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This is to certify that the

dissertation entitled

"Structure of the Proton Unbbund Nucleus 11N"

presented by

AFSHIN AZHARI

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Physics

 

 

was?

Major [irofessor
M. Thoennessen

Date 9/23/96

 

MS U i: an Affirmative Action/Equal Opportunity Institution 0.12771

 

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TO AVOID FINES return on or before date duo.

 

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MSU In An Affirmative Adlai/Emu Opportunity Intuition

 

STRUCTURE OF THE
PROTON UNBOUND NUCLEUS 11N

by

Afshin Azhari

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

1 996

ABSTRACT
STRUCTURE OF THE PROTON UNBOUND NUCLEUS “N
By

Afshin Azhari

A previous experimental study of the proton unbound nucleus llN did not observe
the ground state and theoretical calculations predicted the ground state to be unbound to
proton decay by 1.9 MeV. However, a study of the diproton decay of 12O favored a
sequential proton decay via an intermediate state in 1‘N at 900 keV to the ground state of
loC. Recent theoretical calculations of the ground state of 11N obtained a decay energy of
1.6 MeV. Therefore, a clarification of the structure of 11N was needed.

In the present study, the states in 11N were populated through the reaction
9Be(12N,”N) using a radioactive nuclear beam of 12N. Due to extremely short lifetimes
(~10'2' seconds), the 11N decayed into a 10C nucleus by emitting a proton inside the
target. These decay products were detected in coincidence and momentum vectors were
Obtained for each, thus allowing for a kinematic reconstruction of the decaying states
within “N.

Monte Carlo simulations of the decay energy spectrum of llN were performed. x2
optimizations of the simulations relative to the data yielded a decay energy of 1.45 MeV

and a width of 2.4 MeV for the ground state of llN.

To my mother and the memory of my father

iii

ACKNOWLEDGMENTS

First and foremost I would like to thank God for without Him nothing would be
possible.

I had the privilege of working with one of the most talented scientists I have ever
known. No words can convey the gratitude I feel for all that Michael Thoennessen has
done to make my experiences as a growing scientist as smooth and efficient as possible.
As a mentor and a friend, I could have found none better.

I would like to thank professors Julius Kovacs, David Tomanek, and Gary
Westfall for being on my guidance committee. I thank Prof. Alex Brown for his tutelage
and providing some of the theoretical calculations in addition to being a member on my
guidance committee. I also greatly appreciate the theoretical calculations provided by Dr.
Millener.

I truly enjoyed working with my collaborators Dan Russ and Houria Madani from
Maryland and am indebted for all their help with the setup of the experiment. I also had
the good fortune of working with Tiina Silomijarvi from Orsay who tirelessly helped with

the electronics setup.

iv

My thanks go out to the staff of the NSCL without whom this work would not
have been possible. I wish to especially thank Craig Snow and Dave Sanderson for their
continuos help, especially those last minute lifesavers.

Perhaps the most enjoying part of working at the NSCL was working alongside
members of our group Easwar Ramakrishnan, Shigeru Yokoyama, Bob Kryger, Thomas
Baumann, Peter Thirolf, and Marcus Chromik. I also had the pleasure of interacting with
Jim Brown, Michael Fauerbach, Jon Kruse, Phil Zecher, Barry Davis, Mathias Steiner,
Raman Pfaff, Stefan Hannuschke, Sally Gaff, Jing Wang, Richard Ibbotson, and John
‘Ned’ Kelley who always kept me on my toes.

Closer to home, I would like to thank my beloved wife Nancy for all her patience
and moral support throughout my graduate years. My heartfelt thanks go out to my aunt
and uncle Guity and Ataolah Modir Massihai for all they have done for me and are as
dear to me as my own parents. As a final note, I would like to thank my mother and
sister Mahvash and Maryam for their encouragements and aid, and to my father Nader

who made my dream come true.

TABLE OF CONTENTS

LIST OF TABLES ....................................................................................... viii
LIST OF FIGURES ....................................................................................... ix
CHAPTER 1
INTRODUCTION .......................................................................................... 1
CHAPTER 2
THEORETICAL OVERVIEW
2.1 Calculation of the Ground State ............................................................................... 6
2.1.1 The IMME ......................................................................................................... 6
2.1.2 Potential Model Calculations ............................................................................ 8
2.2 Proton Decay .......................................................................................................... 10
CHAPTER 3
EXPERIIVIENTAL SETUP
3.1 Production of the Radioactive Beam ..................................................................... 13
3.1.1 Introduction ..................................................................................................... 13
3.1.2 Production and Purification of 12N ................................................................. 14
3.2 Detector Assembly ................................................................................................. 21
3.2.1 The Tail of the RPMS ..................................................................................... 21
3.2.2 Proton Detectors .............................................................................................. 23
3.2.3 Fragment Detectors ......................................................................................... 24
3.3 Calibration Beams .................................................................................................. 26
3.3.1 Proton Beams .................................................................................................. 26
3.3.2 Carbon Beams ................................................................................................. 27
3.4 Electronics ............................................................................................................. 27
3.5 Data Acquisition .................................................................................................... 31
CHAPTER 4
DATA ANALYSIS
4.1 Analysis Software .................................................................................... 33
4.2 Energy Calibrations ............................................................................................... 34

vi

4.2.1 Proton Detectors .............................................................................................. 34

4.2.2 Fragment Detectors ......................................................................................... 37
4.3 Position Calibrations .............................................................................................. 39
4.3.1 Fragment Telescope PPAC ............................................................................. 39
4.3.2 Beamline PPACS ............................................................................................. 40
4.4 Particle Identification ............................................................................................. 41
4.5 Contamination ........................................................................................................ 43
4.6 Decay Energy ......................................................................................................... 44
4.7 Monte Carlo Simulations ....................................................................................... 45
4.7.1 Secondary Beam ............................................................................................. 46
4.7.2 Interactions in the Target ................................................................................ 46
4.7.3 The Reaction ................................................................................................... 47
4.7.4 The Decay ....................................................................................................... 49
4.7.5 Detectors ......................................................................................................... 51
4.7.6 Decay Energy .................................................................................................. 51
4.7.7 Efficiency and Resolution ............................................................................... 51
4.8 Fitting of Data ........................................................................................................ 54
CHAPTER 5
RESULTS AND DISCUSSION
5.1 Decay Energies ...................................................................................................... 56
5.1.1 Decay of13N .................................................................................................... 56
5.1.2 Decay of130 .................................................................................................... 62
5.1.3 Decay of 10B .................................................................................................... 65
5.1.4 Decay of “N .................................................................................................... 67
5.2 Comparisons to Theory .......................................................................................... 73
5.2.1 Populations Ratios .......................................................................................... 73
5.2.2 Higher Excited States ...................................................................................... 74
CHAPTER 6
SUMNIARY AND CONCLUSIONS ........................................................... 79
APPENDIX A
MONTE CARLO SIMULATION
A] Description ............................................................................................................. 82
A2 Input File ................................................................................................................ 83
A.3 Output .................................................................................................................... 87
AA Pro_Decay .............................................................................................................. 90
A5 Subroutines .......................................................................................................... 104
A.6 Sample Input File ................................................................................................. 114
BIBLIOGRAPHY ....................................................................................... 1 15

vii

List of Tables

2.1 Parameters used with the IMME for the calculation of the ground state of
11N shown in the last row [Ben79]. The last column contains the mass
excess of each state. .............................................................................................. 8

2.2 11N states calculated by FKL using a Woods-Saxon potential [For95] ................ 9
4.1 Degrader combinations and proton energies obtained during the calibration

of the MFA. The 35 MeV proton beam was used to obtain the values in the

first three rows. The remainder were obtained using the 70 MeV proton

beam .................................................................................................................... 36
4.2 Energy calibration points obtained for the fragment telescope ........................... 38

5.1 Excitation energies and widths used to simulate the 13O decay energy
spectrum. Also included are the relative population ratios calculated from

the simulations .................................................................................................... 65
5.2 The relative population ratios of the states in 11N ............................................... 73
5.3 Parameters for the calculated excited states of llN [Mil96] ............................... 76
A1 Variables in the array ‘Revent’ .......................................................................... 88
A2 Description of variables in the array ‘Ievent’ ..................................................... 89
A.3 Values of Ievent(2) and description of the cause of the ‘bad’ event .................. 89

viii

List of Figures

1.1

1.2

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

Competition between Sm and pm levels [Tal60] ................................................ 3

Spectrum obtained for “N from l4’N(3He, 6He)”N [Ben74]. The solid peak
is a theoretical fit of the first excited state. Indicated on the spectrum is the

proton decay threshold for 11N ............................................................................ 5
Diagram of the A1200 mass separator in the configuration used to produce
and purify thelzN secondary beam .................................................................... 14

A plot of energy loss versus time-of-flight for the fragments seen at the

focal plane of the A1200. Indicated within the figure are fragments with

equal number of protons and neutrons, and the secondary radioactive beam

of N ................................................................................................................. 16

Vertical versus horizontal positions of the fragments in the PPAC at the
focal plane of the A1200. The dashed lines represent the position of the

horizontal slits ................................................................................................... 17
Time-of-flight of fragments seen at the focal plane of the A1200 before and
after the insertion of the slits ............................................................................. 17
The Reaction Products Mass Separator ............................................................ 19
Beam line components behind the RPMS ........................................................ 20
Energy loss versus time-of-flight spectrum of fragments seen in the

fragment telescope prior to the insertion of the vertical slits ............................ 20

Vertical versus horizontal s ectrum of PPAC2. This spectrum shows the
spatial separation between 1 N and 130 fragments and was used to Optimize
the position of the vertical slits for the 12N beam .............................................. 21

Detector setup at the end of the RPMS tail. The A81 and the MFA

provided position, AB, and E information for the protons. The fragment
telescope provided position, AB, and E for the heavier fragments .................... 23

ix

3.10 Schematic diagram of the electronics. All times are in nanoseconds .............. 28
3.11 Logic signal diagram for the creation of the TDC common start signal ........... 31

4.1 Proton energy versus the radial strip of the A81 for 63.96 MeV protons ......... 36

4.2 Energy calibration of the MFA gated on one ASI pixel ................................... 37
4.3 Calibration spectrum for one quadrant of the fragment silicon detector .......... 38
4.4 Calibration spectrum for the fragment telescope PPAC ................................... 39

4.5 Fragment telescope PPAC spectrum gated on fragment telescope silicon
and projected onto the x-axis ............................................................................. 40

4.6 Total energy versus energy loss spectrum (lefi) and long versus short

(right) spectra used to identify the protons ........................................................ 42
4.7 Energy loss versus energy spectra for the fragment telescope. Raw

spectrum (left) and gated on all protons (right) ................................................. 43
4.8 Vector diagram of a proton decay. All distances are in millimeters ................ 45

4.9 Scattering reaction between projectile and target in laboratory and center of
mass frames ....................................................................................................... 47

4.10 Distribution used to simulate the scattering angle in the laboratory frame ...... 48
4.11 Simulated geometric efficiency of the decay of 11N ......................................... 52

4.12 Simulated width of the decay energy spectrum versus the input width of a
1/2+ state at 1.9 MeV decay energy in IN ........................................................ 53

4.13 Simulated decay energy spectra for a 1/2+ state in 11N at decay energies of
1.6 MeV. Decay widths of 200 keV and 2.0 MeV were used to obtain the
left and the right spectra, respectively. The solid lines were obtained at a
geometric efficiency of 100%. The dashed lines include the experimental
geometric efficiency and were normalized to the height of the solid lines
for comparison ................................................................................................... 53

5.1 Raw decay energy spectrum of 13N ................................................................... 57

5.2 12C fragment energy versus decay energy data (lefi) and simulation (right).
Points below the 360 MeV dashed line are background events ........................ 58

5.3 Background spectrum of 13N [Ajz91]. This spectrum was obtained by
gating on 12C fragments with energies lower than 360 MeV ............................ 59

5.4 Background subtracted spectrum of the decay energy of 13N. The error bars
represent the statistical uncertainties only ......................................................... 59

5.5 Level diagram of 13N. The excited state proton decay to 12C is indicated by
the arrow. All energies are in units of MeV ..................................................... 60

5.6 x2 surface plot as a function of excitation energy and width of the

simulated 3/2' excited state of 13N. The dashed lines intersect at the

minimum )8 and the shaded are shows the region within xzmin+l .................... 61
5.7 Background subtracted decay energy spectrum of 13N fitted by a Monte

Carlo simulation of a state at an excitation energy of 3.45 MeV and a width

of 90 keV (solid line) ......................................................................................... 61
5.8 Raw energy spectrum of ”N fragments ............................................................ 62

5.9 Decay energy spectrum for 13O. The data is shown as points with
statistical error bars ............................................................................................ 63

5.10 Energy levels and decay scheme for 13O. The widths of the first two
excited states of 130 were not known. All energies are in units of MeV ......... 64

5.11 Fit of the decay energy spectrum of 13O. The points with the error bars are
the data, the solid line was obtained from the sum of the simulations for the
first three excited states ..................................................................................... 65
5.12 Level structure of loB. All energies are in units of MeV ................................. 66

5.13 Decay energy spectrum for 10B. Contributing states are indicated by the
lines ................................................................................................................... 67

5.14 Raw decay energy spectrum for 11N ................................................................. 68

5.15 Background spectrum for the decay of 11N. The background was obtained

by gating on C fragments with energies below 360 MeV (solid). The
dashed line indicates a simulation of the efficiency of the setup ...................... 68
5.16 Background subtracted decay energy spectrum of 11N ..................................... 70

xi

5.17 Fit of the HN decay energy spectrum. The fit to the data (solid) is a sum of
the contributions from the known 1/2' excited state (short dashes) and a
1/2+ state at Edccay = 1.45 MeV and F = 2.4 MeV (long dashes) ....................... 70

5.18 x2 surfaces as a function of decay energy and width for the 1/2+ state.
Recent theoretical predictions are indicated by the filled square with error
bars [For95] and the filled circle [Bar96]. The Wigner limit on the decay
width is represented by the dashed line. The position of minimum x2 is
marked by the star ............................................................................................. 71

5.19 loC energy spectrum. The arrows indicate energies corresponding to
formation of llN via transfer reactions (Bf) and fragmentation reactions (EQ...74

5.20 Decay scheme for the theoretically predicted [Mil96] 3/2' and 5/2' excited
states of 11N. All energies are in units of MeV ................................................. 75

5.21 Simulation of the decay of the excited states of UN to the ground and first
excited states of IOC ........................................................................................... 77

xii

Chapter 1

Introduction

Although nuclei near and beyond the particle drip lines exhibit extremely short
lifetimes, typical of strong interaction time scale, they play an important role in the
observed abundance of elements in the universe. On the proton rich side of the valley of
stability, the rp-process is a major contributor to nucleosynthesis within supernovae.
Therefore, an understanding of the structure of these exotic nuclei is of high priority to
astrophysical calculations.

The exact location of the particle drip lines is one of the most stringent tests of
nuclear structure models, especially in the extension of mass formulas to and beyond the
drip lines. Beyond the proton drip line, the Coulomb and centrifugal barriers can lead to
relatively long lifetimes (proton radioactivity) [Hof94]. Several of these ground state
proton emitters have been observed [Pag92] and provide a unique probe of the nuclear
structure since their lifetimes are sensitive to the nuclear potential [Pag94, Tig94].

However, the production of these exotic nuclei at rates needed for experimental

studies has posed a major challenge to experimenters. The recent availability of

radioactive nuclear beams has opened a new doorway into the production of unstable
nuclei away from the valley of stability. Since radioactive beams of nuclei can be created
which are already deficient in neutron number, nuclei near and beyond the proton dripline
can be formed at higher rates.

In the absence of the centrifugal barrier, such as an s-wave proton, the lifetime of
the ground state of nuclei beyond the proton drip line can be extremely small. One such
possible candidate is l 1N. A Simple shell model picture of 11N predicts the ground state
to be a 1/2' state, however the ground state of 11 Be (the mirror nucleus of 11N) is a 1/2+
state [Ajz90]. The 0.32 MeV gap between the 1/2+ ground state and the 1/2' first excited
state of 11Be can be accounted for in terms of three distinct physical contributions
[Bro94]. First (I), the Isl/2 single-particle energy is calculated to be about 3.6 MeV above
the 0pm single-particle energy. (ii) There is an extra pairing energy in the 1/2+
configuration due to the two neutron holes in the Op shell which lowers the 1/2+ energy,
relative to the 1/2' configuration, by about 2.2 MeV. (iii) There is mixing with the
[2+® dm](1/2+) configuration which lowers the energy by about 1.5 MeV [Bro94].
Adding the three aforementioned effects, the 1/2+ and 1/2' configurations become
essentially degenerate. Also, a linear extrapolation between the p 1,2 - 51/2 difference in
13C (3.09 MeV) and the corresponding difference between the center Of mass in 12B (1.44
MeV) gives the predicted difference in 11Be [Tal60]. Figure 1.1 shows the result of such
a calculation. Therefore, the ground state of 11N is expected to also consist of 1/2+

intruder state.

51/2

Pm

 

 

Figure 1.1: Competition between S] ,2 and Pm levels [Tal60].

Primary interest in the structure of llN arose from the results of an experiment to
study the two-proton decay of 12O. Goldanskii [Gol61] predicted the existence of ground
state diproton decay in proton-rich even-Z nuclei where the pairing energy between the
last two protons would forbid the one proton decay branch. Based on current mass
measurements [Ajz91], 12O is one such candidate. Previous to the experimental study of
12O, calculations of the ground state of 11N reported the ground state at 1.9 MeV above
proton decay threshold. If the ground state of 11N was located at 1.9 MeV, a sequential
decay of 120 via llN would not be energetically possible. In addition, the sequential
decay through the tail of a broad (F = 1.5 MeV) llN state at 1.9 MeV is strongly
suppressed by penetrability and does not reproduce the measured width of the 120 ground
state. Therefore, the ground state of 12O was believed to decay by a di-proton to 10C since

the sequential proton decay of 120 would have been suppressed by the predicted ground

state energy of 11N. However, Kryger et al. [Kry95] established an upper limit of 7% on
the di-proton decay branch and discovered that the data could be reproduced with the
assumption of an intermediate state in 11N at about 900 keV.

The first experimental study of 11N used the reaction li'N(3He,6He)”N to
reconstruct the levels of 11N by observing the kinematics of the 6He [Ben74]. Figure 1.2
shows the spectrum that was obtained in this experiment. The solid line is a fit attributed
to the 1/2' excited state of '1N. A decay energy of 2.241 0.10 MeV and a width of
740 i 100 keV were obtained from this fit [Ben74]. The 1/2+ ground state was deduced
from the Isobaric Multiplet Mass Equation (IMME) to be at 1.9 MeV [Ben74].

Recently the ground state of 11N has been reexamined by several theoretical
approaches. Sherr [She95] suggested that the isobaric analog states in 11B and 11C could
have been misidentified and used a new set of energies to calculate a decay energy of
1.5-.1- 0.1MeV for the 1/2+ ground state of llN from the IMME. Fortune et al. [For95]
(hereafter referred to as F KL) also pointed out the possibility of the rrrisidentification of

the states in 11B and 11C. However, FKL believe that the IMME does not apply to lightly
bound (or unbound) 2.3-5— states. Instead they carry out a potential model calculation to
obtain Edecay = 1.60 i 0.22 MeV and F =1.58:%.§§ MeV for the ground state. Barker

[Bar96] attempted the same calculations performed by FKL, but could not obtain the
same results. Barker suggests that the potential model calculation of FKL applies only to
single-particle states where the spectroscopic factor is of the order of 1. This is not valid
for the levels in 11Be, the isobaric analogue of llN which was used to obtain the

parameters of the potential well, Since the spectroscopic factor for all levels are

considerably less than one. By incorporating a significant amount of d-wave
contribution, Barker calculated a decay energy of 1.60 MeV and a width of 1.39 MeV, in

agreement with the results obtained by FKL but not using the same method.

Mass Excess (MeV)
23.0 24.0 25.0 26.0 27.0 28.0

I l I I l o I

140

120_
100.. °°

 

00
O

>— 0

Ch
0
o
o
o

y—

Counts
0
O
O

P—IOC +

 

o
000 °

 

 

00
nnnnnnnnnl 1

o 10 210 310 410 50
Position on Focal Plane (Channel Number)

 

Figure 1.2: Spectrum obtained for UN from l‘N(3He,(’l~le)”N [Ben74]. The solid peak is a theoretical fit of
the first excited state. Indicated on the spectrum is the proton decay threshold for 1'N.

These apparent inconsistencies could be resolved by an experimental study of the
ground state of 11N. The recent availability of radioactive nuclear beams has made the
task of studying exotic nuclei such as 11N more feasible. Therefore, we designed an
experiment to study llN using a radioactive beam of '2 N. The detailed structure of the

setup and analysis of this experiment will be provided in the following chapters.

Chapter 2

Theoretical Overview

2.1 Calculation of the Ground State

Although an experimental study of IN has been previously performed, only the
1/2' first excited state was observed and the ground state was calculated to be at 1.9 MeV
above proton decay threshold [Ben74]. As yet, the only information available on the
ground state has come from theoretical calculations using various methods. This section

will present an overview of these calculations and their results.

2.1.1 The IMME

The Isobaric Multiplet Mass Equation was first proposed by Wigner [Wig57] and
was successfully applied to 22 cases by Benenson et a1. [Ben79]. This success prompted
the application of the IMME to the ground state of llN. Due to the quadratic nature of the
IMME, at least three known states are needed to calculate the coefficients within the
equation. Therefore, it can only be applied to a minimum of an isobaric quartet. In

principle all nuclear states with isospin T belong to a group of 2T+1 levels with similar

wavefunctions but different charges as measured by the T2 component of T. As an
example with direct bearing here, llN belongs to the quartet containing 11C, 1'B, and
1‘Be.

A thorough derivation is presented by Benenson et al. [Ben79] which yields the

total energy of a state as

(m,

 

HCI +H'|nTT,)=a+BT,+ny (2.1)
where HC, is the charge independent Hamiltonian, H’ is the two-body perturbation
Hamiltonian, T2 is the z-component of the nuclear isospin defined by

N—Z
T, =7, (2.2)

a, B, and y are constant coefficients and n represents the rest of the quantum numbers.
From Equation 2.1 one can calculate the mass of a state from

M(Tz ) = a + sz + ch. (2.3)
Equation 2.3 is known as the IMME and can be used to find a mathematical equality for

the coefficient of T23 which must be zero. Calculation of this coefficient yields
1
d = g[M(—3 / 2) — M(3 / 2) — 3(M(-1 / 2) — M(1/2))]: o. (2.4)

Table 2.1 contains the parameters used by Benenson et al. [Ben79] to calculate the mass
of the ground state of 11N. The last column contains the atomic mass excesses. Inserting
these values into Equation 2.4 one obtains the ground state mass of llN to be higher than
the mass of the ground state of loC [Ajz88] plus a proton by 1.9 MeV using

M(Tz)=A(Tz)x u+Ex(Tz)+A(Tz)

where A is the atomic number, u is the atomic mass unit, Ex is the excitation energy, and

A is the atomic mass excess of the state.

Table 2.1: Parameters used with the IMME for the calculation of the ground state of IN shown in the last
row [Ben79]. The last column contains the mass excess of each state.

 

 

 

 

 

 

Nucleus TZ Ex (MeV) A (MeV)
‘ IBe 3/2 0.0 20.176
”B 1/2 12.91 21.580
”C -1/2 12.50 23.150
IIN 1/2 0.0 24.98

 

 

 

 

 

The form of the IMME shows a strong dependence on the accuracy of the
excitation energy of the states used therein, especially the states in 11B and HC since these
values are multiplied by a factor of three in Equation 2.4. Therefore, a small error or
misidentification can create large discrepancies in the calculation of the unknown state.
Sherr [She95] stated that the energy of the relevant state in 11C could be lower by 100 -
150 keV which would lower the IMME prediction by 300 - 450 keV resulting in a ground

state energy of 1.5 i 0.1 MeV for 11N.

2.1.2 Potential Model Calculations

FKL [For95] used a Woods-Saxon potential including Coulomb and spin-orbit

forces to describe 11Be and llN. The geometric parameters r0, a, and the Spin-orbit

potential Vspimrbit were obtained from satisfactory fits of the 2s% single-particle energies

of 17O and 17F. Known energies of the states in llBe were used to obtain the potential
depth. These parameters were then applied to llN. It should be noted that this method
applies only to single-particle states, however the use of this method was justified by
FKL by observing that the first three states of 11Be are dominated by single-particle
configurations.

The width of the states were defined by FKL as

4

r (2.5)

 

d(sin 28)]
dB

where 8 is the total nuclear phase shift. This is equivalent to defining the width as the
energy interval over which 5 changes from RM to 31r/4. Table 2.2 shows the results of
this calculation where the observed width of the state was calculated from the single-
particle width incorporating the Spectroscopic factors from 11Be in the relation I‘pmd = S '
F

sp.

Table 2.2: llN states calculated by FKL using a Woods-Saxon potential [For95].

 

 

 

J“ 15:decay (MeV) rs, (MeV) rpm, (MeV)
—
1/2 1.60 5; 0.22 21:32 1583;;
1/2‘ 2.48 1.45 091$ 0,22

 

 

 

 

 

 

Barker [Bar96] argued that the above definition for the width is best suited for
narrow levels and that it could result in large uncertainties for broader states. He also

points out that FKL do not Show how they obtain the resonance energy.

10

Barker introduced two definitions for the energy; the resonance energy Er at
which the resonant nuclear phase shift ,8 goes through 1t/2, and the peak energy Em where
the density of states function p passes through a maximum. The single-particle widths
were then defined as the energy interval over which ,6 goes from n/4 to 31t/4 and the

FWHM of p. These definitions were used in a potential model calculation similar to FKL
using states of 11Be to adjust the Woods-Saxon potential parameters. A decay energy of
1.40 MeV and a width of 1.01 i 0.07 MeV were calculated for the ground state of 11N.
These values do not match those obtained by FKL. The disagreement was attributed to
the uncertainty in the definitions used by FKL [Bar96].

Barker [Bar96] argued that this method is only applicable to single-particle states
and that the states in 11Be are not totally single-particle states. By incorporating d-wave
contributions into the wavefimction of 11Be and calculating the Coulomb displacement
energies for 11N, a decay energy of 1.60 MeV and a width of 1.39 MeV were obtained for

the ground state of 11N.

2.2 Proton Decay

If the resonance energy of a proton unbound state is lower than the Coulomb and
centrifugal barrier, the decay via proton emission can only occur through barrier

penetration. The wave function of the decaying state can be written as

X10.)
1'

w(r.e.¢)= Y...(e.¢). (2.6)

11

The decay rate 7L = 1/1: is defined as the product of the probability of finding the proton at
the surface multiplied by the flux of the proton. In the limiting case of the proton leaving

the surface, the decay rate can be written as

 

hk
2. = {—j lim Llw(r,0,¢)l2 rde. (2.7)
p r—no
where ,u and k are given by
MpMc 21113,,
= —— k = .
” Mp + Mc ’ h

The subscripts p and c have been used to denote the proton and the core nucleus,
respectively. Since the Spherical harmonics are normalized, the integral over the solid

angle is equal to one. This normalization can be inserted into Equation 2.7 to obtain

hk 2
A = who») . (2.8)

The tunneling effect through the Coulomb and centrifugal barriers for a free particle is

given by the penetrability as [R0188]

Moo) 2

P E,R =
L‘ “) xL(R.)

(2.9)

 

where Rn is the distance between the center of the core nucleus and the particle being
emitted and is given by

Rn = Rp + R.
The penetrability can be inserted into Equation 2.8 to give

2

hk
A = Il-PL(E,RD)

 

 

MR.) (2.10)

12

The nuclear oscillator model [Bla91] can be used to obtain

Li

R

n

 

Ix.(R.) 9?, (2.11)

where 0: is the spectroscopic factor. Inserting Equation 2.11 into Equation 2.10 gives

the partial width, defined as FL 2 h}. , of a state as

2h2k
F E,R =

 

PL(E,Rn)ei . (2.12)

Therefore, calculations of the partial width require a knowledge of the
penetrability factor PL(E,R,,). The solution of the SchrOdinger equation, containing the
Coulomb and centrifugal potential terms, for a particle outside the nucleus is a linear
combination of the regular and irregular Coulomb functions fL(r) and gL(r) [R0188] which
can be written as

xr(r) = afar) + bgr(r) (2.13)
where a and b are constant coefficients. Since we are considering a decay, the radial
solution XL (r) must be an outgoing wave which implies that a = i ' b [R0188]. Using this
relation in Equation 2.13 and inserting the result into Equation 2.9 yields

1
fD(E’Rn)+ gL(E9Rn) .

 

PL(E, Rn )= (2.14)

The form of the Coulomb functions is quite complex for the decay of a proton,
especially when considering angular momenta greater than zero. Therefore a pre-existing

code [Kry94] was used to calculate these functions at given energies and radii.

Chapter 3

Experimental Setup

3.1 Production of the Radioactive Beam

3.1.1 Introduction

The experiment was performed at the National Superconducting Cyclotron
Laboratory at Michigan State University. A primary beam of 16O was used in a
fragmentation reaction to produce the radioactive beam of 12N. Purification was achieved
by using the A1200 [She92] mass analysis device in conjunction with the Reaction
Products Mass Separator (RPMS) [Cur86, Har81]. The purified 12N beam was incident
on a thin secondary target to produce the 11N via one neutron stripping.

Due to a very short lifetime (~10'2' seconds), llN decays within the target into a
proton and 10C. The decay products were observed in coincidence and the momentum of
each particle was measured which allowed for a complete kinematic reconstruction of the

originating state within llN.

13

14

3.1.2 Production and Purification of 12N

An 80 MeV/nucleon primary beam of 1606+ was extracted from the K1200
cyclotron at an average current of 750 enA and bombarded a 980 mg/cm2 thick 9Be target
in front of the A1200 mass separator. A schematic diagram of the A1200 indicating the
position of elements used in the production and purification of the beam is shown in
Figure 3.1. The 12 N secondary beam was produced along with many other fragments.
The A1200 uses a series of dipoles, quadrupoles, and degraders to filter the fragments
according to their rigidity. The rigidity is defined as the particle momentum divided by
total charge. Therefore, the A1200 was tuned to improve the purity of the 12N and to
focus the beam at the focal plane. The magnetic field settings required for the tuning of

the A1200 were calculated using the code INTENSITY [Win92].

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

   

5» ~\\ //‘3(// -
~—“.‘;1M1':ti:--. 1&0
4 if “ ihti:::, if
160 beam from the 127 mg/cmZ IZN secondary
K1200 cyclotron Al wedge beam

Figure 3.1: Diagram of the A1200 mass separator in the configuration used to produce and purify the l2N
secondary beam.

15

The 12N fragments were produced with a 10% momentum spread [Win92]. To
reduce the momentum spread, the 3% momentum slits in the dispersive image #2 of the
A1200 were inserted. The fragmentation method used to produce the 12N also produced
other proton rich fragments with rigidities close to that of 12N. Therefore, these
fragments were also focused at the focal plane of the A1200 as contamination.

A 300 um Silicon AE detector in the focal plane along with a thin timing
scintillator were used to identify the incoming particles. Figure 3.2 shows a plot of the
energy loss versus time-of-flight of the fragments at the focal plane. The radio frequency
(RF) structure of the cyclotron was taken advantage of to check for wrap-around in the
time spectrum of Figure 3.2. By observing the down-scaled RF pulses in coincidence
with fragments at the focal plane, an identical image translated in time was obtained.
This effect can be seen by comparing the events in Figure 3.2 on each side of a vertical
line near the center of the spectrum.

The amount of the 12N present at the focal plane was only 0.3%, therefore a 127
mg/cm2 aluminum wedge was placed in the second dispersive image chamber of the
A1200. This wedge acted as a degrader where different fragments lose different amounts
of energy, leading to a larger spread in their rigidity. Due to the changes in the rigidity of
all fragments, the last part of the A1200 was retuned for the new rigidity of 12N. This
resulted in a secondary beam of 40.8 MeV/nucleon 12N fragments at the focal plane of the
A1200, increasing the amount of the 12N to 1.3%. The difference in rigidity resulted in a
spatial separation in the horizontal direction at the focal plane allowing further

purification by the insertion of a pair of slits, driven into the beam in the horizontal

16

direction, at the focal plane. To determine the most effective slit Opening, a position
sensitive Parallel Plate Avalanche Counter (PPAC) was used in the focal plane. A plot of
the vertical versus the horizontal position of the fragments in the PPAC is shown in
Figure 3.3. The dashed vertical lines represent the position of the slits.

Figure 3.4 shows a time-of-flight spectrum for the fragments before and after the
insertion of the slits. Although the slits did not affect the 12N and 11C rates, the amount of

10B and 160 were reduced, thus increasing the amount of 12N to 1.7%.

 

   

 

 

 

180 T I I I I 1 I I I 1 1 1’ T T 1 I r T T I 1 I T
'50 : at a» 1.
i ‘ 1.. ~ 1
A 120 ~ [23.3 a J
:3 .. . -. F : . . _,
£5; ‘ ‘fi-fiiz _ T: u
E 90 “I" ' (N 9" ‘
S “it ‘ “’1
E T “I 1‘ :
T '9 hr “
30 V H, '19! g “‘*
a 5:6 ”I”, -5
: 0.6 ‘1“ g 3* T

0 1 ‘41—'1 l 1 1 ‘Tr‘L1 1 1 l '1 1 1 l 1 1 J_l 1 1 1

0 40 80 120 160 200 240

Time of Flight (a. 11.)

Figure 3.2: A plot of the energy loss versus time-of-fiight of the fragments seen at the focal plane of the
A1200. Indicated within the figure are fragments with equal number of protons and neutrons, and the
secondary radioactive beam of 12N.

 

 

 

 

 

180 I I I I I g I I I I I I I SI I T I I I I I I I

150 L L
120 — —
'3 i 3
6i _ _
g 90 _ _
:5 r _
a _ _.
a _ -
>‘ 60 _ _
T ‘1
30 1.— _
0 I. 1 1 1 l 1 1 1 l 1 1 1 l 1 :1 1 l 1 1 u l 1 1 1 -

0 40 80 120 I60 200 240

x-position (a. u.)

Figure 3.3: Vertical versus horizontal positions of the fragments in the PPAC at the focal plane of the
A1200. The dashed lines represent the position of the horizontal slits.

 

 

 

 

 

 

 

 

 

.I'ITITH‘I ifl’dfil'TTlI’t
C
103 E— "’B #3
_ //‘\X f
102 r l/ ”N z
‘3 l .
o _ 1 J
U

101 r i
1 .1111 W11 1

:3 l“ “.15 "' Z: .i 1
100 l: 11 “hi p11,? 9'; mm a,
l Ltgrl r14111111111111A.

50 100 150 200 250

Tune of Flight (a. 11.)

Figure 3.4: Time-of-flight of fragments seen at the focal plane of the A1200 before and after the insertion
of the slits.

18

To achieve higher purity, the 12N fragments along with the contaminants were
transported to the Reaction Product Mass Separator (RPMS). The RPMS consists of a
quadrupole doublet followed by a Wien filter, a dipole, and a second quadrupole doublet.
This configuration is shown in Figure 3.5. A Wien filter consists of perpendicular
electric and magnetic fields to select particles according to their velocities. The selected
velocity is proportional to the ratio of the electric field to the magnetic field. Particles
with significantly different velocities from the tuned setting are bent away and are
stopped in the walls of the Wien filter. The contaminants entering the RPMS had the
same rigidity as the 12N fiagments, and thus different velocities. Therefore, the RPMS
created a spatial separation between the 12N and the contaminants. However, not all of
the contaminants were stopped in the walls of the RPMS. These contaminants were
transported out of the RPMS parallel to the 12N beam, but spatially separated in the
vertical direction.

Behind the Wien filter a magnetic dipole followed by a quadrupole acted as a
rigidity filter and focused the fragments further down the beam line. These magnets also
served another important role. The beam entering the Wien filter had a non—zero radius,
therefore corrections had to be made to the beam to allow for a good focus on the target
placed further down the line from the second quadrupole doublet. This was achieved by
optirrrizing the magnet settings such that by pivoting the beam line to an angle of 4.5
degrees, at the pivot point indicated in Figure 3.5, the beam would be focused at the
secondary target position. The parameters required for the tuning of the RPMS were

calculated using the code GIOS [Wol88]. Input parameters needed for this calculation

19

were the electric field of the Wien filter, the fragment of interest (UN), and the fragment

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

energy.
Quadrupole Dipole Quadrupole
Doublet Wien Filter Doublet Detector
Bill-T M _____________________ :< (o K m 3535:?!
“ 1’1 “K ILI’T
Pivot Point

 

 

 

Figure 3.5: The Reaction Products Mass Separator.

In addition to the ”N, spatially separated 11C and 130 were also transported
through the Wien filter and had to be identified and eliminated before the target. Figure
3.6 shows a diagram of the tail of the RPMS and the components used therein. The
timing scintillator and the AE Silicon detector in the first chamber were used to identify
the particles coming out of the RPMS by plotting the time-of-flight versus the energy
loss. Although the RPMS successfully removed the 11C fragments from the beam, a
small amount of 13O was still present. Identification of the impurities was performed
using the fi'agment telescope, described in Section 3.2.3. Figure 3.7 shows a plot of the
energy loss versus the total energy of the particles seen in the fragment telescope. The
small amount of contamination by 13O can be seen in Figure 3.7.

The spatial separation between the 13O and '2 N fragments was observed by PPAC
2 and is Shown in Figure 3.8. By observing the fragments in PPAC 2 the vertical slits
were used to block the 130 resulting in a 95% pure 12N beam at an average rate of 15,000

particles per second.

20

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

Timing ‘ ‘ PPAC 1 PPAC 2
Scintillator\ Vertical \
in] Slits
[— ' \ I I I
Degrader AE Silicon U
Ladder 1 —_' J./ Detector
F 1 ' Degrader/ .. 1
Chamber #1 Ladder Chamber #2

Figure 3.6: Beam line components behind the RPMS.

 

 

 

 

 

 

 

 

I80 1 T I I I I I l I I I l I I uj T I 1 I I 1
r. 1
150 l j
._ I30 .1
A 120 ~ —
5' .— .4
93 9o 1 _\. j
#3 __ - 1,; “3",? {Q ~
6‘1 60 l ' L
30 — —
i 1
O 1.1;"1 I 1 1 1 l 1 1 1 l_1 1 1 l 1 1 1 l 1 1 1
0 40 80 120 I60 200 240
Energy (a. 11.)

Figure 3.7: Energy loss versus tirne-of-flight spectrum of fragments seen in the fragment telescope prior to
the insertion of the vertical slits.

21

 

 

 

 

180 T ij F l l l I I I I I 1 T l I I I l I I f
150 L j
: ‘30 :
120 — 1 —
’3 : o :
5; >- - -
1:1 90 _ _
o
:9 ~ —
8 L 5
9. _ -
>\
60
:- IN T
1.. J
1‘
3o 1 l
L- —1
1- —<
1 4
0 1 1 1 [#1 1 l 1 L4 L 1 1 1 I 1 1 4 I 1 1 1
0 40 80 120 160 200 240

x-position (a. 11.)

Figure 3.8: Vertical versus horizontal spectrum of PPAC 2. This spectrum shows the spatial separation
between 12N and 130 fragments and was used to optimize the position of the vertical slits for the 12N beam.

3.2 Detector Assembly

3.2.1 The Tail of the RPMS

Figure 3.6 shows the setup at the tail of the RPMS. Beam trajectory information
was obtained from the two PPACS in the chambers. The copper block in the second
chamber was used to protect the proton and fragment detectors during the tuning of the
12N beam and the calibration beams. Aluminum degraders in the degrader ladders were
used in conjunction with the calibration beams to provide a broad range of energies and

more specific information will be presented later. The target ladder contained a

22

scintillator for tuning the beam, an empty target holder for target out measurements, a 37
mg/cm2 thick 9Be foil as the secondary target, and the last position was occupied by a 94
mg/cm2 gold foil for proton calibrations.

11N was produced by the one neutron stripping reaction 9Be('2N,l 1N) in the
secondary target. Due to the very Short lifetime (~10 '21 ns) llN decayed immediately by
emitting a proton already inside the target. The proton and 10C daughter nuclei were
observed in detectors placed in a large chamber behind chamber #2, (Figure 3.9). The
protons were detected using an annular silicon detector (ASI) backed by the Maryland
Forward Array (MFA) [Llo92] which consists of a ring of plastic phoswich detectors.
This array was placed such that the ASI was at a distance of 20 cm from the target. Due
to their larger mass, the 10C fragments were more forward focused than the protons and
passed through the central hole of the A81 and MFA. The heavy fragments were
collected in a fragment telescope consisting of a position sensitive PPAC, a thin silicon
AE detector, and a thick silicon energy detector. The fragment telescope PPAC

(FTPPAC) was placed 62 cm from the target.

23

     
    

300 “in Silicon . .
Detector (ASI) PPAC 3 [gm Silicon
Protons \ \ etector

 

. .«amrb?

 

   
 
  

2
37 mg/cm 9B6 500 um Silicon

 

Target
Plastic Phoswich Detector
Detectors (MFA)
Heavy Fragments
0 19.9 cm 62.1 cm

Figure 3.9: Detector setup at the end of the RPMS tail. The ASI and the MFA provided position, AE, and
E information for the protons. The fragment telescope provided position, AE, and E for the heavier
fragments.

3.2.2 Proton Detectors

The protons were detected and identified by the ASI and the MFA. The double
sided annular 300 pm thick silicon detector (ASI) had an inner radius of 2.4 cm and an
outer radius of 9.6 cm. The ASI consists of 16 pie shaped segments on one side and 16
concentric strips on the other, providing 256 pixels. This detector was used to measure
the angular distribution of incoming particles along with their energy loss. The MFA,
mounted directly behind the ASI, contains a ring of 16 plastic phoswich detectors. The
front face of each plastic detector has the same dimensions as the pie segments of the

ASI. Each phoswich detector is constructed of a 1 mm thick fast plastic detector optically

24

coupled to a 10 cm thick slow plastic detector. The light from the two plastic detectors
passes through a light guide and into a photomultiplier tube (PMT). Therefore, the signal
from the PMT contains a fast and a slow component arising from the two different types
of plastic detectors.

Particle identification was achieved in a AE - E plot from the ASI versus the total
signal from the PMT (total energy). Alternatively pulse shape discrimination between the

slow and fast components of the signals fiom the PMT was also used.

3.2.3 Fragment Detectors

The fragment telescope consisted of three detectors. A position sensitive PPAC
(FTPPAC) with an active area of 5x5 cm2 was used to measure the position of the
incoming particles. Mounted behind the PPAC was a 5x5 cm2 500 pm thick quadrant
silicon detector. This detector was used to measure the energy loss of the heavy
fragments. The last component was a quadrant 3 mm thick, 5 cm radius Lithium drifted
silicon detector. This detector was thick enough to stop the 10C daughter nuclei within
the energy range of interest. It also stopped other fragments of sirrrilar mass and velocity,
therefore measuring the energy of these particles. Particle identification was achieved by
plotting the energy loss in the thin silicon detector against the energy measured in the
thick Si(Li) detector.

All silicon detectors exhibit a current in the absence of any incoming particles.
This current is due to thermal movement of electrons and holes in the presence of

impurities and lattice defects. Since this is essentially a difiusion effect and ultimately

25

dependent upon temperature, the fragment telescope was cooled to -20° to lower the
thermal noises in the silicon detectors. By reducing the noise within the detector, the
signal to noise ratio was improved which in turn improved the energy resolution of the
detector.

The fragment telescope had to be shielded during the proton calibrations of the
MFA. The fragment telescope was mounted on two rods which would allow one to move
it closer or further from the proton detectors. A thick copper plate was mounted such that
once the telescope was pulled away from the MFA, one could swing the copper plate in
front of the telescope and stop all incoming protons, thus protecting the telescope.

The fragment identification spectra exhibited some smearing of the fragments.
This effect can be seen in Figure 3.7. The horizontal tail on the 12N fragments is due to
incomplete charge collection within the silicon energy detector. A Spread at the bottom
edge of the 12N fragments is also present in Figure 3.7. This spread is due to two effects
within the silicon AE detector. The first contribution is from the incomplete charge
collection effect similar to that of the energy detector. The second contribution arises
from lattice defects within the AE detector which makes it possible for a fragment to pass
through these Sites without interacting completely with the detector, resulting in a spread
in energy loss. This effect did not create a problem for the energy detector because the

fragments were stopped in that detector, thus losing all their energy.

26

3.3 Calibration Beams

3.3.1 Proton Beams

Molecular beams of 70 MeV/nucleon 3(D-H)+ and 30 MeV/nucleon 5(H-He)+
were used to calibrate the proton detectors. The beams were extracted from the K1200
cyclotron and passed through a 5 mg/cm2 aluminum stripper foil breaking the molecular
beam into its individual components. Separation of the protons from the other
components, i.e. He or D, was achieved through the A1200 mass separator and the
protons were then transported to the tail of the RPMS.

The above procedure provided proton beams of 70 MeV and 30 MeV at the
secondary target position. Since the proton energy range of interest extended from 10
MeV to 70 MeV, degrader ladders were placed in the chambers at the tail of the RPMS to
degrade the proton beams to within the desired range. The ladder in the first chamber
contained 600, 1200, and 2400 mg/cm2 thick aluminum degraders and the second
chamber contained 200, 400, and 600 mg/cm2 aluminum degraders. By using these
degraders in combination with each other and the two proton beams, calibration points
were obtained covering proton energies from 12 MeV to 70 MeV. The gold foil in the
target ladder was used to elastically scatter the protons to irradiate all pixels of the ASI

and the MFA.

27

3.3.2 Carbon Beams

Calibration of the fragment telescope was achieved by using beams of IOC at
44.63 MeV/nucleon and 11C at 46.42 MeV/nucleon and 35.89 MeV/nucleon. These
fragments were produced in the fragmentation of the primary 160 beam in the primary
target at the front of the A1200, therefore no additional beams were tuned in the
cyclotron. The 35.89 MeV/nucleon llC beam had the same rigidity as the 12N beam,
therefore only the RPMS had to be retuned. The degraders within the degrader ladder of
the second chamber were used with the above beams to obtain calibration points for

carbon isotopes with total energies from 110 to 510 MeV.

3.4 Electronics

Figure 3.10 shows a schematic diagram of the electronics setup. Two separate
modes were used in the electronics, one for tuning and calibration, and one for the
coincidence runs during the main part of the experiment. The difference between these
two modes was the method the AB], PPACl, and PPAC2 were integrated into the
electronics. The fragment telescope was shielded during the tuning of all beams and the
proton calibration phase. Similarly, the ASI and MFA were shielded during the tuning of
the beams. Therefore an alternate source was needed to generate the master gates and the
“or” of the signals from PPAC], PPAC2, and AEl detectors was used for this purpose.
This signal, represented by the dashed line in Figure 3.10, was taken out during the

coincidence runs of the experiment.

28

 

 

m .5 «OE.—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

mOm mmO

 

 

a

 

 

a

.ZmO EEC

 

 

 

 

 

 

 

 

H<DD <m~m~

 

 

 

 

 

 

 

why—<5 00...

«88.5 8.“
39am 830

 

 

 

 

 

 

 

 

 

32m d fian—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.10: Schematic diagram of the electronics. All times are in nanoseconds.

29

The coincidence mode was used to detect the decay products of the unstable
nuclei, i.e. protons and daughter nuclei. In coincidence experiments it is necessary to
subtract the “random” coincidences from “real” coincidences. In the present
measurement, random coincidences are events where a proton and daughter nucleus are
seen in coincidence but did not originate from the same decay. These random events can
be identified by timing against the radio frequency (RF) of the cyclotron. Uncorrelated
events are evenly spread in time resulting in several equal peaks separated in time by the
RF frequency whereas correlated events result in one peak only. However, this main
peak also contains contributions from the uncorrelated events which can be subtracted by
using the average of the other uncorrelated peaks.

Since the coincidence rate was very small, scaled down proton and fragment
singles were taken simultaneously with the coincidence events to obtain the necessary
statistics for particle identification.

The “or” of the proton signals fiom the MFA created a wide 300 ns signal
indicating the detection of a proton. The “or” of the fragment AE detectors produced a
narrow 50 ns output pulse timed to fall within the proton signal at the input of the
coincidence “and” unit, thus providing a coincidence signal timed on the arrival of the
fragments. The “or” of the coincidence output with the scaled down proton singles
provided coincidence information timed on the fragments, and proton singles timed on
the protons. The “and” of this output with the delayed “or” of the fragments provided a
delayed coincidence Signal timed on the fragments. This signal was then used to start all

the TDCS. Figure 3.11 Shows a logic diagram which depicts the procedure described

30

above. It was necessary to time the coincidence output on the fragments since this
provided the “start” signal for the TDC units which were “stopped” by the arrival of the
protons.

The “or” of the scaled down fragment singles with the “or” of the scaled down
proton singles and the coincidences produced the master gate which was enabled by the
computer “not busy” to generate the mater-gate-live.

The 80 channels of the AE Signals from the ASI were amplified through a
preamplifier and a shaping amplifier and fed into peak sensing ADCS. The 16 energy
signals from the MFA were each separated into three parts. Two of the Signals were
taken to charge integrating ADCS (QDCs). A short gate was applied to one QDC
covering the fast part of the signal and a long gate was applied to the other covering the
whole signal, thus creating AE and E signals which were used for particle identification.
The fast and slow gates for the protons were obtained from the “or” of the coincidence
and the scaled down proton singles. The third signal from the MFA was used to stop the
MFA TDCS to provide timing for the protons.

The coincidence gate and the strobes were generated from the master-gate-live
signal. The RF time was obtained from the “and” of a narrow fragments signal with a
wide master-gate-live signal to produce a wide pulse timed to contain the RF pulses. The
TDC stop for the RF was obtained fiom the “and” of the RF and the wide pulse.

The inset at the top right comer of Figure 3.10 shows the remaining signals such
as the FTPPAC and fragment telescope energy detector which were amplified and fed

into peak sensing ADCS. T1 and T2 were the beam-line timing detectors at the end of the

31

A1200 and in the first chamber at the tail of the RPMS, respectively. These detectors

were used only during the tuning of the beams.

“or” of all Protons

“or” of all Fragments

“and” of Protons with
Fragments

Scaled Down Protons

“or” of Protons with the
66a“ d”

200 ns Delayed Fragments

Delayed Coincidences Timed on

Fragments

_|

300 ns Time

 

A
V

_|"

L—-

"1

 

:1

b---—

 

 

Figure 3.11: Logic signal diagram for the creation of the TDC common start signal.

3.5 Data Acquisition

The CAMAC electronics were used inside the experimental vault for the data

acquisition. A front end code was written

specifically for this experiment to control the

CAMAC units. This code instructed the front end to read specific modules based on an

array of bits received from the two bit registers used in the electronics. One bit register

consisted of 16 bits, one for each detector in the MFA. Upon receiving these bits, the

front end proceeded to read the corresponding channel of the MFA and all 80 channels of

32

the ASI. The second bit register contained information for several different detectors.
Bit 1 was generated by PPAC2 and controlled the readout of the PPACl and PPAC2
signals. Bit 2 was generated by AB] to read AEl. Bits 3 through 7 were generated by the
four quadrants of the fragment telescope AE detector to read the corresponding channel in
the AE detector, the FTPPAC, and all four channels of the fragment telescope E detector.
Bit 8 was generated by the “and” of the protons with the fragments representing the
coincidence bit which triggered the reading of all the proton, and RF TDCS. Bits 9 and
10 were the proton and fragment singles bits respectively. These were present to measure
the ratio of singles to coincidences. Bit 11 was produced by the timing scintillator in the
first chamber at the tail of the RPMS and it was used to read the beam-line timing
detector and the timing scintillator TDCS.

The incoming data stream was recorded on 8 mm tapes for off-line analysis. The
program SARA [She94] was used to create histograms of the parameters within the data

stream which could then be displayed for online monitoring and data analysis.

Chapter 4

Data Analysis

4.1 Analysis Software

The data was analyzed using the programs SARA [She94] and DAMM [M1192].
SARA was used to read the 8 mm tapes containing the data and to create histograms for
viewing spectra of interest. To speed up the data analysis, the coincidence data were
filtered onto disk using SARA. The filtered data was created by imposing certain criteria
based on software gates defined on spectra in SARA. These software gates were imposed
on the protons, fragments, PPAC2, and FTPPAC spectra. A valid event was defined as
an event which satisfied all the aforementioned gates. These events were then filtered
and put on disk for faster retrieval and analysis.

SARA uses a set of predefined parameters based on the electronics used in the
experiment. These parameters can be used to create one and two dimensional histograms.
SARA also allows for the definition of “pseudo” parameters based on the predefined
parameters which can be histogrammed. For example, histograms of the calibrated

energies and the decay energies were created by this method.

33

34

The histograms created by SARA were converted into a format readable by
DAMM. Direct comparisons between the data and simulations were made by converting

the output of the simulations to DAMM format also.
4.2 Energy Calibrations

4.2.1 Proton Detectors

The protons from the molecular beams in conjunction with several aluminum
degraders of known thickness were used to calibrate the MFA. A 94 mg/cm2 Au target
was used to elastically scatter the protons onto the proton detectors. The energy loss of
the protons in the degraders were calculated using the code TRIM91 [Bie80].

Table 4.1 shows the degrader combinations used in the calibration and the
degraded proton energies obtained. The first three rows represent calibration energies
using the 35 MeV proton beam, while the remainder were obtained from the 70 MeV
proton beam. It should be pointed out that the energies listed in Table 4.1 represent
proton energies out of the target and do not include energy loss in the ASI nor the
material used to create a light-tight environment for the MFA. Therefore, the proton
energies measured fi'om the actual decays represent the energy out of the target since the
energy loss within materials after the target are folded into the calibrations.

The peaks obtained during the proton calibrations of the MFA exhibited lower
resolutions than were expected. This was caused by the difference in light collection

within the photo-multiplier tubes, effectively introducing a position dependence on the

35

detection of the protons. Figure 4.1 shows the detected pulse height in channels versus
the position of the event where the radial number 1 corresponds to the outermost ring and
the radial number 15 corresponds to the innermost ring. In the absence of any position
dependence the spectrum in Figure 4.1 would be a horizontal line. This position
dependence worsens the energy resolution of the MFA, therefore a position dependent
energy calibration was performed. This was achieved by calibrating the MFA separately
for each radial strip of the ASI. Figure 4.2 shows the calibration curve obtained for one
of these pixels. Due to the geometry of the MFA, high energy protons near the outside
edge of the plastic detector punched through the detectors leaving only a fraction of their
total energy. These events were identified by gating on the outer three rings of the ASI
and were excluded from the analysis. The proton energy calibration spectra were fitted
using a second order polynomial and the fit parameters were used to convert channel
numbers into proton energies. The average energy resolution of the proton detectors was

about 5%.

36

Table 4.1: Degrader combinations and proton energies obtained during the calibration of the MFA. The
35 MeV proton beam was used to obtain the values in the first three rows. The remainder were obtained
using the 70 MeV proton beam.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Degraders (mg/cm‘) Proton Energy (MeV)
1213 Al+94 Au 11.96
600 A1 + 94 Au 25.49
94 Au 34.25
2400 Al + 600 A1 41.96
2400 Al + 389 Al 44.28
2400 A1 + 94 Au 47.79
1213 Al+600 Al 53.97
1213 A1+389 A1 55.90
1213Al+94 Au 58.83
1213 A1 59.34
600 Al + 94 Au 63.96
600 Al 64.43
389 Al 66.10
94 Au 68.63
_ I 1 I I I I r I I I I I I I I
970 ;_ .1
2 3
960 :— J
’3 C 1
E 950 L 2
'5 940 E_ . j
930 :_ :
1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 _
5 10 15
Radial Strip Number

Figure 4.1: Proton energy versus the radial strip of the ASI for 63.96 MeV protons.

37

 

Energy (MeV)

 

 

 

10 l l l
200 400 600 800 1000

Channel

 

Figure 4.2: Energy calibration of the MFA gated on one ASI pixel.

4.2.2 Fragment Detectors

The thick silicon detector in the fragment telescope, used to measure the total
energy of heavy particles, was calibrated with beams of 10C and 11C. By modifying the
A1200 rigidity two beams of 11C at 35.89 MeV/nucleon and 46.42 MeV/nucleon, and one
beam of ‘0C at 44.63 MeV/nucleon were separately obtained. The same aluminum
degraders as in the proton calibrations were used with these beams to provide additional
calibration points. Table 4.2 shows the degraders and the corresponding total energies
calculated from TRIM91 [Bie80]. Similar to the protons, these energies reflect the
energy of the particles out of the target and do not include the energy loss in the FTPPAC

and the AE detector.

38

Table 4.2: Energy calibration points obtained for the fragment telescope.

 

 

 

 

 

 

 

 

Beam Aluminum Degraders Total Energy (MeV)
395 MeV ”C - 395
51 1 MeV ”C - 511
511 MeV 11C 389 mg/ch 344
511 MeV HC 600 mg/cmz 219
446 MeV luC - 446
446 MeV ”C 389 mg/cmz 268
446 MeV I"C 600 mg/cm‘ 109

 

 

 

 

 

The four calibration spectra, one for each quadrant of the thick silicon detector,
were fitted and the fit parameters were used to convert the channel numbers into particle
energies. Figure 4.3 shows the fit obtained from the calibration of one of the quadrants of
the fragment telescope. The energy resolution of the fragment detectors was determined

to be about 3%.

 

 

 

 

500:__l"""T'IHHI’H'IH'TIILJ

1‘40:— é

g» : :

‘5 300.— _

2005l...il1.1.1.11111111111111111:
400 600 800 10001200 1400

 

Chamel

Figure 4.3: Calibration spectrum for one quadrant of the fragment silicon detector.

39

4.3 Position Calibrations

4.3.1 Fragment Telescope PPAC

The F TPPAC was calibrated using a thin aluminum mask with holes in an
asymmetric pattern. The asymmetric pattern is required to ensure the correct orientation
of the mask relative to the axis and direction of the beam. The calibration was performed
with an alpha source at a distance of about 20 cm in front of the mask. Figure 4.4 shows
the F TPPAC spectrum obtained during the calibration. The holes in the mask had a
diameter of 2.34 mm which was used to obtain the position resolution, and the distance
between each adjacent hole was 6.35 mm which provided the position calibration. A
Monte Carlo simulation was written to fit the individual peaks in the F TPPAC using the
position resolution as a variable of the fit. This yielded a value of 1.0 mm full width at

half maximum (F WHM) for the position resolution of the FTPPAC.

 

11L111

 

160: {T

g 1 1 gr 1 I"?! 1 d

80 140 200 260 320 380 440
x-axis (a.u.)

_4

 

 

90-1L111111111

 

Figure 4.4: Calibration spectrum for the fragment telescope PPAC.

40

The data acquisition recorded events on the condition that the thin silicon AE
detector of the fragment telescope fired. This detector was subdivided into four quadrants
with a thin dead layer between each active surface. Therefore, fragments hitting these
dead layers were not recorded and thus resulted in gaps in the FTPPAC Spectrum. Figure
4.5 shows the spectrum of the FTPPAC projected on the x-axis. This gap was used to
define the center of the x-axis. The same method was applied to the projection of the

FTPPAC spectrum on the y-axis to obtain the center in the vertical direction.

 

300 ,fi.
250_ j
200— L
a C 2
5150— _
8 :
100- j

—1

1
P 1 r 1 1 r 1 1 1 1 1 l 1 1 1 l 1 1 1 l j
(80 140 200 260 320 380 440
x-axis (a.u.)

50L

 

 

 

Figure 4.5: Fragment telescope PPAC spectrum gated on fragment telescope silicon and projected onto the
x-axis.

4.3.2 Beamline PPACS

PPAC2 was calibrated and centered using the same technique applied to the

FTPPAC. The position resolution measured for this PPAC was 1.0 mm FWHM.

41

Position calibration for PPACl was carried out using the same mask as PPAC2
and a position resolution of 1.3 mm F WHM was observed. However, due to the large
distance between PPACl and the fragment telescope, the center of PPACl could not be
found by the same method used for PPAC2 and the FTPPAC. A line was projected
through the center of PPAC2 and F TPPAC which also crossed the center of PPACl at an
unknown channel number on PPACl. A similar line was reconstructed for each beam
particle using the FTPPAC and PPAC2 in the absence of a target. By plotting the relative
position of the beam particles in PPACl to the projected center of PPACl, a spectrum

was obtained with a peak denoting the center of the detector.

4.4 Particle Identification

The protons in the ASI and MFA had to be identified and separated from other
light charged particles entering the detectors. The combination of fast and slow signals
from the MFA allowed fragment identification by pulse shape discrimination. The fast
plastic produced a pulse with a fast rise time corresponding to an energy loss, whereas the
slow plastic produced a pulse with a slow rise time corresponding to the total energy of
the particle. By plotting the total area under the signal versus the area of the fast
component, the AE - E plot shown on the left in Figure 4.6 was obtained. Alternatively,
the AE signal form the ASI could be used instead of the fast plastic signal. Figure 4.6
shows the AE - E spectrum (right) obtained using the ASI signal. Both spectra in Figure

4.6 exhibit excellent separation between the protons, deuterons and tritons (p, d, t).

42

The fragments were identified using the AE - E spectra obtained from the thin
silicon versus the thick silicon detector in the fragment telescope. Figure 4.7 shows the
raw AE - E spectrum (left) and the same spectrum with a proton coincidence condition
imposed on it (right). Due to the large number of 12N and 130 from the beam, other
fragments can not be seen in the raw AE - E spectrum. By gating on protons the number
of Observed 12N and 130 fragments in the AE - E spectrum was reduced dramatically
allowing the rest of the fragments to show more clearly. The 12N in the gated spectrum
was identified from the position of the 12N beam in the raw AE - E spectrum. Carbon

isotopes were identified from the AE - E spectra of the 10C and HC calibration beams.

 

 

   

 

 

 

 

 

 

 

300 T T T J T T T J T T I I T T I I I T ”I q T-T- _. 400 WE]. T T T J T T T I T T T I' T g y '5'] 1 1 r

t ,0" r 1'. 5" ‘. '_-" _ 'qrié. . . . 7| 55:: .1
250 i ‘ ' S 350 _ ~"" " _‘
I : h "’3. I.
200" 1 2? 213:; ' ~ L
’3 i ‘ 3 -.'..;';,-'.-. -
' m ,. ..= -
3 m .' 1
g 150 . _‘ g j
a b d d
U) 1- ‘ .
100 _ _
. 15 _, t .
50 h _ '.:.'*:d. . I _-:
1 5 V p . :J
O 5 ALT l 1 1 1 l 1 1 1 l 1 1 1 l 1 4 Oh I lailflafi‘flf-f [finals-$4” :-

0 50 100 150 200 250 300 0 50 100 150 200 250 300

Long (a.u.) Energy (a.u.)

Figure 4.6: Total energy versus energy loss (left) and long versus short (right) spectra used to identify the
protons.

43

 

 

 

 

 

 

 

 

 

600T—fTITTIJIj—FIIIIIITIIIII-J 6oobllillllltlIIITIIIIIIIIY
C 13 1 L

500 _ 0 I 500_ _
" $1 :
34001 .7 '2': $400:
,3 300'— ' - 3 3001
E3 Z i E :
‘5 200a _. 15 200_
100; j 100;
1- ’1, _ I:

Or+lei"l 1'1 #111 1F‘1"11"l1’1‘111 Ll‘ 05i ”WIN 1 111 L 11111111111 Lid

0 100 200 300 400 500 600 0 100 200 300 400 500 600
Energy (a.u.) Energy (a.u.)

Figure 4.7: Energy loss versus energy spectra for the fragment telescope. Raw spectrum (lefi) and gated
on all protons (right).

4.5 Contamination

One possible source of contamination in the data were events generated not within
the target, but from the target frame or the beam pipes. An empty target frame was used
to reproduce such events. The empty target frame would only reproduce reactions that
were not target related and would result in a spectrum that could be subtracted from the
data. However, the contribution of these events to the data was less than 1% and a

subtraction was not necessary.

44

4.6 Decay Energy

In a proton decay of the form X —) D + p , the laboratory rest mass of the

decaying particle X can be calculated using

 

M; = M: + Mf, + 2(Tp + Mp)(TD + MD) — 2J(Tp2 + 2TpMp)(T,§ + 2TDMD)cose (4.1)
where M is the rest mass (including any excitation energies), T is the kinetic energy of the

particles in the laboratory frame, and 6 is the laboratory opening angle between the
proton and the daughter nucleus D. Equation 4.1 is a relativistic reconstruction of the
mass of the parent nucleus using the experimental observables kinetic energy and opening
angle. The decay energy can be calculated using Equation 4.11 in

E =Mx—Mp—MD

decay
If the mass of the daughter nucleus is known, one can reconstruct the mass of the state
from which the decay occurred. Although this allows the calculation of the decay energy
of a state, it does not provide information about neither the decaying state nor the state
within the daughter nucleus.

The kinetic energy of the protons and fragments were obtained from the energy
calibrated spectra of the proton and fragment detectors respectively. The laboratory
opening angle between the proton and the fragment were reconstructed from three
quantities (i) the position at which the decay occurred, (ii) the position of the protons in
the ASI, and (iii) the position of the fragments in the FTPPAC. Figure 4.8 shows the

geometrical reconstruction of the opening angle. Due to the close proximity of PPAC2 to

the target (~2 cm) the origin (x0,y0,0) was measured directly from PPAC2. The proton

45

position (xp,yp,l99) was obtained from the pixels of the ASI, and the fragment position
(xD,yD,621) was measured by the FTPPAC where all distances are in millimeters. From
these positions, the opening angle 0 can be calculated by using the definition of a vector

dot product to obtain

(Xp - X0)(Xn -Xo)+(yp - poyn - yo)+(199 >< 621)

 

     

 

c050: 2 2 2 2 2 2 . (4.2)
J1<x.—x.> +<y.—y.> +199 Hon—x.) +(yD—y.> +6211
(XO’yOaO) Pl'OtOIl (xp’yp’ 199)
j
Beam 9
- --------------------------- F ragment -----------------------
Target (xD,yD,621)

 

 

Figure 4.8: Vector diagram of a proton decay. All distances are in millimeters.

4.7 Monte Carlo Simulations

A Monte Carlo simulation was written to include the experimental observables
describing the beam, the energy loss within the target, and the detector geometry. In
addition, theoretical treatments of the transfer reaction and the decay were incorporated in
this code. The body of this code and a brief description of each unit therein is given in

Appendix A.

46

4.7.1 Secondary Beam

The first step within the simulation was a description of the 12N beam. The
kinetic energy of the 12N beam was calculated from the final A1200 rigidity to be 40.77
MeV/nucleon. The spread in energy was calculated using the 3% setting of the
momentum slits in the A1200 which corresponds to a 6% spread in energy. Although a
secondary beam has a Gaussian distribution, the FWHM of the distribution was about
10% for 12N [Win92] compared to the 3% cut imposed by the momentum slits.
Therefore, a flat distribution is a good approximation for the energy distribution.

The position spectra from PPACl and PPAC2 were converted into Gaussian
distributions by measuring the position and width of the beam in each PPAC. These

parameters were then used in the simulation to calculate the trajectory of the beam.

4.7.2 Interactions in the Target

Due to the finite thickness of the target, the exact location of the interaction could
not be measured experimentally, however the probability of an interaction in the target
along the beam trajectory is constant and was included in the simulations as such.

The energy loss of the 12N beam, the proton, and all daughter nuclei of interest
within the 9Be target, as a function of distance traveled in the target, were calculated with
the code STOPI [Mil88]. The energy loss for the parent nuclei was not included because

of their negligible distance of travel (~ 10'19 mm).

47

The calculated distances included all the angles as the particles traversed the
target. The angle of the incoming beam was calculated from PPACl and PPAC2 while
the angle of the proton and daughter nuclei were calculated from the scattering angle and

the decay, as described in the next sections.

4.7.3 The Reaction

Figure 4.9 depicts the kinematics of the 12N interaction in the target to produce the
fragments of interest. The scattering angle 0’ was simulated to describe the distribution
shown in Figure 4.10 where 0c was chosen to be 0.05 radians and the slope of the tail of
the probability distribution was set to 37.56 deg". These values were obtained from
experimental studies of transfer reactions for nuclei with similar mass and charge

[Oer70].

P3 ‘1
m ' m 9
ml MK . ml p Lmz
mxfit ; m, '1’
P4 -q
Laboratory Frame CoM Frame

Figure 4.9: Scattering reaction between projectile and target in laboratory and center of mass frames.

48

Log(do/dfl)

 

 

Figure 4.10: Distribution used to simulate the scattering angle in the laboratory frame.

The magnitude of the momentum q in Figure 4.9 is given by

2 _ (W2 -m§ -m§)2 -4m§mi
q _ 4w2

 

(4.3)

where

w2 = m? + m; + Zsz/pim + mi (4.4)

is the square of the total energy in the center of mass frame. Throughout these
calculations c is set to 1. By using the Lorentz transformation parameters for the center

of mass vcm and you, defined as [Jac75]

V = "15MB 'Y =m2+VPiAB+m12 (45)
cm m2+I/pim,+ml2 cm W

one can calculate the laboratory scattering angle 03 from

 

 

 

v sinO'
t 9 = 3 . 4.6
a“ 3 7....(v30089 +v...) ( )
and the laboratory velocity
\/v2 + vim + 2vvcm cosO' - (vvcm sin 0')2
V3 = (4~7)

1+ vvcm cos 0'

49

where

q
q2+m§'

 

v (4.8)

These equations define the momentum of the outgoing proton unbound nucleus.
However, the laboratory angle calculated in Equation 4.6 is relative to the angle of the
incoming beam. Since the beam does not always travel exactly on the center of the beam
pipe, a matrix rotation was applied to transform the fragments to a frame where the z-axis

lies along the center of the beam pipe.

4.7.4 The Decay

The unstable parent nuclei decay instantly into the daughter nuclei by emitting a
proton. The important parameters of the decay are the decay energy and width of the
state. The decay energy is incorporated by adding it to the sum of the masses of the
proton and the daughter nucleus to obtain the mass of the decaying state in the parent
nucleus.

The cross section for barrier penetration has a Breit-Wigner form given by

"N I‘L(E’I{n) 49
°“ (E-E.)2+FE(E.R.)/4 (')

 

where N is a constant used to normalize the distribution within the simulations. Due to

the factor of I“: (E, Rn) in the denominator of Equation 4.9, the center of the peak is

pushed towards lower decay energies as the width of the state increases. The calculation

50

of the width in Equation 4.9 requires the calculation of the partial widths. The reduced

width of a state 7L is defined as

hZ
2uRn

2

 

IxL(R..> (4.10)

'YLE

Note that the reduced width depends only on the nuclear radius and angular momentum
and is a constant with respect to energy. Using Equation 2.11 for the radial wave
function and Equation 2.12 for the partial width of a state Equation 4.10 becomes

__ I‘L(E,,Rn)
' 2k,ML(E,,Rn)

 

n (4.11)

where Er is the resonance energy, the penetrability is defined by Equation 2.14, and the

radius and wave number are given by

 

ZPET l/3
1<,=v h , Rnst+RD=l.4(l+AD).

Once the reduced width has been calculated, the width I‘L(E,Rn) at any energy can be
calculated from a rearrangement of Equation 4.11:
FL<E)= 2kR.vLP..(E,R.). (4.12)
The momenta of the proton and daughter nucleus were calculated using simplified
versions of the equations discussed in Section 4.7.3. However, an isotropic distribution
was used for the decay angle instead of the distribution given for the scattering angle in
Figure 4.10. As in the case of the scattering, a rotation was applied to the final momenta
to transform them into a frame where the z—axis lies along the center of the beam pipe.
The angles calculated for the proton and daughter nucleus were used to calculate the

energy-loss within the target as well as the location of interaction within the detectors.

51

4.7.5 Detectors

By tracing the trajectory of the protons and fragments, only those events were
included when both particles hit their perspective detectors. Defective pixels within the
ASI were identified in the simulations and events where a proton was detected in these
strips were eliminated. The fourth segment of the fragment telescope had insufficient
energy resolution and was not included in the data analysis, and thus excluded in the
simulations. This procedure for incorporating the shape and position of each detector into

the simulation accounted for the geometric efficiency of the detection system.

4.7.6 Decay Energy

Once the decay has been simulated and the proton and the daughter nucleus have
been traced into the detectors, Equation 4.1 was used to calculate the decay energy. By
varying the decay parameters within the input file, decay energy spectra were generated

and fitted to the data.

4.7.7 Efficiency and Resolution

The Monte Carlo simulation code contained the geometry of the detectors, and
thus was used to obtain the geometric efficiency of the experimental setup. By running a
flat decay energy distribution for the decay of llN the spectrum in Figure 4.11 was
obtained. Similarly, the experimental resolution was obtained from the simulations.

Figure 4.12 shows the observed F WHM of the decay energy peak versus the input width

52

of a 1/2+ state at 1.9 MeV decay energy. By observing the FWHM of the observed peak
at an input width of 0.0, the experimental resolution of 400 keV was obtained from
Figure 4.12. The flattening off of the spectrum in Figure 4.12 is a direct effect of the
geometric efficiency.

The effects of the efficiency and experimental resolution on the decay energy
spectrum is shown in Figure 4.13 which was obtained from the simulation of a 1/2+ state
at 1.6 MeV decay energy. Decay widths of 200 keV and 2.0 MeV were used to obtain the
left and the right spectra, respectively. The solid lines in Figure 4.13 represent the decay
energy spectrum at 100% geometric efficiency, whereas the dashed lines represent the
same decay including the geometric efficiency. The dashed lines were normalized to the
height of the solid lines to show the effect of the efficiency on the shape of the decay

energy peak.

 

 

 

 

 

O l I 1 l 1 I I J l L 1 I k l L I 1 I I I I I
0 2 4 6 8 10
Decay Energy (MeV)

Figure 4.] 1: Simulated geometric efficiency of the decay of ”N.

 

 

 

 

 

%‘ .
E ° ‘-
1, I
0 .
g .
m . _.
—° 4
O .
: I l l l l l 1 #4 I l l l l l l l l l :
O l 2 3 4
Input Width (MeV)

Figure 4.12: Simulated width of the decay energy peak versus the input width of a l/2+ state at 1.9 MeV
decay energy in 11N.

 

l I T I r T I V Y I T I I U I I Y Y I

15000'

10000.

Counts

5000

 

 

 

 

 

Decay Energy (MeV)

Figure 4.13: Simulated decay energy spectra for a 1/2+ state in "N at decay energies of 1.6 MeV. Decay
widths of 200 keV and 2.0 MeV were used to obtain the left and the right spectra, respectively. The solid
lines were obtained at a geometric efficiency of 100%. The dashed lines include the experimental
geometric efficiency and were normalized to the height of the solid lines for comparison.

54

4.8 Fitting of Data

Spectra obtained from the data were interpreted by comparing them to the results
of the simulations. The quality of these fits were calculated using the reduced x2 which is

defined as

 

 

1 j=l
2 _
xn — NM; 62 (4.13)

where N is the number of points used in the fit, M is the number of free parameters, yi is

the number of counts in the ith charmel of the data, y 'i,j is the number of counts in the ith

channel of the j‘h simulated peak, and a,- is the uncertainty in yi.

Within the Poisson distribution the mean of the reduced x2 (hereafter referred to
as )8) is equal to 1 with a variance of 1. This implies that a good fit would yield a value
of 1 for the x2 and the uncertainties of the fit can be obtained when the value of the x2
varies by one from the minimum. Values of x2 higher than one denote an inadequate fit,
while values below 1 imply a fit that is better than average. One possibility for obtaining
x2 values significantly different from 1 is in the uncertainties or error bars. If the error
bars are too large, the value of the x2 will be smaller, and vice versa. Also, the value of
x2 can be less than one if there are more parameters than are needed to fit the data.

The Monte Carlo simulations were run using sufficient numbers of events that
statistical error bars within the simulated peaks could be ignored when normalized to the

data. The x2 test was used to optimize the normalization of the simulations to the data.

55

For a general case of fitting a spectrum from the data with M number of simulated
peaks, we would need M normalization constants nj, where j is an integer from 1 to M. A
slight modification to Equation 4.15 can be performed to calculate the x2 for this case.
The y ’1', j in Equation 4.25 must be multiplied by the normalization constants nj thereby
including the normalization constants in the calculation of the x2 to get

[Y1 - in,- WRIT

1 N j=l
2
= . 4.14
X N—M; 0.2 ( )

 

 

The best possible fit would be the one with the lowest x2, therefore the optimum
normalization constants are obtained when the )8, given in Equation 4.16, is minimized.
The minimum of x2 can be found by setting the partial derivatives of the x2 with respect
to each 111' to zero. This results in M equations of the form

M
N (—2y.i,k 3'1—any'm)

5x2 1 '=1
an =N_M; 62’ =0 (4.15)
k 1=

 

 

where the 111' can be solved for using

( 12 1 1 1 1
Ya, Yr, 'Yi, Yr, ’Yi,
X‘s—i; Z :3? 2 Z ‘0’? M (Zyvim \

Y'i, ‘Y'i, Yi,2 y'i, y', n i i
Z_2_2_1_ Z—z Z—ZT'E‘ n: Yi'y'm

2

 

 

 

 

 

 

y'i,M 'Y’m y'i,M 7.1.2 ... y 1.1:
\; 6.2 21: 6-2 Z 6.2 )

 

 

Solving the above simultaneous equations will provide a solution for the normalization

factors that minimize the x2.

Chapter 5

Results and Discussion

5.1 Decay Energies

In addition to the one neutron stripping reaction 9Be(12N, llN), one neutron pickup
reactions populating l3N, one proton pickup reactions producing 13O, and two proton
stripping reactions creating 10B were also recorded. Evidence for the production of these
nuclei can be seen in Figure 4.7 (right) where the daughter nuclei of these fragments are
seen in coincidence with protons. These nuclei all have proton unbound excited states
[Aj288, Aj z90,Ajz91] and were used to test and calibrate the method.

Some of the input parameters of the simulations, i.e. energy resolution, could be
adjusted by comparing the results of the simulations for these nuclei to the data. Once all

the input parameters were optimized for these test cases, they were applied to 11N.

5.1.1 Decay of 13N

Although the ground state of 13N is bound to proton decay, all excited states of

this nucleus are proton unbound. Applying Equation 4.1 to the 12C fragments detected in

56

57

the fiagment telescope in coincidence with protons, the decay energy of l3N was

calculated. The raw decay energy spectrum obtained for 13N is shown in Figure 5.1.

 

150_

so;

  
      

 

 

l J

0 2 4 6
Decay Energy (MeV)

 

Figure 5.]: Raw decay energy spectrum of '3N.

Figure 5.2 shows a comparison of the data (left) to the simulation (right) for the
total kinetic energy of the 12C daughter nuclei versus the decay energy for ”N. The
dashed line at fragment energy of 360 MeV shows the low end cut off energy obtained
from the simulations. However, the 12C energy spectrum obtained from the data (lefi)
exhibits a tail extending into regions below 360 MeV. The absence of such a tail in the
simulations indicates that this tail is due to background events.

By applying a software gate to the fragment energy spectrum to contain all events
with ”C fragment energies lower than 360 MeV, the background spectrum shown in

Figure 5.3 was obtained. This background was normalized to the 4.0 to 7.0 MeV region

58

of the decay energy spectrum, containing events with 12C energies above 360 MeV, and
subtracted. The resulting background subtracted decay energy spectrum for 13N is shown
in Figure 5.4, where the statistical error bars have been included. A prominent peak at a

decay energy of about 1.5 MeV can be seen.

 

500 [IUTIIIIIIIIIIIIIIIIIIIH TIIIIIIIIIIIIIYTIIRTIIIV

400

    

Fragment Energy (MeV)
w
8

llllllll-llll_llllln

 

 

 

 

oTTTTITTTTJTITTJTTJTJTTTT

200
100
o1111L1111l1111l1144|1111 lllllllLlLJllllllllllll
4 6 8 0 2 4 6 8 10
DecayEnergy(MeV)

Figure 5.2: ”C fragment energy versus decay energy data (left) and simulation (right). Points below the
360 MeV dashed line are background events.

Figure 5.5 shows the level diagram for 13N [Ajz91]. The proton decay threshold
is at 1.944 MeV, therefore a decay energy of about 1.5 MeV would correspond to the
decay of the 3/2' and the 5/2+ states. The experimental width of about 400 keV does not
allow the separation of these states. The width of the 3/2' and 5/2+ states are much
narrower than the experimental resolution and could thus be used to calculate the energy

resolution of the detectors.

59

 

 

IIIIIIITUIIIIIIITTTITIIIIITII

 

 

 

 

Decay Energy (MeV)

Figure 5.3: Background spectrum of 13N. This spectrum was obtained by gating on ”C fragments with
energies lower than 360 MeV.

175

 

d
—1
—q
—
—q
—
.1
.—
_
—
c—r
sq
—
c—
—l

IIIIIIIIIIIITIIIIIIIIIII

 

llLlllllllllLLJlllllllll

 

  

twmtm

Decay Energy (MeV)

Figure 5.4: Background subtracted spectrum of the decay energy of 13N. The error bars represent the
statistical uncertainties only.

60

3.547 E0032 5’?

3502 I'=0(Xi2 3Q
2365 r=oo¢7 le‘
1.944 0’

 

 

 

Figure 5.5: Level diagram of ”N [Ajz91]. The excited state proton decay to ‘2C is indicated by the arrow.
All energies are in units of MeV.

Monte Carlo simulations were performed for the 3/2' state. By fitting the proton
energy spectrum an adequate fit was obtained using an energy resolution of 5% FWHM
for the MFA in the simulations. The fragment energy resolution was obtained by fitting
the data using the tabulated values of the 3/2' state [Ajz91] and varying the fragment
energy resolution to obtain the best fit. These simulations revealed an average energy
resolution of 3% for the fragment detectors, measured at FWHM.

By varying the excitation energy and width for the 3/2’ state of 13N in the
simulations, the x2 surface plot shown in Figure 5.6 was obtained. The minimum x2 of

0.8 is indicated by the dashed lines at E. = 3.45 MeV and F = 90 keV and a scale of 0.1 is
used for the x2 lines. The shaded area corresponds to the space within the xfnm +1.0
boundary line. At this boundary the maximum and minimum values of the excitation
energy and width were obtained resulting in an excitation energy of 3.453% MeV and a
width of 90330 keV which are in good agreement with the known values of 3.50 MeV

and 62 keV respectively [Ajz91]. The fit of the data using the excitation energy and

width obtained above is shown in Figure 5.7.

 

     

 

 

   

 

1 I I 1 1 1
600 _ _
E 400 _ _
é _ _
$2 - _
B _ _
200— _
I. .......... :
Q 1 1 1 1 1
3.35 3.40 3.45 3.50 3.55
Excitation Energy (MeV)

Figure 5.6: x2 surface plot as a function of excitation energy and width of the simulated 3/2' excited state
of ‘3N. The dashed lines intersect at the minimum x2 and the shaded area shows the region within xzth.

 

 

 

 

 

12. _* '
10. ‘_
a 7. -
a ;
o _
5. _
2. ’_
t-.. I
0 1 3 4

2 .
DecayFnergy (MeV)

Figure 5.7: Background subtracted decay energy spectrum of 13N fitted by a Monte Carlo simulation of a
state at an excitation energy of 3.45 MeV and a width of 90 keV (solid line).

62

5.1.2 Decay of 13O

The decay energy spectrum of ”O was obtained by requiring a coincidence
between protons and 12N. However, the secondary beam used in the experiment was 12N,
therefore a distinction had to be made between the actual decay fragments and random
coincidences with the beam. When gated on 12N, the fragment energy spectrum shown in
Figure 5.8 contained two distinct peaks. The prominent peak at higher energies is due to
the beam whereas the smaller peak at lower energies is due to decays fi'om 130. Also,
events within the lower energy peak contained no random coincidences. Therefore, the

decay energy spectrum was constructed from events within the low energy peak only.

 

300 I I I I I I I I I I I I I I I I I I I I I I I I

240

180

IIIIIIIIIIIITI

Counts

120

I

 

60

111111111I1111I1111l1111

IIIIIIIII

 

 

        

        

IL11114141111L1 11

O 1 L l
320 360 400 440 480 520
Fragment Energy (MeV)

 

Figure 5.8: Raw energy spectrum of 12N fragments.

63

A search for the background was performed using the same methods that were
applied to 13N. However, no background events were present. Figure 5.9 shows the
decay energy spectrum for 130 including statistical error bars. Three peaks are apparent
at decay energies of about 1.2, 2.8 and 4.5 MeV. Figure 5.10 shows the level diagram of
13O [Ajz91]. The decay of the first three excited states of 13O to the ground state of 12N

exhibit energies close to the center of the peaks in Figure 5.9.

 

 

 

 

 

 

 

_ I l I l I l 17 V T I I I T l I l l l I _

40 '_ .. _2

30 L .1; —.‘

L _ 1

3 . J
g 2 I I
0 O :— i
10 '__ I I J

: 11111-1». .

0 Lil l l l 1 1 1 l I I I T 1- fl? 1+Hmnml_r

 

 

 

0 2 4 6 8
Decay Energy (MeV)

Figure 5.9: Decay energy spectrum for '30. The data is shown as points with statistical error bars.

64

6.02 I" = 1.2 MeV

 

4.21

 

2.75

 

1.52 1
IZN + p

 

 

130

Figure 5.10: Energy levels and decay scheme for 13O. The widths of the first two excited states of 130
were not known. All energies are in units of MeV.

The excited states of '2N are proton unbound, therefore proton decays of l 3O to
excited states of 12N would not be observed. Monte Carlo simulations of the decay
scheme shown in Figure 5.10 were performed. Although x2 minimizations for the
excitation energies and widths of the states in 130 were not performed, a satisfactory fit of
the data using the parameters in Table 5.1 was obtained and is shown in Figure 5.11. The
values in Table 5.1 are in good agreement with the tabulated values [Ajz91] and further
strengthen the feasibility of the applied methods.

The normalization constants used to fit the decay energy spectrum were used to
calculate the relative population ratios of the states in 13 O following the reaction

913e(”N,‘-”0) and are also listed in Table 5.1.

 

 

'I'”1§1,'11i1. I

 

 

 

 

Decay Energy (MeV)

Figure 5.11: Fit of the decay energy spectrum of 13O. The points with the error bars are the data, the solid
line was obtained from the sum of the simulations for the first three excited states.

Table 5.1: Excitation energies and widths used to simulate the 13O decay energy spectrum. Also included
are the relative population ratios calculated from the simulations.

 

 

 

 

 

State (MeV) Width (keV) Population (%)
2.85 400 58.3
4.41 500 26.7
6.0 1200 15.0

 

 

 

 

5.1.3 Decay of “’13

The decay energy spectrum of 10B was constructed by requiring a coincidence

between protons and 9Be fragments. Similar to the 130 treatment, background events

66

were not present. The level structure of 10B above the proton decay threshold is more
complex than in the previous two cases. This structure is shown in Figure 5.12 [Asz8].
Therefore, only the first nine states above the proton threshold were considered.

Figure 5.13 shows the decay energy spectrum calculated for 10B. Due to
the inherent 400 keV resolution of the experiment, the states in 10B could not be

separated. However, the position of each state is identified in Figure 5.13 by the solid

 

 

 

 

 

 

 

 

 

 

lines.
8.894 2+
8.889 3°
8.894 172+
8.070 2,
7.819 7'67 1
7.560§ :
7.478 1+,2+
7.467/ 7.430 2
6.588 372
5’Be + p
/—\—/
/\—_’/
3+
1013

Figure 5.12: Level structure of loB. All energies are in units of MeV.

 

30' 2 "
“H |

 

 

  

I’ll—LI—U'U'LII Iq
.
0 .I. 1 I l I 1 L 1 l A L l A I 1 A l I I

0 1.0 2.0 ‘30 1 i 14.0‘ 5.0 6.0
DeeayEnergy(MeV)

 

Figure 5.13: Decay energy spectrum for 10B. Contributing states are indicated by the lines.

5.1.4 Decay of 11N

The decay energy spectrum for 11N was obtained by observing coincidences
between protons and 10C fragments. Figure 5.14 shows the raw spectrum without
background subtraction.

Simulations of the decay of MN indicated that the 10C fragments must have total
kinetic energies above 360 MeV. Similar to the case of 13O, the 10C energy spectrum
showed the presence of a tail extending below the 360 MeV limit. Therefore, the
background decay energy spectrum for HN was obtained by gating on 10C fragments
energies below 360 MeV. This background spectrum is shown in Figure 5.15 (solid).
Superimposed on the data is the simulated efficiency curve for the setup. This curve was
obtained by simulating a flat decay energy distribution and describes the experimental
background very well, justifying the methods used to obtain the experimental

background.

68

 

 

 

 

 

 

 

 

 

200 1 1
150;. _
£9 - 1
5100_ _
O 1 1
U . .
50L J

’ 1 1n 1 1 1 1 1 1 1 1;
O0 2 4 6 8

Decay Energy (MeV)

Figure 5.14: Raw decay energy spectrum for llN.
401114,....,11..,.1 60
30; I L45

" ‘ >
1. ‘ 0
£920: .30 .3
_. 1 z o
g : ‘ : E
I I e\°
10_ —15
0 1 L 1 1 1 .1 1. 1‘ "r—"hi" 0
0 2 4 6 8

Decay Energy (MeV)

Figure 5.15: Background spectrum for the decay of 11N. The background was obtained by gating on ”C
fragments with energies below 360 MeV (solid). The dashed line indicates a simulation of the efficiency
of the setup.

69

The background was normalized within the 5.0 to 8.0 MeV region of the decay
energy spectrum and subtracted resulting in the decay energy spectrum of Figure 5.16.
Also shown in Figure 5.16 is a simulation of the 1/2' excited state of 11N with the
parameters Edmy = 2.24 MeV and F = 740 keV (dashed line). Although this state is
populated, it can not explain the entire peak observed in the data. The enhancement at
lower decay energy could be evidence for the ground state of 11N, therefore simulations
for a 1/2+ ground state were performed.

By varying the decay energy from 0.1 MeV to 6.0 MeV and the width from 0.1
MeV to 4.5 MeV for the 1/2+ state by 50 keV steps within the simulations, a grid was
obtained. A similar grid was also obtained for the 1/2' excited state, except the variations
in decay energy and width were kept within the known uncertainties of these parameters
[Ben74]. Fits to the data were performed using every combination of these grid points
resulting in a )8 value for each fit. A search for the minimum x2 was performed using the
normalization method described in Section 4.8. The fit using Ed,caly = 1.45 MeV, F = 2.4
MeV for the 1/2+ state, and Edamy = 2.15 MeV, F = 750 keV for the 1/2' excited state
provided the minimum x2 value of 0.838. A fit of the data using the above values is
shown in Figure 5.17. The solid line was obtained by summing the normalized
simulations for the 1/2' excited state (short dashes) and the 1/2+ ground state (long

dashes).

 

 

 

 

 

I I I I I I I I 1 I I I I I I I I
150— _
" ‘fi
__ r _
g 100__ I __
o b I \ —
U P I __
I— I —
.— I —
50— 1
_. I _
_. I _
/
E / l
0 ’I I I I I I
0 2

 

Decay Energy (MeV)

Figure 5.16: Background subtracted decay energy spectrum of llN.

 

150— _.

100_ {i _

 

 

 

a '- I \ ‘
: d I' :
\ I

50 I— 1 I'\ \\ —

r- I \ -

_ I’ \ _

p- , \\ c

.. I, \\ a ..

0 K I 1 l I l l \l 1.:
O 2 4 6 8
Decay Energy (MeV)

Figure 5.17: Fit of the llN decay energy spectrum. The fit to the data (solid) is a sum of the contributions
from the known 10' excited state (short dashes) and a 1/2+ state at EM = 1.45 MeV and F = 2.4 MeV
(long dashes).

71

The uncertainties in the decay energy and width of the 1/2+ state were found by
searching the x2 space for the energies and widths where the value of the x2 deviated
from the minimum by 1. This search was performed using the known parameters and
uncertainties for the 1/2' state [Ben74]. A plot of the x2 surfaces as a fimction of decay
energy and width of the 1/2+ state is shown in Figure 5.18. Lines of constant x2 values

show the boundaries on the energy and width. Particular emphasis has been put on the

X12“. +1.0 line which can be used to determine the uncertainties in the decay parameters.
The position of the minimum x2 is represented by the star. For comparison, the values
obtained by recent theoretical calculations for the 1/2+ ground state are represented by the
filled square with error bars [For95] and the filled circle [Bar96]. These values are in

good agreement with the data.

 

2.5

.N
o

u—
M

Decay Energy (L=0) (MeV)

is

 

 

0.5
0

 

 

r1,-0 (MCV)

Figure 5.18: x2 surfaces as a function of decay energy and width of the 1/2+ state. Recent theoretical
predictions are indicated by the filled square with the error bars [For95] and the filled circle [Bar96]. The
Wigner limit on the decay width is represented by the dashed line. The position of minimum x2 is marked
by the star.

72

An upper limit on the width of the 1/2+ state could not be obtained
experimentally, however the dashed line in Figure 5.18 represents the Wigner limit which
provides an upper bound on the width of a state as a function of the decay energy and
angular momentum of that state. In order to derive the Wigner limit, we must first

introduce the concept of the reduced width. The reduced width yL is defined as

’12
: 2p.Rn

2

 

 

It Ixt(R11) (5.1)

where the XL(R) is the radial wave function defined by Equation 2.11 where the
spectroscopic factor 9: is the probability of the proton being at the surface of the HN
nucleus and is usually measured experimentally. Since 6: is a probability, it has an
upper limit of 1. This is called the Wigner limit. Incorporating this limit in Equation 5.1
gives

hZ
1111. = “R: °

 

'Y L (5.2)

Inserting this into Equation 4.14 we obtain an upper limit on the partial width of a state
given by

2h2k

“=an

 

I1(19R11)

 

PL(k, R, ). (5.3)

As can be seen in Figure 5.18, the Wigner limit provides an upper limit on the width.
The valid space of decay energies and widths is highlighted within the shaded area of

Figure 5.18.

73

The relative intensities for the states included in the fit were used to calculate the
relative population of the states in the parent nucleus. Table 5.2 contains the population

ratios calculated for the 1/2+ and 1/2' states of 11N.

Table 5.2: The relative population ratios of the states in llN.

State Population (%) fl
1/2 44.7 H

n 1/2' 55.3 ll

 

 

 

 

5.2 Comparisons to Theory

5.2.1 Population Ratios

The production of HN has been assumed to follow the one neutron shipping
reaction 9Be('2N,l lN), however theoretical calculations for this reaction result in
population ratios different from the ones in Table 5 .2. Shell model calculations of this
transfer reaction result in a population of only 1% for the l/2+ state [Bro96]. Although
there is a large discrepancy between the theoretical population and that shown in Table
5.2, the shell modeI calculations do not include multi-step processes or fragmentation
reactions that could contribute to the production of the 1/2+ state. Figure 5.19 shows the
fragment energy spectrum for the 10C daughter nuclei. Marked in Figure 5 .19 are the
expected energies of 10C from 11N nuclei formed in transfer reactions (E0 and

fragmentation reactions (13;). Clearly both mechanisms contribute to the population of the

74

states in llN. This mixture of fragmentation and transfer reactions was also observed in
an experimental study of 16B [Kry96]. Therefore, the population of the 1/2+ state in 11N
could be larger than the predicted 1%. Other possibilities for the discrepancy between the
experimental and theoretical population ratios for the 1/2+ state were considered and will

be presented in the following sections.

 

200 ITTTIIIITIIIIjITIrIIIII

IIjI

160

I

Counts

80

        

40

Er E:

TIIIITIIIITIIIIITI

 

IIIIlIIIIlIIIIlIiuhlILJI

 

llnth lLlllllllllll .5111

0
220 210 340 400 460 520
Fragment Energy (MeV)

 

Figure 5.19: 10C energy spectrum. The arrows indicate energies corresponding to formation of UN via
transfer reactions (E0 and fragmentation reactions (Bf).

5.2.2 Higher Excited States

The construction of the decay energy spectra is sensitive to the relative energy
between the parent and daughter states and not the excitation energy of each state alone,
therefore decays from other excited states of 11N to the ground and excited states of 10C

must also be studied. Although excited states above the 1/2' state in llN have not been

75

studied experimentally, theoretical calculations based on the mirror nucleus llBe have
predicted the presence of higher excited states. Aside from the 2.24 MeV state in Figure
1.2, an enhancement is present at about 4.5 MeV decay energy which was interpreted as
higher excited states of 11N [Ben74]. An equivalent enhancement can also be seen in the
decay energy spectrum of 11N obtained in this study at 4.5 MeV, Figure 5.16. Theoretical
calculations predict the existence of 3/2' and 5/2' excited states in 11N at 4.6 MeV and 5.7
MeV, respectively, above the proton decay threshold [Mil96]. The 3/2' state would
contain decay branches to both the ground state and the first excited state of 10C as
depicted in Figure 5.20. The 5/2' state would decay prominently to the first excited state
of 10C since the centrifugal barrier for an f5,2 decay is large. Calculations of the partial
width of each decay branch of the 3/2' state were performed and F(3/2' -) 2+) = 200 keV
and l"(3/2' -) 0+) = 300 keV were obtained [Mil96]. These partial widths are denoted in
Figure 5.20 as percentages for each decay branch. Table 5.3 contains the calculated

energies, partial widths and ratios of the contribution of each excited state of 11N

[Mil96].

 

 

 

 

 

 

5.7 5/2'
4.6 3/2'
3.35 2+
40%

2.24 1/2'

60% 1.45 1/2+

11
0+ N

Figure 5.20: Decay scheme for the theoretically predicted [Mil96] 3/2' and 5/2' excited states of ”N. All
energies are in units of MeV.

76

Table 5.3: Parameters for the calculated excited states of llN [Mil96].

 

 

 

 

 

State Decay Energy (MeV) Width (keV) Ratio (%)
1/2' 2.24 790 24.5
3/2' 4.61 300 31.1
3/2' 1.26 200 20.7
5/2' 2.35 640 23.7

 

 

 

 

 

 

The decay of the 3/2' state to the ground state of 10C by 4.6 MeV is a plausible
explanation for the enhancement at 4.5 MeV seen in Figure 5.20. However, the decay
branch to the first excited state of 10C has a decay energy of 1.25 MeV. This energy is
close enough to the 1.45 MeV decay energy, obtained in this study, that they are
indistinguishable from each other. This raises the possibility that the enhancement in the
decay energy of 11N, which has so far been treated as the 1/2+ ground state, could be
entirely due to the decay of the 3/2' state.

Up to now the use of Equation 4.13 to calculate the decay line-shape was valid
because we have been considering states that have only one decay branch. Therefore, the
total width was equal to the partial width of the state. However, the 3/2' state has two
decay branches of comparable widths and the total width is now the sum of the two
partial widths. Therefore, slight modifications were applied to Equation 4.13 to obtain a
line-shape for each decay branch. The new form is given by

_N, 1‘1(E,Rn) 54
6‘ _ (13-13,)2 +FT2(E,Rn)/4 (')

 

where i represents the ith decay branch, and the total width of the state is defined by

77

FT(E,Rn)=ZFi(E,Rn). (5.5)

The reduced width was calculated using Equation 4.10 and inserted into Equation 4.14 to
obtain the partial width for each decay branch as a function of the decay energy.

Figure 5.21 shows the result of the simulations for the states listed in Table 5.3.
The solid line is the sum of the four components (dashed) marked in the figure, and was

normalized to the data within the 2.5 to 6.0 MeV decay energy range.

 

150

100

Counts

50

 

 

 

 

 

 

 

 

Decay Energy (MeV)

Figure 5.21: Simulation of the decay of the excited states of ”N to the ground and first excited states of
10
C.

An overabundance of the 1.26 MeV decay of the 3/2' state is apparent in Figure
5.21. Although a reasonable fit to the data could possibly be obtained by varying the

parameters given in Table 5.3, without the inclusion of a 1/2+ ground state, one can not

 

78

rule out the possibility of contribution from a 1/2+ state by using arbitrary parameters for
the excited states. Since the population ratio of the ground state is strongly dependent on
the parameters describing the excited states of 11N, a concrete ratio for the UT state can

not be obtained without accurate information on these excited states.

Chapter 6

Summary and Conclusions

In this experiment radioactive nuclear beams were successfully applied to the
study of light nuclei beyond the proton dripline, in particular llN. These nuclei were
formed via transfer reactions and decayed immediately by emitting a proton. By
observing the proton and the daughter nucleus in coincidence, a complete kinematic
reconstruction of the parent nucleus was performed to obtain the decay energy.

A Monte Carlo simulation program incorporated all aspects of the experiment,
especially the decay kinematics and the detector geometry. Testing and calibration of this
code was achieved using the decay of well known nuclei which were produced in
conjunction with the 11N. Fits to the data were obtained at various decay energies and
widths. These fits were used to obtain 12 surfaces which in turn provided the optimum
values of the decay energy and width along with their uncertainties.

A decay energy spectrum was calculated for 13N. Fits of this spectrum yielded an

excitation energy of 3.45 31?: MeV and a width of 90330 keV for the 3/2' second excited
state of 13N. These values compare well with the tabulated values of 3.50 MeV and 62

keV, respectively [Ajz91].

79

80

The decay spectrum for ‘30 was also obtained and fitted. In this case the first
three excited states were observed. Although these states were previously studied, the
width of the first two excited states were unknown. The excitation energies of the first
three excited states were measured at 2.85 MeV, 4.41 MeV, and 6.0 MeV with widths of
400 keV, 500 keV, and 1.2 MeV respectively. Uncertainties for these values were not
calculated since 130 was not the primary nucleus of interest and the obtained decay
parameters were in good agreement with the tabulated values [Ajz91]. A comparison of
the number of events used within the simulations and the normalization factors used to fit
the data allowed for the calculation of the relative population of these states in the
reaction forming 13O.

The decay energy spectrum of 11N was reconstructed by observing the protons in
coincidence with 10C fragments. The previously known 1/2' first excited state was
identified successfirlly. An enhancement was observed in the lower decay energy region
of the spectrum and x2 optimizations were performed to fit this region with a 1/2+ state.
The best fit was obtained for a decay energy of 1.45 MeV and a width of 2.4 MeV.
Uncertainties could not be obtained for these values directly, however a valid region
within the decay energy and width space was calculated. These parameters are in good
agreement with recent theoretical calculations [Bar96,For95].

Relative population ratios were obtained for these states using the normalization
factors obtained from the fits to the data. Although shell model calculations predicted a
population ratio of only 1% for the ground state of 11N [Bro96], a ratio of 45% was

measured from the data. However, the shell model calculations did not include

81

contributions from fragmentation reactions which contributed to the population of the
states of llN. Simulations were performed to observe the possible contribution of
theoretically predicted 3/2' and 5/2' excited states in 11N decaying to the first excited state
of loC. However, the contribution from these states over-predicted the data in the low
decay energy region. Therefore, a population ratio could not be obtained for the 1/2+
state in the presence of the excited states.

The lower decay energy of the ground state of HN, compared to the previously
predicted value of 1.9 MeV, could explain the low branching ratio observed for diproton
decay of 12O. A 11N state at 1.45 MeV would provide the intermediate state needed for
the sequential proton decay of 120 within the constraints of the observed width of the
ground state of 120. Also, further analysis of the 120 data could provide further limits on

the decay parameters of the ground state of 11N.

APPENDIX

Appendix A

Monte Carlo Simulation

A.1 Description

The Monte Carlo simulation code was written in F ortran, except for the random
number generator which was written in C. All routines were adapted to run in double
precision mode. The program, along with all subroutines and firnctions have been
commented, therefore only brief descriptions will be provided.

PRO_DECAY: This is the main body of the program. All inputs, kinematic calculations,
and outputs are handled by this unit.

GASDEV: This function generates Gaussian [Pre92], Lorentzian, and flat
distribution of random numbers between -0.5 and 0.5.

UNIRAN: In conjunction with ERAND48, these functions provide random numbers
between 0 and 1 based on two seeds.

THERMAL: Generates a thermal distribution according to

2 E
£11-11

where E is the energy and T is the temperature.

DAUGHTER]: Allows the simulation of a square grid detector system for the fragments.

82

DAUGHTER3 :

PROTON 1 :

PROTON2:

DAF TAN :

BIN SEARCH:

LINESHAPE:

PENETRATE:

ERAND4 8 :

83

Contains the geometric information for the fragment telescope in the HN
experiment.

Allows the simulation of a square grid detector system for the protons.
Contains the geometric information for the MFA.
A function to calculate the arc-tangent fi'om the division of two numbers.

Uses a binary search pattern to find the array element containing the
desired data.

Calculates the line-shape of the decay based on the width of the state.
This routine incorporates Coulomb and Centrifugal barriers in the line-
shape. There are two versions of this code listed. One is for use with
states that have only one proton decay branch, the second simulates a
state with two proton decay branches.

This subroutine calculates the penetrability at a given energy and nuclear
radius. This subroutine calls on another subroutine, COULL, which
calculates the Coulomb functions. The body of the code for COULL is
not included here.

This code is written in C and uses two seeds to generate two random
numbers.

A.2 Input File

A sample input file is included at the end of the code with a brief description of

each input parameter, however a more expanded description follows.

Line la:
Line 1b:

Line 2:

Line 3a:

First seed needed for random number generation.
Second seed needed for random number generation.

Nmnber of decays to simulate.
Output mode. A value of zero creates a decay energy spectrum only,

whereas a value of 1 creates an event file containing a description of every
event. This event file will be described fully later.

Line 3b:

Line 4:

Line 5a:

Line 5b:
Line 5c:

Line 6a:

Line 6b:

Line 7a:

Line 7b:
Line 7c:

Line 8a:
Line 8b:
Line 8c:

Line 9a:
Line 9b:

Line 10a:
Line 10b:

Line 1 1a:
Line 1 1b:

Line 12a:
Line 12b:

Line 13a:
Line 13b:

Line 14a:
Line 14b:
Line 14c:

Line 15:

84

Factor used to define the width of the binning in the decay energy
spectrum.

Selects either a fragmentation reaction or a transfer reaction to populate
the parent nucleus.

Selects the distribution used to describe the decay. Available distributions
are Lorentzian, penetrability, thermal, and flat.

Sets the angular momentum when using the penetrability distribution.

Sets the temperature for the thermal distribution, or the width of the flat
distribution (MeV).

Switch to turn the target excitation on or off.

Sets the temperature for the thermal distribution used to obtain the target
excitation (MeV)

Switch to turn on the singles mode where the scattering angle and decay
angle of the proton can be fixed.

Scattering angle in radians (CoM).

Decay angle of the proton in radians (CoM).

Atomic number of the beam.
Proton number of the beam.
Mass of the beam including any excitation (MeV).

Beam energy (MeV/nucleon).
% spread in the momentum of the beam (FWHM).

FWHM of the beam in the x-projection of PPAC2 (mm).
FWHM of the beam in the y-projection of PPAC2(mm).

Beam offset in the x direction on PPAC2 (mm).
Beam offset in the y direction on PPAC2 (mm).

FWHM of the beam in the x-projection of PPAC2 (mm).
F WHM of the beam in the y-projection of PPAC2(mm).

Beam offset in the x direction on PPACl (mm).
Beam offset in the y direction on PPACl (mm).

Atomic number of the unstable nucleus.
Proton number of the unstable nucleus.
Mass of the unstable nucleus including any excitation (MeV).

Width of the decaying state (MeV).

Line 16a:
Line 16b:
Line 160:

Line 17a:
Line 17b:
Line 17c:

Line 183:
Line 18b:
Line 18c:

Line 19a:
Line 19a:

Line 20:

Line 21:

Line 22:

Line 23a:
Line 23b:

Line 24a:
Line 24b:

Line 25a:

Line 25b:

Line 26a:
Line 26b:

Line 27:

Line 28a:

Line 28b:

85

Atomic number of the daughter nucleus.
Proton number of the daughter nucleus.
Mass of the daughter nucleus including any excitation (MeV).

Atomic number of the target nucleus.
Proton number of the target nucleus.
Mass of the target nucleus including any excitation (MeV).

Atomic number of the residue nucleus.
Proton number of the residue nucleus.
Mass of the residue nucleus including any excitations (MeV).

Target thickness (mm)
Target thickness (mg/cmz)

Average energy loss of the beam per mm in the target (when not using the
energy loss equations).

Average energy loss of the daughter nucleus per mm in the target (when
not using the energy loss equations).

Average energy loss of the protons per mm in the target (when not using
the energy loss equations).

Distance of the proton detector from the target (mm).
Distance of the fragment detector from the target (mm).

% energy resolution of the proton detectors (F WHM).
% energy resolution of the fragment detectors (FWHM).

Grid size for the proton detectors (mm) (only when using subroutine
PROTONl).

Grid size for the fragment detectors (mm) (only when using subroutine
DAUGHTERI).

Be used in calculating the scattering angle in the CoM (rad.)
Slope of the distribution used to calculate the scattering angle (degfl)

Switch to turn energy loss equations on or off.

Lower limit on the beam energy where the energy loss equations are no
longer valid (MeV)
Upper limit on the beam energy where the energy loss equations are no
longer valid (MeV).

Line 29:

Line 30:

Line 31:

Line 32a:

Line 32b:

Line 33:

Line 34:

Line 35:

Line 36:

Line 373:

Line 37b:

Line 370:

Line 37d:

Line 38a:
Line 38b:

Line 39a:
Line 39b:

Line 40a:

86

Constant term in the quadratic used to fit the energy loss of the beam in
the target.

Coefficient of the linear term in the quadratic used to fit the energy loss of
the beam in the target.

Coefficient of the squared term in the quadratic used to fit the energy loss
of the beam in the target.

Lower limit on the daughter energy where the energy loss equations are no
longer valid (MeV)

Upper limit on the daughter energy where the energy loss equations are no
longer valid (MeV). .

Constant term in the fourth order equation used to fit the energy loss of the
daughter in the target.

Coefficient of the linear term in the fomth order equation used to fit the
energy loss of the daughter in the target.

Coefficient of the squared term in the fourth order equation used to fit the
energy loss of the daughter in the target.

Coefficient of the cubic term in the fourth order equation used to fit the
energy loss of the daughter in the target.

Lower limit of the first region of fit for the proton energy loss equations
(MeV).
Lower limit of the second region of fit for the proton energy loss equations
(MeV).
Lower limit of the third region of fit for the proton energy loss equations
(MeV).
Upper limit of the third region of fit for the proton energy loss equations
(MeV).

Constant coefficient for the first region of the proton energy loss equation.
First order coefficient for the first region of the proton energy loss

equation.

Second order coefficient for the first region of proton energy loss equation.
Third order coefficient for the first region of proton energy loss equation.

Constant coefficient for the second region of proton energy loss equation.

 

 

87

Line 40b: First order coefficient for the second region of proton energy loss
equation.

Line 41a: Second order coefficient for the second region of proton energy loss
equation.

Line 41b: Third order coefficient for the second region of proton energy loss
equation.

Line 42a: Constant coefficient for the third region of the proton energy equation.

Line 42b: First order coefficient for the third region of the proton energy equation.

Line 43a: Second order coefficient for the third region of the proton energy equation.

Line 43b: Third order coefficient for the third region of the proton energy equation.

Line 44: Same as Line 1a. Used for cross checking the input procedure.

A.3 Output

Depending on the output mode selected in Line 3a of the input file one of two
forms of output is generated. If Line 3a contains a zero, a decay energy file is created
which contains the decay energy in the first column and the number of counts per channel
in the second column. However, if Line 3a is set to one then a binary event file is
created. The pattern used to store the information is:

Integer *4 Ievent(4)

Real *8 Revent(40), event(44)

Equivalence (event(l), Ievent(l)), (event(5), Revent(l))
where the arrays ‘Ievent’ and ‘Revent’ contain the parameters created by the simulation.

Table A.1 gives the description of the variables in the array ‘Revent’ and Table A2 gives

similar descriptions for the array ‘Ievent’.

88

Table A. 1: Variables in the array ‘Revent’.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Revent(#) Variable Description
1 Fragment mass + width (MeV)
2 x on PPACl (mm)
3 y on PPACl (mm)
4 x on PPAC2 (mm)
5 y on PPAC2 (mm)
6 Beam radius on PPAC2 (mm)
7 Theta of beam in laboratory (deg)
8 Phi of beam in laboratory (deg)
9 Theta of beam with respect to the average beam angle
10 Phi of beam with respect to the average beam phi
11 Target Penetration (mm)
12 Beam energy at scattering (MeV)
13 Theta scattering in CoM (rad)
14 Velocity of fragment in CoM (v/c)
15 Velocity of fragment in beam coordinates (We)
16 Theta of scattering relative to beam (rad)
l7 Velocity of fragment in the laboratory frame (v/c)
l8 Theta of fragment in the laboratory frame (rad)
19 Velocity of proton in CoM (VIC)
20 Theta of proton in CoM (rad)
21 Velocity of daughter in CoM (v/c)
22 Theta of daughter in CoM (rad)
23 Velocity of proton in laboratory frame (v/c)
24 Theta of proton in laboratory frame (rad)
25 Velocity of daughter in laboratory frame (v/c)
26 Theta of daughter in laboratory frame (rad)
27 Proton energy loss in target (MeV)
28 Daughter energy loss in target (MeV)
29 Kinetic energy of proton including experimental resolutions (MeV)
30 Kinetic energy of daughter including experimental resolutions (MeV)
31 x position of proton in detector not including position resolution (mm)
32 y position of proton in detector not including position resolution (mm)
33 x position of daughter in detector not including position resolution
34 y position of daughter in detector not including position resolution

 

 

89

Table A.l (cont’d)

 

 

 

 

 

 

 

Revent(#) Variable Description
35 x position of proton in detector including position resolution (mm)
36 y position of proton in detector including position resolution (mm)
37 x position of daughter in detector including position resolution (mm)
38 y position of daughter in detector including position resolution (mm)
39 Laboratory opening angle between proton and daughter (rad)
40 Experimental decay energy (keV)

 

 

 

Table A2: Description of variable in the array ‘Ievent’.

Ievent(#) Description
1 Event number

Non-zero if event was dropped

2
3 Iseed(l)
4 Iseed(2)

 

If Ievent(2) is not zero that indicates that a ‘bad’ event occurred. The values of

Ievent(2) and the cause of the ‘bad’ event are given in Table A.3.

Table A.3: Values of Ievent(2) and description of the cause of the ‘bad’ event.

 

 

 

 

 

 

 

Value of Ievent(2) Description
1 Negative decay energy
2 Transfer reaction did not occur
3 Proton stopped in target
4 Daughter Stopped in Target
5 Proton missed detector
6 Daughter missed detector

 

 

 

 

 

90

AA Pro_Decay

OOOOOOOO

PROGRAM PRO_DECAY

this program was written to simulate the proton decay of nuclei
beyond the proton drip line and all the observed effects within
the experiment.

by

Afshin Azhari
l 994- l 996

real‘4 fac

integer’4 iaf(7),eloss_mode,ievent(4),therm,singles,
integer’4 i,k,d,hexact(0: 1000),hexp(0: 1000),flag,icount
integer*4 a_beam,z_beam,a__frag,z_frag,a_dau,z_dau
integer‘4 a_tar,z_tar,a_prod,z_prod,iii,iseed(2),eventn
integer‘4 j,out,itempl,itemp2,itemp3,itemp4,itemp5
integer‘4 itemp6,itemp7,reaction,prod_term,check,l_mom
real‘8 Uniran,erand48,gasdev,ep_res,ed_res,res_p,res_d
real“ 8 e _p_res,al ,a2,vfiag_z,m_beam,e_beam l ,rad_l ,rad__2
real*8 l,temp_r,detz_p,detz_d,x_p,y_p,x__d,y__d,relvel

real‘8 e_beam,dp_beam,p_beam,de_beam,e_d_res,rbin

real“ 8 p_beam l ,x l_beam,y l_bearn,z l _beam,gamma_d,m_fiag
real’8 e_frag,v_fi'ag,phi_d,e_d,pen_tar,b_eloss,e_p,v__pl
real*8 v_fi'ag l ,ke_d,loss_d,m_frag l ,x_p_res,m_dau,theta_dl
real" 8 v_d l ,theta_d,v_d,theta2,phi2,p_2,m_tar,tar_thick
real‘8 d_eloss,p_eloss,m_prod,mar_thd,m_prot,theta_p1,phi_p
real‘8 theta _p,v _p,gamma _p,ke _p,mar_rd,loss _p,pi,w_decay
real‘8 theta1,ll,12,vp_x,vp _y,vp_z,vd_x,vd _y,vd_z,vfrag_x
real*8 phil,w_2,q,gamma,beta,vfrag_y,vp_xl,vp_yl,vp_zl
real‘ 8 norm,cons,a_c,x_d_res,mar_rp,vd_x l ,vd_y l ,vd_zl
real’8 y_d_res,grid_p,grid_d,vp_res,vd_res,revent(40),r_emit
real‘8 templ,temp2,temp3,temp4,event(44),xlres_beam
real'8 ylres_beam,max_rad,th_emit,vfiagy,vfragz,e

real‘8 b,c,mf_exact,mf_exp,cos_exact,cos_exp

real’8 ep_exact,ed_exact,ep_exp,ed_exp,phi_emit,vfragx
real‘8 theta_c,slope, _p_res,mar_thp,fwhm,a,singl,sing2
real" 8 offset(2),v l p(3),v1d(3),v2p(3),v2d(3),temp,xoff
real‘8 e_therrn(0:20000),dafian,rad_beam l ,rad_beam2,yoff
real‘8 xoft2,yoff2,rtemp(7),mgcmsq,v_beam,prod_temp
real‘8 bemin,bemax,beloss(0:2),prod_th(0:20000),x2_beam
real‘8 demin,demax,deloss(0:3),m_prodl,y2_beam

real‘8 plimit(4),peloss(3,0:3),22_beam

real‘8 x2res_beam,y2res_beam,rad_beam3,rad_beam4
real‘8 prob(0: 1000),energy(0: 1001)

equivalence (event(l),ievent(1 )),(event(5),revent( 1 ))
common /daughter/ x_d,y_d,x_d_res,y_d_res,grid_d,pi
common /proton/ x_p,y_p,x_p_res,y_p_res,grid_p

common /all/ ievent,flag

common /seeds/ iseed

“I

91

common /penetrl/ m_frag,a_dau,z_dau,m_dau,m_prot,l_mom
common /penetr2/ w_decay,prob,energy

pi = dacos(-l.d0)

fwhm = 2.d0 * dsqrt(2.d0 * dlog(2.d0))
vlp(l)=0.d0

vlp(2)=0.d0

vlp(3)=0.d0

read(5,*) iseed(l),iseed(2)
read(5,*) eventn

read output mode (out=l eventfile, out=0 masses only)
read(5,*) out,fac

read type of reaction in target (transfer or fiagmentation)
read(5,") reaction

read type of lineshape to use for decay
read(5,"') therm,l_mom,temp

read in the thermal info for target product
read(5,"') prod_terrn,prod_temp

read in the singles info
read(5,"') singles,sing1,sing2

read in beam parameters

read(5,") a_beam,z_beam,m_beam
read(5,*) e_beam,dp_beam
read(5,*) rad_beam l ,rad_beam2
read(5,*) xofi',yoff

read(5,*) rad_beam3,rad_beam4
read(5,*) xoff2,yoff2

read in fragment parameters

read(5,*) a_frag,z_frag,m_fi'ag

read(5,*) w_decay

if ((therm.eq. l).and.(w_decay.le. 1 .d-2)) therm = 0

read in daughter parameters
read(5,*) a_dau,z_dau,m_dau

some data about protons
m_prot = 938.27231d0

read in target parameters
read(5,*) a_tar,z_tar,m_tar
read(5,*) a_prod,z_prod,m_prod
read(5,*) tar_thick,mgcmsq
read(5,*) b_eloss

read(5,*) d_eloss

read(5,*) p_eloss

 

cl

c1.4

92

read detector parameters
read(5,"') detz__p,detz_d
read(5,“) res _p,res_d
read(5,*) grid _p,grid_d

read the scattering parameters
read(5,‘) theta_c,slope

read energy loss parameters
read(5,*) eloss_mode

read(5,*) bemin,bemax
read(5,") beloss(O)

read(5,*) beloss(l)

read(5,*) beloss(2)

read(5,*) demin,demax
read(5,*) deloss(O)

read(5,*) deloss(l)

read(5,*) deloss(2)

read(5,*) deloss(3)

read(5,“‘) plimit(l), plimit(2), plimit(3), plimit(4)
read(5,"') peloss(l ,0),peloss( 1 , l)
read(5,"') peloss(l ,2),peloss(1 ,3)
read(5,*) peloss(2,0),peloss(2, l)
read(5,*) peloss(2,2),peloss(2,3)
read(5,*) peloss(3,0),peloss(3,1)
read(5,"') peloss(3,2),peloss(3,3)
read(5,*) check

if (check.ne.iseed(l)) then
write(“‘,*) ' '
write(“‘,*) 'Warning: Input file does not have the right'
write(*,*) ' number of parameters'
write(",*) ' '
write(",*)
write(“,*)
goto 1034

endif

write(*,"') 'Input completed successfully ....... '

 

 

if (out.eq. 1) then
open(unit=62,file='pro__decay.event',form='unformatted',status='new')
endif

calculate beam energy and momentum

e_beam = e_beam * a_beam

e_beam = e_beam + m_beam

p_beam = dsqrt(e_beam"'*2.d0 - m_bearn”2.d0)
dp_beam= dp_beam " p_beam/ 100.d0

now we can do some calculations to setup for the transfer reaction
first calculate the normalization factor

cons = l.d0 + slope‘theta_c
norm = slope "‘ dlog(10.d0) ‘ dsin(theta_c) + dcos(theta_c)

cl.7

c2

93

norm = norm " dexp(-l.dO‘slope‘theta_c*dlog(10.d0))

norm = norm + dexp(-l .d0*slope*pi‘dlog(10.d0))

norm = norm "‘ dexp(cons*dlog(10.d0)) / (slope * dlog(10.d0))**2.d0
norm = norm + 10.d0 ‘ (1 .d0 - dcos(theta_c))

norm = l.dO / norm

now calculate the critical area and start the picking process
a_c = 10.d0 * norm "' ( l.dO - dcos(theta_c))

write(",*) 'Calculating Line Shapes ...'

now determine the thermal distributions
if (therm.eq. 1) call lineshape(icount)
temp = temp " l.d3
if (therm.eq.2) then
call therrnal(temp,e_therm,0,20000)
m_frag=m_dau+m_prot
endif
prod_temp = prod_temp "' l.d2
if (prod_term.eq. 1) then
call thermal(prod_temp,prod_th,0,20000)
endif
m _per = m_prod

now set the line-shapes for the ppacs
call ppac_xy(3 ,rtemp(0),rtemp( 1))

now start the main loop of events
write("',*) 'Entering the Monte Carlo loop ...... '

do 10000 iii=l,eventn

j = int(iii/5000)
if (j.eq.real(iii)/5.e3) write(*,*) 'Event grsooo

ievent(1) = iii
ievent(2) = 0
ievent(3) = iseed(l)
ievent(4) = iseed(2)

put in the decay width
if (therm.eq.0) then

m_fragl = m_frag + (w_decay * gasdev(iseed,2))
elseif (therm.eq. 1) then

m_frag] = uniran(iseed)

call binsearch(prob,0,1000,m_fragl ,i,rbin)

rbin = (energy(i+1) - energy(i)) "‘ rbin

m_frag] = m_dau + m_prot + energy(i) + rbin
elseif (therm.eq.2) then

m_frag] = uniran(iseed)

 

c3

C4

C45

94

call binsearch(e_therm,0,20000,m_fragl,i,rbin)

m_frag] = m_fi'ag + (float(i)+rbin)/1.d3
elseif (therm.eq.3) then

m_frag = m_dau + m_prot

m_fragl = m_frag +(temp/1.d3 * uniran(iseed))
endif

if(m_fi'ag1.lt.m_prot+m_dau) then
ievent(2) = 1
goto 9999

endif

this section will give the target recoil a thermal excitation
if (prod_terrn.eq. 1) then
m_prod = uniran(iseed)

call binsearch(prod__th,0,20000,m_prod,i,rbin)

m_prod = m _prodl + (float(i)+rbin)/l.d2
endif

revent(l) = m_fi‘agl

now start the tracing:
fn'st calculate the distance penetrated into target
pen_tar = tar_thick * uniran(iseed)

revent(l l) = pen_tar

now calculate position for the beam
x1_beam = uniran(iseed)
yl_beam = uniran(iseed)
call ppac_xy(2,x l_beam,yl_beam)

x1_beam = x1_beam + xoff
yl_beam = yl_beam + yoff
zl_beam = pen_tar

calculate beam position on PPACl
x2_beam = uniran(iseed)

y2_beam = uniran(iseed)

call ppac_xy( l ,x2_beam,y2__beam)

x2_beam = x2_beam + xoff2
y2_beam = y2_beam + yoff2
22_beam =_ -l.85d3

now fold in the position resolution on target
xlres_beam = x1_beam
ylres_beam = yl_beam
x2res_beam = x2_beam
y2res_beam = y2_beam

 

c5

c6

10

95

revent(2) = x2res_beam
revent(3) = y2res_beam
revent(4) = xlres_beam
revent(S) = ylres_beam

now calculate the angles

x2_beam = x2_beam - x1_beam

y2_beam = y2_beam - yl_beam

th_emit = dsqrt(x2_beam“2.d0 + y2_beam"2.d0)
th_emit = dabs(datan(th_emit/22_beam))

phi_emit = pi + dafian(y2_bearn,x2_beam)

revent(9) = th_emit
revent(IO) = phi_emit

now calculate the beam momentum right at the moment of scattering
p_beam] = p_beam + dp_beam "' gasdev(iseed,3)

de_beam = b_eloss * pen_tar / dcos(th_emit)

e_beaml = dsqrt(p_beaml**2.d0 + m_beam”2.d0)

if (eloss_mode.eq. 1) then
rtemp(7) = e_beaml - m_beam
if (rtemp(7).lt.bemin .or. rtemp(7).gt.bemax) itemp5=itemp5+l
de_beam = pen_tar / tar_thick * mgcmsq / dcos(th_emit)
rtemp(7)=rtemp(7)* *2.d0‘beloss(2)+rtemp(7)*beloss( l )+beloss(0)
de_beam = rtemp(7) * de_beam
endif

e_beaml = e_beaml - de_beam
if (e_beam1.le.m*_beam) then
ievent(2)=7
goto 9999
endif
p_beam] = dsqrt(e_beam1"2.d0 - m_beam**2.d0)

revent(12) = e_beaml - m_beam
now the transfer reaction occurs, so find CoM angle of fragment

first we find the area and then theta
if (singles.eq. 1) then
thetal = singl
goto 66
endif
a = uniran(iseed)
if (a.le.a_c) then
thetal = dacos(l.d0 - a / (10.d0*norm))
else
ll=theta_c
12=pi
l=(11+12)/2.d0
a l =slope‘dlog( 10.d0)*dsin(theta_c)+dcos(theta_c)
a1 =al *dexp(-l .d0’slope'theta_c*dlog( l 0.d0))
a2=slope*dlog(10.d0)‘dsin(l)+dcos(l)

15

66

c7

c8

96

a2=a2*dexp(-l.d0*slope‘1*dlog(10.d0))
al=(al-a2)*dexp(cons*dlog(10.d0))/(slope*dlog(10.d0))**2.d0
a 1 =norm*(al+10.d0"'( l .d0—dcos(theta_c)))
if (dabs(a1-a).le.0.00000001d0) goto 15
if (al .ge.a) then
12=l
else
11=1
endif
goto 10
thetal = 1
endif
revent(13) = thetal

now find phi of reaction
phil = 21d0 "‘ pi " uniran(iseed)

convert thetal to lab coordinates
w_2 = m_beam"2.d0 + m_tar"2.d0 + 2.d0"‘m_tar*e_beam1
if (reaction.eq.0)then
first the transfer reaction

q = (w_2 - m_fragl"2.d0 - m _prod**2.d0)"2.d0

q = (q - 4.d0 * m _prod"2.d0 * m_frag1“2.d0)/(4.d0*w_2)
elseif (reaction.eq. 1) then
now the fragmentation (basically v_frag = v_beam)

v_beam = (m_tar " p_beaml)"2.d0 / w_2

v_beam = dsqrt(v_beam / (v_beam + m_beam"2.d0))

q = (m_fragl " v_beam)"2.d0 / (l.d0 - v_beam"2.d0)
endif

if(q.lt.0.d0) then
ievent(2) = 2
goto 9999
endif

e_frag = dsqrt( q + m_frag] **2.d0 )
v_frag = dsqrt(q) / e_frag

revent(14) = v_frag

gamma = (m_tar + e_beam1)/dsqrt(w__2)
beta = p_beaml/(m__tar + e_beaml)

theta2 = gamma*(v_frag"'dcos(theta1)+beta)
theta2 = dafian(v_fiag*dsin(thetal),theta2)

v_frag] = v_frag"2.d0 + beta"2.d0 + 2.d0*v_frag"'beta*dcos(thetal)
v_frag] = dsqrt(v_fragl - (v_frag‘beta‘dsin(theta1))"2.d0)

v_frag = v_fiagl / (1 .d0 + v_frag * beta " dcos(thetal))

phi2 = phil

revent(15) = v_frag
revent(16) = theta2

c8.5

c9

97

now the fragment is in the beam flame, so do a rotation to take
it into the lab flame.

vflagx = v_flag "‘ dsin(theta2) “ dcos(phi2)

vfragy = v_frag * dsin(theta2) " dsin(phi2)

vflagz = v_frag * dcos(theta2)

temp_r = vflagx "‘ dcos(th_emit) "‘ dcos(phi_em it)

temp_r = temp_r - vflagy "' dsin(phi_emit)

temp_r = temp_r + vflagz " dsin(th_emit) " dcos(phi_emit)
vfrag_x = temp_r

temp_r = vflagx " dcos(th_emit) " dsin(phi_emit)

temp_r = temp_r + vfragy " dcos(phi_emit)

temp_r = temp_r + vflagz " dsin(th_emit) * dsin(phi_emit)
vflag_y = temp_r

temp_r = vfragx " dsin(th_emit)

temp_r = vflagz “ dcos(th_emit) - temp_r

vflag_z = temp_r

v_frag = dsqrt(vflag_x"2.d0 + vfrag_y"2.d0 + vfrag_z"2.d0)
theta2 = dacos(vflag_z / v_frag)

phi2 = dafian(vfi‘ag_y,vfrag_x)

revent(l7) = v_frag
revent(l8) = theta2

now everything for the fragment is in lab coordinates, so now
we can do the decay of the fragment.

first in CoM

theta _pl = dacos(2.d0*uniran(iseed) - 1.0d0)
if (singles.eq.l) theta _pl = sing2

phi _p = 2.d0*pi * uniran(iseed)

revent(20) = theta _pl

theta_dl = pi - theta _pl

phi_d = pi + phi _p
if (phi_d.gt.2.d0*pi) phi_d = phi_d - 2.d0 * pi

revent(22) = theta_dl

p_2 = (m_frag1"2.d0 - m _prot"2.d0 - m_dau"2.d0)**2.d0

p_2 = (p_2 - 4.d0"‘(m_prot""2.d0)*(m_dau" *2.d0))/(4.d0*m_frag1 **2.d0)
e _p = dsqrt(p_2 + m_prot"2.d0)

e_d = dsqrt(p_2 + m_dau"2.d0)

v_pl = dsqrt(p_2) / e _p

v_dl = dsqrt(p_2) / e_d

revent(19) = v_pl
revent(21) = v_dl

now go to lab frame

clO

cll

012

CB

78

C

98

first transform to the fragments lab coordinates

gamma = l.d0 / dsqrt(11d0 - v_frag**2.d0)

theta _p=gamma*(v _pl *dcos(theta _pl)+v_frag)

theta _p = daftan(v_pl *dsin(theta_pl),theta_p)

theta_d=gamma‘ (v_d l *dcos(theta_d l )+v_flag)

theta_d = dafian(v_dl *dsin(theta_dl ),theta_d)

v_p = v_pl"2.d0 + v_frag"2.d0 + 2.d0"‘v_pl*v_flag*dcos(theta_pl)
v_p = dsqrt(v_p - (v_pl*v_frag‘dsin(theta_pl))"2.d0)

v_p = v_p/(l.dO + v_pl*v_frag*dcos(theta_pl))

v_d = v_d1"2.d0 + v_flag"2.d0 + 2.d0*v__dl *v_frag*dcos(theta_dl)
v_d = dsqrt(v_d - (v_dl *v_flag*dsin(theta_dl))**2.d0)

v_d = v_d / (1 .d0 + v_dl *v_flag*dcos(theta_dl))

now convert v_p and v_d to their x,y,z coords. using theta and phi
vp_x] = v_p "' dsin(theta_p) "' dcos(phi_p)

vp_yl = v_p * dsin(theta_p) * dsin(phi_p)

vp_zl = v_p * dcos(theta_p)

vd_xl = v_d "‘ dsin(theta_d) "‘ dcos(phi_d)

vd_yl = v_d * dsin(theta_d) "' dsin(phi_d)

vd_zl = v_d "' dcos(theta_d)

now apply the rotation

vp_x = vp_xl "‘ dcos(theta2) * dcos(phi2) - vp_yl * dsin(phi2)
vp_x = vp_x + vp_zl " dsin(theta2) * dcos(phi2)

vp_y = vp_xl "' dcos(theta2) * dsin(phi2) + vp_yl * dcos(phi2)
vp_y = vp_y + vp_zl "‘ dsin(theta2) "' dsin(phi2)

vp_z = vp_zl * dcos(theta2) - vp_xl * dsin(theta2)

vd_x = vd_xl "' dcos(theta2) * dcos(phi2) - vd_yl * dsin(phi2)
vd_x = vd_x + vd_zl " dsin(theta2) ‘ dcos(phi2)

vd_)I = vd_xl " dcos(theta2) * dsin(phi2) + vd_yl "‘ dcos(phi2)
vd_y = vd_y + vd_zl * dsin(theta2) "' dsin(phi2)

vd_z = vd_zl * dcos(theta2) - vd_xl "‘ dsin(theta2)

now convert back to v,theta,phi
v_p = dsqrt(vp_x"‘"‘2.d0 + vp_y"2.d0 + vp_z**2.d0)
theta _p = dafian(dsqrt(vp_x"2.d0 + vp_y**2.d0),vp_z)
if (vp_x.eq.0.d0) then
phi_p = pi / 2.d0
goto 78
endif
phi_p = dafian(VP_y,VP_x)
v_d = dsqrt(vd_x"”"2.d0 + vd_y**2.d0 + vd_z**2.d0)

theta_d = daftan(dsqrt(vd_x"2.d0 + vd_y"2.d0),vd_z)
if (vd_x.eq.0.d0) then
phi_d = pi / 2.d0
goto 79
endif
phi_d = daftan(vd_y,vd_x)

c HERE I WILL INSERT A PART WHERE WE CAN PUT IN INTENSITY [Win92] CALCULATED
c PARAMETERS FOR THE FRAGMENT

C

00000000

99

THETA_P = .298 * GASDEV(ISEED,1)
THETA_D = .065 "' GASDEV(ISEED,1)

V_P = 254.7 * .3 * GASDEV(ISEED,1) + 254.7
V_P = V_P / DSQRT(M_PROT**2. + V_P**2.)

V_D = 3002.2 * .02 * GASDEV(ISEED,1)+ 3002.2
v_o = V_D / DSQRT(M_DAU"2. + v_onz.)

C END OF INTENSITY INPUT

79

CM

revent(23) = v_p
revent(24) = theta _p
revent(25) = v_d
revent(26) = theta_d

Need to find the energy loss for both of these particles so that 1

can get them out of the target. 80 find their KINETIC ENERGIES first.
gamma _p = 1.d0 /dsqrt(1.d0 - v_p"2.d0)

ke_p = (gamma_p -1.d0)" m_prot

gamma_d = l.d0 / dsqrt(].dO - v_d"2.d0)

ke_d = (gamma_d — l.dO) "' m_dau

loss _p = (tar_thick - zl_beam) / dcos(theta_p) * p_eloss
loss_d = (tar_thick - zl_beam) / dcos(theta_d) " d_eloss

if (eloss_mode.eq.l) then

if (ke_d.lt.demin .or. ke_d.gt.demax) itemp6=itemp6+1

if (ke_p.lt.plimit(1) .or. ke_p.gt.plimit(4)) itemp7=itemp7+l
loss _p = (1 - zl_beam/tar_thick) " mgcmsq

loss _p = loss _p / dcos(theta_p)

loss_d = loss _p / dcos(theta_d)

rtemp(7) = ke_d"2.d0"deloss(2) + ke_d‘deloss(1) + deloss(O)
rtemp(7) = ke_d"3.d0*deloss(3) + rtemp(7)

loss_d = rtemp(7) "' loss_d

if (ke_p.lt.plirnit(2)) j = 1

if (ke_p.ge.plimit(2) .and. ke_p.lt.plimit(3)) j = 2

if (ke_p.ge.plimit(3)) j = 3

rtemp(7) = ke_p"3.d0‘peloss(i,3) + ke_p"2.d0"peloss(i,2)
rtemp(7) = rtemp(7) + ke_p‘pelossGJ) + peloss(i,0)

loss _p = rtemp(7) "‘ loss _p
endif

ke_p=ke_p-loss_p
ke_d = ke_d - loss_d

revent(27) = loss _p
revent(28) = loss_d

if (ke_p.le.0.d0) then
ievent(2)=3
goto 9999
endif

c14.5

015

c155

000000

0
a:
5:"
5"

00000000

100

if (ke_d.le.0.d0) then
ievent(2)=4
goto 9999
endif

fold in the energy resolution for the proton and daughter
ep_res = res _p "' ke_p/ 100.d0

ed_res = res_d * ke_d/ 100.d0

e _p_res = ke_p + ( ep_res * gasdev(iseed,l))

e_d_res = ke_d + ( ed_res "‘ gasdev(iseed,l))

revent(29) = e_p_res
revent(30) = e_d_res

now find the x and y of the proton and daughter on the detectors.
temp_r = (detz_p * 1000.d0 - zl_beam) / vp_z

x_p = vp_x * temp_r + x1_beam

y_p = vp_y * temp_r + yl_beam

temp_r = (detz_d "' 1000.d0 - zl_beam) / vd_z

x_d = vd_x * temp_r + x1_beam

y_d = vd_y * temp_r + yl_beam

revent(31) = x_p
revent(32) = y _p
revent(33) = x_d
revent(34) = y_d

Now call the subroutines containing the proton and daughter
detector configurations.
first the detector array for the daughter nuclei

the "infinite" grid

call daughter]

if (flag.eq. 1) then
ievent(2)=5
goto 9999

endif

the flagment telescope for flu experiment
call daughter3
if (flageq. 1) then
ievent(2)=5
goto 9999
endif

some energy cuts to match experimental cuts for 13O decay
if (ke_d.ge.442.d0) then
if (x_d.ge.0 .and. y_d.ge.0) then
ievent(2)=5
goto 9999
endif
endif
if (ke_d.ge.445.d0) then
if (x_d.gt.0 .and. y_d.1t.0) then

0000000000

101

ievent(2)=5
goto 9999
endif
endif
if (ke_d.ge.447.d0) then
if (x_d.lt.0 .and. y_d.lt.0) then
ievent(2)=5
goto 9999
endif
endif

c Now enforce the fragment energy lower limit

00000000

016

if (ke_d.le.3.6d2) then
ievent(2)=5
goto 9999
endif

Now let’s do the protons
now the detector array for the protons
first the "infinite" grid
cal] proton]
if (flag.eq. 1) then
ievent(2F6
goto 9999
endif

now the Maryland Forward Array
grid_p = pi
call proton2
if (flag.eq. 1) then
ievent(2)=6
goto 9999
endif

revent(35) = x_p_res
revent(36) = y _p_res
revent(3 7) = x_d_res
revent(3 8) = y_d__res

now let's analyze the results we have gotten to reconstruct
the mass of the flagrnent
first the exact solution.

calculating the distance travelled by proton
temp] = detz_p " 1000.d0 - revent(] 1)

temp2 = revent(32) - revent(S)

temp3 = revent(31) - revent(4)

a = temp] "2.d0 + temp2"2.d0 + temp3**2.d0

now the distance travelled by the daughter
temp] = detz_d * 1000.d0 - revent(] l)

temp2 = revent(34) - revent(S)

temp3 = revent(33) - revent(4)

b = templ”2.d0 + temp2"2.d0 + temp3"2.d0

c1615

cafsh

102

distance between where the proton and the daughter hit on
the detectors

temp] = (detz_p - detz_d) "‘ 1000.d0

temp2 = revent(32) - revent(34)

temp3 = revent(3 ]) - revent(33)

c = temp] ”2.d0 + temp2"2.d0 + temp3"2.d0

now calculate the cosine of the angle between the path of
the proton and that of the daughter
cos_exact = (a + b - c)/(2.d0 "' dsqrt(a'b))

now use the above information to get the mass of the flagment
ep_exact = ke_p + m _prot + revent(27)

ed_exact = ke_d + m_dau + revent(28)

mf_exact = ep_exact"2.d0 - m _prot“2.d0

mf_exact = mf_exact * (ed_exact**2.d0 - m_dau”2.d0)
mf_exact = 2.d0 "' cos_exact "' dsqrt(mf_exact)

mf_exact = 2.d0 * ep_exact * ed_exact - mf_exact

mf_exact = dsqrt(mf_exact + m _prot‘*2.d0 + m_dau”2.d0)

now the experimental solution (with resolutions folded in).
if (v2p(3).eq.0.d0) v2p(3) == detz_p‘1.d2

temp] = v2p(3) "' l.dl

temp2 = revent(36) - revent(5)

temp3 = revent(35) - revent(4)

a = temp] "2.d0 + temp2"2.d0 + temp3"2.d0

temp] = detz_d * 1000.d0

temp2 = revent(3 8) — revent(5)

temp3 = revent(3 7) - revent(4)

b = temp1“2.d0 + temp2**2.d0 + temp3"2.d0

temp] = (v2p(3)/l.d2 - detz_d) * 1000.d0
temp2 = revent(3 8) - revent(36)

temp3 = revent(37) - revent(35)

c = temp] “2.d0 + temp2"2.d0 + temp3“*2.d0

cos_exp = (a + b - c)/(2.d0 "' dsqrt(a“ b))
revent(3 9) = dacos(cos_exp)

ep_exp = revent(29) + m_prot

ed_exp = revent(30) + m_dau

mf_exp = ep_exp“2.d0 - m _prot"2.d0

mf_exp = mf_exp “ (ed_exp"2.d0 - m_dau"2.d0)
mf_exp = 2.d0 ’ cos_exp " dsqrt(mf_exp)

mf_exp = 2.d0 ‘ ep_exp * ed_exp - mf_exp

mf_exp = dsqrt(mf_exp + m _prot"2.d0 + m_dau**2.d0)

revent(40) = (mf_exp - m _prot - m_dau) * l.d3

now put decay energy out
if (revent(40).gt.2.4d3) goto 9999

c17

9999

l 0000

1033

1034

103

now write out the good information.
j = nint(revent(40)* ] .d-3/fac)

if (j.gt. 1000) goto 9999

hexp(j) = hexp(j) + l '

if (out.eq.]) write(62) (event(j)J=1,44)

if (ievent(2).eq.0) iaf(7)=iaf(7)+1
if (ievent(2).eq. 1) iaf(1)=iaf(l)l~1
if (ievent(2).eq.2) iaf(2)=iaf(2)l-l
if (ievent(2).eq.3) iaf(3)=iaf(3)+l
if (ievent(2).eq.4) iaf(4)=iaf(4)+l
if (ievent(2).eq.5) iaf(5)=iaf(5)+l
if (ievent(2).eq.6) iaf(6)=iaf(6)+l

continue
write(*,"‘) 'Bad Decays = ',iaf(l)
write(‘,"‘) 'Bad Transfers = ',iaf(Z)

write(*,*) 'Proton Stopped in Target = ',iaf(3)
write(‘,*) 'Daughter Stopped in Target = ',iaf(4)
write(‘,*) 'Proton Missed Detector = ',iaf(S)
write(‘,*) 'Daughter Missed Detector = ',iaf(6)
write(‘,*) "

 

write(‘,"') 'Good Events = ',iaf(7)
write(*,*)' '
iaf( 1 )=iaf( l )+iaf(2)+iaf(3 )+iaf(4)+iaf(5)+iaf(6)+iaf(7)
write(‘,"') 'Unaccounted = ',eventn-iaf(])
if (eloss_mode.eq.l) then
write(*,‘) ' '
write(‘,"') 'Energy Loss Results on Min. and Max. Limits:'
write(","') 'Beam Out of Range = ',itemp5

write(*,*) 'Daughter Out of Range = ',itemp6
write("',"‘) 'Protons Out of Range = ',itemp7
endif

if (out.eq.1) goto 1034

open (unit=7 1 ,file='masses.dat',status='new')

do 1034j=],100
write(7 1 ,‘) real(j)*fac,hexp(j)
if (hexp(j).ne.0) write(7l,*) real(j)‘fac,hexp(i)
write(7],"') hexp(j)

continue

stop
end

A.5

0000

104

Subroutines

NOW THE SUBROUTINES AND FUNCTIONS USED IN PRO_DECAY

Function to generate random numbers according to desired distributions.
function gasdev(iseed,dist)

The variable ‘dist’ determines the distribution:
1) Generates a gaussian of sigma=1 [Pre92]
2) Generates a Lorentzian of FWHM=1

3) Generates a flat dist. between —.5 and .5

Real*8 Uniran,erand48,gasdev,daftan
lnteger‘4 iseed(2),lseed(2)

lnteger'2 jseed(3),dist

Equivalence (Lseed( l ),jseed( 1 ))

DATA ISET/O/
if (dist.eq.l) then
IF (ISET‘.EQ.0) THEN
V1=2.d0*uniran(iseed)—].d0
V2=2.d0*uniran(iseed)-1 .d0
R=V]”2+V2"2
IF (R.GE.1..OR.R.EQ.0.) GOTO ]
FAC=DSQRT(-2.d0"‘LOG(R)/R)
GSET=V1"‘FAC
GASDEV=V2*FAC
ISET=1
ELSE
GASDEV=GSET
ISET=O
ENDIF
endif
if (dist.eq.2) gasdev=dtan(dacos(- ] .d0)*uniran(iseed))/2.d0
if (dist.eq.3) gasdev=1rniran(iseed)-0.5d0
RETURN
END

 

C

Random number generator (used with erand48.c)

Function Uniran(iseed)

Real‘8 Uniran,erand48,gasdev,daflan
Integer‘4 iseed(2),lseed(2)

Integer‘2 jseed(3)

Equivalence (Lseed( l ),jseed( 1))

C-UniX

Iseed(l) = iseed(l)
Iseed(2) = iseed(2)
Uniran = Erand48(jSeeD)
iseed(l) = Iseed(l)
iseed(2) = Iseed(2)

 

105

return

end

 

10

l 000

Routine to create a thermal dist.
subroutine thennal(temp,e_therm,jj,kk)

real*8 temp,e_therm(jj:kk),f(5),pi,e,dafian
integer "4 i,j,k,jj,kk
pi = dacos(- 1 .d0)
e_thenn(0)=0.d0
do 1000 i=jj,kk-l
do 10 j=l,5
e=float(i) + float(j-])"2.5d-l
f(j)=dsqrt(e/pi/temp)‘2.d0/temp
f(j)=f(j)"dexp(-e/temp)
continue
e_thenn(i+])=f(l)+4.d0"f(2)+2.d0*f(3)+4.d0*f(4)
e_thenn(i+ l He_therm(i+ ] )+f(5))*2.5d- 1/3 .d0
e_thenn(i+l)=e_therm(i+1) + e_thenn(i)
continue
return
end

 

3023

simulates a grid detector for the flagments
subroutine daughter]

integer‘4 ievent(4),itemp2,itemp4,flag

real‘8 revent(40),grid_d,x_d,y_d,x_d_res,y_d_res,dafian
common lall/ ievent,flag

common /daughter/ x_d,y_d,x_d_res,y_d_res,grid_d

flag=0
if (dabs(x_d). gt. 1 .d4.or.dabs(y_d). gt. 1 .d4) then
ievent(2)=5
flag=l
goto 3023
endif
y_d_res=(float( int(y_d/grid_d))+float( int(y_d/dabs(y_d)))* 5 .d- 1 )" grid_d
x_d_res=(float(int(x_d/grid_d))+float(int(x_d/dabs(x_d)))"' 5 .d- ] )* grid_d

return
end

 

The flagrnent telescope in ] 1N experiment
subroutine daughter3

integer'4 ievent(4),itemp2,itemp4,flag,iseed(2)

106

real‘8 revent(40),grid_d,x_d,y_d,x_d_res,y_d_res,dafian
real‘8 gasdev,r,t,pi

common lall/ ievent,flag

common /daughter/ x_d,y_d,x_d_res,y_d_res,grid_d,pi
common Iseeds/ iseed

flag=0

if (dabs(x_d).gt.2.dl .or.dabs(y_d).gt.2.d]) then
ievent(2)=5
flag=l
goto 3023

endif

if (x_d.lt.0.d0 .and. y_d.gt.0.d0) then
ievent(2)=5
flag=1
goto 3023

endif

r = grid_d * gasdev(iseed,]) ‘
t= 2.d0 * pi "' (gasdev(iseed,3)+.5d0)

x_d_res = x_d + r*dcos(t)
y_d_res = y_d + r‘dsin(t)

 

3023 return
end
c
subroutine proton]
integer‘4 ievent(4),itemp1,itemp3,flag
real‘8 x_p,y_p,x_p_res,y_p_res,grid_p,daftan,rtemp
common /proton/ x_p,y_p,x_p_res,y_p_res,grid_p
common /all/ ievent,flag
c now fold in the position resolution assuming a square detector grid
0 with square elements. If the particle hits right in between two
c elements, then it will be “ignored".

3024

rtemp = dsqrt(x_p"2.d0 + y _p"2.d0)

flag = 0
if (rtemp.lt.2.4dl .or. rtemp.gt. 1 .d2) then
ievent(2)=6
flag=l
goto 3024
endif
x_p_res = float(idint(x_p/grid_p))
x_p_res =(x_p_res+float(idint(x_p/dabs(x_p)))*5.d-1)*grid_p
y _p_res = float(idint(y_p/grid_p))
y _p_res = (y _p_res+float(idint(y _p/dabs(y _p)))*5.d-l)* grid _p

return

107

end

 

The MFA
subroutine proton2

integer*4 ievent(4),flag,ir,ith,icheck

real*8 x_p,y_p,x_p_res,y_p_res,mar_rp,pi,dafian
real*8 mar_thp,mar_thpl

common /proton/ x_p,y_p,x_p_res,y_p_res,pi
common lall/ ievent,flag

icheck = 0

Let's put in the Maryland Forward Array.
first change the x,y into r,theta

mar_rp = dsqrt(x_p"2.d0 + y _p"2.d0)

flag = 0

if (mar_rp.lt.2.4dl .or. mar_rp.gt.4.35d]) then
ievent(2)=6
flag = 1
goto 20

endif

mar_thp is the theta according to detector 2's left side and
follow the numbering scheme (2,3,4,...,15,l6,1)

mar_thp = daftan(x _p,y _p)*1.8d2/pi - 1.4d1
if (mar_thp.1t.0.d0) mar_thp=mar__thp+3.6d2

now check to see where on the detector the hit is (resolution)

mar_thp = float(int(mar_thp / 2.25d1))
ith = int(mar_thp / 4.)
mar_thp = (mar_thp * 2.25d1 + l.]25dl+l.4dl) * pi/].8d2

mar_rp = float(int(mar_rp/ 1.5d0)) * 1.5d0 + .75d0
ir = 16 - nint((mar_rp - 24.75) / 1.5) + ith“ 100

if (ir.eq.l 14 .or. ir.eq.206 .or. ir.eq.207) icheck=1
if (ir.eq.213 .or. ir.eq.216 .or. ir.eq.309) icheck=1
if (ir.eq.3 10 .or. ir.eq.31 1) icheck=1
if (ir.eq.2 ] 5.and.mar_thp.gt.4.3.and.mar_thp.]t.4.4) icheck=1

if (icheck.eq.l) then
ievent(2)=6
flag = 1
goto 20
endif

now convert back to x,y
x_p_res = mar_rp " dsin(mar_thp)

20

108

y _p_res = mar_rp * dcos(mar_thp)

return
end

 

A function to do arctan without the usual error problems
function daftan(y,x)

real*8 a,b,c,x,y,daftan,pi

pi = dacos(-1.d0)

a = dabs(y/x)

dafian = datan(a)

if (x.lt.0.d0.and.y.ge.0.d0) daftan=pi-datan(a)

if (x.ge.0.d0.and.y.lt.0.d0) daftan=2.d0"‘pi-datan(a)
if (x.]t.0.d0.and.y.lt.0.d0) daftan=pi+datan(a)

return
end

 

10

Routine to do a binary search within an array
subroutine binsearch (a,j,k,b,i,r)

real‘8 a(j:k),b,r
integer‘4 i,max,min

if (a(k).le.a(j)) write(‘,"‘) 'There is an error in Binsearch‘
r=000
i = int((j+k) / 2)
if (a(k).le.b) then
i = k
goto 50
elseif (a(j).eq.b) then
i =1
goto 50
endif

max=k
min=j

i = int((max+min) / 2)
if (a(i).gt.b) then
max = i
else
min = i
endif
if (max-min.le. ]) goto 20
goto 10

109

20 if (dabs(a(max)—b).lt.dabs(a(min)-b)) then
i = max
else
i = min
endif

if (max.eq.min) then

r = 0
else

i=mm

r = (b-a(min)) / (a(max)-a(min))
endif

50 return
end

 

subroutine lineshape(icount)
c this routine will calculate the line-shape due to the width of a state including Coul. and Centrifugal
c barriers.

Ctttttttittittlit##‘ttttttttttitttfitltttt*ttttttttttttttttttttttttc

C THIS IS FOR A STATE WITH ONE DECAY BRANCH ONLY C

Cttttit!*lttlttitit!!!itttittttittttttiittittttt#*##**********¥#**C

integer*4 ievent(4),flag,iseed(2),l_mom,a_dau,z_dau
integer‘4 al,zl,icount

real‘8 m_flag,m_dau,m_prot,w_decay,pen,estep
real*8 r,k,rk,rwidth,eres,m_mu,emin,emax

real" 8 prob(0: 100 l ),energy(0: 100 1 ),e_e,gamma
real*8 sigma

common /all/ ievent,flag

common /seeds/ iseed

common lpenetrll m_flag,a_dau,z_dau,m_dau,m_prot,l_mom
common /penetr2/ w_decay,prob,energy

c first let's setup for the penetrability calculation by calculating
c the reduced width

a] = a_dau
21 = z_dau

eres = m_flag - m_dau - m _prot

m_mu = (m_dau*m_prot) / (m_dau+m_prot)
= 1.4 *(a1"(1.d0/3.d0)+ l.dO)

k = dsqrt(2.d0*m_mu‘eres)/ 1.97d2
rk = r * k

call penetrate(l_mom,k,rk,zl ,1,m_mu,pen)

110

rwidth = w_decay / (2.d0*rk"'pen)

emin = 0.d0
emax = l.dl
estep = l.d-2
prob(O) = 0.d0
energy(O) = 0.d0
icount = 0

c now loop over the decay energies
do e_e = emin+estep,emax,estep
icount = icount+l
k = dsqrt(2.d0"‘m_mu‘e_e)/ 1.97d2
rk = k * r

call penetrate(l_mom,k,rk,z] , 1 ,m_mu,pen)

gamma = 2.d0 "' rk " rwidth ’ pen
sigma = gamma/((e_e-eres)**2.d0+(gamma/2.d0)*‘2.d0)

prob(icount) = prob(icount-l) + sigma
energy(icount) = e_e

end do

do i = ],icount
prob(i) = prob(i) / prob(icount)
end do

return
end

 

subroutine lineshape(icount)
c this routine will calculate the line-Shape for a state with two
c proton decay branches.

Ctittttttit.#1iii!itlt##0##titttttttttttttitttti*tttttttttttttttttc

C THIS IS FOR A STATE WITH TWO DECAY BRANCHES ONLY C

Cttttt##ttttttttifi##0##.##tttttlttttfittttttt##tifittltttttittttfit##C

integer‘4 ievent(4),flag,iseed(2),l_mom,a_dau,z_dau
integer‘4 a],z],icount

real*8 m_flag,m_dau,m_prot,w_decay,pen,estep
real‘8 r,k,rk,rwidth,eres,eres2,m_mu,emin,emax
real‘8 prob(0: 100 l ),energy(0: 100 ] ),e_e,gamma
real’8 sigrna,m_mu2,k2,rk2,rwidt112,garnma2

common /a]l/ ievent,flag

common /seeds/ iseed

common /penetrl/ m_frag,a_dau,z_dau,m_dau,m_prot,l_mom
common /penetr2/ w_decay,prob,energy

111

c first let's setup for the penetrability calculation by calculating
c the reduced width

caf

caf

caf

a] = a_dau
z] = z_dau
r = 1.4 " (a]"(1.d0/3.d0) + l.d0)

eres = m_frag - m_dau - m _prot
eresZ=eres-3.354d0
ere52=eres+3.354d0

m_mu = (m_dau‘m_prot) / (m_dau+m_prot)
m_mu2= ((m_dau+3.354d0)*m_prot)/(m_dau+m_prot+3.354d0)
m_mu2= ((m_dau-3.354d0)‘m_prot)/(m_dau+m_prot-3.354d0)

k = dsqrt(2.d0*m_mu*eres)/ 1.97d2
k2= dsqrt(2.d0*m_mu2*ere52)/ 1.97d2
rk = r * k

rk2= r " k2

call penetrate(l_mom,k,rk,zl , l ,m_mu,pen)
rwidth = w_decay / (2.d0*rk"'pen)
call penetrate(l_mom,k2,rk2,zl , 1 ,m_mu2,pen)

total width is included next in the (# - w_decay)
where # is the total width
rwidth2 = (.5d0-w_decay) / (2.d0"rk2"‘pen)

emin = 0.d0
emax = l.dl
estep = l.d-2
prob(O) = 0.d0
energy(O) = 0.d0
icount = 0

0 now loop over the decay energies

caf

do e_e = emin+estep,emax,estep
icount = icount+1
k = dsqrt(2.d0*m_mu‘e_e)/ 1.97d2
rk = k “ r

call penetrate(l_mom,k,rk,zl , ] ,m_mu,pen)
gamma = 2.d0 * rk * rwidth * pen

if (e_e.gt.3.354d0) then

k2= dsqrt(2.d0"m_mu2‘(e_e-3.354d0))/ 1.97d2
k2= dsqrt(2.d0*m_mu2*(e_e+3.354d0))/ 1.97d2
rk2= k2 "‘ r

call penetrate(l_mom,k2,rk2,zl , l ,m_mu2,pen)

112

gamma2= 2.d0 * rk2 " rwidth2 " pen
else

gamma2=0.d0

endif

gamma2= gamma2 + gamma
sigma = gamma/((e_e-eres)*‘2.d0+(gamma2/2.d0)"2.d0)

prob(icount) = prob(icount-l) + sigma
energy(icount) = e_e

end do

do i = l,icount
prob(i) = prob(i) / prob(icount)
end do

return
end

 

Subroutine Penetrate(l,xk,rk,Z 1 ,Z2,xmu,Pen)
c Calculates the Coulomb + Centrifugal penetrability.

integer‘4 1,21,12
real‘8 xk,rk,xmu,pen
real*8 xl,xeta,xrho,xxf,xxfp,xxg,xxgp

if(xk.eq.0.d0) then
Pen = 0.d0
return

end if

x1 = float(l)
xrho = rk
xeta = float(zl‘12)*1.44d0*xmu/(1.97d2**2.d0 " xk)

if(xeta.lt.5.d2) then
Call Coull(xl,xeta,xrho,xxf,xxfp,xxg,xxgp)
Pen = l.0d0/(xxf”2.d0 + xxg"2.d0)
else
Pen = 0.d0
end if

return
end

 

C

c Random number generator written in C to be linked with the rest of the code

113

double erand48_(xsubi)
unsigned short xsubi[3]; /*Arrays passed by reference*/

double r,erand48();
r = erand48(xsubi);

retum(r);

114

A.6 Sample Input File

1 INPUT FILE FOR PRO_DECAY

3685367,4364983

100000
0,1.e-l

0
1,1,]0.d0
1,].7dl

0,0.d0,1 .483529864d0
12,7,l.l 191693d4

4.077d1,3.d0
2.8d0,4.8d0
0.d0,0.d0
2.dl,1.45dl
-9.d0, 1 Ad]

1 1,7,10270.459d0

.2d0

10,6,9330.930d0
9,4,8392.753d0
10,4,9325.506d0
0.2032d0,36.64

104.82d0
77.260d0
2.14d0
.l99d0,.62 1 d0
5.d0,3.d0
l.dO, 1 .d0
5.d-2,37.56d0
1

4.2d2,5.4d2
1.52776d0
-2.87639d-3
1.9453d-6
2.5d2,5.5d2
1.5147100d0
-5.410140d-3
9.0328270d-6
-5.612011d-9
l , 10,30,100

2.30462d-1,-6.20771d-2
7.29768d—3,-3.0]875d-4
7.96004d-2,-5.98858d-3
2.05111do4,-2.56410d-6
3.08083d-2,-7.24017d-4
7.75216d-6,-3.00694d-8

3685367

1 seeds

! number of events

1 output mode (1 eventfile, 0 masses), expansion factor
! 0=Transfer, 1=Fragmentation

! 0=Lor/1=Pen/2=Therm/3=Flat, L (hbar), T (MeV)

! 1=100% Thermal Target Recoils, T (temperature MeV)

! 1=Singles,Scat. Theta, Decay Theta of P (rad)
! A, Z, M of beam

! E/A, and momentum width(%) of beam

! beam spot x-FWHM, y-FWHM on PPAC2(mm)
1 beam x-offset, y-offset on PPAC2(mm)

! beam spot x-radius, y-radius on PPAC1(mm)
! beam x-offset, y-offset on PPAC 1(mm)

! A, Z, M of flagrnent

! decay width (MeV)

! A, Z, M of daughter 9327.576

! A, Z, M of target

! A, Z, M of product

! target thickness(mm), and mg/cmsq

! energy loss of beam / mm (at beam energy)

! energy loss of daughter / mm (at beam energy)
! energy loss of protons / mm (at beam energy)
! proton, fragment detector distances flom target (m)
1 proton and fragment energy resolution (%)

! proton and fragment detector grid sizes (mm)
! theta critical and slope for scattering

! Actual E_loss equations on/off

! Beam : valid energy region

! a0 for beam

1 al for beam

! a2 for beam

! Daughter: valid energy region

1 a0 for daughter

1 a] for daughter

! a2 for daughter

! 33 for daughter

! Protons : divided into three energy regions

! 30, a] region 1

! a2, a3 region 1

! a0, a] region 2

! a2, a3 region 2

1 a0, a] region 3

! a2, a3 region 3

! iseed] for checking of input file

BIBLIOGRAPHY

Bibliography

[Ajz88] F. Ajzenberg-Selove, Nucl. Phys. A490, 142 (1988).
[Ajz90] F. Ajzenberg-Selove, Nucl. Phys. A506, 43 (1990).
[Ajz91] F. Ajzenberg-Selove, Nucl. Phys. A523, 38 (199]).
[Bar96] F. C. Barker, Phys. Rev. C 53, 1449 (1996).

[Ben74] W. Benenson , E. Kashy, D. H. Kong-A-Siou, A. Moalem, and H. Nann, Phys.
Rev. C 9, 2130 (1974).

[Ben79] W. Benenson and E. Kashy, Rev. Mod. Phys. 51, 527 (1979).
[Bie80] J. Biersack and L. Haggmark, Nucl. Instr. and Meth. 174, 257 (1980).

[Bla91] J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, Dover
Publications Inc., New York (1991)

[Bro94] B. A. Brown, NSCL Report No. MSUCL-94], (1994).
[Bro96] B. A. Brown, private communications.

[Cur86] M. S. Curtin, L. H. Harwood, J. A. Nolen, B. Sherrill, Z. Q. Xie, and B. A.
Brown, Phys. Rev. Lett. 56, 34 (1986).

[For95] H. T. Fortune, D. Koltenuk, and C. K. Lau, Phys. Rev. C 51, 3023 (1995).

[06161] V. I. Goldanskii, JETP 39, 497 (1960); Nucl. Phys. 19, 482 (1960); 27, 648
(1961).

[Har81] L. H. Harwood and J. A. Nolen, Nucl. Instr. and Meth. 186, 435 (1981).

[Hof94] S. Hoflnan, GSI Report No. GSI-93-04, (1993); Handbook of Nuclear Decay
Modes, CRC Press, Florida (1994).

115

116

[Jac75] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York
(1975)

[Kry94] R. A. Kryger, private communications.

[Kry95] R. A. Kryger, A. Azhari, M. Hellstriim, J. H. Kelley, T. Kubo, R. Pfaff, E.
Ramakrishnan, B. M. Sherrill, M. Thoennessen, and S. Yokoyama, Phys. Rev.
Lett. 74, 860 (1995).

[Kry96] R. A. Kryger, A. Azhari, J. Brown, J. Caggiano, M. Hellstrtim, J. H. Kelley, B.
M. Sherrill, M. Steiner, and M. Thoennessen, Phys. Rev. C 53, 1971 (1996).

[Llo92] W. J. Llope, R. Pedroni, R. Pak, G. D. Westfall, J. Wagner, A. Mignerey, D.
Russ, J. Shea, H. Madani, R. A. Lacey, D. Craig, E. Gualtieri, S. Hannuschke,
T. Li, J. Yee, and E. Norbeck, National Superconducting Cyclotron Laboratory
Annual Report, 236 (1992).

[Mi188] W. T. Milner, Program STOPI, Oak Ridge (1988).
[Mil92] W. T. Milner, Program DAMM, Oak Ridge (1992).
[Mil96] D. J. Millener, private communications.

[Oer70] W. Von Oertzen, M. Liu, C. Caverzasio, J. C. Jacmart, F. Pougheon, M. Riou, J.
C. Roynette, and C. Stephan, Nucl. Phys. A143, 34 (1970).

[Pag92] R. D. Page, P. J. Woods, R. A. Cunningham, T. Davison, N. J. Davis, S.
Hoflnann, A. N. James, K. Livingston, P. J. Sellin, and A. C. Shotter, Phys.
Rev. Lett. 68, 1287 (1992).

[Pag94] R. D. Page, P. J. Woods, R. A. Cunningham, T. Davison, N. J. Davis, A. N.
James, K. Livingston, P. J. Sellin, and A. C. Shotter, Phys. Rev. Lett. 72, 1798
(1994)

[Pre92] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical
Recipes in F ortran The Art of Scientific Computing, Press Syndicate of the
University of Cambridge, Cambridge (1992).

[R0188] C. E. Rolfs and W. S. Rodney, Cauldrons in the Cosmos, The University of
Chicago Press, Chicago (1988).

[She92] B. M. Sherrill, D. J. Morrissey, J. A. Nolen, N. Orr, and J. A. Winger, Nucl.
Instr. Meth. Phys. Res., Sect. B 70, 298 (1992).

117

[She94] B. M. Sherrill and J. Winfield, MSUCL report no. 449, (1994)
[She95] R. Sherr, private communications.
[Tal60] I. Talmi and I. Unna, Phys. Rev. Lett. 4, 469 (1960).

[Tig94] R. J. Tighe, D. M. Moltz, J. C. Batchelder, T. J. Ognibene, M. W. Rowe, and
Joseph Cerny, Phys. Rev. C 49, R2871 (1994).

[Wig57] E. P. Wigner, Proceedings of the Robert A. Welch Conferences on Chemical
Research 1, 67 (1957).

[Win92] J. A. Winger, B. M. Sherrill, and D. J. Morrissey, Nucl. Instr. Meth. B70, 380
(1992)

[Wol88] H. Wollnik, Gios: A program for the design of General Ion Optical Systems,
(1988)

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