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dissertation entitled

STUDY OF LIGHT NEUTRON RICH NUCLEI

presented by

MICHAEL FAUERBACK

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PhD degree in PHI S I CS

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STUDY OF LIGHT NEUTRON RICH NUCLEI

by

MICHAEL FAUERBACH

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1997

ABSTRACT

STUDY OF LIGHT NEUTRON RICH NUCLEI

By

Michael Fauerbach

One of the fundamental questions in nuclear physics is the exact location of the
proton and neutron driplines. These driplines indicate the point at which nuclei are
no longer stable against spontaneous particle emission. The predictions of the exact
location of the driplines depend strongly on the model used, and the exact location
of the neutron dripline is experimentally only known for the lightest elements. In
the first part of the present work we will expand the knowledge of the exact location
of the neutron dripline by searching for the nucleus 2’60, which has been the center
of controversy for a long time, as different model calculations lead to varying results

about its stability.

The second part of the present work discusses the Coulomb excitation of the one-
neutron halo nucleus 11Be on several secondary targets, ranging from 208Pb to 9Be.
The different targets were used in order to investigate the various contributions on
the cross section, which scale differently with A and Z. The nucleus 11Be is the only
1

-+ ground state and

known halo nucleus with a bound excited state, and only the 2

the {- excited state are bound. Both bound states are neutron halo states with root-
mean-square radii of approximately 7 fm. This fact makes 11Be an excellent test case

for Coulomb excitation with a radioactive beam.

To my family

iii

ACKNOWLEDGEMENTS

First, I would like to thank my advisors Dr. Walt Benenson, and Dr. Dave
Morrissey, for giving me a chance to do research together with them. They gave me
plenty of freedom in my research, but where always there to help me, when help was

needed. It was a great experience working together with them.

I would like to give my special thanks also to Dr. Thomas Glasmacher for giving

me the chance to do research together with his group.

I would like to thank Dr. Wolfgang Bauer, Dr. Suzanne Hawley, and Dr. Jon

Pumplin for being on my guidance committee.

I would also like to acknowledge the many fruitful discussions with Dr. Sam
Austin, Dr. Alex Brown, Dr. Gregers Hansen, Dr. Brad Sherrill, and Dr. Paul

Mantica.

Peter Thirolf helped me a lot to understand more about 7-spectroscopy. The
A1200 postdocs Maggie Helmstrom, and Jim Brown were always there when I needed
help. Especially Jim made the big mistake to stop by at my office on his way home.
He then had to spend some time with me explaining why things didn’t work, what

we should have done, . . . .

I would also like to thank Mike Mohar, who signed me up for graduate school at
MSU — without my knowledge ! He was also the person who told me that there would
not be more snow in Lansing than in Darmstadt. Well, I was young and naive, but I

learned my lesson. I made him pay by choosing him as my best man for my wedding.

Raman Pfaff taught me a lot about Football, and american television. However, he
never told me how to pick a winning team. I also couldn’t live up to his expectations,
as I was only able to watch a small fraction of the TV shows he recommended.

iv

Of course, there are also my office mates who had to ‘live’ with me during the last
years. I admire Sally ‘I might stay in physics’ Gaff for her enthusiasm in analyzing
her data. I would not be brave enough to face her data everyday ! I enjoyed talking to
Jing ‘MBA’ Wang about the asian culture. He was also always there when I needed
some heavy stuff carried around. I’m sure, next time he will check more carefully who
lives close to him ! Don ‘the canadian spy’ Anthony was able to add a new dimension

to our office, although he is ‘only’ a chemist.

Mathias ‘Master’ Steiner and Charlotte helped me to survive the first year —

thanks l We also had much fun shopping for used cars together ll

I enjoyed the help of and many conversations with Chris ‘PPAC-Man’ Powell, Jon
‘the Jeep will run soon’ Kruse, Heiko ‘I know a faster way to do it’ Scheit, Rich ‘1
need more coffee’ Ibbotson, Marcus ‘Discotheque’ Chromik, Shigeru ‘huh’ Yokoyama,

Jac ‘. . .mobil’ Caggiano, John ‘Ned’ Kelley, Bertram, Easwar, Professor T., Barry,

Luke, Boris,
Thanks to the NSCL operations staff for great beams !

I would also like to thank my friends in Germany for not forgetting me, and for

keeping up my spirits.

I would like to thank my brother and his family, and especially my parents. Their
continuing support made this work possible ! I have to appologize to my mom, for

making her come to the USA so often, although she is afraid of flying.

Finally, I would like to thank my wife Lina for her patience and love. She never
complained about the many hours I had to spend at the lab, instead of being together
with her, and she was always there for me when I needed her. My son Kyle helped
me to keep in mind what the most important things in live are. His smile helped me

to forget how tired I sometimes was.

Contents

LIST OF TABLES

LIST OF FIGURES

1

Introduction
1.1 Physics with Radioactive Nuclear Beams ................
1.2 Production Mechanism ..........................

1.3 Purpose of the present work .......................

The Search for 26O

2.1 Motivation and Introduction .......................
2.2 Experimental Apparatus and Analysis .................
2.2.1 The A1200 Fragment Separator .................
2.2.2 Detector Setup ..........................
2.2.3 Electronic Setup .........................
2.2.4 Isotope Identification .......................
2.3 Experimental Results and Discussion ..................

2.3.1 Momentum Distributions and Fragmentation Cross Sections

2.3.2 The Search for 26O ........................

Coulomb Excitation of the One-Neutron Halo Nucleus llBe

3.1 Motivation and Introduction .......................

3.2 Experimental Apparatus and Analysis .................
3.2.1 Detector Setup ..........................
3.2.2 The NSCL 7-array ........................
3.2.3 Position Calibration .......................
3.2.4 Energy Calibration ........................
3.2.5 Efficiency Calibration .......................
3.2.6 Absorption of Photons in the Target ..............

vi

viii

ix

CprV—‘H

13
14
16
I8
20
24
24
30

34
34
39
39
43
45
52
54
60

3.3 Experimental Results and Discussion .................. 68

3.3.1 11Be on 208Pb and 19"'Au targets ................. 69
3.3.2 11Be on "“‘C and 9Be targets ................... 75
3.3.3 Nuclear Breakup of 11Be to the First Excited State in 10Be . . 79
4 Summary and Outlook 85
A Measured Cross Sections 88
B The Coulomb excitation process 91
C Nuclear Transition Matrix Elements 96
D Transition Amplitude for the Nuclear Excitation in 11Be 99

LIST OF REFERENCES 105

vii

List of Tables

3.1 Energy calibration and energy resolution for the eleven detectors that
form the inner ring of the NSCL y-array, using the 8BY and 228Th
calibration sources. ........................... 55

3.2 Parameters used to fit the atomic cross sections ............. 64

3.3 Detection efficiencies for the 320 keV photon, without absorption in—
side the target material (6mm) and with absorption (6am), and the

transmission probability. ......................... 68
3.4 The different sources of uncertainty are added in quadrature to yield

the total error (see text for details). .................. 72
3.5 Measured B(E1) values .......................... 72
3.6 The different sources of uncertainty are added in quadrature to yield

the total error (see text for details). .................. 77

A.1 Isotopic cross sections from the reaction 40Ar + 9Be at 90 MeV/ nucleon. 89

A.2 Isotopic cross sections from the reaction “Ar + 9Be at 90 MeV / nucleon. 90

viii

List of Figures

1.1

1.2

2.1

2.2

2.3

2.4

2.5

2.6
2.7

2.8
2.9

Chart of nuclides illustrating the limits of particle stability. The solid
squares represent the stable isotopes, and the area between the two
solid lines is the region of nuclei synthesized in the laboratory up to the

present. The dashed lines indicate the proton and neutron driplines,
according to Ref. [Mye66]. .......................

Schematic view of the projectile fragmentation process. .......

Measured binding energies of the ground states of oxygen isotopes
(solid circles) compared to several model predictions (solid lines). For
display purposes 10 MeV has been added to the W - WBMB com-
parison and 20 MeV to the SDPOTA comparison. The vertical lines

indicate the filling of the d5/2, 31/2, and d3); neutron orbits, respectively. 11

The effective neutron separation energy as a function of neutron num-
ber according to the shell-model calculations. .............

Two-neutron separation energies for various light isotopes as given
by Ref. [Aud93]. The solid circles indicate measured values whereas
the open circles show estimates from systematics for isotopes that are
known to exist, but for which no direct mass measurements have been
performed. ................................

Schematic layout of the A1200 fragment separator. ..........

Schematic view of the magnetically shielded Faraday cup. The per-
manent magnets are located on both sides at the front of the Faraday
cup, as indicated in the plot. All units are given in inches. ......

A schematic diagram of the electronic setup. .............

Magnetic rigidity versus the horizontal position at the dispersive image
#2. The open circles represent the data, and the solid line is a linear
fit to the data. ..............................

Energy loss versus time-of-flight as observed during the experiment.

Energy and time calibrations for the detectors used. The open circles
with the error bars represent the experimental data, and the solid lines
show the linear fit used to calibrate the data. .............

ix

13

14

15

17
19

2.10

2.11

2.12

2.13

3.1

3.2

3.3
3.4

3.5

3.6
3.7

3.8

a) Measured relative production yield as a function of the magnetic
rigidity, which is linearly related to the fragment momentum, for eight
different oxygen isotopes ranging from 17O to 24O. The solid lines are
Gaussian fits to the data. b) An expanded version showing only the
measured values for 23'240. The dashed line indicates the expected mo-
mentum distribution for 25O, which is unbound against one neutron
decay. The dash-dotted line shows the expected momentum distribu-
tion for 2"O, and the hatched area shows the acceptance of the A1200
at one specific setting (see text for details). .............. 26

Measured parallel momentum width as a function of fragment mass.
The solid line represents the predictions of the Goldhaber model, using
00 = 90 MeV/c. ............................. 29

Measured reaction cross sections for neutron-rich isotopes (solid points),
ranging from 38P to 11B. The full lines indicate the predictions of the
semi-empirical formula EPAX [Siim90]. ................ 31

Measured reaction cross sections for several oxygen isotopes (solid
points), ranging from 180 to 24O. The full line indicates the predic-
tions of the semi-empirical formula EPAX [Siim90], and the dashed
line is an exponential fit to the experimental data. The extrapolated
production cross sections for 260 agree within a factor of two. . . . . 32

Schematic View of the Coulomb excitation process. The minimum im-
pact parameter b is larger than the sum of the radii of the two colliding
nuclei. ................................... 36

The wave function x(r) = rR(r) for the two bound states of 11Be, as
calculated from a Woods-Saxon potential. The rms radii are 7.3 fm,
and 6.3 fm for the 1.31/2 and 0193/2 states respectively. The inset shows

the level scheme of 11Be, and the vertical dotted line indicates the rms

radius of the 10Be core. ......................... 38
Schematic layout of the N SCL facility. ................. 41
Schematic view of the experimental setup used for the projectile Coulomb

excitation experiment. .......................... 42

Energy loss vs. time-of flight (in arbitrary units), as recorded in the
zero degree detector and the beamline timing detector. The left panel
was recorded without the wedge at image #2. The right panel was
recorded with the wedge and the 533 £95 secondary gold target. . . . 44

The NSCL 7-array. ........................... 46

Schematic view of the mechanical setup of the NSCL 7-array. The
support structure and the lead shielding (hatched areas) can be seen.
The target is located in the center of the beampipe. The beam enters
from the left in the right hand picture, and the Figure on the left hand
side shows the array in the beam direction. .............. 47

The N SCL 7-array, with part of the low background lead shielding in
place. ................................... 48

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

3.19

The raw ADC output from the two phototubes of a typical detector (a,
b), as well as the derived energy from combining the two raw signals
(c) and the calculated position (d), for a 60C0 source in the center of
the detector. ...............................

Typical position calibration spectrum for one detector. Shown is the
actual detector position (in cm) versus the calculated position from the
two photomultiplier signals (in channel number). The points represent
the actual measurements, the error bars represent the width of the
distribution, and the solid line shows a cubic fit to the data. .....

Typical energy calibration spectrum for one detector and position slice,
using the 88Y and 228Th calibration sources. The open circles represent
the measured data points, whereas the solid line represents a linear fit
to the data. ................................

Calibrated position versus calibrated energy for one individual detector
using the 228Th source. .........................

Efficiency for a typical NaI(Tl) detector of the NSCl 'y—array, as a
function of detector position for several different photon energies from
the calibration sources. The lines represent the results for different
energies, and are only drawn to guide the eye. .............

Efficiency for a typical NaI(Tl) detector divided into ten position slices.
The open points with error bars represent the measured efficiencies, and
the full lines show a fit to the data, using equation 3.6. The position

50

51

53

54

56

slices are in ascending order from left to right, and from top to bottom. 58

Doppler-shifted 'y-ray energy in the laboratory frame for the 320 keV
photon as a function of emission angle in the laboratory frame. The
vertical lines indicate the angular coverage of the NSCL 7-array (- -)
and the restricted angular coverage used for the analysis ( - ). . . . .

Transmission probability for a 300 keV 'y-ray through the secondary
targets as a function of photon emission angle, for the lead ( - -), gold
(—), carbon (- - -) and berylium (- -) targets used in this experiment.
The vertical lines indicate the angular coverage used for the analysis.

Atomic cross sections as a function of photon energy for the four dif-
ferent secondary targets used. The open points represent the data as
given in Ref. [Sto70], and the smooth curves show a fit to the data.

Same as Figure 3.16, but now taking the finite target size and the
energy dependence of the atomic cross section into account. .....

Comparison between three different transmission probability calcula-
tions for the 53333,”,- Au target. See text for details. ..........

xi

59

62

65

67

3.20

3.21

3.22

3.23

3.24

3.25

D.1

Doppler-shift corrected energy spectra for the 208Pb and 197Au sec-
ondary targets. The data in the top row were summed up over the
eleven detectors that form the inner ring of the NSCL '7-array, whereas
the bottom row shows only the data taken with the three detectors used
for the analysis. The '7—rays were recorded under the condition that a
11Be fragment was detected in the zero-degree detector. The counting
rate is normalized by the incoming particle flux and the number of
scattering centers in the various targets. ................ 70

Comparison between the different experimentally derived B(E1) values
for the 1/2+ —) 1/2' transition in 11Be. The results from the lifetime
measurements (x) and the adopted value (0) [Mi183] are shown together
with the RIKEN measurement (0) [Nak97] and the GANIL data (13)
[Ann95]. The solid symbols represent the present data. ........ 73

Doppler-shift corrected energy spectra for the "“‘C and 9Be secondary
targets. The data in the top row were summed up over the eleven
detectors that form the inner ring of the NSCL 7-array, whereas the
bottom row shows only the data taken with the three detectors used
for analysis. The y-rays were recorded under the condition that a 11Be
fragment was detected in the zero-degree detector. The counting rate is
normalized by the incoming particle flux and the number of scattering
centers in the various targets. ...................... 76

Doppler-shift corrected energy spectra for the 9Be secondary target.The
data were recorded under the condition that a 10Be fragment was de-
tected in the zero-degree detector. The spectrum on the left hand side
has the additional requirement of only a single detected 7-ray. . . . . 81

Doppler-shift corrected energy spectra for the "“‘C secondary target.The
data were recorded under the condition that a 10Be fragment was de-
tected in the zero-degree detector. The spectrum on the left hand side

has the additional requirement of only a single detected y-ray. . . . . 82

Background subtracted counting rate in the 3368 keV photopeak as a
function of detected 7-ray multiplicity — or fold — for the carbon target. 83

The coordinate center originates in the 10Be core with the radius RC,
and the relative coordinate of the halo neutron is given by 1'". The
impact parameter of the target nucleus, moving along the z axis is

given by F. ................................ 101

xii

Chapter 1

Introduction

1.1 Physics with Radioactive Nuclear Beams

One of the fastest growing — and most interesting -— areas of nuclear physics is carried
out with radioactive nuclear beams (RNBs). Up until recently, nuclear physics exper-
iments were limited to beams and targets of stable or near stable isotopes. However,
less than 300 isotopes are stable and can be found in nature, whereas predictions
show that about 7,000 isotopes should be particle stable, which means their lifetime
is long on a nuclear time scale, and they do not decay via spontaneous particle emis—
sion. The properties of these ‘exotic’ nuclei are of vital interest to many areas of
nuclear structure and nuclear astrophysics. This is illustrated by Figure 1.1 in which
the chart of nuclides is shown. The solid squares indicate the stable isotopes, and
all isotopes between the two solid lines have been experimentally observed (February
1997). The dashed lines indicate the proton (Bp=0) and neutron (Bn=0) driplines
according to the predictions of Myers and Swiatecki [Mye66], and the magic proton

and neutron numbers are also indicated on the plot.

One of the fundamental questions in nuclear physics is the exact location of the
so-called proton and neutron driplines. These driplines indicate the point at which

nuclei are no longer stable against spontaneous particle emission. The predictions of

1

 

 

Figure 1.1: Chart of nuclides illustrating the limits of particle stability. The solid
squares represent the stable isotopes, and the area between the two solid lines is the
region of nuclei synthesized in the laboratory up to the present. The dashed lines
indicate the proton and neutron driplines, according to Ref. [Mye66].

the exact location of the driplines depend strongly on the model used. Of course, these
mass models all use information obtained from known isotopes. Information about
the nuclear structure of isotopes farther away from stability has to be extrapolated

from the available data taken closer to stability.

The onset of the proton dripline is determined by the balance between the attrac-
tive nuclear force and the repulsive Coulomb force between the protons in the nucleus.
The repulsive character of the Coulomb force limits the extent of the proton-rich nu-
clei, so that the proton dripline is rather close to the valley of stability. This combined
with the fact that the proton-rich nuclei are more easily accessible experimentally,
leads to the fact that the proton dripline is mapped out to at least intermediate mass
nuclei (Z = 29). On the neutron-rich side, on the other hand, the binding of the
nuclei is basically limited by the filling of non-identical states in the nuclear potential
well by neutrons. Therefore, the neutron dripline extends much further away from
the valley of stability than the proton dripline. The exact location of the neutron
dripline is only known experimentally for the lightest elements, and the theoretical
predictions for heavier elements disagree, depending on the models used, by up to 5
to 10 mass units. Therefore, experimental studies to establish the limits of particle

stability can yield important information.

Nuclei far away from stability are called ‘exotic’ not only because they do not
exist in nature, but also because of the fact that they can exhibit new and ‘exotic’
behaviour. New decay modes, e. g. proton emission and fl-delayed particle decay, have
been observed. Other interesting topics include : the Gamow-Teller fl-decay strength
distribution, as well as the possibility of direct diproton decay for nuclei close to the
proton dripline; ‘superallowed’ fl-transitions observed for even neutron nuclei close
to the neutron dripline, and many more. The nuclei close to the driplines are also of

particular interest as they should show some unique properties due to their very small

one- or two-particle separation energies. This is especially true on the neutron-rich
side, where the influence of the Coulomb force on the mean field is small. Very large
nuclear matter radii have been found for several light neutron-rich nuclei and have
been interpreted as evidence for a halo structure. A more detailed discussion about

some properties of halo nuclei can be found in [Han95].

1.2 Production Mechanism

To perform experiments with radioactive beams, one first has to synthesize them in
the laboratory. Several different techniques are employed to produce them, such as
fusion-evaporation reactions, fission, as well as target- and projectile fragmentation.
An overview about the different techniques can be found in the review articles by AC.
Mueller and BM. Sherrill [Mue93] and H. Geissel, G. Miinzenberg, and K. Riisager
[Gei95] and references found therein. We will only briefly describe the technique of
projectile fragmentation as it is employed for example at the National Superconduct-
ing Cyclotron Laboratory (NSCL) at Michigan State University, and as it was used

for the present work.

In 1947 Serber [Ser47] proposed the idea that peripheral high-energy heavy-ion
reactions can be described by a two-step process in which each step takes place in
a distinct time interval. The first step describes the violent collision between the
constituents of the projectile and target nucleus. This step occurs on the order of
10433 and leads to so—called prefragments, which are excited and lose their excitation
energy via the emission of particles (neutrons, protons, alphas, and sometimes even
light clusters of particles) and 7-rays. This second, deexcitation step usually takes
on the order of 10‘183 to 10‘163, depending on the excitation energy. In Figure 1.2

one can see a schematic view of the process. A heavy ion projectile with mass Ap,

element number Zp, and incident energy Ep interacts during a peripheral collision
with a target nucleus with mass AT and elemental number ZT. The nucleons in the
overlap zone of the two nuclei interact with each other, and in the most simple picture
get sheared off the nuclei. The remnant of the projectile that was not involved in the
collision ~ the so-called prefragment — travels with almost the beam velocity, loses
its excitation energy via the evaporation of particles and 7—rays, and finally ends
up as the observable fragment. For a more detailed microscopic description of the
fragmentation process the reader may refer to [Fau92]. An overview of some of the
experimental details involved in intermediate energy projectile fragmentation can be

found in [Pfa96].

Projectile Prefragment Fragment

 

VF
Fireball

 

Target

Figure 1.2: Schematic view of the projectile fragmentation process.

The different fragmentation products can be analyzed in fragment separators (see
[She91, Mue93, Gei95], and Figure 2.4). The time to analyze the fragments is given
by the flight time through the device, typically on the order of a couple of hundred
nanoseconds. Therefore, in principle one can detect isotopes with a lifetime that is
not significantly shorter than the analyzing time. This is a big advantage over Isotope

Separation Online (ISOL) systems, which use target fragmentation induced by intense

proton or light ion (e.g. l2C) beams. The production mechanism is similar to the one
described above, however the radioactive isotopes produced have to be chemically
released from the target, which takes typically on the order of milliseconds or more.
Another advantage of projectile fragmentation is that the fragmentation products are
forward focussed, travel with almost the same velocity as the incident beam, and are
already ionized. Therefore, a radioactive ion beam of any element is readily available
to impinge on a secondary reaction target, so that within limits — due to the lower
available secondary beam intensities — similar experiments as with stable beams can
now take place with radioactive beams. Again, this is an advantage over the ISOL
facilities, where the radioactive ions have to be reaccelerated before they can be used
for secondary experiments. However, the fact that the beams are produced at high
incident beam energies for projectile fragmentation is not always advantageous. Most
reactions of astrophysical interest take place at rather low energies which are not
accessible by projectile fragmentation, in which the secondary beams typically have
energies between several tens — or even several hundreds — of MeV/ nucleon. ISOL
facilities seem to be better suited for this kind of reaction. Therefore, the two methods

of producing radioactive ion beams are complementary to each other.

1.3 Purpose of the present work

The ultimate purpose of the present work is to study light neutron rich nuclei. This
thesis will concentrate on two different aspects of physics with radioactive beams. In
the first generation of experiments with radioactive nuclear beams, the main purpose
was to understand the production mechanisms involved in projectile fragmentation re-
actions. Therefore detailed studies of momentum distributions and production yields
were performed. These experiments are rather simple and straightforward but are

still of great importance. As was mentioned earlier, the exact location of the neutron

dripline is only known for the lightest elements. Recent advances in technology that
lead to an increase in the available primary beam intensities, enables one to push the
limits ahead further. In the first part of the present work (Chapter 2) we will expand
the knowledge of the exact location of the neutron dripline by searching for the nu—
cleus 26O. This nucleus has been the center of controversy for a long time, as different
model calculations lead to varying results about its stability. Although a previous
attempt to synthesize 26O was not successful [Gui90], many people still speculated
about its existence. In a recent paper by Z. Ren et al. [Ren95] for example, 2"0 was

described as a possible candidate for a four neutron halo nucleus.

Radioactive nuclear beams can of course also be used to perform secondary beam
experiments. In this second generation of experiments, the radioactive nuclei inter-
act with a secondary target, and depending on the experimental setup used, one can
deduce a wide variety of nuclear structure information. These experiments require
a much more sophisticated experimental setup than that used in the previously de-
scribed first generation experiments. Now, the radioactive ions not only have to be
produced and identified before they interact with the secondary target but also after,
and often in coincidence with other particles and / or 7-radiation. In Chapter 3, we will
describe a Coulomb excitation experiment with the one-neutron halo nucleus 11Be on
several secondary targets. During this experiment, the secondary beam was identified
before it impinged on the secondary target, and the scattered particles were measured
in coincidence with the emitted '7-rays. The nucleus 11Be is the only known halo nu-
cleus with a bound excited state; both the %+ ground state and the %_ excited state
are bound. Both bound states are neutron halo states with root-mean-square radii of
approximately 7 fm. This fact makes 11Be an excellent test case for Coulomb exci-

tation with a radioactive beam. There is also the possibility that more subtle effects

will appear when the known lifetime of the first excited state is compared with cross

sections for inelastic nuclear scattering because (a) the Coulomb transition amplitude
approaches unity at small impact parameters so that first-order perturbation theory,
which is usually employed to describe Coulomb excitation, may no longer be valid,
and (b) the extended halo wave functions may lead to nuclear excitation well beyond
distances normally considered safe in Coulomb excitation experiments. However, the
first Coulomb excitation experiment [Ann95], in which a 45 MeV / nucleon 11Be beam
bombarded a lead target, produced an excitation cross section of only 40% of the
expected value for pure Coulomb excitation at this energy. This reduction is much
larger than what could be reasonably expected from higher-order effects (calculated
to be on the order of 10-20% [Ann95, Typ95, Kid96, Ber95]). In chapter 3, the results
from our measurement on various targets will be discussed and compared to previous

experimental data as well as to some theoretical predictions.

Chapter 2

The Search for 26O

2.1 Motivation and Introduction

The exact location of the boundaries of particle stability - the so called driplines - are
interesting from an experimental as well as from a theoretical standpoint. Many the-
oretical models, both macroscopic and microscopic, predict the ground state masses
as well as the one- and two-particle separation energies, and therefore they also pre-
dict the driplines locations. The exact location of the neutron dripline is only known
experimentally for the lightest elements, and, therefore, experimental studies to es-
tablish the limits of particle stability remain important. The nuclei close to the
driplines are also of particular interest as they should show unique properties which
are due to their very small one- or two-particle separation energies. This is especially
true on the neutron-rich side, where the influence of the Coulomb force on the mean
field is small. As an example, 11Be has a very large nuclear matter radius that has
been interpreted as evidence for a halo structure. It is expected that many more of
these one- and two-neutron halo nuclei will be found as one approaches the limits of
particle stability. In chapter 3, a discussion of some more detailed properties of the

one neutron halo nucleus 11Be is presented.

Some disagreements, particularly in the sodium region, between the experimen-

10

tally determined values and the theoretical estimates for the binding energies of the
most neutron-rich isotopes have been noted by several authors [Wou88, Gui90]. With
the use of large scale computer codes the disagreement has lessened in recent years
although there are still several regions where the models do not agree with the exper-
imental findings or with each other. The predictions of several models for the particle
stability of 26O can be summarized in terms of the binding energy curves for oxygen
isotopes shown in Figure 2.1. The experimental data (solid circles) are compared
with several predictions: the global prediction of Méller and Nix [M6188] (MN) and
two different shell-model calculations, labeled as W-WBMB and SDPOTA. W refers
to Wildenthal’s USD interaction [Wil84] obtained from a fit of the two-body matrix
elements to binding energy and excitation energy data in the 031d shell (N,Z=8,20).
SDPOTA [Bro88] is a more recent interaction obtained from a fit of density-dependent
one-boson exchange potential strengths to the same set of energy data. The W inter-
action has effectively 47 parameters whereas SDPOTA is more constrained with 14
potential strength and three single-particle energy parameters. WBMB is an addi-
tion and extension of W into the f p shell [Wab90]. Both W and SDPOTA do about
equally well in reproducing the known experimental binding energies of the oxygen

isotopes.

The region close to the dripline is very sensitive to small differences in the binding
energy since only a very small change in binding energy can determine whether a
nucleus decays via fl-decay or by spontaneous particle emission. Therefore, the lo-
cation of the driplines is a stringent test for mass models and provides a good way
to judge whether a particular model can be used to extrapolate away from stability.
The global model of Miiller and Nix predicts 26O and even the doubly magic nucleus
280 to be stable, while the global systematics by Audi and Wapstra [Aud93] predict

24O as the heaviest bound oxygen isotope with 26O unbound by 50 keV. The W -

200

 

—
—
_.
—
—4
—«
_,
—<

190
180
W—WBMB
170
160

150

Binding energy (MeV)

140

130

 

 

 

 

llllllllllllllllllllllllllllllllILLLlll

 

IIIIIIWWTITIIIIIIIIIIIIllllllllllll—TT
K
2

 

llll lLllmlnlpll

15 20 25

L

 

120

U"
H
O
(.0
O

Neutron number

Figure 2.1: Measured binding energies of the ground states of oxygen isotopes (solid
circles) compared to several model predictions (solid lines). For display purposes 10
MeV has been added to the W — WBMB comparison and 20 MeV to the SDPOTA
comparison. The vertical lines indicate the filling of the d5/2, 31/2, and d3); neutron
orbits, respectively.

l2

WBMB calculations also predict 26O to be bound by about 1 MeV. The SDPOTA
calculation, on the other hand, predicts 26O to be unbound, but only by 20 keV.
These differences are an indication of the rather large model dependence which exists
in the theoretical extrapolation to the most exotic nuclei. In the case of the oxygen
isotopes, the shell-model predictions rely on assumptions in the calculations which
cannot be tested with known experimental data beyond 240. The odd-even nuclei
25370 are known to be unbound against one neutron decay [Aud93], in agreement

with all of the model calculations.

The difficulty of predicting the binding energies of the heaviest oxygen isotopes, for
both global- and shell—model calculations, can be easily understood in the simple j-j
coupling limit. In Figure 2.2 the effective neutron separation energy (3,.) is shown as
a function of neutron number. As can be clearly seen, 3,. changes slowly with neutron
number due to the relatively weak average interaction between the valence neutrons.
Between N =8 and N=14, the neutrons fill the (is/2 orbit, which is bound by about 4
MeV. This, together with the attractive pairing energy, provides the sharp increase in
binding energy observed between N=8 and N=14. At N=15 and N=16 the neutrons
fill the 31/2 orbit which is less bound than the d5/2. Thus one sees a smaller increase
in the binding energy at this point. From N=16 up to N=20 the neutrons fill the
d3/2 orbit which has close to zero binding energy, and the total binding energy curve
becomes flat. This slow variation in separation energy with the neutron number leads
to difficulty in predicting whether a nucleus is bound or unbound. Beyond N220 the
neutrons have to go into the next major shell (pf orbits) which are unbound, hence

the binding energy curve decreases beyond this point.

Figure 2.3 shows the two-neutron separation energies for the known isotopes in
this region from the compilation of Audi and Wapstra [Aud93]. As clearly shown

on this figure, all the elements near oxygen have several bound isotopes with two-

13

 

 

 

 

9 5 I T? I T I I I I Tj I I I I r I
0 - ..
S
v ,. ...
>~
B ' d ‘
O _ 3/2 _
£3
0 O ,_ _.
2 \_¥\'
.2 ' ‘
g )- -l
L.
as _ _
a: 31/2
I _ d _
,2 6/2
an _ __ _
.5 5
m . J
m ,_ a
>
1.3“ _ a
O
Q)
a d
m __ 10 l 1 l 141 L L l l I l l l l I l
5 10 15 20

Neutron number

Figure 2.2: The effective neutron separation energy as a function of neutron number
according to the shell-model calculations.

neutron separation energies that are much smaller than that of 24O. The heaviest
experimentally-known oxygen isotope is 24O, and a previous attempt to synthesize 260
using a 48Ca beam was not successful [Gui90]. In the present work a new investigation
of the particle stability of 2"O was carried out with greater sensitivity than that
previously employed. This was possible due to the unique features given by the
combination of the K1200/A1200, with the available high primary beam intensities,

and the large acceptance of the A1200 for the rigid fragments.

2.2 Experimental Apparatus and Analysis

In the following the experimental apparatus used will be described along with the

analysis procedures employed during the search for the very neutron-rich nucleus

260.

 

SZn (MeV)
48
l

 

 

 

. 2 2 2 .
“Ll 14Be 198 12 c 4N 29F 3 Ne _
A A I A A A l A A A A A A‘

 

5 10 15 20
Neutron Number
Figure 2.3: Two-neutron separation energies for various light isotopes as given by
Ref. [Aud93]. The solid circles indicate measured values whereas the open circles
show estimates from systematics for isotopes that are known to exist, but for which
no direct mass measurements have been performed.

2.2.1 The A1200 Fragment Separator

A 90 MeV/nucleon 4(’Ar beam from the K1200 cyclotron at the National Supercon-
ducting Cyclotron Laboratory at Michigan State University irradiated at 190 GEN-:12 9Be
target located at the medium acceptance target position of the A1200 fragment sep-
arator [She91]. The target thickness was chosen to optimize the production rate of
the heavy oxygen fragments. The average beam intensity — using the superconduct-
ing ECR ion source — was about 20 particle nanoamps (pnA), which corresponds to
approximately 1.25.1011 particles per second (pps). The angular acceptances of the
A1200 were i10 mrad in 0 and $20 mrad in d), centered around 0°. The momentum
acceptance of the A1200 was set to the maximum of Ap/p = $1.570. A schematic

layout of the A1200 is shown in Figure 2.4.

In order to map out the (parallel) momentum distributions of the fragmentation

15

A1200 Radioactive Beam Facility

    
 
 

Dispersive Image #1 Final
Achromatic
9&2 Target Dispersive Image #2 Image

 
 
 
  

TOF .
\ Detector “ .
‘oAr beam from Stop Time 2 PPACs ' 2 PPACs
K1200 Cyclotron X,.Yl,6,¢ Si Telescope
9O MeV/A
- 2 . ; ~ (ligxirtzj‘vulr 'l‘rzj‘éus X2,Y2.AE.EW
Sextupolcs - ‘ 1 ' ' " . Start Time

Figure 2.4: Schematic layout of the A1200 fragment separator.

products and to provide information about the production yields (see Ref. [Pfa95]
for details), the magnetic rigidity of the separator was varied in overlapping steps
over the range Bp = 2.96 to 4.31 Tm. The relative beam current was monitored
continuously by four small PIN diodes mounted around the target position to nor-
malize data obtained at different magnetic rigidity settings. A small irreproducibility
in the absolute beam current was found in the offline analysis. Therefore, a new
magnetically shielded Faraday cup was built, and the beam intensity was carefully
measured in subsequent short runs along with the yields of a few products. The cross
sections for these products were calculated and used to renormalize the relative cross
sections determined in the long experiment. In Figure 2.5 a schematic view of the
new Faraday cup is shown. Two strong permanent magnets were glued to the sides

to prevent secondary electrons from escaping.

The momentum distributions of the lighter oxygen isotopes were monitored online

16

so that the optimum separator setting for 260 could be inferred without reference to
any model calculations, an improvement over the previous experiment. Once the
optimum setting was determined, an energy loss degrader (9Be, 47 332) was inserted
at the intermediate focal plane of the A1200 to cut down the range of atomic numbers
of the fragments reaching the final focus. The fact that 260 fragments would be
centered at the intermediate image and the final focus of the A1200 was also verified
online by measuring the positions of 19C and 20C fragments (which span 26O in A / Q)
at the intermediate focus and by measuring the energy loss of all the fragments versus

their position at the final focus.

2.2.2 Detector Setup

The experiment used a standard detector setup for the A1200 fragment separator.
The time of flight for each particle was measured over the 14 m flight path between a
8 £197 thick plastic scintillator (BC404) located at the first dispersive image and the
time pick-off signal from a totally depleted surface barrier semiconductor detector
located at the focal plane of the A1200. The position and angle of the reaction
products were measured at both the second dispersive image and at the focal plane
with two pairs of x-y position sensitive parallel-plate avalanche counters (PPAC’s)
[Swa94], separated by approximately 40 cm. Combining the position measurement at
the second dispersive image of the separator with NMR measurements of the A1200
dipole fields provided us with information about the magnetic rigidity (Bp) of each

particle. The magnetic rigidity is linearly related to its momentum (p) via the relation:

szezm (2.1,

Q Q

where Q is the particle charge, c is the speed of light, 6 and 7 are the usual relativistic

P
‘l‘l‘

L<‘ZMUG
<—1.5@@——>'

1.00% [<—

6.563

 
 
 
  
 

 

     

.813

Figure 2.5: Schematic view of the magnetically shielded Faraday cup. The permanent
magnets are located on both sides at the front of the Faraday cup, as indicated in
the plot. All units are given in inches.

18

factors (fl :: fi, 7 = W), A is the fragment mass number, and mo is the unified

 

atomic mass number (mo z 931.5%). The reaction products that reached the
A1200 focal plane were implanted into a four-element silicon detector telescope. The
telescope consisted of two thin (100 and 75 pm) AE and two thicker (500 and 1000
pm) E detectors, each with a 300 mm2 active area. All detectors were totally depleted

surface barrier detectors, and were cooled to below +10° C to reduce thermal noise.

2.2.3 Electronic Setup

The standard, permanent A1200 electronics configuration was used for this experi-

ment. A schematic View of the electronic setup is shown in Figure 2.6.

Thirty signals were read by the front end computer for each event trigger. The
signals in detail were, a charge sensing ADC (QDC) and a TDC signal for the image
#1 start detector; an ADC signal from the TAC measuring the flight time from the
start detector to the AEI detector; an ADC signal from the TAC measuring the time
between the cyclotron RF and AEI; one TDC signal and four ADC signals (up, down,
left, right) from each of the four PPAC detectors; an ADC and TDC signal from AEI
and ADC signals from the other silicon detectors. The master gate was constructed
from a valid signal in the fourth silicon detector and a NOT BUSY signal from the
front end CPU. We used the fourth silicon detector, not the usual condition of good
signals in the first two silicon detectors for the mastergate, to enhance our sensitivity
for low-Z isotopes. The data acquisition system, for both hardware and software,
was the standard NSCL data acquisition system [Fox89]. The data were written on

8 mm tapes.

BIT 2 (Dispersive Image)
BIT 3 (Focal Plane)

 

 

Figure 2.6: A schematic diagram of the electronic setup.

20

2.2.4 Isotope Identification

By transporting the primary beam through the production target and recording its
horizontal position as a function of the magnetic rigidity, one is able to establish the
rigidity calibration of the device. This is necessary in order to calculate the exact
flight path of each fragment through the A1200 fragment separator on an event-by-
event basis. The linear relationship between the magnetic rigidity and the horizontal
(x) position at the second dispersive image of the A1200 fragment separator is shown
in Figure 2.7. The open points represent the experimental data, and the solid line

shows the linear fit to the data.

 

 

 

 

316 _ I I 1 I T F I I I I I r I I I I I If] I T I I T I
i- .
3.14 '— -"
.I
.J
E 1
5:. 1
'3 I
3.10 l—' _
)- —1
L ..
3.08 '— _.
" 'i
I. a
L l 1 1 1 A l 1 L 1 1 l n 1 J 1 I 1 1 1 1 l 1 4%; L l 4"
300 400 500 600 700 800

x position [c1113.]

Figure 2.7: Magnetic rigidity versus the horizontal position at the dispersive image
#2. The open circles represent the data, and the solid line is a linear fit to the data.

The initial isotope identification was performed by setting the magnetic rigidity
of the A1200 about 3% higher than that of the primary beam through the produc—
tion target. This way, the primary beam and lighter fragments could be detected

simultaneously for unambiguous identification of the atomic number (Z). In a two

21

dimensional display of the energy loss (AE) versus the time-of-flight (TOF) the dif-
ferent elements form well separated bands. Individual isotopes are well separated in
the same figure as is shown in Figure 2.8. The N = Z line as well as several of the iso-
topic bands are indicated in Figure 2.8. The isotope identification was independently

verified by the absence of unbound or extremely short lived isotopes, like 8Be.

An energy calibration for each silicon detector was obtained by comparison to the
calculated amount of energy deposited in each detector, using the formalism given in
Ref. [Hub89], to the observed ADC channel for various fragments. The total kinetic
energy (TKE) deposited is then determined by summing up the energy loss in all
four silicon detectors. The time-of-flight of the fragments was calibrated in a similar
manner, by comparing the calculated TOF to the observed TDC channel. Figure 2.9

shows the energy and time calibrations obtained for the various detectors.

The simultaneous measurements of AE-E, the magnetic rigidity, and the time-of-
flight for each particle provides an unambiguous identification of the atomic number
(Z), charge state (Q) and mass of the produced fragments (A), via the following

relationships :

Z = a + bI/AEH — 1) (2.2)

3.105Tm * TKE =1: Bp
931.5MeV Bp(7 — 1)

A _ Q * Bp
— fly * 3.105Tm

 

(2.3)

 

(2.4)

Here, a and b are constants determined for each silicon detector, and 5 and 7 are

 

the usual relativistic factors (fl = g, 7 = «1.125), determined from the TOF of the

fragment.

The combination of the high primary beam energy and the low atomic number of

22

 

 

Energy Loss —>

 

 

 

 

Time-of-Flight —>

Figure 2.8: Energy loss versus time-of-flight as observed during the experiment.

23

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

__ :T”'l”FFl"”l'T'fT l
60: 403‘ .
: : g 1'30:P ‘,
E 20;— 4 I .
. 10E— —
Ob 11.111111111111111111’ O: lJlllllJllllllllllJLllJ:
0 100 200 300 400 0 100 200 300 400 500
ADC Channel ADC Channel
300~1'”*’r""" ' "'w'r'w'rIw"'WI"w'L
: 600;—
s L. g- :
s 20"; 1?. 400:-
to I v
a 1007 3 2007
O:111l1111l1111l1111l1111l: O:111l1111l1111l1111l1111l11:
0 100 200 300 400 500 0 100 200 300 400 500
ADC Channel ADC Channel

 

210JIIIIIIITIITIIIIIIIIIPIIIII‘
I- c1
d

u

 

 

 

 

200:
E: 190;
§ 180:
170:
160i-l-unI-u.1.n.n.u..,”,:
220 240 260 280 300 320
TDC Channel

Figure 2.9: Energy and time calibrations for the detectors used. The open circles with
the error bars represent the experimental data, and the solid lines show the linear fit

used to calibrate the data.

24

the fragments assures that we only observe fully stripped fragments. A calculation,
using the code INTENSITY [Win92], predicts the ratio of the fully stripped primary
beam to the hydrogen-like primary beam after the production target to be about
1010 to 1. This fact was verified during the off-line analysis. Unfortunately, one of
the silicon detectors (AE3) was radiation damaged during this experiment, and as
both its absolute efficiency and its energy resolution were deteriorating, it had to
be neglected from further analysis. Therefore, we were not able to use the above
formalism to determine Q, and A for the latter part of the measurement. However, as
is shown in Figure 2.8 the isotopic separation in the two-dimensional AE versus TOF
plot is very good and did not rely on the damaged detector. Therefore the production
yields of the various fragments could be obtained from this plot. It should be noted
again that this was possible because it was determined that all fragments were fully

stripped.

2.3 Experimental Results and Discussion

The main objective of the experiment was to determine whether 260 is particle bound.
To make this determination, the momentum distributions of many neutron-rich frag-
ments were carefully determined. This information was then used to determine pro-
duction cross sections of these fragments and to find the optimum separator setting
to observe 2"O. The results of these yield measurements will be discussed first and

compared to the predictions of a semi-empirical parameterization [Siim90].

2.3.1 Momentum Distributions and Fragmentation Cross Sec-
tions

The momentum distributions of eight different oxygen isotopes in the range from 17O

to “0 from the fragmentation of ”Ar are shown in Figure 2.10. Notice that these

 

25

distributions span approximately seven orders of magnitude in relative production
yield. As can be seen in Figure 2.10, the momentum distributions of the oxygen

fragments are well described by Gaussian functions.

The widths of the individual momentum distributions were obtained from fitting
Gaussian functions to the data. In Figure 2.3.1 we compare the measured momentum
widths for various fragments (symbols) as a function of mass loss, to the formalism
developed by Goldhaber [Gol74] (solid line). The dashed lines connect different iso-
topes (of the same element) and are only drawn to guide the eye. The Goldhaber
model is an independent particle model which is based on the assumption that a
nucleus can be described by a Fermi gas made up of independent nucleons. In the
projectile rest frame the net momentum of the projectile is zero. The sudden removal
of several nucleons, each moving with a mean momentum < p > inside the projectile
nucleus, results in a Gaussian recoil momentum distribution of the fragments. The

width of this distribution in the projectile rest frame is given by the expression :

 

 

0'” : 00\/AF(AP - AF) (25)

where A P.F denotes the mass of the projectile and fragment respectively. The reduced
width 0'0 is the mean quadratic deviation of the momentum projected on one spatial
dimension. Assuming that the width of the momentum distribution is the same in

all three spatial dimension, one can calculate 00 as follows :

1
do=fi<p> (2.6)

where < p > is, as mentioned earlier, the mean momentum of the nucleons. This

26

 

  
      
  

   

 

 

 

 

100 I_Ill=lllllfl|llll IVIIIllIllllIllIlllllllllllII_
‘ x170 .200 ‘
1:180 0210
r E1190 .220 __10—4
0230 E
10‘2 — .240 ‘
E -1
o
53‘ ’ 110—5
a) E
g _
4,310“1 — 4
g .—
\ E
10_6 — ’x‘\_
a) b) I . 14.10-7
_ , I I I
llllllillJ_l_lll l llillllilllllfllllJL’llllLlJllII-j-
3.0 3.5 4.0 3.25 3.50 3.75 4.00 4.25 4.50

Bp [Tm]

Figure 2.10: a) Measured relative production yield as a function of the magnetic
rigidity, which is linearly related to the fragment momentum, for eight different oxygen
isotopes ranging from 17O to 24O. The solid lines are Gaussian fits to the data. b)
An expanded version showing only the measured values for 23340. The dashed line
indicates the expected momentum distribution for 25O, which is unbound against one
neutron decay. The dash-dotted line shows the expected momentum distribution for
26O, and the hatched area shows the acceptance of the A1200 at one specific setting
(see text for details).

27

mean momentum can be expressed in terms of the Fermi momentum (pp) as :

3
< P >= \/;PF (2-7)

The Fermi momentum of the nucleons inside the nucleus has been determined previ-
ously by quasi-elastic electron scattering. Moniz et al. reported a Fermi momentum
of pp z 230 MeV/c for 40Ar, leading to a reduced width of 00 z 100 MeV/c. A
comparison to experimental data taken by Viyogi et al. [Viy79], confirmed the func-
tional form given by equation 2.5. However, the value of the reduced width 00 had
to be decreased to 90 MeV/c to reproduce the data. This discrepancy with the sim-
ple Fermi-gas model assumption has lead to several theoretical efforts which discuss
possible causes for the narrowing of the observed momentum width. G.F. Bertsch
[Ber81] for example pointed out the importance of the Pauli principle, and its effect
on the momentum width. Hiifner and Nemes [Hiif81], as well as W.A. Friedmann
[Fr183], use a geometric approach to take into account the fact that most nucleons are
removed from the periphery of the nucleus. The articles by M.J. Murphy [Mur84],
and H.H Gan et al. [Gan90] take into account that the nucleons that are not removed

have to be bound in the fragment, which also introduces a reduction in 00.

As can be clearly seen in Figure 2.3.1, the measured momentum widths vary
widely, and the heaviest observed isotope of a given element chain has the largest
width. This is in contradiction to the Goldhaber model, which predicts that the
width is only a function of mass loss. However, one can understand this behavior
easily as being due to the target thickness effect. Since the main objective of the
present experiment was to produce a very exotic nucleus, a relatively thick target
was used to optimize the production rate of this nucleus. Dufour et al. [Duf86] have

shown that the target thickness introduces an increase in the width of the momentum

28

distribution due to the difference in the energy lost by the projectile and that by
the fragment in the target material. In the extreme case in which the fragment
is produced at either the front or the back of the target, the differential energy loss
creates an almost rectangular momentum distribution, the width of which increases in
proportion to the target thickness. The width (0T) introduced by the target thickness
can be parameterized as follows :

Z3 2%

0T = CHE; - 71;) (2-8)

where cT is a target specific constant, and Zap, Apy denote the atomic charge and
the mass of the projectile or the observed fragment respectively. Equation 2.8 explains
the observed behaviour of the increasing momentum width for the heaviest observed
isotope of each element chain and shows that this effect is due to the thick production
target used. In principle the two different components can be separated to give a
nuclear reaction momentum width. However, since the unwanted effect from the
target thickness was so large in the present case, the nuclear reaction widths were
not deconvoluted. For similar reasons, we did not analyse the momentum shift of the

fragments, as was done in Ref. [Pfa95].

Because the production cross sections are expected to change very little over the
energy range spanned by the target thickness, the production cross sections were cal-
culated and compared to the semi-empirical parameterization (EPAX) by Sfimmerer,
et a1. [Siim90]. The absolute isotopic cross sections were obtained by integrating
the fitted Gaussian distributions over momentum and normalizing the results by the
incoming particle flux and the acceptance of the A1200. The incoming particle flux
was measured between runs with a Faraday cup, and the beam current was moni—

tored during the run using four PIN diodes mounted around the target position. The

29

 

    

 

 

 

: x P l | l l :
400— 8 SI .9. fl
_ A1 -
_ x Mg 4’ a: _
— ’1“ Na .. , ‘ ‘I' _
_ )1 Ne I I 56+?“ “Mp -
’3’ 300—— + F ’9‘ ‘ I \ ~ *‘ -—
_ 1:1 ‘ _
\ _ o -
> _ u _
H 200—- ——
6— - I
I. —.
100 ——
,
o
o 10 20 30 40

Mass Number (A)

Figure 2.11: Measured parallel momentum width as a function of fragment mass. The
solid line represents the predictions of the Goldhaber model, using 00 = 90 MeV/c.

30

acceptance for each individual isotope was calculated with the code LISE [Baz97].

The absolute production cross sections for 72 neutron-rich isotopes ranging from
11B to 33P were determined and are shown in Figure 2.12. The overall agreement
between the measured data and predictions of the semi-empirical formula EPAX is
good. However, slight deviations between the measured and predicted cross sections
are present. In Figure 2.13, the cross sections for the oxygen isotopes are shown in
more detail. Again, the overall agreement between the experimental data and the
prediction is good although the measured cross sections indicate a slightly different
functional form for the isotopic yield than that used by EPAX. A similar trend can
be seen in Figure 2.12 for all isotopes with Z < 12. These, and other new data for
light residues from fragmentation reactions could be used to refit the parameters of

the EPAX formula to provide even better predictions of the cross sections.

The extrapolation of the expected counting rate for 260 from the cross sections
of the lighter oxygen isotopes is indicated in Figure 2.13. The EPAX prediction is
shown by the solid line. However, due to the systematic difference mentioned above,
a simple exponential fit to the most neutron-rich oxygen fragments was also used.
Even though the curves have clearly different shapes, they predict a similar order
of magnitude for the cross sections for 2"O, namely 2.43 nb (EPAX) and 1.07 nb

(exponential fit).

2.3.2 The Search for 26O

The carefully selected setting of the A1200 during the search for 26O is indicated in
Figure 2.10 by the hatched area, that corresponds to the momentum acceptance of
the A1200 during the long run. After 36 hours of data collection at this one setting,
2132 events were identified as 24O — approximately an order of magnitude more than

the previous experiment [Gui90] — without a single event that could be attributed to

31

 

 

 

 

H
O H
I o
N 0

Cross Section [mb]
8
4L

 

 

5...

O H
l O
N O

 

H
0.
uh

 

 

 

 

IMHLLUIU .1 .il.
2 4 6 B 10 2 4 6 8 10

N — Z
Figure 2.12: Measured reaction cross sections for neutron-rich isotopes (solid points),

ranging from 38P to 11B. The full lines indicate the predictions of the semi-empirical

formula EPAX [Siim90].

1114411141 llllJlJlelJlMlll

 

32

 

      

 

 

 

2
10 I I l I I I I \I\ I I I I I I l I I I I I I I T T T I
\
‘ 4
F-‘fi O ——-4
Q 10
E.
8
13‘ 10—2 —
C.)
<1)
m u
U)
8 0 exp. data
L. 10_4 — —
D -——EPAX
- ——— exponential fit. ~
10‘6 \
I M l l JIJ J l I l I l I J J l 1 J J 1 l L J J 1 L1
16 18 20 22 24 26

Mass Number

Figure 2.13: Measured reaction cross sections for several oxygen isotopes (solid
points), ranging from 18O to 2“O. The full line indicates the predictions of the semi-
empirical formula EPAX [Siim90], and the dashed line is an exponential fit to the
experimental data. The extrapolated production cross sections for 260 agree within
a factor of two.

33

260. Using this yield of 2"O and the two cross section extrapolations (see previous
section) of the yields of the lighter oxygen isotopes, we predict that approximately
400 or 800 events of 260 should have been observed. The present data corresponds
to an upper limit on the production cross section for 260 of 7 pb at a 90% confidence
level. The expected momentum distribution for 260 from the EPAX extrapolation is

indicated by the dash dotted line in Figure 2.10b.

As a large effort was put into assuring that the particles of interest were centered
at both the intermediate and final focus of the A1200, this result indicates that
the lifetime of 260 must indeed be appreciably shorter than the central flight time
through the separator (approximately 188 us). We interpret this to mean that 260 is

not particle stable, and that 24O is, in fact, the heaviest bound oxygen isotope.

Finally, there is the issue of the doubly magic nucleus 2"0. Only the global pre-
diction of Mfiller and Nix [M5188] predict this nucleus to be bound. In a recent
experiment at RIKEN Motobayishi et al. [Mot95] found a sizeable deformation for
the nucleus 32Mg, which indicates the vanishing of the N = 20 neutron shell for exotic
nuclei. A search for 28O, which was not possible during the present experiment, was
recently performed at GANIL [Tar96], and the experimenters concluded that it is

particle unstable.

Chapter 3

Coulomb Excitation of the
One—Neutron Halo Nucleus 11Be

3.1 Motivation and Introduction

Coulomb excitation is a well known and powerful tool to study nuclear structure
effects. If the charge distributions of the two colliding nuclei do not overlap during the
collision process, the measured Coulomb excitation cross sections are directly linked to
the electromagnetic matrix elements that characterize the electromagnetic excitation
of specific nuclear states. Therefore, Coulomb excitation can provide information
on the electromagnetic transition strength, and hence on the nuclear collectivity and
deformation. In the past, Coulomb excitation has been used to study low-lying nuclear
states by bombarding targets with heavy ions at bombarding energies below the
' Coulomb barrier. This technique insures that pure Coulomb excitation is the only
possible process to excite the nuclei. Until recently only stable nuclei were accessible

by this technique.

Recent advances in accelerator technology, together with advances of fragment sep-
aration devices allow the production of intense beams of radioactive nuclei. Therefore,
it is now possible to use Coulomb excitation to study nuclei far from stability. As

mentioned earlier, two different techniques are frequently used to produce the radioac-

34

35

tive fragments. The ISOL technique with subsequent acceleration of the secondary
beams to energies below the Coulomb barrier allows techniques similar to those used
for stable beams. One disadvantage of this technique is that only nuclei with a rather
long half-lives are accessible due to the time it takes to separate and (re-)accelerate

the fragments.

Projectile fragmentation on the other hand enables one to perform experiments
with radioactive nuclei that have a much shorter half-life. The limiting factor here
is the flight—time from the production target to the experimental area (typically on
the order of a few hundred nano—seconds). However, there are two disadvantages
to this approach. The first is that the 7-rays of interest are emitted from a source
moving at a sizeable fraction of the speed of light. Therefore, one has to be able
to correct for the Doppler-shift and keep the Doppler-broadening of the observed 7—
rays small. A more detailed discussion of the consequences these facts have on the
photon detection devices can be found in section 3.2.2. The second disadvantage is
that it is more difficult to assure ‘pure’ Coulomb excitation, as nuclear excitation
channels are now open as well. However, Coulomb excitation can be assured by
requiring extreme forward scattering angles, and by excluding reactions that lead to
fragmentation of the incoming radioactive beam. Therefore it is necessary to identify
isotopically the fragments before and after the interaction with the secondary — or
Coulomb excitation target. One advantage of this method is the possibility of using
thick secondary targets to Coulomb excite the projectile. Another advantage is that
projectile Coulomb excitation on stable high-Z targets is favored by a factor of Z§~ /Z,%.

Figure 3.1 shows a schematic view of the Coulomb excitation process.

A semi-classical description of the Coulomb excitation process that is valid for
relativistic energies can be found in [Win79]. In Appendix B a brief recap of the most

important points of the theory developed by Winther and Alder will be recapped. The

36

Zp'Ap

 

 

ZTvAT

Figure 3.1: Schematic view of the Coulomb excitation process. The minimum impact
parameter b is larger than the sum of the radii of the two colliding nuclei.

basic idea one has to keep in mind is that for pure Coulomb excitation the measured
Coulomb cross section (dam) is proportional to the reduced transition strength B(7r)\),
where /\ stands for the multipolarity and 7r is a placeholder to denote either the electric
(1r —i E) or magnetic (7r —» M) character of the transition. For the present work
we are interested in an E1 transition, and here the Coulomb cross section and the

reduced transition strength are related as follows :

 

0..., = (if—f) B‘flmT’rignom 42mm] (3.1)

where C is the so-called adiabaticity parameter and the functions gm can be evaluated
in terms of the modified Bessel functions (see Appendix B and Ref. [Win79] for

details).

We employed the technique of intermediate energy projectile Coulomb excitation,

37

which has previously been shown to be a useful method to study nuclear structure
effects in nuclei far from stability (see e.g. [Mot95, Sch96]), to Coulomb excite the
one-neutron halo nucleus 11Be. One can describe 11Be as a two-body system consisting
of a valence neutron coupled to quadrupole deformed 10Be core. The ground state is
dominated by the single particle 131/2 intruder state and not — as would be expected
from the simplest shell model -— by the 0p3/2 state. 11Be is the only known halo
nucleus with a bound excited state, in fact only the {I ground state and the %_
excited state are bound. Both are neutron halo states with root-mean-square radii of
approximately 7 fm [Esb94]. Figure 3.2 shows the wave functions of the ground and

excited state, as calculated with a Woods-Saxon potential.

Millener et al. [1], were the first to realize the special character of this system.
They measured the lifetime of the excited state with a Doppler shift technique and
found it to be 166(15) fs corresponding to a B(E1) of 0.116(12) ezfmz. This is the
fastest known E1 transition between bound states in nuclei. There is the possibility
that more subtle effects will appear when the lifetime is compared with cross sections
for inelastic nuclear scattering because (a) the Coulomb transition amplitude ap-
proaches unity at small impact parameters so that first-order perturbation theory may
no longer be valid, and (b) the extended halo wave functions may lead to nuclear exci-
tation well beyond distances normally considered ‘safe’ in Coulomb excitation experi-
ments. The first Coulomb excitation experiment [Ann95] in which a 45 MeV/ nucleon
11Be beam bombarded a lead target produced an excitation cross section of 191(26)
mb, only 40% of the expected value for pure Coulomb excitation at this energy. This
reduction is much larger than what could be reasonably expected from higher-order

effects (calculated to be on the order of 10-20% [Ann95, Typ95, Kid96, Ber95]).

We performed a new experiment in which both heavy and light targets (lead, gold,

carbon and beryllium) were used in order to investigate the various contributions to

38

 

IIIUITIII

   
  

I 1 I I T I I I

’ ' l
504 kev/////////

320 kev——— 1/2'

 

 

 

 

~
‘
~~_
———__,_
-*_—
‘—

 

x(r) [fm’l/zl

   

_ 1s,,2 3n = 504 keV

- - Ops/2 Sn = 184 keV

 

 

I I l l I L I l l l l l l I L l i l l l L
10 15 20 25 30
Radius [fm]

Figure 3.2: The wave function x(r) = rR(r) for the two bound states of 11Be, as
calculated from a Woods-Saxon potential. The rms radii are 7.3 fm, and 6.3 fm for

the 131/2 and 0p3/2 states respectively. The inset shows the level scheme of 11Be, and
the vertical dotted line indicates the rms radius of the 10Be core.

 

39

the cross section, which scale differently with Z and A.

3.2 Experimental Apparatus and Analysis

In the following sections the experimental setup used to produce the secondary 11Be
beam will be briefly described, and the experimental setup for the actual secondary
beam experiment, especially the position sensitive 7-array, will be discussed in some
more detail. Thereafter, the analysis process will be discussed, and we will briefly
mention how the position—, energy—, and efficiency calibrations were performed. As
we are interested in measuring a 7-ray with a relative low energy, we have to take the
absorption of the photon inside the target material into account. In section 3.2.6 the

implementation of this in the analysis will be described.
3.2.1 Detector Setup

A 80 MeV / nucleon 18O beam from the K1200 cyclotron was used to produce the sec-
ondary 11Be beam. The primary 180 beam impinged on a 884 £195 9Be target located
at the mid-acceptance target position of the A1200 fragment separator [She91]. The
magnetic rigidity of the first half of the A1200 was set to a value of Bp = 3.2439Tm
to transport fragments with a mass to charge ratio of 2.75. The energy spread of
the fragments was limited to :l: 1% by using the so—called momentum slits located at
the first dispersive image of the separator. To further separate 11Be from other frag—
mentation products, an achromatic wedge (”AL 425%) was placed at the second
dispersive image of the A1200. The fragments were identified in the A1200 focal plane
using a detector setup consisting of two x-y position sensitive parallel-plate avalanche
counters (PPAC) [Swa94], a 500 pm thick PIN diode to record the energy-loss, and a
thick plastic stopping detector for the total energy measurement. The time-of—flight

was determined with a TAC that measured the time difference between the thick

 

40

stopping plastic detector and the cyclotron RF signal. To center the 11Be fragments
at the focal plane, the magnetic rigidity of the second half of the separator was set
to a value of Bp = 3.1166Tm. Three different fragment species were observed at the
focal plane of the A1200, namely 9Li, 11Be, and 13B. The relative intensities of these
fragment groups were 1%, 20% and 79% respectively. The three groups were well
separated spatially, and in principle one could produce a very pure 11Be secondary
beam using the moveable slits located at the exit of the A1200 fragment separator.
However, as the counting rate was not a limiting factor, the mixed beam was used for
the secondary beam experiment, and individual isotopes were selected on an event-
by-event basis during the off-line analysis. The average energy of the 11Be particles
was 60.1 MeV/nucleon, and the counting rate at the focal plane was about 40,000
particles per second. The secondary beam experiment took place in the N3 vault and
the transmission from the A1200 focal plane to this experimental area was about 50%.
The time-of—flight of each particle was measured on an event-by-event basis over the
approximately 30 m long flight path, between a thin fast plastic scintillator (0.01” =
0.254 mm, BC404) located after the exit of the A1200 and the zero-degree detector
(fast—slow phoswich plastic). A schematic layout of the NSCL facility is shown in
Figure 3.3.

The experimental setup for the secondary beam experiment is shown in Figure 3.4.
After passing through two x-y position sensitive PPAC detectors [Swa94], the beam
impinged on the secondary target. The scattered ions then passed through a third x-y
position sensitive PPAC before they were stopped in the zero-degree detector. The
position information from the PPACs can in principle be used to track the incoming
and scattered particles. However, one of the PPACs was damaged in a previous
experiment, and could not be replaced in time for this experiment. Therefore, the

PPACs were only used to monitor the relative beam position, and to assure that

41

05—. HWMAN 0mm «EOBE vmw
_< «Esme ma. <32). 8 02

9.2:: “/3 I/ /

are

rlbl‘ll'l‘l I‘Po .. / s

t

   

 

 

=3¢> 02

Figure 3.3: Schematic layout of the NSCL facility.

42

the beam position did not shift over time. The zero-degree detector consisted of a
cylindrical fast-slow plastic phoswich detector with a diameter of 10 cm. It was made
out of a 0.6 mm thick layer of BC400 (fast), and a 10 cm thick piece of BC444 (slow).
This detector defined a half-cone opening angle of approximately 0max < 41°. The
secondary target was surrounded by the NSCL 7-array which will be described in
more detail in the following section. In this series of measurements four different

secondary targets (2°8Pb (80 £95), 197Au (533 3:19;), "“‘C (411 gin-97), 9Be (195 5%))
were used to excite the projectiles. All targets had a dimension of 5 cm x 5 cm and

are therefore much larger than the secondary beam spot, which has a dimension of

approximately 1 cm x 1 cm.

 

 

 

[’PAC PPAC
" ’ .. -. _' Zero degree
" Naltfi) detector
Natal)

4—— 1m ——-><—-—- 1m -——><-— .72m-—->

Figure 3.4: Schematic view of the experimental setup used for the projectile Coulomb
excitation experiment.

The energy and time resolution in the zero-degree detector used for the isotope
identification is shown in Figure 3.5. Here we display the energy loss in the zero-
degree detector (from the fast plastic) against the time-of-flight. The left hand side
of Figure 3.5 was obtained without the wedge at image #2, while the right hand side

of Figure 3.5 shows the result with the wedge and the the 533 £97 secondary gold

43

target in place.

One can see the A/Z = 3 line in Figure 3.5a), and the reduction of the number in

fragments that reach the zero degree detector after inserting the wedge at image #2,

in Figure 3.5b).

3.2.2 The NSCL y-array

To measure the deexcitation 7-rays following intermediate energy projectile Coulomb
excitation, a high efficiency photon spectrometer is needed. As the spectrometer is to
be used with a wide variety of radioactive beams, it has to fulfill certain requirements.
Due to the sometimes low beam intensities of the radioactive beams, a large angular
coverage and high efficiency over a broad range of y-ray energies are necessary. Since
the photons are emitted from a moving source - typically ,6 z 0.3 - one also has to be
able to correct for the Doppler-shift of the detected 7-rays, and try to minimize the
Doppler-broadening of the photopeak. Therefore, a high granularity of the detection
system is desirable. These requirements favor a compact array of position sensitive
N aI(Tl) detectors over a less compact array of non-position sensitive high purity ger-
manium (HPGe) detectors. Although HPGe detectors have a far superior intrinsic
energy resolution than N aI(Tl) detectors, their intrinsic photopeak efficiency is lower.
In addition, the intrinsic energy resolution of the NaI(Tl) detectors is comparable to
the limit in energy resolution set by the Doppler-broadening and is therefore sufficient
for our purposes. The NSCL 7-array [Gla97] consists of 38 position sensitive N aI(Tl)
detectors arranged in three concentric rings parallel to the beamline. The detectors
are on loan from a decommisioned PET machine at Washington University in St.
Louis. Each NaI(Tl) crystal is 18 cm long, has a diameter of 5.75 cm, and is enclosed
in a 0.45 mm thick aluminum shield. The crystals are connected to a photomultiplier

tube on each end, and the position and energy of the incident photons can be recon-

44

 

 

Energy Loss —>

 

b
.

0&8“:

 

llBe

 

 

 

 

<— Time-of-Flight

Figure 3.5: Energy loss vs. time-of flight (in arbitrary units), as recorded in the
zero degree detector and the beamline timing detector. The left panel was recorded
without the wedge at image #2. The right panel was recorded with the wedge and
the 533 £95 secondary gold target.

45

structed from the two photomultiplier tube signals. Assuming a coordinate system
with its origin in the center of the crystal and its x-axis along the crystal axis, the

light yield of the two photomultipliers (Ym) is proportional to :

Ym cx ei" (3.2)

The energy and position of the incident photon are then given by :

Energy o< ‘/K*Y2 (3.3)

Position or Iog(;—i-) (3.4)
2

To shield against background 7-rays, the whole array is surrounded by a 16.5 cm
thick layer — this corresponds to about 6 tons — of low background lead. A schematic

side and front view of the mechanical setup is shown in Figure 3.7.

A photograph of the support structure and the detector system is shown in Figure

3.6

The photograph in Figure 3.8 shows the actual NSCL 7-array with part of the

lead shielding in place.

3.2.3 Position Calibration

In the following the procedure used for the position calibration for each of the 38
NaI(Tl) detectors will be described. As mentioned earlier, it is very important to
achieve good position resolution as well as an accurate position determination, as
we have to correct for the Doppler-shift of the photons of interest. As described in
section 3.2.2, each NaI(Tl) crystal is read out by two photomultipliers. The position

of the incident photon is, according to equation 3.4, proportional to the logarithm of

46

 

Figure 3.6: The NSCL 'y-array.

47

  

%%%%NNNN
NNNN%W%/

     

0.4:...

 

    

 

    

  

'— "i_
’f%'%N%N%Nfl

I/////// Isa In ’ ‘I \‘V’/ V7 V7

NNNQ/V/fl/

      

  

    

 

 

 

 

 

 

 

 

 

 

Figure 3.7: Schematic view of the mechanical setup of the N SCL 7—array. The support
structure and the lead shielding (hatched areas) can be seen. The target is located in
the center of the beampipe. The beam enters from the left in the right hand picture,
and the Figure on the left hand side shows the array in the beam direction.

48

 

Figure 3.8: The NSCL 7-array, with part of the low background lead shielding in
place.
the ratio of the two light. The pseudo parameter POSITION was recorded for each

detector, and the following formula was used to calculate it :

POSITION = 2000 + 1000 * log (é) (3.5)
2

This gives the position in channel numbers, and places a photon that hits the center of
the detector into channel 2000 as described below. To calibrate the absolute position
we used a well collimated 60Co source. This source emits two 'y-rays with energies
of 1173.2 keV and 1332.5 keV, respectively. The calibration was performed in a
very labor intensive way, as each detector had to be calibrated individually. For the
calibration, a single detector was placed inside a cavity made out of low background
lead bricks, to shield the detector against background radiation and to collimate the
60Co source. The 'y—rays from the source reached the detector through a small hole

and illuminated about 1 cm of it. Each detector had a length scale (tape measure)

49

attached to the outside, which ranges from 0 cm to 20 cm, and is used as a reference

mark.

In the initial step the detector was placed inside the cavity, so that the center of
each crystal (position 10 cm) was illuminated by the 6"Co source. We made sure that
the actual signal height from each phototube was approximately equal by matching
the amplifier outputs on an oscilloscope. This places the pseudo parameter POSI-
TION around channel number 2000 for the central position (10 cm). Figure 3.2.3
shows the raw ADC output from the two photomultiplier tubes of one crystal — panel
a) and b) — as well as the calculated energy, using both photomultipliers (c) — see
equation 3.3 — and the calculated position ((1) -see equation 3.5— for the case that
the source illuminated the center of the detector. It should be pointed out that the
energy was not yet calibrated this point and that therefore all values are given in

channel numbers.

This measurement was repeated eight more times for each detector in position bins
of 2 cm width (2 cm, 4 cm, . . . , 18 cm), and the centroids of the calculated positions
were recorded. The actual position of the detector inside the cavity was determined
by removing the source, and peeping through cross hairs in the collimator hole onto
the tape measure on the detector. We performed a cubic fit to the data, and the
resulting fit coefficients were written into a lookup table to transform the position
in channel numbers into a calibrated position in centimeters. A typical example of
the achieved position calibration — given as position in centimeters versus calculated
position in channel numbers — for one detector is shown in Figure 3.10, together with
the results from the cubic fit. The vertical error bars shown represent the actual
width of the fitted position distribution (see Figure 3.2.3d). One has to be sure that
the cubic fit to the data does not turn over within the range of the detector. This

would result in an ambiguity of the determined position.

50

 

125 '
100 j

75:

LlA—LJ—LLLJAAIALA

50:

ALA

Counts/(5 chs)

25,

 

ALL].

 

 

0 11111111lu-1_._111l1’
0 500 1000 1500

 

 

Energy[chs]
:....,....,....,‘
125'? I “I
100’— C) —‘

75

iii

 

50

'IlTv—fi"

Counts/(5 chs)

 

h 1

b 1
25- e
b 1

4

 

1
d

 

0 AJ a L A l L A A L “Au“flnAI A

 

0 500 1000 1500
Energy [chs]

 

1257' v v v Ii 7 r v T f r v 1 l’

100} b) -3

 

 

 

 

 

 

 

 

 

O 1 1 1 1 J L 1 1 1 1
o 500 1000 1500
Energy [chs]
250_..+-,.r-.,....,....,‘
E i
200 _- d) 1
E i
150 f- j
100 :— €
50 E ‘1
i1 4
1 AJ ‘
o 1 L4 1 l 1 1J 1 l
o 1000 2000 3000 4000

Position [chs]

Figure 3.9: The raw ADC output from the two phototubes of a typical detector (a,
b), as well as the derived energy from combining the two raw signals (c) and the
calculated position ((1), for a 60C0 source in the center of the detector.

51

 

 

 

 

If T I j If I fl r I r q
3000 [— .—
1 .
3 2500 P ‘
\-‘ h —1
> -
‘07: r ‘
,9. 1- -
5 L 1
O
53 2000 ~—- —-
+ P d
o -t
8 " 1
N __ ']
1500 — ——+
1000 l l l L 1 l l l l J 1 J
0 5 10 I5 20

Position (cm)

Figure 3.10: Typical position calibration spectrum for one detector. Shown is the ac-
tual detector position (in cm) versus the calculated position from the two photomul-
tiplier signals (in channel number). The points represent the actual measurements,
the error bars represent the width of the distribution, and the solid line shows a cubic
fit to the data.

The best position resolution is of course achieved when a very strong correlation
between the calculated position and the actual position exists. The average position
resolution achieved was better than 2 cm. This corresponds to an angular resolution
of better than 10° in the present configuration. After the detectors were placed
in the array support structure and surrounded by the approximately 6 tons of low
background lead, the validity of the position calibration was checked again. This time
we placed a source at the secondary target position in the center of the array and
recorded the correlation between the observed energy and the calibrated position. We
expect to see no correlation between the energy and calibrated position, in this case
of isotropic emission at rest in the laboratory system, and this is demonstrated nicely

in Figure 3.12 for the 228Th source.

52

3.2.4 Energy Calibration

All 38 detectors that form the NSCL 7-array need also to be energy calibrated. This
task was performed after the detectors were placed in the support structure and the
low background lead shielding was in place. We placed different '7-ray sources in
the center of the array, in the same position that was used for the secondary targets
during the experiment. To cover a wide range of photon energies we used three
different sources, namely 88Y, 152Eu, and 228Th. As one can see in equation 3.3, the
deposited energy in each crystal is proportional to the square root of the product of
the light yield from the two photomultipliers. We require a position dependent energy
calibration and therefore divide each detector into ten slices of equal size. This was
done using the pseudo parameter POSITION (see equation 3.5), and not with the
calibrated position pseudo parameter, as we did not want to introduce additional
sources of error. For each position slice we used a linear fit to calibrate the energy.
The nice linear dependence between the calibrated energy (in keV) and the raw energy
(in channels) calculated via equation 3.3 can be seen for one specific position slice in
Figure 3.11. In this plot, we show the calibration points from the 88Y (898.1 keV,
1836.1 keV) and the 228Th (235.6 keV, 2651 keV) source. To check that there is no
correlation between the calibrated position and the calibrated energy, we plot one

against the other in Figure 3.12 for one individual detector, using the 228Th source.

In Table 3.1 we demonstrate the effectiveness of the energy calibration for four
calibration 7-rays spanning an energy range from 235.6 keV to 2615 keV, for the
eleven detectors that form the inner ring of the NSCL 7-array. We also present the
energy resolution achieved for those 7-rays. Typically we obtained a resolution of
better than 15% for the 235.6 keV photon, better than 9% for the 898.1 keV photon,
better than 7% for the 1836.1 keV photon, and near 6% for the 2615 keV photon.

53

 

 

 

 

.. r l T I j I 1 T r r I d

1000 ’— _.

l- -4

£3 1' -1
Q

C: l" _.

1: ’- .,

3. 300 — _

>5 '- _,

on >— ..
Lo

2 I- -1

[I] '- ‘i

, 400 — A

Q '- -4

m .. ..

.- -<

200 '— _

>- -1

.J

l .

o i- 1 I 1 1 1 1 i I l 1 1 I 1 1 l 1 i 1 A l L i 1 1d

0 500 1000 1500 2000 2500

Energy [keV]

Figure 3.11: Typical energy calibration spectrum for one detector and position slice,
using the 88Y and 228Th calibration sources. The open circles represent the measured
data points, whereas the solid line represents a linear fit to the data.

However, one can see in Table 3.1, that some of the detectors had rather poor energy
resolution and were therefore excluded from the subsequent data analysis. This point

will be discussed in the next section on the efficiency calibration, as well as in section

3.3.

During the experiment we performed several so-called ‘fast calibrations’, using a
152Eu source, to check for a possible drift in the energy calibration. For the ‘fast
calibrations’, we did not put the source at the secondary target position — which
would have required venting part of the beamline - but placed the source between the
support structure and the low background lead shielding. Even though the detectors
did not get illuminated equally under this condition, the energy signals were sufficient
to provide us with a few data points to check the stability of the energy calibration.
The detectors turned out to be stable in time, so that no additional changes to the

original energy calibration were needed.

54

 

Energy

 

 

 

 

Position

Figure 3.12: Calibrated position versus calibrated energy for one individual detector
using the 228Th source.

3.2.5 Efficiency Calibration

The efficiency calibration of the NaI(Tl) detectors was performed with a set of cal—
ibrated sources. As with the energy calibration we wanted to cover a wide range of
photon energies. Therefore, we used a calibrated 152Eu source, as well as a calibrated
88Y, and a calibrated 228Th source which emit 7-rays with known intensities in the en-
ergy range between 244.7 keV and 2615 keV. The sources where placed in the middle
of the array in the same position that was occupied by the secondary target during
the experiment. The energy spectra from each detector were again divided into ten
position bins of equal width. The efficiency for each bin was determined by counting
the number of 'y-rays registered in each position bin — corrected for the live-time of
the data acquisition system — and comparing them to the expected number from the

calibrated sources. The result for one specific detector is displayed in Figure 3.13.

55

Table 3.1: Energy calibration and energy resolution for the eleven detectors that form
the inner ring of the N SCL 7-array, using the 8’E‘Y and 228Th calibration sources.

 

 

E7 = 235.6 keV E7 = 898.1 keV E7 = 1836.1 keV E7 = 2615 keV
Detector channel FWHM channel FWHM channel FWHM channel FWHM

01 232.2 33.6 904.8 81.6 1838.9 120.1 2602.0 153.1
02 232.8 34.2 915.0 84.6 1855.1 120.5 2618.4 166.3
03 232.1 33.4 907.4 81.3 1843.2 108.4 2602.2 133.5
04 231.4 38.6 911.3 103.0 1852.8 131.5 2619.2 179.1
05 229.6 31.6 907.2 82.6 1843.1 107.5 2607.3 125.0
06 229.8 30.2 904.3 76.7 1839.5 96.5 2602.1 122.5
07 230.0 32.6 902.0 81.7 1834.6 107.0 2602.2 134.3
08 230.6 30.1 906.1 77.4 1841.3 96.9 2605.0 133.5
09 230.0 40.1 900.7 103.7 1836.2 127.7 2601.9 180.1
10 230.6 35.8 911.0 95.1 1853.1 122.9 2614.0 165.2
11 227.7 32.9 894.0 81.9 1820.6 107.7 2577.0 138.0

 

The solid lines connect the results for specific position bins, and are only drawn to
guide the eye. The energies displayed in this Figure are (from top to bottom) : 244.7
keV, 344.3 keV, 778.9 keV, 964 keV, 1100 keV, and 1408 keV.

The accuracy of the efficiencies determined by this method was checked by a
different method to measure the efficiencies. The 88Y source emits two correlated
photons, and by measuring the coincidence rate, one can deduce the efficiencies at
those energies (898 keV and 1836 keV). The obtained result agreed well with the

result obtained by using the above mentioned method.

To represent the steep drop in the efficiency with increasing photon energy, we fit

the calculated efficiencies with a function of the form :

6 = e-(ao+a1*109(E/(50k€V))) ,.. Jam’s—0W3?) (3.6)

The fit parameters a0, a1, and f were written into a table. The second term in equation
3.6 models the drop in the efficiency for the low energies. In Figure 3.14 the results

from the efficiency calibration (data points) are displayed together with the results

56

 

X2447 keV

, 0344.3 keV _
0.6 '— 0778.9 keV —
X 964 keV
’9 1100 keV
_ . 0 1408 keV s
0.4 — i ' : —

Efficiency [2]

 

 

 

 

 

' 0 5 10 15 20
Position [cm]

Figure 3.13: Efficiency for a typical NaI(Tl) detector of the NSCl 7-array, as a function
of detector position for several different photon energies from the calibration sources.
The lines represent the results for different energies, and are only drawn to guide the
eye.

obtained from the fit using the above formula (solid lines). The results are shown for
each of the ten position slices from one specific detector. As one can clearly see in
this double-logarithmic plot, we can achieve a very good efficiency calibration over
the whole range of calibration energies for the inner six position slices. However, for
the outer two — on each end of the crystal - the energy dependence of the efficiency
cannot be reproduced accurately by equation 3.6. Different fit functions were tested
as well, but did not lead to a better overall agreement to the measured efficiencies.
The extremely steep drop in the measured efficiency for the low photon energies in the
outer position bins, can of course also be seen in Figure 3.13. The steep drop is caused

by two different factors : a) absorption of the low energy 7-rays in the beampipe, and

b) the absorption inside the crystal itself. The latter attenuates the signal from a

57

low energy photon produced close to one phototube too much to permit a trigger of
the discriminator connected to the other phototube. Our main focus is on the 320
keV 7-ray from the %+ —+ {- transition in 11Be. Therefore, it is very important to
determine the detection efficiency for the low photon energies very accurately. Due to
the very good statistics in the photopeak from the transition of interest (see Figure
3.20), we were able to choose only those detectors for which a very good efficiency
calibration over the whole energy range of the calibration sources could be achieved.
This way, we were able to minimize the error introduced by the efficiency calibration
in our cross section calculation. Only three inner ring detectors were used for the
rest of the analysis, and even for those detectors, the two outermost position slices
on each side were omitted. As mentioned above, this could be done because we had

enough statistics in the photopeak of interest.

Our restriction of only using the six inner position slices of each detector reduces
the angular coverage (in theta) of the NSCL 7-array from approximately 45° to 135°
for all ten slices to about 60° to 120°. As we are interested in 7-rays emitted from
a source moving at about 1/3 of the speed of light (3 m 0.34), we have to take the
Doppler-shift of the emitted photons into account and calculate the efficiency at the
Doppler-shifted energy. The Doppler-shifted energy depends on the emission angle of

the photon as follows :

Lam/“$71 —

E7 = 1 — 60030

 

(3.7)

where E, stands for the Doppler-shifted energy observed in the laboratory frame, E70
is the unshifted photon energy in the moving frame (320 keV in our case), B is the
velocity of the projectile with respect to the laboratory frame in units of the speed of

light, and 0 is the 7-ray emission angle in the laboratory relative to the beam axis.

 

 

 

 

 

10‘2 .
5E g:
10‘3 .
-4 ' '
1° 5:: .5
log» 1 1 1 1 1 1 1 11
10 5.E E :
55: NE— E
>x10—4- '
0 SE E i:
5103 1 . .1....1 1 . 1 ..1....L 1. 1 ..1....1 1
"-110— r s- r
o 5;— ;- :-
Lfl _4g 5 F
10 5: z 5
10‘? 1 1 1 1 r 1 1 1 11” 1n.11...1 1 .
10‘ 55; .5 200 500 1000 2000

r—s v—n
°1 on
U1 b O1 CO
l"‘l ' "l""l
l'"‘I "'l"'1

 

 

 

10’5i 1 111....1 1 .i L .111...1 1.
200 5001000 2000 200 5001000 2000

Energy [keV]

Figure 3.14: Efficiency for a typical NaI(Tl) detector divided into ten position slices.
The open points with error bars represent the measured efficiencies, and the full lines
show a fit to the data, using equation 3.6. The position slices are in ascending order
from left to right, and from top to bottom.

 

59

The angular dependence of the Doppler-shifted energies can be seen in Figure 3.15.
For the actual NSCL 7-array the Doppler-shifted energy varies between 396 keV and
222 keV. For the limited angular range chosen for the analysis, the energy only varies

between 361 keV and 256 keV.

 

450
400

350

Energy [keV]

300

250

 

 

 

1 1 i '1 1 1 1 i 1 i 1 l 1 i g L -l
O 50 100 150
Angle [degrees]

 

Figure 3.15: Doppler-shifted 7-ray energy in the laboratory frame for the 320 keV
photon as a function of emission angle in the laboratory frame. The vertical lines
indicate the angular coverage of the NSCL 7-array (- -) and the restricted angular
coverage used for the analysis ( - ).

The error introduced by the efficiency calibration in the cross section calculation

is smaller than 5% for the {f —-> if transition in 11Be.

We were also able to measure the 3368 keV 7-ray from the 2+ —> 0+ transition
from the first excited state to the ground state in 10Be. We populate this excited state
in the nuclear breakup of 11Be on the light (C, Be) targets. The observed Doppler-
shifted energy for this 7-ray ranges from 3816 keV to 2707 keV. At the NSCL, there is
no calibrated 7-ray source available that covers this energy range. Therefore, we had
to extrapolate the efficiency from the measured efficiencies at the highest available
photon energies. This introduces an additional error. Whereas the efficiency for the

1408 keV and 2615 keV photon is reproduced to better than 9% by our fit, we assume

60

— in a conservative approach — that this number doubles for the 3368 keV photon.

The detector efficiency was combined with the angular distribution of the emitted
photons to determine the detection efficiency for the photons of interest. The angular
distribution of the 320 keV 7-ray from the g. -+ if transition in 11Be is isotropic in
the projectile rest frame. For the 3368 keV 7-ray from the 2"" —+ 0+ transition from
the first excited state to the ground state in 10Be we have to take the population
of the m—substates into account as well. This was done by using the formalism as
described in Ref. [Win79] (see also Ref. [Sch97]). A MATHEMATICA [Mat97] code

was developed to handle the above calculations.

3.2.6 Absorption of Photons in the Target

The following section describes the absorption of photons in the target material. As
the 7-ray from the transition of the first excited state to the ground state in 11Be
has only an energy of 320 keV, absorption is an important process, especially since
we used several rather thick secondary targets to Coulomb excite the incoming 11Be
nuclei. The method used follows closely the description found in Appendix G of the
Table of Isotopes [Fir96]. The decrease in intensity of a parallel beam of photons
passing through an absorber of thickness t is, according to Wapstra et al. [Wap59],

given by :

I = 10 1. {Tu—2 (3.8)

where 10 is the incoming photon intensity, and I stands for the photon intensity
after passing through the absorber material. The quantity t1/2 is the so called half-

thickness, and depends on the target material as well as the photon energy. It can

61

be expressed as :

t _ A
1/2_ NA*p*a(E)

 

(3.9)

where A denotes the atomic mass, N A is Avogadro’s number, p is the density of
the absorbing material, and 0(E) is the energy and material dependent atomic cross
section. The atomic cross sections are taken from the tables of Storm and Israel
[Sto70]. The cross sections take all four interaction processes of photons with matter
into account, namely the photoelectric absorption, coherent (Rayleigh) scattering,
incoherent (Compton) scattering, and pair production. At low photon energies the
most important process is the first, in which the photon is absorbed by an atom, that
subsequently emits an electron. The atom is left in an excited state and returns to
the ground state via the emission of Auger electrons or x-rays. Also important at low
energies is coherent— or Rayleigh scattering. In this process, the photon gets deflected
by the atomic electrons without losing energy. In the middle-energy regime, between
approximately 0.2 and 5 MeV, the most important process is the incoherent (inelastic)
Compton scattering. In this process the photon is scattered from an atomic electron,
but this time it transfers momentum and energy to the struck electron, sufficient
to put it into an unbound state. At higher energies pair production becomes an
important process. This process involves the creation of an electron-positron pair in
the field of a massive charged particle. The high energy electron of this pair produces
bremsstrahlung radiation as well as ionization along its path while the positron is
eventually annihilated with the production of new photons. For photon energies
below the rest mass of an e+e' pair (1.022 MeV), the latter process cannot take
place. We also neglect the coherent Rayleigh scattering, as this is only important for

so-called narrow beam experiments, where both the source and the detector are well

62

collimated, and this is not the case for our experiment.

To gain a qualitative understanding of the size of the absorption effects encoun-
tered, we calculate the absorption for a 300 keV photon, emitted from a pointlike
source in the middle of the target. The 7-ray then traverses through a different
amount of target material, depending on its emission angle, so that the constant
thickness t in equation 3.9 has to be replaced by one that is angle dependent, specif-
ically 57533791' The transmission probability for the different targets as a function of
photon emission angle is shown in Figure 3.16. The actual NSCL 7-array covers an
angular range from approximately 45° to about 135°. It is interesting to note that
the 411 £192- carbon target actually attenuates the photon intensity more than the 80

£192— lead target.

 

Transmission [7;]

 

lllllllllLllllilLLLlLlL

 

 

 

0 50 100 1 50
Angle [degrees]

Figure 3.16: Transmission probability for a 300 keV 7-ray through the secondary
targets as a function of photon emission angle, for the lead ( - -), gold (—), carbon
(- - -) and berylium (- -) targets used in this experiment. The vertical lines indicate
the angular coverage used for the analysis.

This first, qualitative attempt, however, needs to be refined, since for the light

targets (C, Be), the finite target size (5 cm x 5 cm) is not sufficient to absorb the 7-ray

completely, even when it is emitted under 90°. Therefore one has to take the finite

63

target size into account for the transmission probability calculations. We calculate
the transmission probability for photons emitted at 90° to be about 55% (C) and 65%
(Be) respectively. We also cannot use a single, fixed energy for the emitted photons,
as was done in our first attempt. As we are interested in 7-rays emitted from a
source moving at about 1/3 of the speed of light, we have to take the Doppler-shift
of the emitted photons into account as well. The Doppler-shifted energy depends on
the emission angle of the photon as shown in equation 3.7. For the actual NSCL
7-array the Doppler-shifted energy varies between 396 keV and 222 keV, and for the
limited angular range we choose for the analysis, the energy varies between 361 keV
and 256 keV. Figure 3.17 depicts the energy dependence of the atomic cross section
(including photoelectric absorption, and Compton scattering) as given by Storm and
Israel [Sto70] for 7-ray energies between 200 and 500 keV for the four target materials
used in the present experiment. The open circles represent the tabulated values, and
the error bars represent the possible error of 5% as given in [Sto70]. The smooth
lines connecting the points are our fits to the data as used in the following evaluation.
To reproduce the very steep drop off of the atomic cross section with the increase in
photon energy for the heavy targets (Pb, Au) we used a third order exponential fit

of the form :

El —:ro E: -.r El-xo

0(E) = yo + A119— ‘1 + A213— ‘2 0 + 1436— ‘3 (3.10)

where E, represents the Doppler-shifted energy of the emitted 7-ray. For the light

targets (C, Be), we use a simple quadratic fit, of the form :

where E, again stands for the Doppler-shifted energy of the emitted 7-ray. The fit

64

parameters are listed in Table 3.2.

Table 3.2: Parameters used to fit the atomic cross sections.

 

 

Target A1 A2 A3 t1 t2 t3 (to yo
Pb 185.4 89.32 62.01 115.7 10.42 106.9 196.58 37.62
Au 155.3 69.36 44.88 118.7 28.47 117.1 199.24 33.52

C -5.02-10‘3 3.7510"6 3.30
Be ~3.47-10"‘3 2.7510"6 2.22

 

Putting all the pieces together, we end up with a transmission probability as

shown in Figure 3.18. The asymmetry of the Figure is caused by the Doppler-shift.

One can further generalize this result and neglect the assumption that all pho-
tons are produced in the center of the target. To eliminate the dependence on the

production place, we integrate equation 3.8 over the target thickness :

t _ I 1
I : IO */ e t1/2elcosel * :dil!
O

t * 6 t o‘cos
_ —Io*————l/2 t'ws l /We-“’d$ (3.12)
0

here, t stands for the target thickness (note, that we have to use the full target
thickness now, and not only half of it, as before). Solving the integral leads to the

following expression for the transmission probability :

We can easily compare this result to the previous one, when we look at a series

expansion of the two. Substituting 3 = one can expand equation 3.13 into :

t
t1/2*|6080l

2 3
I=Io(1_i+s__3_...) (3.14)

65

 

 

 

 

_I I I I I I I I I I I I I I I T I I -_I ITI I I I I I I r T I IfI I I I I
1E1 300 — 20813b _
O I
.._>
(0 ~ ..
\ 200 —- ..
'° " ::
L—l
Cl — ..
o 100 ~— _.
”—4 1. ‘-
g ’- di-
0 1 1 1 1 «-
Uq)) I.1 l I I l l I I I I l % l' I l 1 Ir [jg-”-
2.5 — __
m 12
m ' C ‘-
O _.
i—a ..
U - -~ .
2.0 — —— 1
o » -. ,
ea 1' db
0 ' «1
.1—> 1.5 - —— _
<2 - -1
«L
10.111111111111111111.111111111LLL111111:

 

 

 

 

200 300 400 500 200 300 400 500
Photon Energy [keV]

Figure 3.17: Atomic cross sections as a function of photon energy for the four different
secondary targets used. The open points represent the data as given in Ref. [Sto70],
and the smooth curves show a fit to the data.

66

 

 

 

 

 

E
I:
O
.5 .1
.9.
§ . 1
g '— 4
t 0'4 __ —1
f 1
0.2 — —1
00 P 1 1 1 1 l 1 1 1 j I 1 1 1 1 i T
O 50 100 150

Angle [degrees]

Figure 3.18: Same as Figure 3.16, but now taking the finite target size and the energy
dependence of the atomic cross section into account.

Using the same substitution, equation 3.8 expands as follows :

s .92 s3
I—Io(1-§+§—Z8‘°H) (3.15)

As one can see, the results agree to first order, and the higher order deviations are only
small. Our first attempt (using equation 3.8) leads to a slightly lower transmission.
In Figure 3.19, a comparison is made between the two different results for the 533
£197 Au target, which has the strongest absorption and for which the difference in
the two approaches should therefore be the largest. The dotted line represents the
result calculated by equation 3.13, and the dashed line represents the transmission

probability obtained using equation 3.8.

Also shown in Figure 3.19 is the transmission probability as calculated by a linear

fit — instead of the third order exponential fit used so far — between the discrete

67

 

 

 

 

 

t - - . I . . . s 1 . - - . I .
. - ' - Exponential Fit. Integrated .
1.0 - - - Exponential Fit. Center -
‘_ -—- Linear Fit. Center 1
0.8 —
E >
a b
.9. 0.6 r
m b
.2
E b
m b
‘3 1
2
[__ 0.4 :-
1
F
0.2 *-
0.0 i ‘ ‘ ‘ ‘ l ‘ * ‘ l ‘ ‘ ‘ ‘ l
0 50 100 150

Angle [Degrees]

Figure 3.19: Comparison between three different transmission probability calculations
for the 533% Au target. See text for details.

data points of the atomic cross sections as given by [Sto70]. The difference in the
transmission probability from the two interpolation methods was considered and is
about 2% in the worst case (gold target). The difference between the three curves in
Figure 3.19 can be used to estimate the systematic error we introduce in our cross
section calculation. One also has to take the uncertainty in the atomic cross sections
into account, which is according to Ref. [Sto70] better than 5%. The systematic error
was therefore estimated by using the linear interpolation between the atomic cross
section, and the largest possible uncertainty in the atomic cross section. We found

the systematic error to be on the order of 5%.

This transmission probability — as calculated with equation 3.13 — has to be com-

bined with the detection efficiency as discussed in the previous chapter. This was

done by implementing equation 3.13 into the MATHEMATICA [Mat97] code used to

68

calculate the detection efficiencies. The modified detection efficiencies and transmis-
sion probabilities are given in Table 3.3. As can be seen, the absorption of the low
energy 7—ray plays an important role for all targets, and has to be taken into account.
Especially for the gold target one would get unreasonable results, if one would ignore
the absorption of the photon inside the target material.

Table 3.3: Detection efficiencies for the 320 keV photon, without absorption inside the
target material (emu) and with absorption (econ), and the transmission probability.

 

 

Target crow [%] econ [%] Transmission [%]
Pb 80551 6.54 5.71 86.7
Au 53335, 6.55 3.72 56.7
c 41133, 6.55 5.67 86.5
Be 195% 6.55 6.08 92.8

 

3.3 Experimental Results and Discussion

In the following the results from the experiment will be presented and discussed.
For clarity, the experimental results are broken up into three different sections. The
first one discusses the Coulomb excitation ofxBe to its first excited state using the
208Pb and the 197Au target. For those targets we are certain to be dealing with
Coulomb excitation only, and the results are therefore straightforward to interpret.
The measured Coulomb cross sections will be presented, and the B(E1) strength
will be derived and compared to previous Coulomb excitation experiments, and to
the values deduced from lifetime measurements. The second part also discusses the
excitation of 11Be into its first excited state, this time using the light targets (C,
Be). In this case one can no longer assume pure Coulomb excitation, but instead
a combination of nuclear and Coulomb excitation takes place. The measured cross

sections will be compared to model calculations for both the nuclear and Coulomb

69

excitation processes.

Finally in the third part, the results from the nuclear breakup of 11Be to the first
excited state in 10Be are discussed. This process was also observed during the runs
on the light targets. From the strength of this transition some information on the

ground state wave function of 11Be can be obtained.

3.3.1 11Be on 208Pb and 197Au targets

The Doppler corrected 7-ray energy spectra for the heavy targets, recorded under the
condition that a 11Be fragment was detected in the zero-degree detector (61.11, S 4.1°),
are shown in Figure 3.20. The two spectra on the top were summed up over the eleven
NaI(T1) detectors that form the inner ring of the NSCL 7-array, whereas the two lower
spectra were summed up only over the three detectors we used for the analysis. The
counting rates in all spectra are normalized for the particle flux and the number
of scattering centers in the various targets. In the projectile frame (,8 z 0.34) the
strong photopeaks centered around a 7-ray energy of 320 keV - corresponding to the
y —-> {f transition in the 11Be projectile — are clearly visible for both the 208Pb and
Au targets. The widths of the photopeaks are about 47 keV. According to Table
3.1, this is close to the intrinsic width expected for a 320 keV 7-ray, and shows the
effectiveness of the employed Doppler-shift correction. The obvious difference in the
yield of the photopeaks measured for the two heavy targets is easily understood by
the fact that a fraction of the 320 keV photons are absorbed in the rather thick Au

target (see section 3.2.6).

Assuming Rutherford trajectories, we calculate a minimum impact parameter be-
tween the heavy targets (Pb, Au) and the 11Be fragments, of approximately 13 fm.
This is about 3 fm larger than the distance between the centers of the projectile and

target assuming touching spheres, if we neglect the halo character of the 11Be nucleus.

70

 

 

'fifi'I'IITrIYfi'IWITr TYYY‘ITYVY'VYTY‘IYfV

Pb

      
 

2000

—1

600
1500

400

vIvvv—vfr'vv

1000

500 200

YVVfiIYVV

 

 

l
l

o 1 : - ‘
600
’ 200

100
200
i 50

Norm. Counts / (5 keV)

 

 

 

 

 

 

 

 

o A A 1 A A A A A A A A A A o A A A A A A A A A A A A
200 400 600 800 1000 200 400 600 800 1000

Energy [keV]

Figure 3.20: Doppler-shift corrected energy spectra for the 208Pb and 197Au secondary
targets. The data in the top row were summed up over the eleven detectors that form
the inner ring of the N SCL 7-array, whereas the bottom row shows only the data taken
with the three detectors used for the analysis. The 7-rays were recorded under the
condition that a 11Be fragment was detected in the zero-degree detector. The counting
rate is normalized by the incoming particle flux and the number of scattering centers
in the various targets.

71

As mentioned earlier, the extended matter distribution of 11Be may lead to nuclear
excitation well beyond distances normally considered safe in Coulomb excitation ex-
periments. However, a comparison between the measured angular distribution of the
scattered particles and a coupled channel calculation — using the code ECIS79 [Ray79]
— by Nakamura et al. [Nak97] show, that nuclear contributions to the cross section in
the angular range covered by the zero-degree detector are negligible. As we shall see,
this is not true for the light targets (C, Be), where we estimate that the contributions
are similar in size from nuclear and Coulomb excitation processes. Therefore, we will

treat the heavy and the light targets separately.

The integrated number of beam particles detected in the zero-degree detector
(9165 S 4.1°), the number of scattering centers in the various targets, and the num-
ber of background subtracted 7-rays in the photopeak provides a relative yield. To
determine the Coulomb cross sections, the relative yield was combined with the de-
tector efficiency folded with the angular distribution of the emitted photons (which
is isotropic in the projectile rest frame for a 1/2" —> 1/2+ transition). The absorp-
tion of the photons in the different target materials was also taken into account, as
already mentioned. Although we included only three NaI(Tl) detectors in the final
analysis, the statistics achieved in only a few hours of beam time is sufficient. For the
208Pb target we recorded 1055 :1: 95 counts in the background subtracted photopeak
within approximately 4.5 hours. For the much thicker 197Au target we recorded 1203
:1: 72 counts in the background subtracted photopeak within about 2 hours. From
these numbers we extract a Coulomb cross section of (325 i 39) mb for the 208Pb
target, and (257 :l: 27) mb for the 197Au target, respectively. The quoted errors take
the statistical errors, as well as the uncertainty from the fit / background subtraction,
the error from the efficiency calibration, and the error introduced by correcting for

the absorption probability — all added in quadrature — into account. The individual

72

contributions to the total error can be found in Table 3.4.

Table 3.4: The different sources of uncertainty are added in quadrature to yield the
total error (see text for details).

 

Statistical Fit/Background Efficiency Absorption Total

 

Target 1%] [%l [‘70] [%l [%l
208135 3 9 5 3 11
197Au 3 7 5 5 10

 

Assuming pure first order Coulomb excitation and the formalism by Winther and
Alder [Win79], as introduced in section 3.1 (see also Appendix B), we extracted the
reduced transition strength from the measured Coulomb cross sections. We have
compared our results to those from the lifetime measurements [Mi183] and to those
from the previous two Coulomb excitation experiments [Ann95, Nak97] in Figure
3.21, and in Table 3.3.1. For completeness we show not only the adopted B(E1) value
from the lifetime measurement, but also the three experimental results that led to
the adopted value. A brief derivation on how the lifetime is connected to the reduced

transition strength can be found in Appendix C.

Table 3.5: Measured B(E1) values

 

 

Experiment Ref. Target Em, B(E1) W
[MeV / nucleon] [e2fm2] [%]
Lifetime [Mi183] — — 0.116 :1: 0.012 —
RIKEN [Nak97] 208135 64 0.099 :1: 0.010 85
MSU/NSCL 208% 60 0.098 :1: 0.011 85
MSU/NSCL l97Au 60 0.082 :l: 0.010 71

GANIL [Ann95] 2°3Pb 45 0.045i0.006 39

 

73

 

 

 

 

 

 

0.15 '—
1 " adopted
value
b X
h ’ Pb An
N _
E x__
‘H 0.10 -
N .
m E
l_.J _
A _ -1
H
m _
V
m 0.05 — @
- BNL RIKEN MSU GANIL
- Lifetime Coulomb Excitation
0.00

Figure 3.21: Comparison between the different experimentally derived B(E1) values
for the 1/2+ —> 1/2' transition in 11Be. The results from the lifetime measurements
(x) and the adopted value (0) [Mi183] are shown together with the RIKEN measure-
ment (0) [Nak97] and the GANIL data (D) [Ann95]. The solid symbols represent the
present data.

74

As can be seen in Figure 3.21, the B(E1) value extracted for the gold target is
slightly lower than that from the lead target. Recall that large corrections had to be
applied to the data from the gold target due to the absorption of the low energy 7-ray
inside the target material. As can be seen in Table 3.3 the transmission probability
of the low energy 7-ray, for the gold target is, in our simple model, approximately
57%, compared to 87% for the 208Pb target. Uncertainties in the energy dependent
absorption coefficients introduce systematic errors in the calculation of the transmis-
sion probability which were estimated to be on the order of 5% (see section 3.2.6).
However, this error alone can not explain the discrepancy between the measured val-
ues, and seems to indicate that we underestimated the absorption in the thick gold

target.

In conclusion the extracted value for B(El) of 0.098(11) ezfm2 for the 208Pb target
is in excellent agreement with the recent RIKEN result [Nak97]. The extracted B(E1)
for the gold target is slightly — but not significantly - lower than the two measurements
on the 208Pb targets, and all values agree within error bars. The weighted mean of
all three data sets gives a B(E1) of 0.093(6) ezfmz. Neither our measurements with
several different targets, nor the RIKEN measurement [Nak97] could reproduce the
small transition rate from GANIL [Ann95], which also could not be explained by
several theoretical models. Both the present B(E1) values and the RIKEN results
agree within error bars with the results from the lifetime measurements. The fact
that both recent Coulomb excitation experiments only account for about 85% of
the adopted B(E1) value extracted from three lifetime measurements indicates that
higher order excitation effects, as discussed in Ref. [Typ95, Kid96, Ber95], cannot be

excluded by the present data.

75

3.3.2 11Be on "“‘C and 9Be targets

In Figure 3.22 we show again the Doppler corrected 7-ray energy spectra recorded
under the condition that a 11Be fragment was detected in the zero degree-detector,
but this time using one of the two light targets. As in the previous section, the
spectra on the top were summed up over the eleven NaI(Tl) detectors that form the
inner ring of the NSCL ’y-array, and the spectra on the bottom show the results
for the three detectors we that were used for the analysis. The counting rates were
normalized for the particle flux and the number of scattering centers in the various
targets. One can clearly see the photopeaks centered around a 7-ray energy of 320 keV
— corresponding to the y —+ y transition in the 11Be projectile. As expected, the
photon yield for the light targets is much smaller than the one for the heavy targets
(see Figure 3.20). We obtained photopeaks of similar size for the carbon and the
beryllium target. However, for the beryllium target the photopeak is contaminated
by a strong background, and the distinction between projectile and background '7-rays
becomes difficult. This background is mainly due to nuclear reactions, that were also
observed with the other low-Z target. Therefore, the data taken with the beryllium
target suffer from large uncertainties in the background subtracted counting rate in

the photopeak.

For the low-Z targets we cannot rule out the nuclear contributions to the excitation
process. Specifically, the minimum impact parameter insured by the experimental
setup for the carbon and beryllium target is smaller than the sum of the radii of the
two colliding nuclei. To determine the experimental cross section we followed the
same procedure as described in the previous section for the heavy targets. Again we
included only three detectors into our final analysis, and again the achieved statistical

accuracy is not a limiting factor. We recorded 1173 :l: 117 counts in the background

76

 

 

 

 

Norm. Counts / (5 keV)

 

 

 

 

 

O“‘l““l““l““ 0...
200 400 600 800 1000 200 400 600 800 1000

Energy [keV]

A i A L_L_.A l A a A A l A A A. A_

 

 

Figure 3.22: Doppler-shift corrected energy spectra for the "“‘C and 9Be secondary
targets. The data in the top row were summed up over the eleven detectors that form
the inner ring of the NSCL 7-array, whereas the bottom row shows only the data
taken with the three detectors used for analysis. The 7-rays were recorded under the
condition that a 11Be fragment was detected in the zero-degree detector. The counting

rate is normalized by the incoming particle flux and the number of scattering centers
in the various targets.

77

subtracted photopeak for the carbon target and about 756 :t 151 counts for the
beryllium target. However, as mentioned earlier, because of the large background for
the beryllium target, we introduce a rather large error during the fitting/ background
subtraction process (about 20%). We measured an excitation cross section of (3.4 i
0.4) mb for the carbon target and (1.8 :l: 0.4) mb for the beryllium target. The quoted
errors take the statistical errors, as well as the uncertainty from the fit / background
subtraction, the error from the efficiency calibration, and the error introduced by
correcting for the absorption probability — all added in quadrature — into account. he

individual contributions to the total error can be found in Table 3.6.

Table 3.6: The different sources of uncertainty are added in quadrature to yield the
total error (see text for details).

 

Statistical F it/Background Efficiency Absorption Total

 

Target [%1 [%l [%l [%l [%l
"“‘C 3 10 5 3 12
9Be 4 20 5 2 21

 

These experimentally determined cross sections should be compared to theoretical
predictions for the Coulomb and nuclear process. As in the previous section, we
calculate the (theoretical) Coulomb cross sections according to Ref. [Win79]. For
impact parameters smaller than the sum of the 10Be core and the target radius,
one expects core fragmentation to take place, which will not contribute to the state
of interest. The energy dependent core and target radii were chosen to reproduce
previously measured heavy-ion interaction cross sections [Tan95]. We calculate the
minimum impact parameter to be 5.5 (5.2) fm, and accordingly derive a Coulomb
cross section of 2.56 (1.15) mb for the carbon (beryllium) target. The measured cross

section for the carbon target is about 30% bigger than the theoretical first order

78

Coulomb excitation cross section. For the beryllium target we measure a 60% larger
cross section than predicted by pure first order Coulomb excitation, however, the

measured cross section has large uncertainties.

To estimate the nuclear cross section, a similar technique as described in Ref.
[Han96] was employed. Again one can assume that all reactions with an impact
parameter smaller than the sum of the 10Be core and the target radius result in
fragmentation of the core, and therefore can be neglected. Due to the high secondary
beam energy one can apply the eikonal approximation, and we assume straight line
trajectories for the collision geometry. In the projectile rest frame one can imagine
the target nucleus passing through the halo system, without touching the core and
leaving a wound in the wave function (see Ref. [Han96] and Appendix D for details).
To estimate the nuclear cross section, we have to calculate the overlap of the disturbed
ground state wave function with the I = 1 wave function from the first excited state.
Using Woods-Saxon wave functions, this problem can be solved in the same way
as described in Ref. [Han96], and in Appendix D the interested reader can find
a short recap of some details of the calculation. We will report only the result of
the calculation here. Assuming pure single-particle states, we derive a nuclear cross
section of 3.5 mb for the carbon target. If we assume single-particle amplitudes to
0.9 and 0.95 respectively, we get a nuclear cross section of 3 mb for the carbon target.
Hence, the theoretically predicted nuclear and Coulomb cross sections are of similar
size for the light targets, and comparable to the measured cross section. The fact that
the measured cross section is not the sum of the two contributions seems to indicate

the presence of Coulomb - nuclear interference.

79

3.3.3 Nuclear Breakup of 11Be to the First Excited State in
10Be

In the following we will discuss the nuclear breakup of 11Be into the first excited state
of 10Be(2+, 3368 keV) and its subsequent decay to the 10Be(0"’) ground state. This
reaction channel was observed during the runs on the carbon and berylium targets. It
is of particular interest as one can extract information about the nuclear structure of
the 11Be ground state wave function from the strength of this reaction channel. The
ground state wave function of 11Be is dominated by the single particle 131/2 intruder
state and not — as would be expected from the simple shell model - by the 0123/2 state.
However, the single particle picture only describes the dominant part of the ground
state wave function. One also has to take the coupling of the 0015/2 state to the first
excited state in the 10Be core into account. Therefore, one can write the 11Be ground

state wave function as :

|“Be,1/2+ >= \/§|1°Be(0+) 6; (131,2). > +\/S_d|1°Be(2+) (8) (0515/2). > (3.16)

Where 5 and 5.; are the spectroscopic factors. Zwieglinski et al. used the 10Be(d,p)”Be
reaction at 25 MeV to extract the spectroscopic factor S to be 0.77 [Zwi79]. Therefore

one expects to have about a 20% admixture of 0d5/2 coupled to the 2+ core excitation.

Anne at 01. measured the dissociation cross section of 9Be(“Be,1°Be)n reaction at
41.5 MeV/ nucleon to be (290 :l: 40) mb [Ann94]. This cross section - properly scaled
for the different incident beam energies — can be compared to our cross section for
the 11Be —+ 10Be(2"‘) cross section, to extract information about the 54 spectroscopic
factor. In principle, the necessary information could also be extracted directly from
our data, by comparing the number of incoming 11Be particles to the number of

detected 10Be fragments detected in the zero degree detector. This work is currently

80

in progress. In Fig. 3.23 we show the Doppler corrected ’y-ray energy spectra recorded
under the condition that a 10Be fragment was detected in the zero degree-detector for
the 195 £575 target. The spectra were summed up over the same three detectors used
for the previous analysis to assure a well defined efficiency calibration. The spectrum
on the left hand side of Figure 3.23 was recorded under the additional condition that
only one '7-ray was detected (multiplicity = 1). By multiplicity we mean the number
of simultaneously detected 'y-rays in the NSCL 7-array, and not the multiplicity of
actually emitted 7-rays. The quantity we call multiplicity is also often referred to as
fold. A photopeak at an energy of 3368 keV is clearly visible, as is the first escape
peak, located 511 keV lower in energy. The spectrum on the right hand side of Figure
3.23 does not use the additional multiplicity cut, in other words many 7-rays could
be detected simultaneously. One can clearly see the increase in background counts.
In Figure 3.24 we show the same spectra for the 411 £97 target. Again, we plot the
spectrum with the multiplicity cut on the left hand side. Although, the photopeak
at 3368 keV is still clearly visible, the background — even in the spectrum with the

multiplicity cut — is much higher, due to target 7-rays.

We now have to discuss the influence of the multiplicity — or fold — cut on the
observed 'y—ray yield. In Figure 3.25 we show the background subtracted counting
rate in the 3368 keV photopeak as a function of observed 7-ray multiplicity for the
carbon target. One can clearly see, that the counting rate depends on the multiplicity
cut. Therefore, one has to use the spectrum without the multiplicity — or fold — cut
to evaluate the 7-ray yield. This introduces an additional error, as it is much harder

to get a reliable background subtracted peak from each of those spectra.

We recorded 1230 :1: 188 counts in the background subtracted photopeak for the
berylium target, and 1195 d: 203 counts for the carbon target. Combining the 7-ray

yield with the number of beam particles detected in the zero-degree detector, and

81

 

 

 

 

 

 

 

 

 

BSD-IIIII’IIIIIIIIIITIII‘ I_IIIIIIIIIIIIII IIII1
.. .1 " 7
. 600— —
200 —-—
A . -
> r 1
Q) 1 ..
.3: ¥ . 600
150 fi
5.; r _
v __ ..
\ - a
m . , 400
‘E’: 100— -—~
5 >- A
O 1- -f
U 1- -l
1- -1
50_ __ 200— —
_ J ' '1
.- 4 1- -
ollllllllllllLllllll llllilLlJlllllillll
2000 2500 3000 3500 4000 2000 2500 3000 3500 4001

Energy [keV]

Figure 3.23: Doppler-shift corrected energy spectra for the 9Be secondary target.The
data were recorded under the condition that 3. 10Be fragment was detected in the zero-
degree detector. The spectrum on the left hand side has the additional requirement
of only a single detected 'y-ray.

82

 

 

 

 

 

 

 

 

 

 

50 I I I I I I I W I l l I I I I I VI I I I I l I I Ifi I I I
3 _ 1 I 1 . 100° . 1 r 1 I .
Z 1 E I
_ .. l .,
200 600 —~
A d
> .- -1
Q) _ a
.3: . _
o 150 — 600 __ —~
LO ~ 1 ~
V _ 1. —
\ ~ - -
m _ - ..
B 100 — 400 —
:1 ’ ’
O " "
U .. -
- +
50 — 200 —
o IJlLiJllllLlllillll-i i-L111Jn1LlllllLIJILl 1‘
2000 2500 3000 3500 4000 2000 2500 3000 3500 4001

Energy [keV]

Figure 3.24: Doppler-shift corrected energy spectra for the "“‘C secondary target.The
data were recorded under the condition that a 10Be fragment was detected in the zero-
degree detector. The spectrum on the left hand side has the additional requirement
of only a single detected 7-ray.

83

 

 

 

 

 

 

 

P fir I T—I I j I f T—I I T f I f d
1400 —- -~ —
L .
>- 4
>- -1
1200 ~— _P c) —
I- F .1
1n .- _
A.)
g 1000 —— I ‘_ __
- O ..
O
o 7 CD .
1- q
000 —- CD __
" 1L '1
1- J5 -I
600 ~— A
)- .4
” Fold = 1 2 3 4 5 No Limit 2
l l l 1 J L l 1 J J L l l l 1 l 1 l l L l I l ‘

 

Figure 3.25: Background subtracted counting rate in the 3368 keV photopeak as a
function of detected 7-ray multiplicity - or fold -— for the carbon target.

the number of scattering centers in the various targets provides a relative yield. To
determine the cross section, one has to combine this information with the detector
efficiency folded with the angular distribution of the emitted photons, where one has
to take the population of the m-substates into account as well. The later was done
by using the formalism as described in Ref. [Win79] (see also Ref. [Sch97]). We
extract a cross section of (34.5 i 8) mb for the berylium target, and a cross section
of (22.1 :1: 6) mb for the carbon target. The quoted errors take the statistical errors,
as well as the uncertainty from the fit/background subtraction, and the error from

the efficiency calibration — all added in quadrature — into account.

As mentioned earlier, these cross sections have to be compared to the total dis-
sociation cross section as measured by Anne at al. [Ann94]. However, as their data
were taken at a lower incident beam energy, namely 45.1 MeV / nucleon compared to
our data taken at 60.1 MeV/ nucleon, we first have to scale the measured total dis-

sociation cross section accordingly. If we assume that the dissociation cross section

84

scales like the free neutron-neutron (n-n) cross section, we have to reduce the value
of (290 :l: 40) mb to (221 :l: 31) mb. Therefore, our measured cross section for the
nuclear breakup of 11Be to the first excited state in 10Be is about (16 :l: 4) % of that

of the total dissociation channel.

Chapter 4

Summary and Outlook

The present work studied the structure of light neutron-rich nuclei. Especially the
knowledge about the exact location of the neutron dripline was extended for the
oxygen isotopes. Using the measured cross sections of the lighter oxygen isotopes, we
predict that several hundred events of 260 should have been observed. However, we
could not attribute a single event to the production of 260. An upper limit on the
production cross section for 26O of 7 pb at a 90% confidence level was determined.
As large effort was put into assuring that the particles of interest were centered
at both the intermediate and final focus of the A1200, this result indicates that the
lifetime of 26O must indeed be appreciably shorter than the central flight time through
the separator (approximately 188 ns). This result indicates that 26O is not particle
stable, and that 24O is, in fact, the heaviest bound oxygen isotope. The finding that
26O is particle unbound provides another very stringent test to the predictions of
theoretical models. During the search for 26O the production cross sections of 72
neutron-rich nuclei, ranging from 38F to “B, were also determined and compared to

model predictions.

In the second part of the present work, the technique of intermediate energy pro-

jectile Coulomb excitation was employed to study the one-neutron halo nucleus 11Be.

85

86

The measurement of the excitation cross section of 11Be from its %+ ground state to
the 5' first excited state on both light and heavy targets allowed us to investigate the
various contributions to the excitation cross section. The extracted reduced transition
strength B(E1) for the lead and gold target agrees with a previous Coulomb excitation
measurement from RIKEN [Nak97] on a lead target. Those measurements account
for about 80% of the B(E1) extracted from a lifetime measurement [Mi183]. This indi-
cates that higher order excitation effects, as discussed in Ref. [Typ95, Kid96, Ber95],
cannot be excluded by the present data. Neither the present work, nor the RIKEN

measurement could reproduce the small transition rate reported from a GANIL mea-

surement [Ann95].

For the carbon and berylium targets we predict a contribution of similar size from
the Coulomb and nuclear excitation channels to the measured cross section. From
the fact that the measured excitation cross section is not the sum of the predicted
Coulomb and nuclear processes, we deduce that Coulomb-nuclear interference plays

a role.

The nuclear breakup of 11Be into the first excited state of 10Be(2+, 3368 keV)
and its subsequent decay to the 10Be(0+) ground state was also observed during the
runs on the carbon and berylium targets. The measured cross section for the nuclear
breakup of 11Be to the first excited state in 10Be accounts for about ( 16 :l: 4) % of
that from the total dissociation channel [Ann94], if the later is properly scaled for
the different incident energies. This result is in good agreement with the measured
spectroscopic factor of 0.77 from the 10Be(d,p)“Be reaction at 25 MeV [Zwi79], and

confirms the dominant single particle character of the 11Be ground state.

For an outlook into the future, one can say that although the knowledge and

understanding of light neutron-rich nuclei has increased tremendously in recent years,

87

much work still remains to be done. The recently commissioned S800 spectrograph, as
well as the planned new radioactive beam facilities at MSU and RIKEN will provide
the opportunity to study light neutron-rich nuclei in more detail. The availability
of much increased beam intensities will allow one not only to expand the knowledge
about the exact location of the driplines even further, but also provide sufficient

secondary beam intensities for detailed nuclear structure studies.

APPENDICES

Appendix A

Measured Cross Sections

The following tables show the the individual measured cross sections for 72 neutron-
rich isotopes ranging from 38F to 11B, together with the predictions from the EPAX
parameterization [Siim90]. The listed uncertainty takes the statistical error, as well
as the uncertainty from the gaussian fit to to the momentum distributions — both

added in quadrature — into account.

88

89

Table A.1: Isotopic cross sections from the reaction 40Ar + 9Be at 90 MeV/nucleon.

 

 

Z A UEPAX [mb] O’Exp [mb]

15 38 4.18-10-03 (4.554.42)-10-02
15 37 2.55-10-01 (1.184.08)-10+°°
15 36 1.16-10+°° (1.874.07)-10+°°
15 34 1.17.10+01 (2.154.1).10+°°
14 37 3.24-10-04 (1.344.21)-10-03
14 36 24710-02 (1.684.20)-10-02
14 35 13910-01 (8.404.79).10-°2
14 34 6.19-10‘01 (7.564.38)-10-°l
14 33 2.35-10+°° (1.214.03)-10+°°
14 32 7.41-10+°° (1.344.08)-10+00
13 35 1.95-10-03 (1.054.11)-10-°3
13 34 1.22-10"°2 (1.02:1: .09)~10"02
13 33 6.6910“)2 (1.97:L-.11)-1 10‘01
13 32 3.23-10‘01 (6.59i.3 9)-1 10'01
13 31 1.34-10+°0 (2.114 06) 10+00
13 30 4.67-10+°° (1.76:t .07)-10+00
13 29 1.29-10+01 (3.89:1:.3 5)1 10+00
12 33 8.08-10'0“ (3.504 53) 10-04
12 32 5.3810‘03 (8.05:1: 89) 10‘03
12 31 3.2210-02 (5924.5 7)1 10-02
12 30 17010-01 (4.184. 5)1 10-01
12 29 7.71-10-01 (9.094 42) 10-01
12 28 2.93-10+°° (1034.0 4)1 10“”
12 27 8.90-10+°° (3.764 35) 10+"0
11 31 33110-04 (3.614 56) 10-04
11 30 2.40-10-03 (3.5043 5) 10-03
11 29 1.56-10-02 (5324.2 6)1 10-02
11 28 8.9810")? (3.874 15) 10-01
11 27 4.44.10-01 (1.934.07)-10+°°
11 26 1.84-10+°° (3.004.09)-10+°°
11 25 6.11-10+°° (4.434.42)-10+°°
10 28 1.09-10-03 (1.754.16)-10-°3
10 27 7.69-10'03 (5.95i.48)-10’°3
10 26 4.79-10-02 (2.614.11)-10-°l
10 25 2.57-10-01 (1.124.06)-10+°°
10 24 1.16-10+°° (3.904.16)-10+00
10 23 4.19.10+°° (4.024.20).10+°°
10 22 1.15-10+01 (3.65:1:.36)-10+00

90

Table A.2: Isotopic cross sections from the reaction 40Ar + 9Be at 90 MeV/ nucleon.

 

 

Z A UEPAX [mb] O'Exp [mb]

9 26 50010-04 (3524.6 0)10-04
9 25 3.84-10-03 (1.194.12)10-02
9 24 259-10-02 (1.0040 8)1 10'-01
9 23 wow01 (1.5540 8)10+°°
9 22 732-10--01 (3.29 4.13)10+°°
9 21 2.88-10+00 (7.4843 0)1 10“”
9 20 8.58-10+°0 (3.754.18)10+°°
8 24 2.33-10‘0“ (3.56 :1: .35) -10"°“
8 23 19410-03 (1.404 .12)10-03
8 22 14210-02 (8.16 4 .65) 10-02
8 21 8.8910"02 (6.41 :1:.3 9) 190‘01
8 20 4.67-10-01 (4.1842 0)1 10“”
8 19 1.98-10+°° (5854.2 2)10+°°
8 18 6.42-10+°° (6444.2 6)10+°°
7 22 1.11-10-04 (3.4545 2)10~°4
7 21 90710-04 (8054.8 9)10--°3
7 20 78510-03 (3.8243 6)10-02
7 19 5.31002 (6.2045 1010-01
7 18 3.0110“01 (2.08 :1:.0 8) 810+00
7 17 1.37-10+°° (7.26 4 .29) 10+00
7 16 4.81-10+00 (5.54 i .22)-10+00
7 15 1.16-10"’°l (4.51 :1: .31) 10+00
6 19 5.21-10-04 (1.75 4 35) .10-03
6 18 44310-03 (4.4847 6)1 10-02
6 17 3.2210-02 (8.93 4 .89) 10--02
6 16 1.9610")1 (1.33 i .08) 10+00
6 15 9.62-10-01 (3404.2 0) 10+00
6 14 3.62-10+°° (1.62 4 .09) 10+"1
6 13 9.47-10+°° (5344.2 7)10+°0
5 15 1.98-10-02 (1084.2 41) 10-01
5 14 1.29-10-01 (1.59 4 .27) 10-01
5 13 6.81-10-01 (2.55 4 .25) 10+00
5 12 2.74-10+°° (5.2843 21) 10+00
5 11 7.73-10+°° (5.75 4 .30) 10+00

Appendix B

The Coulomb excitation process

The Coulomb excitation process can be evaluated in a semi-classical picture as de-
veloped by Winther and Alder [Win79]. In this case, one assumes that the motion
takes place along classical Rutherford trajectories, and that the excitation process
itself can be described via quantum mechanical perturbation theory. This of course
is only possible for weak excitations. The cross section for exciting a final state If >

from the initial state Ii > is given by :

do (do)
—— = — Pi; (B1)

The Rutherford cross section is expressed by :

do a?)
— = —— 8.2
(‘19) Ruth 43in4(%) ( )
here 0 is the scattering angle and 2.00 is the distance of closest approach, which can

be calculated as :

_ 2121362

a __ ___.__
0 mo62,32

(13.3)

where, Z13}: is the elemental number of the target (projectile) respectively, mo is the

91

92

reduced mass of the system, and c2 is the square of the speed of light. For reactions
with incident energies below the Coulomb barrier, the minimum 00 is achieved for

central collisions.

Pi; is the probability to excite the state If >, and for weak excitations it can be

expressed via first order perturbation theory as :

R1 = lah'fl2 (8.4)

where the transition amplitude 0,; can be expressed as :

5., = 715/: as!“ < f|V(r(t))|z' > dt (35)

Here, V(r(t)) is the time-dependent electromagnetic potential, that is given by the

Liénard-Wiechert potential, and 1.0;; is given by w],- = %(E, — Ei).

One can expand the electromagnetic potential V(r(t)) in its multipole compo-
nents VAp(r(t)), and express the transition amplitude 11,-; in terms of the electric
and magnetic multipole moments M(EA11) and M(M)\p) — or in general M(1r/\p) —

respectively as :

ZTe < f|M(7rAp)|z' >

A
hca0

 

air = 42 11(4) 03.6)
A

Here, f1(C) is a function that depends on the adiabaticity parameter which, for non»

relativistic energies, is given by :

a
C = wfiTcou = 1421;:0 (8.7)

where Ta,” is the collision time, which is given by 7'60" = 9v“, where v is the projectile

93

velocity, and 2-00 is the distance of closest approach, given by equation B.3. For
C > 1 the collision is adiabatic and the excitation probability falls exponentially with
C. Therefore, the adiabatic cutoff is given for C m 1, and this limits the possible

excitation energies to about 1-2 MeV (see equation B.7).

For intermediate and relativistic energies equation B.7 needs to be slightly mod-
ified. For reactions with incident beam energies below the Coulomb barrier, the
distance of closest approach was reached for central collisions. For higher incident
beam energies, one only can consider pure Coulomb excitation, if the impact param-
eter (b) of the colliding system is larger than the sum of the radii of the two nuclei.
One also has to take the Lorentz contraction of the electromagnetic field into account,

so that the adiabaticity parameter for relativistic energies should be written as :

I b
C: 0107411 = 0111;; (3.8)

where 7 is the usual relativistic factor (7 = 1/ W, ,6 = E). Due to the shorter
collision time Tc'o” in intermediate and relativistic energy collisions, the adiabaticity
parameter is smaller than in the above discussed non-relativistic collisions. Therefore,
the maximum reachable excitation energy is also larger at those higher incident beam

energies.

One can now write the Coulomb cross section for exciting a state If > with A-pole

pure Coulomb excitation as :

ZTe < f|M(1rA;1)|z' > 2
hcb"

 

be do
a, = 24/ d b bdb (13.9)
50

where the lower integration limit is the minimum impact parameter, given by the

 

sum of the two nuclear radii, and the upper integration limit is given by the impact

parameter at which the adiabaticity parameter C (see equation B8) is equal to unity

94

(adiabatic cutoff). This integral can be evaluated using first order time-dependent
perturbation theory. A detailed discussion about this subject can be found in Ref.
[Win79], and goes well beyond the scope of this work. Therefore, we will only quote
the final result, which relates the Coulomb excitation cross section to the reduced

transition strength B(7r)\), namely :

 

ZTC2 2 2(A—l) 1 “U 2
Ucoul : hC E :k ;B(WA)|G1r/\u(2)| gfl(C(bmin)) (B-IO)
«A11

where the functions G“), are tabulated in Ref. [Win79], and the functions gu(() can

be expressed via the modified Bessel functions ( K p) as :

91(4) = 9—0(C) = 7rC2llKu+l(C)|2 - |K11(C)|2 - %&A,u+l(C)Ku(C)l (B-11)

In our case, we are interested in an electric dipole transition from the first excited
state to the ground state in 11Be. Therefore, the above general equation will simplify

for our special application to :

 

3..., = (2;: ) 52411167719154.0441151 (13.12)

It should be pointed out here, that the adiabaticity parameter is specific for each
experiment, as it depends on the beam energy and on the minimum impact parameter
which depends on the experimental setup. One can express the minimum impact

parameter as :

b0 = 99c01(10m.,) (13.13)
7 2

Here, 00 is given by equation B3, 7 is the usual relativistic parameter, and the

95

maximum scattering angle 0mm. is given by the experimental setup (Omar = 4.1° in

our case).

Appendix C

Nuclear Transition Matrix
Elements

In the following, I will briefly describe, how one can derive the reduced transition
strength B(EA) — where A denotes the multipolarity of the transition — from the
measured lifetime. Many intermediate steps in the derivation are left out, and for a

more detailed description of this subject, the interested reader should refer to, e.g.

the textbook of Wong [Won90].

The measured lifetime T, the half-life T1/2 and the transition probability P are

related via :

_ T1/2 _ 1
_ ln2 — P (0.1)

For weak interactions, one can use first oder time-dependent perturbation theory

to derive Fermi’s golden rule, which relates the transition probability to the square

of the nuclear matrix element M f,- :

27r
P = fl < JIMICIH'IJiMiC > FINE!)
21r
= fIMMZMEI) (C3)

96

97

Here p(Ef) is the final state level density per energy interval at the energy E f,
and |J,-M,-€ > and IJfoC > are the wave functions of the initial and final states. H’
represents the perturbation due to the coupling between the nuclear and electromag-
netic fields, and (,5 denote quantum numbers other than the angular momentum,
that are necessary to describe the nuclear states. One can expand the perturbing
Hamiltonian H’ in multipoles Am, as shown e.g. in [Won90], and write the electric

part of H' as :

1(21 +1)11

014E” = _ck*+1()\ +1)

 

J(1")V X (1‘ X V)(jA(kT)YAu(61 (15)) (03)
where the nuclear current density is given by J (r), and j 1(kr) are the spherical
Bessel functions and YM(0, 43) are the spherical harmonics.

The reduced nuclear matrix element is related to the reduced transition strength

B(A) — for a given multipolarity A — via the following relation :

 

3(4) = Z | < JfoCloAulJiMi€ > I2 (C-4)
11M]
__ 2
— N + 11' < JICIIOAHJE >| (05)

After including the level density of the final states, one can now express the
transition rate P(EA) —given by equation C.2 — in relation to the reduced transition

strength B(EA) for electric transitions via the relation :

87r(A+ 1) 1 1

PU“) = “h°1[(2x+1)1!]211(hc)

“HE““B(E/\) (C.6)

 

Here, E, is given in MeV, and B(EA) in units of ezfm“. For the E1 transition

we are interested in, one can combine the constants in equation C6 and end up with

98

the following formula :

P(E1) = 1.59 - 1015E33(E1) (C.7)

Again, E, is given in MeV, and B(E1) in units of (22me. Using equation C7 in
combination with equation Cl, and the measured lifetimes as given in Ref. [Mil83],

we deduce the B(E1) values as shown in Fig. 3.21.

Appendix D

Transition Amplitude for the
Nuclear Excitation in 11Be

To calculate the nuclear excitation of 11Be from its {L ground state to the if first
excited state, we will follow closely the model developed by PG. Hansen to describe
the momentum content of a single-nucleon halo state [Han96, Han97]. Due to the
relative high energy of the secondary beam, one can use the classical impact parameter
(b) and straight line trajectories to describe the colliding system. Only reactions with
a minimum impact parameter (bmgn) larger than the sum of the projectile core radius
(RC) and the target radius (RT) can contribute to the reaction channel of interest.

Reactions with a smaller impact parameter will lead to core fragmentation. The

minimum impact parameter can be calculated as :

bmin = RC '1" RT
: To ° (1410/3 + A;/3)

= (1.7718 — 0.13036 - In[Eb.am]) - (Ag/3 + 41,”) (0.1)

Where AQT are the core and target masses, and we used the energy dependent core

and target radii according to Ref. [Tan95].

99

100

In the projectile rest frame, one can imagine the target nucleus passing through
the halo system without touching the core, but leaving a wound in the undisturbed
ground state wave function (\Ilo). The size of the wound is determined by the range of
the interaction, which one can describe by an effective target radius (Ra). Figure D.1
shows a sketch of the coordinate system and the relevant parameters used. Therefore,
the wave function \Ilo remains unchanged, except in a cylinder of radius Ra, where it is
set to zero. We call the wave function originally contained in this reaction zone 6%,.
This wave function vanishes outside the interaction radius Ra, and it can be chosen to
reproduce the experimental reaction and elastic cross section for free nucleons. One
can express the collision complex (\Il(r)) as the sum of the undisturbed ground state

wave function and the wound as :

‘14?) = ‘110 — 6W0 (D2)

To estimate the nuclear cross section, one has to calculate the overlap of this wave
function with the I = 1 wave function of the first excited state in 11Be, which we will

denote as \Ill. The transition amplitude (001) is therefore given by :

(101 = /W(I‘)*‘I’1d7’

= _ / 611131111311 (0.3)

Where we used the orthogonality of \110 and \III in the last step, and ignore the spin

for now.

It is a good approximation for a narrow reaction zone to replace the wave functions
in the above integrand by their value ‘Ilo,l(b,0,z) along the target trajectory. The

integral over the x-y dimensions can then be done easily and leads to a factor of «RE,

101

 

Figure D.1: The coordinate center originates in the 1"Be core with the radius RC,
and the relative coordinate of the halo neutron is given byj’. The impact parameter
of the target nucleus, moving along the z axis is given by b.

102

which is half of the (free) nucleon reaction cross section. Therefore, we can write the

transition amplitude as :

a0, = —1rRZ/\IIS(b,0,z)\IIl(b,0,z)dz (0.4)

This integral can be integrated numerically using Woods-Saxon wave functions. How-
ever, as the minimum impact parameter is larger than the sum of the core and target
radius, the Woods-Saxon wave functions are given by the spherical Hankel functions
(12;), which represent the asymptotic behaviour of the wave functions. Therefore, we

can write :

9.0-) = 81422105014064) (D5)

Here, the B; represent dimensionless constants of the order unity, used to adjust
the outer and inner solutions to the Schr5dinger equation. The equation is written
in terms of the variable 10;, which depends on the reduced mass (11) and the known

neutron separation energies (Snz) (504 keV, and 188 keV respectively), via the relation:

1 r——
K] = g ZpSnl (13.6)

The lowest order Hankel functions are given by :

h0(if€07‘) = ——€-K°r (D.7)

h1(inlr) (D.8)

II
s
A
i |
H 1—1
‘3
+
Eel
.3 1—‘
M
v
to
l
3‘
g

103

One can now express the transition amplitude as :

am = —7rRzBoBIK3/2K. 3/2/00 h;((iKOT)Y()oh1(ZI€1T)I/1m(0, ¢)dz (D9)

The spherical harmonics Yzm are given by :

Y... = .4
YIO = EcosO = “£7522?! (D.10)
Y1,“ - :2 f—smeo e' "‘d’ = 4&7593

Where the phase factor e‘m‘f’ for the Y141 is equal to one as in the chosen coordinate

system (b = 0. Substituting this into equation D.9 gives :

 

l 1
(101 = 77122 aBOBl( ()ICQI'CI )3/2 f/OO_ e-Kor (_ + 2 2) e-mr

7—:—7db f l 01
l w z d forzm: 041} (0.11)
$71-W Z 01‘ m-

As r = v02 + 22, the first integrand — the solution to the Yio part — vanishes as it is

odd. Therefore, we end up with :

' oo -(no+rc1)r 1
001 = $27rRZBoBlb(nonl)3/2 £7:— E——- (1+ —) dz (D.12)

non1r3 nlr

We can simplify this formula even further, by identifying 27rRa with the total neutron

cross section 010,, and substitute 2 = bv in the above integral. This leads to the

104

following result :

 

a — fiiBoBlatom/Kofll °° e‘“ 1+”1/000_—e“a 1+02
01 -— :F 2 4770 0 (1+v2—_—)_(h)3/2+n13()1+v22

éiBOBlatoM/KOKI [ (g, a) ]
3F\/; 41rb D + nlbD(2’ ,a) (D.13)

 

Where we used the following substitutions :

a 2 (I60 + 161) ' b (D.14)
-00 -am

And the functions D(p, a) need to be evaluated numerically.

The transition amplitude am is related to the transition probability P.) via the

relation :

2
Pif = 5100112 (D16)

where the factor g comes from taking appropriate care of the spins involved. From

the transition probability one can calculate the cross section via :

0;} = £00 P.f(b)27rbdb

 

BgBletztflofll /bma3 ( 3 1 )2 db
= O —— — D.1
84 1...... D(2’° °)+ 141500 °) 0 l 7)

Where the integral is over the impact parameter b, and it can be evaluated numeri-
cally. The numerical calculation gives the nuclear contribution of exciting the %— first

excited state in 11Be from the %+ ground state, as discussed in section 3.3.2.

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