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HUI”IIIIHIIIIHIIIHNIlllllHillllllHlllllililllillllllll

3 1293 013943

This is to certify that the

dissertation entitled

STRUCTURE STUDIES OF 11Li AND 10m

presented by
Brian Matthew Young

has been accepted towards fulfillment
of the requirements for

Ph .D. Nuclear Physics

degree in

 

 

Mada W

M a jor professor

Date 1766 23 /973

 

MSU 13 an Affirmatiw Action /Equal Opportunity Institution 0- 12771

STRUCTURE STUDIES OF 11Li AND 10Li

By

Brian Matthew Young

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

1993

ABSTRACT

STRUCTURE STUDIES OF 11Li AND loLi

By

Brian Matthew Young

The nucleus 11Li, only recently available for extensive study with the advent of fa-
cilities that produce radioactive nuclear beams, has presented nuclear theorists and
experimentalists with an intriguing puzzle. Experimental evidence indicates that 11Li
consists of a 9Li core and a “halo” of two loosely bound neutrons, the matter radius
of which extends well beyond that observed in most nuclei. Theoretical models have
been developed which utilize this picture and predict a very sensitive dependence of
the 11Li two—neutron binding energy on the nature of the n—9Li interaction. The
mass of 11Li has been determined from a measurement of the Q—value of the reac-
tion 14C(nB,uLi)1“O at E /A z 32 MeV. The results, which indicate a two-neutron
separation energy of 5211(11Li) = 295 :l: 35 keV, put this basic quantity on a firm
basis for use in these theoretical models. Experimental measurements and theoretical
predictions of the low—lying structure of the unbound nucleus 10Li have presented
sometimes conflicting, but mounting evidence that 10Li has a ground state consisting
of a low—lying 23:1,- neutron resonance and at least one excited state consisting of a
117% neutron resonance unbound by between 400 and 800 keV. Momentum spectra
have been measured for the reaction 11B(7Li,sB)mLi at E /A z 19 MeV and labora-
tory angles of 5° and 3.5°. The results indicate the existence of a broad state in 10Li,
corresponding to a single p—wave neutron resonance unbound to neutron decay by

538 i 62 keV with width I‘m, = 358 :l: 23 keV. There is also evidence that the 10Li

ground state is either an s— or a p—wave resonance barely unbound to neutron decay

with Sn 2 —100 keV and I‘lab < 230 keV.

Dedicated to my wife, Kathryn.

iv

ACKNOWLEDGEMENTS

During my stay at the NSCL, I have had the privilege to work with many truly
fine individuals. What follows is a woefully incomplete list of folk who have helped

me during my four—plus years here.

I would like to thank my advisor, Walter Benenson, for his help and what now
seem like almost superhuman displays of patience and calm during my work here. I
would also like to thank Alex Brown, George Bertsch, and Mikhail Zhukov for innu-
merable helpful discussions and for entertaining equally innumerable questions about
nuclear structure and resonance calculations. Toshiyuki Kubo, Henning Esbensen,
Ian Thompson, and Rubby Sherr also lent a great deal of assistance without which

the work contained in this dissertation would not have been possible.

Maggie Hellstrom is one of the dearest friends I have ever known. Her support
and encouragement during a particularly murky period of my life have meant more
to me than she will ever know. Maggie is always willing to lend a hand and advice
when asked, and, perhaps more importantly, has the uncanny ability to inject good—
humored sanity into any situation. Such injections were frequently needed and greatly
appreciated, and I consider myself very lucky to know and to have worked with her. I
would also like to thank Sherry Yennello for her encouragement and kindness during

the whole nightmarish postdoc search.

I am especially grateful to Brad Sherrill, Michael Thoennessen, John Stevenson,
David Morrissey, Jim Clayton, and Mike Mohar for teaching me so much about the

right way to conduct nuclear physics experiments.

Among the most enjoyable experiences I’ve had here are the beer-brewing sessions
with Phil Zecher, Mathias Steiner, Damian Handzy, and David McGrew. Phil and

I often colored these sessions with discussions of libertarianism, the philosophy and

V

novels of Ayn Rand, and the financial aspects of chaos theory (or was it the chaotic
aspects of financial theory). For two years I shared an office with Mathias, something
for which I think he deserves a medal. I mercilessly inflicted upon him my infernal
libertarian views, my equally infernal musical tastes, and (even worse) daily doses
of Rush Limbaugh. Although I think he realized the last was predominantly for
theatrical effects: Mathias on an indignant tirade was infinitely more entertaining
than Limbaugh ever could have been. It was quite entertaining to share with Damian
his first reading of Atlas Shrugged; although I’m sure he got as much enjoyment out
of my first reading of Tolkien’s ring trilogy. My basement, in addition to providing
fermenting space for homebrews, also managed to contain the hellish noises that David
and I were wont to make with guitars and other bits of electronic gadgetry. I think
between the two of us we managed to set musical evolution back several millenia.
Michael Fauerbach, Raman Pfaff, Don Sackett, Easwar Ramakrishnan, John “Ned”
Kelley, Jeff Winger, Daniel Bazin, and Nigel Orr have also shared countless shifts,
beers, pizzas, parties, annoying songs, and interminable meetings with me and have
in no small way made lab life tolerable. I would also like to thank the students of the
Physics 231 class that I taught while I was writing this dissertation. I realize that
none of them will ever read this, but it was with them that I learned just how much
fun it is to teach a group of students, to make the connection, and to actually have

them leave the classroom knowing more than they did fifty minutes before.

You have all made my time here incredibly rewarding and enjoyable, and I will

miss you all.

My parents have a picture of me on Career Day in fifth grade. Most of the other
kids wanted to be rock stars, pilots, baseball players, and policemen; then here’s Brian
in a lab coat (actually, it was my mom’s old raincoat) holding a vial of something

(pump oil, maybe, or liquid scintillator). When I told my parents I wanted to be a

vi

physicist, they had no idea what one looked like. Their best guess was the lab coat
and the vial. I must have scared the living daylights out of them. I shudder to think
how much money I cost them with my penchant for taking apart various household
appliances and applying magnets to the television to figure out where the colors came
from. I owe my parents an immeasurable amount of gratitude for putting up with
my antics, questions, and destructive experiments when I was growing up. Thanks a

million, you guys.

Finally, I would like to thank my wife, Kathryn, for everything. For not getting
upset when I came dragging in at all hours of the morning, for fielding phone calls
while I was working, for bringing me dinner at the lab, for tolerating my positively
annoying habit of brainstorming while strolling around our apartment playing the
banjo, for coming up here to this frozen wasteland to live with me for four and a
half years. What can I say? She deserves just as much credit as I do for the work
contained in these pages. Thank you, Ryn, you’ve given me so much, and made it all

worthwhile.

vii

Contents

LIST OF TABLES x
LIST OF FIGURES xi
1 Introduction-Motivation and Overview 1

2 History of 11Li

I Introduction ................................ 4
II Structure of 11Li — Experimental Results ................
III Structure of 11Li — Theory ........................ 16
IV Predicted llLi mass and 10Li ....................... 19
V Previous 11Li mass measurements .................... 20
3 The Mass of 11Li from the l4C(uB,11Li)”O Reaction 24
I Introduction ................................ 24
II Description of A1200 ........................... 25
III Description of experiment ........................ 28
IV Analysis .................................. 31
V Results ................................... 37
4 Structure Studies of 10Li 41
I Introduction ................................ 41
II History of 10Li ............................... 42
III Description of S320 ............................ 51
IV Description of experiment ........................ 53
V Analysis .................................. 58
VI Results ................................... 61
5 Conclusions-The Present and Future of 11Li and 10Li 66

viii

A RELMASS 71
B Calculations of Neutron Resonance Line Shapes 89
C Maximum Likelihood Fitting Procedure 104

LIST OF REFERENCES 124

ix

List of Tables

2.1

3.1
3.2

4.1

4.2

A.1

B.1

C.1

Table of Existing 11Li Mass Measurements ...............

Sources of Experimental Uncertainty in 11Li Mass Measurement. . . .

Table of Existing 11Li Mass Measurements Including Present Work . .

Sources of Nonstatistical Experimental Uncertainty in 10Li Mass Mea-
surement. .................................

Table of Existing Measurements of Low—lying Structure of 10Li Includ-
ing Present Work ..............................

Listing of RELMASS Input and Output File ...............
Listing of RESCALC Source Code. ...................

Listing of FIT.SPEC Source Code. ...................

20

39
40

64

77

94

List of Figures

2.1

2.2

2.3
2.4

2.5
2.6
3.1
3.2
3.3

3.4
3.5

4.1
4.2

4.3
4.4

4.5

4.6

4.7
4.8

The transverse—momentum distributions of 6He and 9Li fragments from
the reactions of 8He and 11Li projectiles, respectively, at E /A = 790

MeV on a C target ............................. 8
Angular distributions of single neutrons from the ( 11Li,9Li) reaction on
Be, Ni, and Au targets at E/A = 29 MeV. ............... II
Schematic diagram of 11Li halo structure ................. 12
Schematic representation and intensities of soft and normal giant dipole
resonances in llLi. ............................ 15
Sensitive dependence of 11Li binding energy on 10Li 1p§ state energy
as predicted by Bertsch and Esbensen. ................. 21
Dependence of 11Li binding energy on 10Li 1p2 and 2.9- state energies
as predicted by Thompson and Zhukov .................. 22
Schematic layout of the A1200. ..................... 26
Particle—identification spectrum from 11Li production run. ...... 30
Momentum spectra from 11Li production and calibration runs. . . . . 32

p vs :5 calibration of A1200 Focal Plane for first llLi production run. 35

Calibration of A1200 dipole magnets ................... 38
Shell structure of the N = 7 nuclei 11Be and 12B. ........... 43
Energy spectra of 8B from the 12C(S’Be,3B)13B and 9Be(9Be,8B)1°Li
reactions at E /A =13.4 MeV and 91am = 14°. .............. 44
Energy spectrum of protons from the reaction llB(1r",p)1°Li ...... 46
Relative velocity spectrum of neutrons collected in coincidence with

9Li fragments from the decay of 10Li. .................. 47
Energy spectrum of 12N from the 9Be(13C, 12N)1°Li reaction at E /A =25.8
MeV and 911.1) = 3. 8°. ........................... 49
Energy spectrum of 17F from the 13C(14C17F)1°L1 reaction at E / A =24 1
MeV and 91.1)" —— 5. 4°. ........................... 50
Schematic layout of S320 magnetic spectrometer. ........... 52

Particle—identification spectrum from 10Li production run at 0131, = 523°. 55

xi

4.9

4.10
4.11

5.1

A.1
A.2

BI
82

Momentum spectra from 10Li production and calibration runs at 01111) =

523° .....................................
Momentum spectrum from 10Li production run at 0161, = 3.73° .....

Theoretical models fitted to the collected spectrum from the reaction
11B(7LI,SB)10L1 at 0131) = 5.230 .......................

Comparison of present data with calculations by Thompson and Zhukov.

Flowchart of magnetic spectrometer calibration procedures .......

Flowchart of Q-value measurement procedures with a calibrated mag-
netic spectrometer. ............................

Calculated line shapes for 23% and lp-é— n—9Li states at several energies.

Six—parameter model fitted to calculated line shape for a Ip-é— neutron
state at 500 keV resonance energy. ...................

xii

56
57

62

69

72

73

91

Chapter 1

Introduction—Motivation and
Overview

Recent advances in accelerator and spectrometer design have made possible the pro-
duction of intense radioactive nuclear beams. This has greatly expanded the region
of experimentally obtainable nuclei to encompass nuclear species far from stability.
The availability of these unstable nuclei has resulted in a large amount of exploration
and mapping of the neutron and proton drip lines, the boundaries marking the upper
limits on neutron and proton numbers beyond which nuclei become particle unstable.
Experiments which involve nuclei near the drip lines have revealed structures and dy-
namic systems very different from those previously studied. These new systems have
presented a challenge to nuclear models which were developed to describe phenomena

exhibited by nuclei closer to the valley of stability.

One of the most interesting nuclei to challenge traditional models is 11Li. A large
body of experimental work has been carried out in an attempt to understand its
structure. These experiments have indicated that llLi consists of a 9Li core and a
“halo” of two loosely bound neutrons, the matter radius of which extends well beyond
that observed in most nuclei. With these ideas as a starting point, several calculations

have been made which treat 11Li as a three—body system comprising a 9Li core and

2 neutrons. Hansen and Jonson have demonstrated in [Hans 87] that in the simplest
model of 11Li, a quasi—deuteron consisting of a 9Li core coupled to a dineutron 2n,
the binding energy is the only variable needed to find the momentum and spatial
distributions of particles within the nucleus. More complex models have also shown
that the radius, and even the existence, of a neutron halo is intimately dependent on
the binding energy of the halo neutrons. For this reason, an accurate experimental

value for the 11Li binding energy is of vital importance to the understanding of the

halo phenomena.

There is however, some uncertainty in the value of the mass of 11Li. The three
measurements to date have yielded two—neutron separations energies ranging from
170 to 340 keV with uncertainties from 50 to 120 keV. The substantial disagreement
between these measurements as well as the magnitude of their uncertainties limits
their usefulness in theoretical calculations. In September of 1992, the 11Li mass was
deduced from the Q—value of the 1“C(11B,11Li)1“O reaction measured with the A1200
fragment separator at the NSCL. In Chapter 2 of this work, a more detailed discussion
of previous experiments pertaining to 11Li is presented along with a discussion of
several two— and three—body models that have been developed to reproduce the results
of these experiments. Chapter 3 contains a detailed description of the NSCL 11Li mass
measurement. The computer code, RELMASS, used in the analysis of the data from

this experiment, is presented in Appendix A.

Central to the models described above is the interaction of a single neutron with
the 9Li core. The fact that 11Li is loosely bound while 10Li, with one less neutron,
is unbound implies that the interaction between the valence neutrons plays a vital
role in the stability and structure of 11Li. For this reason, there is considerable
interest in the unbound nucleus 10Li. The low—lying structure of 10Li is the subject

of much debate, however. There has been a great deal of effort, both experimental

and theoretical, directed towards understanding the structure of this nucleus. These
efforts have resulted in claims that the 10Li ground state, a resonance of a single
unbound neutron in a 9Li well, is unbound by as little as 60 keV and as much as 800
keV. It is expected, in analogy with other N = 7 nuclei, that the 10Li ground state
resonance should be either a 1p; or a 23% neutron state. In addition to significant
uncertainty regarding the resonance energy, there is also a great deal of disagreement
on the spectroscopic nature of the 10Li ground state. In May of 1992, the low—lying
structure of 10Li was determined from momentum spectra which were collected from
the reaction llB(7Li,8B)1°Li at the NSCL. Chapter 4 of the present work contains a
review of the existing measurements of the low-lying structure of 10Li as well as a
detailed description of our measurement. Computer codes for calculating the shapes
of the 10Li neutron resonances and for fitting the resonance shapes to the experimental

data are presented in Appendices B and C respectively.

The experimental results presented in this work will be summarized in Chapter 5.
The three—body models described above predict a sensitive dependence of several
observables on not only the resonance energies of the lowest 10Li states, but also their
spectroscopic nature. Most calculations assume that the 10Li ground state is a 1p;-
neutron resonance. However, very recent calculations by Thompson and Zhukov have
included a 2.9% states as well as a 119% state in the n—9Li interaction. The results
of these calculations and their agreement with our experimental data will also be

discussed.

Chapter 2

History of 11Li

I Introduction

This chapter contains a description of the main trends of experimental and theoretical
research that have been directed towards 11Li. Experimental results are discussed in
section II. Measurements of parallel and transverse momentum distributions of 9Li
fragments from the breakup of 11Li are described as are measurements of neutrons
observed in coincidence with the 9Li fragments, and measurements of Coulomb dis-
sociation cross sections of 11Li. It is seen that the experimental evidence indicates
that 11Li can be described as a three—body system consisting of a 9Li core with a
spatially broad “halo” of two loosely bound neutrons. Based on the three—body pic-
ture described above, several theoretical models have been developed. A sketch of
these models and some of their results are presented in section III. In particular,
it is shown that these models predict that the radius, and even the existence, of a
neutron halo is intimately dependent on the binding energy of the halo neutrons.
A brief discussion is presented in section IV concerning the low—lying structure of
10Li and its effects on the results of these models. A more detailed discussion of this
topic is presented in Chapter 5. The sensitive dependence of halo phenomena on the

neutron binding energy emphasizes the necessity for an accurate measurement of the

11Li binding energy. The three existing measurements of the 11Li mass are detailed

in section V.

II Structure of 11Li — Experimental Results

The existence of 11Li as a particle—stable drip line nucleus was known for quite some
time [Posk 66, Klap 69]. It was not until 1985, however, that evidence was found that
11Li might have a structure radically different from that predicted by traditional nu-
clear models [Tani 85a]. Tanihata et al. report the results of interaction cross section
measurements for several lithium isotopes (6L1 — 11Li) as well as 7Be, 9Be, and 10Be.
These measurements were performed with a beam energy of E /A = 790 MeV and
targets of beryllium, carbon, and aluminum. The lithium and beryllium projectiles
were produced as secondary beams by fragmenting beams (E /A = 800 MeV) of 11B
and 1“’Ne on a beryllium production target. The isotopes in the secondary beam were
separated by magnetic rigidity and identified by velocity, from time—of—flight mea-
surements, and by charge, from pulse height measurements in scintillation counters,
before incidence on the reaction target. The interaction cross section, defined as the
total reaction cross section for changing the proton and/or neutron number in the
incident nucleus, was measured using the high—acceptance spectrometer described in
[Tani 85b]. It was found that the interaction cross section could be separated into

contributions from the projectile and target
0’ = 7r [RT + RP]2 (2.1)

where RT and Rp are the radii of the target and projectile, respectively. The measured
radii of most of the projectiles were between 2.09 and 2.46 fm, a result that is in good
agreement with the empirical relation derived from the liquid drop model of a nucleus

with mass number A, R = (1.2 fm) - A1/3 [Blat 52]. The measured radius of 11Li,

however, was 3.14 fm, a value much larger than that of other nuclei in this mass region.
It is suggested in [Tani 85a] that this large radius indicates a large deformation or a

long tail in the matter distribution of 11Li.

To further understand the structure of 11Li, and in particular the large mat-
ter radius, experiments have been performed to measure the momentum distribu-
tions of 9Li from the fragmentation of 11Li projectiles. It has long been known that
the momentum distributions of fragmentation reaction products exhibit an isotropic
Gaussian distribution in the projectile rest frame [Grei 75]. The width of this distri-
bution can be interpreted in terms of Fermi motion and/or nuclear binding energy
[Gold 74], or in terms of the momentum distribution of the fragment inside the pro-
jectile [Hiifn 81, Shim 87]. Based on these ideas, it is hoped that information about
the momentum distributions from fragmentation of 11Li may shed some light on its

structure.

In the first measurement of the momentum distributions of products from 11Li
fragmentation [Koba 88], Kobayashi et al. fragmented primary beams of 20Ne and
2“Ne at E /A = 800 MeV to produce secondary beams of 790 MeV/nucleon 11Li,
8He, and °He. The secondary beams were then fragmented on carbon and lead tar-
gets. The reaction products were analyzed using the HISS magnetic spectrometer
at the Bevalac in the Lawrence Berkeley Laboratory. The transverse—momentum
(i.e. perpendicular to the beam axis) distributions of several projectile fragments are
presented and analyzed in [Koba 88]. The momentum distributions for most of the
fragments exhibited the expected Gaussian shape. An example of this is shown in
part a) of Figure 2.1, which depicts the transverse—momentum distribution of °He
fragments from 8He projectiles on a carbon target. The width of the fitted Gaussian
is a = 77 MeV/c. It has been shown by Goldhaber [Gold 74], that the width 0 of

fragmentation momentum distributions can be parametrized by a single parameter

0'0 as

2 2AF(AP "' AF)

2 2.2
0' 00 AP __ 1 ( )

 

where Ap and AF are the mass numbers of the projectile and fragment, respectively,

and 00 is the reduced width. The reduced width for the 6He data is 59 MeV/c.

A decidedly different result was observed for the 9Li momentum distribution,
shown in part b) of Figure 2.1. For the 9Li data, there was a two—Gaussian peak
structure. The wide component corresponds to a reduced width of 53 MeV/c, in good
agreement with the values obtained from the °He data as well as the data collected
from other fragmentation reactions. The narrow component corresponds to a reduced
width of 17 MeV/c. According to Goldhaber, the reduced width can be related to the
Fermi momentum Pp of the projectile by 00 = \/5 Pp. In this picture, the existence

of two reduced widths implies two Fermi momenta, an unphysical conclusion.

To interpret their data, Kobayashi et al. relied on the models developed by Hiifner
and Nemes [Hiifn 81] and Shimoura et al. [Shim 87]. In their models, which approxi—
mate the fragment momentum distribution by the Fourier transform of the asymptotic
wave function of the projectile, the reduced width can be related to the fragment sep—

aration energy (a) by

Ap—l

a2 = 11/1,, (5) AP

(2.3)

 

Under this interpretation, the narrow momentum component in the 9Li data corre-
sponds to (e) = 0.34 MeV. When compared with the one— and two-neutron separation
energies of 10Li and 11Li, this result indicates that the 9Li fragments in the narrow
peak come from reactions in which two weakly bound outer neutrons are removed
from 11Li. The broad momentum component comes from removal of two less weakly

bound neutrons, decay of excited 9Li, and neutron decay of 10Li. This observation

 

 

 

 

 

 

 

 

 

50
'3
E
.4
§
\
'13

0

50
3
.3.
€
'0
\
3

o . J 1 i L L .L-l
“-200 ‘100 0 100 200
P, [MeV/c]

Figure 2.1: [Koba 88] a) Transverse-momentum distribution of 6He fragments from
the reaction of 8He at E /A = 790 MeV on a C target. The solid line is a fitted Gaus-
sian distribution with a width 0 z 80 MeV/c. b) Transverse—momentum distribution
of 9Li fragments from reactions of 11Li at E /A = 790 MeV on a C target. The dotted
line is a contribution from the wide component of the 9L1 distribution. The solid line
is the sum of the narrow and wide components. The widths of these components are

a = 23 MeV/c and a = 95 MeV/c.

also provides an understanding of the large 11Li matter radius measured by Tanihata.
et al.. By the Heisenberg Uncertainty Principle, a narrow momentum component cor-
responds to a broad spatial component, and vice versa. Thus, the narrow component
in the 9L1 comes from a broad spatial distribution attributable to a long tail in the
probability distribution for the two loosely bound neutrons in 11Li. It is this long tail

which dominated the matter radius measurements of Tanihata et al..

Of particular importance, given this measurement, is the question of whether the
momentum distributions of projectile fragmentation products reflect the structure of
the projectile only or also depend on the fragmentation reaction itself. A possible av-
enue of inquiry is to study fragmentation reactions from low—Z and high—Z targets, in
which the contributions to the total reaction from Coulomb interactions and nuclear
interactions differ strongly. However, for reactions on a high—Z target, narrow struc-
tures in the transverse—momentum distribution, of the type observed by Kobayashi
et al., would become washed out by Coulomb deflection and multiple scattering.
Parallel—momentum distributions, which, since the total momentum distribution is
isotropic, are known to have widths similar to those of transverse—momentum dis-
tributions for the same reaction, do not suffer from these drawbacks. In 1992, Orr
et al. measured the parallel—momentum distributions of 9Li nuclei from projectile
fragmentation reactions of 11Li on targets of 9Be, 93N b, and 181Ta at E /A = 66 MeV
[Orr 92]. Their results indicate that the momentum widths do not depend strongly
on the Z of the target. For all targets, parallel—momentum widths of 0‘ z 19 MeV/c
were observed. Assuming this momentum component comes from a two—neutron spa-
tial distribution having a Yukawa functional form ( exp(—-r/ p) /r ), this momentum

width yields an rms radius of 6.2 fm.

In addition to the effort described above, experiments have been performed by

Anne, Riisager and collaborators [Anne 90, Riis 92] in which the spatial extent of

10

the outer two neutrons in llLi has been deduced from the angular distributions of
neutrons from the fragmentation of 11Li. In these experiments, secondary beams of
11Li at E/A = 29 MeV were collided with targets of beryllium, nickel, and gold.
The angular distributions of single neutrons from the (”Li,9Li) reactions, reported
in [Anne 90], are shown in Figure 2.2. By assuming that each of the outer two llLi
neutrons is represented by a Yukawa wave function with a range parameter p and
transforming this spatial distribution to momentum space, it was found that the
angular distributions of the neutrons should have a Lorentzian shape. When this
functional form was fitted to the data (as depicted with the solid lines in Figure 2.2),
the range parameter p was found to be approximately 12 fm for all three targets, a
value which corresponds to an rms radius of 8.5 fm. Further analysis of this data,

reported in [Riis 92], support these conclusions.

Based on experiments such as these, the nucleus llLi has come to be viewed as
a three-body system consisting of a 9Li core and a “halo” of two loosely bound
neutrons as depicted in Figure 2.3. While the radius of 9L1 is approximately 2.5 fm,
the experimental evidence has indicated that the rms radius of the two halo neutrons is
between 6 and 10 fm. One of the more interesting phenomena predicted by theoretical
models which incorporate this picture, some of which will be discussed in the next
section, is the “soft” mode of the giant dipole resonance. In nuclei closer to the valley
of stability, the (isovector) giant dipole resonance is an excitation mode in which the
protons and neutrons in the nucleus oscillate collectively about the nuclear center-of—
mass but in opposite phase with each other [Gold 48, Wong 90]. In a nucleus of mass
A, the giant dipole resonance (GDR) energy is found to be roughly proportional to
A‘l/a. For nuclei with mass near A = 10, the GDR energy is approximately 22 MeV.
However, for a halo nucleus, such as llLi, it has been proposed [Hans 87] that two

GDR excitations exist. One, dubbed the “hard” or “normal” mode, involves the

11

 

 

11. A ,9. 1
r L1+Z -v L1+n+X 1
91005
$— 1.
W t
\ [-
1::
E
.0 10s
bun—J E
C.‘ .
“o .
\=
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13 : 1
: Be 1 1
1- d d
1- '——-—' .,

 

 

 

.9
p...-

5 1‘0 115 26 25
Neutron angle [deg]

C

Figure 2.2: [Anne 90] Angular distributions of single neutrons from the (”Lif’Li)
reaction on Be, Ni, and Au targets at E /A = 29 MeV [Anne 90, Riis 92]. The solid
lines are Lorentzian functional forms, derived from the assumption that both of the
outer two llLi neutrons have a Yukawa wave function.

00.

000
o
00

 

9L1 core

0 neutron
Q proton

Figure 2.3: Schematic diagram of 11Li halo structure. Experimental evidence has
indicated that 11Li is a three—body system consisting of a 9Li core and a halo of two
loosely bound neutrons.

13

traditional GDR of the core nucleus only, leaving the halo neutrons unaffected. The
energy of this resonance is expected to be very close to the GDR energy of the naked
core nucleus. The other mode is labelled the “soft” mode, and has the core nucleus
and the neutron halo oscillating against one another. The restoring force for such an
oscillation is very small, and consequently the energy of this resonance is expected to

be quite small (roughly 1.0 MeV for 11Li).

Experimental evidence for the existence of this soft GDR presents a tantalizing
problem. The first possible evidence for such an excitation was reported by Kobayashi
et al. in 1989 [Koba 89]. Secondary beams of 11Li were collided with targets of beryl-
lium, carbon, copper, aluminum, and lead at E /A = 800 MeV. From the dependence
of the interaction cross section, defined as the total cross section for changing the
proton and/ or neutron number of the projectile, on the target proton number, the
electromagnetic dissociation (EMD) cross section was deduced. The measured EMD
cross section for 11L1 on the heavier targets was much larger than that for the less

neutron—rich projectile, 12C.

To explain this result, the EMD cross section was calculated as a product of the
known photo—nuclear cross section 0.,N(E.,) and the theoretically calculated virtual—
photon spectrum NV,(E.,)

”EMD: E NV7(E1)07N(E7)dE1 (2'4)
1h

where Em is the threshold energy for particle emission [Heck 76]. When the photo—
nuclear cross section was assumed to have a simple Lorentzian form, the data were
best reproduced by a 11Li GDR energy of 4.6 MeV. Additional calculations were
performed by assuming that the photo-nuclear cross section consisted of two GDR
peaks. One was located at 22 MeV, as expected for 9Li. The other peak was taken as

a free parameter and was assumed to correspond to the soft dipole mode. When the

14

contributions from both GDR peaks were weighted by the virtual—photon spectrum,
which fell off very rapidly with photon energy, and by the Thomas—Reche—Kuhn sum
rule [Wong 90], it was found that the EMD cross section was dominated by the soft
dipole peak. This is illustrated in Figure 2.4. Under this assumption, the data were

best reproduced by a 11Li soft GDR energy of 0.9 MeV.

Further measurements by Blank et al. [Blan 91], utilizing the same systems as
above, but at a beam energy of E /A = 80 MeV, corroborate the large EMD cross
section for 11Li on heavy targets, as well as the GDR energy of 4.6 MeV as found
in the model described above. It was found, however, that the soft GDR model
predicted a much larger dependence of the EMD cross section on energy than that
which was measured. Very recent measurements by Ieki and collaborators [Ieki 93] of
11Li on a lead target at E /A = 28 MeV also corroborate the large EMD cross section
and the GDR energy of approximately 4.6 MeV. Application of the soft GDR model
yields a soft GDR energy of 0.7 MeV. It is noted in their report that such a GDR
energy corresponds to a classical oscillation period of 1240 fm/c. The measurements
of Ieki et al. are kinematically complete, and it is observed that the 9L1 fragments
from the 11Li breakup suffer a significant Coulomb acceleration in the electric field
of the lead target nucleus. Such a large acceleration implies that the 11Li projectiles
break up very near to the target and consequently that the soft GDR, if it exists,
lives for an extremely short time. A quantitative treatment of the data places the
upper limit on the soft GDR lifetime at 85 fm/c, a value significantly less than the
classical oscillation period. This raises the question of whether the soft GDR exists.
It is possible that the breakup actually took place, not via excitation of a soft GDR,
but through a more direct channel. If not, the question still remains as to whether
it make sense to speak of the existence of a resonance that decays well before one

oscillation period.

 

200

 

 

El Strength (arb)
8
Virtual-photon Intensity [Mgv'l ]

 

 

 

0 10 20 30
Excitation Energy [MeV]

t t

G G)

p n

soft GDR normal GDR

Figure 2.4: [Koba 89] Schematic representation and intensities of soft and normal
giant dipole resonances in 11Li. The solid line in the graph is the virtual-photon
spectrum for 11Li on lead at E /A = 800 MeV. The energies and intensities, as cal-
culated from the TRK sum rule, of the two dipole resonance modes is shown by the
black bars. Even though the normal GDR has a greater intensity as predicted by
TKR, when weighted by the virtual-photon spectrum, the soft GDR dominates.

16

Clearly, llLi presents entirely new nuclear behavior and exhibits phenomena which
challenge traditional nuclear models. It has been shown [Hans 87] that, qualitatively,
the existence of a neutron halo has a very sensitive dependence on the binding en-
ergy of the halo neutrons. For this reason, it is expected that halo behavior should
be exhibited by a large number of drip—line nuclei. Indeed, experimental evidence
[Fuku 91, Tani 92, Riis 92] has indicated halo phenomena in the nuclei 14Be, 11Be,
8He, and °He. It is apparent that neutron halos are by no means isolated phenomena,
and that as further experiments are performed, more nuclei will be found to posses
halos. It is becoming increasingly important that the structures of these nuclei be

understood.

III Structure of 11Li — Theory

Experimental evidence, some of which was presented in the previous section, has led
to the theoretical treatment of 11Li as a three—bodysystem comprising a 9Li core
and a diffuse halo of two loosely bound neutrons. The first, and qualitatively the
simplest, such treatment was presented by Hansen and Jonsen in [Hans 87]. In their
model, 11Li is assumed to be a quasi—deuteron consisting of a structureless 9Li core
nucleus coupled to a dineutron 2n. The nuclear potential for this system is idealized
as a radial square well. Under these assumptions, it was found that, outside the well,

the radial wave function is

 

 

_ -1 2 exp(—r/p) eXp(R/p)
\Il(r)—(27rp) / ] r ][(1+R/p)1/2] (2.5)

where 1" denotes the distance between 9Li and the dineutron, and R the radius of the

square well. The decay parameter p is given by

p = h/(2MEB)l/2 (2.6)

17

where p and EB are the reduced mass and binding energy of the system. The matter
distribution, given by [\Il(r)|2, then decays outside the well as exp(—2r/p)/r2. This
simple model indicates that the existence of the neutron halo, that is, the large
radius of the neutron distribution, arises from the very small binding energy of the
two halo neutrons. The Fourier transform of the wave function given above yields the

momentum probability distribution f (p) [Boyd 93]

1
0‘ [p2 + (W212

By inserting the appropriate values for 11Li (,u = %(931.5 MeV/02) and EB z 0.3

 

f(p) (2.7)

MeV) into these expressions, it is found that p z 6.5 fm and I‘,D z 30 MeV/c. Both

results are in reasonably good agreement with experimental data.

Hansen and Jonsen proceed further with their model and postulate the existence
of a low—energy soft dipole excitation, discussed in the previous section, in analogy
with the deuteron. It was found that the Coulomb dissociation cross section for

collisions with a target ng and relative velocity v is proportional to

23...;
EE’ITZ. (2.8)

and, for low beam energies and heavy targets, should therefore be quite large (e.g.
on the order of 5 barns for E /A = 100 MeV incident on uranium). Such a large
cross section has been confirmed experimentally; however, there is still some question
concerning experimental measurements of the lifetime of, and hence the existence of,

the soft dipole excitation as a means of Coulomb dissociation.

Johannsen, Jensen, and Hansen [Joha 90] extended this model by introducing
structure to the dineutron system. In their model, 11Li is treated as a system of
three interacting particles. More specifically, the dineutron is assumed to be in an

S = 0 state, and any spin—dependent interactions between the halo neutrons and

18

the 9Li core, which is assumed to be structureless, are ignored. Both the neutron-
neutron and neutron—9Li interactions are taken as simple Gaussian radial wells. The
depths and widths of the wells are chosen to reproduce the low—energy n—n scattering
data and the approximate size of the 9Li nucleus. The ground state wave function
is then determined variationally for different n-9Li well depths. These calculations
reproduce the sensitive dependence of the matter and momentum distributions on
the neutron binding energy, but disagree somewhat with experiment. Calculations of
the Coulomb dissociation cross section also reproduce the observed trend but again
disagree somewhat with the data. Perhaps the most interesting result of their calcu-
lations, however, is the prediction of strong correlations between the two neutrons in
11Li. This raises the possibility that, although the isolated dineutron system has no

bound state, such a state may indeed exist in the nuclear medium.

Bertsch, Esbensen and Ieki [Bert 91, Esbe 93] approach the dineutron problem in
a different manner. Their model is conceptually similar to that described in [Joha 90],
but more realistic potentials are used for the 71—71 and n—9Li interactions. The n—9Li
interaction was taken to be a sum of Coulomb and centrifugal terms along with a
Woods-Saxon nuclear well and a Thomas—type spin—orbit term. The parameters of
this potential were adjusted to reproduce the binding energies of other nuclei with
neutron and proton numbers similar to those of 11Li. The single-particle states of this
well were used as the space of basis states for the calculation. The n—n interaction

was taken to be a density—dependent contact interaction

Vnn = 5(1‘1a1'2) (’00 + 'Up(P(P1a 1‘2)» (29)

where r1 and r2 are the positions of the two neutrons. The parameters of this reaction
were also adjusted in a manner similar to that described above. The ground state wave

function and binding energy were found by using the two—particle Green’s function,

19

which was expressed as an expansion in the single particle states of the n-—9Li well.
Their model also exhibits a sensitive relation between the 11Li binding energy and
matter radius, as well as a large Coulomb dissociation cross section. The results of
these calculations are also compared with the kinematically complete measurements of
Ieki and collaborators [Ieki 93]. It was found that the calculations were in agreement
with the measured single—neutron and two—neutron momentum distributions, but
that the predicted 9L1 momentum distributions were narrower than those observed

by Ieki and collaborators.

Many other calculations have been performed with other three—body models sim-
ilar to the ones described above [Tosa 90, Zhuk 91, Bang 92, Thom 93a]. A detailed
review of these models is given in [Zhuk 93]. All of these models have succeeded in
reproducing experimental observables such as the 11Li binding energy, halo radius,
and momentum distribution widths to within a factor of two, but they also show
substantial disagreement with one another. One of the most striking points of dis-
agreement is on the degree to which the halo neutrons are correlated within 11Li.
The discrepancies between these models points up the need for more accurate and
complete experimental data for use as input parameters in the calculations as well as

for comparison with model predictions.

IV Predicted 11Li mass and 10Li

All of the three—body models that have been used to describe 11Li assume some form
for the n-9Li interaction. Only until very recently, it was believed that this system
only has one low—lying resonance state, a 111% neutron state unbound by 800 keV
[Wilc 75, Bark 77]. However, as will be discussed in Chapter 4, there is newer evidence

which calls into question the energy of this state as well as the possible existence of

20

 

 

Reference 52,,(11Li) (keV)
Thibault et al., 1975 [Thib 75] 170 i 80
Wouters et al., 1988 [Wout 88] 320 :l: 120
Kobayashi et al., 1992 [Koba 92] 340 :l: 50

 

 

 

Table 2.1: Summary of existing measurements of the two—neutron separation energy

of 11Li.

a lower—lying 23% state. The calculations of Bertsch and Esbensen have indicated
that the predicted 11Li binding energy is quite sensitive to the energy of the 10Li 119%
state, as illustrated in Figure 2.5. Very recent calculations performed by Thompson
and Zhukov [Thom 93a, Thom 93b, Zhuk 93] have included a 2.9% state as well as a
111-;- state in the n—9Li interaction. Their results indicate that the inclusion of both
states, as well as the energies of these states, has profound effects on the predicted
11Li binding energy, as shown in Figure 2.6, as well as other observables. These issues

will be discussed in more detail in Chapter 5.

V Previous 11Li mass measurements

As discussed earlier in this chapter, the existing models of halo nuclei, and in partic-
ular 11Li, all predict a very sensitive dependence of the signature halo phenomena,
such as the halo radius and momentum distributions, on the binding energy of the
halo neutrons. For this reason, it is essential to the understanding of the structure
of 11Li and halo nuclei in general that the mass of 11Li be known as accurately as
possible. There is some uncertainty on the value of the 11Li mass, as can be seen in

Table 2.1, which lists all of the existing measurements.

In 1975, Thibault et al. [Thib 75] reported the first measurement of the mass of

11Li. In their measurement, lithium ions were produced by 24 GeV protons incident

21

 

 

 

 

 

l r 1 1 I
o 1 1 1 x, ’to
‘0
/

-, _ ./ j
s ./
é’ /
v: -21— ’I -1

N z
w 1’
.I
-3]. /’ _
O”,
/
//
-4_ . ’,I -
I”
II’
-5 ' I 1 1 1 4
-I.O -O.5 O 0.5 I.O |.5
6p ( MeV)

Figure 2.5: [Bert 91] Sensitive dependence of 11Li binding energy on 10Li 1p] state
energy as predicted by Bertsch and Esbensen. The solid and dashed lines indicate
calculations assuming correlated and uncorrelated neutrons respectively. Both calcu-
lations assume the n-9Li system has only a single 11)]; state, the energy of which was

treated as a variable parameter.

22

 

 

EB'uDVfi-V, /,/ .
’ A""""A110 keV1swidth .’ .
0.10 P .l ‘
:G"050 keV ./ ]
A IO—OZS keV I.’ 1
> 0'00 TV-Vtokev ,' .
g ; I'
‘3 010 3
8 C
o D
2’ -O.20 f
'3 ~0.30 :
" i
-0.40 [

l

.15 0.20 0.25 A 0.30 0.35 0.40 0.45 A 0.50
Op", resonance energy (MeV)

 

 

 

Figure 2.6: [Thom 93b] Dependence of 11Li binding energy on 10Li 1p% and 23% state
energies as predicted by Thompson and Zhukov. The ordinate is the predicted two—
neutron binding energy of 11Li. The input parameters of these models are the 1p1
neutron resonance energy of the 10Li first excited state (the abscissa) and the 2.9-:-
resonance energy of the 10Li ground state (shown with various plotting symbols). In
these calculations, two potentials were assumed for the n—9Li system, one for 119%
the interaction, and one for the 23% interaction. Also shown in this figure are the

predicted results assuming the same potential for both interactions (ie. V, = Vp).

23

on iridium foils in a target—ion source. The ions were then accelerated by a DC voltage
through a series of slits and magnetic elements into a shielded counter. The 11Li mass
was deduced by comparing the voltages necessary to transport 9L1, the mass of which
is well known, and 11Li through identical trajectories of the optical system. In 1988,
Wouters et al. [Wout 88] measured the mass of 11Li nuclei produced from fragmen-
tation reactions of 800 MeV protons on a thorium target. The mass of the fragments
was determined using the TOF I spectrometer at LAMPF. The value frequently used
in theoretical calculations is the more recent result of Kobayashi et al. [Koba 92].
In their measurement, the 11Li mass was determined from the measured Q-value of
the pion double charge—exchange reaction llB(71",7r+)”Li. This work, however, has
never been accepted for publication, and the details of the measurement and partic—
ularly the sources of the uncertainty are not known. The substantial disagreement
between these measurements as well as the magnitude of their uncertainties limits
their usefulness in theoretical calculations. Chapter 3 contains a description of a
measurement of the 11Li mass that was performed at the National Superconducting

Cyclotron Laboratory in late 1992.

Chapter 3

The Mass of 11Li from the
14C(11B,11Li)140 Reaction

I Introduction

As indicated in the previous chapter, a central quantity to the understanding of the
structure of 11Li is its binding energy. From the theoretical point of view, the current
state of knowledge of this quantity is unsatisfactory and presents a hindrance to devel-
opment of more accurate models. In September of 1992, the 11Li mass was measured
via the Q-value of the l4C(“B,“Li)1“O reaction analyzed in the A1200 fragment sep-
arator at the NSCL. The details of this measurement are presented in this chapter.
A brief description of the A1200 and its operation as a spectrometer is presented in
section II. Section III contains details of the experimental procedures, including the
calibration of the spectrometer and collection of the production data. The analysis
of the experimental data is discussed in section IV. The analysis relied heavily on the
computer code RELMASS, a description of which is given in Appendix A. Particular
attention will also be paid to calibration of the A1200 dipole magnets. Finally, the

results of the measurement will be presented in section V.

24

25

II Description of A1200

Brought online in 1991, the A1200 is a series of magnets and a standardized array
of detectors integrated into a device used primarily for the separation of radioactive
beams at the NSCL [Sher 91]. The device consists of four superconducting 22.5°
dipole bending magnets and several superconducting quadrupole focusing magnets
grouped into sets of two and three. Four room—temperature sextupole magnets are
also used to correct for optical aberrations. The layout of the magnets in the A1200,
which is located in the K1200 beamline immediately downstream of the K1200 cy-
clotron, is shown in Figure 3.1. The magnets are controlled, singly or in gangs, with
a computer program communicating with the magnet power supplies via ARCNET.
Accurate calibration of the magnetic fields versus the power supply current allows
easy manipulation of the magnetic fields, and hence the magnetic rigidity, of the
device. The dipole fields, the most critical values in determining the rigidity of the
device, are continuously monitored by eight N MR probes located in the four dipole

magnets — two probes per magnet, one for high fields and the other for weak fields.

In addition to the magnets, the A1200 also includes a standardized array of de-
tectors for use in analyzing filtered reaction products. This setup consists of a thin
plastic scintillator detector located at Image 1, four parallel—plate—avalanche—counters
(PPACs), with cathodes segmented to achieve position resolution in a: and in y, lo-
cated in pairs at Image 2 and the Focal Plane, and a 10 cm plastic scintillator detector
located at the Focal Plane. Time-of—flight information is obtained, on a particle—by—
particle basis, from the fast signals of the plastic scintillator detectors located at
Image 1 and the Focal Plane. The signal from the thick plastic scintillator at the F0-
cal Plane is also used to obtain the total energy of the reaction products. The PPACs

provide position information for each fragment particle, and, when used in pairs, can

 

 

 

W/fl/fl/ ///////////// /

Triplet «3%;‘9'. . ‘
an 5’ 0390.08] Sutupolu FINAL

J

 

 

 

 

 

 

 

From IMAGE
Efcolgtgvfiuem 53330301600 I%~ . Amoco '2 /4 :F/ ‘E'
] /// // - . , . 0:11;?!
7 A-IZOO BEAM ANALYSIS
42 DEVICE A

 

Figure 3.1: [Sher 91] Schematic layout of the A1200. The device consists of four su-
perconducting 22.5° dipole bending magnets and several superconducting quadrupole
focusing magnets grouped into sets of two and three. Also used in experiments are
a standardized array of detectors (located at Image 1, Image 2 and the Final Image)
and several “retractable platforms, located at several places throughout the device, for
holding other hardware such as targets, degrader wedges, and scintillator screens.

27

provide information about the angle of a particle’s path off the central axis. This
standard setup is very frequently augmented by one or more energy loss detectors,
usually located at the Focal Plane. Examples of such detectors are 5X5 cm silicon
PIN diodes, silicon surface—barrier transmission detectors, and ion chambers. The
energy loss information from these detectors can be combined with the total energy
information or with the time—of—flight information from the plastic scintillator(s) to

obtain particle identification.

Typically, the A1200 is run in an achromatic optical mode. There are two such
modes, distinguished by the momentum resolving power of the mode and the angu-
lar acceptance of the device. The most commonly used optical mode is medium—
acceptance mode, in which the device subtends a solid angle of 0.8 er as seen from
the target. High-acceptance mode is functionally equivalent to medium-acceptance
mode except that the device subtends a solid angle of 4.3 er and the momentum
resolving power is only :1; that of medium—acceptance mode. This means that, while
high—acceptance mode can provide higher overall beam intensities than medium—
acceptance mode, due to its larger angular acceptance, its lower resolving power will
not provide fragment separation as fine as the medium-acceptance mode. Both of
these modes are achromatic, which means that all momentum components (i.e. frag-
ment species) are focused to the same point at the Focal Plane. In both of these
modes, Image 1 and Image 2 are dispersive, meaning that momentum components
are separated at those points. The dispersion is inverted in the bend-plane (i.e. the
high—rigidity and low-rigidity sides are flipped) at the midpoint of the device. This
inversion, located between the dipole pairs, which bend in opposite directions, al-
lows the dispersions of the dipole pairs to cancel each other out, and produces the
achromaticity of these modes. It is at these locations, particularly Image 2, that

detectors and aperture plates are placed to distinguish and filter reaction products.

28

For the experiment described in this chapter, a new optical mode was developed by
B. M. Sherrill and J. Stetson at the NSCL. The chief requirements for this mode, la-
belled the “high—acceptance chromatic” mode or “spectrometer” mode, were twofold.
To overcome the low cross—section for 11Li production, as high an angular acceptance
as possible was needed; and, in order to use the device as a momentum spectrometer,
a dispersive (i.e. chromatic) image was needed at the Focal Plane. In this mode,
there are no images other than the dispersive one at the focal plane. The dispersion
is inverted inside the second dipole pair. The location of this single inversion allows
the dispersions of the dipole pairs to reinforce each other, thereby producing a disper-
sive image at the Focal Plane. Development of this mode involved extensive optical
calculations, performed by Stetson and Sherrill, as well as several experimental tests

of these calculations.

III Description of experiment

The experiment was performed with an E /A = 32.137 :l: 0.024 MeV, 11B5+ beam from
the K1200 cyclotron, focused onto a self—supporting l"C foil target, 0.450 mg/cm2
thick. The reaction products were analyzed with the A1200 fragment separator set
to a high—angular—acceptance chromatic mode as described in the previous section.
For this experiment, the A1200 detectors were, with the exception of those in the
Focal Plane, identical to those of the standard setup described earlier. The A1200
Focal Plane detectors consisted of one PPAC, a 0.5mm thick silicon position—sensitive
detector, which was located at the focal point of the device, and a 10 cm scintillating
plastic stopping detector. Redundant and unambiguous particle identification was
obtained by combining the energy loss signal from the silicon detector with the to-
tal energy signal from the plastic and with the time—of—flight information obtained

from the scintillator signal relative to the cyclotron rf. Position information at the

29

dispersive focus was obtained from the silicon detector and was combined with posi-
tion information from the PPAC, located 50 cm upstream, to find the angle of each

particle’s trajectory.

llLi production reaction

The 11Li nuclei were produced from the 1"C(11B,11Li)1"O reaction. The experiment
consisted of, in addition to the calibration runs described below, two production runs
of approximately 50 hours each, separated by a period during which the beam was
refocused onto the target and the A1200 magnetic field settings were changed slightly
in order to observe the 14O excited state simultaneously with the ground state. The
cross section for this reaction was determined from the 149 11Li nuclei obtained in

both runs to be 24 i 2 nb/ Sr at 0° in the laboratory frame.

The particle—identification (PID) spectrum from the first run, obtained by his-
togramming the energy loss (AE) signal from the silicon detector versus the time—
of—flight (TOF) information obtained from the 10 cm scintillator detector and the
cyclotron accelerating rf, is shown in Figure 3.2. The PID spectrum from the second
run has identical features. In addition to llLi particles, 10Be3+ particles were seen in
the Focal Plane. In the notation used here, the 3+ indicates that the 10Be nucleus
had a charge of +36, as opposed to a 10Be fully stripped of its electrons, which would
have a charge of +46. These nuclei were produced in the 14C(llB,“’Be)1°N reaction
and were used for calibration of the A1200, as described below. Several other nu-
clear species were also seen in the Focal Plane during these runs, and are labelled in
Figure 3.2. The extreme cleanliness of the PID spectrum is to a large part due to
the very high rigidity of the 11Li particles. Most species produced prolifically have a

much lower rigidity and are not transported to the Focal Plane.

The momentum spectra for the first and second production runs are shown in

30

 

 

. 11B3+

a “Bea"

A.‘ .fg'?a?or?pr 5;:

11u3+

. ”rum.- ‘

 

 

 

 

Time of Flight

Figure 3.2: Particle—identification spectrum from 11Li production run. The spectrum
was obtained by histogramming AE and TOF information taken from the silicon
detector and the thick plastic scintillator detector respectively. In addition to 11Li
nuclei, “’Be5‘+ particles, which were used for calibration of the A1200, and several
other nuclei were observed as well.

31

the bottom part of Figure 3.3. For both runs, there is literally no background in the
spectra. In addition to the primary peaks, corresponding to the ground states of both
11Li and 140, another peak, corresponding to unresolved states in 140 near 6.3 MeV
excitation energy, is seen in the data from the second run. While it is possible that
an excited state of 11Li could also be embedded in this peak, it is clear from the data

that there is no indication of such a state at higher excitation energies.

Calibration reactions

In addition to data from the 11Li production reaction, data were also collected from
the l4C(“B,1°Be)15N reaction simultaneous to the production data. This reaction,
which has a well—known Q—value, was used to calibrate the A1200. The calibration
momentum spectra from both runs are shown in the top part of Figure 3.3. The
ground state and unresolved 5.3 MeV doublet states of 15N were used as the primary
calibration points. Also seen in the calibration spectra are a cluster of 15N and
10Be excited states, corresponding to a total excitation energy between 8.0 and 10.0
MeV, and the 3.37 MeV first excited state of 10Be, which shows marked kinematic
broadening due to the recoil from the isotropic in—flight gamma decay. Also studied
was the 14C(nB,9Li)1°O reaction, the data from which was combined, as described in

the next section, with the calibration data to determine the beam energy.

IV Analysis

Analysis of the experimental data consisted of a multi-step process that culminated
in two experimental measurements of the Q-value of the 1“C(“B,”Li)”O reaction,
from which the mass, or equivalently, the two—neutron separation energy, of 11Li
was deduced. This process, described below, was greatly simplified by the computer

code RELMASS, written by Toshiyuki Kubo with suggestions from Ed Kashy. A

32

 

 

 

 

 

 

 

 

 

 

 

First Run Second Run
4000 b I I I I I I I I rrI FT r I I I I I I I I I I I I I I I I T I l I I I .1 4000
: ,, “C(”B,’°Be°+)’°N 0” “C "B “Be“ “’N E
3000 :- :z 33 ( ’ ’ —; 3000
E " '2‘ 2 3
2000 :— 2 02 g 2000
,, s is. 3
510007 03 -: 1000
e i i... 3
> O: iiii Iii-d III IIIJT: 0
.. _ :
g 15 L'— I‘C(IIB.IILi)I‘O —-:- 25
o L’ I
o _ I | lsan —: 20
1- . -1
10 :— “C(“B.”Li)“0 :9 -10 1 —; 15
5 :_ —; 10
5 — 5
o -1 l l l l l L I l l l l J .l l l 1 d

50 100 150 200 50 100 150 200 0
Channel

C

Figure 3.3: Momentum spectra from llLi production and calibration runs. The data
from the first and second runs are shown in the left and right portions of the figure, re-
spectively. The momentum spectra from the reaction l“C(11B,1°Be3+)‘5N"' are shown
in the top part of the figure. The ground state and unresolved 5.3 MeV doublet states
of 15N were used as the primary calibration points. Other features in the calibration
spectra are a cluster of 15N and 10Be excited states, and the 3.37 MeV first excited
state of 10Be. The momentum spectra collected from the reaction l4C(uB,"Li)”O
are shown in the bottom part of the figure. It is important to note that both the
calibration and 11Li spectra were collected simultaneously.

33

description of the code’s operation is given in Appendix A.

The first run was analyzed first. For a given beam energy, the kinetic energy of
the 10Be ejectile from the 1"C(llB,1°Be)1°N calibration reaction can be found, after
including energy loss effects of the beam and ejectile in the target material, from the
well—known Q—value of the reaction. The magnetic rigidity, defined for a particle
with charge Q and momentum 1) moving in a uniform magnetic field as the product
of bend radius and field strength, can be found by balancing the Lorentz force with

the centrifugal force

Bp : (3.1)

<O|~=3

From this definition, it is clear that the ejectile particle’s magnetic rigidity can also
be found from its kinetic energy. For a given A1200 magnetic field, particles with
different momenta have different bend radii and arrive at different positions in the
Focal Plane. The data from the l“C(11B,1°Be)15N reaction were used to calibrate the
focal plane position as a linear function of bend radius, obtained from the known
rigidities and magnetic field. Since the radius—position (p vs 0:) line is a property of
the spectrometer, independent of the reactions studied, all reaction products must lie

on this line.

Since the l4C(llB,9Li)l°O reaction also has a well-known Q-value, the beam en-
ergy uniquely determines the rigidity of the 9Li ejectile. Utilizing this fact, the beam
energy was obtained by requiring that the measured position and the calculated bend
radius of the 9Li particles lie on the same calibration line as the two states from the
l4C(uB,mBe)15N reaction, as shown in Figure 3.4. The beam energy determined in
this manner was E/A = 32.137 :l: 0.024 MeV. The uncertainty of the beam energy re-
flects the statistical uncertainty of the position measurement of the 9L1 ejectiles. From

the calibration curve, the 11Li rigidity can then be determined from its measured Fo-

34

cal Plane position. The rigidity can then be combined with the beam energy to obtain
the measured Q—value for the production reaction, from which can be deduced the

11Li mass.

The second run was analyzed in a manner similar to that described above. Before
the second run, the beam was refocused onto the target, and the magnetic fields of the
A1200 were decreased slightly. Both of these actions affected the calibration curve.
As the magnetic field was decreased, the momentum spectra shifted to the right on
the Focal Plane, as seen in Figure 3.3, with the result that only one calibration peak,
corresponding to the 5.3 MeV doublet in 15N, remained on the Focal Plane. When
the beam was refocused onto the target, the position of the beam spot was changed.
This change would affect the ofiset of the calibration line, but not the slope, which
reflects a property of the A1200 magnets and optics, namely the degree to which a
change in a particle’s bend radius changes the particle’s Focal Plane position. Thus,
the calibration line for the second run was obtained by keeping the same slope as
before, and determining the new offset from the single calibration peak. This new
calibration curve was then used to determine the Q—value of the 1"C(nB,“Li)1"O

reaction for the second run.

Central to all A1200 measurements is knowledge of the dipole magnetic fields,
for these determine the magnetic rigidity of the particles which arrive at the Focal
Plane. The dipole fields are measured by NMR probes in each of the magnets. These
probes, however, measure the field at only one point in each dipole magnet, whereas,
in actuality, the desired quantity is the average field over the trajectory of a given
particle. The difference between this effective field and the field measured by the
NMR probe is typically less than 0.1% and for most purposes, such as the production
of a radioactive beam or the measurement of a fragment momentum distribution, is

negligible. However, for the measurement reported here, this difference, which varies

35

 

140

120

100

 

O 14C(11B. 10B8)15N

    

80
D 14C(11B Du)160

. Position [channel of 256]

IIITrlIIIIIWIII1IIIITI

60

.]
lJllllLllLllllllJlllllJl

 

 

d

I l l L l l I l l L l L l

3.09 3.10 3.11 3.12
Rho [meter]

 

Figure 3.4: p vs a: calibration of A1200 Focal Plane for first 11Li production run. A line
was fitted to the two points from the l“C(11B,‘°Be)"’N reaction, shown as diamonds
in the figure. The beam energy was determined by requiring that the 1“C(11B,9Li)1°O
points, shown as squares, lie on the calibration line. The error bars for the points are
approximately one fourth the size of the symbols used for plotting.

36

with the strength of the field and is attributable to fringe field effects and to the

magnetic properties of the iron from which the magnets are made, is quite important.

The dependence of the effective field Beg on the field as measured by the NMR
probes BNMR was measured using two techniques. The following discussion assumes
that the two quantities, Beg and BNMR, are related by a correction term 77 which is

dependent on BNMR

Beff = BNMR [1 + 77(BNMR)l- (3.2)

The first technique involved the calibration of the S320 magnetic spectrograph, de-
scribed in the next chapter, using the momentum—matching method detailed in
[Nole 74]. A molecular beam (H—He)2+ and its constituent nuclei, H1+ and He“,
obtained by fragmenting the molecular beam on a thin aluminum foil located at the
target position of the A1200, were then transported through the A1200 and into the
S320. The A1200 magnets were then calibrated by comparing the rigidities of the par-
ticles, as measured with the S320, with the A1200 magnetic fields BNMR, as measured
by the N MR probes in the A1200 dipoles. The three data points obtained in this way
are shown as diamonds in Figure 3.5. In the second technique, a beam of 238U35+ with
energy E /A = 20.08 :1: 0.02 MeV was focused onto a thin aluminum foil at the target
position of the A1200. By interacting with the target material, the uranium particles
lost electrons and emerged with a broad distribution of charge states which were then
transported through the A1200. The A1200 magnets were calibrated by comparing
the known rigidities of these charge states with the measured BNMR values. The data
points thus obtained are shown as squares in Figure 3.5. A polynomial curve was
fitted to the data and is shown as the solid line in the figure. Also indicated, by
the vertical lines, is the range of fields over which the 11Li mass measurement was

performed. It is seen that the effective magnetic field has a weak dependence on the

37

NMR value in this region.

V Results

The deduced values for the two-neutron separation energy of 11Li from the first
and second runs are $2,,(11Li) = 301 :t 24 keV and 32,,(11Li) = 288 :l: 26 keV respec-
tively. The uncertainties in these values reflect statistical uncertainties in deter-
mination of the centroids of the 11Li peaks. The average of these two values is
52,,(11Li) = 295 i 18 keV. The uncertainty here is also only statistical. To take into
account the uncertainty in the beam energy, the data were re—analyzed with the
incident energy increased by one sigma. It was found that this contributed an un-
certainty to the 11Li binding energy of 23 keV. Uncertainty in the calibration of the

A1200 dipole fields was also found to have an 11 keV contribution.

There is an additional contribution from the ,0 vs :10 calibration due to uncertainty
about the relative strengths with which the states of the unresolved 15N doublet were
populated. The two states have excitation energies of 5.270 and 5.299 MeV. In the
analysis, it was assumed that these states were populated equally, and that the peak
corresponded to the average excitation energy of 5.285 MeV. The contribution to
the final uncertainty from this assumption, obtained by re—analyzing the data twice
assuming 100% population of one or the other of the doublet states, was found to be
15 keV. The above analyses were performed assuming the ejectiles were detected at
0° in the laboratory frame. Although the A1200 subtends a finite solid angle in the
forward cone centered about 0° in the laboratory frame, and therefore the average
laboratory angle of the reaction products is not 0°, it was found that this made a

negligible difference in the final result.

The individual contributions to the overall uncertainty are listed in Table 3.1.

38

 

 

 

 

 

 

 

 

 

0.002 I I I I I I I I I 1 I I I I I I I I I I I I I I I T I ‘
F
A D D _
0.001 _ - v 1:1 DLPD _
El D

r 0 Measured with $320 '1
p 0.000 ;_ 0 Measured with mU _:
- charge states -
-0.001 r- —
- Range of fields —=- <— ° -

_0.002 1 l l l I L l 1 LI 1 l l l I l l l L I l l l l I l IJ

0.00 0.25 0.50 0.75 1.00 1.25 1.50
Em (kG)

Figure 3.5: Calibration of A1200 dipole magnets. The effective A1200 mag-
netic field Ben is related to the field BNMR as measured by the NMR probes by
Beg =(1+1])BNMR. The diamonds are those data points obtained by comparing
BNMR with rigidities as measured with the S320. The squares are those data points
obtained from rigidity measurements of uranium charge states. The solid curve is a
polynomial fit to the data. The range of fields over which the 11Li measurement was
performed is indicated by the vertical lines. The central radius of the A1200 is 3.104
meters. The error bars for the points are approximately one tenth the size of the
symbols used for plotting.

39

 

 

 

Source of uncertainty 0 (keV)
statistics 18
beam energy 23
magnetic field correction 11
15N excited state population in calibration 15
total uncertainty 35

 

 

Table 3.1: Sources of experimental uncertainty in 11Li mass measurement. The four
uncertainties listed are added in quadrature to yield the total uncertainty.

When these values are added in quadrature to obtain the final uncertainty, the re-
sulting two—neutron separation energy for 11Li is 32,,(11Li) = 295 :l: 35 keV. The cor-
responding Q—value for the 1“C("B,“Li)14O reaction is —37.120 :t 0.035 MeV. As can
be seen in Table 3.2 this result is in good agreement with the previous measurements
while substantially lowering the uncertainty. Using the existing four measurements,
the weighted best values for the 11Li mass excess and two-neutron separation energy
are 40.802 :1: 0.026 MeV and 295 :l: 26 keV respectively. This places the accuracy of
these quantities, which are essential to the understanding of the structure of 11Li and
halo nuclei in general, on a level high enough to be used with confidence in nuclear

halo models.

40

 

 

Reference 52,,(11Li) (keV)
Thibault et al., 1975 [Thib 75] 170 :l: 80
Wouters et al., 1988 [Wout 88] 320 :t 120
Kobayashi et al., 1992 [Koba 92] 340 :l: 50

 

Present work 295 :l: 35

Average value 295 :l: 26

 

 

Table 3.2: Summary of existing measurements of the two—neutron separation energy
of 11Li, including the present work. Also listed is the weighted average of the four
existing measurements.

Chapter 4

Structure Studies of 10Li

I Introduction

All of the three—body models that have been used to describe 11Li assume some
form for the n—9Li interaction. Until very recently, it was believed that this system
only has one low—lying resonance state, a 111% neutron state unbound by 800 keV.
However, new evidence calls into question the energy of this state and also raises the
possibility that a 2.9% state exists at a much lower energy. Calculations performed by
Bertsch and Esbensen[Bert 91] and Thompson and Zhukov [Thom 93a, Thorn 93b,
Zhuk 93] have shown that the predicted 11Li binding energy is very sensitive to the
number of low—lying 10Li states as well as their energies and spectroscopic structures.
Thus, a thorough understanding of the structure of 10Li will shed a great deal of
light on the structure of 11Li. A review of the existing measurements of the 10Li
ground and/or first excited states is presented in section II. In May of 1992, the
low—lying structure of 10Li was measured from the momentum spectra and Q—value
of the 11B(7Li,8B)1°Li reaction analyzed with the S320 magnetic spectrometer at
the NSCL. Sections III and IV contain a description of the 8320 and the details
of the experimental procedures, respectively. The analysis of the experimental data

is discussed in section V. Part of the analysis consisted of determination of the

41

42

shapes of the 10Li neutron resonances. The theoretical assumptions and computer
codes used in these calculations are presented in Appendix B. The 11B(7Li,3B)’°Li
reaction, populating the low—dying states of 10Li was found to have a cross section
of approximately 9.5 :l: 0.7 pb/Sr at 5° in the laboratory frame. Such a low cross
section resulted in very poor statistics (i.e. less than 15 events per channel) in the
collected momentum spectrum. For this reason, the standard least-squares fitting
technique, the statistical assumptions for which rely on a large number of events per
channel, was inapplicable to the data. A more general maximum—likelihood fitting
technique was employed. This technique, and the computer code through which it
was implemented, is presented in Appendix C. The results of this analysis and the

conclusions that can be drawn from the data are discussed in section VI.

II History of 10Li

If the neutron configuration of 10Li is taken to be that of other N = 7 nuclei, such as
11Be and 12B, then the lowest states should have 6 neutrons in their lowest shell model
orbits and the seventh neutron in the 111% or the 2.9% orbits as illustrated in Figure 4.1.
The combination of these two possible neutron orbits with the 119% orbit occupied by
the third proton in 11Li yields four possible J’r values for the lowest states of 10Li:
1+, 1‘, 2+, and 2‘. In the first measurement of the 10Li neutron separation energy,
Wilcox et al.[Wilc 75] determined the energy of 8B in coincidence with 9Li (from the
breakup of 10Li) produced in the reaction 9Be(9Be,8B)1°Li at 121 MeV. The 8B ejec-
tile and the 9Li recoil daughter were observed in semiconductor detector telescopes
positioned respectively at 14° and approximately 10° in the laboratory and approxi-
mately 50 cm from the target. Their energy spectrum, shown in Figure 4.2, exhibits a
pronounced peak corresponding to a neutron—unbound state with separation energy

5,, = -0.80 :l: 0.25 MeV and width Fem, = 1.2 :l: 0.3 MeV. It was assumed that this

43

 

 

 

 

 

 

.— 23(1/2)

1 d(5/2)

f 1p(1/2)

W W 1 p(3/2)

 

 

_.—p—.——.T1_.— —.';'.——.'n—.— 1 8(1/2)

11139 1213

Figure 4.1: Shell structure of the N = 7 nuclei 11Be and 12B. The 11Be ground state
has the seventh neutron in the 2.9-;- orbit. The 12B ground and first excited states (as
well as the 11Be first excited state) have the seventh neutron in the 11):],- orbit.

44

 

201-10) I I I T I I I 4
121 MeV ’Be "c1’ee.'e)"e
'5 :- Glob. |4. '760 #C 3
IO (3.48-3.71)
"B q-s.
I
11 Fill]! 1

 

 

 

 

 

’Be ( '50 .'B) '°Li
singles
20 c 6950 p0 ‘1
2
S 10 4
o
o
o L g 1 m i
(C) .. - ’Bel’Be. 'e) '°Li
'0 ' L' 3‘” coincidence “
i 69501.1.C
'Lhn
n o - 1’
5 _ L1? 2n 1 .1
I
o, 1 [El—[iii 4 1_ 1 1
60 70 80 90

 

Particle energy (MeV)

Figure 4.2: [Wilc 75] Energy spectra of 8B from the 12C(9Be,3B)13B and
9Be(9Be,8B)1°Li reactions at E /A =13.4 MeV and 01.1, = 14°. 3) Calibration spec-
trum of 8B from the 12C(9Be,8B)13B reaction. b) and c) Energy spectra of 8B from
the 9Be(9Be,°B)1°Li reaction, singly and in coincidence with 9Li from the breakup
of the recoiling 10Li. The coincidence spectrum shows a peak corresponding to a
neutron—unbound state with separation energy 5n = —0.80 :l: 0.25 MeV and width
Fem. = 1.2 i 0.3 MeV.

45

was the ground state of 10Li. Soon thereafter, Barker and Hickey [Bark 77] argued,
from a systematic study of the energies and parities of excited states of known N = 7
nuclei, that the 10Li ground state should be a 112.9% state which is narrow and much
less unbound to neutron decay. They further suggested that the state observed by

Wilcox et al. was an excited le% state.

In 1990, Amelin et al. measured the inclusive spectrum of protons produced in
absorption of 1r‘ by 11B nuclei, shown in Figure 4.3, and reported the observation of
a state in 10L1 unbound to neutron decay by approximately 150 keV [Amel 90]. In
late 1992, Kryger et al. analyzed the relative velocity spectrum of neutrons collected
in coincidence with 9Li fragments from the decay of 10Li [Kryg 93]. Their spectrum is
shown in Figure 4.4. The dotted curve in the figure is the expected spectrum from a
thermal neutron background. The solid curve represents the background added to the
expected line shape from a low—energy decay very similar to that reported by Amelin
et al.. It is important to note that while this curve fits the data quite well, such a
peak in the relative velocity spectrum only indicates the presence of a very low-energy
neutron decay which could either be a low—lying 10Li state decaying to the 9Li ground
state, or a 10Li excited state at Sn 2 -—2.5 MeV decaying to the first excited state of
9Li. The dashed curve is the expected spectrum from a broad state unbound by 800
keV, similar to that reported by Wilcox and collaborators. The two—peaked structure
in this calculated spectrum arises from the fact that, as the 10Li decays in flight, the
acceptance defined by the experimental apparatus allows detection of only those 71—-
9Li events where the decay is parallel or antiparallel to the beam axis. The absence
of such a two—peaked structure in the data fails to corroborate the observation of
Wilcox and collaborators of a broad 10Li state unbound by 800 keV. An observation
similar to that of Kryger et al. has also been reported by Kobayashi and collaborators

[Koba 93].

 

 

 

46

g 3 2 t a 5,. MeV
(n '___I-_I'—-'1"__l"—_
E

1 _ ]

h

8. 2- n“ '

3 " |

2 p ‘ ‘ I ["\

a -“ '

c I . I, \

9. I 1

9 _ I 11

Q [I 1‘ J

'1 a 44/ \ .2
2 92 94‘ 95 Em MeV

Figure 4.3: [Ame190] Energy spectrum of protons from the reaction 11B(1r",p)1°Li.
The dashed curve is a theoretical calculation, a Breit-Wigner form folded with the
resolution of the spectrometer, of the shape of a low—lying neutron resonance. The
solid curve is a sum of the dotted curve and a three—body phase space background, also
folded with the spectrometer resolution. The peak corresponds to a neutron—unbound
state with separation energy 5n = —0.15 :l: 0.15 MeV and width Fem, < 0.4 MeV.

47

 

 

 

 

 

400 II_I ff I I IV I I I T I I—r I VI I I I T I l_
300 '— i _.
| a:
m - I .1
4—3 ' 1' .1
1:: 200 l— . , ._.
:3 ' l ‘
o l- ‘ d
o I 1 j I
100 — r1 \ 3.: , _.
.. ' ." - ..... - -,"”I‘ :
. 5'" 1. * .
. 'I .I‘ I

1- :_._" w,‘ | 45-, .. .‘ .1

0 -4» ‘ ’
_1Ll11L1L1 LL1|IIILLL+L1111

—4 -2 0 2 4
V — VF (cm/ns)

I1

Figure 4.4: [Kryg 93] Relative velocity spectrum of neutrons collected in coincidence
with 9Li fragments from the decay of 10Li. The dotted curve in the figure is the
expected spectrum from a thermal neutron background. The solid curve represents
the background added to the expected line shape from a low-energy state unbound
by approximately 150 keV. The dashed curve is the expected spectrum from a broad
state unbound by 800 keV.

48

Bohlen and collaborators have recently reported on Q-value measurements of the
9Be(13C,12N)10Li and 13C(1“C3,17F)10Li reactions at energies of E/A =25.8 MeV and
E/A 224.1 MeV, respectively [Boh193]. The data from these reactions, collected
with the Q3D magnetic spectrograph at the Hahn—Meitner—Institut, are shown in
Figures 4.5 and 4.6. Both spectra contain broad peaks which Bohlen et al. claim
consist of two 10Li states: one, the ground state, neutron—unbound by 0.420 i 0.050
MeV, and the other, the first excited state, unbound by 0.800 :1: 0.094 MeV. Arguing
from reaction systematics, the authors further assert that both states are 117% neutron
states with J " values of 1+ for the ground state and 2+ for the excited state. Bohlen
et al. speculate that Wilcox and collaborators observed the first excited state and

not the ground state.

The experimental results published to date, summarized in Table 4.2, reflect a
substantial disagreement concerning the energy and spectroscopic nature of the 10Li
ground and first excited states. It seems fairly certain that the state observed by
Wilcox and collaborators is not the 10Li ground state. Several groups [Ame190,
Kryg 93, Koba 93] have presented evidence for a weakly unbound 10Li ground state.
The existence of a weakly unbound 112.9% ground state would be in agreement with the
suggestions of Barker and Hickey, as well as the shell model predictions of Warburton
and Brown[Brow 92, Warb 92]; these shell model calculations, however, also predict
that the V1p§ first excited state should have an excitation energy of only about 100
keV. Also supporting the existence of this state is the observation by Abramovich
et al.[Abra 73] of a T = 2 state in 10Be which, with Coulomb energy systematics,
gives 5n = —60 keV for 10Li. Bohlen and collaborators, however, claim that the 10Li
ground state is neither weakly unbound nor a 112.9% state. In the work reported in the
following sections, the low-lying structure of 10Li was deduced from the momentum

spectra and Q—value of the llB(7Li,SB)l°Li reaction measured with the S320 magnetic

49

 

 

 

 

 

 

 

 

 

 

 

 

 

~71? Tfi'rfv {YTE 4.05Mev
F- _ | C
. 50:
C 50’ gl.s.
: 9Be(‘3c,‘2N)‘°Li 038' C
h- 40E.
1401 336 MeV .
: 3.3°-4.3° 30,?
120: 20,} A
A
i ”i All
100— P
I O 6 «i 2 0
I- 6' [MeV]
2 80r- -4
g . 4.05 MeV j
o - «
p.- v 9.5. fl
0 60; ‘50'->'2N+t l 1
4O;LF:VAV 4
: ‘3N°->‘2N+n :
2°» 0 T
: 'IJN‘—>‘2N-l+-n on ‘ZCImO I
O 30.0 20.0 10.0 o '
E. [MeV]

Figure 4.5: [Bob] 93] Energy spectrum of 12N from the 9Be(13C,”N)1°Li reaction at
E /A =25.8 MeV and 01.5 = 3.8°. The spectrum near low excitation energy contains,
in addition to a peak from a target contaminant, a broad peak which the authors
claim comprises two 10Li states. The peaks are fitted with symmetric line shapes, the
widths of which are determined from R-matrix calculations.

50

 

 

 

 

 

 

 

 

 

10 v F r Y a . T . T J
.. 13C(“C,‘7F)10Ll ..
i- 4
8- 337 MeV -
. .4
_ 3.e°-7.0° .
2 6~ ~
C r- 10Li(2+) q
a » 4
U " 1O - + ‘
4__ 17’:- L'(1 ) .4
>- -+
21- a
‘ l
_ L

1 ii .1

0 10.0 0

5' [MeV]

Figure 4.6: [Bob] 93] Energy spectrum of 17F from the 13C(14C,”F)1°Li reaction at
E /A =24.l MeV and 01.}, = 5.4°. The spectrum contains a very broad peak which
the authors assert consists of the same two loLi states as in the previous figure, and
an excited state (Eex = 0.5 MeV) of 1"F. The peaks are fitted with symmetric line
shapes, the widths of which are determined from R—matrix calculations.

51

spectrometer at the NSCL.

III Description of S320

The 3320 is a QQDMS (Quadrupole — Quadrupole — Dipole — Multipole — Sextupole;
where the Multipole is an octupole) spectrometer used for nuclear reaction analysis
and spectroscopy [Plic 92, Sher 83]. The separating element of the S320 is a 34.4°
bending dipole magnet. The field intensities of the magnets have been accurately
calibrated versus the power supply currents and are controlled via ARCNET in a
manner similar to that employed with the A1200. The dipole field is continuously
monitored by three probes located at one position in the dipole magnet, each probe
sensitive to a different range of field strengths. The spectrometer, depicted in Fig-
ure 4.7, is located in the N1 vault at the NSCL and is mounted on a carriage that
pivots about the target location to allow measurements at laboratory angles from
—4° to 55°. The spectrometer subtends a solid angle that is adjustable via aperture
plates to a maximum of 70 er. The detector box is removable, thereby allowing easy
reconfiguration of the detector setup. There is, however, a standard array of particle
detectors that is used for most experiments. These detectors can be configured to
study fast, light ions (“light—ion mode”) or slow, heavy ions (“heavy—ion mode”).
For the experiment described here, the detectors were run in light—ion mode. In this
mode, the standard configuration consists of a stack of five detectors: a position—
sensitive single wire proportional chamber (SWPC), located at the focal position of
the spectrometer, followed by two ionization chambers (IC), another SWPC, and a
large composite block of scintillating plastic. The SWPC’s are both 0.5 inches thick
and the 10’s are both 6 inches thick. All four detectors operate in the same gas
volume. The scintillator block consists of a thin (0.02 inches) sheet of fast scintillator

followed by a 10 cm block of slow scintillator. The light output from the scintillator

Detector b0!

 

 

 

 

C ryopuxnp
Sextupole
Dipole
Quadmpola
Crmnmv
Wedge extension
Scattering chamber

Figure 4.7: [Plic 92] Schematic layout of S320 magnetic spectrometer. the device
consists of two quadrupole focusing magnets, a 34.4° bending dipole magnet, an
octupole magnet (not labelled in figure, but located just after the dipole), and a
sextupole magnet. While the detector box is removable to allow easy reconfiguration
of the detector setup, there is a standard detector assembly that is used for most
experiments. The entire spectrometer is mounted on a carriage that pivots about the
target location in the scattering chamber to allow measurements at laboratory angles

from —4° to 55°.

 

53

is collected by two phototubes. The SWPC’s provide position information and, when
used in pairs, can also provide information about a particle’s angle of entry into the
focal plane. Redundant energy loss information is obtained from the two IC’s and
total energy information is obtained from the light output of the scintillator block.
Time—of—fiight information is obtained from the fast signal from the scintillator and
the cyclotron rf. Particle identification is found on a particle—by—particle basis by

combining energy loss and time—of—fiight information.

IV Description of experiment

The experiment was performed with an E/A = 18.772 :l: 0.054 MeV, 7Li1+ beam
from the K1200 cyclotron. For the production runs, the beam was transported from
the K1200, through the A1200, which acted as a passive beamline, and focused onto
a 0.125 mg/cm2 thick self—supporting llB foil target, located at the target position
of the S320. A 12C (natural) target, 0.56 mg/cm2 thick, was used for the calibration
runs. The reaction products were analyzed with the S320 magnetic spectrograph with
an overall resolution (FWHM) of 0.23 MeV. The focal plane detectors were config-
ured in the standard light—ion mode described above. The energy loss signal from
the ionization chambers and the time—of—flight, taken from the scintillator signal rela-
tive to the cyclotron rf, provided unambiguous particle identification (PID). Position
information was obtained from the front wire chamber, located at the focus of the

spectrometer.

10Li production reaction

The 10Li nuclei were produced from the 11B(7Li,8B)wLi reaction. Data were collected
at laboratory angles of 5.23° and 3.73° for approximately eight and ten hours, respec-

tively. The PID spectrum from the 5.23° data, obtained from the energy loss in the

 

 

54

first ion chamber and the time—of—fiight information from the scintillator versus the
cyclotron rf, is shown in Figure 4.8. The PID spectrum from the 3.73° data has iden-
tical features. In addition to 8B particles, several other nuclei were observed and are
labelled in the figure. The momentum spectrum for the 5.23° measurement is shown
in the bottom portion of Figure 4.9. The most striking feature of this spectrum is a
broad peak centered near channel 260, corresponding to a neutron separation energy
of approximately —500 keV. Also notable is a weak narrow peak in the spectrum at
a neutron separation energy close to zero. While this is not as well pronounced as
the broad peak, it does not appear to correspond to any likely target contaminants.
As discussed earlier, it is expected that the 10Li ground state might have the valence
neutron in the 23% orbit and the first excited state could have a 119% valence neutron.
Since an s-wave neutron resonance at 5,, S —500 keV is expected to be too broad to
observe, the broad peak in the data is believed to correspond to one or more p-wave
resonances, while the narrower, less unbound peak could be an s—wave resonance
belonging to the 10Li ground state. The cross section for the reaction at 5.23° to
populate these peaks was measured to be 9.5 :l: 0.7 pb/ Sr. The momentum spectrum
for the 3.73° run is shown in Figure 4.10. During this portion of the experiment,
the beam current from the K1200 cyclotron had diminished considerably from ap-
proximately 110 nA, as was measured during the 5.23° run, to approximately 75 nA.
This, combined with a smaller solid angle, due to a smaller aperture placed on the
spectrometer entrance to reduce background at the more forward angle, resulted in
very poor statistics for this spectrum. For this reason, the 3.73° data are not included

in the subsequent analysis.

 

 

 

AE

 

 

 

 

Time of Flight

Figure 4.8: Particle—identification spectrum from loLi production run at 01.}, = 5.23°.
The spectrum was obtained by histogramming AE and TOF information taken from
the first ion chamber and the thick plastic scintillator respectively. Several nuclear
species other than 8B were also observed.

 

 

 

 

 

 

56
60 :T I I I I I I I I I I I II II I I I I I I I I I I1 I:
50E—- a 3 -§

2 a 3
4°;- ~ .5 1
C 12 7 8 11 d 1
30 :- (a) C( Li, B) Be -_
2054- —§

8 E 3

g 10_ .1

.. a. mm =

§ ,- . Hull n... ,

*, - -

g r -Sn (MeV) *

o '- l 1 l L 1 I -‘

0 .. I I I I Ij I .1
15 —- :3 2 1 o ——

I i
i -
10 I: (b) 113(7D,BB)1°H '__‘
u I
H. ‘
5|! , “j
I -
Windmills
0 ll 1 l I l l l l I LJ 1 L I 11
200 225 250 275 300 325 350
Channel

Figure 4.9: Momentum spectra from 10Li production and calibration runs. Only
the 5.23° data are shown here. The top part of the figure contains the momentum
spectrum from the 12C(7Li,8B)nBe reaction. The 1.8 MeV llBe excited state was
used as a primary calibration point. The bottom part of the figure contains the
momentum spectrum from the llB(7Li,8B)1°Li reaction. Both spectra were collected
at the same spectrometer field setting.

 

57

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 L I I I I I I r I I I I I I I I I I I I I 1% I I I I I I -
a L- _‘
.—a h .
Cl - -
d5 6 —- _
.c: - l -
o - -
\ - .
m - -4
'E’: 4 —— __
:1 f_ :
O
U I
J a
o l l l l l l J 1 II I l
200 225 250 275 300 325 350
Channel

Figure 4.10: Momentum spectrum from 10Li production run at 91.1, = 3.73°. Dimin-
ished beam current and spectrometer solid angle at this angle resulted in the very
poor statistics evident in this spectrum. The data shown here were not included in
the final analysis and mass determination of 10Li.

 

58

Calibration react ions

Momentum spectra were collected for the 12C(7Li,8B)“Be reaction, which has a well—
known Q—value, at the same angles and field settings as the production measurements.
The spectrum collected at a laboratory angle of 5.23° is shown in the top part of
Figure 4.9. The ground and first excited states of 11Be are not resolved; however,
the 1.778 MeV second excited state of 11Be is strongly populated. The last state
was used as a primary calibration point. Additional calibration points were also
obtained by setting the spectrometer magnetic elements to step elastically scattered

beam particles across the active area of the detector array.

V Analysis

Analysis of the data was accomplished utilizing the same procedure as described in
Section IV of Chapter 4, but without the benefit of the computer code RELMASS.
The beam energy, determined from the A1200 magnetic dipole fields necessary to
transport the beam through the center of the device, was E /A = 18.772 :1: 0.054
MeV. The uncertainty of the beam energy reflects an uncertainty in the average of
the N MR readings from the A1200 dipoles. The S320 spectrometer was calibrated
by studying two reactions. The first was the 12C(7Li,7Li)12C reaction populating
the 12C ground state (i.e. elastically scattered beam) and 4.44 MeV excited state.
Both states have well—known Q—values. Several runs were taken with this reaction
at different spectrometer field settings so as to step the peaks from these two states
across the focal plane. The rigidities of the 7Li ejectile are easily calculated, and
were used in conjunction with the known spectrometer fields and measured focal
plane positions to amass a set of (p,:c) points. The second calibration reaction was

12C(7Li,8B)llBe, the data for which are shown in the top part of Figure 4.9. Since

59

the ground and 0.32 MeV first excited state of 11Be are not resolved, the 1.778 MeV
second excited state was used to provide an additional (p,:r) point. A second order
polynomial was fitted to this dataset and served as the focal plane calibration. This
calibration curve was then used to deduce the Q—values of the states seen in the
llB(7Li,8B)1°Li production reaction. To a given channel in the momentum spectrum
from this reaction (bottom part of Figure 4.9) can be assigned a rigidity for the 8B
ejectile particle. This rigidity is then used to find a Q—value and hence a mass for
10Li. Thus, a one—to—one correspondence was determined between channel number
and 5,,(10L2'), the neutron separation energy of 10Li. It is known that the ground
and 0.32 MeV first excited states of 11Be are both negligibly narrow, while the 1.778
MeV second excited state has a width of 100 keV. This information, combined with
the measured width of the last state and the fact that the first two states were not

resolved, was used to estimate the (FWHM) resolution of the spectrometer to be 230

keV.

To ascertain the nature and location of the peaks in the 10Li data, the spectrum
was fitted with a multiparameter function which consisted of several components,
the first of which was a constant background. Other components were one or more
p— or s—wave neutron resonances. For these, scattering calculations were performed
to estimate their widths and line shapes. Details of these calculation are given in
Appendix B. It was found that 23% and 119-;- states with resonance energies below
100 keV were significantly narrower than the 230 keV device resolution. For such
states, the functional form used in the fitting was taken to be a Lorentzian with F =
230 keV. At higher energies, s—wave resonance widths increased extremely rapidly
to approximately 2 MeV at a neutron energy of 500 keV. The widths of p—wave
resonances exhibited a more gradual increase with energy. Thus, s—waves above 100

keV resonance energy, calculated to be too broad to observe, were not considered in

 

60

the fitting. For p—waves, the functional forms used in the fitting were parametrizations
of the calculated line shapes. Finally, a 3—body phase space background, attributable
to those reactions which directly produce the 8B-n~-9Li final state, was included in the
fitting. The explicit form for this phase space contribution, as a function of the center—
of—mass momentum of one of the three final particles, is given in [Bloc 56]. To be
applicable to the fitting of the data, the functional form given had to be transformed
into the laboratory frame and the laboratory momentum of the 8B ejectile had to be
expressed in terms of position in the S320 focal plane (i.e. channel number). This
procedure was unfeasable analytically and was done numerically. However, it was
found that the resulting function could be parametrized to an accuracy of 0.03% with

the expression for the upper—right quadrant of an ellipse

 

d031,...(8B): A\/1 4:353:32 ifs: < {tend (4.1)
d2: 0 otherwrse.

In this expression, a: is the position of the 8B, read is the position corresponding to
the kinematic endpoint, and x0 is a parameter determined by fitting the expression
to the calculated functional form. The parameter A is an arbitrary scaling factor and

is treated as a free parameter in the fitting of the 1"Li data.

Because the statistics in the llB(7Li,8B)1°Li production reaction are so low, the
standard least-squares fitting technique was unsuitable. A more general maximum—
likelihood fitting technique was used to treat the data. This technique, presented
in Appendix C, employs a figure of merit L which is similar to X2 in that it is a
measure of the goodness—of-fit, and that its minimization is the objective of the
fitting procedure. However, its analytical behavior is not as well known as that of the
x2 statistic. More specifically, using .C it is possible to compare fits with two different
functional models and to determine whether one fit is better than the other; but it

is not possible to assign to a given value of [I an exact probability analogous to the

61

chi—square distribution as described in [Pres 92].

VI Results

The best fit to the data was obtained with, in addition to a constant background and
a three—body phase space component, a single lp-é- neutron resonance at 5,,(10Li) =
—538 :l: 32 keV and a narrow neutron resonance at 5,,(10Li) 2 —100 keV. This
fit, shown with a solid line in Figure 4.11, yielded a figure of merit L = 106. It
was found that L was insensitive to the location of the narrow resonance between
0 and approximately 70 keV neutron energy. Thus, it is only possible to state with
confidence that the lower state corresponds to a neutron separation energy greater
than —100 keV. A fit was also performed in which the only neutron state was a single
p—wave. The resulting state had a neutron separation energy of 5,,(10Li) = —505
:l: 33 keV. The figure of merit L for this fit, which is shown as the dashed line in
Figure 4.11, was 123. A probability cannot be assigned to a given value of L for a
fit with a certain number of degrees of freedom, and therefore an exact quantitative
comparison between these two models cannot be made. However, further tests with
the fitting procedure have shown that, while the model with the narrow resonance
has one more fit parameter and hence one less statistical degree of freedom than
the model without the narrow resonance, this difference in the number of degrees of
freedom of the two models is not sufficient to account for the difference in the values
of L. This effect was explored by holding fixed the fit parameters for the two-peak
model and increasing the number of channels over which the figure of merit L was
calculated. It was found that in order to increase L from 106 to 123, ten additional
channels (i.e. statistical degrees of freedom) were necessary. This indicates that the
best fit was obtained with one p—wave resonance and one low-lying narrow resonance.

Because the widths of low—energy s— and p—waves are much narrower than the device

62

 

 

 
   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 I I l l r T —I I I I ffi I I I I I j I
- —Sn (MeV) *
' l 1 l l l 1 l d
.. l . I T 1 I I i
15 — 3 2 —(
i- -1
—n h " ‘1
o
g _ “B("Li.'B)’°1.i data .
A __ .
{ 104 p+swavef1t _
3 . _ ------ p wave fit _
g . -
U L d
5 j- _.
- » .1
l l I I I l
200 220 280

Figure 4.11: Theoretical models fitted to the collected spectrum from the reaction
11B(7Li,8B)mLi at 01.1, = 5.23°. The line shape for the broad peak is obtained from
resonance calculations of a p-wave neutron. The narrow state is taken to be an s-wave
neutron state, the line shape of which is assumed to be a Lorentzian with width 230
keV. The details of the resonance calculations and the maximum—likelihood fitting
procedure are given in Appendices B and C, respectively. The best fit, shown with the
solid line, was obtained with one p—wave resonance, one s-wave resonance, a 3—body
background, and a constant background. The dashed line is the fit obtained with
one p—wave, a 3—body background, and a constant background. The two background
terms had similar magnitudes in both models, and their sum is shown with the dotted
line.

63

 

 

 

Source of uncertainty 0 (keV)
beam energy 45
spectrometer angle 27
target thicknesses 6
total nonstatistical uncertainty 53

 

 

Table 4.1: Sources of nonstatistical experimental uncertainty in 10Li mass measure-
ment. The three uncertainties listed are added in quadrature to yield the total non-
statistical uncertainty, which is further added in quadrature with the statistical un-
certainties from the fitting procedure.

resolution, it is not possible to determine the spectroscopic nature of this lower state.

The uncertainties specified above reflect only statistical uncertainties from the
fitting procedure. Other sources of uncertainty were experimental in nature. The
data were re—analyzed with the beam energy increased by one sigma; thus determined,
the contribution to the overall uncertainty from the beam energy was 45 keV. The
spectrometer angle was known to within 0.05°, contributing an additional 27 keV
to the final uncertainty. Finally, the thicknesses of the production and calibration
targets were determined from beam energy—loss to within 1%, yielding an additional
6 keV in the final uncertainty. These contributions, which are listed in Table 4.1,
are added in quadrature to yield the total nonstatistical uncertainty, which is further
added in quadrature with the statistical uncertainties from the fitting procedure. The
best interpretation of the data, therefore, indicates that 10Li has a 1p; neutron state
unbound to neutron decay by 538 :l: 62 keV and a ground state, either a 119% or a 23%
neutron state, unbound by less than 100 keV. The 10Li mass excesses for these states
are 33.563 :1: 0.062 MeV and less than 33.125 MeV, respectively. The Q-values for
the 11B(7Li,8B)1°Li reaction to populate these states are —32.908 :l: 0.062 MeV and

greater than -—32.471 MeV, respectively. The width of the ground state is known to

64

 

 

5,, (MeV) F181, (MeV) Identification

 

Wilcox et al. [Wilc 75] —0.80 :t 0.25 1.2 :l: 0.3 g.s.

Amelin et al. [Ame190] —0.15 :l: 0.15 S 0.4 5%, g.s.

Kryger et al. [Kryg 93] Z —0.15 or g.s.

z —-2.5

Bohlen et al. [B011] 93] —0.42 :l: 0.05 0.15 :l: 0.07 12%, g.s.
—0.80 :l: 0.06 0.30 :l: 0.10 19%

Present work 2 —0.10 < 0.23 g.s.
—0.54 :l: 0.06 0.36 :l: 0.02 19%

 

 

Table 4.2: Summary of experimental data on low—lying structure of 10Li published to
date, including the present work. The neutron separation energy and width are given
for each state. Also given for each state is the identification (if any) claimed by the
experimenters.

be much smaller than the resolution of the spectrometer, 230 keV. The width of the
110-;- excited state, determined from the measured excitation energy and the calculated
width at that energy, is F131, = 358 :1: 23 keV. These values are summarized, together

with the existing measurements, in Table 4.2.

An attempt was also made to fit the broad structure near —500 keV with two
p—wave resonances, as was done by Bohlen et al.. With this model, the best fit
corresponded to a minimum L value of 124 and put both of the resonances at the
same energy as the single p—wave fit described above. Further investigations found
that it was possible to keep L within unity of its minimum value only by placing the
resonances less than 170 keV apart and centered near -520 keV separation energy. It
is important to note that under no circumstances could a reasonable fit be made with a
resonance unbound by more than 650 keV. Both of these results, the possible existence
of a low—lying neutron state, and the fact that the broad peak in the data can only
be fit by a single 119% neutron resonance or by two such resonances separated by less

than 170 keV, do not corroborate the results of Bohlen et al.. A possible explanation

for this is the fact that, although the spectra collected by Bohlen et al. are very
similar to that in the present work, particularly the data from the 9Be(13C,12N)10Li
reaction shown if Figure 4.5, the theoretical p—wave line shapes used in their paper
are symmetric and relatively narrow (I‘lab 2200 keV) whereas the line shapes used
here are asymmetric and fairly broad (Flat, 2400 keV). It seems plausible that a peak
at 5,, = —800 keV is artificially necessitated in order to fit the high—energy tail of
the data with narrow, symmetric p—waves. Although the existence of a low—lying
10Li ground state has been observed here as well as by Amelin et al. [Ame190] and
possibly by Kryger et al. [Kryg 93], it is still unclear whether this state is an s-wave
or a p—wave neutron resonance. However, mounting expectations from systematic
and theoretical considerations indicate that an s—wave contribution dominates the

n—9Li interaction.

Chapter 5

Conclusions—The Present and
Future of 11Li and 10Li

One of the most interesting nuclei that have been made available for study by the
production of intense radioactive nuclear beams is 11Li. This nucleus has exhibited
many remarkable properties which can be explained, both qualitatively and quanti-
tatively, by viewing the nucleus as a three—body system, which comprises a 9Li core
and a halo of two loosely bound neutrons. Measurements of the rms radius of the
halo neutrons have yielded values of between 6 and 10 fm. One of the most basic
quantities used by theoretical models based on this three-body picture is the mass
of 11Li. The measurement of the 11Li mass presented in this dissertation is the most
accurate value to date. The current best estimate of the 11Li mass excess, obtained
by averaging the value reported here with the three previous measurements, is 40.802
:1: 0.026 MeV. This corresponds to a two—neutron separation energy of 295 :t 26 keV.
This quantity, which is basic to the understanding of the structure of 11Li is now

known to sufficient accuracy for precise comparison with theoretical models.

Another important factor in the threebody models of 11Li is the nature of the
n—9Li interaction. Recent attention has focused on the low—lying structure of the

neutron—unstable nucleus 10Li. The experimental evidence pertaining to this nucleus

66

67

does not present a clear picture. Various mass measurements have determined the
10Li ground state to be unbound by as little as 150 keV and as much as 800 keV.
In addition to uncertainty regarding the energy of the ground state resonance, there
is also considerable disagreement over the spectroscopic nature of the ground state,
which is expected to be either a 11% or a 23% neutron state. The existence of a broad
(F 2 300 keV) 10Li state unbound to neutron decay by greater than 400 keV has been
well established. But there is mounting evidence that there is a much lower—lying,
barely unbound state. This evidence includes the observation by Amelin et al. of
a state unbound by 150 :l: 150 keV, independent observations by Kryger et al. and
Kobayashi et al. of a peak corresponding to close to zero relative n—9Li velocity from
the decay of 10Li, and the observation of a narrow state in 10Be by Abramovich et al.
believed to be the isobaric analog of a 10Li state unbound by 60 keV. The experimental
evidence presented in this work is the first simultaneous observation of both a broad,
relatively high-lying 10Li state and a second barely unbound state. Although the
statistics in the present spectrum are rather poor, extensive resonance calculations
and the employment of a statistically accurate maximum—likelihood fitting technique
have allowed quantitative estimates of the low—lying structure of 10Li. The best
interpretation of the data indicates that 1”Li has a ulp§ excited state unbound to
neutron decay by 538 :l: 62 keV (corresponding to a 10Li mass excess of 33.563 :h
0.062 MeV ) with a width of 358 i 23 and a ground state unbound by less than 100
keV (1°Li mass excess less than 33.125 MeV). The spectroscopic nature of the low—
lying state could not be determined from the data. However, theoretical evidence, by
Barker and Hickey as well as Warburton and Brown, indicate that this low—lying state
could be a 23% neutron state. This would not be too surprizing since the neighboring

nucleus 11Be has a 2.9% neutron state for its ground state.

The three—body models that have been used to describe 11Li rely on assumed

68

forms for the particle—particle interactions between the three constituent bodies. In
the initial calculations, the n—9Li interaction was chosen to recreate a lp-é- neutron
resonance at an energy of 800 keV, in accordance with the measurements in [Wile 75].
Recent Fadeev calculations performed by Bang and Thompson [Bang 92], under these
assumptions, failed to reproduce simultaneously the experimentally observed neutron
binding energy, the widths of the 9Li momentum distributions from 11Li breakup, and
the energy of the 3—body breakup peak. It was found [Thom 93b] that the binding
energy and rms matter radius could be reproduced only if the single n-9Li p-wave
state was at approximately 200 keV. The 9Li momentum widths predicted from those
p-wave—dominated Fadeev wavefunctions that reproduced the rms llLi radius were
approximately three times larger than the experimental values. This suggests that
the rms radius measured from the total interaction cross section is smaller than that

deduced from the fragment momentum widths from the breakup of 11Li.

Very recent calculations by Thompson and Zhukov [Thom 93b, Thom 93a] have
included a 23% state as well as a 11% state in the n—9Li interaction. The three—body
Fadeev wavefunctions thus contain admixtures of these two neutron configurations.
The predicted 11Li binding energy is shown as a function of 23% and 1p% resonance
energies in Figure 5.1. The current 11Li mass is indicated in the figure by horizontal
lines, and, although the 10Li p—wave resonance reported here is off the scale in the
figure, the vertical line indicates the lower limit in the uncertainty in this value. To
meet these limits, Thompson and Zhukov’s calculations would require that the s—wave
10Li ground state be unbound by less than 10 keV, a result that is also in agreement
with the data presented here. Their results further indicate that at lower s—wave
energies, the n-9Li and n—n admixtures become increasingly dominated by the 2.9%
and 150 state, respectively. For a 10 keV s—wave, the lowest reported in [Thom 93b],

these admixtures are 64% and 67%. This has several profound effects. Because 5—

69

 

     
    

 

 

 

EGHIDV.=V° .’.’ ‘
0.10 [Am-"A110 keV 15 width .1 :
:e--050 keV ,x’ 1
S [o I 025 keV ’0’ l
0-00 YVH-mGev / ‘->1
3; » ,-
>~ : .’ ...... A]
9 -O.10f l.’ ................ .
5 E ‘ ,0" ......................... , ..-01
g .0207 .I' ................... .-“‘ ”-01
1 fix .............. ,
.3 -0.30 "- - r-Vi
" E A": ,
-O.40 f G i
1 1
-0.50

 

 

 

 

0.15 0.20 0.25 A 0.30 0.35 0.40 0.45 0.50
Op, ,, resonance energy (MeV)

Figure 5.1: [Thom 93b] Comparison of present data with calculations by Thomp-
son and Zhukov. This figure is identical to Figure 2.6 with the experimental results
reported in this work indicated for comparison. The current llLi two—neutron sepa-
ration energy (295 :1: 26 keV) is indicated in the figure by horizontal lines. The value
for the l°Li p—wave resonance reported here (538 :l: 62 keV) is off the scale in the
figure, however, the vertical line indicates the lower limit in the uncertainty in this
value. To meet these limits, Thompson and Zhukov’s calculations would require that
the s-wave 1°Li ground state be unbound by less than 10 keV, a result that is also in
agreement with the data presented here.

70

wave neutron states have no centrifugal barrier, the large 25% admixture in the n-9Li
motion leads to a 11Li wavefunction with a very large rms radius (3.73 fm for the
10 keV s—wave). However, such a large matter radius corresponds to a narrow 9Li
fragment momentum distribution with a width 0 of approximately 24 MeV/c, in good

agreement with the data.

Although the 11Li mass is now well known, and it seems likely that the 10Li ground
state is only barely unbound, the exact nature of the 10Li ground state, and hence
the structure of 11Li, is still unknown. Theoretical models, too, have yet to reach a
consensus on the details of 11Li structure and are still struggling to reproduce all of
the mounting experimental data regarding this puzzling nucleus. However, two main
courses of action seem clear. It appears that the possibility of an s—wave 10Li ground
state is not a remote one, and that more realistic calculations that take this possibility
into account are needed. Also, the experimental data on the low—lying structure of
l°Li are plagued by poor statistics and poor resolution. This situation can only be
remedied by more experimental effort. By repeating experiments such as the ones
presented here on other spectrometers with high resolution and large acceptance such
as the S800 being constructed at the NSCL, the existence of a low—lying loLi state
could be confirmed with high statistical accuracy and its spectroscopic nature could be
determined from its measured width. Nevertheless, the picture of 11Li which is based
on the data presented in this dissertation is a considerable improvement over previous
interpretations, and therefore continued progress is being made on the understanding

of the new phenomenon of halo nuclei.

Appendix A

RELMASS

Analysis of mass—measurement data from a magnetic spectrometer can be divided
into two major steps: calibration of the spectrometer, and measurement of the Q—
value of the production reaction. For each calibration reaction, the beam energy,
reaction Q—value, and ejectile angle, all of which are known, determine the kinetic
energy of the ejectile. This information, when combined with the magnetic field of
the spectrometer and the charge state of the ejectile, will determine the bend radius
p of the particle through the spectrometer. This process, illustrated in the flowchart
in Figure A.1, yields a series of (p,a:) values to which a functional form, typically a
first or second order polynomial, is fit. The Q-value of the production reaction is
then found by reversing the process, as illustrated in Figure A.2. Whereas in the
calibration process, the known Q—value of the calibration reaction is used to relate
the measured ejectile position to the bend radius, the Q-value is here determined

from the ejectile position.

The above processes, while relatively simple conceptually, involve a great deal of
numerical calculation and are prone to error if done by hand. To facilitate analysis
of experiments of this type, Toshiyuki Kubo, with suggestions from Ed Kashy, has
written a computer code, RELMASS, which automates both of the above steps. When

executed, the program searches the current directory for the input file RELMASS . INP.
71

Input Parameters

 

Beam Energy

 

 

 

 

 

 

Reaction Energy Loss
Q‘Value ———" and
Angle Kinematics
Target Thickness
Ejectile
Kinetic
Energy
Magnetic Field (B) _
Ejectile Charge (Q) Bp — p/Q

 

 

 

 

Ejectile Position (at) P

Figure A.1: Flowchart of magnetic spectrometer calibration procedures. The input
parameters are all experimentally measured or calculated quantities. Each calibration
reaction yields one or more sets of (p,a:) values.

73

Input Parameters

 

 

 

 

 

 

 

 

 

 

Ejectile Position (.r) ——-——* Calibration
p
Magnetic Field (B) _
Ejectile Charge (Q) Bp — p/Q
Ejectile
Kinetic
Energy

 

 

Beam Energy
Reaction

Angle

Target Thickness

Energy Loss

——‘ and

Kinematics

 

 

 

Reaction Q~value

Figure A.2: Flowchart of Q—value measurement procedures with a calibrated magnetic
spectrometer. As in the previous figure, the input parameters are all experimentally
measured or calculated quantities. The reaction Q—value is determined from the
measured ejectile position x.

74

Both the code and a sample input file are located in the directory

NSCLLIBRARY: [RELMASS]

on the N SCL VAX cluster. Upon completion, the program creates two files, the
names of which are requested from the user. The first file, given the extension . OUT,
contains numerical details of each step of the calibration and measurement processes.
The second file is a TOPDRAWER file which depicts the p vs a: calibration curve
with the calibration and measurement points. The most straightforward discussion
of the operation of the program is provided by examining the input and output files,

examples of which are included in Table A.1 at the end of this appendix.

Input file: RELMASS . INP

The input file provides the program with the parameters necessary to calibrate the
spectrometer and then to measure the Q—value of a series of production reactions.
The first four lines allow the user to title and comment the input file. The information
in these lines has no bearing on the program operation; they are merely repeated in
the output file. Following the comment lines are groups of calibration data, one group
for each peak in the calibration momentum spectra. Each group of calibration data
begins with the tag-string CALIB_REACTION followed by a label number. The label
numbers are arbitrary and facilitate bookkeeping for the user. The lines following
the tag‘string allow the user to add a comment for that particular set of data and
to specify kinematic and experimental parameters such as the reaction, beam energy,
spectrometer field, target thickness, and excitation energies for the calibration peak.
In addition to the target, it is possible to specify two absorber materials, one before
the target and one after, through which the beam and ejectile, respectively, lose

energy. Finally, for each group of calibration data, the user may define a weighting

75

factor for use in external mode of the least—squares fitting of the calibration curve to
the data, as described below. The user may also instruct the program to ignore any
set of calibration data by placing the characters // before the tagflstring, as is done

for Calibration Reaction 2 in the example file given.

Following the calibration data are sets of data for Q—value measurements. Each
dataset is preceded by the tag—string MASS_MEASUREMENT followed by an arbitrary la-
bel number. The first sixteen lines of each measurement dataset are identical to those
of a calibration dataset. The next three lines pertain to the searching technique used
by the program to find the measured Q—value. When RELMASS processes a dataset
from a production reaction, it consults the MASSPACK table of atomic masses to
obtain an initial estimate of the Q—value. This estimated Q—value, combined with
the rest of the information in the dataset, specifies a point on the (p—x) plane, which
may or may not lie on the calibration curve. The program then iteratively varies
the mass of either the ejectile or residue particle, the choice specified by the user,
to place the point on the calibration curve. These three lines specify the maximum
number of iterations, the variational step size, and the desired absolute accuracy of
the mass search. The next line allows the user to specify which mass, ejectile or resid-
ual, should be varied. The next six lines pertain to various routines in the program
that estimate uncertainties. As of this writing (November 1993), these routines are
not implemented and the parameters in these lines are ignored. As with calibration
datasets, the user may instruct the program to ignore any measurement dataset by

placing the characters // before its tag-string.

The last set of data is preceded by the tag—string LEAST_SQUARE and determines
various aspects of the least—squares fitting of the calibration curve. There are three
possible modes for weighting the calibration data in the fit. The first, equal mode,

gives each data point an equal weight. Internal mode weights each data point by

76

the uncertainties in its peak position. External mode weights each data. point by the

factor specified with its dataset.

Output file

The output file is named by the user and contains numerical details of each step
of the calibration and measurement processes. The first and second pages list the
input data for the calibration reactions and the control parameters for the least—
squares fitting of the calibration points. The third page lists the input data for the
measurement reactions and the control parameters for the mass search. The fourth
page specifies the atomic and nuclear (i.e. sans electrons) masses of each participant
in the calibration reactions as well as the ground-state Q—values. The kinetic energy
of the calibration reaction ejectiles is given on the fifth page. The program includes
energy-loss effects in the target and calculates the ejectile energy as the average
of the values obtained by assuming the reaction takes place at the front and back
surfaces of the target. The sixth page presents all of the calibration points (p,:r)
and lists the polynomial coefficients that provided the best fit to these points. The
polynomial listed here is then used in subsequent steps to measure reaction Q-values.
The seventh page presents the first guesses, based on the MASSPACK table, of the
kinematic parameters for the measurement reaction. Also listed on this page is the
(p,:r) point that is taken as the initial guess in the iterative mass search, the steps of
which are detailed on the eighth page. Finally, the ninth page lists the ejectile mass
and reaction Q—value as obtained from the MASSPACK table and as obtained from
the measurement. Also given on the ninth page is a table of uncertainty estimates.
However, since the relevant subroutines are not fully implemented, these values should

not be used.

Table A.1: An example of the RELMASS input file, RELMASS.INP, is listed on the
two following pages. An example RELMASS output file is listed on the nine pages
following the listing of RELMASS. INP. The contents of both files are discussed in the
text.

 

 

 

78

filename: RELMASS_THESIS.INP
Input data for an analysis or mass measurement
COMMENT DATA (A80)
ExampTe file : Two calibration points and one mass measurement.

CALIB_REACTION 1.1 Read format (T46.A40/T46,A26,15(/T46,Fl$.0))

Comment : 108e3+ (5.3)

Reaction, a + A -> b + B, A(a,b)8 : l4C(llB.lOBe)lSN

Projectile energy, postMeV],neg[MeV/A] : -32.l365

Scattering angle of ejectile[degrees] : 0.0

Excitation energy of ejectile[NeV] 0.0

Excitation energy of residual[MeV] 5.2845

Charge of ejectile 3.0

Magnetic fields of spectrograph[Tesla] 0.921608

Thickness[mg/cm2] of absorber-l 0.0

Atomic number (2) of absorber-1 0.0

Thickness[mg/cm2] of target 0.450

Atomic number (2) of target 6.0

Thickness[mg/cm2] of absorber-2 0.0

Atomic number (2) or absorber-2 0.0

Focal plane position of ejectiletchannel] 60.136

Error(one-sigma) of position[channel] 0.020

Weight £actor(0 to 1,.if zero, ignored) 1.0
CALIB_REACTION 1.2

Comment : 108e3+ (q.s.)

Reaction, a + A -> b + B, A(a,b)B : l4C(llB,lOBe)lSN

Projectile energy. pos[MeV],neg[MeV/A) : -32.1365

Scattering angle or ejectiletdegrees] : 0.0

Excitation energy of ejectiletMeV] 0.0

Excitation energy of residualtHeV] 0.0

Charge of ejectile 3.0

Magnetic fields of spectrographtTesla] : 0.921608

Thicknesstmg/cmZ] of absorber-l : 0.0

Atomic number (2) of absorber-l : 0.0

Thickness[mg/cm2] or target 0.450

Atomic number (2) of target 6.0

Thickness[mg/cm2] of absorber-2 0.0

Atomic number (2) of absorber-2 : 0.0

rocal plane position of ejectiletchannel] : 140.345

Error(one-sigma) of positiontchannel) : 0.069

Height tactor(0 to 1, it zero, ignored) : 1.0
//CALIB_PIACTION 2

Comment : This reaction is commented out

Reaction. a + A -> b + B, A(a,b)3 : l4C(llB,9Li)l6O

Projectile energy, postMeV],neg[Hev/A] : -32.1365
Scattering angle of ejectiletdegrees] :
Excitation energy of ejectiletHeV]
Excitation energy or residual[MeV]
Charge or ejectile

Magnetic fields of spectrographtTesla]
Thickness[mg/cm2] of absorber-l
Atomic number (2) of absorber-1
Thickness[mg/cm2] or target

Atomic number (2) of target
Thickness[mg/cm2] or absorber-2

OOGOOOOUOOO
0

Atomic number (2) of absorber-2 : O

focal plane position of ejectiletchannel] : 101.806

lrror(one-sigma) ot positionlchannel] : 0.074

weight tactor(0 to 1, it zero, ignored) : 0.0
HISS_MIASORIHIUT 1

Comment : 9Li mass meas

Reaction, a + A -> b + B, A(a,b)3 : l4C(llB, 9Li)160

Projectile energy, postHeV],neg[MeV/A] : -32.l365

 

 

Scattering angle of ejectiletdegrees]
Excitation energy of ejectile[Mev1
Excitation energy of residual[MeV]

Charge of ejectile

Magnetic fields of spectrograph[Tesla]
Thickness[mg/cm2] of absorber-l

Atomic number (2) of absorber-l
Thickness[mg/cm2] of target

Atomic number (2) of target
Thickness[mg/cm2] of absorber-2

Atomic number (2) of absorber-2

Focal plane position of ejectile[channel]
Error(one-sigma) of position[channel] .
Maximum number of iteration in mass search:
Energy step in the iteration[MeV] -
Goal of position difference in search[ch]
Ejectile mass(0.) or residue mass(1.) ?
dQ/d(Eproj), d(Eproj) - X'Eproj, X[%] -
dQ/d(Theta), d(Theta) [deg.] -
dQ/d(position), d(position) [channel] - .
dQ/d(Thick,absl),d(Thicx)-X*Thick, xtt] - :
dQ/d(Thick,targ), X[%] - :
dQ/d(Thick.abs2), X[%] - :

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Appendix B

Calculations of Neutron
Resonance Line Shapes

To analyze the data from the 11B(7Li,8B)10Li reaction, resonance scattering calcula-
tions were performed to estimate the widths and line shapes of the n—9Li resonances.
The Schrodinger equation was solved numerically for a single neutron moving in the
potential created by a 9Li nucleus. The potential consisted of, in addition to Coulomb

and centrifugal terms, a Woods—Saxon term

 

-Vo
VWS(T) - 1+ exp(7‘ _ 7.0141/3)/a — -%f(r) (8‘1)
and a Thomas spin—orbit term
Vso(7‘) = W00 ° (141:.(132)
7‘ dr

The parameters To, a, and W0 were taken to be 1.25 fm, 0.65 fm and 15.64 MeV fm2
respectively [Brow 92]. The results of the line shape calculations were insensitive to
small variations of these parameters. Holding Vb constant, the wavefunction \IIE(7')
was found for the neutron {at a given kinetic energy E. To estimate the line shapes

for the neutron decay of 10Li, the wavefunction was normalized inside the nucleus

[Sher 85]

S(E) = [Om W%(r)fl;¥§r2dr. (3.3)

89

90

The behavior of S as a function of the neutron kinetic energy E provided the single—
particle resonance line shape. The energy at which 5 reached a maximum, that is,
the resonance energy, varied with the value chosen for V0. By mapping 5' (E) for
different values of V0 it was possible to explore the behavior of the line shapes, and

in particular the widths, as a function of resonance energy for 25% and 117% neutrons.
.1 '
2.55 line shapes

The function 5' (E) for several 2.3-,1; neutron states is shown in the top part of Fig-
ure 8.]. Depicted are states with peak energies at 0, 30, 150, and 300 keV. The
widths of these states increases rapidly with peak energy. For states with energy
above 100 keV, the width is so large that the state could not be detected above
background. For states below 100 keV, the line shapes are narrow enough to be dom-
inated by the resolution of the spectrometer, and were approximated in the fitting

procedures by a Lorentzian with I‘ = 230 keV.
119% line shapes

The line shapes for several 112% neutron states are shown in the bottom part of F Ig-
ure B.1. Shown here are states with peak energies at 100, 200, 500, and 700 keV.
Although the widths of these states also increases with peak energy, the effect is much
less dramatic than that described above. For p—waves below 100 keV peak energy,
the line shape was also approximated by a Lorentzian in the fitting procedures. For
p—waves with peak energy above 100 keV, the line shapes were parametrized for in-
clusion in the fitting procedures. The parametrization was found by trial—and—error
by examining the shape of the function 3 (E ) On the low—energy side of the peak,
5 (E) exhibits the rapid decay of a Gaussian curve. On the high—energy side of the

peak, the function appears to have two decay components: a Lorentzian—like decay

91

 

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——--——_———
-

 

 

I “'lll lrll

 

-o—
‘-

 

db-
1|-
——l
ur-
-—
-

S(E) (arb. units)

 
  

(b) 1p(1/2)

 

 

lllllllllllllllllllllllLLll I [Ill lllllLlJliJlLllllLlll

 

 

lllll[IIIIIIIIIIIII[IIIIIITTI

 

IIIIIII —l

0.0 0.2 0.4- 0.6 0.8 1.0

Neutron energy (MeV)

Figure B.1: Calculated line shapes for 2.9% and 119:1,- n—9Li states at several energies.
The top part of the figure contains line shapes for 2.9% states with peak energies of
0, 30, 150, and 300 keV. The bottom part of the figure contains line shapes for lp§
states with peak energies of 100, 200, 500, and 700 keV. In both frames, the line
shape for the lowest energy state was reduced by approximately 75% to fit on the
same scale as the other states.

92

and a very long, almost constant, tail. With these considerations in mind, the final

parametrization used for 11% states above 100 keV peak energy was

21‘2

low

Nexp [- M] if E < Eres

5(E) : 1‘2, _ (BA)
N [R + (1 — R)A(E_Er“"“ J if E > 13,...

e.)2+1“;~’nsh

 

This model was fitted to the calculated line shapes at several resonance energies and
in this way, the four parameters I‘low, Flush, R, and A were determined as functions
of the resonance energy Em. An example of this model fitted to a 500 keV resonance
is shown in Figure 3.2. The resolution of the 8320, a Lorentzian with a full—width of
230 keV, was incorporated in the model by adding the S320 half—width in quadrature
with the model widths, I‘ 10,, and Thigh. The final parametrization that was used in the
fitting procedure had two free parameters, the resonance energy Em and the total

amplitude N.

The computer code RESCALC

The resonance calculations described above involved the numerical solution to the
Schrodinger equation for a particle in a potential well. The computer code RESCALC,
the C source code for which is listed in Table 3.1 at the end of this appendix, was
written, with helpful suggestions from Alex Brown and George Bertsch, to accomplish
this. Instructions for compiling and linking the code at the NSCL are given in the

program comments at the beginning of the listing.

The user is first asked to choose between a Woods—Saxon or a Gaussian form
for the nuclear potential, and to specify the appropriate parameters (e.g. depth and
width/diffuseness) for the chosen well. The user is then prompted for the depth of a
Thomas spin-orbit term, the form for which is given above. Next, the user is asked

for the spin, orbital, and total angular momenta for the system and the masses and

 

93

 

S(E) (arb. units)
l

 

 

 

J I I I I I I I j I I I I I I I J I I J I I I I I

0.00 0.25 0.50 0.75 1.00 1.25 1 .50
Neutron energy (MeV)

Figure B.2: Six-parameter model fitted to calculated line shape for a 112% neutron
state at 500 keV resonance energy. Similar fits were made to line shapes at other
energies; four of the six parameters were thus determined as functions of the resonance

energy.

94

Table 3.1: The listing of the RESCALC C source code is given on the nine pages at
the end of this appendix. The operation of the program is discussed in the text as
well as in the program comments.

charges of the target and projectile. The program then calculates several observables
over a range of projectile energies, specified by the user, and writes the calculated

data to the file RESONANCDUT.

At each projectile energy, the Schrodinger equation is solved, using the N umerov
algorithm [Koon 86], to yield the wavefunction \IIE(r), from which the function 5 (E )
is calculated. The scattering phase shift 6 is found by solving for the wavefunction
without the nuclear and spin—orbit terms in the potential and comparing the phases
of the wavefunctions at large radii. The values then written to the output file are
the scattering energy E, the cross—section—related quantities sin2(6) and 9.32791, and
the function 5' (E) The code is very thoroughly commented and is easily modified to
calculate other observables from the wavefunction, potentials, and local wave number,

all of which are stored in global arrays in the program.

95

/..'lt'fl...O.fiiiflfifiififltttiltfltfltfififit.Iit...QifitliiiflfiIttfiflfifltfilitfittfittiiii
tfififlflflfliOi...littifllfiflflttICOOI.'QCQQQI.CIII'COIQIOCCICtfilififiiiiflfllfitlittll.
CflfififitilfltflltI.QififlItiittIfifittfifltflififlIfifitfitfififlfiifitttflitifltttifltfifittit'tfltti

TO COMPILE/LINK THIS PROGRAM ON A VAX AT NSCL:
cc rescalc.c
link rescalc,sysSlibraryzcrtlib/opt

This program solves the radial Schrodinger equation:

 

d2u ( V(r) - E )
--- . ' u(r) - k2(r) ' u(r)
dx2 hbarZ/ZM
where
l- -l
V(r) - C s | sz(r) + Vso(r) l + Vcentrif + Vcoul

The main engine of the program consists of the functions arranged in
the hierarchy diagrammed below:

Calc_Qbservbls . Make_flell
I__ Solve_$chrodinger Diff_eq

Calc _Observbls calls Make _lell which packs an array with V(r) using several
global parameters for the either the Woods-Saxon form or a Gaussian form
for the nuclear potential. It then Calls Solve _Schrodinger

which uses V(r) to find the radial wavefunction u(r). Solve _Schrodinger
uses the function Diff eq to do the mathematical nitty-gritty of solving
the differential equation. Once u(r) is found, Solve _Schrodinger performs
the appropriate normalization, and looks for bound states if applicable.
Calc _0bservbls, if necessary. calls Make _lell with free space parameters
and calls Solve _Schrodinger with the new well. This is so Calc _Observbls
can find the scattering phase shift.

fifiiflfififittltfltitttflitflfififitIfiflttififlfifitfiiiflfiiiflfiififiififlflittifltltfifiiittflttfitfiflfit
Q'IQCCD..‘fl...9"...I...Ilfittifii.fiflflflflflfitfi.flfifitfiflflififlfllfihflflfltfitiI9...0.1...
IfiifltfififllfiQ.D.iflfilfifififltflflfiflflfififlflflfiiflOOICQCOOOICIIODCQIiiifitifitfilIfitittttfiifl

INCLUDE FUNCTIONS BILO'

II.Ott.OIIt.fl.tfiflfltfl...QOQOQII.flflttfifitifififlififififlfiOfifltflfiflitfifi.flittfittflfitttfltt
Itflit...IQt....O...‘DtfitfiiifiitttflfiiiflifltfliIII...t..Q.ittflflflfiitflfltflfllifliifltl
IfifltflififllflflttfiififlfltflflflflD....CfliflflfllitflflflflflfiflflfiflflIflfllfiflfiflttflifiililflflflQfitlttl

Iinclude <math.h>
Oinclude "sysSadmin:[young.cstuff.my_std]bmymath.h'

/*.Qtfiflfifitfifltttfltfififlfifi..fififlI...IQ....D....COOQCIQCfiflflfliifitittflfittiflfltCOO...
OitOt...CI..QQOOOOQOOOOOQ.QQO..OtifltlififitfitiififliififltflflfiIfltfitfifitflitfififii.I'l!
fit.OifififififififlflfiflfifififliflflflfiflCit...tfifififlflfifi.fiflflflhflttfitflfiQtlfififilflflfiiiIflflfitflttfifit

GLOBAL VARIABLIS AND PERIINENT DEFINES BELOW

itififlfifittfifiifittiflfltfiiiiflfltfli.tfltttttitflfififiiifififltflifit.Ititflittfififltflttttflfltti
ififlfliflfiflifit.fififlflfififififlflflfiflflit...i...‘fihfifiifitttfifififitttttfititfiitififltfiiitttfitI.
OtflflflfiflflflflfiflfiiflfiitfltD0.0.l'fiiflfiitflflfllIC.Q'IIGIQQQOQOOQfiflifiifl.lfltfit...‘.‘O'l

IdIrinI uax_annar_srzs 2010

RealType s_proj2,l,j2; /' 2 s, 1. 2'3 for incoming particle '/
Realrype at,zt.ap.zp: /* mass and charge of targ and projectile '/
Realrype Vnorm, adiff, R0: /' Nuclear potential parameters '/

Realrype Vspino; /* Spin orbit parameter '/

96

int potential_type: /' nuke potential selector '/

RealType 'u: /' radial wavefunction '/

RealType *V; /' Total radial potential */

RealType 'des; /' Radial derivative of nulear potential '/
RealType 'KZ: /' Local wave number squared '/

Realrype "EB; /' Array of bound state energies '/

int NMax; /' Number of radial steps in mesh and '/
RealType delr; /' the size of each step. '/
Realrype hbstm; /' h-bar squared divided by 2 ' reduced mass */

/titfifiifi'flfliflflflflfifiiflittflflttQflfiififlflfiiflifltitiflifiififit.fifiiiiifiittfiflfiitifiiflflflfiifl
fltfififli.Itfiifitiittflfiifltfltittfliitttfififitiiltt19".'fifltfitttflfitififliflfil.flfiitfiflt't
tfiflflfliiiflflfltfiflflfiflflflflfliiflfliflfififlflflfiiflflttfiflfliOIROI...lflfliifiitfiiflI'titfifltfltflfltt

FUNCTION DECLARATIONS BELON

it.tittfiflflttifitflfiitflttfiiflfififii.D....fitfiitfififl*fi'flflfittttflfifiifi.Iflfitifitflttfifififitfl
Ittflflfi....fi'.‘ti.Iflflfiflfllfifittifitlfltfltfliflfliiii...IfltfittOtflfitfiiflflitfiitfllflifiiit
flfli...‘I.it.fiflflfltflttfltflfifltflfilflflflti...i...Iii...fiftfifltifltltflltttifiltitififittfl/

void Initialise_Global_Yariables( void );

void Calc_0bservbls( Realrype E, Realrype *dphi,
int *nodes, Realrype *SurfParam );

void Hake_flell( Realrype nuclear_pot_const );
void Solve_§chrodinger( RealType 3, int nsep, Realrype 'phi ):

void Diff_eq( int nfrom. int nto, int nstep,
Realrype 'uu, RealType 'duu. Realrype 'intuz );

void Kill_Global_yariables( void ):

/QQOOQOQDQi.COCO...OOOOQ....Q...‘.OfififlfifititfifittfiflfiflfiflflflflttfittfifififitQ'tiflfliiI
tfifltflfiflifitfitflhfififififlfiifiiifittfifitfiIQQCQQQQCtQifitttfittitiflfiflhfiitfififlflflfitfifiCi...O
Oi...filtfltflfiflfifiififiiflfiflfiitfitfliOI...Qlfiflfltfiitflfltiflifififitfitfi.Q'IOQQOICIO..t'...

FUNCTION DEFINITIONS BELOW

tflfifififiiififliflfiflfififltfififlflfltfittfiflfififittfitfiflfit'fiiitfii'itfiit'.fifltfltfiflfltttfitfitfiifltfl
tltI...IQ.Qtfl.IifltflfiflflflfiflttilflfitififlttitfltflttflfiflfllfitfiflfifltflitfitflfltflttflfltiI...
Iittltfiiiiitttflfltfli....flflfittfifltfitiflttttttttitit.Qt...Qttfifitltflflfittflitfititfil

Oinclude 'sysSadmin:[young.cstuff.my_std]hmymath.i”

int main( void )
{
RealType l,dphi,surf;
int nodes:
Realrype lstart,lstop,dell:
31L! 'outfile;

/ Initialise_Global;Variables();
t

The lines contained in this comment are for the original version of the
program, which asks for a single energy and writes the observables to
the screen.

Start:
[3(0) - 0.0000:
puts("\nlnter I (3 < 0 quits program).");
scanf(Ӥlf".&l):
if (I < 0.0000) goto End:

 

97

Calc Observbls(E,Idphi,&nodes,asurf);
printf("sIn‘2(d) - tlf\n",sin(dphi)'sin(dphi)):
printf("sin‘2(d)/E - §1f\n",sin(dphi)'sin(dphi)/E);
printf("s - tlf\n",surf):

goto Start:

End:

The lines below are for the "eloop" version of the program, which
asks for a max,min.delta values for E and then loops over the
energy values, writing the ovservables out to a file.

*/
if((outfile-{open("resonanc.out","w"))--NULL)
(
fprintf(stderr."Cannot open output file.\n");
exit(1);

l

puts("Enter Estart. Estop, delE -- (MeV) separated by commas please."):
scanf( "%lf, %lf, %lf", EEstart, 58stop, edelE);

for ( E - Estart; E <- Estop; B +- dell )

Calc_0bservbls(£,£dphi,Inodes,&surf);
fprintf( outfile, "tlf tlf %lf tlf\n", E,
sin(dphi)*sin(dphi),
sin(dphi)'sin(dphi)/E,
surf ):

}
fclose( outfile ):

Kill_Global_Variables():
return 0:

}

/'ttttt§t%tt%%%§tttt§*l
/*ttttttttttttttt%ttt'l

void Initialise_Global_Variables( void )
{

RealType redJmass;

u - vIctor(o,mx_amr_srzr);
v - vIcco:(o,Max my srzr):
dvIII - vIctor(o,Rax My 512:),-
x2 - were: (0, m_aliaar_srzr) .-
33 - vector(0,me_AnnA¥_$Izl);

puts("\n');
puts('\n");
puts('\n");
put3("\n”);
puts(”\n"):
NtG_Pot Type:
puts?" nSelect nuclear potential type:"):
puts(”znter l for Gaussian well, 2 for Hoods-Saxon well.");
scanf( 'td', ipotential_type );
if( potential_type - 1 )
l

puts('\nGaussian nuclear potential." );
puts(”\n Vnuke - -VO ' exp{ -(r/a)*2 }" ):
puts('\n R - RO * atarg‘0.33” ):

}

else if ( potential_type - 2 )

{
puts("\nloods-Saxon nuclear potential."):
puts("\n Vnuke - -vo * exp{ (r - R)/a }" );
puts(”\n R - RO * atarg‘0.33" ):

else

98

goto NFG_Pot_Type:

l

puts("\nNuclear potential parameters:"):
puts("Enter V0, a, R0 -- (MeV, fm, fm) separated by commas please."):
scanf( "tlf, tlf, 31f", IVnorm, Gadiff, IRO);

puts("\nSpin-orbit potential parameterz");

puts("\n Vnuke - -V0 ' {(r)" );

puts("\n Vso - WO * (df/dr)‘(l/r)*(L. S) " ):

puts("\n£nter W0 (MeV) "):

scanf( "tit", IVspino );

puts("\nV0, a, R0, W0 - ");

printf( "tlf tlf %lf %lf\n", Vnorm, adiff, R0, Vspino );
puts("\n"):

puts("Angular momentum parametersz"):

puts("NOTE: I dont check for legal combinations of s,l.j.");
puts(" Thats up to you. ");

puts("Enter 2s rojectile, l. 2j -- separated by commas please. ");

scanf( "%lf, % f, %lf", SSIprOJZ, El, 5j2 );
puts("\n2strojectile,l ' "):
printf( "% f tlf %lf\n", s_proj2 l, j2 ):

puts("\n");

puts("Enter at, 2t, ap, 2p -- separated by commas please. ");
scanf( "tlf, tlf, tlf tlf", sat, art, tap, 52p );
puts("\nat, zt, ap, zp - ");

printf( "tlf ilf %lf %1f\n", at, zt, ap, zp ):

puts("\n"):

puts("\n");

delr - 0.1:

red mass - 931.49432 * at * ap / ( at + ap ):
hbs32m - (197.33) ' (197.33) / ( 2.000 ' red_mass ):

return:

}

/'%%%%%%%%$§I%§%%%%§§‘/
/'%%%§§%%%§%%%%§%%§\§‘/

void Nill_§lobal_Variables( void )

(

free vector(u, 0,MAX_ ARRAY _SIZE);
free vector(v 0,MAX ARRAY _srzr);
rrII vectortdWs,O. flax my srzs);
trII «etc: (:2 o. MAX_A§RAY_SIZE) .-
free vector(l3,0,MAX_ARRA¥_SIZE);

return:

}

/'%%%§%t§§§%§§t%%§%%%*/

/'%%§ttt§§§i%t%t%%t§t'l

void Calc _Observbls( RealType B, RealType 'dphi,

/.

int 'nodes, RealType 'SurfParam )

Input parameters: 2 --- Kinetic energy of projectile

Output parameters: dphi --- Phase shift of wavefunction at max r

with nuclear potential with respect to

wavefunction without nuclear potential.
nodes - Number of nodes in wavefunction.
SurfParam - Surface parameter defined as

I r-OO l
I / dV l

S I l I u(r) -- d: I
l / dr I
|

r-O l

99

see R.Sherr and G.Bertsch
Phys Rev C 32 1809 (1985)
equation 2.
Notes:

This function accepts the incident energy 8, sets up (with Make_Well) and
solves (with Solve Schrodinger) the Schrodinger equation and then calcul-
ates some observabIes with the results of this solution.
'/
I

RealType phi_with_pot, phi_without_pot;

int nsep,i;

FILE *outfile:

NMax - 2000; /* Default: resonance, need lotta I'/

nsep - NMax; /' mesh points '/
Make_Nell( 1.000 ); /* Set up well with nuke potential */
if( E < V[NMax]) /' If bound state, we dont need many */
( /' mesh points. See notes on Solve_ '/

NMax - 200: /* Schrodinger for info on nsep '/

nsep - ( RO/delr ) * pow(at,0.33333333);
}

Solve_Schrodinger( E, nsep, Iphi_with_pot );

'nodes - 0: /' Calculate nodes and surface param */
*SurfParam - 0.0:
for( i-l; i<-100: i++ )

if( u[i]*u[i+l] < 0.000 ) (*nodes)++;
'SurfParam +- u[i]*u[i]*des[i];
l

/' BMYoung 11/24/92 Nrite potential and wavefunc to file
if(!(outfile-fopen("resout.dat”,"w”)))
{

printf(”cannot open output file.\n”):
exit(1);

}
for(i-O;i<-Nuax:i++)
fprint£(outfile,"%lf %lf %1f\n",(i'delr),V[i],u[il):

fclose(outfile):
BMYoung 11/24/92 '/

*dphi - 0.000;
if( E >- V[NNax] ) /* If resonance, we need wavefunc '/
/* with no nuke potential. '/
Make_Nell( 0.000 ):
Solve_Schrodinger( E, nsep, ephi_without_pot ):
*dphi - phi_with_pot - phi_without_pot:
}

/‘ PUT IN SON! KIND O! WARNING ABOUT I? E>0 AND l<V[NHAX] INCREASE NMAX '/
return:

}
/*%tttttttttttttttttt*/
/*t§t§§t§%t§ttttttttt'l
void Make_Nell( RealType nuclear_pot_const )
0
Packs the array V[] with the total radial potential.
Packs the array des[] with the derivative of the nuclear potential.

‘/
l.

100

RealType ZZ, Rws, VOws, r, Vcentrif, Vcoul, y, ey, fws,
RealType l_factor,Vso:

int i:

22 - zt'zp:

Rws - R0 * pow(at,0.33333333):

for( i-l; i<-NMax: i++ )
{
r - delr * i:

Vcentrif - l'(1+1.0)*hbsd2m/(r'r);

if( r < Rws )

Vcoul - zz*1.44*(1.S-0.5*r*r/(Rws'Rws))/Rws:

)
else

{
}

Vcoul - ZZ‘l.44/r:

if( j2 > (2.0-1) )
l_factor - 1:
llse if ( j2 - (2.0'1) )
l_factor - 0.000:
else

l_factor - -(l+l.000):

L.S - 0.5 . { j(j+1) - l(l+1) - I(s+1) 1 '/

l_factor - 0.5 * ( j2'(jZ+2.0)/4.0
- 1*(1+1)
- s_proj2*(s_proj2+2.0)/4.0 ):

sz:

if ( potentia1_type - 1 ) /' Gaussian well '/
l

y - r/adiff:
ey - expt-y'y):
vws - -vnorm'ey:

Vso - -(Vspino'2.0/(adiff'adiff))'l_factor'ey:
desti] - 2.0'Vnorm'r'ey/(adiff'adiff):

}
else if ( potential_type - 2 ) /' as well
I

y - (r-Rws)/adiff:

if( y < 13.82 ) /' ie if exp(y) < 1e6 '/

{

ey - exp(y):
fws - 1.0 / ( 1.0 + ey ):
}

else

{
ey - 999999.:

'/

101

fws - 0.000000:

}
sz - -Vnorm*fws:

Vso - -Vspino'fws'fws'ey'l_factor/(r'adiff);

des[i] - Vnorm'fws*fws'ey/adiff:
}

Vti] - nuclear_pot_const*(sz + Vso) + Vcentrif + Vcoul:
return:

}

/'%%%%%%%%%%%%%%%%%%%*/

/'%%%%t%%%%%%%%%%%%%%*/

void Solve_Schrodinger( RealType E, int nsep, RealType 'phi )
/e

Input parameters: I --- Kinetic energy of projectile
nsep - see explanation below in Notes

Output parameters: phi --- Phase of wavefunction at max r
Notes:

Solves the Schrodinger equation using two different methods.

If the energy I is greater than the potential (global array V[]) at

NMax, then Solve_Schrodinger calls Diff_eq to find the wavefunction from
r-delr to r-delr'NHax. The phase of the wavefunction is determined and

the wavefunction is normalized.

If the energy I is less than the potential at NHax, Solve_Schrodinger looks
for a bound state near the given energy. First of all, we cant start at
r~0 and work our way out. The wave function must go to 0 at r-O AND at
r-infinity. So, the routine defines the wavefunction at both extremes

and works its way out from r~0 and in from r~infinity towards nsep which

is arbitrarily defined outside the routine. Once this is done, the
wavefunctions are compared at nsep and a better guess is made on a bound state.
This calculation is repeated until

1) the energy goes above 0, 2) the matchup of the wavefunctions agrees to
within a specified tolerance, 3) more than 20 tries are made.

t

{
RealType u1,du1,inu1,u2,du2,inu2;

RealType k,t,x,sum:
int i,num;bound_guesses, searched_for_resonance:

num_bound guesses - 0: /* Initialize a few things ‘/
x - 0.000000:
searched_for_resonance - 0:

do
1
for( i-l: i<-Nlaa: i++ ) /' Pack K2[) for the current I '/
82(1) - (-I + Vti] )‘delr'delr/hbstm:
}
utl] - 0.00001: /' Define u[] near the origin '/
ut2] - utl] ' pow( 2.0, (1+1.0) ):
if( I > Vt NMax ] ) /* If resonance... '/
I
Diff_eq( l, Nlaa, l, sul, sdul, iinul ): I. Get u[] '/
k - sqrt( -N2( Nuaa ] ): /' Get wavenumber, phi, '/
I'phi - atan( ul * k / dul ): /* and normalization */

t - sqrt( ul'ul + (dul'du1)/(k*k) ) * sqrt( k/delr ):
searched_for_resonance - 1:

102

else /* If bound state... '/
{
Diff_eq( l, nsep, 1, sul, adul, Iinul ): /* Get u[] -- 0 to nsep '/
u[ NMax ] - 0.000000: /' Get u[] -- 00 to nsqp '/

u[ NMax - 1 ] - 0.00001:
Diff_eq( NMax, nsep, -1, auZ, GduZ, 6inu2 );
/e
Match up the two wavefunctions u[]. as per G. Bertsch’s suggestion, and
guess the bound state energy.
'/
x - ul'u2'(dul'u2-du2*ul)/(delr'delr'(inul'uZ'u2+inu2'u1'u1)):
E +- x*hbsd2m:
EB[0] - E: /* The latest b.s. energy guess '/
/' is stored in EB[0] '/
for( i-l: i<-(nsep-l); i++ )

u[i] - u[i] ' u2:
for( i-nsep: i<-NMax: i++ )
u[i] - u[i] ' ul:

-sum - 0.000000:
for( i-l: i<-NMax: i++ )
{
sum +- u[i] ‘ u[i]:
}
t - sqrt( sum ' delr ): /' Find the normalization '/
num_bound_guesses++:

}
}while( ( (fabs(x) > 0.00001) II (numgbound_guesses <- 20) )
56 (!searched_for_resonance) ):

for( i-l: i<-NMax: i++ ) /' Normalize the wavefunciton '/
u[i] - u[i] / t:
return:
}
/*%%%§%§%\§tttttt§§t§*/
/*%§t%tttt%t%t%%t%%tt'/
void Diff_eq( int nfrom, int nto, int nstep,
/ RealType 'uu, RealType 'duu, RealType *intuz )
t

Input parameters: nfrom,nto,nstep --- array index limits and step size
over which to solve the differential

equation.

Output parameters: uu --- value of wavefunction u at array
index nto

duu --- value of du/dr at array index nto

intuz --- integral of u'u from nfrom to nto

Notes: 1

Uses the global arrays u[] and K2[] and solves the differential equation
d2u/dr2 - u * k2

from array elements nfrom to nto stepwise in steps nstep. This function
also calculates the integral of u'u over the requested interval and returns
it as 'intu2. Also, the values of u and du/dr at location nto are returned
as *uu and 'duu respectively. A brief mathematical note is in order here.
The algorithm used to solve the differential equation is the Numerov algorithm.
A full-blown description is given in S. I. Koonin, Computational Physics,
Addison-Nesley 1986, p50ff. The formula used is presented below.

103

NUMEROV ALGORITHM

dZu/er - u ' k2 --- let h be the width of one mesh point. The formula
used is

u(r+2h) - 2u(r+h) + u(r) S
------------------------ - -- * u(r+2h) i K2<r+2h) +
60

S
‘ * ulr+hl ' K2(r+h) +
6

5
-- * u(r) ' K2(r)
60

This is solved for u(r+2h), and since we know K2() everywhere, we can use
u(r) and u(r+h) to step our way over r.

 

 

ALSO USED
du | 4 * ( u(r+h) - u(r) ) - ( u(r+h) - u(r-h) )
a- ‘ . _ -o- __
dr |r+h 2h
'/
(
RealType u1,u2,k2_l,k2_2,k2_3:
int n:
*intuz - 0.0: /' Zero the integral */
ul - u[nfrom]: /' Get u(r) and K2(r) */

k2_1 - K2[nfrom]:
n - nfrom + nstep: /* Get u(r+h) and 32(r+h) ‘/
u2 - u[n]:
x2_2 - KZ[n]:
n +- nstep:
while( n <- nto )
{

k2 _3 - KZIDI;

u[n] - ( 2. 0*u2 - ul + 0. 83333333'u2'k2_ 2 + 0. 083333333'01'k2_ 1 ) /
( 1.0 - O. 083333333'k2_ 3 l:

/* Get K2(r+2h) and find u(r+2h) '/

u1 - u2: /' Get ready to increment u and k2 by h */
k2 _1 - k2 _2: /' and update the integral sum. /
u2 - u[n]:

k2 2 - k2 _3:

'thu2 +- u2*u2:
n +- nstep:

}
/* We are at the end of the interval, find */
'/

n -- (3.0 * nstep): /' du/dr and return it and u.
*duu - ( 4.0*(u2-u1) - (u2-u[n]) ) / ( 2.0'nstep ):
*uu - u2:

return:

l
/'§§§§‘§§§§§§§§i§§§§§'l

Appendix C

Maximum Likelihood Fitting
Procedure

To determine quantitative details of the low—lying structure of 10Li from the data col-
lected from the llB(7Li,8B)1°Li reaction, a parametrized theoretical model, described
in Chapter 4 and Appendix B, was fitted to the data. However, since the statistics
in the collected spectrum are low, the standard least—squares fitting technique was
inapplicable. This is due to the fact that the least—squares technique relies on the
assumption that the measured number of events in a given channel has a Gaussian
distribution about some “true” value (hopefully the value of the parametrized model)
determined by nature. In actuality, the measured number of counting events in a
given channel has a Poisson distribution about the “true” value. Only for a large
number of events per channel (i.e. more than 20), where the Poisson distribution

approaches the Gaussian distribution, is the least—squares technique valid.

A general maximum—likelihood fitting technique, a familiar example of which is
the least—squares method, seeks to fit a model function f(:1:,a) to the data a: by
adjusting the model parameters a to maximize the probability of obtaining the given
data set. Typically, a figure of merit is defined as the negative natural logarithm of

this probability. More convenient than (but equivalent to) maximizing the probability

104

10.5

is minimizing the figure of merit. In the example of the least—squares method, the
figure of merit is the X2 statistic [Pres 92]. With this prescription, if the data in
a given channel, (3,31,), have a Poisson distribution about a model f(:z:,-,a) with
parameters a, then the total probability of obtaining the measured data set is

P = H {f(1;:!a)}y' e—f(r¢,a), (C1)

and figure of merit is

L: = —lnP = Z{f(:r,~,a) + 1n 3],! — yiln f(.r,-,a)}. (0.2)

The [1 statistic is similar to X2 in that it is a measure of the goodness—of—fit, and
that its minimization over the space of model parameters is the objective of the
fitting procedure. However, its analytical behavior is not as well known as that of
the chi-squared statistic. More specifically, using .C it is possible to compare fits
with two different models and to determine whether one fit is better than the other;
but it is not possible to assign to a given value of .C a probability analogous to the
chi—square distribution as described in [Pres 92]. The statistical uncertainties on the
model parameters can be determined, as described in Bevington [Bevi 69], from

2 2

a. : m, (C.3)

0'

The computer code FIT_SPEC

The above maximum likelihood technique is carried out with the program FITSPEC,
the FORTRAN source code for which is listed in Table C .1 at the end of this appendix.
The program will fit an arbitrary function, specified in the user-modifiable function
FITFUNC, to a set of data points read in from a file, either in (x,y) or (x,y,ay) format.

The program will perform a maximum—likelihood fit assuming one of four statistical

106

distributions of the data about the functional form: a Poisson distribution, or a
Gaussian distribution with the uncertainty 0,, in each data point assumed to be unity,
fl, or the value of 0,, read in from the file. In each case, the fitting is performed by
numerically minimizing a figure of merit over the space of model parameters utilizing
the subroutine AMOEBA from [Pres 86]. AMOEBA iteratively searches over the space
of model parameters until either a maximum number (500) of iterations is reached
or until successive changes in the figure of merit become smaller than a specified

tolerance.

The program is designed to be user-modifiable; detailed instructions for such
modification are given in the program comments at the beginning of the code listing.
The chief modifications involve changing the function which is fitted to the data. This
function FITFUNC can be easily located in the code by searching for the character
string BMYFITFUNC. The function accepts as input the value at which the function
is to be evaluated, and a one—dimensional array containing the values of the model
parameters. It is important that the user remember the order in which the parameters
are stored in the array, as the user is prompted, during operation, for initial guesses
at their values. In addition to changing the function, the user must also specify the
number of model parameters in the function. This is accomplished by changing the
value of the parameter NPARAMS. The declarations for this parameter can be located in
the code by searching for the character string BMYNPARAMS. Other parameters which
the user might wish to modify are MAXDATCHAN, which specifies the maximum number
of data points allowable, and TOL, which specifies the relative tolerance on successive

evaluations of the figure of merit necessary for AMOEBA to terminate.

Upon running the program, the user is first prompted for format ((a:,y) or (:c,y,ay))
and name of the data file. If the file is successfully read, the program displays the

filename, number of data points (channels), and the sum of y values in the file.

 

 

107

The user is prompted for the minimum and maximum :1: values over which the fit
should be made, and the assumed statistical distribution (Poisson distribution or
one of three Gaussian forms) of the data points about the model. The minimization
routine requires a set of initial guesses at the model parameters. More specifically,
if the model has 72 parameters, a given set of parameter values represents a vector
in the n~dimensional space of model parameters; initially, the routine requires n + 1
such vectors. The program allows for two ways for the total n(n + 1) initial values
to be supplied. They can all be explicitly specified, or the user can specify one
initial guess and a percent range over which the program will randomly pick other
guesses. The latter option, in addition to being less time consuming, allows the user
to hold any parameter constant for the fitting by specifying a 0% range. Once the
initial guesses are supplied, the program displays a list of values of the figure of merit
corresponding to each of the n +1 initial vectors, and enters the minimization routine.
Depending on such factors as the number of parameters, complexity of the model, and
required convergence tolerance for AMOEBA, minimization typically requires less than
30 seconds. Once convergence is reached, the program has in memory a group of 12 +1
parameter sets, the figures of merit for which all lie within the specified tolerance of
each other. The user is asked for the name of an output file and a. comment string
to go in the output file. This file contains the results of the fitting procedure: the
number of iterations AMOEBA required for convergence, the n + 1 final values of each
parameter as well as their averages and uncertainties, and the value of the figure
of merit evaluated at. the average parameter values. Finally, the user can instruct
the program to write a file containing, in (:c,y) format, the optimized model over a

specified range of a: values.

Table 0.1: The listing of the FITSPEC FORTRAN source code is given on the
fifteen pages at the end of this appendix. The operation of the program is discussed
in the text as well as in the program comments.

0000000 0

000000000000000000000000000000000000000000000000000000000

109

PROGRAM rrr_spzc

This program reads a SARA file and fits an arbitrary function to the

data over a specified region of channels. The fit uses a maximum likelihood
technique that assumes the data have a Poisson distribution about the

"true" functional form.

asan_srscrnan —-
Gsr_panauzrsns -

anbzaa ---------

znupnos --------

srrruuc --------

warrr_rr_par ---

Gsr_snnoas -----

CCCCCCC -- Subroutines:

Reads in a file of two types: (X,Y) or (X,Y,DY)

The data are stuffed into the arrays DATA and DDATA.
Allows the user to specify initial guesses for the para-
meters of the fit. There are two options:

1) Specify one initial guess and a percent range over

which the computer will generate random points to jump

start the fit.

2) Specify all initial guesses to jump start the fit.
The initial guesses are packed into the 2d array P and the
1d array 2 contains the values of the probability funcition
for each set of initial guesses.

Perfoms a simplex minimization. This is a Numerical
Recipes function. It takes P and Y and, by calling
the probability function returns the 2d array P with
values of the function parameters that are within a
specified tolerance of each other.

Probability function. Actually this is the negative
natural log of the true probability function assuming
the data have one of several distributions about the
functional form (see below for more on this). This
function accepts

a ld array x. which contains the parameters to be used
in evaluating the fit function. The fit function is
evaluated many times in a sum over the specified data
points.

The possible distributions of the data about the
functional form are (as specified by STATMODE):

l) Poisson distribution

2) Gaussian distribution with sigma - l

3) Gaussian distribution with sigma - (data)**1/2
4) Gaussian distribution with sigma - dy from file

The actual function to be fit to the data. This function
accepts the 1d array x, which contains the parameters of
the fit, and the data ordinate data: at which the function
is to be evaluated.

writes the parameters and their averages to the screen.
will also, if specified. write a file containing the fit
function in (x,y) format.

finds errors on each parameter in the following manner.
for each parameter we have found, using AHOIBA, several
values all within a specified tolerance of each other.

We take an average of the values AVG and define

OIL - 0.01'AVG. Bevington defines

sigma**2 - 2/(d'*2(lnprob)/d"2x)

CCCCCCC -- A mathematical description:

If we have data points, labelled by i, called (xi, yi), and
we assume the data to have a Poisson distribution about some
functional form f(ai). The probability of obtaining the

data

set (xi,yi) is:

110

I yi I
I [f(xi)] I
Prob Func - L - Product over I ----------- exp[ -f(xi) I I
i I (yi)! I
I_ _I
I ‘I
-ln(L) - Sum over I f(xi) + lnt (yi)! ] - yi'ln[ f(xi) I I
' l

i I_

For more info see

--Leo, Techniques for Nuclear and Particle Physics Experi-
ments. Chapter 4.

--P. R. Bevington, Data Reduction and Error Analysis for
the Physical Sciences. Section 3-2.

--N. 8. Press, 3. P. Flannery, S. A. Teukolsky, N. T.
vetterling, Numerical Recipes (PORTRAN edition).
Section 10.4 and Chapter 14

00000000000000000000000

CCCCCCC -- A note about changing the fitting function and other important
parameters:

First of all, the function PITTDNC must be changed to accomodate the
necessary model. This function can be quickly found by searching for
the character string BHTPITFUNC.

Also there are two PARAHITIRS that may be changed.

1) HAXDATCRAN This gives the largest size allowable for the data array.
This PARAHITIR appears at the beginning of the main function,
in the subroutine READ SPECTRUM, and in the function zLNPROB.
These declarations can-be quickly found by searching for
the character string RHYMAXDATCRAN.

2) NPARAMS This gives the number of parameters in the fit. This
PARAMETER appears at the beginning of the main function,
in the subroutines NRITI_IT_QUT, GIT_;RROR3 and
GIT_PARAHIT!RS. These declarations can quickly be found
by searching for the character string BHYNPARANS.

Also, there is a parameter Ton which specifies the tolerance necessary for
the simplex routine AHleA to boot out. The declaration for TOL appears
in the main function just below the NPARANS declaration.

000000000000000000000000

cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
cccccccc -- main program begins here!!! cccccccccccccccccccccccccccccc
c

1mm renal-cm. new, mutant. mam, summer, rumor
marten

PARADETIIH marcm - 4096 )

comma /oara_smr/ XDATA, Dara, DDATA, NCEAN, mucus, mom

comm! Israr_mos/ sramoos

comma Inns. noon rumour

RIAL‘O man? 0 : mucus-1 )

nave DATA( 0 : marten-1 I

nan-s DDATA( o : MARIAN-l )

00

INTIGIR usaaans. 1T3!
CHNIIPARAHB
PARAHITII( IPAIAIB I 7 )
RIAL P( (NPARAH3+1), “PARAHS ), Y( NPARAH3+1 ), TOL

111

REAL ERR( NPARAMS )
PARAMETER( TOL - 0.000001 )

EXTERNAL ZLNPROB

PRINT ',' ’
PRINT ' :'03/19/93 - fit _spec version. NON READS ONLY x,r FILES'
PRINT '
5 PRINT *,' ’
PRINT * ,’Whats the format of the input file?’
PRINT ',’
PRINT *,' 1) (X,Y)'
PRINT ',’ 2) (X,Y,DY)'

READ *,FILEMODE
PRINT *,' '
IP((FILEMODE.LT.1).OR.(PILRMODE.GT.2)) GOTO 5

CALL asap sacrum:
Ir ( NCNAN .LT. 0) GOTO 9999

IF(FILEHODE.EQ.1) THEN
I I

10 PRINT *
PRINT *,'Do you want 1) Poisson distribution'
PRINT *,' 2) Gaussian distribution: sigma-1'
PRINT *,' 3) Gaussian distribution: sigma-sqrt(y)'

PRINT *,’of data about model?’
READ ',STATNODI

PRINT ',' ’
IP((STATMODE.LT.1).OR.(STATMODI.GT.3)) GOTO 10
ENDIP
IP(PILEMDDB. IO. 2) TEEN
15 PRINT
PRINT ' ,'Do you want 1) Poisson distribution'
PRINT *,’ 2) Gaussian distribution: sigma-1'
PRINT ',' 3) Gaussian distribution: sigma-sqrtIy)'
PRINT ',
+ ' 4) Gaussian distribution: sigma-dy from file'

PRINT *.'of data about model?’

RRAD “,STATNDDR

PRINT *,' '

IFHSTAMDLLTJ) .OR. (STAMDI.GT.4)) GOTO 15
ENDIP

can. our PAMTIRS( 9, r)

can mohair, r, (mama) ,mmnsmrmns. TOL, zmnos, use)
cars. car What P. IRR)

cam. mfr_rr_our( P, 3cm, ITIR, can I

9999 IND
C
CCCCCCCC -- Main program ends here!!! CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
SUBROUTINI RIAD_$PICTRUH

c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c

ccccccc -- Input:

C

C None

C
ccccccc -- Output: Both output arguments are contained in the common
C block DATA_STUPP.

00000000

112

DATA -- ld array of size MAXDATCHAN into which the SARA fil is to be

read.

NCHAN - Number of data channels actually stuffed. if read is successfull.

If there is an error reading the file. NCHAN is assigned a value
of -l and is trapped in the main program.

INTEGER MRXDATCRRN.NCRAN.MINCHRN.MAXCHAN,FILEMODE

CBMYHRXDATCRAN «

17

20

25

27

30

35

40

PARAMETER( MRXDATCRAN I 4096 )

COMMON /DATR_STUFP/ XDATA.DATA. DDATA. NCRAN. MINCRAN. HRXCRAN
COMMON /PILE_MDDB/ FILEMODE

REAL'B DATA( 0 : MRXDATCHAN-l ).DDATA( 0 : MRXDATCRRN-l )
RERL'B XDATA( 0 : MRXDATCRAN-l )

rsrscsa¢2 DUMMY. Rana ox_anG
Rani-8 TOTAL coasts

aannse xurs,2hnx
canancrsa . so rrnasnsm

PRINT * ' '

PRINT * ' '

PRINT * 'lnter the name of the input data file. '
PRINT * ' ’

RRAD(*,'(A80)') PILRNAHI

PRINT *. ' '

PRINT ‘. ' '

\\‘\

OPEN(UNIT-1,?ILI-PILINRMI.STATUs-'OLD')

rrIrrLrsoos.zo.1) Tans
romnr_coasrs-o.o
scans-o
aannInstr-1.tsr-t.ssn-zs.zaa-20) xnnrnIscans).onrn(scans)
rornL_coosrs - romni_coasrs + onTnIscans)
scans - scans + 1
coro 17
rarsr *,'saaoa asnnrsc svzcranu PILBES'
PRINT *. ' '
rarsr *. ' '
scans - -1
coro 40
rsnrr

IP(PILIHDDI.IQ.2) razs
romns_coosrs-o.o
scans-o
asap(cart-1.rur-*.zsn-as.saa-30)
xnnrn(scans), onrniscans), oonrn(scans)
rornn_coasrs - rornn_coosrs + onTnIscans)
scans - scans + 1
coco 27
rarsr '.'saaoa aznnrsc spacrauu rrrsis'
PRINT *. ' '
pars? *. f '
scans - -1
GOTO 40
ssnrr

PRINT '.'Successfully read file’
:gigg :p’PILlNAHI

PRINT PZ'Number of channels read: '
PRINT “.NCRAN

PRINT *.' '

113

PRINT '.’Total number of counts: '

PRINT *,TOTAL_COUNTS

PRINT *,' '

PRINT ',’Enter the min and max x-values for the fit.’
READ *. XMIN,XNAX

PRINT *,' ’
PRINT '.' ’
MINCRAN - 0

DO WHILE ( XMIN.GT.XDATA(MINCRAN) )
MINCHAN - HINCHAN + 1
ENDDO

MAXCRAN - NCRAN-l

DO WHILE ( XMAX.LT.XDATA(MAXCRAN) )
MAXCEAN - MAXCEAN - 1

ENDDO

RETURN

END
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C

SUBROUTINE GET_PARAHETER3( P. Y )
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C

CCCCCCC -- Input:

C

C None
C

CCCCCCC -- Output:

P --- 2d matrix of dimensions (NPARAN3+1.NPARANS) which will
contain the initial guesses for the parameters. The NPARANS+1
rows of this matrix contain a set of NPARANS parameter guesses.

Y --- ld vector of dimension (NPARAN3+1) which will contain the values

00000000

INTEGER NPARAMS
CBHYNPARAHS
PARAMETER( NPARAMS - 7 )
REAL P( (NPARAH3+1), NPARAHS ), ¥( NPARAHS+1 )

INTEGER'Q I

INTEGER DUI, GUESSNUN. PARAHNUH

REAL PIRSTGUE88( NPARAMS ), PERCENTRANGE( NPARAMS ), PG. PR
REAL GUESS

PRINT *.’Ohay. there are ’,NPARAN3.’ parameters in this fit.’
PRINT *.’I need ’,NPARANS+1.’ guesses for these parameters.’
5 PRINT ' ’ ’
PRINT * ’ ’
PRINT ' ’You have 2 choices. 1) Enter all guesses yourself.’
PRINT * ’ 2) Enter a guess and a percent range'
PRINT ' ’ and let me make random guesses.’
PRINT * ’ ’
PRINT '.’Choose!!! ’
PRINT *.’ ’
RRAD '.DUN
PRINT s.’ ’
PRINT *.’ ’
Ir (DUN .30. l) TNHN
DO GUISSNUN - l.NPARANS+l

“‘\\§

of the probability function ZLNPROB at each of the sets of parameters.

114

DO PARANNUN - l.NPARANS
PRINT *,' Guess t: ’.GUESSNUN
PRINT ',’ Param I: ’,PARAMNUN
PRINT '.’Enter guess. ’
PRINT *,' ’
READ *.P( GUESSNUN. PARANNUN )
PRINT *,’ ’
PRINT *,’ ’
ENDDO
ENDDO
ELSE IF (DUN .80. 2) THEN
DO PARAMNUN - 1.NPARANS
PRINT *,'Enter first guess and percent range for param I ’,PARAMNUM
PRINT '.’ ’
READ *.PG.PR
PIRSTGUESSI PARAMNUN ) - PG
PERCENTRANGE( PARANNUN ) - PG * PR / 2.0
PRINT *,’ '
PRINT *,' '
ENDDO
PRINT *.’Enter seed for randum number generator.’
READ *.I
PRINT *,’ ’
PRINT *,’ ’
DO PARANNUN - 1.NPARANS
P( l. PARANNUN ) - PIRSTGDESS( PARANNUN )
ENDDO
DO GDESSNUN - 2.NPARANs+l
DO PARANNUN - l.NPARANS
GUESS-FIRSTGUESS(PARANNUN)+(2.0'RAN(I)-l.0)*
+ PERCENTRANGEIPARANNUN)
P( GUESSNUN. PARANNUN ) - GUESS
ENDDO
ENDDO
ELSE
PRINT ',’INVALID CROICE’
GOTO 5
ENDIP
PRINT '.’ ’
PRINT *,’Values of prob func.’
DO GUESSNUN - l.NPARANS+l
DO PARANNUN - 1.NPARANS
EIRSTGOESS( PARANNUN ) - PI GUESSNUN. PARANNDN )
ENDDO
TI GOESSNUN ) - ZLNPROE( PIRSTGUESS )
PRINT ‘.2( GUESSNUN )
ENDDO
PRINT '.’ ’

ssn
gccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c suaaoarrsz csr_sa3035( P. nan )
gccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
Ecccccc -- Input:

C P --- 2d array of dimension (NPARANS+1.NPARANS) containing the
C final values for the fit parameters.

c
ccccccc -- Output:
c

C ERR - ld array of dimension (NPARANS) containing errors on each parameter

115

C None
C
INTEGER NPARANS. PARANNUN. DUN. GUESSNUN
CBMYNPARANS
PARAMETERI NPARANS - 7 )
REAL P( INPARANS+1I. NPARANS I. AVGI NPARANS I. ERR( NPARANS I
REAL PARANSUN.PARANVAL,DELTA.YO.Y_PLUS_DEL.Y_NINS_DEL
REAL TOP.BOTTON

DO PARAMNUM I 1.NPARAHS
PARAMSUH I 0.0
DO GUESSNUM I 1.(NPARAMS+1)
PARAHSUM I PARAHSUM + P(GUESSNUM.PARAMNUM)
ENDDO
AVG( PARAHNUM I I PARAHSUM/ELOAT(NPARAMS + 1)
ENDDO

DO PARAHNUH I 1.NPARAHS
PARAMVAL I AVG(PARAHNUH)
DELTA I PARAHVAL‘0.01

nvc(rnsnssus) - PARANVAL-DELTA
Y_NIN8_DEL - znsraoaI nvc )

AVG(PARAHNUM) I PARAHVAL+DELTA
Y_PLUS_DEL I ZLNPROB( AVG I

AVG(PARAHNUH) I PARAHVAL
Y0 I ZLNPROBI AVG )

BOTTOM - Y_PLUS_DEL- 2.o*so+r_s1ss_pan
rs (sorrow.rr.o.0001) rats
ERR(PARANNUNI - 999999.o
ans:
sanirnanssus) - soar( nasI 2.0'DELTA / aorros ) I
ssorr
ssooo
RETURN
END
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
suanoarrsz warra_rr_our( P. scans. ITER. ran )

c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c

ccccccc -- Input:

P --- 2d array of dimension (NPARAN3+1.NPARANS) containing the
final values for the fit parameters.

NCEAN -- Number of channels in input spectrum. This is used to
write out a file containing (x.y) values for the fit function.

ITER -- number of iterations needed for ANOEEA to converge.

ERR --- 1d array of dimension (NPARANS) containing sigma for each
parameter.

cccccc -- Output:

NOB.

0000000000000000

INTEGER NPARAHB. NCEAN. ITER. GUESSNUH. PARAHNUH. DOM
CEHYNPARAHS

116

PARAMETER( NPARAMS I 7 )

REAL P( (NPARAMS+1). NPARAMS I. X( NPARAMS ). ERR( NPARAMS )
CHARACTER ' 80 FILENAME.COMMENT

REAL PARAMSUM.Y

REAL XMIN.XMAX.XX.DX.NUMSTEPS

PRINT * ’ '
PRINT * ’ ’
PRINT * ’Enter filename where I should write the results.’
PRINT * ' ’
READ(*.’(A80)'I EILENANE
PRINT * ' '
PRINT * ' ’
PRINT ' 'Enter some comment about the fit.’
PRINT * ' ’
READ(*.'(A80I’I COMMENT
PRINT *,’ ’
PRINT '.' '
OPENIUNIT-l.PILE-PILENANE.STATUs-’NEN’)
NRITE(1.') CONNENT
NRITE(1,*) ’ ’
NRITE(l.*I ’Iterations - '.ITER
PRINT *.’Iterations - ’.ITER
NR1TE(1.*) ’ ’
DO PARANNUN - 1.NPARANS
NRITE(1.*) ’Parameter ’.PARANNUN,’ :’
PARANSUH I 0.0
DO GUESSNUN - 1.(NPARANS+II

‘\“

\“‘

NRITE(1.') ' '.P(GUESSNUH.PARAHNUH)
PARAMGUH I PARAHGUH * P(GUESSNUH.PARAHNUHI
ENDDO

xI PARANNUN I - PARANSUN/ELOAT(NPARANS + 11
PRINT *.PARANNUN.PARANSUN/PLOAT(NPARAN3+1).ERR(PARANNUN)
NRITE(1.*) ’Avg. sigma - ’.PARANSUN/PLOAT(NPARANs+l).
ERR(PARANNUN)
WRITE(l.*) ’ '
ENDDO
Y - ZLNPROE( x I
PRINT *.’-ln(Prob) - ’.!
PRINT ‘.’ '
NRITE(l.*) ’-ln(Prob) - ’.!
CLOSEIUNIT-l)
PRINT '.’ ’
PRINT *.'Do you want me to make a file with the fit function in ’
PRINT '.’xy format? '
PRINT *.’1)¥es Any other integerINo’
PRINT *.’ ’
PRINT '.’Choose!!! ’
PRINT P.’ ’
READ '.D
PRINT P.’ ’
PRINT *.’ ’
1r (DUN .E0. 1) THEN
PRINT '.’Enter filename where I should write the results.’
PRINT '.’ ’
READ('.’(ABO)’) PILENANE
PRINT *.’ ’
PRINT '.’ ’
OPEN(UNIT-l.PILE-PILENANI.STATUs-’NEN’I
PRINT *.’Enter KNIN.XNRx.and number of steps.’
PRINT *.’ ’
READ '.XNIN.XNAN.NUN3TIPS
PRINT fi,’ ’
PRINT fi,’ ’
ox - (XNAx-XNINI/NUNSTEPS
DO xx-xsxs.xsnx,ox

117

Y I FITFUNC( X. XX )
WRITE(1.*) XX.Y
ENDDO
CLOSE(UNITI1)
ENDIF

RETURN

END
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C

REAL FUNCTION ZLNPROB( X )

gCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
gCCCCCC -- Input:

g x -- A ld array of dimension NPARANS containing the parameters for
C the fit.

gCCCCCC -- Output:

The returned funciton value is the value of the probability function
with the input parameters.

0000

aanz. xv)

rs'rscsa marcans. scans. srscans. snxcans. srnrsoos
cammxnnrcans

anansrram Imxnn'rcans - 4096 I

comos lon'rn_srurr/ xnnrn. DATA. oonrn. scans. NINCHAN. snxcans

comes Isrnr soon s'rnmoos

sans-a Dam o : snxnnrcans-l I

REAL'B warm 0 : mucus—1 I

REAL'B xnn'rni o : snxnnrcans-i I

REAL'S E. SUM. TEMP
INTEGER CHAN
REAL'G XX. Y. D!

SUM I 0.0
DO CHAN I NINCHAN.NANCHAN
XXI XDATA( CHAN )
Y I DATA( CHAN )
DYI DDATA( CHAN I
l I EITEUNC( X. XX I

If statmode I 1 then we went Poisson statistics

000

If (STATIODE.EQ.1) THEN
IE (E.GT.0.0) THEN
SUN I SUN + I - Y * LOG(E) + EACLOG(NINT(Y))
ELSE
PRINT ‘.E
ENDIE
ENDIE

If statmode I 2 then we want Gaussian statistics. calculate good ol’
fashioned chi-squared with sigma-l.

If (STATHODE.EQ.2) THEN

TEMP I (Y - I)

SUN I SUN + TENP‘TENP/2.0
ENDIE

0000

00

If statmode I 3 then we want Gaussian statistics. calculate good ol’

118

C fashioned chi-squared with sigma-sqrt(y).

IE (STATMODE.EQ.3) THEN
IF (Y.LE.0.0) THEN
DY I 1.0
ELSE
DY I SQRT(Y)
ENDIE
TEMP I (Y - El/DY
SUM I SUM + TEMP*TENP/2.0

ENDIP
C
C If statmode I 4 then we want Gaussian statistics. calculate good ol’
C fashioned chi-squared with sigma-dy from data file
C
IE (STATNODE.EQ.4I TEEN
TEMP I (Y - FI/D!
SUM I SUN + TENP'TENP/2.0
ENDIP
ENDDO
ZLNPROB I SUN
RETURN
END
c

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C

REAL FUNCTION EACLOG( N I

C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
CCCCCCC -- Input:
C
C N -- Integer argument
C
CCCCCCC -- Output:
C
C The value returned is the natural log of the factorial of the argument.
C

INTEGER I.N

EACLOG I 0.0

It (N .GT. 1 I TEEN

DO I I 2.N
EACLOG I EACLOG + LOG(ELOAT(III
ENDDO

ENDIP

RETURN

END
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CENIEITEUNC
C

REAL EUNCTION EITTUNC(X. XX)
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
CCCCCCC -- Input:

x --- ld array of dimension NPARANS containing the fit parameters.

0000

xx -- x value at which to evaluate the function.

119

c
ccccccc -- Output:

The returned value is fit function evaluated at the input channel number
with the input parameter set.

Functional form has 5 components:
---A constant background K

IIIA 3 body phase space background
ZETA I ( CHAN - X0 I / ( ENDCRAN - X0 I
HACK I A ' SQRT( 1 - ZETA'ZETA I

III2 p-waves
PNAVE I N * EXP( -(E-XRI'(E-XRI/(2*GLOW'GLON) I
if chan > xr

PNAVE I N ' [ R + (l-R) ' GHI * GEI / (FUDGENI'(E-XR)**2 + GRI'GHII I
if chan < xr

R I I0.06874 + 0.38989'EN - 0.18571'EN'EN
GNI I 2.31417 - 6.22500'EN + 3.83333'EN'EN
GLON I I0.09409 + 0.53771'EN - 0.21429'EN'EN
EUDGEHI I 61.96055 - 231.11300'EN + 216.0000'EN'EN

GHI"2 I GHI*'2 + EUDGEHI'DEVKNHM'DEVHNHH
GLON'*2 I GLON'*2 + DEVHNHM'DEVHNHH/ZLNZ

I-IA narrow s-wave at zero energy
SHAVE I B I 9.923 / ( (CEAN-ZEROCEANI**2 + 9.923 I

000000000000000000000000000000000000000000

X(lI I le \
X(2) I Nl > p-wave parameters
X(3) I XRZ /
X(4) I N2 /
X(5) I B > s-wave parameter
X(6) I A > 3body background amplitude
x(7) I E > constant background
REAL NP)
REAL'B xx
REAL EN
REAL DEVNNNN
REAL RE. CLONE. GNIE. PUDGEEIE
REAL R. GLON. GNI. EUDGEEI
REAL EZ.PNAVE1.PNAVE2
REAL SNAVE. ZEROCEAN
REAL ENDCEAN. x0. ZETA. BACKGROUND
REAL E

C
C first P-NAVE
C

C
C DE first convert X(l) to neutron resonance energy
C

C Use the calibration at the bottom of p116 in log. converted to a 1k
C scale. This is only good for 0.1 < EN < 1.25.
C

EN I 10.3621 - 0.03412*X(1)
C
C No. CEICNIECO RE.GLONE.GHIE.FUDGEHIE

00000

000

000000

0000000 000000000 000

000

00000

120

NOTE: In real life DEVENNN I 0.115Nev. Using the above calibration
for a 1k scale. this corresponds to HNNM I 3.370 chan/1k
FWHN I 6.741 chan/1k

DEVHNHH I 0.115

RE I I0.06874 + 0.38989*EN - 0.18571'EN'EN
EUDGEHIE I 61.96055 - 231.11300'EN + 216.0000'EN'EN
GHIE I 2.31417-6.22500'EN+3.83333'EN'EN

GLONE I I0.09409 + 0.53771'EN - 0.21429'EN'EN

Now add the device ENEN

GHIE I SQRT( GHIE**2 + EUDGEHIE'DEVHNHM*DEVHNHN I
GLONE I SQRT( GLONE"2 + DEVHNHN'DEVHNHM/1.3S63 I

Now we need to convert from energies back to channels

PUDGEEIE and RE dont need to he changed
GNIE and CLONE do -> G I GE / (dE/dx)

PUDGEHI I EUDGEHIE

R I RE .
GHI I GEIE / 0.03412
GLON I CLONE / 0.03412

Calculate the resonance shape

22 I XXIX(1I
IE (ZZ.LT.0.0I THEN
PNAVEl I X(2I ' (R * (1.0-RI'GHI'GHI/(EUDGEHI'22*22+GHI'GHI))
ELSE
PNAVE1 I X(2I'EX?(‘(ZE*EEI/(2.0'GLON‘GLO'II
ENDIE

Second P-NAVE

ON first convert X(3) to neutron resonance energy

Use the calibration at the bottom of p100 in log, converted to a 1k
scale. This is only good for 0.1 < EN < 1.25.

EN I 16.3621 - 0.03412IX(3)
Now calculate RE.GLONE.GEIE.EUDGEEIE
NOTE: In real life DEVENIN I 0.115Nev. Using the above calibration
for a 1k scale. this corresponds to ENEN I 3.370 chan/1k
ENEN I 6.741 chan/1k
DEVENEN I 0.115
RE I I0.06574 + 0.30969'EN - 0.18571'lfl'EN
EUDGEEIE I 61.96055 - 231.11300'EN + 216.0000‘EN'IM
GEIE I 2.31417-6.22500'EN+3.63333'EN'EN
GLONE I -0.09409 + 0.53771INN - 0.21429'EN'EN
Now add the device ENNN

GHIE I SOIT( GHIE"2 + fUDGEEIE'DEVHNHN‘DEVHNHN I
GLONE I SOIT( GLONE'*2 + DEVHNHNPDEVHNHI/1.3063 I

Now we need to convert from energies back to channels

EUDGEEIE and RE dont need to be chan d
GRIE and CLONE do -> G I GE I (dE dx)

121

C
PUDGEEI I EUDGEEIE
R I RE
GNI I GNIE / 0.03412
GLON I GLONE / 0.03412
C
C Calculate the resonance shape
C

22 I XX-X(3I
IE (ZZ.LT.0.0) THEN
PNAVEZ I X(4I * (R + (1.0-RI'GHI'GHI/(PUDGEHI‘ZZ'ZZ+GHI‘GHIII

ELSE
PNAVEZ I X(4)'EXP(-(zz*zZI/(2.0'GLON'GLON))
ENDIE
C
C S-NAVE
C
C Note -- 11.360 is 3.370 squared -- the s-wave is the width of
C the resolution
C
C ZEROCEAN I 538.1624 <-- 0 kev
C ZEROCEAN I 536.6970 <-- 50 keV
C ZEROCEAN I 535.2315 <-- 100 kev
C
ZEROCRAN I 536.6970
ZETA I xx - ZEROCEAN
IE ( X(5I .LT. 0.0 I XI5I I 0.0
SNAVE I X(5I ' 11.360 / ( ZETAIEETA + 11.360 I
C
C BACKGROUND
C
C Note -- ENDCNAN is the kinematic limit for the reaction
C x0 is a parameter determined from 3body phase space calculations
C Both are in a 1024 spectrum
C
ENDCBAN I 536.1624
x0 I 237.9954
urn-(xx-XOI/(mcans-XOI
IE ( xx .LT. ENDCEAN I TEEN
I! ( X(6) .LT. 0.0 I X(6I I 0.0
BACKGROUND I X(6) I 80RT( 1.0 - EETA'zETA I
ELSE
BACNGROUND I 0.0
ENDIE
C
C CONSTANT BACNGROUND
C
IE(X(7I.LT.0.0I 3(7)-0.0
C
C NON ADD IT UP
C
EITEUNC I BACEGROUND + SNANE + PNANEl + PNAYEZ + X(7I
RETURN
END

C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

C .

SUEROUTINE MIA". 1.59. NP. NDIN. PTOL. I'UNE. ITERI
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
gCCCCCC -- Input:

C
C

122

CCCCCCC -- Output:
C
C
C
PARAMETER (NMAX-ZO,ALPEA-1.0.BETA-0.Lam-2.0,ITNAX-SOOI
DIMENSION P(NP.NP),¥(MPI,PR(NNAX),PRR(NNAXI,PBAR(NNAX)
NPTS-NDIN+1
ITER-O
1 1Lo-l
I!(Y(l).GT.Y(2))THEN
131-1
INNI-Z
ELSE
181-2
INNIIl
ENDIT
DO 11 I-l.NPTS
1r(¥(1).LT.Y(ILOII ILo-I
II(Y(II.GT.Y(IHIIITNEN
INNI-IEI
INI-I
ELSE IE(Y(II.GT.Y(INUIIITEEN
IE(I.NE.IBII INflI-I
ENDIE
ll CONTINUE
RTOLI2.'ABS(I(ISII-Y(ILOII/(AES(!(IRIII+ABS(T(ILOIII
I!(RTOL.LT.ETOLIRETURN
IT(ITER.EQ.ITNAXI PAUSE 'Anoeha exceeding aaaimum iterations.’
ITER-ITER+1
DO 12 J-1,NDIN
PBAR(JI-O.
12 CONTINUE
Do 14 II1,NPTS
IE(I.NE.IEIITREN
DO 13 41.10qu
PBAR(J)-PEAR(JI+P(I,JI
l3 CONTINUE
ENDI!
14 CONTINUE
DO 15 J-l,NDIN
P3AR(JIIPIAR(JI/NDIN
PR(J)-(l.+ALPEA)*PIAR(J)-ALPEA‘P(IEIIJI
15 CONTINUE
fill-MGR)
Ir(IPR.LE.!(ILOIITNEN
DO 16 J-1.NDIN
PRR(JIIGAIIIA*PR(JI+(1.-GAIIII*PIAR(JI
16 CONTINUE
YPRR-EUNE(PRII
IE(YPRR.LT.¥(ILOIITEEN
DO 17 JI1,NDIN
P(IEI,J)IPRN(JI
l? CONTINUE
HIEII-IPRR
ELSE
DO 18 J-l,NDIN
9 (III. J) IPRM)
10 CONTINUE
2(IEII-YPE
ENDIE
ELSE IE(!PR.CE.!(INEI))TNEN
IT(YPR.LT.T(IEIIITEEN
DO 19 JILNDIN
H181. J) IPRUI
19 CONTINUE
!(IEII-!TR

21

22

23

24

25

123

ENDI!
DO 21 J-IINDIM
PRR(J)-BETA'P(IHI,J)+(1.-BETA)'PBAR(J)
CONTINUE
YPRR-FUNK(PRR)
IF(YPRR.LT.Y(IHI))TEEN
DO 22 J-IINDIH
P(INI,J)-PRR(J)
CONTINUE
Y(IHI)-YPRR
ELSE
DO 24 I-IINPTS
IE(I.NE.ILO)THEN
DO 23 J-IONDI“
PR(J)-O.5‘(P(I,J)+P(ILO,J))
P‘IIJ)-PR(J)
CONTINUE
Y(I)'EUNK(PR)
INDIE
CONTINUE
INDIE
ELSE
DO 25 J-LNDIN
P(INI.J)-PR(J)
CONTINUE
Y(IHI)-YPN
ENDIE
GO TO 1
END

 

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"‘u[]][11771]“