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dissertation entitled

The Halo Nuclei 1IBe and 83 Studied by

Fragmentation Reactions

presented by

John Henry Kelley

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WMA

THE HALO NUCLEI 11Be AND 8B STUDIED BY
FRAGMENTATION REACTIONS

by

JOHN HENRY KELLEY
A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1995

ABSTRACT

THE HALO NUCLEI llBe AND 8B STUDIED BY
FRAGMENTATION REACTIONS

By

John Henry Kelley

The discovery of anomalously large matter radii in some weakly bound nuclei
on the neutron dripline has led to measurements of breakup fragment momentum
distributions aimed at obtaining a qualitative understanding of these nuclei. The
valence nucleons in halo nuclei penetrate the low barrier of the core potential, and
form a diffuse layer around a normal sized core. Momentum distributions of core
fragments following a direct breakup are related to the spatial distribution of the
halo nucleon, via Fourier Transform, and offer a straightforward method to study
halo nuclei. The momentum distribution in the direction parallel with the beam
direction (Pll) is less affected by reaction mechanism effects than the distribution
perpendicular to the beam direction (p i), and is expected to most accurately reﬂect

the halo neutron momentum wavefunction.

The nucleus, llBe, has a one-neutron halo and core fragment momentum distribu-
tions should permit a simple determination of halo characteristics. We measured the
p" distribution of 10Be core fragments in Be, Nb, Ta and U targets and found that
the p” distributions on all targets are in excellent agreement with a projection (onto
the p” axis) of the momentum wavefunction of a 231/2 neutron bound by 500 keV in a

Woods-Saxon potential. This is taken to indicate that reaction mechanism effects do

not signiﬁcantly inﬂuence the 10Be core fragment p” distributions. The corresponding

root-mean-square radius of the halo neutron is 6.5 fm.

Finally, there is a controversy concerning the existence of a proton halo in 8B. We
approached the issue assuming that the p" distributions of 7Be breakup fragments
would reﬂect the spatial distribution of the valence proton, as we had shown for 11Be.
Both p" and p l were measured. The p” distributions are narrow, but are only about
half of the width of a prediction for a proton bound by 140 keV in a Woods—Saxon
potential with a rms radius of 4.24 fm. In this case, it appears that for the smaller halo
of the p-orbital proton the breakup momenta are inﬂuenced by both the nuclear and
Coulomb reaction mechanism effects. When these effects are included, the predictions

agree with the data.

To mom and dad

iv

ACKNOWLEDGEMENTS

I must begin by thanking Sam Austin for his guidance during the past few years.
He has encouraged me to develop my skills as a researcher at all levels, including writ-

ing proposals, planning experiments, data analysis, and peer reviewed publication.

I am also grateful to Michael Thoennessen, Alex Brown, Horace Smith, and Suben-

dra Mahanti for their participation on my guidance committee.

During my stay at the NSCL I have had the opportunity to work with a number of
collaborators to whom I must express my gratitude. Thank you John Winﬁeld, Nigel
Orr, Dave Morrissey, Brad Sherrill, Walter Benenson, Daniel Bazin, Jeff Winger,
Toshiyuki Kubo, Bob Warner, Dave Mercer, Stephan Hirzebruch, Tiina Suomijarvi,
Bob Kryger, Maggy Hellstrom, and Jim Brown. I have also had the privilege of
working closely with Easwar Ramakrishnan, Brian Young, Raman Pfaff, Mathias
Steiner, Michael Fauerbach, Shigeru Yokoyama, Afshin Azhari, Thomas Bauman,

Ken Lamkin, and many others.

On a more personal note, thank you Easwar for putting up with me; thank you
Mike for having an open door (and a clean window); thank you Dave for showing up
in the Data-U at 7:30 AM on most mornings; and thank you Tiina for a chance to

continue my research career.

Contents

LIST OF TABLES ix
LIST OF FIGURES xi
1 Introduction 1
2 Projectile-like ﬁagment Momentum Distributions 7
2.1 Momentum Distributions ......................... 7
2.2 Fragment Transverse Momentum Distributions from 11Li Breakup . . 8
2.2.1 9Li Core P 1 Distributions .................... 8
2.2.2 Neutron Angular Distributions .................. 10
2.3 Breakup Reaction Mechanisms ...................... 10
2.3.1 Stripping ............................. 12
2.3.2 Diffraction Dissociation ...................... 12
2.3.3 Coulomb Dissociation ....................... 14
2.3.4 Reaction Mechanism Effects on Momentum Distributions . . . 15
2.4 The P” Distribution of 9Li Fragments from 11Li Breakup ....... 16
2.4.1 Problems with the 9Li P" Distributions ............. 17
2.5 Need to Study a Simpler System ..................... 18
3 The Neutron Halo of 11Be 20
3.1 The P" Distribution of 10Be Fragments from 11Be breakup ...... 21
3.1.1 The A1200 Fragment Separator ................. 22
3.1.2 Analysis of 10Be Fragment Momentum Distributions ...... 28
3.1.3 Discussion of the 10Be P” Distributions ............. 37

3.2 Comparison with the Transverse Momentum Distributions of Frag-
ments from 11Be Breakup ........................ 45
3.2.1 P J. Distribution of Core Fragments ............... 45
3.2.2 Angular Distribution of Neutrons ................ 46

vi

3.3 Neutron Halo Radius ........................... 48

4 Search for a Proton Halo in 8B 49
4.1 Experimental Measurements of the Nuclear Size of 8B ......... 49

4.2 The P“ and P 1 Momentum Distributions of 7Be Fragments Following

the Breakup of 8B ............................. 52

4.2.1 Experimental Details ....................... 53

4.3 Analysis of the Data ........................... 55
4.3.1 The P1, Distributions ....................... 57

4.3.2 The P" Distributions ....................... 59

4.4 Results ................................... 61
4.4.1 The P” Distribution from Nuclear Breakup Processes ..... 62

4.4.2 The P” Distribution From Coulomb Breakup Processes . . . . 64

4.4.3 Angular Dependence of the Width of the P" Distribution . . . 66

4.5 Valence Proton Radius .......................... 67

5 Summary and Outlook 69
A Theoretical Discussion of Halo Nuclei 71
A.1 Importance of the Binding Energy .................... 71
A.2 Importance of the Coulomb Potential Barrier .............. 73

B Experimental Evidence for Halo Nuclei 75
B.1 Enhanced E1 Transition Strength .................... 75
B.2 Measurements of Reaction Cross Sections ................ 75
B.3 Energy Dependence of the 11Be Reaction Cross Section ........ 77
B.4 Large Electromagnetic Dissociation Cross Section ........... 79
B.5 The 11Li Dipole Excitation Strength Function ............. 80
B6 The 11Be Dipole Excitation Strength Function ............. 82

C Models for Fi‘agment Momentum Distributions 84
CI The Goldhaber Model .......................... 84
C2 The Serber Model ............................. 85
C3 The Friedman Model ........................... 85
C4 The Extended Serber Model ....................... 86

D Coulomb Dissociation 87

vii

E The Inﬂuence of a Limited Acceptance on P" Distributions
E.1 Nature of the Problem ............... . ...........
E.2 Results ...................................
E.2.1 Lorentzian Distribution ......................
E.2.2 Dependence of Width on Angle .................
E.2.3 Complex Momentum Distributions ...............

E.3 Summary .................................

F Coulomb Reacceleration on a Heavy Target

F.1 Coulomb Reacceleration in llLi Breakup Fragments ..........
G Data Tables for 10Be Fragment Momentum Distributions
H Data Tables for 7Be Fragment Momentum Distributions

LIST OF REFERENCES

viii

91
92
93
95
100

102
102

104

110

120

List of Tables

1.1

3.1
3.2

3.3

3.4

4.1

4.2
4.3

4.4

E.1

G.1
G.2
G3
G4

H.1

H.2

The rms radius of valence nucleons compared with their binding energy 4

The nuclei that comprised the RNB in the 11Be breakup experiment. 24

Target thicknesses and momentum resolutions measured for targets in
the 11Be experiment ........................... 36

Widths of parallel momentum distributions ’of 10Be fragments following
the breakup of 11Be on variousrtargets ................. 37

The expected mean momentum and widths of 10Be fragment momen-
tum distributions from fragmentation of contaminant nuclei in the RN B 39

Description of detectors that comprise the two telescopes in the 8B

breakup experiment. ........................... 54
Target thicknesses in the 8B breakup experiment. ........... 55
Widths of the px distributions of 7Be fragments following the breakup

of 8B .................................... 59
Widths of the p” distributions of7 Be fragments following the breakup

of 8B. ................................... 61
Widths of the p" distribution from a Lorentzian momentum distribu-

tion that passes through various aperture sizes ............. 93
The data for the 10Be fragment p" distribution on the Be target. . . . 105
The data for the 10Be fragment p" distribution on the Nb target. . . . 106
The data for the 10Be fragment p” distribution on the Ta target. . . . 107
The data for the 10Be fragment p” distribution on the U target. . . . 109

The data for the 7Be fragment momentum distribution on the Be target
measured in Telescope 1 .......................... 111

The data for the 7Be fragment momentum distribution on the Be target
measured in Telescope 2 .......................... 112

ix

H.3

H.4

H.5

H6

H7

H8

H9

The data for the 7'Be pI fragment momentum distribution on the Be
target. ...................................

The data for the 7Be fragment momentum distribution on the Nb target
measured in Telescope 1 ..........................

The data for the 7Be fragment momentum distribution on the Nb target
measured in Telescope 2 ..........................

The data for the 7Be p1. fragment momentum distribution on the Nb
target. ...................................

The data for the 7Be fragment momentum distribution on the Au target
measured in Telescope 1 ..........................

The data for the 7Be fragment momentum distribution on the Au target
measured in Telescope 2 ..........................

The data for the 7Be p3 fragment momentum distribution on the Au
target. ...................................

113

114

115

116

117

118

119

List of Figures

1.1
1.2

2.1

2.2
2.3
2.4

2.5

3.1

3.2
3.3
3.4
3.5

3.6
3.7
3.8

3.9
3.10

3.11

The rms radii of light neutron rich isotopes ...............

A Woods-Saxon calculation of the proton and neutron density distri-
butions in llLi. ..............................

The transverse momentum distribution of core fragments following the

breakup of 8He and 11Li. .........................
The angular distribution of neutrons following the breakup of 11Li. . .
Schematic illustrating the Coulomb excitation process .........

The p“ distribution of 9Li fragments following the breakup of 11Li . .

The p" distribution widths of 9Li fragments from 11Li breakup . . . .

The nucleus 11Be is comprised of a normal sized 10Be core (2.30 fm)
immersed in a neutron halo that has a rms radius of 6.5 fm. .....

The A1200 fragment separator and beam analysing device ......
A E vs. ToF identifying llBe nuclei ...................
The principle of a dispersion matched energy loss spectrometer . . . .

Reaction products that reach the focal plane of the A1200 when the
Be target is inserted at the Image 2 target position ..........

Conditions that are applied to the position histograms ........
Condition that is applied to the total energy vs. ToF histogram . . .

The condition that is applied to the Time of Flight vs. X focal plane
position histogram ............................

The momentum calibration of the A1200 focal plane ..........

The transmission efﬁciency of the second half of the A1200 vs. position
at the focal plane .............................

The p" distributions of 10Be fragments following the breakup of l1Be .

11
13
16

17

20
22
23
26

29
30
31

31
33

34
35

3.12 The p” distribution of 10Be fragments following breakup in the Ta target 38

3.13

The widths of the 10Be fragment momentum distributions ......

xi

40

3.14

3.15

3.16

3.17

3.18
3.19

4.1

4.2
4.3

4.4
4.5
4.6

4.7

4.8

4.9

4.10

4.11

4.12

A.1

A.2

B.1

The p" distributions of 10Be fragments compared with the a projection
of the wavefunction of a 251/2 neutron ..................

Comparison of the width of the p" distribution of the 231/2 neutron

with the p” distribution expected from dipole excitation of 11Be . . .

The 251/2 intruder state dominates the valence neutron wavefunction.

The transverse momentum distribution of 10Be cores following the

breakup of 11Be ..............................
The angular distribution of neutrons following the beakup of 11Be. . .

The probability distribution of nucleons in 11Be .............

The difference between the rms radii of the proton and neutron density
distributions in the boron isotopes. ...................

The breakup cross section of 8B ......................

A schematic illustration showing the overall layout of experimental
apparatus used in the 8B breakup experiment ..............

AE vs E showing the particle identification of 7Be fragments from 8B.

The pa. distributions of 7Be fragments following the breakup of 8B. . .
The p" distributions of 7Be fragments following 8B breakup. .....

Data from 8B breakup in the Be target compared with the predicted p”
distribution of a p-orbital proton bound by 137 keV in a Woods-Saxon
potential. .................................

The p” distribution of 7Be fragments following breakup in the Be target
predicted by a model that considers the m; sub-states of the p-orbital
proton in 8B. ...............................

The predicted shapes of the momentum distributions from E1, E2 and
E1+E2 Coulomb breakup of 8B. ....................

The x2 minimization to obtain the E2 strength that best reproduces
the observed 7Be p“ distribution from 8B breakup on the Au target.

The 7Be fragment p" data compared with the predicted Coulomb breakup

distribution obtained when x2 is minimized. ..............

The probability distribution of nucleons in 8B ..............

The dependence of the rms radius of valence neutrons'on binding en-
ergy. ....................................

The dependence of the rms radius of a valence proton on binding energy
for a nucleus that has four protons in the core, for example 8B. . . . .

The difference between the proton and neutron density rms radii in
light neutron rich nuclei .........................

xii

42

43
44

45
46
48

50
51

54
56
58
60

62

63

65

65

66
68

77

3.2
B3
B4

E.1

E.2

E.3

E.4
B.5

RI

The energy dependence of the 11Be reaction cross section ........
The decay energy spectrum of 11Li from Coulomb Excitation .....

Coulomb reacceleration observed in 11Be breakup on a heavy target .

The expected p“ distributions from a Lorentzian for various transverse
acceptance limitations ..........................

The expected p” distribution from Lorentzian and Gaussian distribu-

tions compared with data from the breakup of 11Li in the S320.

The transmission of the A1200 spectrograph for a Gaussian distribution
as a function of its standard deviation. .................

The three component Gaussian ﬁt to a Lorentzian distribution . . . .

The unﬁltered and acceptence filtered p” distribution for a 131/; orbital
neutron bound by 500 keV in a Woods-Saxon potential .........

A schematic representation of the Coulomb breakup of 11Li. .....

xiii

78
81
83

100

103

Chapter 1

Introduction

Until recently, experience with stable nuclei was the only way of predicting the neutron
and proton driplines and the nuclear properties of heavy elements that can exist
only in stellar interiors. However, beams of short lived nuclei at radioactive isotope
facilities have recently permitted experiments that probe the structure of dripline
nuclei. These studies permit a determination of the characteristics of these nuclei
that will reﬁne the structure models and improve their ability to predict properties
of heavier nuclei that are and will remain out of reach, even using state-of-the-art

methods for producing unstable nuclei.

Recently, these studies have uncovered a new class of nuclei, halo nuclei. Using
reaction cross sections the nuclear size of light neutron-rich isotopes was determined

[Tan85a, Tan85b, Tan88] using a Glauber model approach,
01 = 7r[R1(target) + R1(projectile)]2. (1.1)

The neutron-rich nuclei 6He, 8He, 11Be, 14Be, and 11Li exhibited anomalously large
matter radii, Figure 1.1, that do not follow the simple 7‘ = “Al/3 rule. These results
have subsequently been explained assuming that the valence neutrons form a diffuse

halo that surrounds a normal sized core.

The explanation for the two-neutron halo of 11Li, given by Hansen and Jonson

1

 

 

 

 

 

35 —r r V T l I F ‘F f r T T T T i T I r I rm 1 I r - 1
’ ' He
_ . I ‘
1': " 9 L1 1
g 3.0 —- __
CU " I Be
E + l
g + x B i ..
m“ 2.5 —- § i n _
L g I f
f I i
h -1
2.0 1 l 1 L i. 1 1 1 L 1 l 1 L 1 l l i 1 1 l l 1 l l I 1 _L
2 4 6 a 10

 

Neutron Number

Figure 1.1: The rms radii obtained from the 790A MeV cross section measurements.
The nuclei 6He, 8He, 11Be, 1“Be and 11Li are noticeably larger than other nuclei in
this region. Data from [Tan88].

[Han87], was based on a simple assumption that the 9Li core provides a square-well
potential that binds a di-neutron (two neutrons that are strongly correlated). In
this model a very low binding energy (295i26 keV [You93]) permits the two weakly-
bound valence nucleons to quantum-tunnel through the wall of the core potential
barrier. Assuming no angular momentum barrier the wavefunction outside the core

is a Yukawa, .
exp(-r/p)
2 ———- 1.2
. we) o< \/ «p r . < >
and falls off exponentially with a decay length p,

h
p — [2" g2“ ,

where p is the reduced mass of the system and $2,, is the separation energy for the

 

(1.3)

valence neutrons.

The di-neutron is unbound when free in nature. However, it was hoped that

the presence of the 9Li core would permit a strong correlation of the two neutrons.
Although studies of the correlation of the neutrons emitted in breakup reactions
[Iek93] are not consistent with the di-neutron model for 11Li, the simple interpretation
of the core-halo neutron system serves as a general guide for understanding halo
nuclei. A more detailed discussion of potential barriers is included in Appendix A. A
realistic prediction of the nuclear densities in llLi, Figure 1.2, based on wavefunctions

in Woods-Saxon potentials, indicates a sizeable tail in the density distribution.

 

I T I T T f f I I I I I T I I I F I I I

11Li Nucleon Density .

 

\ neutrons
10 _ 2 — \ _

protons - - - -

Density
‘3.
l

10‘6 -

 

 

 

 

Radius

Figure 1.2: [She94] A Woods-Saxon calculation of the proton and neutron density
distributions in 11Li.

The rms radius of halo particles can be estimated from a two-body (core-halo)

treatment for halo nuclei [Han87].

Moore + mhalo Moore + mhalo

< r2 > — < r2
mhalo Mcore

>]. (1.4)

 

2 _
< 1‘hazo >-

In Table 1.1 the rms radii of the valence nucleons for a variety of nuclei are calculated.

Table 1.1: The rms radius of valence nucleons (Eq. 1.4) compared with the binding
energy. The root-mean-square radii of the core and halo nuclei are tahen fron the
790A Mev interaction cross section data [Tan85a, Tan85b, Tan88].

 

 

 

Nucleus Valence N ucleons Binding Energy RMS radius
(keV) fm
11Li 2n 295 5.991052
llBe 1n 504 5.66:1:020
6He 2n 970 4.50:1:003
8H6 2n 2137 3.04:}:003
9Li In 4060 1.99:1:0.02
10Be In 6110 1.46:}:0.19
8B 1p 137 3.02:1:0.16

 

 

 

 

 

The values are realistic for nuclei like 11Li and 11Be. However, for tightly bound
nuclei, such as 9Li and 10Be, the two-body assumption leads to unrealistic values; for
example, the rms radius of the valence nucleon is smaller than the rms radius of the

core nucleus.

The simple two- and three-body nature of halo nuclei permits a treatment where
the removal of halo nucleons probes the wavefunction of the halo nucleons. Ap-
pendix B summarizes experiments that show the presence of neutron halos in 11Li

and 11Be, for the interested reader.

In the Serber limit, the momentum distributions of core and halo fragments fol-
lowing breakup are related to the Fourier Transform of the spatial wavefunction of the
halo nucleon [Ser47]. A narrow momentum distribution implies a broad spatial distri-
bution. Reactions that strip off halo nucleons, leaving the core intact, are particularly
interesting since they sample the wavefunction of the halo nucleon primarily when
the core-halo separation distance is large. Hence studies of core fragments should be
sensitive to the details of the halo wavefunction in the region of interest where the

halo extends beyond the core.

The three-body nature and low binding energy of 11Li pushed studies of this nu-
cleus to the forefront in radioactive beam experiments. However, the interpretation
of fragment momentum distributions from llLi was complicated by a combination of
reaction mechanism and ﬁnal state interaction effects. For example, the Serber model
assumes a direct breakup which implies that the two neutrons are emitted simulta-
neously. In this case, measurements probe the ground state of 11Li. However, some
breakup events proceed by a sequential emission of the two neutrons and are sensitive
to the properties of the unbound nucleus 10Li. Because the relative contributions of
simultaneous and sequential breakups are not well known, uncertainties exist in the

interpretation of results.

To resolve issues of the inﬂuence of reaction mechanism effects we measured the
momentum distribution of 10Be core fragments following the breakup of a simple
two-body system, 11Be. In Chapter 2 issues relevant to measurements of momen-
tum distributions are discussed, and in Chapter 3 the distribution of 10Be fragment
momentum parallel to the beam direction (pH) following 11Be breakup is discussed.
The advantages of this measurement are two-fold. The p" should directly reﬂect the
momentum wavefunction of the halo neutron [Ber92], and the method utilized a high
resolution spectrometer technique to measure the momentum distribution. We ob-
served narrow distributions for a wide range of breakup targets (from Be to U) and
found excellent agreement when the data was compared with a single particle calcula-
tion for a 231/2 neutron in a Woods-Saxon potential which has a rms radius of 6.5 fm
[Esb94]. This appears to indicate that the halo neutron momentum wavefunction is

measured.

In Chapter 4 measurements of the momentum distribution of 7Be fragments fol-
lowing the breakup of 8B are presented. A recent measurement of the 8B quadrupole

moment suggests that it possesses a prominent proton halo [Min92]. However, the

earlier reaction cross section measurements had not found evidence for a proton halo
[Tan88]. Therefore, the momentum distributions of breakup 7Be core fragments pro-
vide a measurement of the proton rms radius from a perspective that is different
from previous measurements. The distributions are narrow; however, they are not in
agreement with a simple prediction based on the wavefunction of a p-orbital proton
in a Woods-Saxon potential [Esb95]. In this case, it is necessary to include reaction
mechanism effects; this leads to good agreement with the data. The wavefunction has
a rms radius of 4.24 fm and is signiﬁcantly larger than the 7Be [Esb95] core. However,

the size of the proton halo of 8B is small when compared with the neutron halo of

"Be

Chapter 2

Pro jectile—like Hagment
Momentum Distributions

Momentum distributions of projectile-like fragments (fragments with velocities simi-
lar to that of the incoming projectile) have been measured for a wide range of stable
projectile nuclei, i.e. 12C, 14N, and 16O [Kid88, Gla90, Van79]. The momentum dis-
tribution widths are generally broad, on the order of 180 MeV/c and larger. Core
and halo fragments following the breakup of halo nuclei have narrow momentum dis-
tribution widths (~45 MeV/c), in reasonable agreement with the Fourier transform
of simple predictions for the halo neutron wavefunctions based on the low binding
energies. It appears that the narrow momentum distributions reﬂect, via the uncer-
tainty principle, the large spatial distribution of valence nucleons. Thus momentum

distributions provide convincing evidence for neutron halos.

2.1 Momentum Distributions

When a projectile interacts with a target nucleus it can be broken apart or fragmented.
The breakup fragments can be described in terms of two momentum components, one
parallel with the beam velocity (PM) and the other perpendicular to the beam velocity

(p l). Experience with stable nuclei has led to the following general observations.

When Coulomb deﬂection contributions are taken into account the p“ distributions
of fragments have widths in relatively good agreement with the p 1 distributions
[Van79]. The p” distribution widths observed for light targets are generally the same
as the p” distribution widths for breakup on heavy targets [Gla90]. A brief summary

of fragmentation models is in Appendix C.

The nucleus 11Li is interesting because of its three-body, n-n-9Li, structure. How-
ever, a review of measurements of the fragment momentum distributions from llLi

breakup, in the following sections, shows the need to study a simpler two-body system.

2.2 Fragment Transverse Momentum Distributions
from 11Li Breakup

2.2.1 9Li Core P 1 Distributions .

A narrow width was observed in the transverse momentum distribution of 9Li frag-
ments from llLi breakup showing, via Fourier transform, the presence of the halo
[Kob88]. The distribution was initially ﬁt with a two-component Gaussian shape,
Figure 2.1. The ﬁrst component was broad, 2242f:28 MeV/c Full Width at Half Maxi-
mum, while the other component was much narrower, 54:1:28 MeV/c F WHM [Kob88].
The broad component, which has a width similar to that expected from fragmenta-
tion of normal nuclei, was attributed to the removal of tightly bound neutrons from
the normal sized 9Li core of 11Li, while the narrow component is associated with the

removal of the weakly bound valence neutrons from the halo.

However, there is a problem with this interpretation since the relative contri-
butions from each of the components (broad 70%, narrow 30%) is different from
what is expected from a halo nucleus; the component associated with the removal of

halo neutrons should dominate the cross section. A Gaussian (or sum of Gaussians)

 

(J'
O
I

da/dpl [am]

 

 

 

 

da/dpj [07b]

 

 

 

 

200

PJL [MeV/cl

Figure 2.1: [Kob88] The transverse momentum distribution of core fragments follow-
ing the breakup of 790A MeV ions in a carbon target. In (a) is a relatively broad
distribution, 188 MeV/c F WHM, observed for 6He fragments from 8He. The p l dis-
tribution of 9Li fragments following the breakup of 11Li is shown in (b) and is ﬁt with
two Gaussian components, 54 MeV/c FWHM and 224 MeV/c FWHM.

10

momentum distribution is not expected on any a priori grounds; therefore, further
consideration of the shape of the fragment momentum distributions from halo nuclei

was necessary.

As has been shown earlier, Hansen and Jonson [Han87] suggest that the wave-
function of halo nucleons outside the core will decrease exponentially with the shape

associated with a Yukawa wavefunction, Equation 1.2. Since the Fourier transform

WP) = gm (2.1)

r = 2;/2,152,,, (2.2)

a Lorentzian description of the core and halo fragment momentum distributions is a

of a Yukawa is a Lorentzian,

and

reasonable ﬁrst approximation.

2.2.2 Neutron Angular Distributions

The angular distribution of neutrons from the breakup of 11Li, Figure 2.2, indicated a
width of 24i4 MeV/c FWHM [Ann90]. This was much narrower than any previously
measured fragment momentum distribution width and less than half the width of the
p 1 distribution of the 9Li cores. The narrow width is attributed both to the presence
of the halo and to the fact that the neutrons are uncorrelated. This raised issues of

the effects of the reaction mechanism and ﬁnal state interaction effects.

2.3 Breakup Reaction Mechanisms

Because there are concerns that perturbations associated with the reaction mecha-
nisms inﬂuence the distributions, a discussion of the most signiﬁcant breakup pro—

cesses (stripping, diffraction dissociation and Coulomb dissociation) is necessary.

11

 

.L J

“L'H-‘Z -> °Li +n4—X

   

100 -

 

don/d0 [barn/sr]
8

 

 

0-10 5 1'0 1‘5 27) 25
Neutron angle [deg]

Figure 2.2: [Ann90] The angular distribution of neutrons following the breakup of 11Li

are compared with Lorentzian shapes. The deduced widths are small, 24i4 MeV/c
FWHM.

12

2.3.1 Stripping

In 1947 Serber interpreted the momentum distribution of neutrons produced when a
190 MeV beam of deuterons impinged on a target [Ser47]. A direct breakup process,
stripping, was assumed. Stripping removes nucleons from the projectile when they
come into contact with the target nucleus. The process can also be termed absorption
since nucleons are absorbed or removed from the projectile by the target. This model
assumes that the halo particle can be removed by an interaction sufficiently weak
that the core momentum is unaffected by the breakup. It is also necessary that
the core fragment momentum is not greatly affected by ﬁnal state interactions while
leaving the scene of the breakup. When halo nucleons are stripped away from the
projectile the remaining core nucleons continue in their original motion. Therefore,
the trajectory of the fragment is determined by the collective motion of the surviving
nucleons at the instant of the breakup. In the case of a halo-core system the simple
two-body nature implies, by conservation of momentum, that the core momentum

reﬂects the momentum wavefunction of the halo nucleon.

2.3.2 Diffraction Dissociation

Another mechanism that induces breakup reactions is diffraction dissociation [Gla55].
Glauber explained that in diffraction dissociation the wavefunctions of a nucleons in a
nucleus are localized as it approaches a target nucleus (assumed to to be a black disk).
Portions of the wavefunctions become restricted, and the reduction in the volume of
the wavefunctions increases the total energy of the system so that the nucleus is no
longer in the ground state. The resulting excited nucleus is comprised of an admixture

of excited states which decays.

13

(a)

 

 

 

 

 

 

 

Figure 2.3: [Ber88] As a projectile approaches a target nucleus, the electric ﬁeld lines
are contracted into the plane which is perpendicular to the direction of travel. The
resulting electric and magnetic ﬁeld lines, that can be approximated with two plane
waves, Coulomb excite the passing nucleus.

14

2.3.3. Coulomb Dissociation

The ﬁnal breakup mechanism to be considered is Coulomb dissociation. The very
low binding energy of halo nuclei leads to a large Coulomb dissociation cross sections,
and the breakup mechanisms for halo nuclei are dominated by Coulomb processes
for heavy targets [Bla91]. Coulomb excitation processes are discussed, for example
in Winther and Alder [Win79] and the Electrodynamics text of J .D. Jackson [Jac75];
a straightforward development by Bertulani and Baur [Ber88] is summarized in Ap-

pendix D.

Coulomb excitation occurs when a projectile enters the Coulomb ﬁeld of a target
nucleus. The electric ﬁeld of the target nucleus is contracted into the plane which
is transverse to the incident direction of travel (chosen as the Z axis in Figure 2.3).
The resulting electric and magnetic ﬁelds are associated with a spectrum of virtual
photons which excite the projectile. With these considerations, the probability for
excitation via dipole (E1) photons can be calculated, and the momentum distribu-
tion of fragments from Coulomb dissociation reﬂects the dipole excitation strength

function.

Esbensen and Bertsch [Esb92] ﬁnd that the dipole operator does not change the
momentum of the breakup fragments much, and thus in halo nuclei the ground state
halo particle momentum distribution should be similar to the momentum distribu-
tion of the dipole excitation strength. Therefore, although the reaction mechanism
changes when considering breakup reactions on light and heavy targets the momen-
tum distributions remain similar and primarily reﬂect the wavefunction of the halo

nucleons.

Dipole excitation is the primary contributor to breakup at intermediate energies.

Because excitation by E2 photons is generally much smaller than that of E1 excitation,

15

the E2 contribution is often neglected.

The predicted Coulomb breakup energy distribution in the projectile rest frame is
obtained using this method. Further consideration of experimental conditions leads
to the p” and p _L distributions of breakup fragments. When the target Z is sufficiently
large, the Coulomb breakup cross section is much larger than the nuclear breakup
cross section in halo nuclei, and there is little need to consider inﬂuences from nuclear

processes.

2.3.4 Reaction Mechanism Eﬂ'ects on Momentum Distribu-
tions

Reaction effects inﬂuence measurements of neutrons and core fragments and measure-
ments of the parallel and transverse components of fragment momentum in different
ways. Absorption strongly affects neutrons since it is possible for the neutron to col-
lide with a target nucleus and then scatter with a momentum that no longer reﬂects
the initial state wavefunction. When a core fragment strikes a target nucleus it will
likely be disintegrated as a result. Therefore, the momentum distribution of core
fragments is less affected by absorptive effects than the momentum distribution of
neutrons. In an extreme limit, absorption could lead the p J_ distribution to become
a Fraunhofer diffraction pattern, while the effect on the p” component narrows the

distribution slightly [Ber92].

Diffraction inﬂuences the neutron and core fragments equally. However, since
diffraction localizes the wavefunctions in the transverse direction, and not in the

longitudinal direction, the p” distributions are only weakly perturbed [Bar95].

Finally, the initial and ﬁnal state interactions associated with high Z targets are
mainly from Coulomb deﬂection and multiple elastic scattering effects. These act in

the direction perpendicular to the beam direction.

16

2.4 The P” Distribution of 9Li Fragments from 11Li
Breakup

Bertulani and McVoy showed that p” of core fragments measures the internal mo-
mentum of the halo particle and is relatively insensitive, to the details of the in-
teraction for both nuclear and Coulomb processes [Ber92]. A measurement of 9Li
fragments following the breakup of 11Li fragments, at the NSCL, found that the p”
distributions for Be, Nb and Ta [Orr92] and U [Orr95] targets are similar in width
(around 45 MeV/c FWHM). The data agree with a prediction based on the momen-
tum wavefunction of the two halo neutrons, Figure 2.4. Therefore, it appeared that

p” distributions accurately reﬂect momentum wavefunctions.

 

 

§

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P- 1 I I r I v r 1 I ' I v v r r 1 V Y r r I f r v r I r T.T T r1
no} a' - 20.9 (0.0) MeV/c (a) j ”i a - 10.7 (0.0) MeV/c (c) i
I 1 : 2
r- l- ..
too- I d ”.7 1
a : 1 = s
r- d «I- —
r: . , : :
C: “P 1 : :
:1 C ”r 1
. t E :
lo oh: I 1 LI 1 Pl 1 1 1 l o'rl’n’L LL14 l_l 1 LI‘Y‘I
3- am an an also :10. 31.0 as an.
3 I. t t T r I v r I r I r I I I I ‘ j” v Y 7 v
i- (b) 1
C‘. : a = 21.2 (0.7) MeV/c :
a 1up _: 100:-
3 E l .3
b 1m:- 4 .
N l- " 30:—
‘U - j' :
”F- d :
I I ”7
r i
a 1 A L A L 1 l A L l A 1 1 1 l ‘ O i . r .
81“ 3150 m 3’. 31°“
- p (MeV/c)

Figure 2.4: [Orr92] and [Orr95] The p“ distribution of 9Li fragments following the

breakup of 66A MeV 11Li in 9Be, 93Nb, 181Ta and 238U targets, (a) through ((1)
respectively. In (c) the dashed line is a dipole excitation calculation, Section 2.3.3,
that is in excellent agreement with the data.

17

2.4.1 Problems with the 9Li P“ Distributions

Data from the 9Li p” distributions following the breakup of 11Li [Orr95] have provided
room for discussion of how well we understand the reactions. The p” distribution from
breakup on the Ta target is in agreement with the calculated momentum wavefunction
of the halo neutrons and supports the Serber model approach. However, the widths
of the distributions from breakup on other targets decrease slightly with increasing

target mass.

 

 

 

 

ao-ITIITITTITIIIITFITITIIITf7o
E I High Acceptance E
A 25 . 1 30
S : I Medium Acceptence : g
o) C 3 z
2 20 P 1 q 50 A
V l- } : g
b 7 “
: if; g : 4° i
15 : : 8/
E : 30
l- d
10 L 1 i L l l l l 1 l 1 LI 1 _1_ 1 l L I 1 1 LL 1
O 20 40 60 80 100

Target 2

Figure 2.5: [Orr95] The p” distribution widths of 9Li fragments from 11Li breakup
have a slight target Z dependence.

Three-body Effects

There are complications in 11Li breakup due to reaction mechanism effects or, possibly,
three-body effects. For example, breakups on the light target may proceed mainly
via sequential emission of the neutrons while breakups on the heavier target would be

expected to proceed by simultaneous emission of the two neutrons [Bar93]. This would

18

lead to different momentum distribution widths for light and heavy targets. Accurate
predictions require a precise knowledge of the structure of the nuclei that participate
in the breakup. However, existing uncertainties in experimental measurements, which
are quite large in some cases, limit our understanding in complicated three-body

systems.

Essential issues in the problem, for 11Li, are uncertainties in the ground state prop-
erties of 11Li and 10Li. Measurements of the 11Li binding energy were not in agreement
( ranging from 170:}:80 keV [Thi75] to 340:1:50 keV [Kob92] ), and the accepted value
of 295i26 keV [You93], which is a weighted result based on all measurements to date,

has a sizeable uncertainty.

The ground state properties of 10Li are critical in sequential breakups. However,
since 10Li is unbound it is difficult to study. Young et al. [You94] suggested that
the low-lying structure is made up of a p-orbital state at 538 keV and a s-orbital
state in the range of S 100 keV. This just unbound s-orbital state was corroborated
by a 0° sequential neutron decay spectroscopy measurement [Kry93]. This indicates
that s-orbitals comprise a signiﬁcant percentage of the wavefunction for the valence
neutrons of 11Li. Because the angular momentum barrier’is critical to the long range
behavior of the wavefunction, this discovery signiﬁcantly inﬂuenced the interpretation

of some measurements.

2.5 Need to Study a Simpler System

These results for 11Li have led to a cross-road in the study of two-neutron halo nu-
clei. Evidence of the halo is observed. However the sensitivity to reaction mechanism
effects, initial and-ﬁnal state interactions, and uncertainties in the ground state prop-

erties of the participating nuclei leads to an unacceptable level of uncertainty in the

19

results. The interplay between halo effects and reaction effects mandates studies on
simpler two-body (one nucleon halo) systems to provide a basis for complete under-

standing of the more complicated three-body systems.

Chapter 3

The Neutron Halo of 11Be

 

Figure 3.1: The nucleus 11Be is comprised of a normal sized 10Be core (2.30 fm)
immersed in a neutron halo that has a rms radius of 6.5 fm.

In the early 19608 11Be attracted attention because the parity of its ground state
is inverted when compared to the simple shell model expectation [Sag93]. Therefore
many of the important nuclear model parameters have been measured, and 11Be is well
characterized. The parity inversion leads the ground state to be primarily determined

by a single particle 281/2 neutron state instead of the 1p3/2 state. Although 12Be and

20

21

1“Be are more neutron rich than 11Be, they are more tightly bound because of pairing
effects. Therefore, the rms radii “of their valence neutrons are not as large as the rms
radius of the valence neutron of 11Be. Since 11Be breakup is not troubled by three-
body issues it should provide a less complicated determination of halo characteristics

than measurements on llLi.

The binding energy of the valence neutron in 11Be has been measured precisely,
504:1:6 keV [AszO], and the lack of an angular momentum barrier for the valence neu-
tron makes the potential shape rather simple. A measurement of the spectrosc0pic
factor indicates that the 231/2 intruder state comprises nearly 77 % of the wave func-
tion [Ajz90]. Therefore, in the treatment of Hansen and Jonson [Han87] the extent
of the halo is proportional to the square root of the binding energy with few other
inﬂuences. Using a more realistic Woods-Saxon potential, with the depth adjusted
to reproduce the binding energy [Esb94], Esbensen ﬁnds that the rms radius of the
halo neutron is 6.5 fm when the radius of the core potential is chosen to match the

measured radius of 10Be (2.30 fm [Tan88]).

3.1 The P“ Distribution of 10Be Fragments from
11Be breakup

We chose to study the p" distribution of 10Be fragments following the breakup of
11Be, because the p" distribution of core fragments should most accurately reﬂect the

momentum wavefunction of the halo nucleon [Ber92, Ban93, Sag94].

22

3.1.1 The A1200 Fragment Separator

Production of the 11Be beam

At the N SCL the method for producing beams of neutron rich nuclei near the drip-line
is fragmentation [She92]. In this experiment the Radioactive Nuclear Beam (RN B)
was produced when 80A MeV 180 ions, from the K1200 cyclotron, impinged on a 790
mg/cm2 natural Be target. Just after the target, the resulting RN B consisted of par-
ticle stable isotopes ranging from 17O to the lighter particles like protons, deuterons

and tritons. The makeup of this beam can be easily understood in the treatment of

Goldhaber, discussed in Section C.1.

A1200 Radioactive Beam Facility

 
  
  

Dispersive Image #1 Final
Achromatic
Be Target Dispersive Image #2 Image

"target" position

\Primarybeamfrom

K1200 Cyclotron

Figure 3.2: The A1200 fragment separator and beam analysing device.

The members of a RN B travel with nearly the beam velocity, and are ﬁltered by the
A1200 fragment separator. Most light ions produced in this manner are fully stripped
of electrons, and by passing this beam through a magnetic dipole and then an aperture
at a momentum dispersive image (see Figure 3.2) nuclei are separated by p/ q. For
fully stripped ions traveling with exactly the same velocity, this gives separation by

(m/q) or (A / Z). However, because the fragments actually have a spread in velocity

23

centered around the beam velocity, or may not be fully stripped, the separation is

not complete.

 

Delta E

 

 

 

 

 

V ‘ , ‘f' .w . . i . ‘. ‘ ‘
K." "H - Ir .,..,-.. . 5‘ > ‘
W". "\-'- 4' -‘-' -’ ' . I -‘ ' ”V I ' 7‘7 ' . ya _ g4. .
. . ., . . . . . _ . sa..:::=;:.;.;... . .. _-:v-t:§7.;;-’:1-'-:‘r'?
I ime 0f F light —I)

Figure 3.3: A plot of A E vs. ToF which shows the isotopic separation of fragments
that are analyzed in the A1200. The 11Be beam group is circled.

After the aperture A/ Z groups are dispersed by different velocities (or times of
ﬂight through the A1200, ToF). Isotope groups are commonly identiﬁed by measuring
both the energy (or energy loss) and the velocity of beam particles. Figure 3.3 shows
AE vs ToF for a particular spectrometer setting. In the ﬁgure the straight vertical
line of isotopes is the A/Z=3 line. Observable in this group, whose members all have
the same ToF, are 18C, 15B, 1"’Be, 9Li, 6He and 3H, from the top to the bottom.
Two other groups of isotopes that have identical velocities are present; they are the

A/Z=8/3 (16C and 8Li) and the A/Z=5/2 (15C and 10Be) groups.

24

Table 3.1: The isotopes comprising the produced RNB after the ﬁrst set of A1200
dipoles. Protons, deuterons tritons and 0’s are omitted.

 

 

Isotope Percent of the RNB
6He 2.55
8Li 7.39
9Li 1.33
loBe 14.02
11Be 2.43
12Be 0.30
13B 14.35
14B 0.81
15B 0.10
15C 0.49
16C 21.23
17C 1.27
18C 0.01

 

 

 

 

The nucleus of interest is 11Be. The beam is selected by adjusting the ﬁrst set of
dipoles in the A1200 to maximize the rate of 11Be ions that pass through the mo-
mentum aperture located at Image 1. The rate was optimized at 4300 11Be particles
per second with a ﬁeld strength of 1.03034 Tesla. The central radius of the dipoles,
p, at this ﬁeld setting is 3.105 meters, and the energy of the 11Be beam is 63A MeV.
The size of the aperture at the image permitted a :l:0.5% spread spread in the beam
momentum which translates into approximately a i1% spread in the beam energy.
The purity of the beam was not very good (2.4% 11Be ). However, as will be dis-
cussed later, the contaminants do not signiﬁcantly affect the measurement of the 1"Be

fragment momentum distribution.
The Principle of a Dispersion-matched Energy-loss Spectrometer

A dispersion-matched energy-loss spectrometer makes it possible to compensate for

the energy spread of a beam with the correct achromatic optical conditions. Thus, in

25

spite of the relatively large spread in momentum of a produced RN B (up to 3% in the
A1200) it is possible to measure, for example, transfer reactions with a momentum

resolution of 0.05% [Sh692].

In this mode the ﬁrst object position is located at the RNB production target,
at the start of the A1200 Spectrograph. The produced fragments pass through a
set of superconducting 22.5° dipoles, where the fragments are separated by p/ q. At
the exit of the dipole set is amomentum dispersed image in the horizontal direction,
Image 1. The momentum aperture located at this image permits, in our case, a spread

of i0.5% in momentum for the produced 11Be beam.

An achromatic device must have an even number of images between the dipoles,
and the A1200 is operated with two. One is at Image 1, and the other is at Im-
age 2. The optics following the center pot are nearly a mirror reﬂection of the optics
preceeding the center pot of the A1200, so that in the second half of the A1200 the
beam returns to an achromatic focus at the ﬁnal image. As mentioned above, under
optimum conditions the optics correct for the spread in momentum of the produced

RN B so that all beam particles focus at a point at the ﬁnal image.

A simple application of the Dispersion Matched Energy Loss Spectrometer tech-
nique is the measurement of target thicknesses. As described above when fragments
are produced at the target position they are focused to a point at the focal plane.
When a target is placed at the second dispersive image, the position of this focus point
shifts on the focal plane because of the energy loss in the target. By adjusting the
ﬁelds of the magnets that follow the target, the position of the focus can be returned
to the center of the focal plane. This high resolution measurement of the energy loss

of particles passing through a target permits a deduction of the target thickness.

Breakup fragments can be studied in a similar manner. When a breakup target

26

Dispersion-Matched Energy-Loss Spectrometer

 

 

 

 

lst 2nd
Dipole ,1, Secondary Targe Dipole
Stage - I: Stage
We. spread \—
B-rhol=3.20 T-m 34110241201911?” M:
= . ~m

 

 

xx

Production Focal Plane
Target

Figure 3.4: The principle of a dispersion matched energy loss spectrometer. A RNB
has a momentum spread that is introduced by the production mechanism. However,
the optics of the device return the beam to an achromatic focus at the focal plane.
The spread in momentum of fragments produced at the secondary target is projected
onto the focal plane.

27

is inserted at the Image 2 position, the magnetic ﬁelds can be adjusted to position
the centroid of a particular fragment momentum distribution at the center of the
focal plane (Figure 3.4). Since the momentum distribution of breakup fragments
has a spread, it is projected onto the focal plane. This method measures the total
momentum of breakup fragments. However, when the components (PII and p 1) of
the momentum are considered for a narrow distribution centered near 3400 MeV/c,
pm“, = W 2 p“ within 0.05%. Therefore, measurements of fragment momen-

tum distributions with this technique serve as measurements of the p” distribution.

Detector System

A standard A1200 detector setup identiﬁed the particles reaching the Focal Plane. A
gas ionization chamber provided the energy loss signal, while a thick plastic stopping
scintillator provided both a total energy signal and a start signal for timing infor-
mation. The Time of Flight (ToF) measurement is made relative to the cyclotron
rf-signal which has a period of 55 us. The Z of a particle can be easily determined
from a plot of AE vs E, and as seen in Figure 3.3 isotopic resolution is obtained from
a plot of AE vs ToF when the A1200 spectrometer is operated in a simple trans-
mission mode. A pair of two—dimensional position-sensitive Parallel Plate Avalanche
Counters (PPAC) located at the Focal Plane (separated by 37.4 cm) measured the
trajectories of the incoming particles. The second of these PPAC3, historically known
as PPAC3, was located at the focus and measured the position at the Focal Plane,
which is related to the momentum of the particles. Four Si diodes located near the

production target monitored the primary 180 beam intensity.

28

3.1.2 Analysis of 10Be Fragment Momentum Distributions

In order to identify the 10Be fragments following 11Be breakup in the Image 2 target
a straightforward approach is used. The 10Be reaction products travel with nearly
the same speed as the incoming llBe particles. When their speed is the same, the
energy loss of 10Be ions in the Ionization Chamber is nearly the same as the energy
loss of 11Be ions. For example, the 1"Be breakup fragments have a larger energy loss
per nucleon than 11Be particles that pass through the Ionization Chamber. Therefore
to a good approximation 10Be fragments produced from 11Be breakup in the Image 2
targets have the same AE and TOP as the 11Be group in Figure 3.3. Small changes in
the velocity due to the energy lost in the breakup target lead to a slightly longer ToF
and a slightly higher energy loss in the ionization chamber. The same contour that
showed the 11Be group in Figure 3.3 is shown in Figure 3.5. It is important to note
that Figure 3.3 was produced when a 3% momentum aperture was at Image 1, while
all breakup data was collected using a 1% momentum aperture. The momentum
spread of the secondary beam is important for determining the spread in TOP of
fragments reaching the Focal Plane. Therefore, in Figure 3.5 the size of the spot
for the 10Be breakup fragment group, collected with the 1% momentum aperture, is

roughly three times smaller than in Figure 3.3.

Scattered Beam Contaminant

It is clear from the ﬁgure that this simple approach is not sufﬁcient for selecting the
10Be breakup fragment group since tWo groups fall within this region. When certain
reactions are studied in the A1200 problems can arise due to beam particles that
scatter off the sides of the dipoles and reach the focal plane. Most RNB particles that
do not have a reaction in the Image 2 target are stopped in the dipoles. In Figure 3.5

seven groups are labled. The ﬁrst of these groups is the 10Be breakup group, while

29

 

  
      

Scattered 10Be
Beam particles

Delta E

 

l- , l7

 

Time of Flight

Figure 3.5: The reaction products that reach the focal plane of the A1200 when the
Be target is inserted at the Image 2 target position. A 1% momentum aperture was
used during the breakup runs. Therefore, the 10Be fragment group from llBe breakup
(circled) appears smaller than the 11Be group in Figure 3.3.

Counts

30

the other six are identiﬁed as 8Li, llBe, 13B, 14B, 16C and 1.,C ions that are scattered

in the dipoles and ultimately reach the focal plane.

The scattered beam made it necessary to require conditions other than AE and
ToF. The ﬁrst conditions were on the vertical PPAC positions, Figure 3.6, to eliminate
trajectories that could indicate scattering. The next condition was on a histogram
which compares the total energy measured in the stopping scintillator with the ﬂight
time through the A1200 (E vs ToF), Figure 3.7.

 

 

200 I T V I I I I 1 I T UT Y—l' I 7 V 1 r1 r1 rt I I 1 I I I I T I T T ‘7 I V I I l I I Y T 1"
h <
; ~ 300 — ~
-4 D q
150 ; A) - . B) .
.. _ 4
: I f ‘
. . 200
100 r 1 I
P d I-
I- 1-
50 j j 100 _
p

 

 

 

 

 

 

 

 

   

o I- q‘ 0 )-
0 200 400 600 800 1000 0 200 400 600 800 1000
Y position PPACZ Y position PPAC3

Figure 3.6: Conditions are applied to the position histograms to eliminate trajectories
that may indicate beam scattering. The data in these ﬁgures are gated on the 10Be
reaction products. We accept only 10Be fragments that lie within the vertical lines.

The ﬁnal condition, which was found to be the most selective, was on a his-
togram which compares the Time of Flight with the horizontal position at the focus
(ToF vs X), Figure 3.8. Because the momentum distribution was obtained from the
X Focal Plane position, this condition could directly inﬂuence the distribution. How-

ever, the condition was made loose to avoid these problems. Furthermore, the width

of the distribution was not sensitive to small changes in this condition.

31

 

E-Plastic

 

 

 

 

Time of Flight -—)

Figure 3.7: The total energy vs. ToF histogram is also used to isolate the 10Be

reaction products. Again, this histogram is gated on the 1"Be reaction products in
Figure 3.5.

 

 

 

 

(— Time of Flight"

PPAC Position

Figure 3.8: The most selective condition is on the Time of Flight" vs. X focal plane
position histogram. Time of Flight‘ is a pseudo-parameter, based on the real Time of
Flight, that removes the position dependence of the Time of Flight for particles that
have different path lengths through the dipoles (related to different radii of curvature).

32

Acceptance Limitations for the 1°Be Reaction Products

Breakup fragments produced in the Image 2 target are restricted in two ways by the
acceptance of the device. The transverse acceptance of breakup fragments is approx-
imately A0 = 21:20 mrad and A¢ = :l: 10 mrad. Even with this small acceptance
10Be breakup events from llBe with transverse components as large as 68 MeV/c
were accepted by the device. The second restriction is attributed to the longitudinal
momentum acceptance of the dipoles which accept only fragments with a momentum

of roughly i1.5% of the central value.

To measure the longitudinal momentum acceptance of the A1200 dipoles for
breakup fragments, a 10Be beam, produced in the production target at the Target
Pot, was transported to Image 2 and the ﬁelds in the second half of the A1200 were
varied to position the beam at different places on the focal plane. Both the efﬁciency
of transmission from the Image 2 to the Focal Plane and the momentum calibration
of the Focal Plane as a function of position (bending radius) were determined from

this measurement (Figures 3.9 and 3.10).

Determination of the Momentum

The momentum of 10Be fragments is determined based on the simple relation,
Force = qv x B = mvz/r r. (3.1)

In our case, this leads to,

mv=p=qu. (3.2)

Using the momentum calibration, Figure 3.9, the momentum of 10Be breakup frag-

ments was related to the bending radius by,

MeV/c

=Z .
p Bpx(29980Kgm/s

 

) z 3415 MeV/c, (3.3)

33

 

 

 

 

- T I T l T I I r I T
3.14 r— ._
7,? i 3
L 3.12 ;- -:
Q) j. u
.7.) _ .
m '- -4
E 3.10 — —
V " .4
l- .
s 1 *
S... 3.08 '- .—
F- -I
3.06 —- _
r- . j
l l l 1 J 1 L I I i l 1 l 1 l I J L I l l L l-4

0 200 400 600 800

X focal plane

Figure 3.9: The momentum calibration of the A1200 focal plane.

Z is 4 for a fully stripped 10Be fragment, B is the dipole magnetic ﬁeld (roughly 0.917

Tesla), 299.8 is a units conversion factor obtained from

 

MeV/c _19 [931.5 MeV/(uc2) 6.022 x 1026 u/Kg]
. — = , I
2998CKg(m/s) 1602x 10 Cou omb 2.9979x108(m/s)/c ,
- (3.4)
and p is the radius of curvature in the dipole,
p = 3.039 + 1.283 X 10‘4X meters. (3.5)

In this formula X, the focal plane position measured in the PPAC, varies from 0 to

1024, and the pewter value is 3.105 meters.

The p" distributions required corrections for various experimental effects namely
the efﬁciency of transmission to the focal plane, momentum resolution and energy

straggling in the thick production targets.

34

 

 

 

 

 

1.25 _ l l l l __
1.00 -
P
a +-
C 0.75 r-
.2 l- -1
.2 -
i: L e 'l
Q) 0.50 — .1
r- 4
r 1
0.25 -— A
+- -4
L- .1
)— u
000 l l l l L 1 L IL 1 LL 1 l L l I 1 l l l L l I
0

200 400 . 500 000
X focal plane

Figure 3.10: The transmission efﬁciency of the second half of the A1200 vs. position
at the focal plane.

Corrections to the Data for Experimental Effects

The transmission efﬁciency decreases at the edges of the focal plane and is asymmetric,
perhaps because the quadrupole magnet ﬁelds were. not optimized for transmission.
The dipoles accept roughly :i:1.5% of the central momentum. However, only the
central :l:1% of the distributions, where the transmission is relatively ﬂat, is shown.
The 10Be fragment distribution from breakup on the Ta target, where the central 5%
of the distribution was measured, is discussed later. Figure 3.11 shows the momentum
distributions after a correction for the transmission efﬁciency to the focal plane. To
correct for the transmission efﬁciency the number of counts in a given bin is divided
by the transmission efﬁciency. A quadratic ﬁt to the data in Figure 3.10 is used as

the efﬁciency curve,

6 = 0.2048 + 2.105 x 10‘3X — 1.398 x 10'6X2. (3.6)

35

The maximum transmission of the linear ﬁt to the data is normalized to 1.0. The
uncertainty in the number of counts per bin in the distribution is also divided by the

transmission efﬁciency in order to maintain the same percentage of error in the data.

 

 

 

 

 

 

 

 

 

 

_Trrrr[rrrvIrrtrrrtrrlrrrr‘ :[rrrrltitty—FTIIIIIIIIIrrr
A I - 800 ' '1
U) 2000 " ‘ Z ,
a 1500 ~’- n" "" J 60° -' ‘i' i ‘-
5 I n. I r I :
C I‘ I L .1
E 1000 t’ 1|. 775 1 400 : j! i'. 1
E .- ' E : I I 1
.33 500 _— ' - zoo .- :
.o : 1 ; :
f6 0" l‘llillLLlLLllllleLllll 711111111111111111111111113
v 3350 3375 3400 3425 3450 3350 3375 3400 3425 3450
E 600 I U V T I i U U V r Y MT, 1 I I T T 600 Eﬁ—r' I I I TTI V I r T I h I F I ‘ ' f I I r:
O :— “I
C: 5 0 Ta 500 E U ﬁgs!!! 3
“o 400 i 400 .- 5 Fl, —.
v : :
g 300 5‘ 300 :— I‘i“ -:
m E I E
—u 200 200 :- g 1 3
100 \ 100 :— i
1 1L1 i 1 l 1 1 1 1 l 1 1 : 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 I 1 1 1 1 l 1 j

 

 

 

 

0
3300 3350 3400 3450 3500 3375 3400 3425 3450 3475
Momentum (MeV/c)

Figure 3.11: The efﬁciency corrected p” distributions of 10Be fragments following the
breakup of 63A MeV 11Be ions.

Momentum resolution and energy straggling effects were determined simultane-
ously by passing a 10Be beam through each of the breakup targets and measuring the
momentum spread at the focal plane. At the same time a comparison of the change
in momentum of the 10Be beam provided a measure of the energy loss in the targets,
which is related to the target thicknesses. The details of the target measurements
are in Table 3.2. The resolution functions were taken to be Gaussian in form. The
correction for the resolution of the spectrometer and the energy straggling in the thick
breakup targets is made by subtracting, in quadrature, the width found in column 3

of Table 3.2 from the widths of the transmission efﬁciency corrected distributions.

36

Table 3.2: Target thicknesses and momentum resolutions for the reaction targets.

 

 

Target Target thickness Momentum Resolution
mg / cm2 MeV/ c
9Be 191 9.3
93Nb 249 11.6
l’E’lTa 301 14. 7
238‘U 275 13.5

 

 

 

 

 

In Appendix E an effect is discussed that indicates the size of the angular ac-
ceptance for breakup fragments can inﬂuence the width of the observed distribution.
The momentum distribution of breakup fragments is not known a priori, and a correc-
tion to the data for the inﬂuence of the angular acceptance requires this distribution.
Therefore, a procedure is developed that ﬁlters theoretical predictions of the momen-
tum distribution through the acceptance of the device, which leads to a distribution
that can be compared with the data. It appears that the change in width is small
(decreased by ~ 6%).

The data is characterized in terms of the full width at half maximum height
(FWHM). This avoids the association of the distribution with simple wavefunction
shapes that cannot account for the subtle inﬂuences of perturbations or other reaction
effects that can distort the observed momentum distributions. In order to ﬁnd the
widths of the distributions the maximum height of the distributions was ﬁrst deter-
mined from the average of ﬁts of Gaussian and Lorentzian line shapes to the data.
Straight lines were ﬁt to the sides of the distributions, and the width (FWHM) is the

distance between the lines at half the maximum height.

37

Table 3.3: Widths (:l: uncertainties) of parallel momentum distributions of 10Be frag-
ments following the breakup of 11Be on various targets. All widths are F WHM.

 

Target Uncorrected Efﬁciency System Resolution 11Be Rest
Corrected Resolution Corrected Frame
[MeV/c] [MeV/c] [MeV/c] [MeV/c] [MeV/cl
9Be 44.5(2.0) 45.3(2.0) 9.3(l.0) 44.3(2.3) 41.6(2.1)
93Nb 48.3(1.9) 50.1(1.9) 11.6(1.0) 48.7(2.2) 45.7(2.0
181Ta 48.2(1.9) 50.3(2.0) 14.7(1.0) 48.1(2.2) 45.2(
233U 45.8(2.1) 46.3(2.1) 13.5(1.0) 44.3(2.3) 41.6(

 

 

 

 

 

 

 

 

 

Transformation into the 11Be Rest ﬂame

The ﬁnal procedure is to transform the distribution into the 11Be rest frame which

reduces the width by the relativistic factor 7 = ‘/1 / (1 — 62), roughly 5%. Corrections

to the widths and the transformation are detailed in Table 3.3.

3.1.3 Discussion of the 10Be P” Distributions

The results for the Be, Nb and U targets show the central 2% of the measured mo-
mentum distributions. However, three different spectrometer magnetic ﬁeld settings
were used to measure the central 5% of the distribution from the Ta target. The data
from each setting with the Ta target are normalized by the beam current measured in
the beam monitor Si PIN diodes, and the ﬁnal distribution is obtained by adding the
three data sets together and properly taking into account the overlap of the settings.
The distribution on the Ta target does not show any indication of a two component
structure as has been suggested by earlier measurements of the transverse momentum

distribution of 10Be cores from the breakup of 11Be [Kob89].

38

 

600 T I Y T T I I I 7 I I
l T l

I I

1 L411

500

.1”
Int—1!

run-ET,
111

400

300

200 L 11}:
100 ‘- #11 .3
: 1 gig}! :
.. IRE .1
b 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l L41. &

o
3300 3350 3400 3450 3500
Momentum (MeV/c)

Counts

”I. .rl.
M
WW

1 .1

 

 

 

Figure 3.12: The pl distribution of 10Be fragments following breakup in the Ta target.
The background is low, and the asymmetry hints that a dissipative tail may be present
on the low momentum side.

10Be from Contaminants in the RNB

It is possible that 10Be fragments could be produced by nuclei in the RN B other than
llBe. For example, as can be seen in Figure 3.3, the nuclei 14B and 16C have a similar
velocity to that of 11Be. Therefore, these nuclei could also produce 10Be fragments
that would arrive at the focal plane with a ToF similar to that of the 10Be fragments
produced by 11Be breakup events. It is possible to determine the centroid of the
10Be fragment momentum distribution from contaminants, and using the Goldhaber
model, Section C.1, the widths of the distributions produced from these contaminants;

see Table 3.4.

The momentum distribution measured for breakup in the Ta target has a narrow
width with a relatively ﬂat background. The contaminant nuclei produce broad 10Be

fragment distributions that are peaked at momenta quite different from the peak in

39

Table 3.4: The expected mean momentum and widths of 10Be fragment momentum
distributions from nuclei in the RN B that could produce 10Be fragments. Predicted
widths, FWHM, are from the Goldhaber model with 00:80 MeV/c, Section C.1.

 

Isotope Centroid Width
MeV/c MeV/c
13B 3616 297
1“B 3341 330
15C 3760 355
16C 3511 376
17C 3289 393

 

 

 

 

 

 

the observed distribution. This indicates that breakup reactions from the contami-
nants do not signiﬁcantly contribute to the observed distribution. There is a slight
asymmetry on the low momentum side of the distribution that may originate from
the 14B contaminant which would produce 10Be fragments peaked at 3341 MeV/c.
However, it is also possible that this asymmetry is rooted in dissipative reaction pro-
cesses that produce low momentum tails in the fragmentation of nuclei in this energy

range[Sou92L

The transverse acceptance for breakup fragments plays a significant role for de-
termining the transmission of a distribution with a given width. The one-neutron
breakup of 11Be has a very large cross section and produces 10Be fragments with a
very narrow momentum distribution width (~ 43 MeV/c); thus the bulk of this dis-
tribution is transmitted. On the other hand, the contaminants will have a relatively
small cross section for producing 10Be fragments, and the associated 10Be fragment
distribution widths are quite large. Thus very few l"Be fragments produced by con-
taminants will reach the Focal Plane. For these reasons loBe fragments from the

breakup of 11Be dominate the distributions we observe.

40

Distribution Widths Independent of the Reaction Mechanism

The Be, Nb, Ta and U breakup targets provide the opportunity to investigate the
effects of the different reaction mechanisms. While breakup in the Be target is domi-
nated by nuclear mechanisms (approximately 6% Coulomb [Ann94]), breakup in the
U target is essentially all from Coulomb mechanisms [Ann94], and the breakup in the
Nb and Ta targets is induced by a mixture of nuclear and Coulomb mechanisms. It
appears that the width of the parallel momentum distributions is not strongly related
to the target mass indicating that the widths of the momentum distributions are not

sensitive to the reaction mechanism.

 

55

 

 

 

_ I I I I l T I I I l I I f I l I I I I FI I I f _
A l: I
i 5 f i E

45 —_ . . 1 I , . . ‘—
Ei El ii
2 40 i— 41
IE 5 g
E 35 L— __

3O : 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 j

0 20 4O 60 80 100

Target Z

Figure 3.13: The corrected widths of the 10Be momentum distributions in the 11Be

rest frame. The widths are in agreement, showing no systematic dependence on the
target. The weighted average (shaded region) is 43.6:1:1.1 MeV/c.

Prediction for the 2s1/2 Neutron Momentum Wavefunction

The momentum wavefunction of the halo neutron in 11Be is obtained by assuming
a 231/2 neutron bound in a Woods-Saxon potential with the parameters adjusted

to reproduce the binding energy and the rms radius of the 10Be core. The parallel

41

momentum distribution is the projection of this spherically symmetric momentum
density distribution ( |\Il‘\II|) onto one axis [Esb94]. In the spirit of the Serber model
we compare this prediction with the data for all breakup targets, and the prediction is
transformed into the laboratory frame and convoluted with the momentum resolution

of the spectrometer and the energy straggling in the targets.

A ﬁnal step is to correct the shape of the distribution for a change that occurs
due to the limited transverse acceptance (discussed in Appendix E). The changes
in shape arise when a non-Gaussian distribution passes through a limited transverse
acceptance. An approximate method was used to estimate the effect of the limited

acceptance and the changes are small. The width of the prediction is reduced from

45.4 MeV/c to 43.7 MeV/c (6%).

The momentum distribution obtained from the projection of the 231/2 neutron
wavefunction onto the parallel momentum axis is in good agreement with the dis-
tributions from all of the targets, Figure 3.14. However, this method does not take
into account any reaction mechanism effects and is a reasonable approximation for

breakup on a light target where stripping dominates.

On the heavy target, where Coulomb dissociation is the dominant breakup mech-
anism, subtle changes to the distribution could occur. For example, at this energy
the momentum distribution from a dipole excitation calculation [Esb94] is slightly
narrower (39.8 MeV/c) than the width of the momentum distribution of the 231/2
neutron wavefunction (45.4 MeV/c), Figure 3.15. This is likely because of complex
dynamical factors in the dipole excitation process. Our data does not show a sys-
tematic change in the momentum distribution width with increasing target mass.
Therefore, we choose to neglect these reaction mechanism effects and compare all

data with the wavefunction of the halo neutron.

42

 

 

 

 

 

 

 

 

 

 

C' I I I I I I V U I I U I I ' ‘ I U I I I I 7 7 Id -1 U FT 1 l t Y 1 U I I U U I I U I 7 U I T U I I
A » r 800 1 1
m .. . I- ..
r: 2°°° E Be ' g :L Nb 3
5 1500 :- -5 50° : ?
l: I C :
E“ 1000} -5 400 E ‘2
.42 500 :- -j 200 E €
.0 : : E :
S o L11111111111114111J_1J1141'l 0 l111111111111111111111111'l
V 3350 3375 3400 3425 3450 3350 3375 3400 3425 3450
a 600 1 U U T I I T 1 U I t I f ﬁ" 7 f ‘7 U 600 :17 T Y I 1 Y Y I I 1 U U T I U U I U I I U U 1 I I:
“U 500 500 E— U -3
G 5 a
“U 400 400 .- 1
v : -
g soo 300 g- -:
"—u zoo zoo ;- -:
100 100 :- ‘3
1 1 1 1 I 1 1 1 1 I 1 1 :1 1 1 I144 1 l 1 1 1 1 l 1 MLI 1 1 1 1 I :

 

 

 

 

 

0 0
3300 3350 3400 3450 3500 3375 3400 3425 3450 3475
Momentum (MeV/c)

Figure 3.14: The p" distributions of 10Be core fragments compared with the a pro-
jection of the wavefunction of a 231/2 neutron, bound in a Woods-Saxon potential by
500 keV, onto the p" axis [Esb94].

43

 

 

 

 

_ T FT I I I I I l r I T I I T l I I T I l I I [j I I-l
A ..
A
0 ~ -
H p— —4
E ’ 1
1. I: i
(U
V 0.010 _ —
A l" ‘
On 1 a
"U _ _
C I. -.
"O 0.005 —- —
v b .4
\
. b " ~
02 _ ..
"C3 _ 4
0.000 L l I l 11 l 14 l l I ll 11 L1 1

—75 —50 -25 0 25 50 75
Momentum (MeV/c)

Figure 3.15: [Esb94] The width of the p” distribution of the 231/2 neutron is slightly
broader than a perturbarion theory prediction of the 10Be p” distribution expected

from dipole excitation of 11Be.

44

P” Distribution to Determine Ground State

The p" distribution is sufﬁcient to establish that the valence neutron is in the 231/2
orbital rather than the 1p3/2 orbital. Figure 3.16 shows the p” distributions that
correspond to these orbitals [Br095a] and the data from the Be target. The data
are transformed into the 11Be rest frame, and agree with the 231/; wavefunction. To
account for the inﬂuence of the acceptance, Appendix E, the corrected theoretical
distribution for the 231/2 orbital was obtained. The data were scaled by the difference
between the corrected and the uncorrected distributions, point by point, to account

for the small difference in transmission.

 

    
    

Y I I T j Y I I Y T I I 1 I I I T I I I T

§ Kelley et a1. 1 Be -‘>1°Be

1111

51)

1

S1/2

 

L0

(15

IIVTIII

I

0£!—-‘

 

 

 

Momentum Wavefunction (arb. units)

01llilILJlllllllILlLllllll

20 40 60 80 100

Momentum (MeV/c)

Figure 3.16: The p“ distribution sufﬁcient to show that the 281/2 intruder state dom-
inates the valence neutron wavefunction. The momentum wavefunctions are from

[Bro95a].

45

3.2 Comparison with the Transverse Momentum
DistributionS-of Fragments from 11Be Breakup

3.2.1 P ‘L Distribution of Core Fragments

The transverse momentum distribution of 10Be fragments from the breakup of 790A
MeV 11Be [Kob89] was measured and was ﬁt with a two-component Gaussian distri-
bution, Figure 3.17. As with 11Li, the broad component, 257 MeV/c FWHM, was
associated with the removal of core neutrons, and the narrow component, 59 MeV/c
FWHM, was associated with the removal of the more peripheral halo neutrons. Be-
cause diffraction and other reaction mechanism effects inﬂuence p i, this distribution
is not expected to reﬂect the momentum wavefunction of the halo neutron as accu-

rately as the p” distributions.

 

m U I I Ij‘jj I I U U III I I V l U TTTII I

400

100

- l A l l l l l l l
0 -400 -2W 0 M 400
m W [HOV/c]

Figure 3.17: [Kob89] The transverse momentum distribution of 1°Be cores following
the breakup of 790A MeV 11Be.

46

3.2.2 Angular Distribution of Neutrons

The angular distribution of neutrons following the breakup of 41A 11Be was measured
by [Ann93, Ann94] in an attempt to isolate and study the different reaction mecha-
nisms. The neutrons were detected in an array that covered the lab angles from 0°
to 97° while a AE-E detector telescope at 0° measured the charged fragments. With
gross assumptions about the reaction mechanisms, this information made it possible

to separate the data into two categories.

The ﬁrst category, exclusive reactions, includes breakups with both a fast neutron
and a 10Be core in the ﬁnal state. These reactions are assumed to result from dis-
sociation reactions, Section 2.3. On a heavy target the reactions are from Coulomb

dissociation while on a light target diffraction dissociation is the breakup mechanism.

 

F n m l I I l j
,0 41MeV/u “Be ‘-
Be target

.0
gun

 

doldﬂ (barn/3r)

0.01 g -2 v

0.001F 1 m l I f I; J
0

10 20 I!) 40 50 00 7O
Neutron Lab. Angle (deg)

Figure 3.18: [Ann94] The angular distribution of neutrons following the beakup of
41A MeV 11Be is in agreement with a prediction based on a 2s-orbital neutron (ls in
some notation) in a simple square-well potential. A narrow width (I‘=60 MeV/c) is
extracted from the restricted inclusive data ( circles, n+ no 10Be). Also shown are
the distributions for n+He (triangles) and n+Li (squares).

The second category was termed restricted inclusive and includes events with a

47

neutron detected in coincidence with any fragment other than a 10Be core, Figure 3.18.
These events are assumed to result from stripping reactions in which the core collides
with a target nucleus and is broken apart. This type of reaction is claimed to, leave
the motion of the halo neutron mostly undisturbed so that the resulting angular

distribution should reﬂect the initial state wavefunction.

It was apparent, from the restricted inclusive events, that the neutron distributions
from violent core-target collisions contain information about the halo wavefunction.
A Lorentzian width parameter F=60 MeV/c was extracted from the data and is con-
sistent with the simple interpretation of [Han87] where I‘ = m = 58 MeV/c. The
differential cross sections derived from the data were found to be in good qualitative
and quantitative agreement with the predictions for the various breakup reaction

categories.

A discrepancy arises, however. A Lorentzian width parameter of F =60 MeV/c is
inferred from the angular distributions of neutrons from 11Be [Ann93, Ann94] while a
width of 43.6 MeV/c (average of the four targets) is obtained from the present work.
The difference in the width of the parallel momentum distribution of core fragments
and the width derived from the angular distribution of neutrons may arise from differ-
ences in the effects of the reaction mechanism (and in initial and ﬁnal state reactions)
on the parallel momentum of core fragments and the transverse momentum of halo
fragments. In contrast to the case for angular distributions where a particular line-
shape must be assumed to interpret the data, a distribution width can be measured
directly from the parallel momentum distribution. Furthermore, a Lorentzian distri-
bution shape results from the assumption that the 10Be core provides a square-well
potential, and that outside the core the valence neutron wavefunction falls off as a
Yukawa. More realistic potential shapes, for example a Woods-Saxon, should be used

for comparison with the data.

48

3.3 Neutron Halo Radius

The agreement between the data and the calculation of Esbensen [Esb94], using a
Woods-Saxon potential, supports the use of the rms radius, 7.2 fm, to determine the
neutron halo radius in 11Be. This is the rms radius for the relative motion of the
neutron and the core. In the 11Be rest frame the halo neutron rms radius is 6.5 fm
(7.2 X 10 / 11) and is consistent with the value 6.4i0.7 fm that is required to reproduce

the E1 strength observed in the Coulomb breakup of 11Be [Nak94] (and Section B.6).

 

3 rrr.1jtrrtIrTr

‘ .
‘ 11Be
I

— Core Neutrons

   

IAJAJAL

Core Protons

P(r)

I I I I I I I I I I I 1 I r

 

 

 

Figure 3.19: [Bro95] The probability distribution of nucleons in 11Be.

By comparing the probability distribution of nucleons in the 10Be core with the
distribution of the valence 231/2 neutron [Br095a], which has a node near 2 fm, the
extent of the neutron halo is plainly seen. All measurements to date on the nucleus
llBe are consistent with the presence of a prominent neutron halo, and the present

result ﬁnds excellent agreement with the wavefunction of the halo neutron.

Chapter 4

Search for a Proton Halo in 8B

The discovery of neutron halos in weakly bound neutron rich nuclei such as 11Li and
11Be raises the issue of the possible existence of proton halos in proton rich nuclei.
It had been argued that a loosely bound proton would be localized by the Coulomb
barrier, hindering the formation of a halo [Han93]. The valence proton in 8B is less
tightly bound than the neutrons in 11Li and 11Be. However, the Coulomb barrier
and the angular momentum barrier for the p-orbital proton increase the effective
barrier height and decrease the extent of the proton wavefunction. The issue has been
resurrected by recent measurements and theoretical models that suggest 8B, bound
by only 137 keV, may possess a proton halo. The present experimental situation is
unclear since the interpretations of experimental results are not in agreement on this
issue. The earliest evidence of a proton halo was in 1992, but there are only a few

measurements to date that would be sensitive to a halo in 8B.

4. 1 Experimental Measurements of the Nuclear Size
of 8B

The interaction cross section measurements of light nuclei at 790A MeV that found

evidence for neutron halos [Tan85a, Tan85b, Tan88] showed no enhancement in the

49

50

83 reaction cross section. The rms radii obtained for the boron isotopes indicate
that the nuclear size does not change signiﬁcantly in the range from 8B through ”E.
It is interesting in this case to compare the difference between the rms radii of the
proton and neutron density distributions; Figure 4.1 shows that the proton density

distribution in 8B does extend beyond the neutron density distribution.

 

1.0 TI I I I I I I I Ij I I I I I I I I I I I I I I I I

I j I I I
1

0.5

1 1 1 1

fit!
I—H
g

 

L 1 1 1

 

l

-0.5

er S (protons) — ers (neutrons)

 

 

 

oﬁrillirti
p-
LL11

_1.0 1 1 1 I l 1 1 J J l l 1 1 I 1 1 1 1 I 1 l 1 LJ 1 1
2

4 0 a 10
Neutron Number

Figure 4.1: The difference between the rms radii of the proton and neutron density
distributions in the boron isotopes, obtained from the 790A MeV cross section data.
Data from [Tan88].

The 790A MeV reaction cross section data [Tan88] indicates that the rms radius
of the valence proton, obtained from Equation 1.4 and the radii of 8B (2.38:l:0.04 fm)
and 7Be (2.31:l:0.02 fm), is only 3.02 fm. This value is much smaller than the rms
radius of the neutron halo in 11Be (6.5 fm), and appears to indicate that the nuclear

distributions in 8B are similar to those of normal nuclei.

A recent measurement of quasielastic scattering of 40A MeV 8B on a 12C tar-
get [Pe095] found that the small angle cross section was not easily reproduced in a

coupled-channels calculation and probably reﬂects the loose binding of the valence

51

proton. However, the proton removal cross section was also measured and showed no
signiﬁcant enhancement of the breakup cross section.

The ﬁrst evidence of a proton halo came from a measurement of the quadrupole
moment of 8B (68.3:l:2.1 mb); it was found to be much larger than that of the mirror
nucleus 8Li (32.7:l:0.6 mb) [Min92]. The observed value was reproduced using shell I
model wavefunctions generated in Woods-Saxon potentials. The rms radius of the
ﬁve protons in this model was considerably larger, 2.99 fm, than the rms radius of the
three neutrons, 2.20 fm, and was taken as an indication of a prominent proton halo in
8B. However, these results do not necessarily provide deﬁnitive evidence of a proton

halo. For example, the quadrupole moment may be enhanced by contributions from

a deformed 7Be core [Rii93b, C3093].

 

4 I IIIIIIII I ITIIIIII I IIIIIII

‘1 d
U Microscoplc
Model Predictions “l

     
   
 

 

1" 14
l' N

  
  

 

1 _.

if mmiluul I ulmil l Iiluw
1 10 100 1000
EM (MeV/nucleon)

Figure 4.2: [War95] The breakup cross section of 8B compared with a microscopic
calculation that shows the trend expected if 8B has a proton halo. Also shown are
calculations for the two neutron halo nucleus 11Li and the normal sized nucleus 14N.

Finally, the 8B cross section in active targets was recently measured at the NSCL.

A stack of silicon AE detectors measured reaction rates in the range between 20A and

52

60A MeV [War95]. The increase of the nucleon-nucleon cross section at low energies
leads one to expect that, if present in 8B, halo effects should dominate breakups. A
comparison of the data with a microscopic calculation which assumes a proton halo in-
dicates that the observations are consistent with a extended proton distribution. The
cross section data are shown in Figure 4.2 with the microscopic calculation showing

the expected enhancement from the halo that would dominate at lower energies.

4.2 The P” and P I Momentum Distributions of
7Be Fragments Following the Breakup of 8B

In an attempt to understand the existing discrepancies, we have approached the
problem from an independent direction by simultaneously measuring the p" and p i
distributions of 7Be fragments produced in the breakup of 8B on 9Be, 93Nb and
197Au targets. The p” distributions of 10Be fragments following the breakup of 11Be
ions indicated that the breakup reactions accurately probed the wavefunction of the
halo neutron. Therefore, it was expected that the 8B breakup data could also be
understood by a direct comparison with the momentum wavefunction of the valence

proton.

Narrow distribution widths are observed (~75 MeV/c). However, as will be dis-
cussed later, the reaction mechanisms appear to strongly inﬂuence the observed dis-
tributions. Thus, it seems premature to conclude that 8B has a halo. None the less,

realistic predictions of the core fragment momentum distributions, based on the wave-
function of a proton bound by 137 keV in a p-orbital Woods-Saxon potential, include
reaction mechanism effects and are found to agree with the data. The rms radius of
the theoretical wavefunction is 4.24 fm and indicates that the valence proton has a

much larger radial extent than the 7Be core (2.31 fm [Tan88]).

53

4.2.1 Experimental Details

To obtain a high quality relatively pure 8B beam both isotope separation devices at
the NSCL, the A1200 fragment separator and the RPMS Wien-ﬁlter, were used to
remove contaminants from the 8B beam. A beam of 60A MeV 160 ions, from the
K1200 cyclotron, was fragmented in a thick Be production target. Products passed
through the A1200 fragment separator, with an Al absorber (wedge) placed at the
second dispersive image. Isotopes reaching the focal plane were dispersed because of
differential energy losses in the wedge, and a moveable aperture at the focus eliminated
most contaminants. The resulting beam consisted of the N23 isotones, with a large

34.3A (MeV 7Be component (8Bz7Be=1:20).

This beam then passed through the RPMS Wien-ﬁlter which separated the iso-
tones, leaving a 95% pure 41.2A MeV beam of 300 8B particles per second at the
breakup target. The 8B beam momentum spread was limited to i0.25% by the aper-
ture located at the ﬁrst dispersive image of the A1200, and the relative velocity of
ions reaching the breakup target was determined by their time—of-ﬂight over a 40 me-
ter ﬂight path between a thin plastic scintillator and the particle detector telescopes.
This insured that the detected 7Be particles came from reactions of 41.2A MeV 8B

in the target, instead of the 34.3A MeV 7Be component of the incoming beam.

After the RPMS, a pair of two-dimensional position sensitive Parallel Plate Avalanche
Counters (PPACs) located upstream of the target permitted a reconstruction of the
incoming particle trajectories. Downstream of the target position, breakup products
were detected in two 5 cm by 5 cm AE-AE-E telescopes. The ﬁrst AE detector
was a position sensitive Si detector that was segmented into 16 vertical strips and
16 horizontal strips. This corresponds to a strip spacing of 3.125 mm. The second

was a Si PIN-diode, and the E signal was provided by a stopping CsI detector. Tele-

54

Table 4.1: Description of detectors that comprise the two telescopes, and the energy
loss of a 40A MeV 7Be ion (provided for reference). Note that in the telescopes the
order of the detectors was Strip Detector, PIN-diode and the stopping CsI detector.

 

Detector Thickness Thickness 7Be E-loss
(#m) (mg/cmz) (MeV)

 

Telescope l

 

 

 

 

 

 

Strip Detector 502 117.0 22.8
PIN-diode 504 117.4 22.9
Telescope 2

Strip Detector 304 70.8 13.6
PIN-diode 518 120.7 23.5

 

scope l was 60 cm from the target and covered angles 0=3.3° to -1.4° (horizontal)
and 06 = i2.4° (vertical). The second was 50 cm away from the target and covered

the angles 0 =-2.6° to -8.3° and o = i2.9°.

 

.. -H mm H “Ii;

 

 

 

 

I \ / I
Thin plastic Position-sensitive Position-sensitive
time detector parallel plate dE-dE-E
avalanche counter Telescopes
16 I , s | 3
60 MeV/A 0 RNB N=3 Isotones (5% B) 95% pure B

Figure 4.3: A schematic illustration showing the overall layout of experimental appa-
ratus used in the 8B breakup experiment.

The target thicknesses were chosen to make the p” momentum distribution broad-
ening effects, caused by the differential energy loss of 7Be and 8B, similar to the
momentum resolution. The average size of this effect is approximately 10 MeV/c.
This is insigniﬁcant when subtracted in quadrature from the measured fragment dis-

tribution widths.

55

Table 4.2: Target thicknesses in the 8B breakup experiment.

 

 

Target Target thickness
(mg/cm”)
9Be 47
93N b 85
23‘3U 97

 

 

 

 

4.3 Analysis of the Data

The 7Be reaction products detected in the telescopes were-selected using AE vs. E
signals. Because channeling, an effect related to the crystal structure of the silicon,
occurred in the Si PIN-diodes it was also necessary to consider the energy signal
from the position sensitive Si strip detectors. The 7'Be ions were easily identiﬁed
using redundant AE—E selections. The relative ﬂight time of incoming ions was used
to identify 8B projectiles that impinged on the target, and the ﬁnal condition was
applied to the PPAC located immediately in front of the target and deﬁned the beam
spot on the target (5 mm by 7 mm F WHM) With these requirements it was possible

to cleanly identify 7Be fragments following the breakup of 8B, Figure 4.4.

Angular Deﬂection Calibration

The angular deﬂection of breakup fragments was determined from the measured po-
sition in each of the position sensitive detectors. Incoming trajectories were obtained
from the two PPACs which were separated by 1.22 meters. Using this trajectory, the
position on the target was extrapolated. The separation of the ﬁnal PPAC and the
target was 2.5 cm. After the target, the scattering trajectory was determined from the
position on the target and the position in the strip detector. The angular resolution,

10 mrad, was obtained from a target out run where there was no deﬂection.

56

 

 

 

 

 

L
LL] t
a .— , Elastic and inelastic
.0 ° ' . scattered 83
Q
Z , _,,Be fragments from8 ..
, E Li fragments Q, brealfup
f ~; . Hefragments _ . _
l I , \l . . I I ‘ I ‘ I '
CsI E

Figure 4.4: AE vs E showing the 7Be fragments from 8B breakup measured in Tele-
scope 2. Li, and He fragments from 8B breakup are also identiﬁed.

57

Momentum Calibration

An energy calibration was obtained using 7Be beams at six different energies. These
beams were produced with two A1200 magnetic rigidities that provided 7'Be at en-
ergies of 51.47A MeV and 41.41A MeV. Two aluminum targets, 190 mg/cm2 and
398 mg/cmz, located in a target ladder degraded these beams so that 7Be beam ener-
gies of 51.5A MeV, 47.3A MeV, 42.3A MeV,.4l.4A MeV, 36.3A MeV and 30.1A MeV
were obtained. A position dependent response in the CsI detectors required that
these detectors be sectioned into 256 sub-detectors using the position sensitive detec-
tors. The total momentum was determined from the sum of the energies measured in
the telescope. The overall momentum calibration of each telescope indicated that at

2000 MeV/c the resolution , Ap/ p, was better than 0.5%. The measured deﬂection

angle was used to determine the p”.

4.3.1 The P, Distributions

The p” and p l of breakup fragments was measured. However, it is necessary to
assume a distribution shape, for example Lorentzian or Gaussian, to interpret p i
distributions. In comparison, a. projection of the p i distribution onto one axis removes

this complication and provides a distribution that is simpler to interpret.

The p, distributions are the projection of p 1 onto the horizontal axis. Although
a set of PPACS was present, to permit ray tracing of the incoming 8B and outgoing
7Be trajectories, it was found that the angular resolution for deﬂections (19 MeV/c)
was comparable to the angular spread of the incoming beam (26 MeV/c). Therefore,
the data are binned according to the horizontal position of the strip that detects
them. These distributions are corrected by subtracting in quadrature the widths of

the angular spread of the incoming 8B beam and the contribution due to multiple

58

 

 

 

 

 

 

800 l l T l I l lTj I l l I I F! l l l I ﬁd
: §§ I
_ §§ § _
600; Be target § 2 3
400:— 5 E
: a :
zoo;— nn, -E
I ‘wggu‘m “we 2
ZOBW‘VHHHHHHHHE
150; Nb ii {Hi .3
100E HE E
— —1
E §§ Hi 3
50:. §§ § :.
_ §§ _
I a §n§§§ E
oﬂ—P‘PHIHHHHHHHHL
15o- -
e l 2
100 3 Au {H i} E
- i m -
I § §§ i § i Z
50 :- § §§§ § i i j
.. §§ . _
' §§§§§§ ‘
O 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 I 1 1—
—300 —200 -100 o 100

Momentum (MeV/c)

Figure 4.5: The p; distributions of 7Be fragments following the breakup of 8B.

59

Table 4.3: Summary of the pz distribution results.

 

Target Uncorrected Resolution and Multiple Corrected
F WHM Beam spread scattering FWHM
(MeV/c) (MeV/c) (MeV/c) (MeV/c)

 

 

 

 

 

 

 

Be 912t8 26i1 8 87i8
Nb 162:}:15 261—1 34 l56:l:15
Au 242i20 26:}:1 56 234i20

 

scattering from the measured widths (FWHM) of the pa, distributions.

Like the p” distribution, p, distributions result from a projection of the momentum
density distribution onto one axis. Therefore, in the absence of reaction mechanism
and ﬁnal state interaction effects these distributions could be directly compared. For
example, the p” and p3 distributions are almost the same for the 9Be target. On the
other hand, the p” distribution for breakup'in the Au target is narrow while the pa,
distribution has a width near 240 MeV/c, apparently because of a sideways Coulomb
deﬂection in the ﬁeld of Au. Since p“ is less affected than p i by reaction mechanism

details and ﬁnal state interactions, such as diffraction and Coulomb deﬂection [Ber92],

p” is the focus of the remaining sections.

4.3.2 The P“ Distributions

The 7Be fragment p” distributions measured in Telescope 1 and Telescope 2 are shown
in Figure 4.6. Experimental effects that broaden the observed p" distributions are as—
sumed to have Gaussian resolution functions. The widths of these functions (FWHM)
are subtracted, in quadrature, from the widths of the p” distributions. These include
the small, but not negligible, spread in the momentum of the incoming beam; the
momentum resolution of the telescopes and the energy spread associated with the

thick targets. Finally, the transformation into the 8B rest frame reduces the width

60

Telescope 1

 

Telescope 2

 

50° fT T1 I r T T T l l T T l l fl T T :T I T T I l T T I I T r1 l I l T T T:
400 Be target 5533! so? —
300 55 i mi Iii j 5
200 i! : {if} i 3

h § 20:. 3f; i i} l
:3 “I... ‘11..." ‘WIEKI I I IiII I I l I I IiIglgﬁIé

 

 

 

 

 

 

... E .2
E {El 3
150 L H {l __
100 E H E
I- H -2
5° :- if}; :
0 E I
150 :- so :_ _5
:Au target {if}
mo:— If 1 {Hill
E 5 40 E i If g
50 :— I} if 20 :— i If -:
0M11411!—-—- ownllllw
1700 1000 1900 2000 2100 1700 1000 1900 2000 2100

Momentum (MeV/o)

Figure 4.6: The p" distributions of 7Be fragments following the breakup of 8B. The
narrow distribution shown with the Telescope 1 Be target data shows the momentum
resolution (~0.5%) obtained from one of the energy calibration measurements.

61

Table 4.4: Widths (uncertainties) of the p” distributions of 7Be fragments following
8B breakup. All widths are FWHM.

 

 

 

 

 

 

 

 

 

 

 

Target Uncorrected Resolution Differential Corrected 8B
FWHM and Beam E-loss FWHM Rest
Spread 7Be and 8B Frame
(MeV/c) (MeV/c) (MeV/c) gMeV/c) (MeV/c)
Telescope 1
Be 86i4 13.5:121 8 85:1:4 81i4
Nb 73:1:4 13.5:l:1 10 71:1:4 68:1:4
Au 67:4:3 13.5:l:1 8 65:1:3 62:1:3
Telescope 2 _
Be 140:1:20 12.1i1 8 139i20 133:1:19
Nb 87i10 12.1i1 10 86:]:10 82:1:10
Au 80:1:6 12.1:t1 8 79:1:6 75:1:6
by 4.4%.

Target out runs showed that the background was smaller than 0.2 counts per MeV/c

for all targets. Therefore, there was no background subtraction. The distributions

are displayed in 8 MeV/c bins.

4.4 Results

The present results yield p” distribution widths around 62-81 MeV/c in Telescope 1,
and show a systematic dependence of the widths on the target Z. These widths are
much smaller than those of normally bound nuclei, and based on this information
alone, one would conclude that 8B has a proton halo. On the other hand, the mo-
mentum wavefunction of a proton bound by 137 keV in a p-orbital Woods-Saxon
potential, that is adjusted to reproduce the proton separation energy and the 7Be

core radius, yields a momentum distribution width of 160 MeV/c [Esb95].

62

4.4.1 The P" Distribution from Nuclear Breakup Processes

In the Serber interpretation, the data from breakup in the Be target, where nuclear
process dominate the breakup, could be compared with the momentum wavefunc-
tion of the valence proton. However, the two are not in agreement (81 MeV/c vs.
160 MeV/c). Since the Serber approach fails to reproduce the data it appears that
the reaction mechanisms inﬂuence the ﬁnal momenta of the core fragment following
the removal of the p-orbital valence proton. Therefore, the details of the breakup

reaction must be included in the calculations.

 

*
5001- a
b 4

   
  

Be target.

200
p

 
 

 

I <
l 1 L 1 A l :L‘ * Iii!-

1700 1.00 1000 8000 2100

Momentum (MeV/c)

 

Figure 4.7: The data from breakup in the Be target compared with the predicted
p] distribution of a p-orbital proton bound by 137 keV in a Woods-Saxon potential.
The solid(dashed) line represents the distribution without(with) an absorptive cut-off
[Br095a].

In the spirit of the Friedman model and the Extended Serber model, discussed in
Appendix C, Brown et al. have explored the sensitivity of the predicted p" distribution
to an absorptive cut-off radius [Br095b]. When the cut-off radius is deﬁned as the

radius where the density of the valence proton exceeds the density of the 7Be core, only

63

the wavefunction outside this limit is considered. The resulting width of the proton
p“ distribution is 96 MeV/c, nearly in agreement with the data. This approach
is not rigorously correct, since it does not attempt to imitate the situation where
the wavefunction is sampled only at large impact parameters, but the method does

demonstrate that the observed distribution should be sensitive to these effects.

 

I ﬁ f T V T

0001- '

    

10°F Be target

Counts (arbitrary)
i

 

    

m 1 L L

30” 8100

 

 

 

I 1:1! 1 l 1 1 1 L l
1000 1000

Momentum (MeV/c)

predicted by Esbensen’s model that speciﬁcaly considers the m;

Figure 4.8: The PI
hsorptive processes are included [Esb95].

sub-states. Only a

In comparison, Esbensen [Esb95] is developing a model that considers the orien-
tation of the quantum sub-states of the p3,; wavefunction. The general model was
described by [Bar93] though reaction effects that could alter the distribution will be
added. Speciﬁcally, an interaction of the target with a proton in a mg=0 state (ori-
ented along the beam axis) would usually lead to a collision of the core with the target
nucleus, and possibly a disintegration of the core. Therefore, absorption of protons
that are in the m; = 0 sub-state is highly suppressed when measuring core fragments.
Esbensen ﬁnds that the m¢=0 sub-states contribute only 18% of the total breakup

cross section, instead of 33%, and this leads to a distribution of width 83 MeV/c.

64

In the future, diffractive effects, that broaden the distribution, will be added to the

model.
Measurement of 7Be ﬁagment P” from 1471A Mev 8B in a Dispersion
Matched Energy Loss Spectrometer

Recently, a group at GSI measured the P“ momentum distribution of 7Be fragments
following the breakup of 1471A MeV 8B. The resulting momentum distribution from

breakup in a light (carbon) target [Sch95] has a width of 81:1:6 MeV/c, in excellent
agreement with the present data.

4.4.2 The P” Distribution From Coulomb Breakup Processes

The magnitude of the valence proton wavefunction in 8B at large radii is of interest for
determining the rate of the 7Be(p,7)sB reaCtion that leads to high energy neutrinos
in the sun. A longstanding problem for solar models is an inability to reproduce the
8B formation rate in the sun, which is deduced from the observed rate of high energy
neutrinos reaching the earth. In an effort to solve the solar neutrino problem, recent
exclusive measurements of Coulomb breakup have been used to determine this rate,
via Detailed Balance, for energies of 600-1000 keV [Mot94]. The E2 contribution
to the breakup cross section must be known to extract the relevant E1 part of the

reaction rate from these experiments, but the size of the contribution is debatable

[Lan 94] .

The inclusive momentum distributions that we measure are sensitive to the in-
terference of the E1 and E2 contributions to the breakup process. The distribution

from the Au target has a noticeable asymmetry, the signature of the interference, and

permits an estimate of the E2 contribution.

The Coulomb breakup is calculated in a perturbative model similar to [Esb91], and

65

 

_ I T T ' l ' I l l .
2-5— —4
21)_— __
t. . :
a L5-—- __
L. 1- .
7:3 _ .
13 -
in
< _
r— -l
- «1
(15*— .E
r-
I.
f l 1 l 1

 

 

 

 

-100

Momentum (MeV/c)

Figure 4.9: [Esb95] The predicted shapes of the momentum distributions from E1
(long dashes), E2 (short dashes) and E1+E2 (solid) Coulomb breakup. An asymmetry
in the shape of the distribution is the result of the E1+E2 interference.

 

 
 

 

 

 

_ l T I l’ 1 T HTT I 1 F I T [m T 1 l '1
- 1
12 — ~
1— -1
r- —1
10 — —
1— -4
>- -1
N 8 — —
>< .. _.
1— cl
6 —- _
i- -1
- 4
4 — —
— L l l 1 i 1 l l 1 i l 1 l J i l l l J -
—1.0 -0.5 0.0 0.5 1.0

Figure 4.10: The X2 minimization to obtain the E2 strength that best reproduces the
observed 7Be p” distribution from 83 breakup on the Au target.

66

by varying the E2 strength ( |E1+ kE2|2 ) it was possible to adjust k to reproduce the
data [Esb95]. Negative values of k represent the mirror'reﬂection of the asymmetric
distribution which gives a distribution sloped in the opposite direction. The X2 is
minimized when k=0.25 in this model; this corresponds to an E2 contribution that
is 0.7% of the total breakup cross section. In Figure 4.11 the resulting distribution

is transformed into the lab frame, convoluted with experimental effects and shown in

comparison with the data from breakup in the Au target.

 

Au target

150

V Tr TY—F
4

 

 

 

INC 1000 1000

Momentum (MeV/c)

I data compared with the predicted distribution

Figure 4.11: The 7Be fragment p
obtained when k=0.25 [Esb95]. The corresponding E2 strength is 0.7% of the total

strength.

4.4.3 Angular Dependence of the Width of the P” Distribu-
tion

Since the Cartesian components of the momentum wavefunction may not be sepa-
rable into independent functions of each coordinate [Rii93a], the measured parallel
momentum distribution could be sensitive to the angular coverage of the detectors,

as described in Appendix E. By measuring the p” and p l of the fragments simul-

67

taneously we observed these effects. The p i perturbations associated with the Nb

and Au targets appear to distort the 7Be angular distributions. Therefore, we brieﬂy
comment only on the Be target data.

When a Lorentzian distribution (I‘=95 MeV/c) is assumed the limited acceptance
leads to p” distribution widths of 80 MeV/c and 180 MeV/c in the two telescopes.
Although the measured distribution widths are 81:1:6 MeV/c and 133:1:19 MeV/c
the dependence is similar to that expected for a Lorentzian momentum distribution,

though not as extreme. Poor statistics in Telescope 2 limit discussion on this topic.

In contrast, the fragment p" distribution widths from 250A MeV 12C ions [Kid88]

remain constant out to at least 4.5°, and indicate a different, more Gaussian-like

behavior in the fragmentation of tightly-bound stable nuclei.

4.5 Valence Proton Radius

The p“ distributions of 7Be from 8B on both light and heavy targets appear to be
signiﬁcantly inﬂuenced by reaction mechanism effects. Predictions that agree well
with the data assume a single-particle p-orbital proton bound by 137 keV. The cor-
responding rms radius for the valence proton is large (4.24 fm) [Esb95] and is taken
to indicate that 8B possess an extended proton distribution for the valence proton.
A comparison of the probability distribution of core nucleons with the distribution of

the valence proton shows that at large radii the density of the proton is dominant.

68

 

N
d
'a
I
i
—1
d
d
i
q
—
q
q
.1

Core Protons 1

  
  
 

Core Neutrons

8B

 
 

”

 

 

 

E
0.. 1
Valence Proton 1
o ‘ L A L L
0 I I. I [I

Radius (fm)

Figure 4.12: [Br095a] The probability distribution of nucleons in 8B.

. Chapter 5

Summary and Outlook

The two measurements of breakup fragment momenta in the direction parallel with
the beam direction described in Chapters 3 and 4 are in good agreement with pre-
dictions based on the loosely bound nature of the valence nucleons. However, the

fragment momentum distributions are found to have very different sensitivities to the

reactions that lead to their formation.

In the ﬁrst case, 10Be fragments following the breakup of 11Be, the data agree with
a simple projection of the spherically symmetric wavefunction of a 231/2 neutron bound
by 500 keV in a Woods-Saxon potential. The inﬂuences of reaction mechanism effects
appear small and are not included in our comparisons with the data. For example,
the p” distribution expected from Coulomb dipole excitation leads to a width that is
roughly 14% narrower than that of the projected momentum wavefunction [Esb94]. A
similar sized decrease in width is found by [Bro95a] when a radial cut-off, described in
Section 4.4.1, is included in the projection of the momentum wavefunction. Therefore,
it appears that the observed momentum distributions reﬂect the momentum of the
halo neutron of 11Be with only small inﬂuences arising from the reaction mechanisms.

The situation for 8B is somewhat different. The momentum distributions are

narrow, which indicates a different behavior from the fragmentation of normal nuclei

69

70

where widths of 180 MeV/c FWHM or larger are expected [(30174]. However, the
simple projected wavefunction of a p-orbital proton bound by 137 keV in a Woods-
Saxon potential is not in agreement with the observed p” distributions. When reaction
mechanism effects are included in the calculations the predicted widths are decreased
substantially, and are in much better agreement with the data. In fact, it appears that
the p“ distribution observed for breakup on the Au target reveals the contribution of
E2 strength in the Coulomb breakup calculation. In the future, Esbensen will make
more detailed dynamical calculations in order to account for the higher order effects

of the reaction mechanisms [Esb95].

The results of the two measurements indicate that when the extent of the halo
is large, as in the case of 11Be (rms radius=6.5 fm), the reaction mechanisms do not
strongly inﬂuence the breakup core fragments, and that simple measurements of the
core fragments reﬂect the properties of the halo nucleon. In the case of 8B, the valence
proton does extend considerably (4.25 fm). However, since the proton density is not
as prominent at large radii as the neutron halo of 11Be, the reaction mechanisms

inﬂuence the 7Be fragment momentum distributions from 8B breakup.

Appendix A

Theoretical Discussion of Halo
Nuclei

Theoretical studies of halo nuclei have progressed, and the inﬂuence of various po-
tentials on the wavefunction of halo nucleons has been studied by [Rii92]. There is
a clear dependence of the extent of the wavefunction on the binding energy. How-
ever, when the rms radius of the wavefunction of a neutron in a square-well potential
was compared with that of a neutron in a Gaussian shaped potential, the wavefunc-
tion associated with the Gaussian shape extended slightly further. In this case the
”softer” wall of the Gaussian potential permits a somewhat larger halo. Because the
dependence on the potential shape can be observed in such a simple calculation, it is
clear that realistic potentials should be used for comparisons with data. Riisager et

01. approach their studies using Gaussian shaped potential shapes,
U(r) = 50 exp(—r2/bz). (A.1)

The value b=2.1 fm is realistic for a nuclear core radius.

A.1 Importance of the Binding Energy

Consideration of a valence s-orbital neutron shows that as the binding energy in~

creases the wavefunction of the valence neutron is ”squeezed” into the core, Fig

71

72

ure A.1. Since other potential barriers, such as the angular momentum barrier and
the Coulomb barrier, also play key roles in determining the long-range characteristics

of the wavefunction, these potentials must also be considered.

 

8 T v rvvvvv' V 1 V YIYY'I v 1

 

 

 

 

 

-E 0’ (Mev m!)

Figure A.1: [Rii92] The dependence of the rms radius of valence neutrons on binding
energy (b2 2.1 fm).

Importance of the Angular Momentum Barrier

The halo nucleus 11Be is not complicated by the angular momentum barrier. However,
to understand the long range behavior of the wavefunctions in other weakly bound
nuclei with valence nucleons in non-s-orbitals, for example 8B (13:1), this potential is
important. The angular momentum barrier is,

’12

2mr2'

 

V(r) = «13 +1) (A.2)

It appears, from Figure A.1, that signiﬁcant penetration can occur only for s- and

p-orbital neutrons that have small binding energies. The wave function diverges as

73

E‘l/2 for an s-wave neutron, and as E'l/4 for a p-wave neutron. Higher angular
momentum orbitals remain ﬁnite regardless of the binding energy. Although there
have not yet been any studies of neutron halo nuclei that have a single halo neutron in
a non-s-orbital a few candidates do exist (”B,15C and 17C). When studies of a broader
range of nuclei have been completed, our detailed understanding of the inﬂuences of

the angular momentum barrier will be greatly enhanced.

A.2 Importance of the Coulomb Potential Barrier

The Coulomb potential barrier is large in high Z nuclei. However, it appears that light
proton rich nuclei like 8B can possess an extended proton distribution. Much like the
angular momentum barrier, the Coulomb barrier inhibits a halo in in weakly-bound
proton-rich nuclei. Assuming that the charge distribution is Gaussian, the Coulomb

potential is,

Z a OZCOTC 2
U(T')Couzomb = —_h—l_7—?_erf(r/bcorc)- (A3)

Here er f is the error function. As the binding energy decreases the rms radius of
the wavefunction increases. However, near 100 keV fm2 in Figure A.2 the barrier
penetration of the proton wavefunction is dominated by the Coulomb potential, and

below this value there appears to be little sensitivity to the binding energy.

74

 

 

 

 

 

 

 

0
[=1 .......... ..
2 ........
11 14 1111 1 1
0 1 -l 1
IO 1 10

—E 132 (MeV fm’)

Figure A.2: [Rii92] The dependence of the rms radius of a valence proton on binding
energy for a nucleus that has four protons in the core, for example 8B.

Appendix B

Experimental Evidence for Halo
Nuclei

Simple problems of barrier penetration are well understood, and the current mea-
surements on 11Li and 11Be taken independently are consistent with an interpretation
that includes a halo. When considered collectively, the measurements overwhelmingly

favor the existence of neutron halos.

B.1 Enhanced E1 Transition Strength

The ﬁrst excited state of 11Be is only 320 keV above the ground state , and the ﬁrst
observation indicating a neutron halo was in 1983 when a very short gamma decay
lifetime (1.7 x 10-13 seconds) of this state was observed [Mil83]. Such a small lifetime
requires that large distances, on the order of the size of a halo, must be involved in

the transition.

B.2 Measurements of Reaction Cross Sections

Reaction cross section measurements appear to be a straightforward method of mea-
suring nuclear sizes, since the cross section is nearly proportional to the square of the

rms radius. In the Berkeley experiments [Tan85a, Tan85b, Tan88] isotopic identiﬁ-
75

76

cation was obtained for reaction products from interactions of 790A MeV He, Li, Be
and B isotopes in a range of targets. Therefore, reactions that changed either the A
or Z of the projectile were detected. The ”interaction” cross sections were initially

analyzed using a simple ”hard sphere” Glauber-model approach,
01 = 7r[R1(target) + R1(projectile)]2. (B.1)

However, the radii extracted for the 3He and 4He isotopes indicated that a different
approach was necessary to interpret the data. Using a model developed by Karol
[Kar75] that treats nuclei as ”soft” spheres rather than ”hard” spheres or black disks, _
values of the rms radii of the proton, neutron and matter density distributions were

obtained.

The systematics of the rms radii provided a measure of the growth in nuclear size
with increasing atomic number, and a comparison with the well known 7' = roAl/3
rule. It was observed that the rms radius of most light p-shell nuclei was nearly the
same, lying between 2.3-2.4 fm. However, the neutron-rich nuclei 6He, 8He, 11Be,
l4Be,_and llLi exhibited nuclear radii that were considerably larger than other more

tightly bound nuclei, see Figure 1.1.

The information provided by the difference between the rms radius of the neutrons
and the rms radius of the protons, seen in Figure 3.1, clariﬁes that this group of nuclei

possesses thick neutron skins or halos.

The reaction cross sections of 6He (Sgn=970 keV) and 8He (S2n=2100 keV) showed
indications of extended neutron distributions. However, a deformation due possibly
to clustering could lead to erroneous values for the rms radii derived from cross section
data for the proton and neutron density distributions. For example, a thick neutron
layer surrounds the nucleus 8He, though the binding energy of the valence neutrons is

large. In this case, it appears that the neutron layer is simply the result of having two

 

 

 

 

 

 

.0 l rm! r T r I Y T T I I I r rm r r 1 T r
1 _ l I I l I

_ ° He

_ ° Li -
0
i5: : K B .
5 _ i 1‘"
m _
I 0~° 2% f T

I
”.7 ~ 5 I f I -
g L' i I 4
o.)
2 ~ 4
3 ~ §
5 -o.5 - § —1

a: _ .

’ -1

_10 1 I l J._L l l l 1 l l l l 1 i l l l l l 1 1 l L l 1 J
o 2 ' 4 . a a 10

Neutron Number

Figure B.l: The difference between the proton density rms radius and the neutron

density rms radius obtained from the 790A MeV cross section measurements. Data
from [Tan88].

protons and six neutrons in the nucleus. This leads to rather different Fermi energies

for the protons and neutrons.

B.3 Energy Dependence of the 11Be Reaction Cross
Section

A two- or three-body approach for halo nuclei is permitted only by an assumption that
there is a well deﬁned distinction between the core and the halo. To investigate this
assumption Fukuda et a1. [Fuk91] measured the reaction cross section of 33A MeV
11Be and compared with the 790A MeV data of Tanihata [Tan88]. Consideration
of the energy dependence of the nucleon-nucleon cross sections, permitted a more
accurate determination of the proton and neutron density distributions than could

be obtained from the 790A MeV data alone. As can be seen from the increase in the

78

 

‘ - ----- ﬁned (Gaussian + tail)

 
   

 

 

--- fitted (Gaussian only}
Al target deformatioMB-O- 5)

2.0 ‘

:5;
1 6 ‘

a:

D
1.2J
0.8

 

33 A MeV 790 A MeV

Figure B.2: [Fuk91] The energy dependence of the 11Be reaction cross section. The
two models that ﬁt the trend of the data both include a large halo, while a deformation
in 11Be does not reproduce the data.

79

nucleon-nucleon cross sections,
app(33MeV)/app(790MeV) = 2.0 (B.2)

and,

apn(33MeV)/apn(790MeV) = 7.2, (B3)-

at lower energies the reaction cross section is sensitive to the neutron distribution at

a density that is much lower than is the case for the cross section data of Tanihata.

An increase in the reaction cross section, similar to that expected for a neutron
halo structure, was observed. However, because the increase in the cross section
could also be produced by deformation, for example due to clustering in the nucleus,
the data were compared to predictions from several halo models and cluster models.
The data favors a Hartree-Foch calculation with a well depth that gives the neutron
separation energy. Because the change in the cross section is most sensitive to the tail
of the density distribution, the observed behavior shows the presence of the extended
tail in the wavefunction of the valence neutron. However, the method is not very

sensitive to the details of the wavefunction.

B.4 Large Electromagnetic Dissociation Cross Sec-
tion

The nucleus 11Li became the focus of many experiments because of its three-body
nature and very low binding energy (295 i26 keV). The low binding indicates that the
Coulomb breakup cross section on heavy targets should be enhanced [Han87], and it
has been suggested that a soft dipole mode could exist that is ﬁguratively represented
as an oscillation of the core against the halo neutrons. This oscillating mode would
have an energy much lower than the giant dipole resonance, which represents the

oscillation of all of the protons in the nucleus against all of the neutrons.

80

The Soft Dipole oscillation is easily excitable via electromagnetic excitation when
a halo nucleus passes near a heavy nucleus at relativistic velocities. A study of the
Coulomb breakup cross section of 70A MeV 11Li showed a strong target Z dependence
[Bla91]; the reaction cross section changed from 1.24 barns on C to 7.23 barns on
Pb. This supported the neutron halo model for 11Li. However, the data were also
consistent with breakup cross sections expected from excitation of the giant dipole
resonance. Therefore further experiments were necessary in order to fully probe the

wavefunction.

B.5 The 1I1Li Dipole Excitation Strength Function

To determine what breakup mechanism accounts for the reaction cross section data,
soft dipole excitation or giant dipole excitation, a kinematically complete measure-
ment of the full three-body systematics for Coulomb dissociation of 11Li was per-
formed at the NSCL [Sac93, Iek93]. A beam of 30A MeV llLi nuclei impinged on a
Pb target and the decay energy spectrum of 11Li from Coulomb Excitation was mea—
sured using a position sensitive 9Li fragment detector and a wall of neutron detectors.
The dipole excitation strength function was deduced from the relative energy spec-
trum of the 9Li-n-n system and showed a low energy peak near 1 MeV. Since the giant
dipole resonance energy should correspond to an excitation energy near 77A‘1/3 MeV

(34 MeV) this observation strongly favored the neutron halo interpretation.

Early interpretation of the data assumed that a soft dipole resonance state par—
ticipated strongly in the process. However, the shape was also consistent with that
expected from breakup into the continuum, Figure B3 The observation of Coulomb
reacceleration of the heavy core after breakup (see Appendix F) provided evidence

that the breakup time scale was very short and was consistent with a direct breakup

81

 

4000 —— ﬁI

2000 *-

 

Yield (Counts/MeV)

 
 

 

 

   

{i

lllllllllllllllllll

0.0 0.5 1.0 1.5 2.0

111.1 Decay Energy (MeV)

Figure B.3: [Sac93] The decay energy spectrum of 11Li from Coulomb excitation. A
Monte Carlo simulation that accounts for the acceptance of the measurement indicates
a peak near 1 MeV in the dipole excitation strength function. The low energy peak
is associated with a large separation distance for the 9Li core and two neutrons that
comprise llLi

82

into continuum states. Because the breakup processes are direct, measurements re-

ﬂect the ground state properties.

3.6 The 11Be Dipole Excitation Strength Function

The large halo-core separation distance can be directly related to a concentration
of strength at low energy in the dipole-excitation strength function. Therefore, the
10Be-n relative energy spectrum from breakup on a Pb target [N ak94] was measured
which permitted a determination of the E1 strength function of 11Be. In the model

used by Nakamura et al. the rms radius of the halo neutron that reproduces the data

is 6.4:l:0.7 fm.

The measurement of the 10Ben center of mass scattering angle permitted a com-
plete investigation of the Coulomb post-acceleration effect (Appendix F), which im-
plies that for a direct breakup charged fragments will emerge with a higher average
energy than neutrons. This effect is strongly related to the impact parameter, and as
can be seen in Figure 3.4, this trend is followed. Therefore, the presence of Coulomb
reacceleration is further evidence that the breakup process is direct, into continuum

states, a key assumption in the Serber model.

83

 

 

  

 

     

 

 

 

. 1°I*"'I‘ 1*"'1""1
1- i
O I
> °T l
0 I I
a 1 II
--* O;- .
4
A I t
b 1
0'..- -b'- l mBe "
D
I
V b 1
’1- -1 1-11 -1 mnl--- 1-:::
I I T"'TI'T'TT""I
.— C
O , -l
\ 0 - neutron
D
’7. >
:I ’ l
I
h—' OI ‘
A r
r
- ’
Q. -6 7
p
V .
)-
-AL----l--ax1---xl----1x---
-tO “
0 0.5 1 1.5 2 8.6 3

Scattering Angle 8 [deg]

Figure B.4: [Nak94] The mean momentum of breakup neutrons and 10Be cores plotted
vs. the center of mass angle. The trend expected from Coulomb reacceleration is
observed. This strongly favors a direct breakup interpretation of the data from 11Be

Coulomb breakup.

Appendix C

Models for Fragment Momentum
Distributions

C.1 The Goldhaber Model

The standard model for projectile-like fragment momentum distributions in interme-
diate and high energy fragmentation of stable nuclei is the Goldhaber model [Gol74].
In this model the width of fragment momentum distributions is related to the Fermi
energy of the projectile; a Boltzman energy distribution is assumed. When a reaction
strips away a portion of the projectile, the momentum distribution for the remaining

part, the fragment, is

Pfragment = 2 Pi; (Cl)

Afragment

p; is the initial momentum of the fragment nucleons in the projectile prior to the
breakup. The distribution is statistically related to the number of nucleons in the
initial projectile and the number of nucleons in the ﬁnal fragment. The momentum

distributions are Gaussians with widths,

 

0' : aoJrAfv-agmendApv-ojcctile — Afragment) . (C2)

Aprojectile -' 1
The parameter 0'0 is related to the Fermi momentum and is generally near 80 MeV/c.

Hence for one nucleon removal in a fragmentation reaction the fragment momentum

84

85

distribution width 0:80 MeV/c is anticipated; this leads to a width of 188 MeV/c
(Full Width at Half Maximum, F WHM).

0.2 The Serber Model

In a two-body system the Serber assumption [Ser47] implies that the momentum dis-
tribution of the core fragment directly reflects the momentum wavefunction of the
halo neutron. Since the Serber model does not include reaction effects, the momen-
tum distribution of halo and core fragments should be identical to the momentum

wavefunction of the halo neutrons.

C.3 The Friedman Model

The Friedman model includes inﬂuences of reaction mechanism effects [Fri83]. In ﬁrst
order the width of the momentum distribution is related to the cluster separation
energy and the reduced mass of the halo neutrons, a or m . This is related to
the Yukawa parameter p (Equation 1.3) which describes the exponential fall-off of
the wavefunction. However, the Friedman model assumes an absorptive cut-off limit
which distinguishes it from the Serber model. It is thought that central collisions,
where the core impacts the target, will result in a disintegration of the core. Collisions
that remove only a few nucleons are generally peripheral, and this assumption leads to
the absorptive cut-off. The absorptive cut-off limits consideration of the wavefunction

only to impact parameters that are larger than the cut-off radius.

An additional effect related to the transfer of potential energy into kinetic energy
as the projectile exits the Coulomb ﬁeld of the target nucleus is termed Coulomb
distortion. When the breakup occurs the potential energy will be divided among the

projectile nucleons based on their charge. As the fragments accelerate out of the

86

Coulomb ﬁeld of the target the p” distribution is distorted.

 

The widths predicted by the Friedman model or MAL-3115,.) are similar to

. However, the treatment of

 

those predicted by the Goldhaber model (cc
Friedman moves a step beyond that of Goldhaber by including speciﬁc details about
the structure of the fragment, the separation energy, that may have a strong inﬂuence

on the fragment momentum distributions.

CA The Extended Serber Model

The extended Serber model [Uts85] was developed by Utsunomiya and includes both
Coulomb distortion and an adsorptive cut-off limit. A key difference between the
extended Serber model and the Friedman model is that the absorptive cut-off limit is
determined by a critical radius where the constituents of the projectile take on a two-
body nature. The core halo separation distance is characterized such that either the
halonucleon can be absorbed without disturbing the core (R > He) or an interaction
with the target would result in a destruction of the core (R < RC). Although the
model was ﬁrst applied to the stable nuclei 12C, 13C, and 1“N prior to the discovery of
halo nuclei, this approach seems well suited for explaining the fragment momentum

distributions from halo nuclei.

Appendix D

Coulomb Dissociation

Coulomb excitation occurs when a projectile enters the Coulomb ﬁeld of a target
nucleus. A summary of [Ber88] follows. At high velocities the electric ﬁeld of the
target nucleus is contracted into the plane which is transverse to the incident direction
of travel (chosen as the z axis in Figure 2.3). The resulting electric and magnetic ﬁelds
that the projectile nucleus will encounter, assuming straight line trajectories, are given

by the equations,

_ Ztargct e7vt

 

 

E” = [b2 + (MP/2 (”'1’
_ —Ztargete7b
15* ‘ [122+ (WP/2 (D2)
8; = ,6 X E; (13.3)
and,
B; = 0. (D4)

Where v is the speed of the projectile, Bzv/c, 1/7 = \/1 — ﬂ? and t is time (t=0 at
the distance of closest approach).
When the interaction time is sufﬁciently short (At ~ b/7c) the electric ﬁelds can

be approximated with two plane waves, one traveling parallel with the trajectory of

the projectile and the other transverse to the trajectory of the projectile. The energy

87

88

incident on the projectile nucleus is,
I — —c—|E x Bl (D 5)
— 47r ’ °

and the spectrum of virtual radiation, ie. the number of equivalent photons of energy

ha) is given by,

22 2
N(.,.)- wet—2w

1r 7v)(%)2[K12(wb/7v)+7713K§(wb/7v). (us)

Here 021/137, and K,(wb/7v) are Modiﬁed Bessel functions. The probability for

excitation via dipole (E1) photons is,

P(=b) /I(w, b)()a,hw (1(a)) (13.7)

= / N(w, b)a,(hw)¥-. (D.8)

The momentum distribution of fragments from Coulomb dissociation reﬂects the

dipole excitation strength function, which is related to 0.,(Ebrcakup).

Appendix E

The Inﬂuence of a Limited
Acceptance on P“ Distributions

It has been pointed out that observed widths of the p” distribution might be affected
by an incomplete acceptance for the breakup products [Rii93a], and the necessary
corrections are not negligible for most measurements to date. In this section the
effects that an incomplete angular acceptance for breakup fragments can have on the
parallel momentum distributions are discussed, and analytical results for some simple
cases are obtained. Finally, a correction procedure that is useful in the more general

case is described.

E.1 Nature of the Problem

The momentum distribution of the halo particles is not known a priori; however, two
limiting forms, Lorentzian and Gaussian, are in common use. Hansen and Jonson

[Han87] assume a Yukawa spatial wavefunction (Equation 1.2). When I‘ = 25 / p, and

P = pxéx + me} + Pzéza (El)

89

90

the Fourier transform of the Yukawa radial wavefunction is a Lorentzian momentum

wavefunction for the neutrons,

r 1
‘I’(p) = l/E;p2 + F2/4- (E2)

Therefore, a Lorentzian description of the halo and core fragment momentum distri-

 

butions is a reasonable ﬁrst approximation.

On the other hand, measurements of the fragment momentum distributions from
tightly bound normal nuclei such as 12C, 14N and 16O, [Kid88, Gla90, Si188, Van79], are
well described by Gaussian distributions. The Goldhaber model Section C.1, relates
the width of the fragment momentum distribution to the Fermi momentum of the
incoming projectile, and assumes a Boltzman energy distribution for nucleons. This
leads to Gaussian momentum distributions with a near 100 MeV/c (FWHM=2.355

a).

Momentum distributions following the breakup of lightly bound halo nuclei have .
often by described by Gaussians or sums of Gaussians; however this description must
be regarded as phenomenological, with no theoretical basis. More realistic distri-
butions are generally intermediate in shape between Gaussian and Lorentzian, so
that these distributions provide good starting points for dealing with the acceptance

problem.

The situation for a Gaussian distribution is extraordinarily simple. Because it is

separable into independent functions of the Cartesian components,

ﬂ) ' (13.3)

1w*(p)w(p)l = Wad 2,.

_ Pg

——1 ("”3”) ('2) (”3)
_ Wexp 202 exp. 202 exp 202 ,

 

 

91

a limited acceptance in p, and p, does not change the shape of the distribution of
p,. Thus measurements of breakup momenta from normal, stable nuclei are at most

weakly inﬂuenced by acceptance effects.

However if the momentum distribution is non-Gaussian (e.g. Lorentzian or the
sum of Gaussians) the incomplete acceptance for fragments can change the observed
momentum distribution. The effect is rooted in the inseparability of the Cartesian
components of the momentum density distribution. Key points relevant to measuring
non-Gaussian momentum distributions are the following, an analytical solution for
the predicted line shape of a Lorentzian distribution after passing through a limited
acceptance, evidence that a strongly limited transverse acceptance (pinhole) leads
to a increase in the width of the p” distribution when the observation angle is large
enough, and a comparison with recent observations of the breakup of 11Li on Be and
U targets. Finally a simple approach to estimating transverse acceptance effects when
the analytical form of the density distribution cannot be integrated in closed form is

outlined.

E.2 Results

The measured measured momentum distributions are related to the momentum den-

sity distribution which is the square of the momentum wavefunction,

P(PL) = I‘I’I(PL)‘I’(PL)|- (B5)
In the lab system,
PL = p38} + Pyéy + ((1/7)Pz + Pave.)€n = Pie“; + P119311, (36)

when pm,c is the fragment average momentum, and 7 is the relativistic correction

factor. Observed momentum distributions reﬂect a projection onto a particular axis,

92

i.e. an integration of the momentum density distribution over the other coordinates

in momentum space.

E.2.1 Lorentzian Distribution

For a Lorentzian distribution

F dpx dpy

10(1):.) = W [p, + 1.2/4],. (13.7)

 

The transverse (.L) and parallel (II) momentum distributions of this distribution are,

assuming complete acceptance:

Poo.) = /_:P(PL)dP|| = ZED, +11 (113/2’ (E8)

 

and
l
(Pu - pavc)2 + III/4°

Notice that in the lab frame the p" distribution is broadened by 7, since p“ — pm,e =

 

P(pu) = f: f: P(pL)ddePy = 2% (E9)

(1 / 7)pz. When the entire momentum space is not sampled, the integration limits are
ﬁnite, and P(p”) is modiﬁed. To simplify the integrations the transverse acceptance
limits are taken to be rectangular in shape, and are given by p31,". = ipaue sin(A6/ 2)
and pyum = ipav, sin(A¢/2). Then the observed fragment parallel momentum distri-

bution is [Mat94],
F/1r x
(pll " pave)2 + F2/4

 

Pobs(P||) =

. Pylim . ‘
172:1thth (\/(pII-pa.e)’+piz.m+F2/4) + pyhm Atan (Q (PII-P°v¢)2+Pitim+r2/4)

\/(PII - Pave)2 + Pin-m + I‘2/ 4 \/(Pu - pm)2 + 1931..., + 1‘2/4

 

] (13.10)

The inﬂuence of a incomplete acceptance is demonstrated in Figure E.l, where the
parallel momentum distribution of a Lorentzian distribution (F=50 MeV/c) is given
for complete acceptance and for an acceptance similar to that of the medium accep-

tance mode of the A1200 Spectrograph at the National Superconducting Cyclotron

93

Table E.1: Widths of the p” distribution from a Lorentzian momentum density dis-
tribution (I‘=50 MeV/c) that passes through various aperture sizes.

 

 

 

 

 

 

 

Aperture Angular Acceptance Width Transmission
(mrad) (FWHM) [ % ]
Complete A0 = A¢ 2: in I‘ 100.
A1200 A0 = i20 A¢ = :l: 10 0.86 F 58.2
Line A0 = in A¢> = $0.5 0.77I‘ 4.59
Pinhole A0 = A¢ = :l:0.5 0.64I‘ 0.313

 

Laboratory. The solid line is for complete acceptance, (while the long dashed line
shows the change in shape and transmission efﬁciency for the A1200 acceptance
(A0 = 40 mrad and A925 = 20 mrad). In the center of mass (cm) frame the full
width at half maximum (FWHM) with complete (A1200) acceptance is 50 MeV/c
(43.1 MeV/c). While the decrease in width is only 15% there is a marked decrease
in the tails of the distribution. This corresponds to the parameters for the p” dis-
tribution of 10Be fragments from 11Be in the case where simple assumptions lead to
a distribution which appears to be close to a Lorentzian momentum distribution.
For completeness the distributions observed when the acceptance is a horizontal line
(A0 = in mrad and A45 = :l: 0.5 mrad) and when the acceptance is a pinhole
(A0 = A43 = :t 0.5 mrad) are included, as suggested by [Rii93a]. Details are found
in Table E.1.

E.2.2 Dependence of Width on Angle

Although an incomplete acceptance can inﬂuence the observed distribution this fea-
ture can be used to characterize the nature of the momentum distribution even with
low statistics experiments. For a Lorentzian density distribution, the width of the
p” distribution depends strongly on the scattering angle when p _L > I‘. This effect is

most easily explained for a pinhole transverse acceptance. Then the p" distribution

94

 

 

 

 

A 1.2 1 r I r 1 1 171 1 1 1 r 1 1 1 1
m ’- . l l l
+> I l
.E _ _4
1.0 —- .__
:5 3 Z
a I I
0.8 — —‘
(U — f\ 1
‘4 _ I -1
4-9 C /\\ ﬂ
"" / \ I
”D 0.6 ’_ ’3'. \ __‘
5" l: l“ .'\\\ 8
(U 1" I]: -(
v 1- ’." -.\\ -1
0.4 _ / .. '.\ —1
Q1 I //’;' \\ 2
PC _ // .\\ ..
— _' \ -4
\ 0'2 .— // \\\ —..
b )- //l '. \ -
N _ // .' \\ a
)— / -' '. \\ -4
PO ‘441 " 1 1 l 1 1 "1‘ PF 1 1
-200 --100 0 100 200

Momentum ( MeV/c )

Figure E.l: The expected p” distributions for a fragment with pav¢=3400 MeV/c
and with I‘=50 MeV/c are shown for complete acceptance (solid), for an acceptance
similar to that of the A1200 Spectrograph medium acceptance mode (long dashes),
for a pinhole acceptance (short dashes X10), and for a horizontal line acceptance
(dots X100). See Table l for details.

95

is given by evaluating |\II'\IJ| at a value of p J_ corresponding to the position of the

pinhole.
F I .
2W2 [(PII ‘ pave)2 + 19.21. + I‘2/4l2 ,

p i = pave sin(0(ab) and 01a), is the detector angle with respect to the beam. The width

 

Pobs(pll) = (13.11)

W of the p” distribution is,

W = 27005 —1(/p1 + r2/4 FWHM. (13.12)

When the pinhole is placed at 0° (i.e. p _L = 0 MeV/c) the width is (cm frame)
W=0.64I‘ F WHM, while for large angles the width of the parallel momentum distri-

bution depends linearly on p _L.

Figure E.2 shows the p” distribution width (cm frame) expected for a small an-
gular acceptance (A0 = A¢=12 mrad). The solid line is for a Lorentzian momentum
distribution with I‘ = 50, while the dashed line is for a Gaussian momentum distri-
bution (no acceptance dependence is expected). The data shown are from a recent

experiment [She95] on the break up of 37A MeV llLi in Be and U targets.

The Be target data follow the trend predicted for a Lorentzian momentum density
distribution although the increase in width is somewhat smaller than predicted. This
is consistent with the conclusion that the measured 9Li momentum distribution is
similar to that for a Lorentzian. The distributions from breakup in the U target are
in better agreement with a Gaussian momentum density distribution. This may result
from effects of the Coulomb-excitation mechanism [Esb94] that dominates breakup

in the U target and/or from ﬁnal state interactions.

E.2.3 Complex Momentum Distributions

The data of [She95] show that in some instances the shape of the p" momentum

distribution is sensitive to the collection of breakup fragments. However, these data

Width of PH Distribution (MeV/c)

Figure E.2: The expected p" distribution from a Lorentzian density distribution
(solid) and a Gaussian density distribution (dashed) compared with data from the
breakup of 11Li in the S320 [She95]. For this case p 1=paue sin(0) where pavc=2300

300

96

 

250

200

150

100

50

ITII

 

IIIIIIIIIIIIIIIITTTIIIIII

——(
—1

 
 

1:1 0 ..
I1> ED ‘
C CD

   

LL

lllllllllllliulJiJJ

 

 

OF-
)—

MeV/c. Details are given in the text.

Angle (Degrees)

97

also suggest that the effect is not as pronounced as expected from fragments having
a Lorentzian shaped distribution. This indicates that fragments are emitted with a
distribution that is more complex than either a Gaussian or a Lorentzian shape. In
order to estimate the effects of incomplete acceptance on complex (non-Lorentzian,
non-Gaussian) momentum distributions, a technique based on the superposition of
Gaussian distributions, to ﬁt more complex shapes is developed here. The basic
assumptions are that the momentum density distribution is symmetric in three di-
mensional space and that the density is continuously decreasing as |p| increases. It
appears that such a distribution can be well described by a linear combination of

Gaussian distributions.

~ _|_-p¥.| -|pi.|
p(p1,)~ Aexp —T + Bexp 2 + (E.13)
20A 203

If the original distribution and the linear combination of Gaussians have closely the
same probabilities for all important momenta in space, P(px,py,p”), then the two
distributions will give essentially the same observations, regardless of the nature of

the acceptance.

The inﬂuence of an incomplete acceptance on a superposition of Gaussian distri-
butions is simple to determine. Integration of each Gaussian distribution over the
acceptance limits provides the transmission coefﬁcient, 6, for each of the component

Gaussian distributions. For example, for the ﬁrst component above,

(MA) ([27:03,“ pl—___ (pt: —_p_§,y )p’ld dp"(E'14)

The ﬁnal step necessary to obtain the p" distribution is to superimpose the Gaussian

 

distributions taking into account the transmission efficiencies.

 

 

Pob,(p”) z 6(0A)Aexp(— (p02; zpave)2 ) + C(03)Bexp(_(p"2;§avc) ) + (E.15)

This approach has the advantage that the dependence of the transmission coefficients

on the acceptance can be represented as a single curve, see Figure E.3. Thereafter,

98

any theoretical result can be ﬁtted by a series of Gaussians, the corresponding trans-
missions taken from the curVe, and the ﬁltered result compared to the experimental

data.

 

 

 

 

 

 

A I f I I
3 : l l ‘ I:
C: 1.0 -—
(1) C I
"—0 (_ -1
2 _ ..
«H 0.8 -— __
q... ~ ..
CD ~ _
g 0.6 T —-
"—0 "
m p .1
(I) b a
”-0 - .4
a 0.4 — —
m C I
c: g 2
2 0.2 —— _.
4—3 : .
V _ 3
w 0.0 i- l l l l l l l L l l L l l l .4
5 10 20 50 100 200

0 (Gaussian Width, MeV/C)

Figure E3: The transmission of the A1200 spectrograph for a Gaussian distribution
as a function of its standard deviation. The curve is calculated for 10Be fragments
with pave=3400 MeV/c produced in the A1200 medium acceptance mode, (A0 = 0:20
mrad and A¢ = :l:10 mrad).

This approach has been tested for a Lorentzian distribution where the analytical
results of Equation E.10 are available. In Figure E.4 are shown the results of fitting
a Lorentzian distribution with three Gaussians (two Gaussians were not adequate).
The resulting ﬁltered distribution agrees well with the analytical result, except in the
far wings of the distribution, and predicts the width of the observed p” distribution
within 1%. This approach is adequate at 0°; however, at detector angles other than

0° it appears that a signiﬁcantly larger number of Gaussians are necessary in the ﬁt.

99

 

 

      
  

 

 

 

 

 

 

 

 

 

1.2 FVY'IIHITYIIIYVIV'[‘IVIYVITYV': YI'TVTTI 1‘ I [V r' fl
" I 1.0007
t'. _. : 1
mg A) i 0.500:- B) «1
0.8:— 1 Z I
0.63- 30.1005- 1:
2 50.050:- i
”a? 0.4: 1 ~ .
32’. E E ' i
c: 0.2.- 30.010; I \ 3
:1 I 4 :- / \
>‘O.O"11A 1 111111111 1 -11" 0.005P111111111l1111l11111111111111‘
S -300 -200 -100 0 100 200 300 -300 -200 -100 0 100 200 300
L1
’53 12
. hvtffIYYY—VTVTTTTTYTYITTTWTW YMYq T' I Y ‘1 I I VYTrW—fTT
'9. E I 100 g a
3 mg C) i E 1
g, L .1 —1
_° 0.8;. 310 E’ .8
E 0.6:— «1 2 : :
m : 1 -
'0 0.4:- 510 E i
t : : i
0.2:— i -3
: :10 f E
"11111111: 111l1111" b1111]111111111114111111111111"

 

 

 

 

 

 

.0 . .
-300 —200-100 0 100 200 300 -300 -—200-100 0 100 200 300

Momentum (MeV/c)

Figure E.4: The three components of the Gaussian ﬁt to a Lorentzian are shown in A),
and a comparison of the three Gaussians (dashes) with the Lorentzian (solid) is shown
in B). The three Gaussian distribution, corrected for the incomplete acceptance using
transmission efﬁciencies from ﬁgure 3, is shown in C). Note that the broad component
in A) is almost eliminated after correction for the acceptance. A comparison of
the analytical solution obtained from Equation E.10 (solid) and the three Gaussian
estimate (dashes) is in D).

100

The results for a more realistic wavefunction are shown in Figure B.5. The p”
distribution is that of a 281/2 orbital bound by 500 keV in a Woods-Saxon poten-
tial [Esb94] (appropriate to describe the dominant conﬁguration for the 11Be ground
state). In this case the effects of the incomplete acceptance are smaller than for a
Lorentzian wavefunction. The width (FWHM) of the p” distribution is reduced by

6% from 45.4 MeV/c to 42.5 MeV/c.

 

 

 

 

 

 

 

 

 

:5 * "r""'***"'ﬁoozoowr'www'vvvww
\ - j.

% 0'0”): 30.0100: _:
2 '- .. " :1
5 3 I .
V I 10.0020 _
30.005: .‘

P : 20.0010E -:
”230.000. ‘0.0005'~ 111.1..1....1-..:

 

—50 0 50
Momentum (MeV/c)

Figure E5: The p" distribution for a 131/2 orbital neutron bound by 500 keV in a
Woods-Saxon well (solid), and the corresponding ﬁltered p” distribution using the
three Gaussian method to estimate the inﬂuence of the A1200 in medium acceptance

mode (dashes) (A0 = 21:20 mrad and A45 = 21:10 mrad).

E.3 Summary

An analytic result for the parallel momentum distribution expected when a Lorentzian
momentum distribution is observed with a device having an incomplete angular ac-
ceptance is given. Coupling between the Cartesian components of the Lorentzian
distribution causes the measured p“ distribution to deviate signiﬁcantly from that of

a device that collects all breakup fragments.

The apparatus detecting the breakup fragments can inﬂuence the observed distri-

101

bution, since the width of the 9Li p” distribution from 11Li breakup increases at large
p 1- However, the increase in width is not as large as expected from a Lorentzian

distribution.

Finally, an approximate method based on a superposition of Gaussian distribu-
tions is developed in order to make simple estimates of the inﬂuence of an incomplete
acceptance on the p” distribution from a theoretical model when no straightforward

analytical solution exists.

Appendix F

Coulomb Reacceleration on a
Heavy Target

Coulomb reacceleration occurs when a breakup occurs near to a target nucleus. In this
case, the potential energy that was stored in the Coulomb ﬁeld of the target is divided
between the breakup fragments. If the charge to mass ratio of the two fragments is
signiﬁcantly different, then the energy (per nucleon) is unevenly divided, and the mean
energy of breakup fragments can reﬂect information revealing the breakup timescale.
The Coulomb reacceleration of 9Li following the breakup of 11Li was explained by

Bauer and Bertulani [Bau92] and is summarized here.

F.1 Coulomb Reacceleration in 11Li Breakup Frag-
ments

By assuming straight line trajectories and neglecting the recoil of the target nucleus
the solution of the reacceleration problem becomes simple. Relying on the conserva-

tion of momentum,
P110) = P9(t)+P2(t) (F-I)

and the conservation of energy,

Elab " VlR(t)l " Q6“ _ tbrcakup) = Pfl(t)/(2m11), (F2)
102

103

it is possible to determine the ﬁnal energies of the breakup fragments when Q, the
energy transfer to the 11Li, is at the breakup threshold value. R(t) z (b2 + v3t2)1/2,
V[R(t)] is the potential energy stored in the Coulomb ﬁeld, and 9(t — tbrcakup) is the

step function.

11.
L1.

 

 

 

 

l target
Figure F .1: [Bau92] A schematic representation of the breakup of 11Li.

At the instant of breakup the the 9Li core and the two neutrons continue with
equal speeds. However,.because the two neutrons have no nuclear charge, the 9Li core

inherits all of the potential energy. As a result the 9Li energy is given by,
9 2
E9 = ﬁlEIab - Q] + ﬁVlRUbreakupH- (R3)

When The 9Li emerges far from the breakup site, the energy is shifted by,

9 9 2
11 ll 11
In a similar manner,
2 2

The term that reﬂects the post acceleration of the 9Li (neutrons) is the +(-) V[R(tbnakupﬂ

term.

Appendix G

Data Tables for 10Be Fragment
Momentum Distributions

104

105

Table G.1: The data for the 10Be fragment p“ distribution on the Be target.

 

 

 

 

 

Momentum Counts Uncertainty
(MeV/c)
3385.5 662.3 29.
3387.5 720.9 30.
3389.5 848.3 32.
3391.5 884.2 33.
3393.5 1016.4 35.
3395.5 1125.1 36.
3397.5 1181.0 37.
3399.5 1434.9 40.
3401.5 1439.5 40.
3403.5 1518.7 41.
3405.5 1580.5 42.
3407.5 1754.4 43.
3409.5 1845.9 44.
3411.5 1921.3 45.
3413.5 1905.5 45.
3415.5 1798.9 43.
3417.5 1932.9 45.
3419.5 1816.4 43.
3421.5 1795.8 43.
3423.5 1646.6 41.
3425.5 1563.6 40.
3427.5 1529.6 39.
3429.5 1333.9 36.
3431.5 1284.4 36.
3433.5 1096.6 33.
3435.5 1014.2 31.
3437.5 934.9 30.
3439.5 853.2 29.
3441.5 752.2 27.
3443.5 592.6 24.

 

 

106

Table G.2: The data for the 10Be fragment p" distribution on the Nb target.

 

 

Momentum Counts Uncertainty
(MeV/c)
3383.5 251 . 18.
3385.5 313. 20.
3387.5 318. 20.
3389.5 369. 21.
3391.5 426. 23.
3393.5 398. 22.
3395.5 479. 24.
3397.5 491. 24.
3399.5 549. 25.
3401.5 560. 25.
3403.5 591. 26.
3405.5 661 . 27.
3407.5 629. 26.
3409.5 623. 26.
3411.5 658. 26.
3413.5 734. 28.
3415.5 790. 29.
3417.5 711. 27.
3419.5 680. 26.
3421.5 662. 26.
3423.5 650. 26.
3425.5 638. 25.
3427.5 575. 24.
3429.5 582. 24.
3431.5 497. 22.
3433.5 494. 22.
3435.5 381. 19.
3437.5 399. 20.
3439.5 357. 18.
3441.5 330. 18.
3443.5 263. 16.

 

 

 

 

 

Table C.3: The data. for the 10Be frag-

ment p" distribution on the Ta target.

 

 

 

 

 

Momentum Counts Uncertainty
(MeV/c)
3307.5 20.4 9.1
3309.5 27.3 10.
3311.5 40.7 12.
3313.5 31.9 10.
3315.5 13.5 6.7
3317.5 29.5 9.8
3319.5 19.0 7.7
3321.5 30.7 9.7
3323.5 26.8 8.9
3325.5 34.7 10.
3327.5 36.5 10.
3329.5 27.3 8.6
3331.5 26.7 8.4
3333.5 31.3 9.0
3335.5 43.4 10.
3337.5 35.0 9.3
3339.5 51.5 11.
3341.5 45.7 10.
3343.5 40.2 9.7
3345.5 39.6 9.6
3347.5 36.6 9.1
3349.5 38.4 9.3
3351.5 46.8 10.
3353.5 59.4 11.
3355.5 48.9 9.1
3357.5 57.9 7.8
3359.5 69.0 8.3
3361.5 58.2 7.6
3363.5 80.7 8.8
3365.5 70.6 8.1
3367.5 82.9 8.7
3369.5 76.8 8.3
3371.5 104.0 9.6

 

 

107

 

 

 

 

 

Momentum Counts Uncertainty
3373.5 118.6 10.
3375.5 114.8 10.
3377.5 102.9 9.4
3379.5 122.7 10.
3381.5 137.4 10.
3383.5 155.0 11.
3385.5 189.5 12.
3387.5 206.3 13.
3389.5 201.5 12.
3391.5 231.7 13.
3393.5 252.8 - 14.
3395.5 281.7 14.
3397.5 315.3 15.
3399.5 320.7 15.
3401.5 360.2 16.
3403.5 371.7 16.
3405.5 408.8 18.
3407.5 400.0 19.
3409.5 449.7 20.
3411.5 446.3 19.
3413.5 480.4 20.
3415.5 427.1 18.
3417.5 424.8 22.
3419.5 452.1 23.
3421.5 466.2 24.
3423.5 434.7 22.
3425.5 487.9 23.
3427.5 460.0 23.
3429.5 471.1 23.
3431.5 400.9 21.
3433.5 330.4 18.
3435.5 315.6 18.
3437.5 303.5 18.

 

 

108

 

 

 

 

 

Momentum Counts Uncertainty
3439.5 261.8 16.
3441.5 244.9 15.
3443.5 216.9 14.
3445.5 172.3 13.
3447.5 158.7 12.
3449.5 137.7 11.
3451.5 119.5 10.
3453.5 101.2 10.
3455.5 88.1 9.1
3457.5 81.4 8.8

. 3459.5 77.6 8.9
3461.5 59.8 7.6
3463.5 58.3 7.6
3465.5 34.8 5.5
3467.5 30.3 5.3
3469.5 37.7 9.7
3471.5 37.5 9.6
3473.5 29.8 8.6
3475.5 12.3 5.5
3477.5 39.3 9.8
3479.5 2.4 2.4
3481.5 12.1 5.4
3483.5 19.4 6.8
3485.5 7.2 4.1
3487.5 12.0 5.4
3489.5 12.0 5.4
3491.5 14.4 5.9
3493.5 4.8 3.4
3495.5 2.4 2.4
3497.5 16.8 6.3
3499.5 14.4 5.9
3501.5 7.2 4.1
3503.5 4.8 3.4

 

 

Table G.4: The data for the 1”Be fragment p" distribution on the U target.

109

 

 

 

 

 

Momentum Counts Uncertainty
(MeV/c)
3393.5 184.4 15.
3395.5 171.6 14.
3397.5 220.4 16.
3399.5 224.7 16.
3401.5 244.0 17.
3403.5 256.8 17.
3405.5 293.2 18.
3407.5 319.0 19.
3409.5 337.6 19.
3411.5 392.0 21.
3413.5 370.5 20.
3415.5 435.0 21.
3417.5 455.4 22.
3419.5 488.6 23.
3421.5 471.0 22.
3423.5 451.8 21.
3425.5 446.0 21.
3427.5 485.5 22.
3429.5 461.4 21.
3431.5 426.3 21.
3433.5 494.8 22.
3435.5 449.3 21.
3437.5 383.1 19.
3439.5 358.2 19.
3441.5 361.0 19.
3443.5 327.7 18.
3445.5 294.8 17.
3447.5 235.0 15.
3449.5 229.7 15.
3451.5 172.4 13.

 

 

. Appendix H

Data Tables for 7Be Fragment
Momentum Distributions

110

Table 11.1: The data for the 7Be fragment
momentum distribution on the Be target
measured in Telescope 1.

 

111

 

 

 

 

 

 

Momentum Counts Uncertainty
(MeV/c)
1700.5 9. 3.0
1708.5 9. 3.0
1716.5 8. 2.8
1724.5 10. 3.1
1732.5 16. 4.0
1740.5 6. 2.4
1748.5 13. 3.6
1756.5 13. 3.6
1764.5 16. 4.0
1772.5 13. 3.6
1780.5 20. 4.4
1788.5 27. 5.1
1796.5 33. 5.7
1804.5 30. 5.4
1812.5 37. 6.0
1820.5 28. 5.2
1828.5 49. 7.0
1836.5 58. 7.6
1844.5 64. 8.0
1852.5 94. 9.6
1860.5 100. 10.
1868.5 124. 11.
1876.5 187. 13.
1884.5 178. 13.
1892.5 261. 16.
1900.5 316. 17.
1908.5 397. 19.
1916.5 442. 21.
1924.5 397. 19.
1932.5 430. 20.
1940.5 435. 20.
1948.5 352. 18.

 

 

 

 

 

 

Momentum Counts Uncertainty
1956.5 300. 17.
1964.5 256. 16.
1972.5 207. 14.
1980.5 161. 12.
1988.5 129. 11.
1996.5 101. 10.
2004.5 82. 9.0
2012.5 66. 8.1
2020.5 61. 7.8
2028.5 37. 6.0
2036.5 43. 6.5
2044.5 31. 5.5
2052.5 41. 6.4
2060.5 23. 4.7
2068.5 20. 4.4
2076.5 15. 3.8
2084.5 12. 3.4
2092.5 7. 2.6

 

Table H.2: The data for the 7Be fragment
momentum distribution on the Be target
measured in Telescope 2.

 

 

 

 

 

Momentum Counts Uncertainty
(MeV/c)
1604.5 3. 1.7
1612.5 5. 2.2
1620.5 4. 2.0
1628.5 3. 1.7
1636.5 4. 2.0
1644.5 3. 1.7
1652.5 5. 2.2
1660.5 4. 2.0
1668.5 7. 2.6
1676.5 3. 1.7
1684.5 13. 3.6
1692.5 10. 3.1
1700.5 8. 2.8
1708.5 7. 2.6
1716.5 10. 3.1
1724.5 10. 3.1
1732.5 9. 3.0
1740.5 11. 3.3
1748.5 10. 3.1
1756.5 12. 3.4
1764.5 11. 3.3
1772.5 14. 3.7
1780.5 15. 3.8
1788.5 13. 3.6
1796.5 13. 3.6
1804.5 13. 3.6
1812.5 22. 4.6
1820.5 18. 4.2
1828.5 16. 4.0
1836.5 18. 4.2
1844.5 26. 5.0
1852.5 23. 4.7

 

 

112

 

 

 

 

 

Momentum Counts Uncertainty
' 1860.5 30. 5.4
1868.5 26. 5.0
1876.5 27. 5.1
1884.5 40. 6.3
1892.5 44. 6.6
1900.5 46. 6.7
1908.5 45. 6.7
1916.5 33. 5.7
1924.5 44. 6.6
1932.5 46. 6.7
1940.5 33. 5.7
1948.5 38. 6.1
1956.5 26. 5.0
1964.5 18. 4.2
1972.5 20. 4.4
1980.5 21. 4.5
1988.5 24. 4.8
1996.5 16. 4.0
2004.5 11. 3.3
2012.5 6. 2.4
2020.5 10. 3.1
2028.5 5. 2.2
2036.5 7. 2.6
2044.5 4. 2.0
2052.5 6. 2.4
2060.5 4. 2.0
2068.5 1. 1.0
2076.5 0. 0.0
2084.5 0. 0.0
2092.5 1. 1.0
2100.5 1. 1.0
2108.5 0. 0.0

 

 

Table H.3: The data for the 7Be p3 fragment momentum distribution on the Be

target.

113

 

 

 

Momentum Counts Uncertainty
(MeV/c)

-271. 19.1 3.9
-259. 17.5 3.8
' -247. 12.5 3.2
-236. 17.5 3.8
—223. 22.5 4.3
-212. 30.8 _ 5.0
-200. 34.1 5.3
-188. 36.6 5.5
-176. 41.6 5.8
-164. 57.5 6.9
-152. 50.8 6.5
-140. 69.1 7.5
~128. 74.1 7.8
-116. 90.0 8.6
-104. 104.1 9.3
-92. 110.8 9.6
-26. 573. 23.
-16. 640. 25.
-6. 674. 25.
4. 704. 26.
14. 648. 25.
24. 605. 24.
34. 498. 22.
43. 425. 20.
54. 306. 17.
64. 202. 14.
74. 182. 13.
84. 155. 12.
94. 103. 10.
104. 95. 9.7
114. 88. 9.3

 

 

 

 

Table 11.4: The data for the 7Be fragment
momentum distribution on the Nb target
measured in Telescope 1.

 

 

 

 

 

Momentum Counts Uncertainty
(MeV/c)
1700.5 3. 1.7
1708.5 2. 1.4
1716.5 4. . 2.0
1724.5 2. 1.4
1732.5 7. 2.6
1740.5 3. 1.7
1748.5 4. 2.0
1756.5 10. 3.1
1764.5 7. 2.6
1772.5 7. 2.6
1780.5 6. 2.4
1788.5 7. 2.6
1796.5 5. 2.2
1804.5 11. 3.3
1812.5 14. 3.7
1820.5 18. 4.2
1828.5 15. 3.8
1836.5 12. 3.4
1844.5 17. 4.1
1852.5 31. 5.5
1860.5. 44. 6.6
1868.5 45. 6.7
1876.5 69. 8.3
1884.5 81. 9.0
1892.5 110. 10.
1900.5 134. 11.
1908.5 164. 12.
1916.5 163. 12.
1924.5 147. 12.
1932.5 160. 12.
1940.5 149. 12.
1948.5 122. 11.

 

 

114

 

 

 

 

 

Momentum Counts Uncertainty
1956.5 92. 9.5
1964.5 40. 6.3
1972.5 30. 5.4
1980.5 30. 5.4
1988.5 9. 3.0
1996.5 ' 15. 3.8
2004.5 11. 3.3
2012.5 8. 2.8
2020.5 11. 3.3
2028.5 6. 2.4

1 2036.5 4. 2.0

. 2044.5 5. 2.2
2052.5 9. 3.0
2060.5 10. 3.1
2068.5 7. 2.6
2076.5 4. 2.0
2084.5 2. 1.4
2092.5 1. 1.0

 

 

Table H5: The data for the 7Be fragment
momentum distribution on the Nb target
measured in Telescope 2.

 

 

 

 

 

Momentum Counts Uncertainty
(MeV/c)
1604.5 0. 0.0
1612.5 0. 0.0
1620.5 0. 0.0
1628.5 0. 0.0
1636.5 1. 1.0
1644.5 1. 1.0
1652.5 0. 0.0
1660.5 1. 1.0
1668.5 2. 1.4
1676.5 1. 1.0
1684.5 5. 2.2
1692.5 1. 1.0
1700.5 4. 2.0
1708.5 1. 1.0
1716.5 3. 1.7
1724.5 3. 1.7
1732.5 1. 1.0
1740.5 2. 1.4
1748.5 2. 1.4
1756.5 3. 1.7
1764.5 2. 1.4
1772.5 2. 1.4
1780.5 6. 2.4
1788.5 7. 2.6
1796.5 7. 2.6
1804.5 5. 2.2
1812.5 9. 3.0
1820.5 11. 3.3
1828.5 9. 3.0
1836.5 16. 4.0
1844.5 14. 3.7
1852.5 18. 4.2

 

 

115

 

 

 

 

 

Momentum Counts Uncertainty
1860.5 23. 4.7
1868.5 39. 6.2
1876.5 38. 6.1
1884.5 49. 7.0
1892.5 47. 6.8
1900.5 47. 6.8
1908.5 48. 6.9
1916.5 37. 6.0
1924.5 42. 6.4
1932.5 29. 5.3
1940.5 29. 5.3
1948.5 21. 4.5
1956.5 20. 4.4
1964.5 10. 3.1
1972.5 11. 3.3
1980.5 6. 2.4
1988.5 7. 2.6
1996.5 5. 2.2
2004.5 2. 1.4
2012.5 3. 1.7
2020.5 0. 0.0
2028.5 0. 0.0
2036.5 0. 0.0
2044.5 1. 1.0
2052.5 3. 1.7
2060.5 0. 0.0
2068.5 1. 1.0
2076.5 0. 0.0
2084.5 0. 0.0
2092.5 0. 0.0
2100.5 0. 0.0
2108.5 0. 0.0

 

 

Table H.6: The data for the 7Be p, fragment momentum distribution on the Nb

target.

116

 

 

 

Momentum Counts Uncertainty
LMeV/c)

-271. 5.8 2.2
-259. 4.1 1.8
-247. 9.1 2.7
-236. 11.6 3.1
-223. 15.8 3.6
-212. 12.5 3.2
-200. 23.3 4.4
~188. 22.5 4.3
-176. 24.1 4.4
—164. 37.5 5.5
-152. 42.5 5.9
-140. 43.3 6.0
-128. 62.5 7.2
-116. 68.3 7.5
~104. 70.8 7.6
-92. 78.3 8.0
-26. 165. 12.
-16. 169. 13.
-6. 179. 13.
4. 167. 12.
14. 166. 12.
24. 163. 12.
34. 138. 11.
43. 124. 11.
54. 123. 11.
64. 103. 10.
74. 99. 9.9
84. 85. 9.2
94. 81. 9.0
104. 89. 9.4
114. 55. 7.4

 

 

 

 

Table H.7: The data for the 7Be fragment
momentum distribution on the Au target
measured in Telescope 1.

 

 

 

 

 

Momentum Counts Uncertainty
(MeV/c)
1700.5 3. 1.7
1708.5 2. 1.4
1716.5 0. 0.0
1724.5 4. 2.0
1732.5 2. A 1.4
1740.5 3. 1.7
1748.5 3. 1.7
1756.5 1. 1.0
1764.5 4. 2.0
1772.5 4. 2.0
1780.5 5. 2.2
1788.5 4. 2.0
1796.5 12. 3.4
1804.5 5. 2.2
1812.5 5. 2.2
1820.5 12. 3.4
1828.5 10. 3.1
1836.5 5. 2.2
1844.5 9. 3.0
1852.5 9. 3.0
1860.5 8. 2.8
1868.5 18. 4.2
1876.5 28. 5.2
1884.5 41. 6.4
1892.5 78. 8.8
1900.5 101. 10.
1908.5 107. 10.
1916.5 122. 11.
1924.5 126. 11.
1932.5 110. 10.
1940.5 121. 11.
1948.5 96. 9.7

 

 

117

 

 

 

 

 

Momentum Counts Uncertainty
1956.5 50. 7.0
1964.5 25. 5.0
1972.5 14. 3.7
1980.5 4. 2.0
1988.5 8. 2.8
1996.5 7. 2.6
2004.5 3. 1.7
2012.5 3. 1.7
2020.5 4. 2.0
2028.5 8. 2.8
2036.5 5. 2.2
2044.5 5. 2.2
2052.5 7. 2.6
2060.5 7. 2.6
2068.5 3. 1.7
2076.5 2. 1.4
2084.5 1. 1.0
2092.5 2. 1.4

 

 

118

Table H8: The data for the 7Be fragment
momentum distribution on the Au target

measured in Telescope 2.

 

 

 

 

 

Momentum Counts Uncertainty
(MeV/c)
1604.5 0. 0.0
1612.5 2. 1.4
1620.5 1. 1.0
1628.5 0. 0.0
1636.5 0. 0.0
1644.5 2. 1.4
1652.5 0. 0.0
1660.5 4. 2.0
1668.5 2. 1.4
1676.5 1. 1.0
1684.5 2. 1.4
1692.5 1. 1.0
1700.5 1. 1.0
1708.5 4. 2.0
1716.5 0. 0.0
1724.5 0. 0.0
1732.5 2. 1.4
1740.5 1. 1.0
1748.5 1. 1.0
1756.5 2. 1.4
1764.5 2. 1.4
1772.5 1. 1.0
1780.5 3. 1.7
1788.5 3. 1.7
1796.5 4. 2.0
1804.5 4. 2.0
1812.5 6. 2.4
1820.5 5. 2.2
1828.5 7. 2.6
1836.5 4. 2.0
1844.5 10. 3.1
1852.5 18. 4.2

 

 

 

 

 

 

 

Momentum Counts Uncertainty
1860.5 11. 3.3
1868.5 24. 4.8
1876.5 33. 5.7
1884.5 49. 7.0
1892.5 55. 7.4
1900.5 ' 58. 7.6
1908.5 60. 7.7
1916.5 65. 8.0
1924.5 56. 7.4
1932.5 63. 7.9
1940.5 39. 6.2
1948.5 41. 6.4
1956.5 28. 5.2
1964.5 21. 4.5
1972.5 8. 2.8
1980.5 4. 2.0
1988.5 0. 0.0
1996.5 3. 1.7
2004.5 1. 1.0
2012.5 1. 1.0
2020.5 0. 0.0
2028.5 2. 1.4
2036.5 1. 1.0
2044.5 0. 0.0
2052.5 0. 0.0
2060.5 0. 0.0
2068.5 0. 0.0
2076.5 0. 0.0
2084.5 0. 0.0
2092.5 0. 0.0

 

 

119

Table H9: The data for the 7Be pr fragment momentum distribution on the Au
target.

 

 

Momentum Counts Uncertainty
(MeV/c)

-271. 19.1 3.9
-259. 15.8 3.6
-247. 15.8 3.6

-236. 19.1 3.9 ,
-223. 19.1 3.9
-212. 17.5 3.8
-200. 39.1 5.7
-188. 32.5 5.2
-176. 28.3 4.8
-164. 50.0 6.4
-152. 46.6 6.2
-140. 46.6 6.2
-128. 64.1 7.3
-116. 50.8 6.5
-104. 61.6 7.1
-92. 55.0 6.7
-26. 141. 11.
-16. 99. 9.9
-6. 101. 10.
4. 101. 10.
14. 87. 9.3
24. 93. 9.6
34. 103. . 10.
43. 69. 8.3
54. 75. 8.6
64. . 64. 8.0
74. 74. 8.6
84. 77. 8.7
94. 49. 7.0
104. 67. 8.1
114. 50. 7.0

 

 

 

 

 

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"‘lllltlll"121111111“