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THESIS
Z
Zeal

This is to certify that the

dissertation entitled

Single-Neutron Knockout Reactions:
Application to the Spectroscopy of 15’16’17’19C

presented by

Valentina Maddalena

has been accepted towards fulfillment
of the requirements for

Ph.D Physics

degree in

 

 

7 Major professor

P. Gregers Hansen

 

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

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I University

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PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

1 DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 c/CIRC/DateDuepss-sz

SINGLE-NEUTRON KNOCKOUT REACTIONS:
APPLICATION TO THE SPECTROSCOPY OF l5’16'17’19C

By

Valentina Maddalena

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy
and
National Superconducting Cyclotron Laboratory

2000

ml

14‘

a
Y).
41..
I

ABSTRACT

SINGLE-NEUTRON KNOCKOUT REACTIONS:
APPLICATION TO THE SPECTROSCOPY OF 15’16"7*19C

By

Valentina l\*Iaddalena

The technique of one-nucleon removal reactions has been recently developed at
the NSCL to study the single-particle stucture of nuclei far from stability, close to

both the proton and the neutron drip lines.

Among those, the neutron-rich carbon isotopes 16'17’19C have been investigated
to understand their essentially unknown structure. In particular, 17”C are both
characterized by a low neutron separation energy, one of the typical signatures of the
halo character. The technique consists in the measurement of partial cross sections,
and their associated momentum distributions, corresponding to the final states of
the 15"6‘1sz residues. These are compared with predictions based on shell-model

calculations and an eikonal model of the reaction mechanism.

As a test case, the well—known nucleus 15C has been investigated. The abso-
lute validity of the spectroscopic factors extracted by our method will be discussed.
The measured momentum distributions of the residues have revealed a characteristic
broadening and low-energy tail, observed already for the 6’ = 0 halo of 11Be. More-
over, an interdependence between scattering angle and momentum has been observed

and interpreted theoretically as arising from the diffractive channel.

Spectroscopic factors and F-value assignments for all the isotopes have been ex-

. . + + _ .
tracted. The ground-state spins of 3 and % have been assigned to 17C and 19C,
respectively. An estimate for the one-neutron separation energy for the ground state

of 19C has been obtained, suggesting that it be revised upwards.

To my family, my friends, and my love Bruno!

iv

Vt ‘

ACKNOWLEDGMENTS

Many people have contributed in making my four years at Michigan State Uni-

versity unforgettable.

First of all, I am deeply grateful to my advisor Gregers Hansen, my continuous
source of wisdom. He has led me through the intricacies of physics with much patience

and always with a smile.

I wish to thank Thomas Glasmacher, S. D. Mahanti, C. P. Yuan, and Vladimir

Zelevinski, for serving in my committee and for nice and fruitful discussions.

I am also indebted to Alex Brown and Jeff Tostevin, my theory guides. Thank you
for providing me with the calculations, and even with the opportunity (and thrill) of

running some programs myself!

Thank you Navin, for being such a good friend and always there to guide me in

the data analysis and interpretation. It was great working with you!

My gratitude goes also to the 8800 collaboration, the operators and all who helped
in the experiments, making them successful, and especially Brad Sherrill, Daniel

Bazin, and Mathias Steiner.

Thank you to Joachim, with whom I could share the ups and downs of the analysis,

and for his helpful hints for my thesis.

Thank you to the ones who helped me to come. here, without whom I could have

‘7

never gained such a wonderful experience. They are Bill Lynch and Betty Tsang from
the N SCL, and Massimo Di Tom and Salvatore Sambataro from Catania, Italy. Their
help and support was very important during my transition from Italy to Michigan

State University.

Thank you also to Rich and Tom, for their prompt help anytime I asked them,
and for all the happy moments we shared together. For this I thank also Sharon,
Anuradha, Heather and Greg, Stephane, Anca, Takashi, Thomas, Andreas, Marielle,
Wilhelm, and all the graduate students! Thank you to Giuseppe, who has always
been someone I can count on, since I started working with him in Catania. Thank
you to my friends officemates: Joann, Katie and Barry, always there to listen, help
or just have fun together! Thank you to all my Italian friends who alleviated the
nostalgy of being far from home. Thank you to my foreign friends who enriched my

life in a unique way.

Thank you to the Computer Department and all the NSCL staff, always friendly
and so efficient! Thank you also to the Physics Department secretaries, for always

being so helpful to me.

I am very grateful to Joe and Ann who took me in their family as their own child.

They gave me a great support in the last months of my PhD.

Thank you to Bruno, for being my greatest supporter, for his endless help, and for
loving me! Finally, my gratitude and love to my parents, my sister, my grandmothers,
and all my relatives, for the support they gave me while far away from home, for

coming to visit me, and for patiently waiting for my return home.

Thank you, thank you, thank you to all of you!

vi

 

 

 

 

 

Ll

LI

3

Contents

LIST OF TABLES

LIST OF FIGURES

1

2

Introduction
1.1 Motivations for the study of nuclear structure .............
1.2 Properties of Halo Nuclei .........................

1.3 Experimental approach ..........................

Experimental technique and setup

2.1 Single-nucleon knockout reactions ....................

2.2 Experimental setup ............................

2.3 The 8800 spectrograph ..........................
2.3.1 Cathode Readout Drift Counters ................
2.3.2 Ionization Chamber ........................
2.3.3 Scintillation Detectors ......................
2.3.4 Particle Identification .......................

2.4 The NaI array ...............................
2.4.1 Simulations ............................
2.4.2 Analysis of the measured y-ray spectra .............

2.5 The experiments .............................

Theoretical Analysis
3.1 Theory of the spectroscopic factors of 15’16'17’19C ............

3.2 Theory of the single-particle cross sections ...............

The test case: 15C
4.1 Previous theoretical and experimental work ..............

4.2 Present shell-model results ........................

vii

ix

xi

Niki—aid

9

9
11
12
13
15
16
16
20
23
25
29

33
35
37

42

43

4.3 Experimental results ........................... 45

The structure of the heavier carbon isotopes 61
5.1 Spectroscopy of 16C ............................ 61
5.1.1 Previous theoretical and experimental work .......... 61
5.1.2 Present shell-model results .................... 62
5.1.3 Experimental results and discussion ............... 63
5.2 The projectile 17C ............................. 68
5.2.1 Previous theoretical and experimental work .......... 68
5.2.2 Present shell-model results .................... 68
5.2.3 Experimental results ....................... 69
5.3 The projectile 19C ............................. 78
5.3.1 Previous experimental and theoretical work .......... 78
5.3.2 Present shell-model results .................... 80
5.3.3 Experimental results ....................... 81
Summary 90
Data Analysis 94
A1 Summary of the procedure ........................ 95
All The analysis in the case of 17C .................. 97
A.1.2 The analysis in the case of 15C .................. 99
A13 The analysis in the case of 16C .................. 102
A14 The analysis in the case of 19C .................. 103

viii

List of Tables

2.1

4.1

4.2

CI]
p—A

5.2

5.3

Experimental parameters and settings. .................

Experimental and calculated energy levels of 14C and associated spec-
troscopic factors for the different components of the 15C ground state

wave function, calculated with WBP, WBT and WBP+. There is no
combination of the 14C 3‘ state with a p — sd shell neutron that can

1
2
calculated with WBP, except for the excited 0+ state and the last 2+
state, which appear around 13 and 15 MeV, with WBP and WBT,
respectively. For the other levels the three different calculations give
similar energies. The experimental energies are taken from [Aszl].

couple to the 15C round state spin +. The theoretical ener ies are
S g

Comparison between the experimental cross sections and the theoreti-
cal predictions with spectroscopic factors calculated with different in-
teractions (WBP and WBP+, denoted with the subscript +) for the
reaction 9Be(15C,14C)X at E = 54 MeV/nucleon. The level energies
are taken from previous work [Aszl]. .................

Partial cross sections a and branching ratios b for the final states I 7’
in the residues produced in 9Be(16C,15C)X at E = 62 MeV/nucleon.
The theoretical cross sections 0th are calculated from Eq. (3.1) using
the WBP spectroscopic factors C23 and the single-particle cross sec-
tions 03p. The values of an, include overlap factors of 0.897 and 0.948,
respectively (see section 3.2). ......................

Partial cross sections a and branching ratios 1) for the final states I1r
in the residues produced in 9Be(17C,16C)X at E = 62 MeV/nucleon.
The theoretical cross sections at), are calculated from Eq. (3.1) us-
ing the WBP spectroscopic factors 025 and the single-particle cross
sections 03p. " The components of this group have been analyzed to-
gether. We identify them with three states predicted by theory in the
range 4.9-5.7 MeV. The main contributions were given by at least two
components. ...............................

The reaction 9Be(17C,16C)X at E = 62 l\Ie\"/nucleon, for a 17C ground-
state spin J7r = 5/2+ or 1/2+, see Table 5.2. " The components of this
group have been analyzed together. We identify them with three states
predicted by theory in the range 4.9-5.7 MeV. The main contributions
were given by at least two components. ................

ix

44

65

72

5.4 Partial cross sections 0 and branching ratios b for the final states I 7’ in
the residues produced in 9Be(19C,18C)X at E = 57 MeV/nucleon. The
theoretical cross sections a”, are calculated from Eq. (3.1) using the
WBP spectroscopic factors 025 and the single-particle cross sections
a,,,. The neutron separation energy of 19C was assumed to be 0.8 MeV.
"‘ All excited states in 18C were analyzed as one group. The energy
1.6 MeV for the 2+ is the experimental value; the WBP calculation
gives 2.16 MeV. .............................

List of Figures

1.1

2.1

2.2

2.3

2.4

2.6

The chart of nuclides. The valley of stability is indicated by the black
dots representing the stable nuclei in nature. The limits of nuclear
stability are indicated by the proton and neutron drip lines, behind
which no bound nuclei can exist. The double lines indicate the magic
numbers for the stable nuclei. ......................

Experimental setup. The dispersion-matched beam line delivers a sec-
ondary beam of radioactive ions on the target arrangement shown
schematically in the inset. The target is surrounded by an array of
38 N aI(T l) detectors, 20 cm long and 5 cm in diameter, which detect 7
rays in coincidence with projectile-like fragments measured in the 8800
spectrograph. ...............................

Energy loss (AE) vs. Time of Flight (TOF) spectra for two different
field settings. The top plot was obtained with the spectrosgraph ad-
justed to the momentum of the 16C beam particles, identified by the
gate drawn. The contaminants 14B and 18N (left and right, respec—
tively) were also present. The bottom plot was obtained at a reduced
magnetic field to identify the 15C reaction residues from 9Be(16C,15C)X,
selected by the gate drawn. (See text for details on the identification
of the other fragments which appear in the plot.) ...........

Time of flight vs. 3:, position in the first (top) and in the seCond (bot-
tom) CRDC for 9Be(16C,15C)X. The scattered beam particles included
in the gate drawn in the bottom plot of Fig. 2.2 can be well identified
and separated from the 15C residues (selected with the gates drawn).

'y-ray energy Elam", vs. 003(0) before (Ezab, top) and after (Eam, bot-
tom) Doppler correction of the ”y rays from the deexcitation of the 16C
residues produced in one-neutron removal reactions on 17C. .....

Measured and simulated 'y-ray energy spectra for the 88Y source, nor-
malized to the number of 'y rays emitted during the electronics live
time. The observed deviations between data and simulation are at-
tributed to the natural background. ..................

Experimental laboratory (top) and em. (bottom) y-ray spectra from
the residues 11Be and 15C for E7 > 1 MeV. ..............

xi

12

18

19

22

24

2.7

2.8

2.9

4.1

4.2

4.3

4.4

4.5

4.6

4.7

Experimental c.m. (full dots) and laboratory system (open dots) 7-
ray spectra representing the continuum distribution lying beneath the
measured 7-ray spectra for E7 > 1 MeV. The two spectra were obtained
as an average of the 12Be and 16C data. ................

Experimental laboratory system angular distribution of the 7 rays with
E, > 1.2 MeV measured from one-neutron removal reactions on 16C.
The distribution is focused at forward angles, demonstrating the ori-
gin of the continuum distribution from interactions induced by the
neutrons removed from the beam. ...................

Experimental laboratory system angular distribution of the 7 rays with
1.6 < E, < 2.0 MeV, corresponding to the decay of the 16C fragments,
produced in one-neutron removal reactions on 17C, from the 2+ state
at 1.77 MeV to the ground state. The solid line is a simulation of the
angular distibution of Lorentz-boosted 7 rays of 1.77 MeV. .....

Doppler corrected 7-ray spectrum from 9Be(15C,14C)X. The curves are
the simulated response functions for the observed 7 rays and the expo-
nential measured continuum distribution extracted from the 12Be and
16C data (Fig. 2.7). Inset: Simplified level scheme of 14C. .......

Longitudinal momentum distributions of ”Cg, (top), and of l“Cl—,0-

(bottom), after one-neutron removal from 15C. The curves are calcu-
lated eikonal momentum distributions for I? = 0, 1. ..........

Laboratory and c.m. system 7-ray spectra measured in coincidence
with 14C residues produced in one-neutron removal reactions on 15C.
The 0.98 MeV peak in the laboratory spectrum corresponds to a 7 ray
emitted by 8Li, produced in reactions in the target, and is smeard out
by the DOppler correction in the c.m. spectrum. ...........

Laboratory 7-ray spectra from one-neutron removal reactions on 12Be
and 16C. The 0.98 MeV peak coresponds to a 7 ray emitted by 8Li,
produced in reactions in the target. ..................

Experimental cross sections and calculated diffraction cross sections
with CDCC, for 15C (top) and 11Be (bottom). .............

Longitudinal momentum distributions of the ”C93,, at different scat-
tering angles. Top: The curves are calculated with CDCC, and scaled
to reproduce the measured cross sections. Bottom: The curves are the
sum of the calculated CDCC distributions for the diffractive part and
the eikonal distributions for the stripping part. ............

Longitudinal momentum distributions of the ”Beg,“ at different scat-
tering angles. Top: The curves are calculated with CDCC, and scaled
to reproduce the measured cross sections. Bottom: The curves are the
sum of the calculated CDCC distributions for the diffractive part and
the eikonal distributions for the stripping part. ............

xii

46

49

51

CI!
CI!

4.8

4.9

4.10

5.1

5.2

5.4

Longitudinal momentum distribution of the ”Cg“, residues from one-
neutron removal reactions on 15C. The curves are CDCC (solid line)
and eikonal (dashed line) calculations. The CDCC curve is scaled to
reproduce the measured cross section, the eikonal curve is arbitrarily
scaled. ' ..................................

Longitudinal momentum distribution of the ”Beg, residues from one-
neutron removal reactions on 11Be. The curves are CDCC (solid line)
and eikonal (dashed line) calculations. The CDCC curve is scaled to
reproduce the measured cross section, the eikonal curve is arbitrarily
scaled. ..................................

Projectile center of mass longitudinal momentum distributions of the
14C and 10Be residues in their ground state after one-neutron removal
from 15C and 10Be. The 10Be distribution has been broadened by the

square root of the ratio of the separation energies. ..........

Laboratory system 7-ray spectra from 9Be(16C, 15C+7)X (filled circles)
and 9Be(”Be, 11Be+7)X (open circles), normalized to the number of
projectile fragments. The solid lines are simulated response functions
for the 0.74 MeV 7 ray from the decay in flight of 15C“ and for the
0.32 MeV 7 ray from 11Be". The inset shows experimental 7-ray spectra
from 11Be and 15C for E, > 1 MeV. The “tails” above the 7 lines, the
only discrete lines present, are the continuum distributions discussed
in the text. ................................

Inclusive longitudinal momentum distribution of the 15C fragments
produced in one-neutron removal reactions of 16C on a 9Be target. The
experimental errors are smaller than the size of the points. The full-
drawn line is a fit with the short-dashed lines showng the individual
theoretical components. The thin lines indicate the error limits allowed
by the fit. The broad component corresponds to 58 :l: 6% d—wave and
the narrow one to 42 IF 6% s—wave, with the contributions from the
(unobserved) tails of the d—wave distribution taken into account.

Doppler-corrected 7-ray spectrum measured in 9Be(17C, 16C+7)X. The
black curve is a fit to the spectrum using an exponential curve for
the background and response functions (grey curves) for each of the
7-ray transitions shown in the simplified level scheme of Fig. 5.5. The
dashed line corresponds to an estimated upper limit of 2% for the direct
transition from the J = 2 level at 3.99 MeV to the ground state. Inset:
7 spectrum gated on the transitions between the levels at 2 4 MeV
and the 2+ level at 1.77 MeV. The spectrum was fitted using the same
procedure as described above. ......................

Comparison between the 7-ray spectrum from the excited 16C residues
after one-neutron removal reactions on 17C, and the measured contin-
uum distribution extracted from the analysis of the ”Be and 16C data.
The solid line is the result from the fit of the spectrum. .......

xiii

58

60

64

66

70

5.5

5.6

5.7

5.8

Longitudinal momentum distributions corresponding to the states in-
dicated in the simplified level scheme of 16C. (a), (b): The solid curves
are the calculated momentum distributions with a mixture of s-» and
d—waves shown as dashed and dot-dashed lines, respectively (8% s and
92% d in (a), 26% s and 74% d in (b)). (c): The solid curve is the
calculated momentum distribution of a pure d—wave. .........

Longitudinal momentum distribution corresponding to the ground state
of the 18C residues after one-neutron removal from 19C on a 9Be target.
The coincidences with 7 rays have been used to correct the inclusive dis—
tribution for contributions from excited levels. The momentum distri-
bution corresponding to the extracted separation energy 5,. = 0.8 MeV
is represented by the solid line. The dashed lines represent the mo-
mentum distributions corresponding to separation energies of 0.5 and
1.1 MeV. The dot-dashed curve is calculated for a d—state for a sepa-
ration energy of 0.8 MeV. ........................

Inclusive longitudinal momentum distribution corresponding to 18C
residues after one-neutron removal from 19C on a 9Be target. The
solid lines represent the calculated inclusive momentum distributions
corresponding to 8,, = 0.5 MeV and 8,, = 0.8 MeV obtained as a
least-squares fit assuming the branching ratios given by the theoretical
spectroscopic factors of Table 5.4. These values represent approxi-
mately the 3:10 limits of the allowed interval and correspond to a x2
of 8 and 9, respectively for 14 degrees of freedom. (For 5,. = 0.3 MeV
and 8,, = 1.1 MeV X2 increases to 18 and 14, respectively.) The dashed
lines labeled with s and d represent the contributions from the Z = 0,2
excited states for 5,. = 0.8 MeV. Their contributions for 5,, = 0.5 MeV
would be almost identical. ........................

Permitted regions in the space of spectroscopic factor and one-neutron
separation energy for the ground state of 19C. The hatched areas result
from different information: measured nuclear and Coulomb breakup
cross-sections (cam, 00%;), and analysis of the ground state momen-
tum distribution (do/(1PM). Also shown are the result from the Coulomb
dissociation experiment in [Nak99], deduced from da/dQ, and the sep-
aration energy value in [Aud93, Aud97]. A consistent description of
the experimental results is given for values of 025' between 0.5 and 1
and of 3,, between 0.5 and 1 MeV. Note that this graph does not dis-
play the two analyses of inclusive momentum distributions discussed
in the text. ................................

xiv

74

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Chapter 1

Introduction

1.1 Motivations for the study of nuclear structure

Radioactivity was found by Henri Antoine Becquerel in 1896 and the subsequent
discovery of readioactive elements by Pierre and Marie Curie earned them the 1903
Nobel Prize for Physics. With Rutherford’s discovery of the atomic nucleus the basis
had been laid for all subsequent research on nuclear physics and for a development
that still continues, one hundred years later. Although the present work is concerned
only with basic aspects of nuclear structure, it should be pointed out that there are
many practical applications of nuclear physics in various fields, such as medicine,

biology, archeology, national security and energy generation.

The prOperties of matter are determined by the number of protons and neutrons in
a nucleus. In particular, the number Z of protons characterizes the different elements,
from hydrogen to uranium; depending on the number N of neutrons, each element
can be present in nature in a variety of isotopes. The stable elements that exist in
nature are 80 and there are almost 300 stable isotopes. When displayed in the chart of
the nuclides (Fig. 1.1), where the number of protons Z is plotted against the number
of neutrons N, the stable nuclei lie approximately along the diagonal from the lower

left to the upper right, called the valley of stability.

 

 

 

 

 

 

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Figure 1.1: The chart of nuclides. The valley of stability is indicated by the black dots
representing the stable nuclei in nature. The limits of nuclear stability are indicated
by the proton and neutron drip lines, behind which no bound nuclei can exist. The
double lines indicate the magic numbers for the stable nuclei.

 

 

 

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However, thousands of radioactive isotopes are stable in time scales that are long
compared with the time of nuclear intrinsic motion (10“?2 5). Their properties offer
important clues to nuclear structure, and to astrophysics as well. In fact, they are
continually created in the cosmos and play a crucial role in the evolution of the stars
and in the origin of the elements in our solar system 4.5 billion years ago. Most of
them cannot be found naturally on earth because they decay by different mechanisms
into stable species with half-lives that range between less than a thousandth of a
second to billions of years. These unstable isotopes lie away from the line of stability,
in the region within the neutron and proton drip lines, which define the limits beyond
which nuclei become unstable to nucleon decay. Due to their distance from the valley
of stability, nuclei close to the drip lines are often referred to as “exotic” nuclei,

indicating entities different from the most ordinary ones, available in nature.

The drip lines have experimentally been reached only for the lightest 8 elements
and their position in the nuclear chart is not exactly known yet. Therefore, in order
to test existing nuclear structure models, one of the most important topics of research
in nuclear physics is the exploration of the nuclear chart to find out where the limits
of nuclear binding are. One of the most remarkable results of these studies was the
discovery of novel nuclear structures in nuclei far from stability in the last decades
of the 20‘” century. Being characterized by very different properties than ordinary
nuclei, these exotic nuclei have a thin cloud of nucleons orbiting at large distances
from the others, forming the core. In analogy with the luminous ring around the sun
or the moon seen under certain meteorological conditions, these nuclei are referred to

as “halo nuclei”.

Besides the search for the exact position of the drip lines in the nuclear chart,
other motivations to investigate nuclei far from stability are the quests for which
combination of protons and neutrons can make up a nucleus; what are the properties

3

 

 

 

of

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of nuclei with an extreme proton-to-neutron ratio; or how existing models should be

modified in order to describe rare isotopes.

The study of nuclei far from stability is not intrinsic to itself, but it has important
implications in the understanding of many astrophysics phenomena. An example is
the solar neutrino problem, related to the 7Be(p,7)8B reaction taking place in the
sun, or the CNO—cycle in the nucleosynthesis process and all nuclear reactions taking

place in the stars, under extreme nuclear matter conditions.

Thus, the investigation of the properties of exotic nuclei has become one of the
most important goals in nuclear physics and it is now possible thanks to the develop-
ment of modern technologies available for radioactive beam production and heavy-ion

accelerators.

In an effort of fine-mapping the nuclear chart, the essentially unknown structure
of the heavy carbon isotopes 16’17’19C has been investigated in this work. The ex-
perimental approach used for this purpose is a new technique originally designed to
study halo nuclei, but that was found to be a powerful tool to probe the structure of
any exotic nucleus. The technique was tested using a beam of 15C, the structure of
which is well known. Before describing the technique in detail, a brief overview on

the properties of halo nuclei is given below.

1.2 PrOperties of Halo Nuclei

Nuclei at the drip lines are filled with nucleons up to the limit set by the combination
of the nuclear kinetic energy and the depth of the nuclear potential well, resulting
in a bound state close to the continuum. The binding energy for this state is very
small and, due to the short range of the nuclear force, threshold effects appear.

The last nucleons undergo the quantum-mechanical effect of tunneling, so that their

probability of being at large distances from the core is appreciable. This is what
was referred to as “halo” state for the first time in 1987 [Han87] and since then the
term halo has been referred to all exotic nuclei manifesting those properties. The
manifestation of the halo phenomenon is less evident if the halo nucleons are in a
state of large angular momentum (t’ > 1), in which case the centrifugal barrier lowers
the probability of tunneling far from the core. Analogously, the Coulomb barrier
hinders the formation of proton halos, because the repulsion between the protons and
the nuclear core makes it difficult for the nucleons to tunnel. The Coulomb repulsion
also affects the absolute binding, so that the proton drip line actually lies closer to

the valley of stability than the neutron drip line.

The most simple halo nucleus is the deuteron, although when it was first studied
it was not labeled as such. Its binding energy is only 2.22 MeV and the average
distance (root-mean square) between the neutron and the proton is 3.9 fm. Other
halo nuclei observed so far include neutron halos as well as proton halos. Neutron
halo nuclei exist in a variety of configurations, where the halo may be formed by one,
two or four neutrons, as is the case for 11Be (l-n halo), 6He, 11Li, 1“Be and 17B (2-n
halo), 8He (4-n halo). An established case of one-proton halo is 8B. The hypertriton
is an interesting example of halo nucleus, where the halo is not formed by a nucleon,
but by the A particle, orbiting around a deuteron core. The estimated separation
energy is as small as 0.08 :l: 0.02 MeV [Boh70, Boh68]. More recently halo-like states
have been discovered in deeply bound pionic states in heavy nuclei [Yam96]. At the
atomic scale examples of halo states may be negative ions or two-electron atoms,
although these are bound by the long range Coulomb potential. Systems subject to
a potential which decreases with distance faster than the Coulomb potential, such as
an electron bound in the field of a dipole, or noble gas molecules subject to the weak

van der W'aals force present analogies to the nuclear halo states as well.

 

 

 

I.
(fiends,
”M'IL’C
flu in;
”brill:
{in ‘li

,-

lilo“! a

Several are the experimental signatures of a halo nulceus. Tanihata et al. [Tan85,
Fuk91] measured surprisingly large interaction cross sections in a series of experiments
on the neutron-rich 6”He and 11Li. This result was interpreted as due to a long tail
in the wave functions of those nuclei. It was also found [Arn87] that the proton
distributions in 9Li and 11Li are the same, which led to the conclusion that the broad
spatial distribution in 11Li was to be attributed to the neutrons. l\*‘Ioreover, while the
interaction cross sections were found to increase with the mass number of the 8*9’11Li
isotopes, those were shown to be constant as a function of mass in reactions which

changed the proton number [Bla92].

Coulomb dissociation cross sections were also expected [Han87, Ber88] and later
observed [Kob89, Nak99] to be surprisingly large. As a direct consequence of the
Heisenberg principle, extended spatial distributions manifest themselves as narrow
momentum distributions of the fragments produced in breakup reactions of halo nu-
clei. Observations of this effect were found for a number of nuclei [Kob88, Ann90,

Orr92, Baz95, Ba298, Bau98, AumOO, Gui00, Nav00].

Is any of the carbon isotopes studied in this work a halo nucleus? Those are
certainly good candidates, especially 15C, 17C and 19C, with their low separation
energies (1.218, 0.74, 0.8i0.3 MeV, respectively, the last value from this work), but
more details on their wave functions need to be known. Much information has been
obtained indeed with the experimental approach used for this work and the results
are discussed in Chapters 4 and 5. We shall see that 19C is definitely a halo state,

from what we know as good a case as 11Be.

 

 

 

E‘llf
{Pill
111v

PfU‘.

1.3 Experimental approach

The most traditional method used to study ordinary nuclei consists in bombarding
the nucleus of interest in the form of a target with a stable beam. Obviously such a
method can not be used for the study of short-lived nuclei, as it would require the use
of a radioactive target. The experimental approach to study short-lived radioactive
isotopes is then reversed with respect to the traditional method, in that radioactive
beams of the nuclei of interest are produced and react on a stable target. In these
experiments the kinematics is referred to as “inverse”, as the measurements are done

with respect to the projectile’s frame, rather than the target’s.

Information on the various properties of exotic nuclei can be extracted from differ-
ent kinds of experiments, such as Coulomb excitation, transfer reactions and breakup
reactions. In a Coulomb excitation experiment the incident nucleus is excited by
the Coulomb field of a heavy target and the 7 rays measured from the de-excitation

provide information on the transition probabilities for the excited states.

Single-nucleon transfer reactions at low beam energies identify rather directly
single—particle configurations of the nuclear wave function [Boh75, Sat90, Fe592]. The
classic tools have been stripping and pickup reactions, such as the (d,p) and (p,d)
reactions, and the analogous reactions for probing proton states. For medium mass
and heavy targets these light ions have a short mean free path inside the nucleus.
The reactions are therefore surface dominated. They can also be described usefully
as one-step processes involving the transfer of a nucleon to or from a given single-
particle state. The development of theoretical methods such as the distorted waves
Born approximation (DWBA) has facilitated the use of transfer reactions to make
angular-momentum assignments from the shapes of angular distributions, and to

deduce spectroscopic factors from the magnitudes of measured cross sections.

 

 

and T.
(If Ili'
It‘t'lzil
lie-en
1"C.
Di
Chap?
inter};
llit‘illl!
of ”(f
ClielIiY
on ”C
Tilt“ ('i
'31: ii!!! r<

lNir’
~ . HIM

Breakup reactions have been extensively used to measure interaction cross sections
and the momentum distributions of the emerging fragments resulting from the removal
of the halo nucleon(s). With the additional information from coincident 7 rays, this
technique allows the identification of the various final states in the residues and it has

been recently applied to a variety of exotic nuclei, including the carbon isotopes 15C,

16C, 17C and 19C which are the subject of this work.

Details of the technique and of the experimental setup used can be found in
Chapter 2. The theories for the structure and the reaction mechanisms used to
interpret the experimental data are illustrated in Chapter 3. The validity of the
method used as a spectroscopic probe is demonstrated in Chapter 4 through the study
of 15C, which was intended to serve as a test case, being its structure known. Finally,
Chapter 5 presents detailed discussions of previous theoretical and experimental work
on 16C, 1'7C and 19C, as well as the results from the experimental data of this work.
The conclusive Chapter 6 offers some comments and a perspective on the potential
of knockout reactions for precise single-particle structure studies with beams of rare

isotOpes. Details on the data analysis procedure are given in Appendix A.

Chapter 2

Experimental technique and setup

2.1 Single-nucleon knockout reactions

A new technique suited for spectroscopic studies of rare nuclei produced with low
intensity as beams from fragmentation reactions has been recently developed at the
NSCL. The projectile residues formed by removing a single nucleon in the interac-
tion with a light target are observed in inverse kinematics. The final states of the
heavy residues are identified by their 7 decay [Nav98, Tos99b, Aum00, Gui00, Nav00,
MadOO]. The 7 rays tag reactions leading to individual discrete final levels and allow
a determination of differential and integrated partial cross sections. It is possible to
extend this technique to unbound final states by reconstructing the invariant mass (or
other parameters of the intermediate state) from observations of the breakup prod-
ucts, see the recent work of Chen et al. [Che00] dealing with proton knockout from

11Be leading to the unbound systems 10Li and 9He.

Analogously to the reactions with light particles, e.g., the classical (p,2p) knock-
out reactions [Sat90], where the angular correlation between the emerging protons
carries information about the momentum of the proton removed from the target,
in single-nucleon knockout reactions the momentum of the removed nucleon can be

reconstructed from the recoil of the observed heavy residue. The shape of the lon-

 

ngltii
II'IIIHV
Ni‘opli'
formal
511li 21
imidll'
llllll (1
ill The
of lilf‘

.. ,1
(‘lllglt‘

gitudinal momentum distribution identifies the orbital angular momentum f of the
removed nucleon, while the absolute removal cross section determines the spectro-
scopic factors. The transverse momentum components carry essentially the same in-
formation, but they are more sensitive to contributions from the reaction mechanism
such as Coulomb deflection and diffractive scattering. The momentum components
parallel to the beam direction are those that carry a clean signature of the momen-
tum content related to the single-particle state in question. The results are shown
in the laboratory system, as distributions of the quantity P”, which is the projection
of the measured total momentum onto the beam axis. Since the residue’s deflection
angle is small, typically a few degrees, the difference between the total momentum
and the parallel momentum is small. The laboratory distributions are broadened by

the relativistic 7-factor, which has to be included in the comparisons with theory.

The principal virtue of this technique for the spectroscopy of rare isotopes is its
high sensitivity, which is of paramount importance in experiments aimed at explor-
ing nuclei at the drip lines. This is illustrated in this work, where the results from
reactions with an incident 19C beam intensity of less than one particle per second
are presented. The special experimental strength of the technique lies in the high
energy of the beam particles and the detection of only the heavy residue. The high
energy allows the use of thick targets and gives a strong forward focusing and hence
a detection efficiency close to unity. It also allows the secondary beam and “tertiary”
fragments to be tracked particle by particle, so that there is essentially no back-
ground. There are also important theoretical advantages. The high beam energies
allow the use of reaction models, based on the sudden and the eikonal approximations
[Tos99b, Hen96], which have high predictive power. These methods can be used to
relate the measured single-nucleon removal cross sections to theoretical spectroscopic

information using a fixed set of theoretical input parameters, as discussed in detail

10

 

 

in C

1!] c
(if 3!

TM

the," l

lifif‘y I
6H \r
liltin-

aft! 4‘; f

51
f1"4:111
"limp
If; F}:

rifltly ‘
" Hit!“

f:

in Chapter 3.

The first applications of the knockout reaction method were aimed at clarify-
ing specific features of exotic nuclei that are otherwise well understood, such as
the €~assignments and spectroscopic factors for the presumed proton halo states
of 26’27’28P [Nav98], and the spectroscopic factors linking known states in 10’11:”Be
[Tos99b, AumOO, Nav00], and in 13“B [Gui00]. The method was then extended to
the neutron-rich carbon isotopes 16’17’19C about which much less is known. However,
they have been the subject of a number of recent theoretical and experimental studies
[Ba295, Baz98, Mar96, Rid97, Rid98, Bau98, Ban98, Oza98, Nak99, Sme99, T0899a,
Chu00, Cha00, DesOO, Sau00, KanOO]. The validity and accuracy of the method was
also tested on the well-known nucleus 15C. This work will focus on the results ob-
tained on the carbon isotopes. In particular (in agreement with [Nak99]), it will be

shown that the 19C ground state is similar to 11Be and has a well developed halo.

2.2 Experimental setup

The experiments were performed at the National Superconducting Cyclotron Labora-
tory (NSCL) at Michigan State University, on 15’16‘17’19C, at energies of approximately
60 MeV/ nucleon. The radioactive beams were produced by fragmentation of a pri-
mary beam on a thick 9Be target. The primary and secondary beam characteristics
are given in Table 2.1, page 32. These secondary beams were purified in the A1200
fragment separator [She92] by the combination of magnetic analysis and an interme-
diate degrader. The resulting beam was delivered to the experimental setup shown
in Fig. 2.1, consisting of three parts: a dispersion-matching beam line, a target sur-
rounded by an array of position-sensitive NaI(Tl) 7 detectors [Sc1199], and the 8800

spectrograph [Cag99], used for detecting the projectile residues from the reaction.

11

If.

m.\ .,

 

5+“

NaI(Tl) array and target

 

 

 

 
 
     
   

 

 

 

 

Beam —>

_, ., _~ ,
Extras SQ
\/,_

 

 

 

 

 

 

   
 

       

       
   

\\ a”?
a , \\\ \‘\ J?!-
”SI—mm witt/
Iii] Dispersion matched beamline 8800 spectrograph

Figure 2.1: Experimental setup. The dispersion-matched beam line delivers a sec-
ondary beam of radioactive ions on the target arrangement shown schematically in the
inset. The target is surrounded by an array of 38 NaI(Tl) detectors, 20 cm long and
5 cm in diameter, which detect 7 rays in coincidence with projectile-like fragments
measured in the S800 spectrograph.

2.3 The 8800 spectrograph

Spectrographs are very versatile and their performance can be customized as needed
for different experiments. For the experiments discussed in this work, for instance, the
technique of single-nucleon knockout reactions described above required the detection
of coincident 7 rays. For this purpose a position-sensitive NaI(Tl) array was placed

around the target.

The analysis beamline of the 8800 [Cag99] is very similar to the A1200 fragment

separator [She92], but with better resolution and larger solid angle acceptance.

Since the A1200 separator has been designed to accept a large momentum bite
(up to 3%), a dispersion-matched system was used in order to achieve high resolution.
With this technique the spread in incident momentum is compensated by dispersing
the secondary beam on the reaction target and using the spectrograph to cancel its
dispersion at the focal plane. The 8800 spectrograph has been designed to operate

in this way. Due to the large dispersion of the S800, the beam must. be limited to a

12

 

sprf‘

prtjrt

Solit'l
dilil
ment
i‘C‘Rl
HMWH
Sprite
Hlf‘tth“

part it,

a!” Fat
I0 ”121‘
dim,”
FIFO-ff 7':
("Jilin—n1
kv\njyh

lvlf filth (x'

spread in relative momentum of 0.5%. In this case, it is possible to study reaction

products at a relative momentum resolution of 0.025%.

The spectrograph is characterized by a large angular acceptance, (up to 20 msr
solid angle: at 5° in the horizontal, i 35° in the vertical, dispersive direction),
and by a momentum acceptance of i: 2.5%. The position and angles of the frag-
ments are determined by two :r/y position-sensitive cathode-readout drift chambers
(CRDC) [Yur99] separated by 1 meter at the focal plane of the spectrograph. Im-
mediately following the CRDC’s a segmented ionization chamber 41 cm deep and a
series of plastic scintillators of different thicknesses (5, 10 and 20 cm respectively)
measure the energy loss, total energy and time of flight of the residues, used for

particle identification purposes.

2.3.1 Cathode Readout Drift Counters

The two CRDC’s measured the position coordinates of the fragments at the exit of
the spectrograph, providing angle information both in the a: (dispersive) and in the
y direction. The detectors are filled with 80% CR, and 20% C4H10 at 140 Torr.
The advantages of this gas are low aging deterioration, high drift velocity and low
avalanche Spread. The detectors have an active area of 30x 59 cm2 and an active depth
of 1.5 cm. Along the 59 cm long side (corresponding to the dispersive direction) there
are cathode pads separated by 2.54 mm. The principle of operation is analogous
to that of a single-wire drift detector. When a charged particle goes through the
detector it induces ionization in the gas. A constant vertical electric field drifts the
electrons toward an anode wire, where amplification takes place and the charge is
collected. The cathode pads placed in front of and behind the anode wire collect the
positive charge induced by the anode. The position :r is then taken as the centroid

of the Gaussian fit of the charge distribution. The position along the y direction is

13

 

(ii)?
iii?

(if I

L'l'52

. “
“"l.‘

I
lellT
and :

l'illi iv;

Trajr

obtained from the drift time of the electrons to the anode wire. Typical drift times
are of the order of 0-20 us, while the response time from the scintillators is 3—4 orders
of magnitude smaller. The drift. time can then be measured as the time between
the scintillator signal and the anode wire signal, from which the y position can be
deduced. The resolution in both the .73. and y position was about 0.3 mm, tested with
a beam of 22N e at 80 MeV/u. Since the S800 momentum dispersion is 9.6 cm/% and
the S800 has been designed to have a momentum resolution of 1 part in 20,000, a
position resolution of better than 0.48 mm is necessary and the two CRDC’S are thus

well suited.

The calibration of the detectors is done by inserting a mask with holes and slits in
front of the detectors, exposed to the beam particles. The known positions of holes
and slits provide y calibration points, while the :1: position is calibrated using the

known spacing between each pad.

Trajectory reconstruction

The measured coordinates of the fragments at the focal plane were the input for the
code COSY INFINITY to reconstruct the trajectory of the fragments and obtain their
position before entering the spectrograph after the reaction. As the particles pass
through the magnetic elements of the spectograph their positions change, due to the
action of the magnetic fields. If the magnetic fields are known at all points, the
trajectory of the ions passing through is governed by the Lorentz force law dp/dt =
q(E + v x B) and can be thus calculated analytically, as a matrix transformation
from the inital to the final coordinates. In the particular configuration of the S800,
the four coordinates (If, 9;, yf, 99;) measured at the focal plane (position and angle
in the dispersive and non-dispersive directions, respect.i\-'ely) can be expressed as a

transformation of the coordinates at the target position (9,, 3],, «pt, 6,), where 6 is the

14

 

Ti
"d2 ‘5; ID. i
from 1
DH 'ti‘m

Illti‘ ('ii

dllf‘! if

fractional kinetic energy difference from the central ray, 6 z: (E — EO)/E0, through a

matrix S, i.e.,

:17, 6t
6f : S yt
9f Wt
99] 6t

This assumes that the initial spot size in the dispersive direction :1: is small and
approximated to zero. To reconstruct the particle coordinates at the target position
from the knowledge of the coordinates at the focal plane an inversion procedure is
needed. This is rather complicated because the matrix S is not invertible. However,
the code COSY INFINITY [Ber93] has been developed for this purpose, including third

order optics and all optical aberrations. The code calculates the inverse matrix R,

such that:
at IIIf
ya = R 9;
(Pt 31f
0t 99]

and it can be used to deduce energy and angles in the target chamber.

2.3.2 Ionization Chamber

The ionization chamber is a standard Frisch Gridded Ion Chamber segmented into
16 one-inch anodes perpendicular to the ions’ trajectory. The chamber is filled with
P-10 (90% argon and 10% methane) at 300 Torrs. This detector provides precise Z

identification up to Z 2 50 [Yur99].

 

2.3.3

'flui st
Illa: hi
aili‘zuir
timin:

will (w

“as It

2.3.4

Alllm‘.
llle fr,
[lie r,
lli‘lIIlS
We Ih
Sun's, ._

lf'llir-

2.3.3 Scintillation Detectors

The scintillation detectors provide energy loss, total energy and timing information.
The first one is also used as the exit window of the ionization chamber, with the
advantage of reducing the straggling normally associated with an exit window. The
timing information is derived from the signals of the photomultipliers at each end of

each scintillator. The time resolution obtained for a test beam of 16O at 60 MeV/u

was 160 psec FWHIVI.

2.3.4 Particle Identification

Although the radioactive beams produced by in-flight fragmentation were purified in
the fragment separator A1200, where the magnetic settings were chosen to allow for
the transport of mainly the isotope under investigation, not always all the contami-
nants could be eliminated by the magnetic analysis. Therefore, particle identification
was the first important step in the data analysis. For this purpose a series of different
gates on the measured variables was determined. First of all, the data corresponding
to the setting of the spectrograph adjusted to the full beam momentum were analyzed
to identify the beam particles of interest. This was done by selecting the events with
the appropriate coordinates in the energy loss (AE) vs. time of flight (TOF) plane,
where particles occupy specific regions depending on their mass and charge. The
energy loss information was given by the ionization chamber, while the time of flight
was measured between the first scintillator and a beam-line timer (BLT), a scintilla-
tor placed at the end of the A1200 separator. The signal from the first scintillator
was also the trigger condition for the data acquisition. Therefore, the BLT signal
was delayed and, as a result, the TOF appears reverted, so that an increase in TOF

corresponds to an increase in velocity. As an example, the plot observed for the case

16

 

ofalv
the H‘.‘

at the

I.
1115 si-
\Vitli
bet-an:
essvmi
rem-tit
lift" in}
.‘Ilorm.
fiagnn
While 1
l’dflit'l

Com a 1;

fl”

lillft'II 1'1
Tht‘se

l7 .
[ilk-l })(’I

: I-
In

of a beam of 16C is shown in the top panel of Fig. 2.2. The 16C beam particles occupy
the central part of the plot. and were selected using the gate drawn. The contaminants

at the left and at the right of 16C were 1‘B and 18N, respectively.

To identify the residues from the reaction, the magnetic field of the spectrograph
was scaled by a factor (A-1)/A to take the smaller mass of the residues into account.
With this setting, the fragments had the same time of flight as the incident beam,
because the removal of one neutron from each beam particle left the fragment velocity
essentially unchanged. This allowed a simple identification of the fragments from the
reaction by just gating on the same region of the AE vs. TOF plane, occupied by
the beam particles with the previous settings, as shown in the lower panel of Fig. 2.2.
Moreover, since the TOF is directly proportional to the ratio A/ Z, the groups of
fragments immediately below the 15C residues could be identified as 13B and 12B,
while the lower group of fragments corresponds to 10Be. Analogously, the groups of
particles below 17N are 15C and 14C fragments produced in reactions on the 18N beam

contaminant.

However, other conditions were necessary to select only the events of interest, as
obvious from the bottom panel of Fig. 2.2. Events satisfying the first gate included
also (slower) beam particles scattered in the S800, which needed to be excluded.
These appear at smaller TOF and higher AE in the left part of the gate drawn in
the bottom plot of Fig. 2.2. A clear differentiation was possible using the T OF vs. :1:
(dispersive position) information for each of the two CRDC detectors. The gates used

for 16C are shown in Fig. 2.3. Similar gates were defined for the other carbon isotopes.

The intensities of the beams and residues, obtained using the gates discussed
above, were normalized using the signals of the BLT scintillator, placed at the end of

the A1200. The cross sections for one—neutron removal reactions were calculated as the

17

 

1500

10 3
1250 '
.1- 10 2

750 h '

 

 

33 500
'E 10
:3
g 250
g 1500
103
1250
1000 102
750
10

 

 

250

 

750 1000 1250~ 1500 1750 2000
TOP (arb. units)

Figure 2.2: Energy loss (AE) vs. Time of Flight (TOF) spectra for two different field
settings. The top plot was obtained with the spectrosgraph adjusted to the momen-
tum of the 16C beam particles, identified by the gate drawn. The contaminants 1“B
and 18N (left and right, respectively) were also present. The bottom plot was obtained
at a reduced magnetic field to identify the 15C reaction residues from 9Be(16C,15C)X,
selected by the gate drawn. (See text for details on the identification of the other
fragments which appear in the plot.)

 

 

ltIi1——~

Win

Hi“

111111

'l'( )F (4111). units)

1611‘)

l 4110

l3”)

1 (”if .I 1

 

 

 

 

10
1

A

Q

I:

3

.o'

3;

n.

O

I— 10 2

10

 

 

 

 

Figure 2.3: Time of flight vs. :5, position in the first (top) and in the second (bottom)
CRDC for 9Be(16C,15C)X. The scattered beam particles included in the gate drawn in
the bottom plot of Fig. 2.2 can be well identified and separated from the 15C residues

(selected with the gates drawn).

19

 

{316‘
a ("(1) l

l
deH

are e

2.4

1119 t
p1 nit
in f1
[“111
lHlPII

llit’sw

{11:15

{he I

.07 T"!

His

rate of detected fragments divided by the rate of incident projectiles and taking into
account the thickness and number density of the 9Be target, as well as the electronics
dead time. Details on the derivation of the cross section for each individual isotope

are given in Appendix A.

2.4 The NaI array

The excited states of the residues were detected by an inner ring of 11 cylindrical
position-sensitive NaI(Tl) scintillators [Sch99] surrounding the target, as illustrated
in Fig. 2.1. The complete array has a total of 38 detectors with 27 arranged in
two outer rings. However, only a marginal improvement of the peak to background
intensity was gained when the outer detectors were included in the analysis, therefore

these were not used.

Each scintillator was read out by two photomultiplier tubes, one at each end,
thus allowing for the determination of both the energy and the interaction point of
the photon in the detector. This was possible because the measured light outputs
E13 from the two photomultipliers at each end depend on the interaction point as
E13 o< E6445”), where :r is the interaction point from the center of the scintillator,
,u is the attenuation coefficient, L is the detector length and E is the total energy
deposited. Therefore, the total energy E and the position a: can be obtained as

E or \/E1E2 and :1: or log(E1/E2), respectively.

A collimated 60Co source was used for the position calibration. The collimator
consisted of two heavy met (95% tungsten, 3.5% nickel and 1.5% copper) cylinders
arranged coaxially with a 4.6 mm gap in between, where the source was placed. A
long ruler attached to the collimator served as a handle to insert the source inside

the beam pipe and to read the actual position at which the source was placed. The

20

source was moved in steps of 1/4 inch, and for each position data were recorded on
tape for 5 minutes. For each detector the reconstructed position was plotted in func-
tion of the position read from the ruler and the data were fitted with a polynomial of
order 3, to accurately describe the non-linear behavior at the edges of the scintilla-
tors. The polynomial was finally inverted and used to extract the calibrated position

information from the recorded light signals.

For the energy calibration different sources of known energies were used, covering
a wide range of energies. More precisely, 7 rays of 0.511 and 1.274 MeV from 22Na,
of 0.898 and 1.836 MeV from 88Y and of 0.239, 0.583 and 2.614 MeV from 228Th
were used. The data for each source were recorded for about 2—6 hours. A position-
dependent energy calibration of the energy spectra was derived at ten different points

along the detectors in the direction parallel to the beam.

The position information provided by the array made it possible to correct for
the Doppler shift in the energy of the 7 rays emitted by the fast (,8 2 0.34) residues.
This is illustrated in Fig. 2.4, where the energy dependence on the angle disappears

after the correction.

The back transformation to the center of mass system (c.m.), however, does not
generate the spectrum that would have been observed from a source at rest, due to
events in which radiation has escaped from the crystal. Examples of these are anni—
hilation radiation and Compton-scattered photons. Since the reconstruction cannot
identify these features, the part of the response function that does not belong to
the full-energy peak gets smeared. This may seem unimportant since the full-energy
peaks obviously are reconstructed correctly. However, an accurate understanding of
the measured envelope of the 7 spectrum requires knowledge also of the shape of the

continuum distributions underlying the peaks. For the decomposition of the measured

21

 

 

‘1!

5

lb!

12”,, 1 1.1-V 1

I]:

l [fl I

181111;

m Ikt'VI

lJIVI

If
L

l‘”

-1 Ii

lrliiij

”(mphfl

II] OI] (1‘ f

.
J..;

 

 

    

' 8
"-
3:1
I-_ 7
_ 1'.
$27;
'5 6
"-I
5‘ .-.' 5
0 -
a '2".
'3 —-_ 4
[if '3
3
2
1
o
8
:.-_—-_- 7
._..
:'_ 6
'--l -
-‘I=
' 5
.— '..I-
> Jra'r
3 .r
H. f'_-. 4
E. PL--
0 .u"
m .I‘:-' 3
.-.'-.
:"r
7:“: 2
I'-
E-
'I'J 1
-:-_--.:S.-..::
h...— ___ 0

0
cos(0)

Figure 2.4: 7—ray energy Elam", vs. cos(0) before (Em, top) and after (Ecm, bottom)
Doppler correction of the 7 rays from the deexcitation of the 16C residues produced
in one-neutron removal reactions on 17C.

22

 

 

spvtifu
2.4.1

For a ‘

a with"
general?
(“.1110 ('1
\i‘é‘ll as .
million
outmm
sale 11'.
PUIIPSEH
on the

signals 1
energy i
th‘shaI
Pliminat
filling I}

and 3.

The 1

I

\ ll (1! r

~A"f

9, >
‘Ilf‘ Illf‘l‘1\

spectrum, complete response functions were constructed in a numerical simulation.

2.4.1 Simulations

For a 7 ray of a given energy, assumed to be isotropically emitted in the c.m. system,
a sequence of Lorentz-boosted 7 events with the appropriate angular distribution was
generated in a Monte Carlo procedure. These were subsequently used in the Monte
Carlo code GEANT [Gea94], which simulated the energy deposited in the detectors as
well as losses generated by interactions with chamber walls and detector mounts. One
million primary 7 rays were generated for a given energy. For each event the (random)
outcome was randomly folded with the energy resolution, which was assumed to
scale with the square root of the energy and was fixed to the measured resolution
corresponding to a full width at half maximum (FWHM) of 7.5% at 1.33 MeV. Based
on the observed spatial resolution of 1.5 cm FWHM, the sequence of simulated 7
signals was corrected event by event for the Doppler shift to construct the apparent
energy in the c.m. system. Histograms of the simulated events created the reference
line shapes. The resulting shapes were approximated by smooth analytical curves (to
eliminate statistical fluctuations from the Monte Carlo procedure) and were used for
fitting the observed spectra. Examples of these fits will be encountered in Chapters 4

and 5.

The reliability of the simulations had been previously verified by comparing mea-
sured and simulated 7-ray spectra from (necessarily stationary) calibration sources.
An agreement to within 10% in the absolute intensity was found. As an example,
the measured and simulated spectra from the calibration source of 88Y are plotted in
in Fig. 2.5. Noting that the data include also natural background, the comparison

between the two spectra shows a remarkable agreement.

23

_>u§\br§_<x::_:. 1 5.2.3::

 

 

 

 

 

 

 

 

 

 

 

 

 

 

”TT" F"— ! T T" T" T’TU—‘fi T I
88Y
, ----- Simulation

" . i
> :1
g :
>5 -
N r —
0 I
a) .
'U .
\ -
if.
S:
:3
O
E
3;. 1
To"
1: .
8 '.
::
h—t

II
0 1 l l A
0.5 2.5

 

Figure 2.5: Measured and simulated 7-ray energy spectra for the 88Y source, normal-
ized to the number of 7 rays emitted during the electronics live time. The observed
deviations between data and simulation are attributed to the natural background.

24

 

 

2.4.2

For earh '
D1_1IlIlli‘f—'
the axis
arthIE I!
final 9pm
I
asstiriatu :
l
the hits 11.
|
Wire retz;
llt‘lf‘t'li 1r '
in each p\
data wer-
IhI‘cc spin

this my 1 .

11 Sperm,

Ah alt-
Ihventrgg
primary :
inilllIJlH}_
llli‘ ffihllli’;
~11» 1m, ,,
ll SIT Par-It

l1] Tll .-

( Elf-If

 

 

2.4.2 Analysis of the measured 7-ray spectra

For each event the 7 rays detected by each of the 11 scintillators were recorded and
DOppler-corrected. Since the events are expected to be equally distributed around
the axis of the NaI(Tl) array, only one 7-ray spectrum was analyzed, obtained by
adding the spectra from the individual 11 detectors. The procedure to obtain the
final Spectrum was the following. Since over 99% of the detected fragments was
associated with a multiplicity of 7 rays m, g 3 in the first ring of NaI(Tl) detectors,
the hits were sorted according to their energy and only the three highest energy hits
were retained. This reduced the number of variables from 11 (one energy per each
detector) to 6 (detector number and energy for the three highest energy hits measured
in each event), producing a data file essentially equivalent to the original, where the
data were just sorted differently. The final spectrum was obtained by summing the
three spectra corresponding to the three highest energy hits. In this particular case,
this was equivalent to the spectrum that would have been obtained by summing the

1 1 spectra from the individual detectors.

An alternative procedure to analyze the 7-ray spectra would be the add back of
the energies event by event. In the add back, since the Doppler effect acts only on the
Primary 7 ray, while it does not affect secondary 7 rays (from Compton scattering, for
example), the energies should be summed before applying the Doppler correction and
the resulting energy should be corrected using the angle of the primary (usually the
One with the highest energy) 7 ray. However, the situation becomes more complicated
if for each event more than one 7 ray is produced (i.e., if the excited residues decay
to the ground state via a cascade of two or more 7 rays). In that case an unambigous
identification of primary and scattered 7 rays is not possible. These considerations led

to the choice of the first procedure. For a quantitative analysis, the same procedure

25

was applied to the simulated spectra.

A complication in the data analysis was the presence of a continuum distribution
varying approximately exponentially with energy. This was attributed to neutrons, 7
rays and charged particles produced in the target and to their secondary interactions
with construction materials and the scintillators. This distribution has been seen
consistently in previous experiments [N av98, Aum00, Gui00, Nav00] with an intensity,
for 7 energies above 0.25 l\=-’IeV, of approximately 9% per outgoing fragment. Although
it reduces the sensitivity to detect weak transitions (the 17C analysis shows an example
of this, section 5.2.3), it does not significantly interfere with the determination of the

7-ray energies and intensities.

In particular, accurate estimates of this continuum could be obtained by compar-
ing the ”Be and 16C data. In both cases only one 7 ray of energy 0.32 and 0.74 MeV,
respectively, was observed. At energies higher than 1 MeV the two spectra (both
in the laboratory system and in the c.m.), normalized to the number of outgoing

fragments, agree in shape and intensity to within 15%. This is shown in Fig. 2.6.

By averaging the ”Be and 16C data, two experimental continuum distributions
for E, > 1 MeV were obtained, one in each reference system. The two distributions

were compared and found to be equivalent, as demonstrated by Fig. 2.7.

It will be seen in Chapter 5 that the result for the exponential continuum obtained
from the fit of the 7-ray spectrum measured in coincidence with 16C residues from

reactions on ”C, was consistent with the measured distribution.

The assumption on the origin of the continuum distribution as mainly arising
from reactions induced by the neutrons removed from the beam is demonstrated by
the observed angular distributions of the 7 rays in the laboratory and in the c.m.

systems. As shown in Fig. 2.8, the 7 rays of the continuum distribution measured

26

 

 

 

 

 

 

 

 

 

E0; fi fi lat: T I 0 (”Ci 15'C +17) I
- ..9° 0 0 (”Be, l1Be+y)
“80090 ¢ 1
102E 09002¢$¢¢¢$ 3 1
a 41 ¢ 0 fl g g E
i if I++$ ”Hi: II I
*1 11+ it
,9; Egéo . [cm I T o (:C, :15C+Ty) ' T
if; . .... 29¢: ¢ 0 ( Be, Be+y)
10 ’2: . 2 ¢ 1
' Wifeiitittfi I i 1 i
$ +f fi+++iIIlffli+ :
1031.— 1 :4 L j i . I i . lJ in

BY [MeV]

Figure 2.6: Experimental laboratory (top) and c.m. (bottom) 7-ray spectra from the
residues 11Be and 15C for E, > 1 MeV.

27

Llrslpls

0
_>.42}2.2.:1:...L\r..::2. V~ 43.22.52:

‘ .
.

 

L Fl!\

10

1

MI.

Figllff‘

In I“

SI’JF‘I‘T

 

O c.m. -
B, 0 lab.

Intensity [Counts/Fragment/MeV]
'5
I l I l I to
-O-O-
—O—
-.-
-o-o—
*5 i
+

 

-3
10 — “

 

 

l l l l l l 1 l l l l l l l l l
1 2 3 4 5
E, [MeV]

O\

Figure 2.7: Experimental c.m. (full dots) and laboratory system (open dots) 7-ray
spectra representing the continuum distribution lying beneath the measured 7-ray

spectra for E, > 1 MeV. The two spectra were obtained as an average of the ”Be
and 16C data.

28

 

 

l'. "
{11.1111 ' f.

.A >111
l0 invest 1
The 1111;!
thine-d ‘1‘
{the Whey
9.111 17C,
fiUETlldl‘ 1
terminus
911011.111 g
huth (use
in the ('31;
we Ihm
A”) and
angular I]:
( ling“

“he

 

 

l6

from C appear focused forward in the laboratory system.

A study of the angular distribution of 7 rays from the residues was also performed,
to investigate whether the possible alignment in the fragments could be identified.
The angular distribution of the 7 rays emitted by excited fragments which are not
aligned is expected to be isotropic. However, this was found [HanOO] to be the case
also when the fragments have some degree of alignment. In fact, for the cases of 15C
and 17'C, using the calculated cross sections for fully aligned states as weights for the
angular distributions characteristic of each projection of the angular momentum, the
resulting angular distributions are isotropic. The angular distributions of the 7 rays
emitted by the 14C and 16C residues in their first excited state (most populated in
both cases) were obtained and show a forward focusing in the laboratory system, while
in the c.m. they are isotropically distributed. The measured angular distributions
were then compared to the distributions predicted by the simulation of the 6.09
MeV and 1.77 MeV 7 rays, respectively. As an example, F ig.2.9 shows that the
angular distribution obtained from the 17C data, corresponding to the decay of the
16C fragments in the laboratory system presents the same behavior as the simulated

0119..

2.5 The experiments

The 15’16’17’19C reacted on a 228 mg/cm? 9Be target. The characteristics of the incident
15’16'17'19C beams are listed in Table 2.1. The secondary beam energy is the average

energy at the target mid-plane.

29

r--

“my;

 

l25 I- A

100 r (“(3, ”C +0 _

 

 

 

 

 

75— .#..6 ‘3-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Counts
4

 

25P+ -

 

l l L 4; 1 1 I 1 L L l 4 l l l l l l l 1 L l
-O.4 -0.2 0.2 0.4

 

0
cos(0)

Figure 2.8: Experimental laboratory system angular distribution of the 7 rays with
E, > 1.2 MeV measured from one-neutron removal reactions on 16C. The distribution
is focused at forward angles, demonstrating the origin of the continuum distribution
from interactions induced by the neutrons removed from the beam.

30

 

 

 

 

l“ _

p

100 >—

7§ __
z
2:
-
~
H
A

'V b
V

l

50

 

.,1I1.‘*~III‘-IIII

't-
'3‘ (lip.

fhl,‘\ r»

. (If i

 

I I I I I I I I I I I I I I I I I fl

125 -— 1
17 16

' ( C. C(2+)+Y)

1

-— 1.77 MeV Simulation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o + f: l
75 _ f
o
(I)
‘5 9
:3
63 ~ 4
sol s
I + +
25 L. —
O 1 l l l l L l l l l l 1 1 l I l l l l L L
04 02 o 02 04
003(9)

Figure 2.9: Experimental laboratory system angular distribution of the 7 rays with
1.6 < E7 < 2.0 MeV, corresponding to the decay of the 16C fragments, produced in
one-neutron removal reactions on 17C, from the 2+ state at 1.77 MeV to the ground
state. The solid line is a simulation of the angular distibution of Lorentz-boosted 7
rays of 1.77 MeV.

31

Table 2.1: Experimental parameters and settings.

 

 

 

 

 

 

 

 

 

 

 

 

Primary beam 180 fiNe jQNe 22Ne ééNe
Intensity of primary beam
[pnA] 330 330 330 330 330
Energy of primary beam
[MeV/ nucleon] 80 80 80 80 80
Secondary beam 15C EC 17C 19C 19C
Intensity of secondary beam '
[particles/ sec] 90 310 230 0.5-1 0.5-1
Energy of secondary beam
[MeV/nucleon] 54 62 62 57 56
Secondary target 9Be 9Be 9Be 9Be 197Au
Target thickness
[mg/cm2] 228 228 228 228 518
Data acquisition time 3.5 h 6 h 18 h 3 d 1 d
Main beam impurities - 14B, 18N 15B, 19N 1BC 18C

 

 

 

 

 

 

 

3‘2

 

 

Cha

The

lint u, l;- .11
Ut'l+'l 1”
{ll N‘IVM l
of the lm
Pllbimul a
girls. gay:

Frnm [hi

lmtits-n n

llflmg dill?

SF}

'u-rrat in

AlHIP

fill lllff‘ h

pligjwf-T ils
L.

f‘HJ kll‘ ML

Tl

alli‘ T(J‘(11

kit,
Mm

Chapter 3

Theoretical Analysis

Knockout reactions have been often used for spectroscopic studies of neutron-halo
nuclei [Han95], where the large cross sections and narrow momentum distributions
observed in the breakup of neutron halo systems provided evidence for the large size
of the halo. It was shown by Bertsch et al. and others [Ber90, Yab92, Hen96] that the
eikonal approximation, previously used for nucleon-nucleus scattering at high ener-
gies, gave a good description of the cross sections for such reactions on light targets.
From this also follows that the outgoing fragment’s longitudinal momentum distri-
bution reflects the momentum content of the wave function in the volume sampled
by the projectile’s interaction with the target [Han95, Han96, He596]. The cross sec-
tions and momentum distributions are very sensitive to the angular momentum and

separation energy of the nucleon in the initial state.

More recently it has been shown that there are also appreciable cross sections
for the removal of a nucleon from the occupied non-halo single-particle states in the
projectile. An example of this is furnished by the light phOSphorus isotopes, where
the knockout of the halo proton from the ground state constitutes only 30-55% of
the total measured knockout cross section [Nav98]. Even for the very pronounced

single-neutron halo nucleus 11Be, one finds by comparing the original inclusive exper-

33

 

 

imem :li'
lllf’ilfillft’l
ulates ex
wit h 1%
Nil“? iH
Slml
stairs I
St‘t‘llt »[1

Written

iment [Ke195] on the 9Be(”Be, 10Be)X reaction at 66 MeV / nucleon with the exclusive
measurement with 7 rays in coincidence [AumOO], that 22% of the cross section pop-
ulates excited levels of the 10Be residues. The momentum distributions of the heavy
residues arising from knockout from deeply bound states can be calculated using the

same techniques as for the halo states.

Similarly, extending the eikonal approximation as applied to halo nucleus ground
states to treat the removal of a non-halo nucleon from the initial state, the cross
section a¢h(I") for populating a given final state I’r of the residue or core, may be

written [TosQQb] as
0M1") = 2 025(1", n€j)0,p(5n, ncj). (3.1)
2'

Here C 25' , the spectroscopic factor for removal of a nucleon with given single-particle
quantum numbers (nt’j), expresses the parentage of this configuration in the initial
state with respect to the specific state I 7' of the remaining nucleons. Following nu-
cleon removal this is assumed to be the final state of the residue, which is therefore
assumed to behave as a spectator particle and to interact at most elastically with the
target [Hus85]. The sum in Eq. (3.1) is taken over all configurations which have a non-
vanishing parentage. The 03,, are the single-particle removal cross sections, which are
strongly dependent on the orbital angular momentum f and the neutron separation

energy Sn. The calculation of these quantities is discussed below.

The approach of the present work and its predecessors [Nav98, AumOO, Nav00,
GuiOO] has been adopted in recent work by Sauvan et al. [SauOO], who have measured
inclusive one-neutron removal cross sections and momentum distributions for 23 nuclei
in the p — 3d shell. The results are in good agreement with calculations based on
techniques that are essentially identical to those used here. In particular, the results
for the inclusive absolute cross sections suggest that the method may actually be more

34

 

 

BCCUIBlP

 

interests;
tress N'"('l
total mu
truss SVV
(it‘llslly .
bit-m n,
hUW'lhr
Di Slu>c1
of how

tfilf [tan

rpllltn'u

3.1

A mm
(Tarbl ,I
and El
lrllt I l
filt1115‘

accurate than the i20% conservative estimate proposed in the following. It is also
interesting to compare this and our results with the measurements of charge-changing
cross sections (ace) reported by Chulkov et al. [ChuOO]. In their analysis they obtain
total neutron-removal cross sections 0-1m by taking the difference between interaction
cross sections a, and ace. While the 01 and ace can be discussed in terms of global
density distributions, Chulkov et al. find that the a-“ show a more complicated
behavior suggesting the influence of nuclear structure effects. The present work shows
how these can be accounted for in the one-neutron removal channel through the use
of spectroscopic factors calculated from a many-particle wave function. An example
of how these effects can show up in the two-neutron removal channel is offered by
our recent experiment on 12Be [NaVOO], where a sizeable fraction of the one-neutron

removal cross section populates the unbound 0035/2 state in 11Be.

3.1 Theory of the spectroscopic factors of 15’16’17’19C

A number of recent papers cited above have discussed the properties of the heavy
carbon isotopes within the framework of particle-core-coupling models, cluster models
and global density distributions. Although such models provide qualitative insight
into the structure and resulting cross sections of the ground states, they do not
furnish a unified description of all states. Shell-model calculations have been used
instead. The natural shell-model space for these nuclei is the complete set of basis
states spanned by the neutrons in 131/2, 0d5/2 and Grim (sd—shell) orbits together

with protons in 0193/2 and 0101 /2 (pshell) orbits.

The Hamiltonian for the neutrons in the sd—shell is well established by Wilden-
thal’s USD interaction [Br088]. The pwshell Hamiltonian is also well established

[War92]. The proton-neutron (p — 3d) part of the Hamiltonian is based upon the

35

 

l “v l l

”1'”.

W l

lllllt

work of Warburton and Brown [V'Var92]. They considered all of the known data
(165 levels) in the mass region A = 10 — 20 which could be associated with the p — 3d
Hamiltonian. Among the 165 energy-level data considered were those of 15C: {L and
g+; 16C: 0+, 2+, 3+, and 4+; 17C: 3+; 18C: 0+ and 2+; and 19C: {a

Two types of p — 3d Hamiltonians were developed: (1) WBT, modeled on a set of
two-body matrix elements (TBME) obtained from a bare G matrix, and (2) WBP,
modeled on a one-boson exchange potential (OBEP) which includes the one-pion
exchange potential (OPEP) (fixed at its known strength) and a long-range (monopole)
interaction. For W BT, 28 linear combinations of the 95 p — 3d TBME were adjusted
to fit the data. For WBP, ten parameters associated with the strength of the OBEP
terms were varied. The root mean squared (rms) energy deviations for the 165 p —
501 data were 389 keV for WBP and 330 keV for WBT. Comparisons between the
predictions made with WBP and WBT will give an indication of the theoretical error

in these type of calculations. W BP is an evolution of the Millener-Kurath potential

model for the p — 3d interaction which was developed earlier [Mil75].

The WBP and WBT interactions have been used to predict many properties of
nuclei in the A = 10 — 20 mass region [Br095, Br098, AumOO, NavOO]. In general,
the wave functions and spectroscopic properties with WBP and VVBT are similar,
but there are differences in the energy-level ordering, especially when the levels are
spaced more closely than the 350 keV rms deviation established in their derivation.
In the present context, the calculations provide both level energies and the required
spectroscopic factors CQS. These are presented in Tables 4.2, 5.1, 5.2, 5.3, and 5.4,

and discussed in detail in Chapters 4 and 5.

For both 17C and 19C theory predicts a triplet of low-lying levels with spin-parity

+, g‘“ and 3+. The ordering of these levels differs between WBP and WBT. WBP

MI!—

36

 

gives {HI

{girdled ~
sum.
0.3 .‘xli-V

ground 5'

 

this tesp
with the
Deiaih ;
'iirergm
{Actors .
Y\Illl l)!»
Sii‘livs -(
I0 new;
all flu,

all f: (A

gives for 17C: 3+ 'rround state), 5” at 0.03 MeV and {I at 0.30 MeV; and for 19C: %+

(ground state), 3+ at 0.19 MeV and 3+ at. 0.62 MeV. VVBT gives for 17C: 3+ (ground

state), %+ at 0.08 MeV and {r at 0.27 MeV; and for 19C: 3+ (ground state), + at

NIH

0.5 MeV and 3+ at 0.40 MeV. The present experimental results give §+ for the 17C
ground state and {I for the 19C ground state. Thus the WBP interaction is favored in
this respect. However, it does not rule out WBT since the required levels associated
with the experimental spins are within the nominal 350 keV deviation expected.
Details about the wave functions will be discussed in Chapters 4 and 5. The WBP
interaction will be used for the spectrosc0pic factors. Generally, the spectroscopic
factors obtained for W BP and WBT are very similar, and any important differences
will be discussed. The spectroscopic factors between the low-lying positive parity
states are related to pickup from s— and d orbitals. Pickup from the p—orbitals leads
to negative parity states at excitation energies above the neutron-decay threshold in

all the carbon isotopes studied in this work, with the exception of 15C, for which they

are observed in the present experiment.

3.2 Theory of the single-particle cross sections

The cross sections 05,, in Eq. (3.1) were calculated in the eikonal model [Tos99b].
The same input parameter set was used as in the reported analyses of Refs. [Nav98,
AumOO, Gui00, Nav00]. The calculation of each single-particle cross section assumes
that the removed nucleon is described by a normalized single-particle wave function
with quantum numbers (716 j ) moving with respect to the core of remaining nucleons
in state c E I 7’. Such configurations are written |e3,,,), where J is the magnitude and

M the projection of the projectile’s ground-state total angular momentum, f: f+ 5".

Since only the residue is detected, and not the neutrons, these single-particle

37

cross sections are a sum of the contributions from removal of the neutron due to
elastic breakup (d1ffract1on dleOCldthIl) and absorption (strlpping) [HusBo], asp —
03;,” + 03}. These two contributions are computed separately, as integrals over the

projectile’s center of mass impact parameter, using [Tos99b]

 

 

03;” Z 2] l/dfi Z<¢SMI l(1—c)85n '2 “bf/M) Z l<JA1'l((1—Sc8rill¢3.w>l2
+ M M M’
(3.2)
and
97' 1 -' C C
0e; : 2J +1/db§(¢“,|(1 — l5nl2ll5cl2l¢1iul (3-3)

Here the quantities 8C and 5,, are the elastic S-matrices, or profile functions [AlK96,
Tos97], for the core-target and removed neutron-target systems, expressed as func-
tions of their individual impact parameters. These are calculated using the optical
limit of Glauber theory [Gla59]. The neutron-core relative motion wave functions
lqfijM) are calculated in a Woods-Saxon potential with radius and diffuseness param-
eters 1.25 and 0.7 fm. The depth of the potential was adjusted to reproduce the
separation energy of the nucleon in the (initial) state with given n6 j . In those cases
where a more strongly bound nucleon is removed from a system which also binds
a weakly bound and delocalizcd neutron, this few-body composite structure of the
residue upon SC was taken into account explicitly, as in [Tos99b]. An example of this

is the case of 16C (section 5.1.3).

Equation (3.3) allows a simple interpretation. It is the integral over impact pa-
rameter, and average over M substates, of the joint probability of the core being left
intact by the reaction (given by the quantity ISCI2) and of the neutron being absorbed
(given by the quantity (1 — |8,,|2)). The diffractive cross section, Eq. (3.2), represents
the probability that both the core and the neutron are not absorbed in the reaction.

For halo states, Eqs. (3.2) and (3.3) make roughly equal contributions to the single-

38

particle cross section. For more strongly bound states the contribution from Eq. (3.2)
is typically a factor of 2—3 smaller than that of Eq. (3.3) and may be smaller. It will

be interesting to test this assumption experimentally.

The essential parameters in the calculation of the functions 8 are an effective
nucleon-nucleon interaction and the rms matter radii of the assumed Gaussian mat-
ter distributions for the core and target nuclei. The 9Be matter radius was taken as
2.36 fm. The effective interaction, also assumed a Gaussian, used the free nucleon-
nucleon cross sections [Cha90] and the real-to-imaginary ratio for the forward scatter-
ing amplitude tabulated by Ray [Ray79] for 100 MeV nucleons. The interaction range,
of 0.5 fm, was chosen [T0399b] so that the calculated reaction cross sections are con-
sistent with measured values. Specifically, calculated reaction cross sections for the
12C—IZC and 27Al—‘QC systems at 83 MeV/ nucleon [Kox87], and for the proton—gBe

system at 60 MeV/ nucleon [Ren72], were consistent with experiment.

The point-particle rms matter radii for the carbon isotopes were first reported
in [Sai89, Lia90]; the results of the recent paper by Ozawa et al. [Oza98] are used here.
Our results are not very sensitive to the matter radius; for 19C an overall increase of
10% reduces the calculated cross sections for removal of the halo or of more bound
neutrons by 8% and 16%, respectively. The calculated single-particle cross sections,
defined as the sum of the stripping and diffraction dissociation contributions, are given
in Tables 4.2, 5.1, 5.2, 5.3, and 5.4. The use of alternative microscopic descriptions
of the neutron-target interaction, and corresponding 8“, has been shown to calculate

very similar 03,, [T0899c].

Theoretical calculations of the longitudinal momentum distributions of the core
fragments were made in a simpler model, based on a black-disc approximation. In

this, SC and 8,, are assumed to be unity outside of a cutoff impact parameter and zero

39

inside [Han96]. These impact parameter cutoffs were chosen to reproduce core-target
reaction cross section systematics [Kox87] and the neutron-target reaction cross sec-
tion of 306 mb at 60 MeV / nucleon. (The corresponding values are 286 and 298 mb
when calculated for 62 and 57 MeV, respectively, with the parameters used for the
partial cross sections. The widths of the momentum distributions are insensitive to
the precise choice of target radius.) The neutron relative motion wave functions were
calculated in a Woods-Saxon potential, as above. In this model the profile functions
affect the limits of impact parameter integrations, and the momentum distribution
takes the form of a one-dimensional Wigner transform of the wave packet produced
in the reaction [Han96]. The integrated cross sections obtained with this procedure
agree well with those calculated using the more realistic profile functions. Test calcu-
lations show that the shapes of the momentum distributions are much less sensitive
to the choice of the impact-parameter cutoff than are the absolute values of the cross
sections. In the following, these calculated shapes are presented scaled to fit the
observed intensity, the idea being that the shape, taken separately, is the quantity
that carries information on the €~assignment. The absolute value of the partial cross

section then leads to the spectroscopic factor.

An alternative treatment, by Bonaccorso and Brink, has also been applied to the
longitudinal momentum distributions of neutrons from the breakup of halo states
[Bon98a, Bon98b, Bon99]. They use a semi-classical (constant velocity, straight line)
approximation for the relative motion of the core and target, with a lower impact
parameter cutoff, but a (non-sudden) quantum-mechanical treatment of the interac-
tion of the neutron with the target. The treatment, which deals with the diffractive
and the stripping parts in a consistent way, gives an excellent description of the an-
gular distribution of neutrons following the breakup of 11Be [Bon98a]. The breakup

contribution to the heavy residue longitudinal momentum distributions discussed in

40

the present work can be inferred from that of the neutron in the rest frame of the
projectile. For (”Be, 10Begs) breakup it was found [Aum00] that the resulting shapes
are essentially indistinguishable from those of the eikonal calculation. A recent appli-
cation to the carbon isotopes [Bon99] reported cross sections somewhat larger than
those given in Table 5.4, however, depending on the chosen set of optical parameters
for the n+9Be system, the agreement is better. Bonaccorso finds [Bon00] for the case
of 19C at 60 MeV/nucleon and an assumed neutron separation energy of 0.5 MeV
single-particle cross sections, given as (stripping, diffraction in mb), of (100, 76) in
agreement with our (99, 71) for the ground state. For an assumed 13 = 2 cross section
to a 1.62 MeV excited level, she finds (21, 11) as compared with our values of (25,
11). Reference [Bon99] reports an interesting feature arising from the inclusion of the
spin-dependence of the neutron interaction in the analysis. It turns out that different
momentum signatures arise from the breakup of the Dds/2 and 0d3/2 spin-orbit part-
ners. The present data are not good enough to reveal this effect, but this prediction

should certainly be kept in mind and investigated in future experiments.

41

Chapter 4

The test case: 15C

4.1 Previous theoretical and experimental work

The structure of 15C is well known from (d,p) transfer reaction studies on 14C [Gos73].

Spins and parities of {L and 3+ were assigned to the ground state and to the first

excited state at 0.74 MeV, respectively. There are no other states below the neutron

separation energy 5,, = 1.218 MeV. The simplest levels that can exist in 15C are one-

particle two-hole states and have positive parity. These correspond to an s — d shell

neutron coupled to a 14C core in the ground state. The low neutron separation energy
1

and the ground state J 7‘ : 5+ suggest the possibility of a halo structure, where the

valence neutron is loosely bound to the semi-magic nucleus 14C.

The momentum distribution from nuclear breakup of 15C was studied in an earlier
experiment [Ba298] and evidence of important contributions from excited states in
14C was found. Interaction and charge changing cross sections (01, ace) have been
recently measured [ChuOO] for several carbon isotopes (IQ—2°C). Interaction cross sec-
tions include all possible reactions, excluding elastic and inelastic scattering, while
charge-changing cross sections represent only the channels corresponding to a change
in the atomic number of the projectile. The difference between the two cross sections

is then equal to the neutron-removal cross section 0.1”, and has been found to be

42

similar for the neighboring nuclei 15C and 14C. This has been taken as evidence for
the absence of a halo structure in 15C. However, the neutron-removal cross section
includes contributions from the removal of any number of neutrons, thus prevent-
ing a discussion of any specific channel. More interesting is the comparison between
the one-neutron removal channel for 14C and 15C. As a matter of fact, an enhanced
one-neutron removal cross section for 15C has been observed [Sau00] and found to be
in close agreement with theoretical estimates based on the same model used in this

work.

4.2 Present shell-model results

According to shell-model calculations [Bro88], the wave function of the ground state
of 15C is predominantly given by an s—wave neutron coupled to a ground state 14C
core configuration (1.91 ,12 (8) 0+), while the components with 14C in its excited states
are small. For 15C, WBP and W BT give similar spectroscopic factors, for all possible
configurations (cf. Table 4.1). However, the calculated level scheme of 15C is different
in the two cases, although within the expected nominal deviation of 350 keV: with
WBP the 13+ excited state is at 0.38 MeV, with WBT it is at 0.66 MeV, closer to the
experimental value of 0.74 MeV [Aj291]. For 14C both WBP and WBT interactions

reproduce well the experimental level scheme [Aszl], as illustrated in Table 4.1.

The set of basis states used to calculate the spectroscopic factors indicated with
WBP and with WBT in Table 4.1 includes only Ohw and 1hw configurations, i.e., the
two lowest energy configurations of the harmonic-oscillator configuration space. The
extension of the configuration space, by including higher energy configurations 2hw
in the WBP calculation (denoted by W BP+ in Table 4.1 and in the following), gives

slightly smaller spectroscopic factors for the first, three largest components (1.91/2 ®0+,

43

Table 4.1: Experimental and calculated energy levels of MC and associated spectro-
scopic factors for the different components of the 15C ground state wave function,
calculated with WBP, WBT and WBP". There is no combination of the 14C 3‘
state with a p — 3d shell neutron that can couple to the 15C ground state spin y.
The theoretical energies are calculated with WBP, except for the excited 0+ state
and the last 2+ state, which appear around 13 and 15 MeV, with WBP and WBT,
respectively. For the other levels the three different calculations give similar energies.
The experimental energies are taken from [Aszl].

 

 

 

 

E,,,,[MeV] E,,,[MeV] 1th [[5] 023.53,) C‘ZSWBT 025W,»
0.0 0.0 0+ 0 0.98 0.98 0.95
6.09 5.48 1- 1 1.18 1.20 0.96
6.59 636 0+ 0 - - 0.02
6.73 6.45 3- - - - -
6.90 7.15 0- 1 0.46 0.46 0.39
7.01 7.77 2+ 2 0.017 0.015 0.017
7.34 6.69 2- 1 0.001 0.001 0.001
8.32 8.32 2+ 2 - - 0.027

 

 

 

 

 

 

 

 

0131/23” 69 1‘ and 0p1 /2 <8) 0‘). For the Dds/2 <8) 2+ component the spectrOSCOpic factor
does not vary. In addition, another 2+ state in 14C comes down in energy at 8.32 MeV
(with WBP and WBT it is at around 13 and 15 MeV, respectively), with a significant
spectroscopic factor. Even though it corresponds to an unbound level in 14C, its
energy of 8.32 MeV is very close to the neutron separation energy of 8.18 [Aszl],
therefore it has been included in the analysis. These two last components have been
treated together, and their spectroscopic factors have been added, in the calculation
of the cross sections and the comparison with the experimental results. The three

different shell-model outputs are listed in Table 4.1.

For a consistent analysis with the other carbon isotopes (see also section 3.1), the
WBP values will be used in the following. The results obtained by using the WBP+

spectroscopic factors will also be discussed and are given in Table 4.2.

44

4.3 Experimental results

Two excited states in 1“C are expected to be the most populated in the reaction (inset
in Fig. 4.1): the 1‘ state at 6.09 MeV, which decays directly to the ground state, and
the 0’ state at 6.9 MeV, decaying to the ground state through the 1”, in a cascade

of two 7 rays of 0.81 and 6.09 MeV.

The 7—ray spectrum observed from the decay of the excited 14C fragments (Fig. 4.1)
was fitted using the simulated line shapes of the 0.81 and 6.09 MeV 7 rays. The
additional continuum distribution measured from the 12Be and 16C data, as discussed
in section 2.4.2 and shown in F ig.2.7, was included in the fit. With these three
components, while reproducing accurately the experimental data up to E7 ~ 6.4 MeV,
the fit somewhat underestimated the yield of 7 rays at higher energies. Although its
associated spectroscopic factor is predicted to be very small, an additional response
curve for a 7 ray around 7 MeV was included in the fit, to account for the 2+ level.
The possibility of another 7 ray at 0.5 MeV from the decay of the 0+ state at 6.59
MeV to the 1‘ was also examined and estimated to be negligible. However, the
presence of 0.511 MeV annihilation photons from the higher energy 7 rays made it

difficult to analyze the low energy part of the spectrum with good accuracy.

As discussed in section A.1.2, the measured total cross section is 137i16 mb, to
be compared to the theoretical prediction of 197.5 mb. Using the coincident 7-ray
information, the partial cross sections to the three excited states and to the ground

state were extracted and are given in Table 4.2.

In agreement with theory, the largest component of the cross section corresponds

to the removal from 15C of the 181/2 neutron coupled to a ”C9, core.

The results from W BP and W BP+ are not sensibly different, and although they

are consistent with the data within two standard deviations, a comparison between

45

 

 

 

 

 

 

 

 

 

 

 

 

 

T I 1 ' ‘ I n T I I I
1 BY [MeV]
2* 7.01
. 0 6.90 ~
_, 1- .31! 6.09
10 ~— 1
+ .
_ 0+ 0 ‘
'— 14
> g C
Q)
E t l: A
1; "
8 g- i
555 .2 l \‘i‘
a 10 j o 1
E t! l
a ll 3
5 p \ ’V’
8 fix, 1
hi 1.1!", \ , 34M .
>‘ is .\ x’ilkrvplfl’lkNJH". Y‘MVW/W‘V J ‘\\ § _
r: 1 ‘ \\ I
U)
8 _ l \ ‘
H - \\
,5, l \
1
-3 l
10 5 l ' j
l' \ /\ -
r- ‘ f
_ J
\‘WN w 67‘3“”)! A \
e , a! ‘
- ”Arm-“‘«I‘J‘W-MV' MW . l J
-4 K
10 . m 1 . . 1 1 1 1 l 1‘ ‘
2 4 6
E7 [MeV]

Figure 4.1: Doppler corrected 7-ray spectrum from 9Be(15C,”C)X. The curves are
the simulated response functions for the observed 7 rays and the exponential mea-
sured continuum distribution extracted from the 12Be and 16C data (Fig. 2.7). Inset:
Simplified level scheme of 14C.

46

Table 4.2: Comparison between the experimental cross sections and the theoretical
predictions with spectroscopic factors calculated with different interactions (WBP

and WBP+, denoted with the subscript +) for the reaction 9Be(15C,14C)X at E =
54 MeV/ nucleon. The level energies are taken from previous work [Aszl].

 

 

 

 

 

 

 

 

E In 6 0'31) C25 02 3+ 0'th 0th,+ 0exp bth bth,+ bexp

[MEI/l lhl [0] l’mbl lmbl [mbl [mbl l%l l%l l%l

0.0 0+ 0 144 0.98 0.95 141.5 137.1 109:1:13 71.7 74.5 80:1:2

6.09 1- 1 34 1.18 0.96 40.5 33.0 22:1:3 20.5 17.9 16:1:2

6.90 0_ 1 32 0.46 0.39 14.9 12.6 3i1 7.5 6.8 2i1

7.01 2+ 2 34 0.02 0.017 0.6 1.5 3:l:1 0.3 0.8 221:].

8.32 2+ 2 34 - 0.027 - 1.5 -

sum 197.5 184.2 137i16

 

 

 

 

 

 

 

 

the predicted and measured cross sections and branching ratios indicates a qualitative
agreement. The comparison also favors the WBP+ spectroscopic factors. Why the
structure of 15C seems to present a different behavior than the heavier carbon isotopes
remains to be understood. One explanation for the observed smaller cross sections
may be the manifestation of the quenching of the spectroscopic factors in nuclei closer
to the valley of stability. Dieperink and de Witt Huberts [Die90] find that the general
occupancy of single-particle proton orbitals below the Fermi surface, as determined
from the charge densities, is only 75 j: 10% rather than unity and that for (e,e’ p)
single-proton removal reactions, the reduction can be even more drastic. For the
nucleus most relevant to those discussed in the present work, 12C, the spectroscopic
factors to the g and %7 states of 11B are only 65% and 50%, respectively, of the
values predicted by a shell-model calculation similar to the one used here. It may be

possible that 15C lies in a transition region where this phenomenon is still present.

With the 7-ray information, the measured momentum distribution of the 14C

residues was decomposed into the two components corresponding to the 1”1C fragments

47

in the ground state and in the excited states 1‘, 0’, shown in Fig. 4.2.

In particular, the momentum distribution of the residues in the 1‘30” states
was obtained by requiring a coincidence with 7 rays in the energy range between
4.4 and 6.4 MeV, where the dominating contribution to the 7 spectrum is given by
the 6.09 MeV response curve. The 6.09 MeV level is fed by the 6.90 MeV level,
therefore the obtained distribution includes the contribution from the 0‘ state as
well. The resulting distribution was corrected for the additional contribution from
the continuum. This was extracted from events in coincidence with 7 rays of energies
between 1 and 2.5 MeV, where the continuum is the major component of the spectrum.
The momentum distribution of the ground state residues was obtained by subtracting
the excited states momentum distribution from the inclusive. Therefore, it includes
also the contribution from the 2+ state. However, its intensity is very small and
within the uncertainty on the ground state intensity, thus it. was neglected. A clean
discrimination between the momentum distributions of the individual excited states
in 14C was not possible due to the presence of a 7 ray at 0.98 MeV, interpreted as
arising from the deexcitation of the 1+ state in 8Li produced in the target. This is
demonstrated in Fig. 4.3, where the laboratory and c.m. spectra are plotted together,
showing how the 0.98 MeV peak is smeared out after Doppler correction, extending
under the 0.81 MeV peak. A peak at that energy was observed also in the laboratory

system spectra from the 12Be and 16C data, as shown in Fig. 4.4.

A comparison with the calculated curves for K = 0,1 clearly indicates the s—wave
character of the ground state distribution and the p—wave character of the excited
states of 14C, as expected (Fig. 4.2). This result reaffirms the spectroscopic power of

one-neutron knockout reactions.

However, while the momentum distribution of the excited states is in good agree-

48

 

1500 I 1 I

 

15 14 ,- -.
( C, C .) x x
gs. , O x 1" 0
’I ‘ —
I s
I s
I s
2 I O \
U I \
3 1000— ’l \\ _.
\
Q) I, \
O ’ \
‘< ,’ 0 \
.0 I . \
E I \
I \
h—d I \
= I \
ea ' ' ‘~
’ O
I \
B 500- ’l \‘ —*
13 , ~
’I . . ‘\
I, s‘
” . . ‘s‘
x’ 0 x
\
x'o' ' ' ‘~
4!. Q .
L l l
O [ l l
200

ISO

do/dl’II [mb/(GeV/c)]

50

 

 

 

l L

 

4.4 4.45 4.5 4.55 4.6
P” (”C) [GeV/c]

Figure 4.2: Longitudinal momentum distributions of ”C9,, (top), and of ”Cl-,0—
(bottom), after one-neutron removal from 15C. The curves are calculated eikonal

momentum distributions for € 2 0, 1.

49

0.3

 

0.2 *

Intensity [Counts/Fragment/MeV]
O
'o

0.04 ~

0.03 ~

 

0.02 ‘

c.m.

0 0 lab.

 

1 I l l 1 L L L 1 1 l l l 1 l J

 

0.2

0.4 0.6 0.8 1 1.2
E7 [MeV]

Figure 4.3: Laboratory and c.m. system 7-ray spectra measured in coincidence with
14C residues produced in one-neutron removal reactions on 15C. The 0.98 MeV peak in
the laboratory spectrum corresponds to a 7 ray emitted by 8Li, produced in reactions
in the target, and is smeard out by the Doppler correction in the c.m. spectrum.

 

0-3 I : T I T l

“ +17

“97+ 0 (”Be, “Be)

0 (16C, 15C)

_o
l

.o .o
8

Intensity [Counts/Fragment/MeV]
§ § 53
.__...._..
___.__
_.._

p
C
UI

 

 

A
1 I ,-

0.03 ~ (b

 

 

 

 

 

 

0.02 1 l 1 1 l 1
0.4 0.6 0.8 l l .2

E71511 [MeV]

Figure 4.4: Laboratory 7-ray Spectra from one-neutron removal reactions on 12Be and
16C. The 0.98 MeV peak coresponds to a 7 ray emitted by 8Li, produced in reactions
in the target.

ment with the i = 1 calculated curve, small deviations in the shape of the ground
state momentum distribution, especially at low energies, are apparent. These could
be related to the use of the sudden approximation (which does not require energy
conservation), on which the eikonal model calculations rely. Recently, fully quan-
tum mechanical calculations have been developed to investigate this possibility with
a coupled discretized continuum channels approach (CDCC) [TosOO]. In fact, with
this approach the calculated diffraction momentum distributions appear asymmetric
and can explain the observed deviations. This effect has been observed also in the

one-neutron removal data from 11Be, and has been investigated [TosOO].

An interesting feature of the CDCC momentum ditributions is that their asymme-
try becomes more pronounced progressing from small to large scattering angles. This
effect is present in the data as well, as shown in Figs. 4.6 and 4.7. Four momentum
distributions for the 14C residues in their ground state were obtained by grouping the
data according to scattering angle, from 0° to 4°, on 1° intervals. The size of the
intervals was limited by the experimental angular resolution, which was estimated to
be 1°, as discussed in detail in the second part of section A.1.2. Their asymmetry
increases with the the scattering angle as predicted by CDCC. However, the measured
momentum distributions include both diffraction and stripping contributions, which
cannot be separated, as the experiment was not designed to detect the outcoming

neutrons.
Two assumptions on the stripping distribution can be made:
(i): it has the same shape as the diffractive;
(ii): it has the shape calculated in the eikonal model.
The calculated ground state cross section has been compared to the measured

value in both 15C and 11Be. For this purpose the measured cross section has been

52

divided into stripping and diffraction components, in the ratio of the predictions of
the eikonal calculation. The obtained diffraction cross section is in good agreement
with the results from the CDCC calculations. Fig. 4.5 shows the experimental and

CDCC diffraction cross sections in each angle bin.

The comparison between the measured and calculated distributions using assump-
tion (1) and assumption (ii) is illustrated in Fig. 4.6 for 15C and in Fig. 4.7 for 11Be.
When assumption (1) is taken, the curves superimposed to the data represent the
CDCC calculations, scaled to the measured cross sections. For assumption (ii) the
stripping (eikonal) and diffraction (CDCC) curves were added in the proportions

obtained from Fig. 4.5.

The comparison with the data suggests that the stripping distribution has a similar
shape to the one corresponding to diffraction. In fact, with assumption (i) a good
description of the data is obtained at all scattering angles. Moreover, the shape of
the measured inclusive ground state momentum distribution is similar to the one
calculated for the diffractive part of the cross section in both 15C and 11Be. (Figs. 4.8
and 4.9). (The instrumental resolution in momentum is not included in the theoretical
curves. However, its value measured for the incident 15C beam was of 6 MeV/c

FWHM and the consequent broadening of the folded distributions would be less than

1%..)

With assumption (ii), the contribution from the eikonal stripping momentum dis-
tribution introduces sizable deviations from the observed behavior, especially at scat-

tering angles between 1° and 3°.

Besides improving the interpretation of the reaction mechanism, these calcula-
tions prove the accuracy of the experimental ground state momentum distributions,

obtained by subtracting the events in coincidence with 7 rays from the inclusive

53

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. 15
l C
60
* l — EXP
_ O CDCC
40 '" T
t
. O '1'
20 —
3 ~ 0
T
E 7 ;
.5 e 1
g 0 1- 1 1 J 1 1 g L 111 1 #
(I)
:2 125 L I Be
8 1
U h l l — Exp.
100 -
T O CDCC
75 — _|_
_ O
50 h
_ . T
25 ~ 6
_ .____.____

65C [deg]

Figure 4.5: Experimental cross sections and calculated diffraction cross sections with
CDCC, for 15C (top) and 11Be (bottom).

 

    
 
 

 
  

 

 

E 0 0-1 5

I I 1-2 2

7 A 2‘3 .1

105:? 0 3-4 5

E 0 5

:1» ’ - :

104 ‘ '1

:5 I

> -

8 103:- '2

1,3 E E

‘2 ' 1 1 1 1 1 1 ‘

: 106 :- 9sc [deg] 1-«

9: 5 o 0-1 5

% : I 12 :

5 - A 2-3 .-
73 105 O 34

    
 
 
    

I IIIIIIII
l Llill I

 

 

104

L lJJlll

CDCC + Eikonal

1 l 1 l 1 l
4.4 4.45 4.5 4.55

P”(MCg's.) [GeV/c]

l I 111111

 

 

 

Figure 4.6: Longitudinal momentum distributions of the 14C9,, at different scattering
angles. Top: The curves are calculated with CDCC, and scaled to reproduce the
measured cross sections. Bottom: The curves are the sum of the calculated CDCC
distributions for the diffractive part and the eikonal distributions for the stripping
part.

 

 

 

  
 
 
 
 
 
 

 

 

    

 

_ l ‘ l ' l ' l_1
106 _05c [deg] 4
E a 0-1 1
- I 1-2 . f
C A 2-3 . .
105 E“ 0 3-4 '2
: I
’ I 1
- .
10“
g" :n
d) 3 ¢
0 10 ,— CDCC 1, _:
3 ~ .
1.8.. . IT ' 1V 1 l i I 4
g 10‘5 56“ [deg] -:
= E o 0-1
g - I 1-2 :
E, _ A 2-3
105 O Q?
104

 

lo 3 CDCC + Eikonal f 2 T

1 . 1 . 1 . 7 -
3.25 3.3 3.35 3.4

P”(10Beg.s) [GeV/c]

Figure 4.7: Longitudinal momentum distributions of the ”Beg,“ at different scatter-
ing angles. Top: The curves are calculated with CDCC, and scaled to reproduce the
measured cross sections. Bottom: The curves are the sum of the calculated CDCC
distributions for the diffractive part and the eikonal distributions for the stripping
part.

 

l

l250

l

1000

do/dPII [mb/(GeV/c)]

' 1 1 1
15 14
( C, Cge.) — CDCC
x "a
----- Eikonal
:' 0
500 " '1' 1‘
‘. o
.. " \‘
o I: ‘
250 _ ,’
| . OI,
o
.. C I"
1 .

 

 

~..
‘-

 

4.5
P”(14Cg.s‘) [GeV/c]

cross section, the eikonal curve is arbitrarily scaled.

Figure 4.8: Longitudinal momentum distribution of the 1“C9,, residues from one-
neutron removal reactions on 15C. The curves are CDCC (solid line) and eikonal

(dashed line) calculations. The CDCC curve is scaled to reproduce the measured

 

 

 

 

4000 I I I I I I I I
_ 11 10 .
( 136» Beg...) — CDCC
)- ’.‘ cl
‘1, ----- Eikonal
1- .. “‘ -1
3000 — : ‘. —
P .0. 1
1'
7: ’ ,~' ‘
o 0
\ .‘
> _ - ,
d) f ,
O ; '.
i“ ,-
9 2000 1- . 1, -
.5. ;' 1
1a.: - .' i, .
'U ; .
\ - |
b ,' .‘
“c _ .' z .
l000 ’— i ‘0 —‘
I. ‘\‘
I" ‘\‘
I" “‘\
1- ”! \‘4. fl
.- . ,"'l ‘“\ .
. “"" 7“..~
0 , ........ 1' . 1 . 1 . 1
3.25 3.3 3.35 3.4
10
P“( Begs) [GeV/c]

Figure 4.9: Longitudinal momentum distribution of the ”Beg, residues from one-
neutron removal reactions on 11Be. The curves are CDCC (solid line) and eikonal

(dashed line) calculations. The CDCC curve is scaled to reproduce the measured

cross section, the eikonal curve is arbitrarily scaled.

distributions.

Finally, an interesting feature of the ground state momentum distributions for
15C and 11Be is that they are identical in shape, when appropriately transformed into
the center of mass system, taking into account the relativistic 7 factor, provided that

the 11Be distribution is broadened by the square root of the ratio of their separation

 

 

energies (\/S,,(15C)/Sn(”Be) = \/1.218/0.504), as shown in Fig. 4.10. This is in
fact the expected dependence with separation energy of the width of the momentum
distribution for an s—-wave neutron and a Yukawa wave function. The wave function
has the form exp(—7‘/p)/r, where the decay length p = h/m, and 11 is the reduced
mass. The Fourier transform of this wave function gives a Lorentzian distribution of
width F = 211/ p = 2%. The result shows a consistency between the data, and

the systematics of the asymmetry effect.

59

 

 

 

 

' T I I l V I f l r r I r r 7 Tfi r l I 1 r 1 If I r
b 1
. 15 o q
. C185 0 9
1200 — o _
~ 0 “Beg_s_(x 0.3) 9 .
. o 0. fl
1000 — ...
l" . d
_ o o E
L5 800 _ ' —
7» - o .
g o
\ 1' ..
.D
.9. ~ .
%= 600 — . _
\ __ o .
8 o
- 0 d
400 — . 9 _
i- o .1
1- ° -4
1. 0°. C _,
O
200 '" q . “
- o no ,
O C
'- 0. Po ..
. O o.“ O O o O .
9 o
0 4 L l l L 1 1 1 1 I 1 1 1 l 1 1 1 1 l 1 4 4 L l’nlol.o 0.
0.] —0.05 0 0.05 0.1
P” [GeV/c]

Figure 4.10: Projectile center of mass longitudinal momentum distributions of the
14C and 10Be residues in their ground state after one-neutron removal from 15C and
10Be. The 10Be distribution has been broadened by the square root of the ratio of the
separation energies.

60

Chapter 5

The structure of the heavier
carbon isotopes

5.1 Spectrosc0py of 16C

5.1.1 Previous theoretical and experimental work

The structure of the low-lying levels in 16C has been investigated [For77, For78,
Ser78, Ba177] in the reactions 14C(t,p)16C and 14C(t,p7)16C. Tilley et al. [Ti193] discuss
properties and the level scheme, that will be referred to in more detail in section 5.2.
The structure of 16C is expected to correspond to 14C (8) 180. Since 14C is a near-magic
nucleus, the simplest conjecture is that the neutron pair of 16C should be similar
to that of 180, which has a relatively pure s2+al2 two-particle configuration with
spectroscopic factors [Law76] C25(151/2) : 0.38 and 025(0d5/2) : 1.44, and where
the missing part is a 4p—2h collective contribution. The wave functions in Ref. [Law76]
were later used in the study of the mirror nucleus 18Ne [She98] and successfully
reproduced the observed Coulomb energy shifts. When the LSF model [Law76] is
applied to 16C, the s2 and al2 components are about equal and the spectroscopic factors
would be about 023(131/2) = 0.93 and C25(0d5/2) : 1.07 [For78]. As discussed in
the next section, the origin of the difference between the 180 and 16C spectroscopic
factors arises mainly from the change in single-particle energies.

61

The experimental situation for the next lighter N = 10 isotone 14Be is interesting
but less clear experimentally. The breakup reaction to 12Be [Zah93] leads to a narrow
momentum distribution indicating a halo structure, presumably arising from a sub-
stantial 32 component. The same is suggested by the beta-delayed neutron decay of
14Be, which shows [Be195, Ber99] an almost superallowed branch (log ft m 3.7) to a
1+ state (not directly observed) at 1-2 MeV excitation energy. The theory of the 14Be
beta decay has been discussed by Timofeyuk and Descouvemont [Tim96]. A recent 2n
pairing model calculation [Lab99] suggests, somewhat surprisingly, a negative-parity

ground state of 13Be and spectroscopic factors of 0.9, 0.6, and 0.5 for single-neutron

l

2‘ ground state, the %+ and 3+ s~-—- and d—states, respectively.

breakup to the

5.1.2 Present shell-model results

For 180, WBP and WBT are equivalent to the sd—shell USD results with 025(131/2) =
0.30 and C2S(0d5/2) = 1.58 (with the remaining in CZS(0d3/2) = 0.12 leading to
a state at high excitation energy). For 16C, WBP gives C25(131/2) = 0.60 and
CQS(0d5/2) = 1.23, and WBT gives 025(131/2) = 0.78 and 023(0d5/2) = 1.07. One
reason for the difference can be related to the 15C spectrum with the 3+ excited
state at 0.38 MeV with WBP and at 0.66 MeV with WBT, compared to the exper-
imental energy at 0.74 MeV; and on this basis the WBT results are preferred. The
spectroscopic factors depend upon the spacing of the single-particle energies and, in
particular, the crossing of the single-particle energies between 170 (where the {r is
0.87 MeV above the 3+) and 15C, which gives rise to the large change between 180

and 16C.

62

5.1.3 Experimental results and discussion

The neutron knockout reaction on 16C (neutron separation energy 5,, = 4.25 MeV
[Au(193, Au(197]) leads to the two (only) bound levels of 15C, the %+ ground state and

the + state at 0.740 MeV [Aszl]. The latter has a long half-life (2.60 :1: 0.07 ns),

which combined with the high velocity of the residues causes the 7 rays to be emitted
at a mean distance of 34 cm from the center of the detectors, which are only 20 cm
long. This means that the Doppler correction scheme described in section 2.4 fails. In
fact, most 7 rays are emitted outside of the apparatus reducing the detected intensity.

Fig. 5.1 shows the 7-ray spectrum measured in coincidence with 15C residues without

the Doppler back-correction.

Fortunately, accurate estimates of the continuum distribution are available from
the previous experiments on llBe [AumOO] and 12Be [Nav00], cf. section 2.4.2. The
16C data in Fig.5.1 show a clear excess above the 12Be background in the region
0.5-0.8 MeV. A simulated response curve of the NaI array to the isomeric decay was
generated in a Monte Carlo procedure in which the 7 events were assumed to appear
downstream along the beam axis with the appropriate exponential distribution. The
response was then simulated by the GEANT code, as before. The resulting efficiency
turned out to be reduced by a factor of 4 as compared to instantaneous emission.
A combined fit to the components in Fig. 5.1 gave an absolute branch of 30 :1: 10%
to the excited 3+ state with the error determined by the statistics alone. However,
this result is almost certainly an underestimate, corresponding to an overestimate of
the detection efficiency. This is because the simulation did not include the size and
divergence of the incoming beam and the angular spread of the outgoing residues,
nor did it include the way that these affect the absorption of the low-energy 7 rays

emitted in the backward direction. This analysis provides a semi-quantitative but

63

 

 

 

 

 

Intensity [Counts/Fragment/MeV]

 

 

 

 

 

 

 

Ev [MeV]

Figure 5.1: Laboratory system 7-ray spectra from 9Be(16C, 15C+7)X (filled circles)
and 9Be(lzBe, 11Be+7)X (open circles), normalized to the number of projectile frag—
ments. The solid lines are simulated response functions for the 0.74 MeV 7 ray from
the decay in flight of 15C“ and for the 0.32 MeV 7 ray from 11Be". The inset shows
experimental 7-ray spectra from 11Be and 15C for E, > 1 MeV. The “tails” above the
7 lines, the only discrete lines present, are the continuum distributions discussed in
the text.

64

Table 5.1: Partial cross sections 0 and branching ratios b for the final states I 7’ in the
residues produced in 9Be(16C,15C)X at E = 62 MeV/nucleon. The theoretical cross
sections a”, are calculated from Eq. (3.1) using the WBP spectroscopic factors C 2S
and the single-particle cross sections asp. The values of 0m include overlap factors of
0.897 and 0.948, respectively (see section 3.2).

 

 

(16C , l ’C ) E 1’r £1 C 25 05,, a”, 0m, bu, be“,
[MeV] [’11 [h] [mbl [mb] 1me 1%] 1%]
. “L 0 0.60 64 34 3216 44 42 i 6
+

1

2
0.74 3 2 1.23 37 43 45:1:7 56 58i6
Um: 77 77:1:9

 

 

 

 

 

 

 

 

direct indication of the contribution of the appearance of the (12 component in the
16C ground state. The statistics in Fig. 5.1 are insufficient to permit an extraction of
the momentum distribution to the excited level, but since the assignments in the 15C

nucleus are well established, this is probably of little importance.

A more accurate estimate of the branching ratio was obtained from the mea-
sured inclusive longitudinal momentum distribution of the residues shown in Fig. 5.2.
Experience from previous experiments has shown that the shapes of the theoretically
calculated momentum distributions are well reproduced by experiment. As these,
furthermore, are very different for s-- and d—state knockout, it is easy to arrive at the
overall fit (envelope) shown in the figure. The criterion adopted to fix the limits of
the fit was to consider the momentum acceptance range of i2.5%. The geometrical
loss for the inclusive spectrum was estimated to be 2% by the method described in
section A.1. The extrapolation of the envelope gave momentum acceptance losses of
4%. Both corrections were applied to the measurement to give the total (inclusive)

cross section of 77 i 9 mb reported in Table 5.1.

This agrees reasonably well with the inclusive one-neutron removal cross section

of 65 :1: 6 mb for 16C recently measured at 55 MeV/nucleon by Sauvan et a1. [SauOO]

65

dCI/dPll [mb/(GeV/c)]

Figure 5.2: Inclusive longitudinal momentum distribution of the 15C fragments pro-
duced in one-neutron removal reactions of 16C on a 9Be target. The experimental
errors are smaller than the size of the points. The full-drawn line is a fit with the
short-dashed lines showing the individual theoretical components.
indicate the error limits allowed by the fit. The broad component corresponds to
58 i 6% d—wave and the narrow one to 42 q: 6% s—wave, with the contributions from
the (unobserved) tails of the d—wave distribution taken into account.

200

 

h

 

P"('5C) [GeV/c]

66

The thin lines

 

and their calculated value of 75 mb is essentially identical with ours. The resulting
intensity of the broad (d—wave) component gave 58 :l: 6% excited state contribution
to the cross section. This is twice the value obtained from the analysis of coincident
7 rays and suggests that the simulation of the delayed events overestimated the 7

detection efficiency.

Table 5.1 compares the measured partial cross sections with the theoretical results
obtained as the product of the spectroscopic factor and the single-particle cross section
following Eq. 3.1. For the case of 16C, it was also necessary, as in [NaVOO], to take
into account the radial mismatch factor arising from the difference in single-particle
potential between the two nuclei. This is not included in the shell-model calculations
used here. The effective neutron separation energies (to which the wave function must
be adjusted) are for 15”C, respectively, 1.22 and 4.25 MeV for the s—state and 0.48
and 4.99 MeV for the d—state. In the table the correction has been included in the
theoretical partial cross sections with the values 0.897 and 0.948. The non-overlap
effect is less important for the 1.7 = ‘2 state, which is already spatially constrained
by the angular momentum barrier. For the direct comparison with the shell-model
occupancies of the 16C neutron pair, the experimental cross sections must be divided
by the corresponding single-particle cross sections and mismatch factors, to obtain
the spectroscopic factors 025* corresponding to the quantities defined in [Nav00].

exp

The resulting values and experimental error limits are O.56:l:0.10 (g) and 1.28:1:0.20

(V

2 ). They are in excellent agreement with the theoretical spectroscopic factors 0.60

and 1.23, respectively.

67

5.2 The projectile ”(3
5.2.1 Previous theoretical and experimental work

The levels of 17C have been studied by Fifield et al. [Fif82] in the multi-nucleon transfer
reaction 48Ca(180,17C)“9Ti. The lowest state, interpreted as the ground state, has a
neutron separation energy of 0.729 :1: 0.018 MeV [Aud93, Aud97] based on this and
a previous measurement. F ifield et al. found a cross section five times larger to

a level at 0.395 MeV. The analysis by Warburton and Millener [VVar89] interprets

+

this as the g state, expected to be favored in a two—step transfer reaction; see

for example [Cat89]. Their analysis of the beta decay data for 17N [Duf86, Cur86]

é

2+ ground-state assignment to be “eliminated

supports this conclusion and allows the
model-independently”. Of the remaining likely spin-parity assignments for the ground
+

% and %+, they prefer the latter, but both remain “quite possible”. Several

state,
theoretical papers [Sme99, Cha00, DesOO] have dealt with the one-neutron removal

reactions on 17C.

5.2.2 Present shell-model results

As discussed in section 3.1, the W BP and WBT interactions both present a triplet
of low-lying states for 17C. The present experimental results are in agreement only
with the spectroscopic factors based upon the 3+ ground state. The WBP interaction
gives a %+ ground state and this will be used for further comparisons. However, for
the given spin-parity of %+ the spectroscopic factors are very similar between WBP

and WBT.

With the WBP interaction, the largest components of the 17C 3+ state are 32%
fOI' [(Op3/2)8, (0])1/2)2, (0(15/2)3] and 31% {OT [(0p3/2)8, (0])1/2)2, (0d5/2)2, (181/2)l], \VlI-h
the remaining 37% in small components. The 0(13/2 component is small resulting in

68

CQS(0(13/2) : 0.035 for the 17C 3+ state to the 16C 0+ ground state. As discussed in
the next section, the strongest 5d spectroscopic factors are to the excited 2+ state in

16C. The consequences and interpretation of this unusual situation will be discussed.

5.2.3 Experimental results

The Doppler-corrected 7-ray spectrum from the decay of the 16C residues produced
in one-neutron knockout reactions from 17C is shown in Fig. 5.3. The simplified level

scheme of 16C, based on [Bal77, Ti193], is sketched in Fig. 5.5.

The 7 peak at 1.77 MeV arises in the decay from the first 2+ level at 1.77 MeV to
the 16C ground state. The broad peak near 2.3 MeV is assumed to represent decays
from the three (unresolved) levels near 4.1 MeV to the 2+(1.77 MeV) state. The
background was parameterized as an exponential, as in Fig.5.1 and Refs. [Aum00,
Nav00]. The resulting curve was compared to the measured continuum from the 12Be
and 16C data (section 2.4.2, Fig. 2.7) and found to be in agreement, as shown in

Fig. 5.4.

The total experimental spectrum was fitted with the individual response functions
obtained from the Monte Carlo simulations superimposed on the background. This

leads to the branching ratios be”, given in Table 5.2.

The branch to the ground state of 19i9% was obtained from an intensity balance;
since this relatively large cross section disagrees with the shell-model calculations, it
has been examined whether the number would be consistent with zero. However,
from the following analysis a substantial branch to this state has been found to be

definitely present.

An alternative explanation for the relatively strong cross section to the ground

state would be the presence of unobserved 7 rays, which would distort the intensity

69

 

 

I

I

)t

 

-1
10 l

 

 

 

rrrr
*
*

 

 

4 111111

 

 

C

10 "

l l l l

Intensity[Counts/Fragments/MeV]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IO 7 "u

 

 

 

 

 

 

 

 

 

 

Figure 5.3: Doppler-corrected 7-ray spectrum measured in 9Be(17C, l6C+7)X. The
black curve is a fit to the spectrum using an exponential curve for the background
and response functions (grey curves) for each of the 7-ray transitions shown in the
simplified level scheme of Fig. 5.5. The dashed line corresponds to an estimated upper
limit of 2% for the direct transition from the J = 2 level at 3.99 MeV to the ground
state. Inset: 7 spectrum gated on the transitions between the levels at 2 4 MeV
and the 2+ level at 1.77 MeV. The spectrum was fitted using the same procedure as
described above.

70

 

 

 

 

I I I T T T I I I I I I I I I I' I I I
+ .1
r d
~ 1
7
h 0.. . 2 l d
'9' .o 0 (1 Be, 6C)
-1
10 L- '2'
I 9 I
Z f ... :
E .- ‘
a _ ~ 0 .
E -,
DD 0
e -.
E ° _ e 1
8 0-0
5 e
O -2
810 :- ¢¢ 1
b " ¢¢ + -
.5; '- -4
8 . c -
r:
- l ”+ $ f+ 'l
l l f n .. '
-3
'0 r ‘ .a
f
l l l l L A l I l l L l l l l I 1 l l
l 2 3 4 5 6
E’7[MeV]

Figure 5.4: Comparison between the 7-ray spectrum from the excited 16C residues
after one-neutron removal reactions on 17C, and the measured continuum distribution
extracted from the analysis of the 12Be and 16C data. The solid line is the result from
the fit of the spectrum.

71

 

Table 5.2: Partial cross sections a and branching ratios (2 for the final states I1r in the
residues produced in 9Be(l7C,16C)X at E = 62 MeV/ nucleon. The theoretical cross
sections at}, are calculated from Eq. (3.1) using the WBP spectroscopic factors C 25
and the single-particle cross sections 03p. * The components of this group have been
analyzed together. We identify them with three states predicted by theory in the
range 4.9-5.7 MeV. The main contributions were given by at least two components.

 

 

 

 

 

(WC, 16C) E I’r 6’ C25 05,, at}, am, bu, beg,p
[MeV] [711 [Ill [mbl [mbl [me 1%] 1%]

0.0 0+ 2 0.03 53 2 22i11 2 19 :t 9

1.77 2+ 0 0.16 75 12 16:1:7 12 14 :t 6

2 1.44 37 53 44i11 53 38 :t 8

sum 65 60:1:12 65 52 :l: 8

 

4.1" 2,3”),4‘L 0 0.22 50 11 221:2 11 2 :t 2
2 0.76 29 22 31:1:7 22 27 :l: 5
sum 33 33:1:7 33 29 :l: 5
(7,0, 100 115i14

 

 

 

 

 

balance. Two possibilities were examined. The first would be a small direct branch
to the ground state from the J = 2 level near 4 MeV. An upper limit of 2% was
estimated as shown by the dashed curve in Fig. 5.3. This is consistent with shell-
model calculations. Another possibility would be that part of the intensity in the
broad peak near 2.3 MeV would arise from a state of this energy decaying directly to
the ground state. (No such state is known or expected [Ti193].) Such a 7 ray clearly
would not be in coincidence with the 1.77 7 ray, while the other 2.3 MeV 7 rays are
followed by this to 100%. This possibility has been examined by searching for triple
coincidences (residue-7—7) in the data. With a gate on E7 2 2.3 MeV the spectrum,
shown as an inset in Fig. 5.3, was obtained and fitted with the response function for
the 1.77 MeV 7 ray and an exponential background. The result was then compared
on a quantitative basis with an event-by-event simulation generated with a 7 ray from

the first excited level at 1.77 l\~"IeV produced simultaneously with a 7 ray from one of

72

the three levels at E :3 4.1 MeV. The ratio between the experimental intensity for the
coincident events and the intensity obtained in the simulation was 105 :l: 15%. The
error limit would allow for at most a 5% (absolute) feeding of the ground state via
such a mechanism, rather unlikely in the first place. These results support the level
scheme assumed in the right side of Fig. 5.5 and the branching ratios for the knockout

cross section given in Tables 5.2 and 5.3.

Yet another experimental effect that, at least in principle, might call the normal-
ization into question and explain the enhanced cross section to the 16C ground state
would be the presence of the spin-parity %+ state of 17C as a contaminant isomer in
the beam. This state has a reaction cross section that goes predominantly to the
ground state of the residue; see Table 5.3. In connection with the analysis of the 16C
experiment (section 5.1.3), where a half-life of a few ns gave a mean flight path for
the residues of 0.34 m, a half-life that were a factor 100 or more longer would allow
isomers produced in the primary production target of the A1200 fragment separa-

tor to react with the target and reach the S800. Since the position of the + state

_1_
2
is unknown but presumably low in energy, such a long half-life is entirely possible.
However, as will be discussed below, the momentum distribution belonging to this
component would have a very characteristic 6 = 0 shape in contradiction with the

experiment, which gives l’ = 2, as shown in Fig. 5.5. Hence also this explanation can

be excluded.

The inclusive longitudinal momentum distribution of the 16C residues was mea-
sured and found to be consistent with earlier measurements [Baz95, Ba298, Bau98].
The estimated angular acceptance correction was 3.6%. From the 7 coincidence in-
formation, the distribution could be separated into three components corresponding

to feeding of the ground state, the 2+ level, and the 4.1 MeV group of levels (Fig. 5.5).

73

 

I fi l I I I I l I I I I I I I In EY [MeV]
_ (a)

 

 

 

 

 

 

 

 

   

 

\399
§ i
> :
Q) n
g E
3 s
E. —g— 1.77
g E
b i
“o 1
100 —-—) 0+ ___i..':"_ 0

 

 

 

14Ll_L+:L 1 L4; L414

5.3 5.4 5.5
P"(16C) [GeV/c]

Figure 5.5: Longitudinal momentum distributions corresponding to the states indi-
cated in the simplified level scheme of 16C. (a), (b): The solid curves are the calculated
momentum distributions with a mixture of s— and d—waves shown as dashed and dot-
dashed lines, respectively (8% s and 92% d in (a), 26% s and 74% d in (b)). (c): The
solid curve is the calculated momentum distribution of a pure d—wave.

74

In view of the relatively large error on the intensity of the ground-state branch, ob-
tained by subtracting an 81% correction from the inclusive Spectrum, it is important
to note that the shape has been observed to remain stable within the error limits given.
The reason for this is that the shapes for the excited levels are very similar, all three
being dominated by f = 2 components. The distributions were fitted with theoretical
momentum distributions as described in section 3.2 assuming 8 = 0,2 components,
in the momentum range corresponding to the :1: 2.5% instrumental momentum ac-
ceptance. The most interesting result was found for the distributions to the excited
states, which are an admixture of s— and d~waves, with a dominant d-wave character
in both cases (92 :1: 8% for the 4.1 MeV group of levels, and 74 i 10% for the 2+
state). This is the second case of a cross section with mixed f-values observed in our
experiments. (The reaction 9Be(1“B,1:‘Bgs)X was found [GuiOO] to be predominantly
(3 = 0 with an 11 i 3% E = 2 admixture.) Finally, the cross section to the 16C ground
state is essentially f = 2. The total inclusive cross section, after extrapolation of mo-
mentum acceptance losses (estimated to be 2.4%), is 115:1: 14 mb, about two standard
deviations above the value of 84 :1:-9 mb measured [SauOO] at 49 MeV/ nucleon. The
information supplied by the momentum distributions has made it possible to subdi-
vide the measured cross section to the 1.77 MeV level into the 6 components given in
Tables 5.2 and 5.3. This evidence is now compared with the theoretical calculations.

- . - ~ . + , +
The llterature basmally leaves two OptIOIIS for the 1‘C spin, % and g , as summa-

rized in the previous subsection. The absence of an 6 = 0 reaction to the 16C ground
state is basically enough to fix the spin as %+ This is actually the assignment that is
proposed here and the one used in the discussion of the structure and spectroscopic
factors in Table 5.2. However, to demonstrate that the spectroscopic factors mea-
sured in the present work allow on their own a unique determination of the spin, the

predicted cross sections and branching ratios for assumed spin-parity assignments of

75

Table 5.3: The reaction 9Be(17C,16C)X at E = 62 MeV/nucleon, for a 17C ground-
state spin J7r = 5/2+ or 1/2+, see Table 5.2. * The components of this group have
been analyzed together. We identify them with three states predicted by theory in the
range 4.9-5.7 MeV. The main contributions were given by at least two components.

 

 

 

 

 

Jn(17C) E1M61/1 117 6 C25 03p 0th 0”,, bth be“,
[MeV] in] 1711 1me [mbl [mbl 1%] 1%]
53* 0.0 0+ 2 0.70 53 37 22111 35 19 :t 9
1.77 2+ 0 0.10 75 8 1617 7 14 :1: 6
2 0.22 37 8 44111 7 38 :1: 8
sum 16 60i12 14 52 a: 8

 

4.1* 2,3(+),4+ 0 0.39 50 20 2:1:2 19 2i2
2 1.16 29 34 31i7 32 27:1:5

 

 

 

 

 

 

 

 

 

 

 

 

sum 54 33i7 51 29 d: 5
0,0, 107 115:1:14
5+ 0.0 0+ 0 0.64 148 95 22111 57 19 i 9
1.77 2+ 2 0.39 37 14 60:1:12 9 52 :t 8
3.03 0+ 0 0.29 59 17 — 10 —
4.1”“ 2, 31+) 2 1.39 29 40 33i7 24 29 :t 5
am 166 115:1: 14
1+ 5+ -
5 and 5 are shown 1n Table 5.3.

The former is clearly excluded by the predicted dominance of s—wave knockout to
the ground state with only weak branches to the excited states, both contradicted by

+, also listed in Table 5.3, are also in

experiment. Calculations for the case J1r = %
disagreement with experiment, which has the main cross section to the 1.77 MeV 2+
level and smaller branches to 0 and z 4.1 MeV. The pattern predicted for an initial
spin-parity of 3+ is exactly opposite. Only the J7r = %+ assignment for 17C explains
that the main part of the cross section goes to the 1.77 MeV 2+ and about half as
much to the 4.1 MeV group. Contrary to the statement made in [Sau00], an f : 2
momentum distribution and the inclusive cross section are insufficient for determining
§+

the ground-state spin and parity of 17C as J’r = .2 . The only remaining difficulty

76

is that the cross section to the ground state of 16C of 22 mb is roughly one order of
magnitude larger than expected. It has been argued above that this does not appear

to be an experimental problem.

It is probably useful at this point to sum up in simple language some of the struc-
tural information conveyed by the 17C results in Table 5.2. The -g-+ ground state may
be thought of as having three components, of which the main one is 0015/2 <8) [Orig/212+.
This accounts for the dominant 6 = 2 knockout to the 2+ state. The smaller 6 = 0
component to the same state arises from a small admixture of 181 )2 ® [0dg/2]2+. There
is excellent agreement between experiment and theory for both components. The
predicted small cross section to the 16C ground state comes from a small amount of
0d3 mg [011% /2]0+ in the 17C ground state and a simple explanation for the experimental
result would be that theory for some reason underestimates this component. There

are, however, other possibilities.

The theory for calculating the cross sections, outlined in section 3.2, assumes ex-
plicitly that the only reaction mechanism is the direct removal of a bound nucleon
from a core of nucleons, which is otherwise a spectator. It is, however, possible to have
contributions from other (higher order) mechanisms, such as the collective contribu-
tions, of order 10 mb, invoked to account for part of the (llBe,1°Be) cross sections
in [AumOO]. Another possibility, recently investigated by Al-Khalili [AIKOO] is to allow
the nucleon-target interaction to induce transitions between different single-particle
states or between different m components of the same state. This mechanism, for the
main Dds/2 ® [Orig/212+ component discussed here, requires a spin-flip 0+ recoupling of

the two unstripped neutrons, and is estimated to contribute less than 1 mb.

77

5.3 The projectile 19C

5.3.1 Previous experimental and theoretical work

The isotope 19C occupies a position in the nuclear chart similar to that of 11Be,
and it has attracted much interest as a possible second candidate for a well devel-
oped one-neutron halo state. This was suggested by the low adjusted value [Aud93]
of its one-neutron separation energy, 5,, = 0.16 :1: 0.11 MeV. This number repre-
sents the weighted average of measurements carried out at Los Alamos and GANIL
[Wou88, Orr91]. The value of 0.24 MeV often encountered in the literature includes,
in addition, two earlier and less precise measurements by the same groups in the
weighted average.The adjustment [Aud97] normally excludes such results from its
recommendation. Indirect evidence discussed below suggests that a value larger than

0.16 MeV, i.e., 0.5-1.0 MeV, is required to interpret the data in a consistent way.

Several previous experiments have investigated the structure of 19C by measuring
the longitudinal momentum distributions of the core fragments 18C [Ba295, Ba298,
Bau98]. The narrow widths observed in these experiments resemble those found for
11Be and were suggestive of a halo structure. They were, however, wider than the
adjusted neutron separation energy value would allow, which prompted speculations
that the 19C ground state is dominated by complex-structure components. Bazin
et al. [Baz95, Baz98], however, pointed out that the momentum distribution in the
Coulomb breakup of 19C could be understood if the neutron separation energy was
about 0.6 MeV. An experiment by Marques et al. [Mar96] observed neutrons from
19C breakup reactions in which neutrons were detected in coincidence with charged
fragments with charge five and lower. They found a broad component in the angular
distribution, which they associated with nucleus-nucleus collisions. A narrow compo-

nent with a Lorentzian width parameter F of 55 MeV/c was taken as evidence for a

78

halo neutron present as a spectator in the collision.

A recent experiment on the Coulomb dissociation of 19C on a lead target by Naka-
mura et al. [Nak99] represents a decisive step forward. They found that the angular
distribution of the decaying 18C+n system required a neutron separation energy 5,,
of 0.53 :1: 0.13 l\»’IeV. With this energy they could also understand the differential cross
section as a function of the relative energy of the dissociation products, which was
not the case for a value of 0.16 MeV. The absolute Coulomb cross section (neglect-
ing possible contributions to excited levels, discussed below) leads to a spectroscopic
factor of 0.67, thus showing that the dominant character of the 19C ground state is
131/2 <8) 0+. Since this analysis does not correct for branches to excited states, which
are expected to be present, also for the Coulomb part of the cross section, the results

should be taken as qualitative.

Interaction cross section measurements have also been reported for 19C on 12C at

960 MeV/ nucleon [Oza98]. The analysis of these data in a few-body Glauber theory

1+
2

approach has shown [Tos99a] that the measured a; are consistent with a J1r =
19C ground state, while excluding J7r = %+ and J" 2 3+ assignments. In particular,
the a, datum was found to be consistent with a dominant 131/2 (8 0+ configuration
for separation energy values ranging between 0.12 and 0.65 MeV, with spectroscopic
factors between 0.7 and 1.0. This result is consistent with the separation energy

value found from the Coulomb dissociation experiment [Nak99] and with the results

reported below.

A number of papers have discussed the structure and reactions of 19C, primarily
in the light of particle—core coupling models [Rid97, Rid98, Ban98, Sme99, Bon99,
Cha00, DesOO, KanOO]. Smedberg and Zhukov [Sme99] attempted to account for a

perceived difference in longitudinal momentum widths observed at 77 MeV / nucleon

79

[Ba295, Ba298] and at 910 MeV / nucleon [Bau98]. They invoked an additional reaction
mechanism involving an unspecified intermediate resonance just above the 18C-l-n
threshold. This hypothesis does not find support in the present work, where the 62
and 910 MeV/ nucleon data are analyzed in more detail below. In another analysis,
Kanungo et al. [KanOO] found it difficult to reconcile the 19C momentum distributions
at the two energies with the measured interaction cross section. As a remedy they

proposed that the core of 19C is considerably larger than that of free 18C.

5.3.2 Present shell-model results

Shell-model calculations [Bro88] predict the presence of three bound states above the
ground state of 18C. There are two 2+ levels at 2.1 MeV (observed experimentally at
1.62 :1: 0.02 [Fif82]) and at 3.4 MeV. A 0+ level at 4.0 MeV, just below the neutron
threshold of 4.2 MeV, is expected to decay by a cascade of 1.6 and 2.4 MeV 7 rays,
and it has a large 6 = 0 spectroscopic factor that would contribute noticeably to
the Coulomb cross section. There are two more states (2+, 3+) close in energy, near
4.9 MeV, which are included in the analysis, observing that the calculations tend
to overestimate the level energies by several hundreds of keV. The lowest levels in
19C are predicted to be ($20.00), (€10.05), ($040) with energies in MeV. In the
following the spin-parity assignment for the ground state is assumed to be established
by the Coulomb dissociation experiment [N ak99]. It will be seen below that the same
conclusion can be reached independently from our data. The spectroscopic factors
obtained with the WBP parameters [Bro88] are given in Table 5.4, where the 3.4 MeV

level, for which the spectroscopic factors are small (total 0.10 for f = 2), is left out.

As‘discussed in section 3.1, the WBP and W BT interactions both present a triplet
of low-lying states for 19C. The present experimental results are in agreement only

with the spectrosc0pic factors based upon the 315+ ground state. The WBP interaction

80

. + . . .
gives a % ground state and this Will be used for further comparisons. However, for
. . . + . . .
the given spin-parity of £- , the spectroscopic factors are very s1milar between WBP

and W BT.

With WBP the largest component of the 19C ground state + is 48% for the

5
configuration [(0p;,/2)8, ((1p1/2)2, (0d5/2)4, (1.91/2)1], with the remaining 52% in smaller
components. The [(0015 )2)“, (181/2)1] configuration appears at an excitation energy of
1.33 MeV in 210 with the USD interaction [Br088], and may be associated with an
experimental state observed at the same energy [Cat89]. In the framework of WBP

(and W BT) its energy is lowered in 19C due to the 1.6 MeV downward shift of the

181/2 state relative to 0(15/2 between 17O and 15C.

5.3.3 Experimental results

In spite of the low intensity of the incident 19C (2 0.5 — 1 particles/sec), enough
information was collected in different reaction channels to confirm that its ground
state is a well developed halo state. This evidence is discussed in the following, first
the 7-coincidence information leading to the ground state partial cross section and
exclusive momentum distribution, second the inclusive momentum spectrum, and,
third, the exclusive cross section for Coulomb dissociation to the 18C ground state.

The evidence combines to give a consistent set of parameters for the halo state.

The 7-ray spectrum in coincidence with projectile residues had too little statistics
for it to be analyzed with the peak—fitting procedure used in the case of 17C. Instead,
all 7 rays above 0.25 MeV were used as a tag identifying 7-coincident events and
a correction based on the average 7 efficiency to the residual noncoincident events
was applied. (This was the experimental approach taken previously in our work on
the phosphorus isotopes [Nav98].) The branching ratio listed in Table 5.4 and the

ground-state momentum distribution shown in Fig. 5.6 were then obtained as follows.

81

 

 

 

dO/dP” [mb/(GeV/c)]

 

 

 

 

I I T I I I I
§
I’ ‘
I
l
I
' I
' I
' I
' I
’ I
' I
' I
' I
' I
. I
. I
l
2000 - - ‘ d
I
I
l
0 ,~
I \ .
l I I .
I O I .
l I 9
l 9 ‘ ‘
' l I I
' 0 I I
. p I. I
I I
. ' ' S — 0'81-003 MeV
n I n
l I
n I I
o . I I
n ‘ I
l I
b ‘ q
I
I
-— —
1000 , .
l
l
O
l
l p “
D g I
a , I I
I g 9‘I
0 0 | I
a. ' 9 ‘I
l I I
' ---------- v ‘
I -. .O
a ' 0 ............ ‘I ‘- ‘
' ‘ ' o ‘ I. ~
I O ‘.
.o I ' I I .
.o l ' I I ~
' I ' I I 9‘
I- " I I . -1
I ’ " ‘ \
o I \
o , l ‘ I Q
I 0 ' ‘ I ~
o r 9 ‘ I '~
' I ’ I ‘ s
- I I
' ’ o ‘ I 9
. l
1 I ' “ \ .
o I ‘ I ~,
I o ‘ I s1
0 I ‘\
I
’ "‘ I I
o ' \ I
' I I
o , s \
o , s \
o ' s 9
o ' s I
'0 o I.
0 ~. g
‘‘‘‘‘‘‘
...........
.
0 .7. L I 1 I 1 I 1“
5.75 5.8 5.85 5.9 5.95

P”(18C (g.s.)) [GeV/c]

Figure 5.6: Longitudinal momentum distribution corresponding to the ground state of
the 18C residues after one-neutron removal from 19C on a 9Be target. The coincidences
with 7 rays have been used to correct the inclusive distribution for contributions from
excited levels. The momentum distribution corresponding to the extracted separation
energy Sn 2: 0.8 MeV is represented by the solid line. The dashed lines represent the
momentum distributions corresponding to separation energies of 0.5 and 1.1 MeV.
The dot-dashed curve is calculated for a d—state for a separation energy of 0.8 MeV.

82

Table 5.4: Partial cross sections a and branching ratios b for the final states I 7’ in
the residues produced in 9Be(19C,18C)X at E = 57 MeV/nucleon. The theoretical
cross sections 0”, are calculated from Eq. (3.1) using the WBP spectroscopic factors
023 and the single-particle cross sections 051,. The neutron separation energy of 19C
was assumed to be 0.8 MeV. * All excited states in 18C were analyzed as one group.
The energy 1.6 MeV for the 2+ is the experimental value; the WBP calculation gives
2.16 MeV.

 

 

(19C,18C) E I7r 6 C 25 03,, 0th 0e35,, bu, be“,
[MeV] [’1] [I’ll [mbl [mbl [mbl [%l [‘76]
0.0 0+ 0 0.58 136 79 148i50 46 56 :l: 9
1.6” 2+ 2 0.48 34 16
4.0“ 0+ 0 0.32 45 14
49* 2+,3+ 2 2.44 26 63
sum 93 116i45 54 44 :l: 11
am 172 264i80

 

 

 

 

 

 

 

 

As discussed in section 2.4.2, a structureless continuum distribution that depends
approximately exponentially on the energy, lies below the experimental 'y-ray spec-
tra. With an integral cutoff at 0.25 MeV, the intensity of this per fragment is ap-
proximately 9%, consistent in all the cases studied, and the corresponding coincident
momentum distribution is very similar to that of the inclusive spectrum. The average
detection efficiency for the excited levels was calculated in the GEANT Monte Carlo
simulations and gave the results of section A.1.4. The resulting branching ratio to
the ground state was 56 :l: 9%, consistent within the error with the result that would
be obtained if the theoretically predicted levels near 4 MeV were not populated, i.e.,
if they were above the neutron threshold. The branching ratio has been corrected
for the momentum acceptance of the spectrometer, which eliminates the “tails” of
the momentum distributions, especially for the 6 = 2 component. The corrections
are based on the theoretical momentum distributions for a neutron separation energy

of 0.8 MeV and assumed the theoretical branching ratios of the excited states. This

83

leads to the partial cross section of 148 i 50 mb, a large value typical of a halo state.

The analysis based on integral-bias 7' tagging also leads to the ground-state mo-
mentum distribution shown in Fig. 5.6, narrow and consistent with an s—state halo
structure. The calculations are based on the black-disk model discussed above. To-
gether with the large partial cross section this proves the ground-state spin-parity of
19C to be J7r = y, in agreement with the analyses of [Nak99, Oza98, T0399a]. The
width of the momentum distribution is quite sensitive to the separation energy; a

least-squares adjustment suggests a value 5,, = 0.8 i 0.3 MeV.

An alternative way of testing the dependence on the assumed value of 8,, is to
fit the inclusive momentum distribution, i.e., without the '7 coincidence requirement,
of the 18C residues. For this analysis the theoretical spectroscopic factors given in
Table 5.4 are assumed. The adjusted momentum distributions for the assumed values
5,. = 0.5 and 0.8 MeV are shown in Fig. 5.7. The result of the least-squares analysis
was 8,, = 0.65 :l: 0.15 MeV, which gives a branching ratio to the ground state of
48 i 2% in good agreement with the 56 i 9% obtained in the 7 coincidence analysis.
This result is in quantitative agreement with the coincidence analysis of the ground

state momentum distribution.

Another inclusive spectrum of the projectile residues has been obtained for 19C on
a 12C target at 910 MeV/ nucleon in a GSI experiment [Bau98]. This distribution is
close to identical to that of Fig. 5.7, and both are marginally consistent with that given
by [Ba295, Ba298], which has much poorer statistics. For this result, adjustment of
a theoretical momentum distribution similar to that in Fig. 5.7 leads to a somewhat
lower branch to the ground state, 40% as compared with the 48% found at our
energy in the same analysis. The smaller value is to a large extent accounted for by

smaller nucleon-nucleon cross sections and real-to-imaginary amplitudes at the higher

84

 

 

 

 

 

  
 

[mb/(GeV/c)]

do/dPII

8n = 0.8 MeV

  

 

 

-..
5 duo...
-

 

5.8 5.85 5.9
P"(‘8C) [GeV/c]

Figure 5.7: Inclusive longitudinal momentum distribution corresponding to 18C
residues after one-neutron removal from 19C on a 9Be target. The solid lines represent
the calculated inclusive momentum distributions corresponding to 8,, = 0.5 MeV and
5,, = 0.8 MeV obtained as a least-squares fit assuming the branching ratios given
by the theoretical spectroscopic factors of Table 5.4. These values represent approx-
imately the ilo limits of the allowed interval and correspond to a x2 of 8 and 9,
respectively for 14 degrees of freedom. (For 3,. = 0.3 MeV and 5,, = 1.1 MeV X2
increases to 18 and 14, respectively.) The dashed lines labeled with s and d rep-
resent the contributions from the K = 0,2 excited states for 5,, = 0.8 MeV. Their
contributions for Sn 2 0.5 MeV would be almost identical.

energy. An analysis for 910 MeV / nucleon on a carbon target and with the theoretical
spectroscopic factors of Table 5.4 reduces the theoretical ground-state branch to 40%
from the 46% obtained for a beryllium target at 57 MeV/nucleon. Therefore, the

experiment of Baumann et al. [Bau98], is entirely consistent with ours.

Finally, data that had been taken for reactions of 190 on a Au target were also used
to provide a constraint on the parameters. A detailed analysis is given in the second
part of section A.1.4. The measured inclusive (19C,18C) cross section on the gold
target was 1.35 :t 0.18 b at 56 MeV/ nucleon. This is close to the value of 1.34 :l: 0.12
b observed in the (19C,18C+n) channel on a lead target at 67 MeV/ nucleon [Nak99].
In order to compare more precisely, an (unobserved) absorptive part, assumed to be
0.15 b, should be added to the cross section of N akamura et al. [Nak99] (their estimate)
and their Coulomb part of the cross section should be scaled by the inverse of the
beam energy and by the square of the target charge number. This yields an equivalent
cross section of 1.53 :l: 0.14 b under our conditions in excellent agreement with our
value of 1.35 :t 0.18 b. Since our experiment observed 7 rays in coincidence with the
projectile residue, the 7-ray tagging method described above could be used for finding
that 85 :l: 7% of the total cross section connects to the ground state corresponding to
an absolute value of 1.15 :l: 0.18 b. The contribution from the continuum distribution
is considerably higher from a gold target than from a beryllium target, especially at
low energies. For this reason the bias setting for the tag was increased to 1 MeV. The
background yield was then 12% as could be deduced from the Au(‘6C,15C)X data (it
would have been 4% with a Be target) and the detection efficiency was estimated (in
the same fashion as for the beryllium target) to be €10: = 24.3%. The deduced ground—
state cross section could now be compared with theoretically calculated single-particle
cross sections based on the assumption that nuclear and Coulomb contributions are

additive. The nuclear part was calculated as in [TosQQb] and the electromagnetic part

86

as in [Ka196].

It will be clear that the neutron separation energy and the ground-state spectro-
scopic factor both must be considered unknown parameters. The data discussed above
are now used to place constraints in the Sn—C2S plane as shown in Fig. 5.8, where the
boundaries corresponding to the five sets of input data indicate limits corresponding
to plus or minus one standard deviation. Two cross-hatched areas represent limits on
Sn. One is from the analysis of the momentum width shown in Fig. 5.6 and the other
from the differential cross section g—g, as a function of the center-of-mass deflection
angle [Nak99]. Two other regions of the Sn-CQS plane, marked with vertical and
horizontal lines, denote limits obtained from the absolute partial cross sections on
beryllium and gold, respectively. The Coulomb cross section is based on our value;
the result of Nakamura et al. would place the curve slightly higher but still within
the error band. Four of the allowed bands point to a single consistent solution corre-
sponding to a neutron separation energy of 0.5-1.0 MeV and a spectroscopic factor of
0.5-1.0 (theoretical value 0.58). This allows the conclusion that the ground state of
19C has a well developed halo, similar to that of 11Be. The fifth band, representing

the direct mass measurement, does not allow a solution consistent with the other

experimental input.

Since the lack of precise knowledge of the 19C mass has been a main obstacle to
our understanding of this case, it should be clear that it would be extremely valuable
to have an accurate direct measurement. Still, it is probably useful at this point to
summarize the indirect evidence. The analysis based on integral 7 tagging (Fig. 5.6)
is free from evident systematic errors, but suffers from low statistics. This leads to the
rather imprecise value of the neutron separation energy, 3,, = 0.8 i 0.3 MeV, which,
nevertheless, has been used for the analysis of the cross sections in Table 5.4. The
values obtained from the analysis of the inclusive momentum spectrum (Fig.5.7),

87

 

@ Mass Measurement

dcsnucl/dPll [Ell] o

nucl

Coul

 

Q doCoulldQ E o

I I

    

 
    
  

 

1 r . ':
N8 E ”V I
K i ‘
H l/ :3' -
8 “O -
0 J»:
«3 -
LL
.9. ' "
On
8 —1
g 10 e
t
b _
0
a) _
D‘ .-
m _

I

 

 

 

—1
10

SI] [MeV]

 

 

 

 

Figure 5.8: Permitted regions in the space of spectroscopic factor and one—neutron
separation energy for the ground state of 19C. The hatched areas result from different
information: measured nuclear and Coulomb breakup cross-sections (Omar, cowl),
and analysis of the ground state momentum distribution (do/dP“). Also shown are
the result from the Coulomb dissociation experiment in [N ak99], deduced from da/dQ,
and the separation energy value in [Aud93, Aud97]. A consistent description of the
experimental results is given for values of C23 between 0.5 and 1 and of 3,, between
0.5 and 1 MeV. Note that this graph does not display the two analyses of inclusive
momentum distributions discussed in the text.

88

0.65 i 0.15 MeV, and from the Coulomb dissociation experiment by Nakamura et
al. [Nak99], 0.53 :l: 0.13 l\=‘leV, are both more appealing. They suffer, however, from
uncertainties concerning the contributions from excited levels, which were taken from
theory in the former case and neglected in the latter. The absolute nuclear and
electric cross sections of Fig. 5.8 are again consistent with the three values given here,
and all approaches give definitely larger values for the separation energy than the
0.16 :l: 0.11 MeV based on the direct mass measurements. It would probably be

premature to propose a combined value at this moment.

89

Chapter 6

Summary

In this work and in previous papers knockout reactions have been proven to offer a
very promising spectroscopic tool that can test nuclear structure theory in consid-
erable detail. This technique also has the high sensitivity that is a prerequisite for
experiments with beams of rare isotopes. The best example of this, so far, is Fig. 5.8,
for which the main part of the data was obtained in reactions induced by an incident
(secondary) beam of 19C of slightly less than one atom per second. (The results for
25F recently presented by Sauvan et al. [SauOO] used a similar beam intensity.) This
information has been sufficient for determining the spin and parity of the ground state
and for showing that it is a neutron halo state with a spectroscopic factor approaching
unity. This is the second established case of a pronounced single-neutron halo beyond
‘1 1Be, although 14B and 15C might also be considered as qualifying for this epithet.
All the measurements reported here have been limited by counting statistics and by
the mediocre resolution of the NaI(Tl) 7 detectors. Very soon better measurements,

based on a more powerful radioactive-beam facility and on segmented germanium

detectors, should become possible.

Theory is the second essential ingredient in the kind of studies presented here. One

Cannot overestimate the importance of basing experiments and analysis on rigorous

90

theoretical models with a predetermined set of parameters and, if possible, offering a
clear choice of alternatives. In the field of exotic nuclei, where only a limited amount
of experimental information is available, it may be dangerous to work with flexible
models that can be tailored to the needs of each individual nucleus and that, seen
in isolation, may seem plausible. The version of the many-particle shell-model used
in this work accounts very well for both single-particle and collective variables, and
Tables 4.2, 5.1, 5.2, 5.3, and 5.4, demonstrate how a combination of f—assignments and
spectroscopic factors can provide very detailed tests of nuclear structure. In the case
of 17C a §+ spin-parity is assigned, cf. the predictions for the excluded alternatives
given in Table 5.3. Note also that the experiment successfully confirms the predicted

20% admixture of E = 0 in the predominantly K = 2 knockout to the 2+ level.

The absolute precision of the method still is an open question. Until now, in-
vestigated €——values and spectroscopic factors in about 20 partial cross sections have
been investigated for proton and neutron removal reactions in the p— and sd—shells,
and consistent results have been found. Preliminary estimates of the experimental
and theoretical errors have been offered in [NaVOO], and arrive in both ways at a
relative value of i20%. The well known 15C nucleus has been investigated as a test
of the method. The results indicate an overall agreement (70%) with theory. Fully
quantum mechanical calculations of the elastic breakup explain the observed asym-
metry of the measured momentum distribution (not present in the eikonal curves)
and the dependence on the scattering angle. These calculations have also been per-
formed to successfully interpret the momentum distribution of 11Be, which appears
also asymmetric. However, the observed deviations do not affect the identification
of the angular momentum value of the removed nucleon. Moreover, the calculated
cross sections for the elastic breakup are similar in both cases, demonstrating the

reliability of the reaction model. It remains to be seen whether this holds (or im-

91

 

proves) as more evidence accumulates, or whether there will be a need for fine-tuning
the theory. Our current impression is that the knockout reactions show promise of
becoming an interesting precision complement to the classical pickup reactions at low

beam energies.

A more fundamental question is why there is such good agreement between exper-
imental and theoretical cross sections. The connection between the two is provided by
equation (3.1), which supplies the heuristic link between two seemingly unconnected
theories. The spectroscopic factors are defined in a severely truncated Hilbert space
with nucleons assumed to be the fundamental building blocks. These are subject
to effective interactions, which take values adjusted to compensate for the neglected
degrees of freedom. The reaction theory used for calculating the stripping and diffrac-
tion dissociation cross sections also, as it seems successfully, starts from a picture of
quasi-free nucleons, generally believed to be valid at very high energies. Essential
input parameters are nucleon densities and free nucleon-nucleon scattering cross sec-
tions. In the present work, this version of Glauber theory is applied well below the
energy at which it is usually assumed to become a good approximation. In both
calculations pre-existing parameter selections have been used and good agreement on

an absolute scale has been obtained.

It is tempting to speculate that this agreement is not a mere coincidence. The
reason could be that the reactions are surface-dominated [Tos99b, Tos99c], and that
they sample predominantly the nuclear exterior. In this region, where the density
is low, the nucleons of the effective-interaction theory have properties close to those
Of a free nucleon. To give a rough quantitative scale for the average size of the
exterior sampled in the experiments, consider, for the case of a beryllium target, the
ratio of a typical single-particle stripping cross section of about 30 mb to the free-

11 entron reaction cross section at the same energy of 300 mb. This means that the

92

observed cross sections represent the outer 10% of the single-particle wave function.
The same argument is the key point in the analysis of the momentum distributions
[Han95, Han96, Hes96], namely that the reactions sample just the momentum content
of this external region and are blind to contributions from the (unexplored) interior.
As was alluded to in the introduction to the experimental technique, section 2.1, the
dominance of the nuclear surface is a general characteristic of nuclear spectroscopy

via transfer reactions at lower energies.

The experiment involving the excited level of 15C showed that isomeric states
produced in the knockout reaction can give rise to experimental complications and
serves as a reminder that it may be prudent (as well as rewarding) in general to carry
out a separate search for possible unknown isomers in the outgoing residues (what one
could call the “tertiary beam”). The presence of isomers in the secondary beam from
fragmentation has already allowed Grzywacz et al. [Ger8, Ger5, Ger7] to discover
a. number of interesting new isomers. This suggests as another possibility the use of
the techniques discussed in the present work for investigating spectroscopic reactions
of isomers. if the outcome of the reactions were sufficiently different, the presence of
two species in the beam may not be an unsurmountable obstacle, provided that the

ratio of the intensities were known from direct experiments.

93

Appendix A

Data Analysis

The data were analyzed using the program package PAW (Physics Analysis Worksta-
tion) [PAW]. It is an instrument that provides statistical and mathematical analysis,
as well as interactive graphical presentation for versatile multi-parameters data sets.
The data are arranged as arrays of variables, called “Ntuples”, on an event-by-event
basis. From these variables, or any combination of them, it is possible to define gates
for selecting or discarding events. PAW can be used interactively or in batch mode

and dedicated programs can be written to conveniently manage the analysis.

A schematic outline of the general procedure is given in the following section.
However, the details of the analysis differ in the different cases, depending on specific
physics characteristics of the nuclei studied, or on the available statistics. Therefore,
the sections following the outline contain the details regarding the separate cases. The
analysis of 17C offers the most comprehensive case to illustrate the analysis, therefore

it will be discussed first. Then the cases of 15C, 16C and 19C will be described in the

order.

94

A.1 Summary of the procedure

1. The preliminary inclusive cross section 0pm,.) was derived as the ratio between
the rate of detected fragments and the rate of incident projectiles, appropriately
scaled by target thickness (15 [g/cn12]) and live time and multiplied by the target
number density (At/NA). The gates are assumed to be as wide as possible to

avoid any losses.

An uncertainty on the stability of the incident beam rate was estimated by
comparing the rates in the beam-line timer (BLT) of the different fragments
produced in the reaction, run by run. Any fluctuation indicated a variation in
the beam intensity and composition or even a shift in trajectory. In the case of
the 15’16'17C this was estimated to give at most a 7% uncertainty, which com-
bined with the uncertainties on target thickness, acceptance, events selection
(all estimated to 5%), gave a 12% total uncertainty on the total cross section.
For 19C, while the uncertainties on target, acceptance and event selection were
still estimated to 5%, the very low statistics prevented the assessing of the
beam stability with the same confidence as in the other cases, and a total 30%

uncertainty on the cross section was determined.

2. The full momentum distributions were reconstructed with the ion optics code

COSY INFINITY [Ber93], described in the second part of section 2.3.1.

The inclusive momentum distribution was extracted and represented in absolute
scale as da/dP” (i.e., scaled so that its area represented the measured cross

section as calculated in step 1).

3. The 7-ray spectrum was analyzed and the partial intensities b and the partial

momentum distributions corresponding to each final state were derived, taking

into account the 7 efficiency, possible feeding from other states and background

events.

The measured branching ratios deduced from the “,'-ray spectra analysis (with
indirect feeding taken into account) are given in Tables 4.2, 5.1, 5.2, 5.3,
and 5.4. The results obtained for 15’16’17'19C are described in detail in sec—

tions 4.3, 5.1.3, 5.2.3 and 5.3.3, respectively.

. Possible losses due to the finite geometrical acceptance were estimated using
simulations of the performance of the 8800 spectrograph for a distribution of
events generated by a Monte Carlo procedure, using the measured transverse
momentum distributions. The corrections were applied to the distributions

obtained in step 2 (inclusive) and to the ones from step 3 (exclusive).

. Theoretical momentum distributions were calculated for the expected E values

of the removed neutron orbitals, following the model discussed in section 3.2.

. The curves were used to fit the intensity of the corrected measured momentum

distributions in an appropriate momentum range.

. The spectrograph acceptance provided complete momentum distributions for
the narrow distributions corresponding to low angular momentum (€ 2 O, 1)
of the removed nucleon. The tails of higher E distributions were lost, due to
both the angular and the momentum acceptance. The additional contributions
from the (unobserved) tails were estimated from the theoretical curves used to
interpret the measured data. The final total and partial cross sections were

then obtained.

In the case of the reaction of 17C leading to the first excited level of 16C the

momentum distributions of the residues observed in coincidence with 7 rays

96

corresponded to a mixture of the K values 0 and 2. This has served to subdivide
the experimental branching ratio further, corresponding to the two 5 values. A

similar case was found in 14B [GuiOO].

A.1.1 The analysis in the case of 17C

NSC/BLT x A, x 1027
0' inc 2 ' ' __——— Z
P’ l 17C/BLT x livetime NA X t

 

_ 38982/82172340 X 9 x 102‘7
_ 0.2909 x 0.98365 6.02 x 10‘23 x 0.228

 

 

= 109(13) mb

Since the statistics were high and fluctuations were observed in the rate of frag-
ments run by run, only the two closest runs to the elastic run were taken to derive
the cross section. Three gates were determined on AE vs. T OF , on TOF vs. x801
(position in first CRDC), and on TOF vs. r802, (position in second CRDC). The
data used to extract the cross section were those satisfying the first gate and either

one of the two other gates.

3. The analysis of the v-ray spectrum (shown in Fig.5.3) gave the following

branching ratios (see also level scheme in Fig. 5.5):
b(g.s.) 2 19(9)%
b(2+) = 52(8)%
b(2,3<+>,4+) = 29(5)%

4. The geometrical acceptance correction was estimated to be 3.6% and applied

to the momentum distributions.

7. The momentum distributions were fitted using the calculated momentum dis-

tributions. To reproduce the intensities given by the result of the fitting procedure,

97

 

and account for the missing tails of the distributions, the following corrections needed
to be applied to the experimental cross sections for each final state: 1.046 for the cross
section to the ground state (09”,), 1.051 for the cross section to the 2+ state (02+),

and 1.083 for the cross section to the group of levels 2,3(7‘),4+ (02‘3(+),4+).
The estimated total average correction was then:
0.19 x 1.046 + 0.52 x 1.051 + 0.29 x 1.083 2 1.06
Therefore, the final inclusive cross section was:
0m, = 109(13) x 1.06 : 115(14) mb
and the absolute partial cross sections were:
09,, 2 19(9)% of 115(14) mb =2 22(10)(3) mb = 22(11) mb
02+ 2 52(8)% of 115(14) mb 2 60(9)(7) mb 2 60(12) mb
02,394+ : 29(5)% of 115(14) mb 2 33(6)(4) mb 2 33(7) mb

l\-*Ioreover, the 2+ momentum distribution is a mixture of 6 = 0, 2 values. A fit of
the experimental distribution with the calculated 3— and d~wave momentum distribu-

tions (step 6) yielded 26(10)% s~wave component and 74(10)% d—wave component.
Therefore, the absolute branching ratios were:
02.33 2 26(10)% of 52(8)% 2 14(5)(2)% 2 14(6)%
by”, = 74(10)% of 52(8)% 2 38(5)(6)% 2 38(8)%
and the corresponding cross sections were:
0293 = 16(7)(2) mb 2 16(7) mb
029d 2 44(9)(5) mb : 44(11) mb

The momentum distribution corresponding to the residues in the group of excited

states (2,3(+),4+) is also a mixture of€ : 0, 2 values. The fit of the measured distribu-

98

 

tion with the s— and d—wave distributions (step 6) yielded 8(8)% s—~wave conmonent

and 92(8)% d—wave component.
Therefore the absolute branching ratios were:
b23949, = 8(8)% of 29(5)% 2 2(2)(1)% = 2(2)%
b2‘3+,4+,d 2 92(8)% of 29(5)% = 27(2)(5)% 2 27(5)%
and the absolute cross sections were:
02,394+“, = 2(2)(0.2) mb 2 2(2) mb

02,3+,4+,d : 31(6)(4) mb 2 31(7) mb

 

‘2.

A.1.2 The analysis in the case of 150

14C/BLT x A, x 1027
0' inc : ' ° ' —— =
p‘ l IaC/BLT X livetlme NA X t

 

_ 1042690/2.2029 x 109 X 9 x 1027
_ 0.2756 x 0.8233 6.02 x 1023 x 0.228

 

 

= 137(16) mb

The gates applied in this case were analogous to the ones used for 17C.

3. The v-ray spectrum (F ig.4.1) was analyzed as described in section 4.3 and
the following branching ratios were obtained for the ground state and for the excited
states of the 14C residues:

by, = 79.7(2.0)%

bm = 20.3(2.0)%

In particular, three excited states were identified and the following branching

ratios were found:

bl" = 1P..7(16)(/O
99

120- = 2.5(0.5)%
02+ 2 2.1(0.5)%

With 2+ it is referred here to the two unresolved excited states (both of spin parity

2+) predicted by theory at energies of 7-8 MeV.

4. The geometrical acceptance correction was estimated to be smaller than 1%

and was thus neglected.

7. The momentum distributions of the 14C residues were wider than expected by

theory (Fig. 4.2), therefore no correction was estimated for the momentum acceptance.

The absolute partial cross sections were then:

09.3. : 79.7(2.0)% of 137(16) mb -— 109.2(2.7)(13.1) mb 2 109(13) mb

06m = 20.3(2.0)% of 137(16) mb 2 27.8(2.7)(3.3) mb 2 28(4) mb.

The cross section to the excited states could be further subdivided into the indi-
vidual components:

01— = 15.7(1.6)% of 137(16) mb 2 21.5(2.2)(2.6) mb 2 22(3) mb

00— : 2.5(0.5)% of 137(16) mb 2 3.4(0.7)(0.4) mb 2 3(1) mb

02+ = 2.1(0.5)% of 137(16) mb = 2.9(0.7)(0.3) mb 2 3(1) mb
Derivation of the angular distributions for (15C,14C)

As discussed in detail in Chapter 4, the measured momentum distributions of the
1“C residues from one-neutron removal reactions on 15C showed a characteristic de-
pendence on the scattering angle. The same feature was observed in previous data
from one-neutron removal reactions on 11Be. This section illustrates how the angu-
lar distributions of Fig. 4.6 were obtained, in the specific case of 15C. An analogous

procedure was used to derive the angular distributions of Fig. 4.7, for the case of 11Be.

100

 

0 Four y-ray spectra were obtained for different scattering angle bins (from 0°
to 4° in step of 1°). The size of the steps was fixed by the measured angular
resolution, which was calculated in the following way. The scattering angle 03.:

was given by:

 

630 : arctanh/tan2 19 + tan2 (0),

where 19 and (b were the angles measured with respect to the beam in the dis-

persive and non-dispersive direction, respectively.

For small angles the following approximation holds:

6,c = W92 + (p2.

The above expression could be derived with respect to 19 and d) and expressed

in terms of finite increments, as:

 

 

192292 + ¢2A¢2
A6,. = .
i/ .92 + .2

For a fixed 636 with an isotropic distribution, 19 and 05 may be expressed as

19 = 036 cosv and d) = 630 sin 0.

Then,

(A61)

 

”/2 63C cos2 "09.192 + 03,. sin2 UA¢2
_1/0 ~ ' dv
—’/T/2 63c

7r 7r . 1 1
402 —A 2] = 202 —A 2.
4 + 4 f 2 +2 “5

 

’IT

The measured FWHM resolutions for 19 and 05 were 23 mrad and 10 mrad,
respectively. Recalling that FWHM = 2.35 a, the corresponding standard de-

viations were (A6)”2 2 9.8 mrad and (1305)”2 2 4.2 mrad.
101

 

WT.”

This gave an experimental standard deviation for the scattering angle (A630)1/2 =

7.5 mrad 2 04°. The FWHM resolution was then approximately 1°.

0 The gated 'y-ray spectra were interpreted analogously to the total spectrum as
described above and in section 4.3. The results of the fits gave the intensity of

the excited states, from which the ground state intensity was derived.

0 For each scattering angle bin, using the ’y-ray information, the momentum dis-
tribution corresponding to the ground state of the 14C residues was obtained

and divided by the solid angle subtended in that angular bin.

A.1.3 The analysis in the case of 16C

a g __ [ISO/BLT x At><1027 _
PW “ 16C/BLT x ivetime NA xt ‘

 

_ 147753/441306528 9 x 1027

0.321 x 0.9539 x 6.02 x 1023 x 0.228 7 (9) m

 

 

Analogous gates to the ones used in the above cases were applied.

3. As discussed in detail in section 5.1.3, the y-ray spectrum could not be used to
derive the partial cross sections and momentum distributions, therefore the partial

Cross sections were obtained from the analysis of the inclusive momentum distribution.
4. The geometrical correction was estimated to be 2%.

6. The inclusive momentum distribution was subdivided into the two expected
Components corresponding to the removal of a neutron with angular momentum € =
0’ 2- This was done by fitting the measured distribution with the theoretical curves
and a mixture of 42(6)% s-—wave and 58(6)% d—wave components was found.

'7. From the fit a correction of 6.4% to the total cross section was obtained.

102

The absolute total and partial cross sections were then:
0mg: 2 72(9) x 1.064 mb 2 77(9) mb
0, : 42(6)% of 77(9) mb 2 32(5)(4) mb 2 32(6) mb

04 2 58(6)% of 77(9) mb 2 45(5)(5) mb : 45(7) mb

A.1.4 The analysis in the case of 19C

The 9Be target

18C/BLT x At x 1027
0 inc : - . ———_ 2
p’ I 19C/BLT x livetime NA x t

 

_ 131/16591729 x 9 x 1027
_ 2.14 x 10-3 x 0.985 6.02 x 1023 x 0.228

 

 

= 245(74) mb

As pointed out in section A.1, since fluctuations were observed in the rate of frag—
ments run by run, only the three closest runs to the elastic run were taken to derive the
cross section, as those were the most stable ones. Only the gate on AE vs. TOF was
applied. An uncertainty of 30% was determined, which included uncertainty on the
incident projectiles rate, as well as the uncertainties on target thickness, acceptance

and statistics.

4. The geometrical acceptance correction was found to be small in this case, in

the momentum range (5.7-6 GeV/c) covered by the measured distribution.

6. The inclusive momentum distribution is a mixture of 8 = 0, 2 values. To ex-
tract the momentum distribution for the ground state (corresponding to an s—wave
neutron), the distributions obtained with and without 7 coincidences were combined,
appropriately weighted by the efficiency and background rate. A ground state branch-

ing of 60(10)% was deduced.
103

 

The efficiency was estimated from the simulations for the expected 7 rays (see

section 5.3 for a detailed discussion of the level scheme of 19C).

Two 7 rays at 1.6 and 2.4 MeV are expected from the decay of the 2+ state at 1.6
MeV and from the cascade from the 0+ state at 4 l\-’IeV, with efficiencies of 23% and
22.6% respectively; two other excited states (2+ and 3+) are expected from theory at
E 2 4.9 MeV. Due to the tendency of the shell-model to overestimate the energy of
the levels, and with the neutron threshold energy being equal to 4.2 MeV, those two
were also taken into account. The cascade efficiency for the decay of the 7 rays with
energies around 4 MeV was cc = 1 — (1 — €1)(1 — 62) 2 41.2%. The total estimated

efficiency was then bth(2+)€1 + (bm(0+) + bth(3+) + bth(2+))eC = 38.1%.

As discussed in section 2.4.2, a background intensity of 9% with an integral cut-
off at 0.25 MeV was estimated from the analysis of (1°C,15C) and (12Be,11Be), which
have no 7 rays above 0.74 and 0.32 MeV, respectively, and consistently with the

(”Be,1°Be) data.

7. The correction factors for the partial cross sections obtained from the calculated
distributions were 1.002 for the ground state, 1.17 for the 2+ state, 1.02 for the 0+

state and 1.23 for the 2+,3+ states, which were analyzed together.

The deduced average correction for the total cross section was then 7.6%. There-

fore the absolute inclusive cross sections was:
am, = 245(74) x 1.076 mb = 264(80) mb
and the absolute branching ratios and partial cross sections were:
09.3, 2 56(9)% of 264(80) mb 2 148(24)(44) mb 2 148(50) mb
Oexc. = 44(11)% of 264(80) mb = 116(29)(35) mb 2 116(45) mb

Here, the branch to the excited states was assumed to result from different con-

104

tributions (taken from theory) by the 2+ state at 1.6 MeV, the 0+ state at 4 MeV,

and the 2+,3+ states predicted around 4.9 MeV.

The 197Au target

18C/BLT x At X 1027
0' inc 2 ' . _ =
p. l 19C/BLT x livetime NA X t

 

__ 120/18 x 107 X 197 x 1027
“ 3.12 x 10-3 x 0.99 6.02 x 1023 x 0.518

 

 

=1.35(18) b

Note that the flux of incident projectiles was stable in the two elastic runs recorded
before and after the breakup runs. No significant fluctuactions were observed in the
rate of produced fragments, therefore the data from all the runs were taken to calculate
the cross section, with the usual gates. An uncertainty of 13% was assigned, which
included the uncertainties on target thickness (5%), on gates (5%), incident rate (7%)

and number of counts (10%).
4. The angular acceptance correction was found to be smaller than 2%.

6. The inclusive momentum distribution is a mixture of f = 0, 2 values. Analo-
gously to the 9Be target case, the distributions obtained with and without 7 coinci-
dences (appropriately weighted by the efficiency and background rate) were combined
to extract the ground state and excited states momentum distributions. A ground

state branch of 85(7)% was extracted.

The efficiency estimate was derived analogously to the 9Be target case. The only
difference was that the cut-off was increased to 1 MeV, because the gold target gave
more contribution to the background. The efficiencies for the two 7 rays at 1.6 and

2.4 MeV were then 13.3% and 15.5%, respectively; the cascade efficiency for the decay

105

 

of the 7 rays at 4-5 MeV was so = 1 — (1 —— €1)(1 — 62) = 26.7%. The total estimated

efficiency was then b,,,,,(2+)61 + (b,h(0+) + bth(3+) + bth(2+))eC 2 24.3%.

With the integral cut-off at 1 MeV, a background intensity of 12% was estimated
from the analysis of (1°C,1‘°C) and (128e,11Be), and consistently with the (”Be,10Be)

data.
7. The absolute partial cross sections were then:
09, 2 85(7)% of 1.35(18) b = 1.15(9)(15) b : 1.15(18) b
06m 2 15(7)% of 1.35(18) b = 0.20(9)(3) b = 0.20(10) b

the 13% error on the partial cross section, respectively.

 

t3

106

Bibliography

[Ajz91]

[AIK96]

[AIKOO]
[Ann90]
[Arn87]

[Aud93]
[Aud97]
[Aum00]
[Bal77]

[Ban98]

[Bau98]
[Baz95]

[Baz98]

F. Ajzenberg-Selove et al. Nucl. Phys. A, 523:1, 1991.

J.S. Al-Khalili, J.A. Tostevin, and I.J. Thompson. Phys. Rev. C, 54:1843,

1996.

J.S. Al-Khalili. Private communication, 2000.

R. Ann et al. Phys. Lett. B, 250219, 1990.

E. Arnold et al. Phys. Lett. B, 197:311, 1987.

G. Audi and AH. Wapstra. Nucl. Phys. A, 565:1, 1993.

G. Audi et al. Nucl. Phys. A, 624:1, 1997.

T. Aumann et al. Phys. Rev. Lett., 84:35, 2000.

D.P. Balamuth et al. Nucl. Phys. A, 290:65, 1977.

P. Banerjee, I.J. Thompson, and J.A. Tostevin. Phys. Rev. C, 5821042,

1998.

T. Baumann et al. Phys. Lett. B, 4392256, 1998.

D. Bazin et al. Phys. Rev. Lett, 74:3569, 1995.

D. Bazin et al. Phys. Rev. C, 57:2156, 1998.

107

 

 

[Bel95]
[Ber88]
[Ber90]
[Ber93]
[Ber99]
[Bla92]
[B01168]
[801170]

[Boh75]

{80119821}

[Bon98b]

[Bon99]
[Bon00]

[Bro88]

[Bro95]

[Br098]

MD. Belbot et al. Phys. Rev. C, 5122372, 1995.

GA. Bertulani and G. Baur. Nucl. Phys. A, 4802615, 1988.

G. Bertsch, H. Esbensen, and A. Sustich. Phys. Rev. C, 42:758. 1990.

M. Berz et al. Phys. Rev. C, 472537, 1993.

U.C. Bergmann et al. Nucl. Phys. A, 658:129, 1999.

B. Blank et al. Z. Phys. A, 3432375, 1992.

G. Bohm et al. Nucl. Phys. B, 4:511, 1968.

G. Bohm and F. W’isotzky. Nucl. Phys. B, 1.52628, 1970.

Aa. Bohr and BR. h-Iottelson. Nuclear Structure. pages Vol. I p. 420, Vol.

II p. 243. Benjamin, New York. 1969 and 1975.

A. Bonaccorso and D.M. Brink. Phys. Rev. C, 57'R22, 1998.

A. Bonaccorso and DM. Brink. I’hys Rev. C, 582864. 1998.

A. Bonaccorso. Phys. Rev. C, 602054604, 1999.

A. Bonaccorso. Private communication, 2000.

BA. Brown and BB. \Nildentlial. Ann. Rev. Nucl. Part. Sczl, 38:29,

1988. the USD energy levels are giver. at.

BA. Brown. In ENAM95. 1995.

BA. Brown. In Infernatzonal Schnol of Heavy-1071 Physics, 4'” Course:

Exotic N'uclez, 1998.

108

 

[Cag99]

[Cat89]
[Cha90]
[Cha00]

[CheOO]

[ChuOO]
[Cur86]
[DesOO]

[Die90]

[Duf86]

[F es92]

[Fif82]

[For77]
[For78]
[Fuk91]

[Gea94]

J.A. Caggiano. Spectroscopy of exotic nuclei with the S800 spectrograph.

PhD thesis, h'licliigan State University, 1999.

W.N. Catford et al. Nucl. Phys. A, 503:263, 1989.

S.K. Charagi and SK. Gupta. Phys. Rev. C, 41:1610, 1990.
R. Chatterjee et al. Nucl. Phys. A, 675:477, 2000.

L. Chen et al. Evidence for an (:0 Ground State in 9He. Submitted to

Phys. Rev. Lett., 2000.

L.V. Chulkov et al. Nucl. Phys. A, 674:330, 2000.
MS. Curtin et al. Phys. Rev. Lett., 56:34, 1986.

P. Descouvemont et al. Nucl. Phys. A, 675:559, 2000.

A.E.L. Dieperink and P.K.A. de Witt. Huberts. Ann. Rev. Nucl. Part.
Sci, 402239, 1990.

J. P. Dufour et al. Z. Phys. A, 3242487, 1986.

H. Feshbach. Theoretical Nuclear Physics: Nuclear Reactions, page 455.
Wiley, New York, 1992.

L.K. Fifield et al. Nucl. Phys. A, 385:505, 1982.
HT. Fortune et al. Phys. Lett. B, 702408, 1977.
HT. Fortune et al. Phys. Rev. Lett., 40:1236, 1978.
M. Fukuda et al. Phys. Lett. B, 2682339, 1991.

Geant. Cern Library Long VVriteup. Technical Report W5013, CERN,

1994.
109

 

[Gla59]

[(36573]
[(317.95]
[GI-297]
[Gr298]
[(16100]

[Han87]

[Han95]

[Han95]

[Han96]
[HanOO]
[Hen96]
[Hes96]
[Hus85]

[K al96]

R.J. Glauber. Lectures in Theoretical Physics, volume 1, page 315. Inter-

science, New York, 1959.

JD. Goss et al. Phys. Rev. C, 82514, 1973.

R. Grzywacz et al. Phys. Lett. B, 3552439, 1995.

R. Grzywacz et al. Phys. Rev. C, 5521126, 1997.

R. Grzywacz et al. Phys. Rev. Lett., 81:247, 1998.

V. Guimaraes et al. Phys. Rev. C, 611064609, 2000.

RC. Hansen and B. Jonson. The neutron halo of extremely neutron-rich

nuclei. Europhys. Lett., 4:409, 1987.

PG. Hansen. Nuclear halos: Structure and reactions. In Proceedings of
the International Conference on Erotic Nuclei and Atomic Masses, Arles,

France, June 1995.

PG. Hansen, A.S. Jensen, and B. Jonson. Ann. Rev. Nucl. Part. Sci,

45:591, 1995.

PG. Hansen. Phys. Rev. Lett., 77:1016, 1996.

PG. Hansen. Private communication, 2000.

K. Hencken, G. Bertsch, and H. Esbensen. Phys. Rev. C, 54:3043, 1996.

H. Hesbensen. Phys. Rev. C, 53:2007, 1996.

MS. Hussein and KW. McVoy. Nucl. Phys. A, 4452124, 1985.

D.M. Kalassa and G. Baur. J. Phys. C, 22:115, 1996.

110

 

[Kan00]
[Ke195]
[Kob88]
[Kob89]
[Kox87]
[Lab99]

[Law 76]

[Lia90]

[M ad00]

[M ar96]
[Mil75]
[Nak99]
[Nav98]
[NavOO]
[Orr91]
[Orr92]

[Oza98]

R. Kanungo et al. Nucl. Phys. A, 6772171, 2000.
J.H. Kelley et al. Phys. Rev. Lett., 74:30, 1995.

T. Kobayashi et al. Phys. Rev. Lett., 6022599, 1988.
T. Kobayashi et al. Phys. Lett. B, 232251, 1989.

S. Kox et al. Phys. Rev. C, 35:1678, 1987.

M. Labiche et al. Phys. Rev. C, 602027303, 1999.

RD. Lawson, F.J.D. Serduke, and HT. Fortune. Phys. Rev. C, 14:1245,

1976.
E. Liatard et a1. Europhys. Lett., 132401, 1990.

V. Maddalena et al. Single-neutron knockout reactions: Application to

the spectroscopy of 162172190 Submitted to Phys. Rev. C, 2000.
F.M. Marques et al. Phys. Lett. B, 3812407, 1996.

D.J. Millener and D. Kurath. Nucl. Phys. A, 2552315, 1975.
T. Nakamura et al. Phys. Rev. Lett., 83:1112, 1999.

A. Navin et al. Phys. Rev. Lett., 81:5089, 1998.

A. Navin et al. Phys. Rev. Lett., 85:266, 2000.

N. Orr et al. Phys. Lett. B, 258:29, 1991.

NA. Orr et al. Phys. Rev. Lett., 6922050, 1992.

A. Ozawa et al. Measurements of interaction cross sections for Carbon
isotopes at relativistic energies and the halo structure of 19C. Technical

Report RIKEN-AF-NP-294, RIKEN, 1998.
111

 

[ruewq
[Ray79]
[Ren72]
[Ifid97]
[rud98]
[Sai89]

[Sak99]
[Sat90]

[Sau00]

[Sch99]
[Ser78]

[She92]
[She98]
[861699]
[Tan85]
[Ti193]

[Tin196]

[Tos97]

Physics Analysis Workstation. http://wwwinfo.cern.ch/asd/paw/.
L. Ray. Phys. Rev. C, 2021857, 1979.

P.U. Renberg et al. Nucl. Phys. A, 183:81, 1972.

D. Ridikas et al. Europhys. Lett., 372385, 1997.

D. Ridikas et al. Nucl. Phys. A, 6282363, 1998. i
M.G. Saint-Laurent et al. Z. Phys. A, 3322457, 1989.

A. Sakharuk and V. Zelevinsky. Phys. Rev. C, 61:014609, 1999.

 

G.R. Satchler. Direct Nuclear Reactions. Oxford, University Press, 1990. .5

E. Sauvan et al. One-neutron removal reactions on neutron-rich psd-shell

nuclei. Submitted to Phys. Lett. B, 2000.

H. Scheit et al. Nucl. Instr. Methods A, 4222124, 1999.

RR. Sercely, R.J. Peterson, and ER. Flynn. Phys. Rev. C, 1721919, 1978.
BM. Sherrill et al. Nucl. Instr. Methods B, 702298, 1992.

R. Sherr and HT. Fortune. Phys. Rev. C, 5823292, 1998.

M.H. Smedberg and MLV. Zhukov. Phys. Rev. C, 5912048, 1999.

I. Tanihata et al. Physical Review Letters, 55:2676, 1985.

DR. Tilley et al. Nucl. Phys. A, 564:1, 1993.

N.K. Timofeyuk and P. Descouvemont. J. Phys. C, 222L99, 1996.

J.A. Tostevin and J.S. Al-Khalili. Nucl. Phys. A, 616:418c, 1997.

112

[Tos99a]
[T0899b]

[Tos99c]

[T0800]

[War89]
[War92]
[Wou88]
[Yab92]
[Yam96]
[Yur99]

[Zah93]

J.A. Tostevin and J.S. Al-Khalili. Phys. Rev. C, 59:5R, 1999.
J.A. Tostevin. J. Phys. C, 252735, 1999.

J.A. Tostevin. In Fission and Properties of Neutron-Rich Nuclei, Pro-
ceedings of the Second International Conference, page 429, St. Andrews,

Scotland, 1999.

J .A. Tostevin. Single-nucleon knockout reactions at fragmentation beam
energies. To be published in Nucl. Phys. A, 2000. Paper presented at
NS2000, International Conference on Nuclear Structure, East Lansing,

MI, USA, August 13-19, 2000.

E.K. Warburton and DJ Millener. Phys. Rev. C, 3921120, 1989.
BK. Warburton and BA. Brown. Phys. Rev. C, 462923, 1992.

J. Wouters et al. Z. Phys. A, 3312229, 1988.

K. Yabana, Y. Ogawa, and Y. Suzuki. Nucl. Phys. A, 5392293, 1992.
T. Yamazaki et al. Z. Phys. A, 3552219, 1996.

J. Yurkon et al. Nucl. Instr. Methods A, 4222291, 1999.

M. Zahar et al. Phys. Rev. C, 482R1484, 1993.

113

 

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