barrier and give an extended charge distribution.The effect is especially pronounced if the valence orbit is an s-shell, when theproton wavefunction can stretch far out. This will reduce the Coulomb energywhich has a 1/r dependence, and the state appears at lower energy than theanalog state for a valence neutron.

As the proton comes close to the top of the barrier, its wavefunction will bedistorted due to the change in Coulomb force and the eigenvalues will change,introducing level displacements relative to the mirror partner. This change in thewavefunction can also introduce breakings of the mirror symmetries of thestructure of the states, such as reduction of the admixtures of other configurationsin single-particle s-waves [76].

56 Results of 10,11,13N resonance experiments

This relative level displacement is called Thomas-Ehrman shift and is indicated infig. 4.1 for the 13N -13C pair. The first excited state in 13N is unbound, and it isseen that this resonance is 720 keV lower than the corresponding 1/2+ level in 13C.The effect is generally well reproduced in single-particle potential-modelcalculations.

From the above, one can conclude is that proton halo nuclei should have enhancedTES, and that the effect can be considered as a property of proton halo states orof mirror nuclei to neutron halos. This could be used as a tool to identify suitablecandidates for proton halo states. For example, the first excited state in 17F isbound in its s orbit by only 105 keV, and has a TES of 376 keV when comparingto its mirror 17O. That this state can be considered as a proton halo level hassuggested from β-decay measurements of 17Ne [97, 98].

In the past decade it has been suggested that the TES contains severalcontributions, and the interest in understanding the phenomenon has naturallybeen spurred by the halo research. Predictions of proton resonances have beenmade using potential models [99] and mass systematics [100]. The authorsof [101, 102] use the complex scaling method and suggest that in addition to areduction of Coulomb energy, the TES gets a contribution from the kinetic energybeing reduced in a spatially extended s–wave.

4.4 Results for 13N

The aim of this series of experiments was not to add to the knowledge of 13N.Instead, its well-known resonances were used as a check of calibrations and possiblesystematic errors. The level scheme of 13N is known from many experiments usingdifferent methods, not least elastic scattering. A number of references are givenin [103], where a re-analysis of elastically scattered protons on carbon is done usingoptical potentials. In the simple shell model, 13N is described as a valence protonoutside a closed 12C core. This would place the valence proton in the 0p1/2 orbitfor the 13N ground state. Exciting the last proton to the 1s1/2 and 0d5/2 orbitswould produce excited states of single particle (SP) nature. As is seen from thelevel scheme in fig. 4.1, this seems to be a valid picture, as the 13N ground statehas Iπ = 1/2−, and the first excited state is Iπ = 1/2+ with a strong single-particles–wave component. A 5/2+ state with SP-character is found at 3.54 MeV, whilethe 3/2− state at 3.50 MeV has a more complicated structure [52].

In fig. 4.2, the excitation function of 13N measured using the ERSIT method isshown. The spectrum covering the large energy region is from the second GANILrun. The two overlapping resonances at 3.50 and 3.54 MeV are clearly seen as one

57

peak. At higher energies, the resonances at 6.38 MeV (width 11 keV) and possibly6.90 MeV (115 keV) are within the reach of the experiment. Some peaks are seenin this region, an maybe a background measurement has to be done to concludewhich ones are true 13N resonances. At this occasion, no background run with CO2

was made, and no subtraction is done. The cut at low energy is due toexperimental thresholds.

The inset shows the region around the doublet peak, recorded in the first GANILrun. From this data a flat background obtained from the CO2 target andamounting to about 10% of the total counts is subtracted. No 13N information wasobtained above ∼3 MeV due to an α contamination in the 12C beam. Sincetelescopes were not used in this run, this contamination could not be removed.The doublet peak at 3.5 MeV was fitted by coherently adding the amplitudes of

0

250

500

750

1000

3 4 5 6 7 8

0

2.36

3.503.55

1/2-

1/2+3/2-5/2+

MeVIπ

Sp=1.94

200

400

600

800

1000

1200 dσ/dΩ (mb/sr)

3.0 3.5 4.02.5

Eex. (MeV)

counts

6.385/2+

Figure 4.2: The excitation function for 13N as measured by 12C+p elastic scattering in inversegeometry using a thick gas target. The large scale was measured in the second GANILexperiment. No background has been subtracted as no measurement was dome. Theinset shows the part around the doublet from the first GANIL run. The resonancesare fitted as described in the text, and the vertical lines are only a guide for the eye.The energy is given relative to the ground state.

the two resonances, using the potential-model program for the 5/2+ resonance anda Breit-Wigner shape for the 3/2− state. The width of this peak for the first(second) GANIL run is 50 (60) keV, and the precision in energy is found to be 25(50) keV. The uncertainty in energy is taken as a measure of systematic error inenergy, and is included in the error bars given. Note that the results for the secondGANIL run are still preliminary.

58 Results of 10,11,13N resonance experiments

4.5 Results for 11N

In the following sections, the 11N results obtained in three independent ERSITmeasurements are discussed and compared with results obtained using differentmethods. Some theoretical work is also mentioned.

4.5.1 Interpretation of the data

The excitation functions were extracted from the data as described in section 4.1,and the resulting spectra from the three experiments are shown in fig. 4.3. Thecircles denote data points from the first GANIL experiments the squares are datafrom the NSCL run. The inset is the preliminary data from the second GANILmeasurement, which agrees well in position and shape with the other two.UUnfortunately, the experimental cut is a little higher in energy than desired, andcomes at the position of the ground state.

The solid curve in fig. 4.3 is the excitation function calculated using the potentialmodel described in section 4.2. The potential parameters are fitted to the first setof GANIL data, which have the best statistics. In a first approach, conventionalvalues of the parameters were chosen, r0 ,`s,C =1.20 fm and a0 ,`s=0.53 fm. Thepotential depths were varied separately for `=0, 1, 2 until the calculated excitationfunction agreed with the data.

Solving the Schrodinger equation in a potential of the type in eq. 4.2a andstandard parameter values will give the level ordering 0p1/2, 1s1/2, 0d5/2 above the0p3/2 subshell. However, all attempts to fit the 10C+p scattering data with thislevel ordering fail miserably. Pulling the 0p1/2 level down in energy makes itnarrow and produces a sharp peak at low energies of which there is no hint in thedata. Simultaneously, as the 1s1/2 state is moved high above the Coulombbarrier3it becomes very broad and its amplitude diminishes, which gives anunderestimation of the cross section between 2 and 3 MeV. The best fit to thedata that retains the conventional level ordering is shown in fig. 5 in paper 4,clearly demonstrating the impossibility of describing the data with a 1/2− as thespin of the lowest resonance. Considering the well known level inversion in 11Be, itis hardly surprising that the 1s1/2, is the ground state also in 11N.

It is worth to stress that already with standard parameters and only varying thewelldepth, these data definitely prove the level ordering 1/2+, 1/2−, 5/2+ for 11N.The level inversion on the neutron side for N=7 is thus mirrored in Z=7. It willbe interesting to investigate if the trend of further lowering of the 1s1/2 orbit

3The Coulomb barrier is around 1.3 MeV in 11N.

59

relative to the 0p1/2 that has recently been reported for 10Li [70] and 9He [71] isalso reproduced in 10,9N.

To get a better description of the data and more exact values for ER and ΓR,further attempts were done to fit the data. We now tried to vary the r0 and a0

parameters independently for the resonances. A remarkable stability of theresonance positions was found when experimenting with different parametervalues, while the widths fluctuated slightly. In the final fit, all radii parameters arethe same as the improvement from separating them was not that large, but thediffusities are different. The need for various a0 (`j) to some extent reflects theindividual spectroscopic factors of the states, giving them widths other thancalculated in the pure single-particle model used here. The potential model

Table 4.1: Parameters used for the best fit to the 11N excitation function measured in the firstGANIL experiment. This fit was made with the additional amplitude 1.25/(4.5-E), asdescribed in the text.

Potential parameters Resonances

V0 (`j) r0 (`j), r`s, rC a0 (`j) V`s a`s ER ΓR(MeV) (fm) (fm) (MeV) (fm) (MeV) (MeV)

1s1/2 -47.544 1.40 0.65 5.5 0.30 1.27 1.440p1/2 -31.500 1.40 0.55 5.5 0.30 2.01 0.840p3/2 -32.592 1.40 0.53 5.5 0.30 -1.33 —0d5/2 -57.960 1.40 0.53 5.5 0.30 4.45 1.270d3/2 -45.570 1.40 0.35 5.5 0.30 3.73 0.60

approach by definition gives single-particle states and hence also single-particlewidths. This approximation is justified by observing the high degree of SP-naturein the lowest states in 11Be [95], but we made an attempt to investigate how theoverall fit is affected by changing only the width of the states. This was done bycomparing the best fits for r0 =1.0, 1.2 and 1.4 fm. For the 1/2− and 5/2+ states,r0 =1.0 makes the states too narrow while r0 =1.4 produces too broad resonances,keeping the same diffusity. For the 1/2+ state the effect is not obvious, but thelargest width seems most appropriate. Since r0 =1.4 fm gives the best agreementalso to the 11Be levels, this was used for the final fit, and diffusities and welldepthswere adjusted.

The parameters of the best fit is given in table 4.1. Changing the proton to aneutron while exactly the same potential parameters are kept reproduces the threelowest levels in 11Be within 150 keV (+67 keV, -150 keV and -150 keV,respectively).

60 Results of 10,11,13N resonance experiments

The degree of single-particle nature of a level is expressed by the spectroscopicfactor S=ΓR/ΓSP . An estimate of S(`j) was made using the ΓR from the best fit tothe data. Since the true shell-model potential for 11N is not known, anapproximation was made by finding a potential that simultaneously fits thepositions of 11Be (as given in the literature) and 11N (as determined by thepotential in table 4.1), gives a reasonable description of the measured 11Nexcitation function and gives widths larger than ΓR(`j). This was achieved usingthe potential parameters radii parameters of 1.4 fm, and increasinga0(`j) = 0.7 fm, and modifying the V0(`j) to fit the positions. The exactparameters are given in paper 4, table I:f. The ratio of ΓR to those widths isaround 0.9, 0.8 and 0.6 for the three levels, but the errors are naturally ratherlarge due to the uncertainty in the SP potential. The first GANIL experiment does

1 2 3 4 5

0

200

400

600

800

1000

1200

0 1 2 3 4 5

Eex. (MeV)

dσ/d

Ω(m

b/sr)

GANIL 1996

MSU 1998

(GANIL 2000)

Figure 4.3: The excitation function of 11N measured by elastic resonance scattering. The solidcurve is the excitation function calculated by the potential model described in sec-tion 4.2 using the parameter set given in table 4.1. The inset is the preliminary resultof the second GANIL experiment.

not show the “dip” at 1.8 MeV predicted by the calculation. However, the shape ofthe data taken with telescope set-ups agrees better with the potential model,indicating that the apparent discrepancy could be due to background effects.

Below 1 MeV, the cross section theoretically arises mainly from Coulombscattering. The experimental data in this region also includes contributions frompositrons that cross the detector at angles, positron pile-up and some beam ionsreaching the detector, which explains that the experimental cross section is largerthan the pure Coulomb scattering.

In the first GANIL experiment, a substantial amount of cross section was located

61

in the high-energy part of the excitation function. In paper 4, it is speculated overpossible origins of this amplitude. However, including particle identification by themeans of ∆E −E telescopes reveal that the part of the spectrum above 4 MeVdoes not stem from scattered protons as it vanishes if a condition is set on thoseevents in the ∆E −E plot. The different appearances of the excitation functionafter applying different ∆E −E selections are shown in fig. 4.4, which identifiesthe origin of the cross section at high energies (cuts a and b), while the protonevents p are only at lower energies. Also, no evidence of higher lying states wasseen in the angular detector in the first GANIL experiment. Four additional

0

200

400

600

800

1000

0 10 20 30

Elab. (MeV)

p

p

a

a

b

b

c

c

∆E − E

Figure 4.4: The 11N excitation function after applying different ∆E −E cuts. Cut p is set on thescattered proton events, while cuts a,b,c investigate different contaminating particles.The data are from the NSCL experiment. The cut a is couloured white and is seen athigh energies. The b cut has dotted borders, and seems to correspond rather well tothe structures seen in the first GANIL measurement. It can be noted that the spikeat 20 MeV is an electronics artefact.

resonances are experimentally verified in an energy interval of 2.2 MeV above the5/2+ state in the 11Be spectrum. The influence of high-lying resonances were inpaper 4 simulated by a decaying tail of the form f=b/(4.5-E). The value 1.25 wasused for the parameter b. This introduced amplitude is small in the energy regionof the resonances, but improved the fit. The difference in the calculated excitation

62 Results of 10,11,13N resonance experiments

function for b = 0 and b = 1.25 is shown in fig. 4 in paper 4. The partial wave

0

200

400

600

800

1000

0 1 2 3 4

Eex. (MeV)

dσ/d

Ω(m

b/sr)

total fit

s wave

p wave

d wave

ER(1s

1/2)

ER(0p

1/2)

ER(0d

5/2)

Figure 4.5: The amplitudes of the 11N par-tial waves 1s1/2, 0p1/2 and0d5/2 together with the totalexcitation function, calculatedusing the parameters in ta-ble 4.1.

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5

Eex. (MeV)

δ(deg

)

90o

s wave

p wave

d wave

Figure 4.6: The phase shifts of the 11Npartial waves 1s1/2, 0p1/2 and0d5/2, calculated using the pa-rameters in table 4.1.

amplitudes for the three resonances and the corresponding resonant phase shiftsgiven by the above parameters are shown in figs. 4.5 and 4.6.

As previously stated, we define ER as where the amplitude of the partial wave at1 fm has its maximum and ΓR as the FWHM of this function. Figs. 4.5 and 4.6show that for ` > 0, this definition of ER gives the same resonance energy asdefining it as where the phase shift passes π/2, within the given error bars. For` = 0 the phase shift never reaches π/2. Defining the resonance position as theenergy for which dδ/dE has its maximum gives a lower resonance position for the1/2+ state (ER ≈ 1.1 MeV) while the 1/2− and 5/2+ positions are only changedwithin the error bars. As the phase shift of the s-wave does not behave as for atrue resonance, the definition of the resonance parameters which is independent ofδ was used.

Comparing the partial waves to the total excitation function in fig. 4.5 gives afeeling of the importance of interference between the partial waves. Due to theinterference effects, the resonance positions can not be determined directly fromthe peak positions of the experimental data. Also, the overlapping, broad stateswould anyhow make this difficult.

The excitation function of the angular detector at 12.5 is shown in fig. 4.7. Thesolid curve in the figure is the calculation using the parameters that were fitted for

63

the data at 180c.m., without further adjustments. It should be remembered thatthe angle to the detector is determined particle by particle. As an example, theangles for the data in fig. 4.7 range from 150c.m. to 95c.m. for different protonenergies. The change of the calculated and measured excitation functions follow

0

100

200

300

400

500

0 1 2 3 4

Ec.m. (MeV)

dσ/d

Ω(m

b/sr)

Figure 4.7: The 11N excitation function from the angular detector, measured in the first GANILexperiment.

the same trend, which strongly supports our interpretation of the 11N structure.

Last, we compare the energy levels of the mirror nuclei 11N and 11Be in fig. 4.8.From the comparison we can note a manifestation of the TES as the levelsplittings 1/2−–1/2+ and also 5/2+–1/2+larger in 11N than in 11Be. Thesesplittings have been estimated from systematics for example by Fortune [104], whoreports E(5/2+)−E(1/2+) = 2.2− 2.3 MeV, in good agreement with our data.The TES of the s1/2 and p1/2 resonances is discussed in [105], whose results also fitours. Coulomb shifts in halo nuclei are investigated in [106], where the authorscalculate a shift in the energy difference between the E(1/2+) and E(1/2+) levelsto be 0.570 keV. Our data gives 0.42 MeV. Treating the Coulomb potential as afirst order perturbation accounts for 75% of the shift [106].

4.5.2 Comparing with other experiments

As is seen from table 4.2, not many experiments treat 11N, but the existing resultsare summarized below. The results of paper 1 are not included in table 4.2 as themore detailed analysis in paper 4 build on the same data with the additionalinformation from the NSCL experiment.

The three-neutron pick-up reaction 14N(3He,6He)11N was used in the pioneering

64 Results of 10,11,13N resonance experiments

11N experiment [77]. The spectrum showed mainly one peak at 2.24 MeV, but thedata also have a peak which could well be the 5/2+ resonance. Due to lack of 1s1/2

admixture in the ground state of 14N, the 2.24 MeV peak was assigned to the firstexcited 1/2− state. The authors also note that the 1/2+ ground state wouldappear asymmetric due to the lower part of the broad resonance being moreattenuated by the Coulomb barrier. The same reaction is used in [107], whichclaim to have resolved this peak in two states of which the angular distributionsupports the spins 1/2+ and 1/2−, but the spectrum is not shown, and no energiesor widths are given.

Subsequent experiments have utilized the experimental possibilities opened by RIBfacilities. The one-neutron stripping reaction from 12N on a 9Be target wasemployed in [75], simultaneously registering the energies of the proton and 10Cfragment as well as the angle between the products. The mass of the decayingnucleus, and from this the decay energy, is extracted using kinematic relations.The spectrum could not be explained only by the 2.24 MeV state from [77] thatwas used as input in the analysis. The data were consistently described byincluding a 1/2+ state at 1.45 MeV. Shell model calculations are stated to predictthe ground state at 1.35 MeV above threshold. Both these values are within theerror bars of our result for the ground state. Since the absolute energy is notmeasured in [75], it could not be excluded that the extra lew-E amplitude did notoriginate in excited states in 11N decaying to the bound 2+ level at 3.35 MeV in10C. Using a shell model calculation for three exited states in 11N, the spectrumwas also well reproduced. The predicted levels were, apart from the 1/2− state

E (MeV)

4

3

2

1

0Sp

Sn

1/2+

5/2+

1/2-

1/2+

5/2+

1/2-

1.78

0.32

1.27

2.01

3.75

2

1

0

Figure 4.8: Comparison of the energy levels of the mirror nuclei 11N and 11Be. The 5/2+ levelshave been put on the same level to facilitate comparisons. The level splittings in 11Nare larger than in 11Be.

65

(which can only decay to the ground state in 10C), a 3/2− level at 4.6 MeV and a5/2− resonance at 5.8 MeV. It seems that the 5/2+ resonance is not predicted bythe calculations. Inclusion of this state would probably not change the conclusionsmuch since it is situated just where the 10C(2+)+p channel opens, and due to itsstructure should decay to the ground state of 10C.

The two remaining papers in table 4.2 are authored by the same group, using twodifferent multi-nucleon transfer reaction. For their first experiment, the 15Cejectiles from the reaction 12C(14N,15C)11N were detected [108]. The energyresolution is given as 270 keV, taken from the width of the bound ground state of13N. The spectra are fitted by Breit-Wigner shapes on top of a background fromthree-body sequential decay. These amplitudes are added without interferenceeffects. By comparing with the simultaneous reactions producing 12N and 13N, theauthors claim a strong hindrance of 1s1/2 population as the reason for notobserving the 1/2+ ground state. The resonance energies for the 1/2− and 5/2+

states agree with our results, but the widths in [108] are smaller. To detect the 11Nground state, a second experiment employed the reaction 10B(14N,13B)11N [109].The same analysis methods were used, now for the 13B ejectiles. The targetcontained impurities of 11B, 12C and 16O. Background runs with these targets weredone, and the data from the 11B and 16O targets were subtracted from the 11Nspectrum, while the 12C reaction is much higher in energy. Again, all widthsobtained are more narrow than our results. They place the 1/2+ state at1.63±0.05 MeV and claim a width of a mere 0.4 MeV, extraordinarily small for anproton in an s-wave orbit that lies above the Coulomb barrier. It can be comparedfor example with the calculated width of ref. [110].

Indirect proof of the low-lying 1/2+ is given by the 12O decay, as investigatedin [74, 111]. From the predictions of the ground state of 11N at 1.9 MeV abovethreshold, a diproton branch in the decay was expected as Q2p(

12O) = 1.79 MeV.However, the data in [74] were consistently described by only sequential decays.This clearly indicates that the 11N ground state energy is smaller than Q2p(

12O).The data from [74] are re-analyzed in [111] where the limits0.70 < Egs(

11N) < 1.45 MeV were deduced from the 12O decay, in contradictionwith the result of ref. [109].

4.5.3 Comparing with theory

Different theories roughly agree with each other and experiments on the positionsof the states in 11N, as is seen in table 4.2.

In [113], a WS potential is used to calculate Coulomb shifts of the A=11 quartet

66 Results of 10,11,13N resonance experiments

Table 4.2: Experimental and theoretical results for 11N, in chronological order within each section.ER and ΓR denote the resonance position and width, respectively. Note that the defin-ition of ER and ΓR varies between the references. All energies are given relative to the10C+p threshold.

1/2+ 1/2− 5/2+

Ref. ER ΓR ER ΓR ER ΓR

Experimental results

[112]a 1.27±0.180.05 1.44±0.2 2.01±0.15

0.05 0.84±0.2 3.75±0.05 0.60±0.05

[77] not observed 2.24±0.1 0.74±0.1 not observed[108] not observed 2.18±0.05 0.44±0.08 3.63±0.05 0.40±0.08[75]b 1.45±0.40 >0.4 2.24±0.1 0.74±0.1 not observed[109] 1.63±0.05 0.4±0.01 2.16±0.05 0.25±0.08 3.61±0.05 0.50±0.05

Theoretical results

[113]b 1.4 0.6 2.24 0.74 3.7 0.3[110] 1.60±0.22 2.1±0.75

0.07 2.48 1.45 3.90 0.88[114]c 1.4 1.31 2.21 0.91 3.88 0.72[115] 1.1 0.9 1.6 0.3 3.8 0.6[116] 1.2 1.1 2.1 1 3.7 1[105] 1.29 1.39 2.12 0.95 not calculatedaPaper 4bThe result from [77] was used for the 1/2− state in the analysis.cThe results for r = 4.57 fm and ER defined as the peak of the density of states are cited.

relative to the 1/2− levels. The author finds that a 20% inclusion of |2+ ⊗ 0d5/2〉better reproduces the known 1/2+ levels in 11C and 11B, and this value is given intable 4.2. If the admixture is increased to 45%, the 1/2+(11N) is moved up to1.8 MeV which is definitely too high compared to experimental results.

The authors of [110] use the same model as we do, and predict the 11N states byreproducing the known 11Be levels and then adding a Coulomb term to thepotential. Spectroscopic factors from 11Be data are used to predict widths, whichare all much larger than measured or predicted by other models. The very largewidth of the 1/2+ gives S = 1.58, while the others are more similar to the ones wededuced.

The same model and procedure are used in [114], where it is pointed out that it isvital to know how the resonance parameters are defined in order to compareresults. Two different definitions of the resonance energy are compared: Where

67

δ = π/2 and where the density-of-states function peaks. The spectroscopic factorsare within ±0.1 of our estimates. The author also investigates an R-matrixapproach, which is not limited to SP-states. The 1/2+ state then appears at1.60 MeV, after fitting the 1/2− energy to [77].

A WS potential is also used in [116], with the addition of a surface term deducedfrom 11Be. The surface term is related to the admixture of core-excited states inthe wave function. A simultaneous description of both nuclei in the mirror pair isachieved.

The Generator Coordinate Method, a microscopic cluster model, was used in [115]to investigate the mirror pair 11Be–11N. The results are in qualitative agreementfor both nuclei. The calculated 10C+p phase shifts are very similar to the ones weobtain.

The authors of [105] use the Complex Scaling Method [101] to solve theSchrodinger equation for the 10C+p system. The potential parameters aredetermined by fitting the two bound states in 11Be. The calculated levels in 11Nare in good agreement with our results, both for the positions and widths.

4.5.4 Summarizing 11N

We are now in a position to summarize the present status of the knowledge of theunbound 11N. There is no doubt from neither theory nor experiment that the levelordering is the same for the three lowest states in the mirror nuclei 11Be–11N:1/2+, 1/2− and 5/2+.

However, the agreement of positions and widths is not very satisfactory. It is bestfor the 5/2+ state, which is natural since this state is the most narrow due to thelarge centrifugal barrier, but also here do the widths vary wildly. Due to the natureof the 1/2+ state, being broad and smeared, it seems unlikely that any experimentcan pinpoint this state very exactly. An additional complication is the difficulty todefine the position of an unbound level of which the phase shift does not pass π/2.The position of the 1/2− state depends on the parameters of the positive paritystates through interference. It should be kept in mind that different, and notseldom undeclared, definitions can make comparisons treacherous.

4.6 Preliminary results for 10N

Research is a continuous work, which this section is an example of. The analysis ofthe 10N data is still under way, but a few remarks on the analysis procedure and

68 Results of 10,11,13N resonance experiments

preliminary excitation functions can be reported here.

4.6.1 Points of the analysis

The crucial point in this experiment is to separate the positrons and β-delayedprotons from the decaying 9C from the elastically scattered protons. This wasdone by placing a ∆E detector, 100 µm thick, inside the chamber and close to theentrance window. The ToF between this detector, in the following called theentrance detector, and the telescope detector shows a clear peak for the scatteredprotons. The raw proton energy spectrum (summed ∆E − E) is shown in fig. 4.9.The first condition is set on the 9C peak in the entrance detector, and this gate isthen combined with the ToF-gate. The β-delayed protons, which are in channel 0of the ToF-spectrum, are well discriminated by the ToF-cut.

1

10

102

4 6 8 10 12 14

10

102

103

3000 5000 7000

10

102

103

104

3000 5000 7000 9000

∆Eentrance

ToF(entrance-detector)

9C

(MeV)

(ch)

(ch)

Figure 4.9: The proton energy spectrum, and how it changes after cut are applied first on the 9Cpeak in the entrance detector and then also on the ToF(entrance − detector). Theinsets show the selections made.

Applying the above described ToF-cut and discriminating ions other than 9C inthe entrance detector, scattered protons can be selected from the identificationmatrix in fig. 4.10. Apart from scattered protons, other mass-bands are seen in the∆E − E plot. There is a substantial amount of α’s, but there are also traces ofhigher Z nuclides. One can suspect that the entrance detector induces reactions inthe 9C projectiles.

In fig. 4.11 the raw energy spectrum of protons resulting from 9C scattering is

69

2000

4000

6000

8000

10000

2000 4000 6000 8000 10000

protonsE(chan

nels)

∆E (channels)

α

Figure 4.10: The ∆E − E identification for 9C+p scattering. In addition to protons scatteredα’s are seen, as are traces of d and 3He. At higher energies, there are even heaviernuclides.

shown, as is the background run on CO2 gas, and the spectrum resulting fromsubtraction of the background after normalizing the number of projectiles and thenumber of carbon-like nuclei in the gas. These spectra are shown to give a hint onwhat the excitation function will look like. Correcting for smaller solid angles forthe higher energies will increase the amplitude at high energies as compared to thelow-lying part. The energy range of the excitation function will be ∼ 1.5− 7.

The spectra were projected with conditions on the 9C peak in the entrancedetector, on the ToF peak corresponding to scattered protons, and on the protonbanana in the ∆E − E plot. Already from the lab. energy spectrum in fig. 4.11, itis clear that it will be very difficult to draw any conclusions on the 10N structurefrom the broad structures. An analysis with a proper model might be able to atleast exclude suggestions for the structure that are incompatible with the data.However, the rather structureless function is obviously consistent with broad,overlapping states.

4.6.2 Reviewing earlier results

Data for this nucleus are scarce, to say the least. The Complex Scaling Methodthat was successful in reproducing our 11N data have been used also to predictlevels in the mirror pair 10N–10Li [117]. This work places a 2− ground state of 10Nat 1.5 MeV, Γ ≈ 3.5 MeV, assuming there is an s-state on the threshold of beingbound in 10Li. The core+nucleon potential is found by reproducing the 1+ and 2+

levels in 10Li as observed in [81]. The same potential for 9C+p predicts these

70 Results of 10,11,13N resonance experiments

positive parity states, corresponding to a p-orbit of the valence nucleon, at 2.84and 3.36 MeV in 11N (Γ = 1.89 and 2.82 MeV), respectively. Such broad s stateswould be very hard to observe and interpret experimentally as they would overlapand interfere with each other.

At the Enam2001 conference, experimental results for 10N were presented [118].The system was populated in the reaction 10B(14N,14B)10N, and investigatedthrough the energies of the 14B ejectiles. The authors find a peak consistent with a2− ground state at ER = 2.6 MeV, Γ = 2.2 MeV.

0

10

20

30

40

0 5 10 15 20 25

2.5

5

7.5

10

5 10 15 20

0

10

20

30

0 5 10 15 20 25

9C + CH4

Ep (lab.) (MeV)

9C + CO2

subtracted: 9C + p

Ep (lab.) (MeV)

Figure 4.11: Laboratory energies of 9C+CH4 and 9C+CO2 scattering, and the background sub-tracted spectrum. The ordinates are in arbtiraray units.

71

5 Break-up reactions: what, why and how

In this section, some aspects of break-up reactions at high energies are brieflyreviewed. The emphasis is put on reactions with halo nuclei and concepts ofimportance for the experiments described in papers 2, 3, 5 and 6.

5.1 What are break-up reactions?

The term “break-up” can in the literature be used with slightly different meanings.In the following I will use it for high-energy reactions where the projectile isbroken up into its constituent parts after target interaction. Defined in this way,“break-up” is a general term which includes several specific processes. Therestriction to high beam energies makes sure that the interaction between targetand projectile takes place on a time scale which is much shorter than that of theinternal motion of the nucleons. This justifies use of the sudden approximation,which assumes the internal degrees of freedom of the participating nuclei to befrozen during the interaction. The assumption naturally gets better at higherenergies, but in practice it is used analyzing all experiments performed with IFSbeams, Ebeam & 40 MeV/u. For halo nucleons, the sudden approximation is alsobetter at lower beam energies than for normal-density nuclei, since their largespatial extension results in lower internal momenta, as is seen from the Heisenberguncertainty relation.

Break-up reactions are well suited to study loosely bound, highly clusterizedsystems such as halo nuclei. If the momentum transfer k to the detected cluster issmall it will be approximately unperturbed by the excitation and carry informationabout the ground state wavefunction of the projectile to the detector systems.

This is of course an oversimplified picture, and the question is to what degree thereaction will modify the properties. Interaction between one cluster and the targetwill also influence the other clusters. The presence of a target nucleus and also thegeometrical presence of other clusters, restricts the possible locations of thedetected cluster. This reduction of the wavefunction volume corresponds to theprojectile being in a superposition of excited states. Another way of saying thesame thing is that we will not sample the complete wavefunction, and thus notobtain the true ground state properties in a measurement. These reaction effectswill be discussed further in sections 5.3.2 and 5.3.4.

Another long-standing question is to what degree Final-State Interactions (FSI)affect the measured variables. While the discussion above concerns the influence ofthe reaction mechanismon the properties of the projectile, FSI modifies the

72 Break-up reactions: what, why and how

observables due to interactions in the final system after reaction but beforedetection. A specific example is 11Li, where the removal of one neutron leaves theother two constituents, 9Li and a neutron. The question is to what degree thesewill interact via resonances in the 10Li system or via the continuum. Due to thehigh beam energy, the decay of a formed resonance will take place far from theinteraction zone, and consequently be unaffected by the target nucleus. Theinterpretation of data should therefore also consider the inner structure of theprojectile used and its possible final states, or use a set-up that measures it.

The mixing of reaction mechanism effects and internal structure effects is apparent.Luckily enough, halo nuclei come close to the ideal case of cluster systems on thelimit of being bound. This makes it possible to use the picture of inert clustersthat remain unperturbed by the interaction in a first approximation. The idea ofdirectly probing the halo wavefunction by break-up reactions give reasonableresults in many cases, but the inclusion of reaction mechanism effects and FSIimprove descriptions and give detailed information. However, the basic conclusionsdrawn from the simpler approximations often hold. For other cases, the datarequires inclusion of reaction mechanisms for the results to make any sense.

5.2 Why do break-up reactions work as probes?

As is understood from the previous section, a good theory of the reactionmechanisms is indispensable when trying to extract information about theproperties of halo nuclei from the experimental data.

Approximations often used when treating high-energy reactions are the previouslymentioned sudden approximation and the eikonal approximation. They bothassume the beam momenta to be much greater than the internal momenta, so thatthe projectile internal coordinates can be considered as frozen during theinteraction and the particle trajectories can be treated as straight lines. Theeikonal approximation allows a classical impact parameter b to be definedperpendicular to the direction of projectile momentum, which is usually taken tobe the z-axis. The magnitude b is the distance between target and projectilethrough b2 = x2 + y2. The FWHM of p// fragment distributions for one-nucleonremoval from stable nuclides are a few hundred MeV/c, corresponding to maximalinternal momenta of 100-150 MeV/c. This is usually less than a few percent of thebeam momentum1

The eikonal approximation assumes so high projectile energies E that (i) a/λ < 1and (ii) |U | < E. Here, a is the range of the nuclear field U and λ is the the

1For a 40Ar beam at 60 MeV/u, pbeam is about 13500 MeV/c.

73

wavelength of the initial wave, which has the momentum p = ~/λ. Under theseassumptions, the outgoing wave after scattering of a structureless particle on thenuclear field U can be written as a product of the incoming plane wave ψi = eiqz

and a phase factor S(b), also called scattering function or profile function (see forexample [119–121]).

ψs = S(b)eiq0z , S(b) = e2iδ(b) , δ(b) = − 1

2~v

∫z

U(√b2 + z2 )dz (5.1)

A complex potential U accounts for absorption reactions. By choosing the form ofthe potential, and thereby the scattering function, different aspects of theinteraction can be studied. One example is the black disc model, or extremeabsorption limit. In this case, the scattered wave is 0 if the target and projectileoverlaps geometrically, expressed as:

S = 0 b < Rt +Rp

S = 1 b > Rt +Rp

(5.2)

The outgoing wavefunction thus has a hole in it even if no particles were absorbed,causing diffractive break-up. For a more realistic treatment, diffuse S functions areused. Extending the formalism to complex projectiles, each part of the projectileexperiences a phase shift. The complete wavefunction Ψ0 can be written as aproduct of a plane wave and a wave function depending on the internalcoordinates.

Ψs = S1(b1)S2(b2)...Ψ0 = S1(b1)S2(b2)...eiq0zψ(r1, r2...) (5.3)

The eikonal approximation implies straight line trajectories, and that theprojectile momentum along z is not changed during the interaction. The limits ofthis approximation were recently investigated by the authors of ref. [122] whofound it to be robust to within a few percent for as low beam energies as20 MeV/u, especially for relative cross sections and p//.

Fig. 5.1 depicts the possible break-up reactions for halo nuclei. These differentprocesses are halo nucleon stripping (5.1:c,d), core stripping (5.1:e,f), diffractiondissociation (5.1:a) and Coulomb dissociation (5.1:b). From a theoretical point ofview, stripping denotes the removal of one or several halo constituents withoutregard to what happens to the removed nucleons. However, from the experimentalside, the reactions are often described in terms of the detectable signature. Theterm absorption then means removal of the core or/and halo nucleon(s) so thatthey can not be detected, while the signal of core break-up is detection of halonucleons in coincidence with fragments from a shattered core.

In diffraction dissociation, neither the core nor the halo nucleons are stripped bythe target, so the impact parameter must be larger than the sum of the target and

74 Break-up reactions: what, why and how

projectile radii. Experimentally, it implies that all constituents of the beamnucleus can be detected. But the target scans a cylindrical volume in the projectilewavefunction, and thus effectively reduces the available space for the haloconstituents. This change in the projectile wavefunction shifts it from the pureground state to a superposition of excited states, which causes the break-up of theprojectile. The cut and the impact parameters are illustrated in 5.1:a. For2n-halos, the resulting neutron momentum distributions will have two components,one wide from the diffracted neutron, and one narrow from the unperturbedneutron. The core distributions are thought to be rather unaffected by theinteraction, at least in the longitudinal direction. As in optics, the diffractionpattern signals the degree of blackness of the diffracting object, in this case thetarget nucleus.

Coulomb dissociation is an important reaction process for high-Z targets which is

pbeam

pn//

pf//

p//

p|__

pf//

a b

c d

e f

bcbn

Figure 5.1: Cartoon descriptions of different break-up channels and concepts for halo nuclei. Parta) illustrates diffractive break-up where all constituents escape the target that createsa cylindrical cut in the wavefunction. It also shows the impact parameters b and theshadow effect, as the neutron can not be detected while being shadowed by the core.Part b) shows Coulomb break-up that also leaves all clusters to be detected. Parts c)and d) exemplify one-neutron absorption for 2n- and 1n-halos, respectively. In part c),the FSI is indicated, and in part d) the principle of measuring the internal p// issketched. Parts. e) and f) shows the core-stripping processes of core absorption andcore break-up, respectively.

75

strong enough to interact with the core at larger b than the nuclear field, while theprocess can usually be neglected for low-Z interactions. The process is illustratedin fig. 5.1:b, showing that projectile constituents can be detected also in this case.The core receives a momentum transfer while under influence of the Coulomb fieldof the target nucleus. Most halo nuclei have no bound excited states, so theinteraction excites the halo nucleus into the continuum from where it breaks upinto core fragment and halo neutrons. In the simplest picture, the halo neutronscontinue their paths unperturbed after the dissociation, as they are unaffected ofthe Coulomb field. A measurement of the neutron momentum distribution wouldthen reflect the ground state wave function of the projectile. It is also possible tomeasure the p// core distributions, since the perturbation in this direction isassumed to be small. The main contribution to the Coulomb dissociationcross-section comes from low-energy dipole strength, in contrast to the case fortightly bound nuclei where the giant dipole resonance, much higher in excitationenergy, carries most of the E1-strength.

In stripping reactions, at least one constituent is removed from the projectile sothat it can not be measured. It can for example be absorbed by the target ordeflected at angles larger than the acceptance of the detection device. Thepossibilities are neutron stripping for a 1n-halo (5.1:c) and a 2n-halo (5.1:d), andcore-stripping (5.1:e,f). For a two-neutron halo nucleus, the unabsorbed neutroncan be detected and has a narrow momentum distribution. For a one-neutron halo,obviously no neutrons are detected. When detecting the core, the diffraction andabsorption channels are summed. Experimentally, core-absorption processes areunfavourable since the experimental condition is the non-detection of the core incoincidence with one (or two) neutrons. This makes efficiency and backgroundknowledge extremely important. In principle, it is hard to get conclusive resultsfrom any experiment that relies on anti-coincidences.

The core break-up channel demands b < Rc +Rt, a sum which lies in the range5-10 fm. The halo wave function extends much further out than this, and it hasbeen proposed that core break-up reactions would therefore leave the halo neutronsrelatively unaffected, continuing from the interaction zone with momenta reflectingthe ground state momenta in the projectile. The core does not survive the violentcollision, and so this reaction channel is characterized by forwardly focused haloneutrons and charged fragments lighter than the core, see fig. 5.1:e. The chargedparticles must be identified to separate the many possible outgoing channels of thecore break-up. For practical reasons, the core break-up condition is often taken asneutron in coincidence with a fragment with Zf < Zp instead of Af < Ap.

The data from the processes above will include effects of shadowing and fragmentsurvival. When detecting the core fragment, the impact parameter has to be larger

76 Break-up reactions: what, why and how

than the sum of core and target radii, or the fragment would have been stripped,at least in the black disc limit. This condition of fragment survival is introducedinto theoretical models the by a cut-off radius. This to some extent reduces thetheoretical problem of constructing the wave function, since there is need to knowit in only the volume where r ≥ rcut, and asymptotic approximations can be used.The concept of shadowing is important for example when detecting halo neutrons.The probability to detect the halo neutrons is reduced as the neutron can be in theshadow of the core. When interpreting the experimental results or reproducingthem in theoretical models, the particular restrictions for the reaction imposed bythe geometry of the process and by the detection system always have to beconsidered.

Calculations of p// and cross sections based on the eikonal approximation has beensuccessfully applied to the break-up of halo nuclei (see refs. [123–126] andreferences therein for a few examples). This type of theories are often referred to asGlauber or Serber models, since those authors treated the subject of stripping [127]and diffractive break-up [128] of the deuteron using these approximations.

There are naturally theoretical methods that treat break-up reactions withoutbuilding on the eikonal approximation. One example is DWBA methods, wherethe basic idea is to excite the system above the cluster threshold and thereforebreak it up ([129] and references therein). Another way is to solve thetime-dependent Schrodinger equation numerically and obtain wave functions forthe final state. One example of such a calculation is found in ref. [122]. Thecontinuum states are obviously important for reactions with loosely boundsystems. Methods which include the coupling to the continuum explicitly are thetransfer-to-continuum model ([130] and references therein) and ContinuumDiscretized Coupled Channels method ([131] and references therein).

5.3 How are break-up reaction measured?

There are a multitude of experimental possibilities involving break-up reactions,and different observables demand specialized set-ups. In this section, a few ofinterest for halo nuclei are mentioned.

5.3.1 Cross section measurements

The cross sections of break-up reactions are interesting observables, and theprinciples of the measurements are rather simple. By knowing the beam currentand target parameters, and detecting the number of reaction products, the cross

77

section can be deduced. The beam current is usually monitored experimentally toget the integral number of incoming beam particles. In a realistic experiment, anumber of correction factors should be considered. Some examples are detectionefficiencies, acquisition deadtime and the transmission efficiency of the reactionproducts through the separator system.

Different cross sections can be measured, and using clever set-ups sometimessimultaneously. For halo cases, interesting observables are the probabilities forremoval of the halo nucleons, σ−1n and σ−1p for neutron and proton halos,respectively. Also the total interaction cross section σI , which includescontributions from one-neutron and one-proton removal reactions, is importantand is the observable from which radii are deduced. For halo nuclei, surrounded bya cloud of low density nucleonic matter, these cross sections are drastically largerthan for normal-density nuclei with similar massnumbers.

The evolution of cross sections along isotopic chains can indicate onsets of haloregions and indirectly give information on structure. A sudden increase of the crosssection can for example mark the filling of loosely bound s states. It can thus beused as a tool to identify candidates which are interesting for closer investigation.

5.3.2 Momentum distributions

A widely used experimental probe of halo properties are momentum distributionsof core fragments or halo nucleons. Measuring the fragment momentumdistributions after projectile fragmentation have for several decades been animportant tool in investigating the internal velocity distributions in nuclei. Themethod, as the fragmentation process itself, is thus rather well understood forstable nuclei, an advantage for any method that is taken to the unexploreddriplines.

The technique is built on the fact that the spatial wave function is related to itscounterpart in momentum coordinates by a Fourier transform, eq. 5.4:

Ψ(p) =1

(2π~)3/2

∫e−ip·r/~ Ψ(r) d3r (5.4)

It is evident from eq. 5.4 that a pronounced spatial effect will influence themomentum distribution. From Heisenbergs uncertainty relation we canqualitatively understand that any object which is extended in space will be small inmomentum space. This means that the nuclear fragments after break-up of a halonucleus into its constituent clusters will exhibit narrower momentum distributions,since they have more extended wave functions than the nucleons in a harder boundnucleus. This is obviously valid only if the momentum transfer to the observed

78 Break-up reactions: what, why and how

fragments in the break-up reaction is small so that the internal properties areconserved. When detecting the halo nucleons, the distribution will always have abroad component from core nucelons, as there is no way to discriminate a neutron(or proton) which is knocked out from the core from the halo nucleon. Thisdistorting contribution can be subtracted by measuring the neutron (proton)momentum distribution using the core of the halo nucleus as beam.

Fragmentation of stable nuclei has been studied since high-energy beams of heavyions became available some thirty years ago. The fragments from 12C and 16Obeams at 1-2 GeV/u were shown to have Gaussian distributions of longitudinalmomenta (p//) in the rest frame of the projectile. Since the Fourier transform of aGaussian returns the same shape, this implies a Gaussian shaped wavefunction inspatial coordinates. The widths of longitudinal and transverse distribution werefound to be equal, σ// = σ⊥, to within 10% [132]. The width parameter σ was onthe order of σ =50–200 MeV/c for the different fragments, which corresponds toFWHM’s of 100–500 MeV/c. It was also seen that the shapes and widths of thep// distributions, as well as isotope production cross sections, were roughlyindependent of target material and beam energy, and only dependent on thefragment. Similar results were obtained for other stable nuclei such as 40Ar and48Ca [133, 134].

The conclusion is that the specific nuclear structure of the projectile and fragmentare dominant in determining the isotope production cross-sections [135]. Anotherobservation was that the fragments are emitted with a velocity roughly the sameas the beam velocity and that the distributions thus were peaked at an onlyslightly lower p// than pbeam. This indicates that the momentum transfers in thereactions are small, supporting the assumption that the measured momentumdistributions are closely related to the internal momentum distribution in thenucleus before interaction and implying peripheral reactions. A model to describethe characteristics of the longitudinal momentum distributions from fragmentationprocesses was developed by Goldhaber [136]. Statistical considerations gave thedependence of the width parameter on fragment mass and Fermi momentum kF :

σ// = σ0

√Afrag(Aproj − Afrag)

Aproj − 1, σ0 =

kf√5≈ 90 MeV/c (5.5)

This model has due to its simplicity been widely employed. However, someessential physical features are missing, as pointed out by Friedman [137].Neglecting demand of fragment survival and Coulomb effects rends the modelincapable of explaining different widths for the same masses or differences betweenp⊥ and p//. The model in [137] relates the width to the separation energy of theremoved nucleon and includes absorption via a cutoff radius, but fails as the

79

driplines are approached. A large collection of data on momentum distributionswere reanalyzed in ref. [138] and presented in terms of longitudinal velocities androot-mean-square longitudinal momenta. Important conclusions were that thelongitudinal momenta are independent of target material and that three differentreaction mechanisms can yield the observed dependence of 〈p2

//〉1/2(prms) on massloss of the projectile.

There are two momentum observables, as there is a choice between transverse orlongitudinal directions. It has been shown that p⊥ is more sensitive than p// tonuclear and Coulomb diffraction, requiring a good reaction theory [139]. This is forexample seen from the fact that the eikonal approximation in which onlytransverse momentum is transfered works well.

For halo nuclei, there are also choices of which particles to detect: the halonucleon(s), the core or pieces of a smashed core. The detection system effectivelydecides which reaction(s) are studied. The first momentum distributions fromneutron-rich projectiles were measured in 1988 [140]. The transverse coredistributions were investigated for 6,8He and 11Li at 0.79 GeV/u. The 9Li fragmentdistribution from 11Li break-up in a carbon target showed two components, onebroad with FWHM = 224 MeV/c, comparable to that of a stable nucleus, and onenarrow component with FWHM = 54±12 MeV/c [140]. Using a lead target, onlyone component was seen whose width was about three times broader than thenarrow component in the case of light target. The target dependence and observedtwo-component structure implies that the reaction mechanisms and final stateinteractions are important in this experimental situation. Since the firstmeasurements, many investigations have been performed under differentconditions, mainly studying 11Li, 11Be and the two helium isotopes 6,8He. Forneutrons and charged fragments, p⊥ and p// have been investigated with energiesfrom 30 MeV/u to above 1 GeV/u.

It can be noted that the pronounced halo nuclei core distributions have Lorentzianshapes. A Fourier transform of this form gives the Yukawa wavefunction shape inspatial coordinates, eq. 1.3.

5.3.3 In-beam γ spectroscopy

As previously stated, the ideal halo ground state consists of inert clusters so thatonly the interactions between the clusters determine the halo properties. In reality,the presence of low-lying excited states in the core will influence the halo structurethrough contributions to the ground state. The core-excited state and the valencenucleon can couple in various ways, of which some can contribute to the ground

80 Break-up reactions: what, why and how

state of the halo nucleus. The weight of components corresponding to suchconfigurations can be measured in halo nucleon removal reactions if a γ-detectionsystem is set up around the break-up target. If the core is excited as the halonucleon is removed, it will γ-decay emitting a photon which can be detected incoincidence with the core fragment. Comparison of the cross sectionsσ(A+ xX→AX+x) and σ(A+ xX→AX∗ + x) gives an estimate of the admixture ofthe configuration of the core excited state X∗. This approach assumes that thecore fragment is not excited in target interactions.

Since the γ’s are emitted in flight the spectrum has to be Doppler corrected. Careshould be taken to suppress background radiation from γ’s and neutrons. Theefficiency of the γ-detection system is crucial for deducing the cross section of theexcited core state. It has to be accurately calibrated, taking energy dependenciesinto account. It is useful to simulate the response of the γ-array, which is oftendone in a Monte Carlo program as Geant.

Measurements using inbeam γ spectroscopy to investigate core excited states havebeen performed at NSCL for beam energies around 60 MeV/u, applied to forexample 11Be [67], 28P [141] and 19C [142]. In paper 6, the technique is applied tothe one-proton removal reaction of 8B at highly relativistic energies, ≈930 MeV/u.

5.3.4 Other techniques

There are several ways to study exotic nuclei using break-up reactions which arenot directly relevant for the papers in this thesis. In this section, a few of thosewill be commented on.

A type of break-up reaction of special interest for halo nuclei is core break-up. Theunderlying idea is to select events with small impact parameters b where the coreinteracts with the target while the halo nucleon(s) continue approximatelyunperturbed. Since the halo properties are connected to the tail of thewavefunction (large b), the initial thought was that the neutron angulardistribution, transformed to transverse momentum distribution, would show lessinfluence of reaction mechanisms than what was seen for the halo removalchannels. Removing the core obviously deletes all possible FSI in the core+nsystem. The core break-up events are experimentally characterized by detection ofthe halo neutron(s) in coincidence with charged fragments having Z < Zcore.However, this break-up channel also presents certain complications. Detection ofneutrons is inherently difficult, requiring large setups which introduce possiblecross talk effects and efficiency calibration dependencies. The neutron angulardistribution will apart from halo neutrons contain a broad component from tightly

81

bound core-neutrons. To get the true halo-neutron distribution, measurementsshould also be done for the core nucleus, i.e. 10Be in the case of 11Be. When thisbackground is subtracted from the data, the statistical errors become large whichcan make the interpretations more uncertain. The shape of the distributions willbe influenced by the shadow effect, explained from the possibility for the haloneutrons to be in the “core shadow” from where they can not be detected. Thisexclusion of the central part of the wavefunction corresponds to excludinghigh-momentum neutrons from the sampling. Examples of core break-upexperiments, their difficulties and results, are given in [143, 144].

Detection of the momenta of all reaction products after a break-up constitutes socalled complete kinematics experiments. After the reaction in the target, theresulting nuclei are typically deflected by a dipole magnet. Their momentumvectors p are measured using position sensitive detectors and ToF, and totalenergies are measured in scintillator walls and neutron detector walls. For theset-up to be truly complete, the momenta of the recoiling target nuclei should bemeasured, but for realistic situations this is not possible. However, γ-detectionaround the target area should be included to see wavefunction components ofcore-excited states. By using different targets, effects of Coulomb excitation can beseparated from nuclear dissociation phenomena. These type of experiments can beused to determine reaction mechanisms and FSI, and turn these effects intoexperimental possibilities. As an example, 10Li [84] and 13Be [145]have beenstudied in complete setups, and conclusions could be drawn about their structure.Investigations of heavy helium isotopes and their unbound subsystems in theLAND–ALADIN set-up are discussed in [146]. The escape of the α + 4n channelfrom 8He break-up discussed there highlights the importance of having trulycomplete set-ups.

An technique recently applied to halo nuclei is the intensity interferometry methodoriginally developed by astronomers Hanbury-Brown and Twiss to measure starsizes [147]. A correlation function C(p1, p2) is introduced that reflects the influenceof FSI and quantum statistical symmetries on the relative momentum of theemitted particles. The deviation of C(p1, p2) from 1 is a measure of the structureof the source of emission, and the source size can be extracted. A possible problemis the need for input of the uncorrelated distribution, which can obviously not bemeasured but has to be reconstructed for example through event mixing. A reviewof the general technique is found in [148], while an application to an experimentalmeasurement of halo neutrons is found in [149].

82 This page is intentionally blank. Feel free to fill it.

83

6 Break-up experiments of halo nuclei at GSI

The experiments reported in papers 2, 3, 5 and 6 treat the one-nucleon removalprocesses from halo nuclei. The measurements were aiming at determining p// coredistributions of neutron rich sd-shell nuclei and 8B. In the last runs thelongitudinal momenta were registered in coincidence with γ-rays from the projectilebreak-up at the target. In addition, nucleon removal cross sections were deduced.

6.1 The FRagment Separator FRS

The Fragment Separator FRS at GSI was constructed to be used with the SIS,ESR and the experimental caves at GSI. This extension of the facility was ready in1990. The layout of GSI is shown in fig A.6. The SIS can accelerate all ions fromH to U, and has the same maximum Bρ = 18 Tm as the FRS. The maximummagnetic field strength in the FRS dipoles is 1.6 Tm, and their effective radii areabout 11.26 m. The FRS momentum acceptance is ∆p/p = 2%. A completetechnical description of the FRS is found in ref. [150]. Below, the FRS is describedas it is used in the experiments relevant for this thesis, also shown in fig. 6.1. At

TPC TPC

SCIMUSIC

TPC (TPC)TPC

(IC) (NaI-array)SCI

TPC

TA

F1

F2

F3

F4

Figure 6.1: The layout of the FRS and the detector set-up used. Note the additional detectorsincluded for the runs measuring γ coincidences. These detectors are marked by theirname in parenthesis.

the entrance of the FRS, the production target (TA) is hit by the SIS beam andsecondary beams are produced, for our purposes by projectile fragmentation. The

84 Break-up experiments of halo nuclei at GSI

FRS consists of four independent 30 dipole stages, with quadrupole sets in frontof and behind each dipole to obtain the wanted ion-optical beam properties.Higher order magnets correct for aberrations in the system. Between each dipolesection is a focal plane, denoted F1–F4. When running in the achromatic orenergy loss ion optical modes, each focal plane has a waist in the x–direction,which is the direction of dispersion1 (see fig. 6.4), while point–to–point imaging inthe y-coordinate in required only at the intermediate and final focal planes F2 andF4. The achromatic and energy loss modes have ion-optics that fulfill theconditions that the beam image in F4 and angular divergence in F2 and F4 areindependent of the initial momentum spread of the beam, setting the optic transfermatrix elements(x′|δp)TA−F2 = (x|δp)TA−F4 = (x′|δp)TA−F4 = (x|x′)TA−F2 = (x|x′)F2−F4 = 0 (seeappendix A.4). The achromacy is a key issue in the p// measurements, since itcancels the initial momentum spread of the beam and thus allows for extraction ofinternal momentum distributions of the investigated nuclei, even though this issmall compared to the momentum width of the secondary beam.

At the intermediate focal plane, a reaction target can be inserted that effectivelydivides the FRS into two stages. The task of the first stage is to separate out thedesired exotic ion from the projectile fragmentation at the production target, forexample 19C, and to transmit that secondary beam to the reaction target at F2.The first stage is thus set to a Bρ value corresponding to the mass-to-charge ratioand velocity of the desired ion, see eq. 1.4. The second stage is then set totransmit either the secondary beam, to experimentally determine the resolution ofthe system and the dispersions, or tuned to the Bρ that corresponds to thefragment that is to be momentum analyzed, i.e. 18C. The spectrometer isconsequently used both as secondary beam provider and analyzing tool.

Other uses of the FRS is simply as a separator which provides clean, secondarybeams for transport to the experimental set-ups such as ALADIN–LAND or thestorage ring ESR (see fig. A.6). An achromatic degrader can then be used in F2,resulting in the Bρ−∆E − Bρ selection that has previously been introduced forthe LISE3 and A1200 spectrometers.

6.2 Detectors and calibrations

A rather large detector setup is needed to extract all parameters necessary for thep// extraction. The detectors and their use in summarized below together withsome notes on the calibration of the system.

1Dispersion: the dependence of the position coordinate x on the change in momentum δp, seeappendix A.4.

85

6.2.1 Time of Flight (ToF)

ToF data is needed to deduce the relativistic β and γ coefficients, needed due tothe high kinetic energies of ions. The ToF is measured using plastic scintillatordetectors placed at F1, F2 and F4, which gives ToF(F2-F4) and ToF(F2-F1).Both are required due to the reaction target in F2, which decreases the beamenergy in the second stage as compared to the first. The time signals arecalibrated with pulser signals of short time intervals, so called time calibrators.The scintillator time signals are read out at both ends, and the ToF is taken as theaverage. The flight path s24 between F2 and F4 is about 33 m and s12 ≈ 16 m. Inthe expression below, (i,j) are either (1,2) or (2,4).

ToF(Fj − F i) =(SCIjleft − SCIileft) + (SCIjright− SCIiright)

2(6.1)

Note that the ToF is not defined as inverse time, so fast ions with large β willcorrespond to short ToF’s. The TOF is thus proportional to the reciprocal of β,and plotting ToF·β versus β should yield a straight line,seen from eqs. 6.2– 6.4.

ToF =A

β+ B (6.2)

ToF · β = A + B · β (6.3)

β =A

ToF− B(6.4)

The ToF parameters are calibrated using the primary beam. By inserting differentamounts of matter into the beamline, a set of Bρ values with corresponding ToF’sare generated. The Bρ’s are converted to β using eq. A.52, and according toeq. 6.3, ToF·β vs. β is fitted with a line. When the slope and offset are extracted,eq. 6.4 can be used to deduce β particle by particle. The β together with theknown Bρ values gives A/Z as shown in eq. 6.5.[A

Z

]F4

= Bρ34

(1− xF4,foc −MF2F4 · xF2,foc

1000 ·DF2F4

)1

β γ c · 10−9(6.5)

The expression in parenthesis is to get the correct Bρ for non-centered particles.In eq. 6.5, DF2F4 is the dispersion coefficient between F2 and F4, MF2F4 is themagnification for the same stage, while xF4,foc and xF2,foc are the x positions atthe focal planes in F2 and F4, respectively. Bρ34 is the average magnetic rigidityof the second FRS stage, (Bρ3 +Bρ4)/2. The A/Z is used with the charge forparticle identification. For the run where TOF(SCI1–SCI2) was available, thesame recipe can be used. However, simply using Bρ2 without the correction gavegood enough resolution.

86 Break-up experiments of halo nuclei at GSI

6.2.2 Charge

For light ions at high energies, the charge of the beam ions is equal to the protonnumber Z. It is measured in F2 and F4 for particle identification. In F4, Z isdetected in MUSIC detectors (MULtiple Sampling Ionization Chamber), largevolume chambers filled with P10 gas (90% Ar, 10% CH4) at 1 bar [151]. Energysignals are read out from four anodes, and the average of the four is used. Toimprove the MUSIC resolution, the energy signals should be corrected forinteraction position and velocity. Measurements of these corrections is done withthe primary beam. By defocusing the beam, the whole MUSIC area is illuminated,and a correlation between x position and average MUSIC energy can bedetermined as a polynomial in x. The x position is found using the TPC’s andextrapolating the track into the MUSIC. As slow ions lose more energy than fastones, a velocity correction is determined by plotting the position corrected MUSICaverage energy versus β and fitting with a polynomial in β. This is done with thesame set of Bρ’s as the ToF–calibration. The ZF4 is the determined from eq. 6.6.

ZF4 = Zprimary

√position − corrected energy√velocity correction factor

(6.6)

For the charge in F2, the scintillator SCI2 was used for charge detection in theearlier experiments. For the 8B run, a small ionization chamber was installedbefore the reaction target, giving well resolved charges without any correctionfactors. With A/Z and Z determined both in F2 and F4, unambiguous particleidentification is achieved throughout the spectrometer. A typical identificationplot from F4 is shown in fig. 6.2. However, the identification in the earlierexperiments should not be deemed inadequate. The different Bρ settings for theselight ions correspond to a change in central momentum that is larger than themomentum acceptance of the FRS (±1%). This makes a selection in ZF2

combined with [A/Z vs. Z]F4 enough to ensure that core fragments arriving at thefinal focal plane only arise from one-nucleon removal reactions in F2.

6.2.3 Positions and angles

Positions and angles are measured using TPC’s (Time Projection Chambers) [152].These gasfilled detectors (P10 at 1 bar) give the y–position from the electron drifttime to four anode wires. The x–coordinate is determined from a delay line, wherethe time difference between the signals from the right and left side yields theposition. All TPC signals are calibrated in ns with a pulser. To discard falseevents, for example where only one delay line signal is registered, a control sum(cs) for each anode i is constructed, eq. 6.7, and only anode events that fulfill the

87

3.5

4

4.5

5

5.5

6

6.5

7

7.5

2.6 2.8 3.0 3.2 3.4 A/q

Z

18C 19

C

B

12Be

15

N21N

B

Be

22

11

14

Z = 6

A/q = 3.0

Figure 6.2: Identification plot in F4 using ToF to extract A/Z and the MUSIC for Z.

condition are accepted.

TPC(i)cs = TPCdl,right + TPCdl,left − 2 · TPC(i) = constant (6.7)

The x and y positions are given as:

TPCx = xslope(TPCdl,right− TPCdl,left) + xoffset (6.8)

TPCy =

∑i yslope · TPC(i) + yoffset

number of anodes in cs−gate(6.9)

The position resolution is approximately 0.2 mm in both directions for lightions [152].

Four TPC’s are placed pairwise before and after the reaction target in F2, and twoadditional TPC’s are located in F4. By pairing the TPC’s, not only the positionbut also the angle of the ions are given and the straight particle paths are deduced.Using this, the ion position can be extrapolated to any position in the same focalplane (neglecting the thin matter in the ion paths). The position detection in F4 iscrucial for the p// measurements, since the position distribution at the final focalplane is transformed to momentum distribution in the projectile rest frame, whichis the desired observable. If the distribution is not measured exactly at the focus,the achromacy condition is not fulfilled. The width of the distribution will then beinfluenced by the initial spread of the secondary beam, and it will be misleadinglybroad. The position zfocus of the final focal plane can be determined by finding the

88 Break-up experiments of halo nuclei at GSI

waist of the beam using TPC tracking along the z direction, and fitting the widthas a function of z with the beam profile function that is given by the ion-optics.Another way is to look at the dependence of the position in F4 (xF4) on the anglein F2 (x′F2) as a function of z. Due to the achromacy condition, at the focal planeall values of x′F2 give the same xF4. These two methods naturally yield the samezfocus, but the first method was mainly used in this analysis.

The dispersion is also measured using the TPC’s (see section 6.3). In F2, trackingis vital for the Doppler correction of the measured γ’s emitted by the corefragments in flight.

6.2.4 γ–detection

For the measurement of γ’s in coincidence with the longitudinal momentumdistributions, a NaI–array was installed after the target in F2. The array consistsof 32 NaI crystals, mounted as shown in fig. 6.3, approximately 90 cm behind thetarget. The array covers 1% of the solid angle, but at the relativistic energies used,the geometrical efficiency is 10%. The total detection efficiency was simulated by

Figure 6.3: The γ-detection array after the reaction target consists of 32 NaI crystals, each withan area of 32.7 cm2.

Geant [153] and was found to be 3% for Eγ ≈ 430 keV. Energy and efficiencycalibrations were performed using 56,60Co and 88Y sources. To later be be able toseparate γ’s from neutron events, the time signals from the NaI detectors wereneeded, and so they were calibrated with a pulser.

89

6.2.5 Beam and system diagnostics

The primary beam intensity is monitored by a SEETRAM (SEcondary ElectronTRAnsmission Monitor) before the production target. It consists of three thin Alfoils, where the outer two are at +80 V and measure the secondary electrons thatthe beam current produces in the middle foil. The resulting current in the middlefoil is transformed to a voltage and digitized. The sensitivity and offset of theSEETRAM can be varied depending on the beam intensity. The detection has noupper intensity limit. Since irradiation decreases the secondary electron emission,the SEETRAM is calibrated using a scintillator in each run.

At F1 and F3, MWPC’s (MultiWire Proportional Chambers) can be inserted tocenter the beam also in these positions. Slits at all focal planes allow for intensityreduction or collimation of the beam. Since the different isotopes are spatiallyseparated by the dipoles, the slits can in principle be used to select a certainfragment.

To reduce systematic errors it is important to experimentally determine as manyinfluential factors as possible. Therefore, each experiment starts with using theprimary beam which is first centered at all focal position of the FRS. Then theeffective ρ values of the four dipoles are determined. By inserting differentproduction targets, the ToF’s and MUSIC are calibrated as described in previoussections.

The dispersion matrix elements (x|δp) are determined by scaling the magneticfields by for example ±0.5− 1% from the centered setting in the stages TA–F1,F1–F2, TA–F2, F2–F3, F3–F4, F2–F4 and TA–F4. The values are checked withthe predictions by the Monte Carlo program Mocadi [154]. Note that any scalingof the whole FRS, TA–F4, should not induce a change in position at F4, as aresult of the achromacy condition. If the system was ideal, the positiondistribution at the final focus would be a δ–function. The width of this peak is ameasure of the resolution of the method in the real FRS.

6.3 Longitudinal momentum distributions p//

The principle of the longitudinal momentum measurements is schematically shownin figs. 6.4 and 5.1. The basic assumption is that the core fragment keeps theinternal momentum it had as the halo nucleon was removed, a picture which isvalid in the sudden approximation if the momentum transfer k is negligible. Thecore fragments will then have a distribution of momenta pf which corresponds tothe momentum distribution inside the halo nucleus. The magnetic force

90 Break-up experiments of halo nuclei at GSI

experienced by the ions in the dipole fields is proportional to the projection of themomentum on the z-direction and is directed along the x-axis. Having slightlydifferent momenta, the core fragments will be displaced in the direction ofdispersion, and thus be registered at different positions x at the final focus. Topass undeflected through the dipole, the Lorentz and centrifugal forces must bebalanced, a condition which gives the correlation between Bρ and A/Z, eq. 6.10.

F = qvB =mv2

ρ⇒ (6.10)

Bρ =mv

q∝ v

A

Z

For each secondary beam setting, the secondary beam is first transmitted to F4 viathe reaction target and centered there, so that an ion with an energy exactlycorresponding to the set Bρ ends up at x = 0 at the focal plane in F4.

The zfocus is determined as previously described. The position distribution willlater be used as a measure of the resolution. The dispersion DF2F4 is determinedby scaling the second FRS stage by ±0.5−1% in several steps. A linear fit is madeof the shift in position as a function of change in Bρ, and the slope is thedispersion coefficient which gets the unity cm/%. D depends on z, so a fit ofDF2F4(z) = d0 + d1z is also done.

After these preparations are made, the Bρ setting for F2–F4 is scaled to the

y

ρ0

B

z

x

xx0

Bρ < Bρ0

x<

x>

Bρ = Bρ0Bρ > Bρ

0

F = q(v r B)

Ä z v r (-y B) = x F/q^ ^ ^

v

F

Figure 6.4: Schematic illustration of the principle of p// measurements. The x-axis is along thedirection of dispersion, which by definition is perpendicular to the momentum directionwhich defines the z-coordinate. The y axis is perpendicular both to x and z. Thecoordinate system follows the beam ions along the flight path s, creating a right-handed, curvilinear coordinate system.

91

fragment to be momentum analyzed. For halo nuclei this is usually the core,choosing to study halo-nucleon removal reactions in the F2-target.

The position distribution for the core fragment is obtained by particle tracking tozfocus. If the distribution is broad, several Bρ settings have to be used to scan thecomplete peak. Obtaining the correct shape and magnitude of the tails isimportant to compare the data with calculated p// curves. Different admixtures ofconfigurations result in variations which are often mainly visible in the tails.

In the offline analysis, the position of each event is transformed into longitudinalmomentum in the rest frame of the projectile (the secondary beam). Below,subscripts p and f denote projectile and fragment, respectively. If the secondarybeam was not centered in F4 it did not have the nominal Bρ, denoted Bρ0, and wecorrect for this in order to obtain the correct energy of the projectile frame,eq. 6.11.

Bρp = Bρ34,p

(1− xmean

DF2F4

)(6.11)

xmean is the centroid of the position distribution of the secondary beam in F4. Therelativistic γ and β parameters are calculated from the relations in the appendix,eq. A.52.

γp =

√(c · 10−6BρpZp

AmAp

)2

+ 1 (6.12)

βp =

√1− 1

γp(6.13)

To get the p// distribution of the fragments, we now need the energy of thefragments. We again use the relation between position in F4 and momentum afterthe breakup target. Here we use the measured x position extrapolated to the focalplane in F4 to get the Bρf particle by particle, eq. 6.14.

Bρf = Bρ34,f

(1− xF4,foc

DF2F4

)(6.14)

From the Bρ value we can deduce the momentum and energy of the fragments.This is shown in more detail in A.5.

pf = BρfZfc · 10−6 (6.15)

Ef =1

c

√p2f + (AfAm)2 (6.16)

92 Break-up experiments of halo nuclei at GSI

Finally, we transform pf from its rest frame to the rest frame of the projectile. Inour case, we obtain p// as:

p// = γp(pf − βpEf ) (6.17)

The distribution is broadened by the aberrations from perfect achromacy and fromenergy and angular straggling in the target. The magnitude of this contribution isgiven by the width of the position distribution of the secondary beam. This will inthe following be referred to as the response function of the system or as theexperimental resolution. However, the ratio of the widths in momentum space isapproximately proportional to the square of the ratio in spatial coordinates:(Γp(p)

Γf (p)

)≈(Γp(x)

Γf (x)

)2

≈( 4 mm

26 mm

)2

≈ 2% (6.18)

The values of the width in eq. 6.18 are taken from the case of 19C and 18C, and itis seen that inclusion of this broadening effect does not influence the widthdrastically.

The break-up of the projectile nuclei is distributed along the path in the reactiontarget, a fact which induces an extra energy broadening. This effect is obviouslynot included in the response curve. As a result of the difference in energy loss ofthe projectile and fragment in the reaction target, the fragments will have slightlydifferent energies in the second half of the FRS, depending on their reactionposition in the target. This effect is negligible for neutron removal reactions, sincedE/dx is proportional to Z2, but it has to be taken into account for protonremoval where the projectile and fragment have different charges. Since thereaction probability is equal throughout the thickness of the target, the effect ismodelled by a step function and was be simulated in Mocadi.

6.4 Cross section measurements

One nucleon removal cross sections are easily measured at the same time as thelongitudinal momentum distributions. First, one has to find the number ofinteresting ions which irradiate the reaction target in F2. In the measurementsreported in paper 5, only charge identification through SCI2 was available in F2.The wanted number is then found by measuring the composition of the secondarybeam in F2, by removing the F2 target and studying the [Z vs. A/Z]F4

identification plot. The relation between the selected projectile and the primarybeam intensity, measured by the SEETRAM, is deduced. After inserting thereaction target, the second stage of the FRS selects the fragments corresponding to

93

one-nucleon removal from the chosen projectile, and the number of fragmentsarriving in F4 are counted. Before extracting σ−1N , corrections for transmissionlosses, reactions in the beamline matter and acquisition dead time are made.

6.5 γ Coincidences

Since the detected γ’s are emitted by relativistic fragments, their energies asmeasured will be heavily Doppler shifted. The transformation to the rest frame isdone using eq. 6.19.

Eγ = Elab.γ

√1− β2

1− β cos θγ(6.19)

The angle θγ is the angle between the emitted photon and pf , which is given byion tracking of the TPC’s after the reaction target. The solid angle of theNaI–array is larger in the moving frame, giving a larger geometrical efficiency forγ-rays emitted in flight than for a stationary source. The relation for the solidangles is given by eq. 6.20 [155], where also the specific NaI-array is investigated.

dΩ = dΩlab. 1− β2

(1− β cos θγ)2(6.20)

The coincidence conditions for γ’s and fragments in F4 are set offline, and p//distributions corresponding to events where the core fragment was emitted in anexcited state can be extracted and compared with the inclusive data. Theprocedure of this extraction with the necessary background reductions, is describedin the next section.

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7 Results of the FRS break-up experiments

The results of the FRS break-up experiments are reported in papers 2, 3, 5 and 6.In this section, some background to the specific measurements will be giventogether with some comments on the results.

7.1 Physics motivations

In the following, some background information on the 17,19C and 8B is given tomotivate the break-up studies reported in the appended papers. Furtherdiscussions and comparisons of theoretical calculations and other experimentalresults are given in the sections the presentation of the FRS results.

7.1.1 Reasons to study 17,19C

The first verified 1n-halo, 11Be, served as test case for basically all theoreticalmodels for over a decade. With data from only one nucleus, it is difficult to knowwhat properties that are common for all 1n-halos and which are isotope specific.Obviously, it is also impossible to check any model predictions if 11Be is used toadjust parameters. For these reasons, the experimental search for other 1n-halosgot a high priority.

The odd-A neutron-rich carbon isotopes, and in particular 19C, are interesting asone-neutron halo candidates. From table 7.1, it is seen that Sn for 17,19C are small,even if it should be noted that the one-neutron separation energy for 19C still isknown with very bad precision. Small separation energy of the valence nucleon(s)is a necessary but not sufficient condition for halo formation. Another requirementis a low angular momentum barrier for the valence nucleon(s). However, theground state spins and parities for 17,19C are not known. It can also be noted thatthe core nuclei 16,18C are much more tightly bound than 17,19C, suggesting that theodd-A isotopes can be described in two-body models.

The naive shell model gives a 5/2+ assignment for the ground state 19C, and thisdominance of ` = 2 relative motion would hinder the development of a neutronhalo. Shell model calculations based on the Warburton-Brown interaction [156] aswell as relativistic mean field calculations [157] give practically degenerate 1/2+,3/2+ and 5/2+ levels. Both favour a ` = 0 motion of the valence neutron in 19C,but the calculated spacings of the levels are generally less than the error bars.Both the shell model calculation in [156] and the relativistic mean field modelof [157] predict Iπ = 3/2+ for 17C, while another shell model calculations deemed it

96 Results of the FRS break-up experiments

unlikely that 5/2+ was the ground state [158]. Consequently, theories alone can notpredict if the ground state of those nuclei are 1/2+, 3/2+ or 5/2+. The degeneratelevels from shell-model calculations indicate the possibility of s-wave intruderstates, analog to the situation in 11Be, a much more favourable situation for haloformation than a valence neutron in a d-orbit. 15C is sometimes used as a test case,

Table 7.1: Some properties of neutron rich carbon isotopes. Some are investigated in this work,and others are shown to for systematics.

Isotope Sn (keV) g.s. Iπ E∗ (MeV) Ref.

14C 8176±0.1 0+ 6.093 , 1− [159]15C 1218±0.8 1/2+ 0.740 , 5/2+ [159]16C 4250±4 0+ 1.77 , 2+ [159]17C 729±18 (1/2+, 3/2+, 5/2+) 0.295 , – [159]18C 4180±30 0+ 1.62 , 2+ [159]19C 242±95a (1/2+, 3/2+, 5/2+) – – [160]

aThis value is based on four mass determinations. Using only the two latest valuesgives 160±110 keV. A recent experiment indicates 530±130 keV [161].

see for example refs. [126, 162, 163]. It has a rather small Sn compared to that of14C. Calculations agree with experiments on the ground state spin 1/2+. Anexcited state of 5/2+ at 0.74 MeV is also verified, and the core 14C is well known.

The four mass measurements performed for 19C gives the averageSn=242±95 keV [164]. The Nubase evaluation of 1997 uses only the two latestvalues and lists Sn=160±110 keV, while a recent Coulomb dissociation experimentrequires a higher value, 530±130 keV, to be consistent with a 1/2+ ground stateassignment [161]. Also 17C has a small Sn, making it interesting in the search for1n-halo nuclei.

The first measurements of the p// fragment distributions from one-neutronstripping of 17,18,19C at were done at 77 MeV/u [160]. This work reported verynarrow momentum distributions for both 16C and 18C fragments. A laterexperiment by the same group, now at 88 MeV/u, revised the FWHM for the 16Cfragments to 145±5 MeV/c, but retained a small FWHM width, 42±4 MeV/c, forthe 18C p// distribution [162]. A core break-up experiment using the large neutrondetection array DEMON deduced widths of the radial momentum distributionfrom the neutron angular distribution. To subtract the broad component fromknocked-out core neutrons, the background isotope 24F was used. The narrowcomponent, associated with the halo neutron in 19C was found to have a FWHMof 42±12 MeV/c and a neutron multiplicity Mn=0.41±0.2. The broader Gaussian

97

had a width consistent with a more tightly bound nucleus, 165±17 MeV/c. Thetotal FWHM of the radial distribution was 64±17 MeV/c. Both experimentsshowed markedly different shapes of the distributions. The p//(

18C) has a formclose to a Lorentzian, while the other fragment distributions are Gaussian.

The small FWHM for p//(18C) indicate a 1/2+ ground state in 19C. As for 11Be,

one can expect that couplings between the 2+ core-excited state and a 0d5/2

neutron will contribute to the ground state wave function (see eq. 7.1). Coupledchannel calculations investigating this scenario were performed in [165] which alsosummarizes the theoretical efforts on the carbon isotopes up to that time.

7.1.2 Reasons to study 8B

The properties of the lightest boron isotope have been investigated in animpressive list of publications. Several intriguing aspects are reasons for thisconcentration of work on one specific nucleus.

Firstly, it is a proton dripline nucleus, in itself an interesting property. Itterminates the A = 8 isotopic chain for which there are no stable nuclei. This massgap acts as a barrier for the nucleosynthesis, and knowledge of the nuclearstructure in the region can help explain the bridging of the gap.

The last proton in 8B has a very small separation energy, Sp = 137 keV. This factmakes it a good candidate for a one-proton halo even though the last proton in ashell-model picture is situated in a 0p3/2 orbit and hence has ` = 1. The firstmeasurements of a narrow p// distribution of the 7Be fragments (81±4 [166],81±6 [167] and 93±7 [168]) supported the extended radial wavefunction suggestedby a measured large quadrupole moment, Q(8B) = 68.3± 2.1 mb [169]. Theinteraction cross section for 8B [10, 170] are not much larger than for neighbouringnuclei, but a p-state proton halo can due to the combined Coulomb and centrifugalpotential barriers not be very large. However, the one-proton removal cross sectionis significantly larger than for more tightly bound nuclei (see i.e. tab. 7.4). This isnot a contradiction, but rather it can be expected that the cross section of achannel that is connected to the specific structure of the system is more a sensitiveprobe than the global σI . A re-examination of the 8B size taking cluster propertiesinto account gives a rms radius of 2.50 fm [11], which is smaller than what isdeduced from the quadrupole moment but larger than the radius of its mirror 8Li.

Theoretical calculations using different models have been performed aiming atdescribing the different aspects of 8B, see for example [18, 124, 171]. Commonconclusions of these investigations is that the internal structure of the 7Be coreshould be taken into account, since couplings between its first excited state at

98 Results of the FRS break-up experiments

429 keV and the valence proton will contribute to the ground state. Probably,clustering of the core as 3He+4He also plays a role, as suggested by three-bodymodels [18]. Most models predict that the proton radius is larger than the neutronradius, but not all draw the conclusion that 8B has a proton halo structure. This isnot surprising, since a proton halo in 8B necessarily is a less clear-cut case thanneutron halos in s states, and the definition of a proton halo is even less agreedupon than for the neutron case.

8B is also a nucleus which has become almost synonymous with the solar neutrinodeficit problem, which concerns the large discrepancy (30-50%) between thetheoretical and experimental values of the solar neutrino flux. The β-decay of 8B isresponsible for all high-energy neutrinos from the sun, and of the detectors usedsome have been sensitive only to those neutrinos. It has seemed that results fromdetectors that also take some lower-energy ν’s into account have smallerdiscrepancies, indicating a possible problem with the 8B reaction rates. However,the neutrino data have large statistical errors, and it has been difficult to draw anycertain conclusions except from the obvious contradiction between theory andexperiment.

When the one-proton halo structure of 8B was discovered, it was realized that thelong tail of its wavefunction should influence the cross sections, and manyexperiments on photo-dissociation of 8B and the inverse process, proton-absorptionof 7Be, have been carried out. The recent experimental evidence from the SNOdetector that the neutrinos oscillate between flavours [43] on the way from the sunseem to explain the solar neutrino deficit, but does not make detailed knowledge ofthe 8B reaction rates less important. A complete understanding of the underlyingsolar processes will allow reductions of the error bars on the oscillation results. Fora review of 8B in the context of the solar neutrino production, see ref. [172] andreferences therein.

7.2 Results of p// measurements

Brief summaries of the obtained results are given below, together with discussionsand comparisons with other theoretical and experimental results. The set-ups arepreviously shown in fig. 6.1, and a summary of the results is given in table 7.2.

7.2.1 p// distributions of 40Ar fragments

As a test of the set-up and for use in checks of the offline analysis, the p//distributions of the fragments resulting from one-nucleon removal from the

99

primary beam used to produce 17,19C, 40Ar at 1 GeV/u, were recorded. 40Ar hasSn=9.87 MeV and Sp=12.5 MeV, and the distributions of 39Ar and 39Cl are goodexamples of the momentum distributions of fragments from tightly bound nuclei.They are shown in fig. 7.1 to give a comparison with the corresponding curves fordripline nuclei. The Goldhaber model would give a Gaussian with width σ0, i.e.

0

50

100

150

200

250

300

350

400

450

-300 -200 -100 0 100 200 300

0

50

100

150

200

250

300

-300 -200 -100 0 100 200 300 MeV/c

counts

Cl39

FWHM=

280 MeV/c

Ar39

FWHM=

207 MeV/c

Figure 7.1: Longitudinal momentum distributions of the fragments resulting from one-nucleon re-moval of 40Ar. The figure is intended as an example of 1N-removal from stable nuclei,and the distributions are not corrected for broadening effects.

≈210 MeV/c for both distributions, see eq. 5.5. From fig. 7.1, we see that thefragments indeed have Gaussian momentum distributions, and roughly agree withthe width from the simple Goldhaber model. The FWHM for 39Cl is larger thanfor 39Ar, reflecting the fact that Sp > Sn. It can be noted that the 39Cldistribution is also broadened by the different energy losses of 40Ar and 39Cl in thereaction target (see discussion in section 6.3), but the effect will be small since thedistribution is intrinsically broad. Measuring the response of 40Ar gave FWHM ofthe position distributions of 3.2 mm, to be compared with 41 mm and 53 mm for39Ar and 39Cl, respectively.

7.2.2 p// investigations of 17,19C

The carbon isotopes were produced by projectile fragmentation of 40Ar at1 GeV/u in a 6.33 mg/cm2 Be target. In F2, a 4.45 mg/cm2 thick carbon break-uptarget was placed. The measurements were carried out as described in section 6.After calibrations with the primary beam and the 1n and 1p removal reactions for40Ar were done, the setting was tuned to 17C/19C. For each secondary beam, thedispersion was determined together with the position of the final focal plane andthe response function, before the spectrometer was set to analyze the 1n-removal

100 Results of the FRS break-up experiments

reaction. The broadening of the fragment distributions from effects included in theresponse function was found to be about 3% for both cases. The widthcontribution from varying reaction positions in the target was estimated usingMocadi and turned out to be negligible.

The p// distributions of 16,18C fragments are shown in fig. 7.2. To record thecomplete high-momentum tails for 18C, two Bρ settings were used. 16C wasrecorded in only one setting, and the transmission cuts at higher momenta areapparent. After correction for the width of the response curve, the FWHM of the

0

50

100

150

200

250

-200 -100 0 100 200

counts

0

50

100

150

200

250

300

350

400

-150 0 150 (MeV/c)

C18

FWHM =

69± 3

MeV/c

C16

counts

FWHM =

141± 6MeV/c

Figure 7.2: The p// distributions for the 16,18C fragments after one-neutron removal from 17,19Con a C target. The 18C distribution is fitted with a Lorentzian, while a Gaussian isused for 16C. These curves are folded with the response of the system. See also figuresin paper 2 .

distributions were obtained as 141±5 MeV/c for 16C and 69±3 MeV/c for 18C.

What can these distributions tell us? It is immediately seen that the 16C width isseveral times larger than anticipated for a halo nucleus, while the 18C p// is moreintriguing. It is about 50% broader than the corresponding 10Be distribution, eventhough it should have smaller separation energy. This indicates that the groundstate configuration is not a pure s-wave neutron, but has a contribution fromcoupling of the core-excited state 18C(2+

1 ) = 1.62 MeV to a d5/2 neutron. Thelow-lying 2+

1 state in 16C implies that core-excitation should be important also for17C.

We applied the coupled channels calculation of ref. [165] to our results. This modelinvolves a non-spherical nucleus where the potential deformation gives couplingsbetween the degrees of freedom for the core and the motion of a valence neutron.Two core states were included in the calculations, the 0+ ground states and 2+

1

101

states. For all both 17,19C, all three Iπ possibilities were checked. The inputparameters separation energy, deformation parameter β and spin-orbit strengthwere varied to explore the sensitivity of different configurations. The p//distributions are calculated in the transparent limit of the Serber model and in theblack disc model. For a halo s-state, these should not differ drastically as thewavefunction extends way beyond the core.

For 19C, calculations were performed for Sn=0.24 and 0.50 MeV, and for β= 0.0(no coupling) and 0.5. It was found that including absorption indeed only makesthe p// distributions marginally more narrow. The calculations conclude a maincontribution from relative s-wave motion with the core either in its ground state orexcited state. The channels included are shown in eq. 7.1, with small weights forthe n(1d3/2) coupling and the main weight (around 75-85%) on the first component.

|19Cg.s.(1/2+)〉 = αs|18C(0+)⊗ n(1 s1/2)〉+αd5|18C(2+)⊗ n(1d5/2)〉+

αd3|18C(2+)⊗ n(1d3/2)〉(7.1)

The 18C distribution is much broader than was reported from NSCL [162], but in amore recent experiment at the same lab. and roughly the same energy [142], theauthors state that their measured width agree with the one we measure.

Unlike for 19C, the p// measurements at different energies and targets agree verywell for 17C, which is a reason to exclude systematic differences as the reason ofthe variation in 19C results. The broad 16C distribution seems to exclude relative` = 0 motion as the dominating mode for the valence neutron in 17C. The coupledchannels calculations described in ref. [165] exclude a 1/2+ ground state, and bothpure ` = 2 configurations, |16C(0+)⊗ n(0d3/2,5/2)〉, seem too broad. The bestagreement to the experimental width, 141±7 MeV/c, was obtained for the 3/2+

state using a deformation parameter β = 0.2 and relative weights of ≈ 20 and 80%the neutron s1/2 and d3/2 orbits, respectively, eq. 7.2. It can also be noted that thedifference between the transparent and absorption limits in this case was muchmore pronounced than for 19C which also indicates a more spatially restrictedwave.

|17Cg.s.(3/2+)〉 = αd|16C(0+)⊗ n(0d3/2)〉 + αs|16C(2+)⊗ n(1 s1/2)〉 (7.2)

To clarify the unresolved points of (i) Iπg.s. of 17,19C and (ii) Sn of 19C, several newexperiments have recently been performed.

The Coulomb dissociation of a 67 MeV/u 19C beam at a Pb target was explored inref. [161]. The nuclear contribution was assumed to be equal to the cross sectionmeasured on a C target, which was subtracted from the Pb data after scaling withthe ratio of the projectile+target radii. The behaviour of the Coulomb dissociation

102 Results of the FRS break-up experiments

observables should be sensitive to different ground state configurations [165].In [161], the Coulomb dissociation cross section as a function of relative energypeaked at low energies, 300 keV, a typical behaviour of a halo system. Anotherfeature is the large B(E1) strength observed, 0.71±0.07 e2fm2. The integrated σCfor all states was measured to 1.19±0.11 b. The shape was consistent with a 1/2+

ground state, but to reproduce the experimental spectrum, Sn = 530± 130 keVhad to be assumed. With these parameters, a spectroscopic factor α2

s of 0.67 wasdeduced. This experiment does not take branches to excited states into account.

An inbeam γ-ray experiment was performed at NSCL to directly investigate thefraction of core-excited states in 16,17,19C [142]. This work also includes shell modelcalculations for all nuclei, while the spectra are fitted using a black disccalculation. For 17C, the NaI(Tl)-array registered γ’s from the 2+

1 state and froman assumed group of states around 4 MeV. The inclusive p// is stated to be“consistent with earlier measurements” [162, 173]. Conclusions from this work isthat the ground state Iπ=3/2+ has three components: the main contribution isproposed to come from a |16C(2+)⊗ n(0d5/2)〉 coupling with small admixtures of|16C(0+)⊗ n(0d3/2)〉 and |16C(2+)⊗ n(1 s1/2)〉. Data was also taken on a Autarget to investigate the Coulomb vs. nuclear break-up mechanisms [174]. Anobservation was the nuclear cross section being comparable to the Coulomb crosssection for reactions leading to the ground state of 16C, but dominating forbranches to excited states. For 19C, the inclusive p// distribution is stated to be“close to identical” with the one reported in paper 2. From the figure, it seems tobe about 55 MeV/c. The work had a low 19C intensity, and no γ spectrum for theone-neutron removal reaction is shown. The branch to the 18Cg.s. is determinedindependently from both γ analysis and fits of the inclusive momentumdistribution to be about 50%. The best fit of the inclusive p// curve, was obtainedfor a 1/2+ ground state with Sn = 0.65 MeV. Data taken on a Au target gives aσC in agreement with the one reported in [161].

A interaction cross section measurement at 960 MeV/u reports to be consistentwith 19Cg.s.=1/2+ while excluding 3/2+ and 5/2+ as possibilities. It indicates adominant |18C(0+) ⊗ n(1 s1/2)〉 configuration [175].

Apart from the new shell model calculations reported in [142], a flood oftheoretical papers have investigated the carbon isotopes recently, using a multitudeof models to resolve the ambiguous spin assignments. A few are mentioned below,and the reference lists in [142, 176] give more examples. In addition to the coupledchannels calculations already mentioned [165, 177], the authors of [178] investigatethe influence of a speculative low-lying resonance in 19C on the p// distribution.To simultaneously reproduce cross sections and momentum distributions atdifferent energies in a Glauber analysis, the authors of [179] suggest that the core

103

Table 7.2: Summary of p// widths (FWHM) obtained in the FRS runs. Eproj is the projectileenergy before the reaction target, taken from the Bρ setting, and the p// value is theFWHM of the distribution.

Results from this work Comparable results

Reaction Eproj. p// Paper Eproj. p// Ref.

(MeV/u) (MeV/c) (MeV/u) (MeV/c)

40Ar+C ⇒ 39Ar 905 207±10 –40Ar+C ⇒ 39Cl 910 280±15 –

19C+C ⇒ 18C 910 69±3 2 88 42±4b [162]30 64±17a [164]57 b,c [142]

17C+C ⇒ 16C 904 141±6 2 84 145±5b [162]62 b,c [142]

8B+C ⇒ 7Be 1440 91±5 3 1471 81±4 [166]8B+C ⇒ 7Be 936 94±5 6 41 81±6b [167]

38 93±7 [168]8B+C ⇒ 7Be∗ 936 110±15 6

aThe neutron pr distribution from core break-up reactions was fitted with two Gaussians.These had widths of 42±12 MeV/c and 165±17 MeV/c, respectively.

bThe data are taken on a Be target.cNo widths are given, but they are stated to be consistent with the values in paper 2.

of 19C is larger than a free 18C. A microscopic study is done in [163], while atheory using a method based on transfer to the continuum is applied in [176].

Comparing theories and experiments, the consensus seem to converge at 17Chaving a 3/2+ ground state and a dominant ` = 2 relative motion between thevalence neutron and the core which hinders halo formation. For 19C, theuncertainty in one-neutron separation energy in does for many cases prohibitdefinite conclusions to be drawn. However, the collected evidence point at a 1/2+

ground state dominated by relative s motion, giving a halo state, although not asdeveloped as for 11Be. It also seems as if many theories fit the data better usingthe higher value of Sn suggested by the Coulomb dissociation experiment [161].

104 Results of the FRS break-up experiments

7.2.3 p// and γ investigations of 8B

We have measured p//(7Be) from one-proton removal of 8B in two different

experiments. The first one used the same set up and analysis procedure as thecarbon experiment. The primary beam was 12C that with an energy of 1.5 GeV/ustruck a 8.0 g/cm2 Be target to produce a beam of 1.44 GeV/u 8B. Theone-proton removal reactions took place in a 4.41 mg/cm2 C target. Three Bρsettings were used to correctly measure the high-momentum tails of thedistribution. In addition to the centered setting Bρ0, 1.01Bρ0 and 1.02Bρ0 wererecorded. As the energy loss of the 8B projectiles is different from the 7Befragments, the broadening due to differing reaction positions in the target is takeninto account as well as the resolution measured by the width of the secondarybeam at the final focal plane. Together, these effects gave a correction of theFWHM of 2%. The width was determined to be FWHM = 93± 5 MeV/c.

The experimental data was compared with a theoretical three-body modeldescribed in [18, 180]. 8B is here treated as a α+3He+p system, with explicitinclusion of the 7Be+p channel. Relative motion wave functions between the coreand valence proton are obtained by projecting the 8B ground state wavefunctiononto the 7Be wave functions. The main characteristics of 8B and 7Be aresummarized in table 7.3.

Table 7.3: Some important data on 8B and 7Be. For 8B, the quantity gives as Sα+3He is reallySα+3He+p

Isotope Iπg.s. Sp (MeV) Sα+3He (MeV) E∗1 (keV) Iπ∗1

8B 2+ 0.137 1.725 774 ?7Be 3/2− 5.606 1.587 429 1/2−

Taking the 429 keV excited state of 7Be into account, the 8B 2+ ground state canconsist of the contributions shown in eq. 7.3. The relative weights of these threeconfigurations are in ref. [18] given as 68.5%, 10.1% and 15.6%, respectively.

|8Bg.s.(2+)〉 = αa|7Be(3/2−)⊗ p(0p3/2)〉+αb|7Be(3/2−)⊗ p(0p1/2)〉+

αc|7Be(1/2+)⊗ p(0p3/2)〉(7.3)

The calculation for one-proton stripping was done in the transparent and opaquelimits of the Serber model. As expected for a p-orbit, absorption reduces thecalculated width compared to the value in the transparent limit. The wavefunctiongives a p// distribution which fits the shape of the experimental spectrum. Since

105

diffraction dissociation was not included in the calculation, the cross section isunderestimated. Scaling the peak value of the theoretical curve to theexperimental reproduces the experimental curve very well.

To probe the ground state structure of 8B, a new experiment was performed at936 MEV/u, but now with the possibility to detect γ-rays emitted at the reactiontarget using in-beam spectroscopy (see sections 5.3.3, 6.2.4 and 6.3). The idea isthat after removal of the valence proton, the 7Be core is left in the state it occupiedin the 8B projectile. If this was the excited 1/2− state, a γ will be emitted inflight, and this quantum is what we try to detect. The measured γ-spectrum afterDoppler correction is shown in fig. 7.3. The peak at Eγ = 429 keV correspondingto the excited state in 7Be is clearly seen on top of a smooth, exponentialbackground. The inclusive p//(

7Be)1, containing all 8B ground state components

Eγ

counts

0

200

400

600

800

1000

1200

0.2 0.4 0.6 0.8 1.0 1.2

429 keV

0 keV

1/2-

2+

Sp137 keV

3/2-

E1

7Be8B

γ energy (MeV)

Figure 7.3: The Doppler corrected γ spectrum recorded after the reaction target. The peak from7Be∗ ⇒7Be+γ at Eγ=429 keV is clearly seen. Conditions used in the analysis areindicated by dashed lines. The solid line is a fit with a Gaussian and a decayingexponential. The Gaussian is also shown as a shaded area under the peak.

has a width of 94±5 MeV/c, in excellent agreement with the previous result. Todeduce the relative weight of the configuration |7Be(3/2−)⊗ p(0p1/2)〉, thefragment momentum distribution is detected in coincidence with γ’s that haveenergy corresponding to the excited state in 7Be. To extract the longitudinalmomentum distribution of the 7Be∗ fragments, the following steps were taken.

1The term inclusive (exclusive) means without (with) selection, and in this specific case no (a)selection in γ energies

106 Results of the FRS break-up experiments

First, a p// distribution was deduced with the condition that a 429 keV γ ray wasemitted. The background contribution to the 429 keV-peak is 69%. To eliminatethis contamination, the corresponding longitudinal momentum distribution withevents coming only from the background was created by selecting events with highEγ. This background distribution was then subtracted from the one obtained incoincidence with the 429 keV peak. The resulting curve represents the longitudinalmomentum distribution of the 7Be in the excited 1/2− state. This distribution hasa FWHM of 109±7 MeV/c. The p// distributions are shown in fig. 7.4. The

0

0.05

0.1

0.15

-150 -100 -50 0 50 100 150

0

0.2

0.4

0.6

0.8

1

1.2

FWHM = 109 ± 7 MeV/c

FWHM = 95 ± 5 MeV/c

p//(MeV/c)

dσ -

1p/dp(m

b/M

eV/c)

7Be(1/2-) ⊗ p(0p3/2)

8Bg.s.(2+)

Paper 3

Paper 6

Paper 6

Figure 7.4: The ptotal// and pexc.// distributions, corresponding to the 7Be fragments being emittedin all states and the 429 keV state, respectively. In the top panel, the results from thetwo experiments are shown. The triangles correspond to the data from paper 3 andthe circles to paper 6.

increase in width for the pexc.// as compared to ptotal

// is expected, since the excitationof the core effectively increases the binding energy of the last proton with 429 keV.For a pure s-wave halo, the width scales with the square root of the binding energy(exact for a Yukawa function), but this is no longer true for the p-wave case.

The same conditions as for the p//, were used to find the cross sections σ−1p forthe inclusive and exclusive channels. The total one-proton removal cross section

107

obtained in this way is σtotal−1p = 94±9 mb, in excellent agreement with earlier

results (see table 7.4). The exclusive one-proton removal cross sectionσexc.−1p = 12±3 mb was deduced by selecting fragments that were measured in

coincidence with the 429 keV gamma rays, subtracting the background andcorrecting the number of 7Be in F4 for NaI and transmission efficiencies.

The relative weight of the component of the core-excited configuration in theground state of 8B, can be obtained from the ratio of σexc.

−1p/σtotal−1p , which is 13±3%.

The error includes statistical errors and the precision of the Geant simulation.

This same theoretical model as described above was used for comparison withtheory, with the difference that realistic profile functions were used (see paper 6 fordetails). The widths of the distributions are reproduced with 99 and 130 MeV/cfor the inclusive and exclusive curves, respectively. The absolute cross sections arecalculated to σexc.

−1p= 11.5 mb and σtotal−1p = 82 mb. These values are somewhat

smaller than the experimental numbers, but the ratios of σexc.−1p/σtotal

−1p , 14% for thetheory and 13% from the data, agree very well.

Corresponding data was also taken on a Pb target, and those results will bepublished elsewhere.

So many experiments have been performed on 8B, especially on the reactionsrelevant for solar fusion, that it is virtually impossible to summarize them in a fewsentences. The same goes for the theoretical effort put into this nucleus. Inconnection with the work of this thesis we can just note a series of experiments onCoulomb break-up at intermediate energies, aimed at deducing the E2 and M1components of this break-up ([181–183] and references therein). The solar capturerate 7Be(p,γ)8B depends on the contribution of these multipolarities in addition tothe dominant E1 transition. Interference of the E1 and E2 amplitudes shouldproduce a marked asymmetry in the longitudinal momentum distributionsmeasured on Pb targets [184]. The p//(

7Be) and p//(p) distributions measuredin [181–183] confirm such skewed distributions. It can be noticed that noreminiscence of this asymmetry is seen in our distributions on Pb targets.

7.2.4 p// and γ studies of 18-24O

In a subsequent experiment, the setup used for the 8B measurement was employedto make a study of neutron rich oxygen isotopes 18-24O. Longitudinal momentumdistributions, cross sections and γ spectra were measured for one neutron removalreactions, and for 2n-removal in the case of 24O. Comparing the inclusive widths ofthe p//(fragment)−1n, a drastic reduction is seen for the 22O fragment from 23O-1nreactions. This fragment has a width ≈140 MeV/c while the other FWHM are on

108 Results of the FRS break-up experiments

the order of 200 MeV/c.

The γ spectra for the even oxygen isotopes could be compared to GANILdata [185], and after the preliminary analysis agree well. 22O has also been studiedat NSCL [186]. The FRS results at relativistic energies, with inclusive andexclusive p// distributions and cross sections, will be reported in comingpublications.

7.3 Cross section measurements

One-nucleon removal cross sections have been deduced from the data collected inthe above described FRS runs. The procedure was previously described insection 6.4. The results are given in table 7.4 below, together with σ−1n(12C)which is shown for comparison. The one-neutron removal cross section of 19C is

Table 7.4: One-nucleon removal cross sections measured at the FRS. N denotes “nucleon”, that isneutron (n) or proton (p). The low value for 10B is probably connected to the fact that9Be has no particle stable excited states.

Reaction N Eproj. σ−1N(mb) Paper

12C+C ⇒ 11C n 1050 44.7±2.8 [187]

19C+C ⇒ 18C n 910 233±51 519C+Pb ⇒ 18C n 910 1967±334 5

17C+C ⇒ 18C n 904 129±22 58B+C ⇒ 7Be p 1440 98±6 58B+C ⇒ 7Be p 936 94±9 68B+C ⇒ 7Be∗ p 936 12±3 6

8B+Pb ⇒ 7Be p 1440 103±17 58B+(CH2)n ⇒ 7Be p 1440 687±117 5

10B+C ⇒ 9Be p 1450 17±2 5

around five times larger than for the stable 12C. Maybe even more supportive ofthe halo structure is the doubling of σ−1n when going from 17C to 19C. When alsoconsidering the narrowing of p// for the same step, it is easy to draw theconclusion that a halo structure is developed in going from 17C to 19C. It can benoted that ref. [142] report σ−1p(

17C)=115±14 mb and σ−1p(19C)=264±80 mb

(≈60 MeV/u on Be targets), in total agreement with our results.

The one-proton removal cross sections are well accounted for by the three-body

109

model and an eikonal approximation of the Glauber model. The results are givenin the previous section.

The interaction cross section σI(8B) could also be deduced since all reaction

channels involve a change of charge. For this case, the surviving 8B are monitoredinstead of the reaction products. For the carbon target, we find σI=831±10 mb, inagreement with previous measurements on light targets; σI=809±11 mb(1471 MeV/u) [166] and σI=798±6 mb (790 MeV/u) [170]. This can be comparedwith σI=768±9 mb (790 MeV/u) [8] for 8Li which is the mirror of 8B, havingSn=2.03 MeV. The halo character of 8B is clearly reflected better in σ−1p than inσI .

110 This page is intentionally blank. Feel free to fill it.

111

8 Outlook

It is easy to think of research in terms of projects to finish and deadlines to meet.But in fact, nothing could be more wrong. Rather than coming in well definedchunks of work, research is an ongoing process that lives its own life and oftenmakes turns along the way that are totally unanticipated to the scientist.

This thesis is a good example of the continuous nature of research. Several of theprojects described were initiated before I started my studies, and most of them willbe continued after the 26 October 2001.

For example, the program of break-up experiments at GSI took a step forward afew weeks ago when 8He and 14Be were studied in a complete kinematicsexperiment using a liquid hydrogen target, and including detection of γ-rays andknocked out protons. Future beamtime is approved and ensures a continuation ofthese investigations for a range of nuclei. Using the FRS, cross sections and p//can be measured for series of isotopes which helps in choosing candidates forcomplete kinematics experiments in the ALADIN–LAND set-up.

Analysis of the 8B and 18-24O data from the previous experiments is under way andthe results promise to shed new light on the structure of these nuclei. Continuedbreak-up experiments of neutron rich nuclei will increase the available information,making it possible to deduce systematic trends concerning the onset of exoticphenomena such as halo structures and intruder states.

Elastic resonance scattering of RIB’s on thick gas targets has proved to be areliable tool for investigations of proton unbound nuclei. Many interestingphenomena appear at the proton dripline, such as n–p pairing. The ERSITtechnique gives a simple way to investigate the unbound proton rich partner inmirror pairs, opening for a deeper understanding of mirror symmetries and shellstructures at the driplines. The method is for example well suited for studies ofastrophysically interesting reactions. Another field which could benefit from suchstudies is the search for two-proton emitters. The decay has to be simultaneous ifthere is no intermediate resonance energetically available for the sequential decay.The situation has been exemplified in this thesis with the case of the two-protonemitter 12O. This nucleus was a strong candidate for simultaneous 2p-emission,but the determination of the 11N ground state explained why no such branch hadbeen observed. Vice versa, the non-observation of 2p-branches gives upper energylimits for the resonances in the intermediate nuclei.

The constant upgrades of experimental facilities play an important role in theextension of the nuclear chart to more extreme N/Z-ratios as well as heaviernuclei. As the basic properties of nuclei far from stability change from hot news to

112 Outlook

well-known facts, the experimental set-ups become increasingly complicated inorder to extract more detailed information. The complexity of the data is alsoincreasing, making the conclusions more difficult to extract. The trend is similaron the theoretical side, where initial qualitative models have paved the way forintricate techniques with predictive power.

However, this increased complexity is not only of good. Interpretations becomeless obvious than for simple methods, making it harder for outsiders to judge thevalue of the results. Sometimes it is important to return to the simple experimentsand estimates in order not to lose perspective. Understanding comes from puttingthe detailed pieces of information together and see the image of the jig-saw puzzle.

Despite the explosive development of the last decades, the field of exotic nuclei isstill an area where questions are numerous and answers often are contradictory. Inthis situation, it is vital to remember that explaining unexpected results andresolving contradictions is what brings science forward, produces new questionsand opens doors to previously unimaginable futures.

Quand on me contrarie, on esveillemon attention, non pas ma cholere:je m’avance vers celuy qui me con-tredit, qui m’instruit. La cause dela verite, devroit estre la cause com-mune a l’un et a l’autre.

When someone contradicts me, itawakens my attention, not myanger: I approach the person con-tradicting me, who teaches me. Thequest for truth should be a commonconcern for all.

Michel de Montaigne, 1595 [188]

113

9 Acknowledgements

Simply by looking at the lists of authors on the appended papers, you realize thatthe work summarized in this thesis could not have been done without a largeinternational collaboration. I have been lucky enough to work with people who,apart from teaching me a lot about physics, have turned nightshifts, conferences,coffee breaks, travels and pub rounds into memories worth keeping.

It really is a hopeless task to name all those who have part in making these sixyears as fun and rewarding as they have been, but I want to thank everyone withwhom I have worked at different times and places:

Arhus: Hans, Karsten, Vera, HenrikMadrid: Olof, Maria, Luis, Inma, YolandaGSI: Hans, Lola, Thomas, Maggie, Gottfried, Ivan, Haik, LeonidGANIL: Stephane, Francois, Olivier, Marek, Marie-GenvieveKurchatov: Vladilen, GrishaCERN: Thomas, Havar, Katarina, Andreas, JoakimNSCL: Michael, Matthias, Eric, Mattiasand: Jim, Faical, Leonid ... as well as those I forgot here!

Special thanks goes to the members, past and present, of the Subatomic Physicsgroup at Chalmers:

Bjorn: for never lacking time or support for meGoran: for always happily sharing your knowledgeMichail: for theoretical explanations whenever I need itKate, Martin, Halina, Lasse, Orjan, Yulia, Maria, Marie, Heini, Kenny:

for creating a good working environment at ChalmersFredrik: for carrying me down the French mountain when I was crying...Mikael: for the way you radiate unbreakable optimismChristian: for all chats and your computer help in times of troubleUffe: for all the good beers we had, on and off dutyLeif: for your part in this thesis

And last, but not least, many thanks to:All friends: for making me forget about physics every now and thenMamma, Pappa, Mia, Morfar: for believing in me, especially when I don’tThomas: for listening

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115

A Appendices

A.1 Abbreviations of facilities and accelerators

The abbreviations of facilities and instruments of nuclear physics are often used asnames, but the acronyms obviously stand for something. This meaning can bemore or less obvious. In the table below, a list of the meanings of abbreviationsmentioned in this thesis are given together with a short description of what theyare. By “IFS facility” or “ISOL facility” is meant the target–separator complex,and “accelerator facility” refers to the whole laboratory area with accelerators anddifferent experimental areas is.

Table A.1: Abbreviation, full name, location and type of facility for the acronyms used in the text.

Abbr. Name Location Type

A1200 Mass separator at NSCL IFS facilityALADIN A LArge DIpole magNet at GSI Reaction product

separator in Cave BBigRIPS see RIPS entry at RIbf IFSCERN Conseil Europeen de

Recherche NucleaireGeneva,Switzerland

Research instititute

CRC Centre de Recherchesdu Cyclotron

at UCL ISOL withpost-acceleration

DEMON DEtecteur MOdulairede Neutron

moving Neutron detectionarray

EURISOL EURopean IsotopeSeparation OnLine

not decided ISOL

FLNL Flerov Laboratory ofNuclear Reactions

at JINR Accelerator facility

FRS FRagment Separator at GSI IFS facilityGANIL Grand Accelerateur

National d’Ions LourdesCaen, France Accelerator facility

GSI Gesellschaft fur Schwer-Ionenforschung mbH

Darmstadt,Germany

Accelerator facility

IGISOL Ion Guide IsotopeSeparation OnLine

at JYFL ISOL facility

Continued on next page

116 Appendices

Continued from previous page

Abbr. Name Location Type

ISOLDE Isotope Separation OnLine at CERN ISOL facilityJINR Joint Institute for Nuclear

ResearchDubna,Russia

Research institute

JYFL Jyvaskylan YliopistoFysiikan Laitos

Jyvaskyla,Finland

Accelerator facility

KUL Katholieke UniversiteitLeuven

Leuven Accelerator facilty

LAND Large Area NeutronDetector

at GSI Neutron detectionwall in cave B

LBNL Lawrence BerkeleyNational Laboratory

Berkeley, Ca.USA

Accelerator facility

LISE3 Ligne d’Ions SuperEpluches

at GANIL IFS facility

MAFF Munich Accelerator forFission Fragments

Munich,Germany

Radioactive beamsthrough fission

MSU Michigan StateUniversity

East Lansing,Mi., USA

Location of NSCL

NSCL National SuperconductingCyclotron Laboratory

East Lansing,Mi. USA

Accelerator facility

REX-ISOLDE

Radioactive beamEXperimentsISOLDE

at CERN ISOL with post-acceleration

RIA Rare Isotope Accelerator not decided IFS and ISOL withpost-acceleration

RIKEN Institute of Physical andChemical Research

Saitama,Japan

Research institute

RIPS RIKEN Projectile fragmen-tation Separator

at RIKEN IFS facility

RIbf Radioactive Ion BeamFactory

at RIKEN Accelerator complex

RPMS Reaction Product MassSeparator

at NSCL Separates by a Wienfilter and dipole

SHE SuperHeavy Elements — Nickname forelementswith Z > 100.Continued on next page

117

Continued from previous page

Abbr. Name Location Type

SHIP Separator of Heavy Ionreaction Products

at GSI Velocity filter forSHE production

SIS SchwereIonen Synchotron at GSI Synchotronaccelerator

SISSI Source d’Ions secondaires aSupraconducteurs Intense

at GANIL IFS production

SPIRAL Systeme de Productiond’Ions Radioactif etd’Acceleration en Ligne

at GANIL ISOL withpost-acceleration

UCL Universit Catholiquede Louvain

Louvain-la-Neuve

Accelrator facillity

UNILAC UNIversal LinearACcelerator

at GSI Linac, first beam in1976

118 Appendices

A.2 Some basic concepts and definitions

It the following table, some important terms and concepts are listed.

Table A.2: Some basic concepts in nuclear physics.

Name Symbol Definition

massnumber A The number of nucleons in a nucleus, A = N + Zproton number Z The number of protons in a nucleusneutron number N The number of neutrons in a nucleusisobar Nuclei with the same A, as 12

6 C6 and 125 B7

isotope Nuclei with the same Z, as 126 C6 and 13

6 C7

isotone Nuclei with the same N, as 126 C6 and 13

7 N 6

binding energy B The amount of energy a nucleon system gains byforming a nucleus

separationenergy

Sx The amount of energy required to remove a boundnucleon or cluster x from a nucleus

mass deficit ∆ The mass difference between a nucleus AX and Afree nucleons

Q-value Q The net energy result of a reaction or decaycross section σ A measure of reaction probabilitymirror nuclei – Nuclei that transform into each other if protons

are changed into neutrons and vice versa, as 136 C7

and 137 N 6 (they are “mirrored” in the N = Z line)

atomic mass unit u A weight measure, defined as 112

of the 12C masselectron volt eV An energy unit. 1 eV = 1.6·10−19 J. In nuclear

physics, MeV (106 eV) is more often used

leptons L Pointlike particles as electrons and neutrinos.quarks q Pointlike particles, builds composite particle,

hadrons. No free quarks exist, and they only ap-pear bound together into particles

baryons B Hadrons built of of three quarks, i.e. protons andneutrons

mesons – Hadrons consisting of quark-antiquark , i.e. π±,0

neutrino ν Leptons without charge, emitted in β-decayα−particle α A 4He nucleus, emitted in α decayβ−particle β An elecron or positron, emitted in β decayγ quantum γ A foton, emitted in de-excitation

119

A.3 Kinematic relations

In this section, kinematic relations for elastic scattering of a projectile nucleus Moff a target nucleus m are given for convenience. For our case, M is a carbon ionand m is a proton in the target. The relations are derived in the non-relativisticlimit as this is enough for the low energies in our experiments. The relations forthe volume element, and some formulas used in the discussion of the differencesbetween inverse and conventional geometries are also given.

A.3.1 Transformation c.m. ⇔ lab.

The situations in fig. A.1 define the variables used and visualize the process viewedin the laboratory (lab.) and center of mass (c.m.) systems, and the systemsoverlaid (g). Our aim is to transform between these systems and express theexcitation energy of the compound system Eex in terms of projectile energy E ordetected energy Em

lab. We are also interested in the dependence of the target mrecoiling angle θlab. A more general treatment of the kinematics of reactions andscattering is given for example in [55].

In the lab. system, the center of mass moves with constant velocity vcm, parallellto v, while it is stationaty in the c.m. frame. As the relative velocitiy between Mand m is the same in both frames, we see:

vmcm = vcm and vMcm = v − vcm (A.1)

From fig. A.1(g), we immediately recognize the following relations between anglesin the two frames:

φcm = π − θcm (A.2)

φcm = π − 2θlab (A.3)

θcm = 2θlab (A.4)

First, we derive relationships between the c.m. and projectile quantities.Conservation of momentum between figs. A.1:a and b gives:

Mv = (M +m)vcm ⇒ vcm =M

M +mv (A.5)

Eq. A.5 can directly be translated into a relation between energies:

Ecm =M +m

2v2cm = A.5 =

M

M +mE (A.6)

120 Appendices

φlab

θlab

M

m

Em , vm

M m m

M+m M+m

Ecm , vcm Ecm , vcm

φcm

θcm

EM, vM

Em , vmcm cm

E , v Em , vmcm cmEM , vMcm cm

vmcm

vMcm

vcm

vcm

φlab

φcm

θcm

vmlab

lab

EM , vMlab lab

lab

vMlab

θlab

Laboratory system C.M. system

Lab. and c.m.

a

b

c

d

e

f

A B

CD

E

g

M

M

m

Figure A.1: The elastic scattering viewed in the lab. (a–c) and c.m. (d–f) frames. The subfigures(a) and (d) show the system before interaction, during interaction (b) and (e), andafter the scattering (c) and (f). The part (g) clarifies the relations between the anglesin the two frames. Equal angles are indicated by hatches. All energies are total kineticenergies.

121

This relation between energy of the recoiling nucleus m and energy of the incomingprojectile M is obtained using the cosine theorem on the triangle ECD, and notingthat ED and DC have equal length vmcm = vcm.

(vmlab)2 = v2

cm + v2cm − 2v2

cm cos φcm

= 2v2cm(1− cosφcm) = A.5

= 2

(M

M +mv

)2

(1− cosφcm) (A.7)

= 2

(M

M +mv

)2

· 2 cos2 θlab

We now write Em after scattering as a function of projectile energy and angle:

Emlab =

m

2(vmlab)

2 = A.7 =m

2

(M

M +mv

)2

4 cos2 θlab

=4M ·m

(M +m)2E · cos2 θlab (A.8)

To calculate the excitation energy from the projectile energy, we first need the c.m.energies of M and m after the scattering process. Remembering the relation A.1we write:

vMcm = v − vcm = A.5 = v

(1− M

M +m

)=

m

M +mv (A.9)

This immediately gives:

EMcm =

M

2(vMcm)2 = A.9 =

M

2

(m

M +mv

)2

=

(m

M +m

)2

E (A.10)

and further:

Emcm =

m

2(vcm)2 = A.5 =

m

2

(M

M +mv

)2

=Mm

(M +m)2E (A.11)

The excitation energy is subsequently obtained through adding the two quantitiesin the equations (A.10, A.11):

Eex = EMcm + Em

cm =m(m+M)

(M +m)2E =

m

M +mE (A.12)

The excitation energy can be derived from the lab. energy of the recoiling targetnucleus by using equation (A.8):

Eex =m

M +m· (M +m)2

4M ·m · cos2(θlab)Em =

M +m

4M · cos2(θlab)Em (A.13)

122 Appendices

From these results all other relations can easily be derived. For bound nuclei (like13N), the binding energy implies that the ground state has a mass lighter than thesum of the projectile and target masses. The mass of the nucleus is M +m−∆m,where ∆mc2 = Ethr = Sp and Sp is the proton separation energy. The generalexcitation energy is then calculated with the threshold (thr.) energy added, i.e.:

Eex = EMcm + Em

cm + Ethr (A.14)

In the case of an unbound system, the threshold energy Ethr.=0 and all excitationenergy will become kinetic energy of the projectile and target. A summary of themost useful results for θlab = 0 is given below:

(A.6) : Ecm =M +m

2(vcm)2 =

M

M +mE

(A.8) : Em(0) =4M ·m

(M +m)2E

(A.12) : Eex =m

M +mE + Ethr

(A.13) : Eex(0) =M +m

4MEm + Ethr

A.3.2 The volume element in lab. and c.m.

When calculating the differential cross section dσ/dΩ, we need the expression inthe c.m. frame instead of the lab. system. The only relevant parameter thatchanges value between the systems is the solid angle, as it depends on the angle ofthe recoling proton. We have already seen that θcm = 2θlab (A.4), which gives:

dΩlab

dΩcm=

sin θlab dθlabsin θcm dθcm

= A.4 =1

4 cos θlab(A.15)

Eq. A.15 immediately gives:(dσ

dΩ

)cm

=1

4 cos θlab

(dσ

dΩ

)lab

(A.16)

A.3.3 Additional useful relations

It can be enlightening to compare the relevant energies in inverse and conventionalgeometries. By inverse geometry is meant a situation where the heavier nucleus isthe projectile. The notations introduced in the preceeding sections are used, withthe addition that T refers to energies in the conventional measurement and E tocorresponiding quantities in inverse geometry [55].

123

• Conventional measurement :

TMcm = TmM

(m +M)2(A.17)

Tmlab = T

(m

M +m

)2cos θlab +

√(M

m

)2

− sin2 θlab

2

⇒

Tmlab(0) = T

(m

M +m

)2(1 +

M

m

)2

(A.18)

Tex = TM

(M +m)(A.19)

• Inverse measurement :

EMcm = E

(m

m+M

)2

(A.20)

Emlab = E

4mM

(m+M)2cos2 θlab. ⇒

Emlab(0

) = E4mM

(m+M)2(A.21)

Eex = Em

(M +m)(A.22)

The ratio between E and T is obtained from A.17 and A.20, together with the factthat TMcm = EM

cm. This ratio is denoted K = E/T = M/m. Equations A.18and A.21 lead to the following ratio between the energy of the detected particle min conventional and inverse geometry for M m:

Em(0)

Tm(0)= 4

K2

(1 +K)2∼ 4 (A.23)

This results in a factor of four higher energy for the light scattered particles ininverse compared to conventional geometry for the same resonance energy, which isan important gain for a study of low lying resonance states.

124 Appendices

A.4 Some ion optics

The transmission of a beam through a magnetic system is often described inanalogy with the transmission of light-rays through a system of optical elementssuch as lenses, hence the name “ion optics. Each ion optical element can bemodelled with a matrix T that transfers a vector of the incoming beam parametersXi to a vector Xf which describes the beam after the element. These state vectorscan be thought of as composed of six parameters:

X =

xx′

yy′

sδp

=

x coordinate

angle in the x− z planey coordinate

angle in the y − z planepathlength

momentum spread

The coordinate system is the same as used before, i.e. x is in the direction ofdispersion, z is in the direction of the beam momentum (parallel to s) and y isperpendicular to both x and z. A general transfer matrix for a system consistingof focusing and dispersive elements has the form [189]:

T =

(x|x) (x|x′) (x|y) (x|y′) (x|s) (x|δp)(x′|x) (x′|x′) (x′|y) (x′|y′) (x′|s) (x′|δp)(y|x) (y|x′) (y|y) (y|y′) (y|s) (y|δp)(y′|x) (y′|x′) (y′|y) (y′|y′) (y′|s) (y′|δp)(s|x) (s|x′) (s|y) (s|y′) (s|s) (s|δp)(δp|x) (δp|x′) (δp|y) (δp|y′) (δp|s) (δp|δp)

The matrix elements can be denoted in many ways, of which one common is usedin the T–matrix above. The different elements also have specific names and aredenoted after these. For example, (x|x) is the magnification M , (x|x′) is theangular magnification (MA) and (x|δp) is the dispersion D.

We are particularly interested in how positions and momenta in F2 are transformedinto F4. For our case, the x position in F4 can be written as (to the first order):

xF4 = MxF2 +MAx′F2 +DF2F4δp (A.24)

In eq. A.24 xFi denotes the s position at focal plane i, while the primed coordinatestands for the angle in the x− z plane. M and MA are the magnification andangular magnification, respectively, and DF2F4 is finally the experimentallymeasured dispersion between F2 and F4. At the focal plane, both M and MA areequal to zero by definition, and the x position in F4 is then completely determinedby the dispersion coefficient.

125

An interesting property of a beam consisting of charged particles is that itsparticle density in phase space is constant under action of conservative forces[189, 190]. This is known as Liouville’s theorem, something most students havebeen confronted with in thermodynamics. Electromagnetic forces are conservative,so a result of this theorem is that it is not possible to change the phase spacevolume of a beam by acting on it with electromagnetic fields. A practicalconsequence of Liouville’s theorem is that the determinant of the transfer matrixmust equal one. Even if we can not change the particle density, it is possible toinfluence the shape of the phase space area.

If on the other hand matter is introduced into the beamline, non-conservativeslowing-down forces act on the beam, something that will change its phase spacevolume. Depending on the specific force, the density can increase or decrease [189].

126 Appendices

A.5 FRS messhutte FAQ

There are some relations between energy, momentum and magnetic rigidity thatare very useful when working with magnetic energyloss spectrometers. The ones Ifound most indispensable during the work with the experiments and the analysisdescribed in this thesis are gathered here for convenience.

A.5.1 Momentum ⇔ kinetic energy

We start with deriving the momentum of a particle as a function of its kineticenergy. Equations A.25 and A.26 are well known relations between the relativistictotal energy E, the kinetic energy T and the momentum p of a particle with massm (see for example [1].

T = E −mc2 (A.25)

E2 = p2c2 +m2c4 (A.26)

Squaring eq. A.25 and rearranging, we obtain:

E2 = T 2 +m2c4 + 2Tmc2 (A.27)

Setting the right-hand sides of equations A.26 and A.27 equal we get:

p2c2 = T 2 + 2Tmc2 (A.28)

The quantity u is the mass of one atomic unit. We often want the energyexpressed as MeV/nucleon, so we define:

T [MeV ] = TA [MeV/u · u] where A is the mass number (A.29)

Then we notice that:

m = AAm (A.30)

The constant Am = 931.494 MeV/c2. Inserting equations A.29 and A.30 in A.28we get:

p2c2 = (TA)2 + 2TA(Au) = T 2A2 + 2AmTA2 (A.31)

By division with c2 and taking the square root of both sides we obtain themomentum p:

p = A

√T (T + 2Am) MeV/c (A.32)

Inverting eq. A.32, we instead get the kinetic energy per nucleon as a function ofmomentum:

T =

√A2m +

p2

A2− Am MeV/u (A.33)

127

A.5.2 Magnetic rigidity ⇔ kinetic energy

The dependence of magnetic rigidity on kinetic energy is an extremely usefulformula. From the equilibrium condition that the magnetic and centrifugal forcesmust be equal for a particle transmitted through the dipoles, we have:

F = BvZ =mv2

ρ⇒ Bρ =

p

Z(A.34)

An analysis of dimensions of eq. A.34 results in Bρ being equal to p/Zc · 10−6:

Tm =V s

m=J

C

s

m=Nm

C

s

m=kgm2

s2

s

m=kgm

C s(A.35)[ p

Z

]=MeV

c

1

C=J

C

s

m

1.602 · 10−13

1.602 · 10−19

1

2.9979 · 108(A.36)

Using the result from eq. A.32 we get:

Bρ =A

Zc · 10−6

√T (T + 2Am) Tm (A.37)

T =

√A2m +

[Bρ

Z

Ac · 10−6

]MeV/u (A.38)

A.5.3 Magnetic rigidity ⇔ momentum

Giving the magnetic rigidity as a function of momentum is now quite simple. Bycomparing T (p) and T (Bρ) (equations A.33 and A.37), we can identify:

p2

A2=

[Bρ

Z

Ac · 10−6

]2

MeV/cu2 ⇒ (A.39)

p = ZBρc · 10−6 MeV/c ⇒ (A.40)

Bρ =p

Z

1

c · 10−6Tm (A.41)

128 Appendices

A.5.4 Velocity ⇔ kinetic energy

We will now derive the dependence of β on the kinetic energy of a particle. Firstsome general relativistic relations are given for convenience:

β = v/c (A.42)

γ =1√

1− β2(A.43)

β =

√γ2 − 1

γ(A.44)

The kinetic energy is the difference between the total relativistic energy and therest energy of a particle with restmass m0:

T = mc2 −m0c2 = m0c

2(γ − 1) (A.45)

From eq. A.45 we get two practical relations:

T = m0c2

(1√

1− β2− 1

)(A.46)

γ =T

m0c2+ 1 = 1 +

T

Am(A.47)

Squaring eq. A.47 and substituting eq. A.43 we obtain:

1

1− β2=

(T

m0c2+ 1

)2

(A.48)

Now we get the expression for β from eq. A.48 and change the kinetic energy intokinetic energy per nucleon:

β =

√√√√1− 1(1 + T

m0c2

)2 =

√√√√1− 1(1 + T

Am

)2 (A.49)

A.5.5 Velocity ⇔ magnetic rigidity

Finally, we write the expression for Bρ as a function of γ, using the relations wehave already obtained. We start with eq. A.46 and substitute for T :

T = Am(γ − 1) (A.50)

129

This and eq. A.37 immediately give us the desired connection as:

Bρ =A

Z

Am

c · 10−6(γ − 1)

√1 +

2

γ − 1(A.51)

Using eqs. A.33, A.40 and A.47, we get a relationship between the relativisticparameter γ and the magnetic rigidity.

γ =

√1 +

(BρZc · 10−6

AAm

)2

= (A.43) =1√

1− β2(A.52)

130 Appendices

A.6 Layouts of accelerator facilities

In this section the layouts of the GANIL, NSCL and GSI accelerator complexes areshown, as well as drawings of the LISE3 and A1200 spectrometers.

A detailed view of the LISE3 spectrometer is shown in fig. A.2.

D4D3

D6

3

First dispersive plane

Target boxFirst achromaticfocal point

Final focal point

Second achromaticfocal point

Second dispersive plane

Wien filter

Scatteringchamber

Figure A.2: A detailed figure of the LISE3 spectrometer. The scattering chamber is placed in D6.

Fig. A.3 shows the accelerator facility GANIL with the two coupled cyclotrons,CSS1 and CSS2, the RIB production device SISSI, and the newly started SPIRALfacility for post-acceleration of ISOL beams. The experimental areas are alsoshown, with the separator LISE3 on the left side of the beam corridor.

The illustration in fig. A.4 shows the NSCL complex, situated at the MSU campus.The two cyclotrons (K=500 and K=1200) are in the upgraded facility coupled, andthe separator A1200 is here upgraded to the A1900.

In fig. A.5, the A1200 spectrometer is drawn as it was at the time of theexperiment, together with the cyclotron providing the primary beam. The RPMSis placed after the final focal plane, as is seen the NSCL layout (fig. A.4).

The complete GSI facility is pictured in fig. A.6. To the left is the ion source area,UNILAC and the low energy research areas. The UNILAC beam is injected intothe SIS from which it can be extracted to the FRS and through to the ESR or theexperimental caves in the bottom of the figure.

131

α

ECR

ECR

SISSI

C01 CSS1 CSS2 SPIRALI

NAUTILUS

ORION

SPEG

G4

C02SME

SIRa

LISE3

INDRA

D6

Figure A.3: The accelerator facility GANIL with experimental areas.

132 Appendices

Figure A.4: The accelerator facility NSCL after the upgrade.

K1200 cyclotron

Target pot

Finalachromaticplane

Intermediateachromatic plane

Figure A.5: Layout of the A1200 spectrometer at NSCL.

133

PENNING,CHORDIS andMEVVAion sources

UNILAC

ECR ion sourceHLI

The complete GSIaccelerator complex

Low energyexperimental area:ISOL, SHIP

FRS

FromUNILAC

SIS

Final FRSfocusBeam dump

Profileddegrader

Productiontarget

ESR

KAOS

4π ALADIN

LAND

Medical

Therapy

Project

Figure A.6: The accelerator facility GSI. The inset shows the whole complex, while the largerpicture is zoomed in on the high-energy area after the SIS.

135

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